# Computational Study of Low-friction Quasicrystalline Coatings via Simulations of Thin Film Growth of Hydrocarbons and Rare Gases

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Quasicrystalline compounds (QC) have been shown to have lower friction compared to other structures of the same constituents. The abscence of structural interlocking when two QC surfaces slide against one another yields the low friction. To use QC as low-friction coatings in combustion engines where hydrocarbon-based oil lubricant is commonly used, knowledge of how a film of lubricant forms on the coating is required. Any adsorbed films having non-quasicrystalline structure will reduce the self-lubricity of the coatings. In this manuscript, we report the results of simulations on thin films growth of selected hydrocarbons and rare gases on a decagonal Al$_{73}$Ni$_{10}$Co$_{17}$ quasicrystal (d-AlNiCo). Grand canonical Monte Carlo method is used to perform the simulations. We develop a set of classical interatomic many-body potentials which are based on the embedded-atom method to study the adsorption processes for hydrocarbons. Methane, propane, hexane, octane, and benzene are simulated and show complete wetting and layered films. Methane monolayer forms a pentagonal order commensurate with the d-AlNiCo. Propane forms disordered monolayer. Hexane and octane adsorb in a close-packed manner consistent with their bulk structure. The results of hexane and octane are expected to represent those of longer alkanes which constitute typical lubricants. Benzene monolayer has pentagonal order at low temperatures which transforms into triangular lattice at high temperatures. The effects of size mismatch and relative strength of the competing interactions (adsorbate-substrate and between adsorbates) on the film growth and structure are systematically studied using rare gases with Lennard-Jones pair potentials. It is found that the relative strength of the interactions determines the growth mode, while the structure of the film is affected mostly by the size mismatch between adsorbate and substrate's characteristic length. On d-AlNiCo, xenon monolayer undergoes a first-order structural transition from quasiperiodic pentagonal to periodic triangular. Smaller gases such as Ne, Ar, Kr do not show such transition. A simple rule is proposed to predict the existence of the transition which will be useful in the search of the appropriate quasicrystalline coatings for certain oil lubricants.

Another part of this thesis is the calculation of phase diagram of Fe-Mo-C system under pressure for studying the effects of Mo on the thermodynamics of Fe:Mo nanoparticles as catalysts for growing single-walled carbon nanotubes (SWCNTs). Adding an appropriate amount of Mo to Fe particles avoids the formation of stable binary Fe$_3$C carbide that can terminate SWCNTs growth. Eventhough the formation of ternary carbides in Fe-Mo-C system might also reduce the activity of the catalyst, there are regions in the Fe:Mo which contain enough free Fe and excess carbon to yield nanotubes. Furthermore, the ternary carbides become stable at a smaller size of particle as compared to Fe$_3$C indicating that Fe:Mo particles can be used to grow smaller SWCNTs.

Another part of this thesis is the calculation of phase diagram of Fe-Mo-C system under pressure for studying the effects of Mo on the thermodynamics of Fe:Mo nanoparticles as catalysts for growing single-walled carbon nanotubes (SWCNTs). Adding an appropriate amount of Mo to Fe particles avoids the formation of stable binary Fe$_3$C carbide that can terminate SWCNTs growth. Eventhough the formation of ternary carbides in Fe-Mo-C system might also reduce the activity of the catalyst, there are regions in the Fe:Mo which contain enough free Fe and excess carbon to yield nanotubes. Furthermore, the ternary carbides become stable at a smaller size of particle as compared to Fe$_3$C indicating that Fe:Mo particles can be used to grow smaller SWCNTs.

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COMPUTATIONAL STUDY OF LOW-FRICTION

QUASICRYSTALLINE COATINGS VIA SIMULATIONS

OF THIN FILM GROWTH OF HYDROCARBONS AND

RARE GASES

by

Wahyu Setyawan

Department of Mechanical Engineering and Materials Science Duke University

Date: Approved:

Dr. Stefano Curtarolo, Supervisor

Dr. Teh Y. Tan

Dr. Laurens E. Howle

Dr. Xiaobai Sun

Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in the Department of Mechanical Engineering and Materials Science in the Graduate School of

Duke University

2008

ABSTRACT

COMPUTATIONAL STUDY OF LOW-FRICTION

QUASICRYSTALLINE COATINGS VIA SIMULATIONS

OF THIN FILM GROWTH OF HYDROCARBONS AND

RARE GASES

by

Wahyu Setyawan

Department of Mechanical Engineering and Materials Science Duke University

Date: Approved:

Dr. Stefano Curtarolo, Supervisor

Dr. Teh Y. Tan

Dr. Laurens E. Howle

Dr. Xiaobai Sun

An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in the Department of Mechanical Engineering and Materials Science in the Graduate School of

Duke University

2008

Copyright c© 2008 by Wahyu Setyawan All rights reserved

Abstract

Quasicrystalline compounds (QC) have been shown to have lower friction compared

to other structures of the same constituents. The abscence of structural interlocking

when two QC surfaces slide against one another yields the low friction. To use QC

as low-friction coatings in combustion engines where hydrocarbon-based oil lubri-

cant is commonly used, knowledge of how a film of lubricant forms on the coating is

required. Any adsorbed films having non-quasicrystalline structure will reduce the

self-lubricity of the coatings. In this manuscript, we report the results of simula-

tions on thin films growth of selected hydrocarbons and rare gases on a decagonal

Al73Ni10Co17 quasicrystal (d-AlNiCo). Grand canonical Monte Carlo method is used

to perform the simulations. We develop a set of classical interatomic many-body

potentials which are based on the embedded-atom method to study the adsorption

processes for hydrocarbons. Methane, propane, hexane, octane, and benzene are

simulated and show complete wetting and layered films. Methane monolayer forms

a pentagonal order commensurate with the d-AlNiCo. Propane forms disordered

monolayer. Hexane and octane adsorb in a close-packed manner consistent with

their bulk structure. The results of hexane and octane are expected to represent

those of longer alkanes which constitute typical lubricants. Benzene monolayer has

pentagonal order at low temperatures which transforms into triangular lattice at high

temperatures. The effects of size mismatch and relative strength of the competing

interactions (adsorbate-substrate and between adsorbates) on the film growth and

structure are systematically studied using rare gases with Lennard-Jones pair poten-

tials. It is found that the relative strength of the interactions determines the growth

mode, while the structure of the film is affected mostly by the size mismatch between

adsorbate and substrate’s characteristic length. On d-AlNiCo, xenon monolayer un-

iv

dergoes a first-order structural transition from quasiperiodic pentagonal to periodic

triangular. Smaller gases such as Ne, Ar, Kr do not show such transition. A simple

rule is proposed to predict the existence of the transition which will be useful in the

search of the appropriate quasicrystalline coatings for certain oil lubricants.

Another part of this thesis is the calculation of phase diagram of Fe-Mo-C sys-

tem under pressure for studying the effects of Mo on the thermodynamics of Fe:Mo

nanoparticles as catalysts for growing single-walled carbon nanotubes (SWCNTs).

Adding an appropriate amount of Mo to Fe particles avoids the formation of stable

binary Fe3C carbide that can terminate SWCNTs growth. Eventhough the formation

of ternary carbides in Fe-Mo-C system might also reduce the activity of the catalyst,

there are regions in the Fe:Mo which contain enough free Fe and excess carbon to

yield nanotubes. Furthermore, the ternary carbides become stable at a smaller size

of particle as compared to Fe3C indicating that Fe:Mo particles can be used to grow

smaller SWCNTs.

v

Contents

Abstract iv

List of Figures ix

List of Tables xiv

Acknowledgements xvii

1 Introduction 1

2 Methods 5

2.1 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . 5

2.1.2 Slater determinant . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 Hartree-Fock equations . . . . . . . . . . . . . . . . . . . . . . 7

2.1.4 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . 10

2.1.5 Local density and generalized gradient approximations . . . . 11

2.1.6 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Vienna Ab initio Simulation Package . . . . . . . . . . . . . . . . . . 13

2.3 Classical interatomic potentials . . . . . . . . . . . . . . . . . . . . . 14

2.4 Simplex method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Grand canonical Monte Carlo . . . . . . . . . . . . . . . . . . . . . . 17

3 Noble gas adsorptions on d-AlNiCo 18

3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Simulation cell . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.2 Gas-gas and gas-substrate interactions . . . . . . . . . . . . . 19

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3.1.3 Adsorption potentials . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.4 Effective parameters . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.5 Test rare gases . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.6 Chemical potential, order parameter, and ordering transition . 25

3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Adsorption isotherms . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.3 Order parameters (ρ5−6) . . . . . . . . . . . . . . . . . . . . . 32

3.2.4 Effects of �gg and σgg on adsorption isotherms . . . . . . . . . 37

3.2.5 Effects of �gg and σgg on 5- to 6-fold transition . . . . . . . . . 39

3.2.6 Prediction of 5- to 6-fold transition . . . . . . . . . . . . . . . 40

3.2.7 Transitions on smoothed substrates . . . . . . . . . . . . . . . 42

3.2.8 Temperature vs substrate effect . . . . . . . . . . . . . . . . . 43

3.2.9 Orientational degeneracy of the ground state . . . . . . . . . . 46

3.2.10 Isosteric heat of adsorption . . . . . . . . . . . . . . . . . . . . 47

3.2.11 Effect of vertical dimension . . . . . . . . . . . . . . . . . . . 48

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Embedded-atom method potentials 52

4.1 Stage 1: Aluminum, cobalt, and nickel . . . . . . . . . . . . . . . . . 54

4.2 Stage 2: Al-Co-Ni potentials . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Stage 3: Hydrocarbon potentials . . . . . . . . . . . . . . . . . . . . . 59

4.4 Stage 4: Hydrocarbon on Al-Co-Ni . . . . . . . . . . . . . . . . . . . 63

5 Hydrocarbon adsorptions on d-AlNiCo 66

5.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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5.2 Adsorption potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Molecule orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4 Adsorption isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.5 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Embedded-atom potentials for selected pure elements in the pe- riodic table 81

7 Effects of Mo on the thermodynamics of Fe:Mo:C nanocatalyst for single-walled carbon nanotube growth 89

7.1 Size-pressure approximation . . . . . . . . . . . . . . . . . . . . . . . 90

7.2 Fe-Mo-C phase diagram under pressure . . . . . . . . . . . . . . . . . 93

7.3 Fe4Mo particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.4 FeMo particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8 Conclusions 99

Bibliography 101

Biography 111

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List of Figures

3.1 (color online). Computed adsorption potentials for (a) Ne, (c) Ar, (e) Kr, and (g) Xe on the d-AlNiCo, obtained by minimizing V (x, y, z) with respect to z. The distribution of the minimum value of these potentials is plotted in (b, d, f, and h) respectively: the solid line marks the average value 〈Vmin〉, the dashed lines mark the values at 〈Vmin〉±SD. . . . . . . . . . . . . . . 22

3.2 Computed adsorption isotherms for all the gas/d-AlNiCo systems. The ranges of temperatures under study are: Ne: T = 14 K to 46 K in 2 K steps, Ar: 45 K to 155 K in 5 K steps, Kr: 65 K to 225 K in 5 K steps, Xe: 80 K to 280 K in 10 K steps. Additional isotherms are shown with solid circles at T � = 0.35: T = 11.8 K (Ne), T = 41.7 K (Ar), T = 59.6 K (Kr), and T = 77 K (Xe). Isotherms above the triple point temperatures are shown as dotted curves. . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Density profiles and Fourier transforms of the outer layer at T � = 0.35 for Ne/d-AlNiCo (T = 11.8 K) and Ar/d-AlNiCo (T = 41.7 K), corresponding to points (A) through (F) of Figure 3.2. . . . . . . . . . . . . . . . . . . 30

3.4 Density profiles and Fourier transforms of the outer layer at T � = 0.35 for of Kr/d-AlNiCo (T = 59.6 K) and Xe/d-AlNiCo (T = 77 K), corresponding to points (A) through (F) of Figure 3.2. . . . . . . . . . . . . . . . . . . 33

3.5 (color online). Order parameters, ρ5−6, as a function of normalized chemical potential, μ�, (as defined in the text) at T � = 0.35 for the first four layers of (a) Ne, (b) Ar, (c) Kr, and for the first layer of Xe (d) adsorbed on d-AlNiCo. A sudden drop of the order parameter in Xe/QC to a constant value of ∼ 0.017 at μ� ∼ 0.8 indicates the existence of a first-order structural transition from fivefold to sixfold in the system. . . . . . . . . . . . . . . 34

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3.6 (color online). Xe on d-AlNiCo at T = 77 K. (a) Adsorption isotherm, ρN , versus the normalized chemical potential, μ�. (b) Nearest neighbor distance derived from the first peak of pair correlation function, rNN , (black line), and average spacing between neighbors at equilibrium, d¯NN , (red line). (c) Order parameter ρ5−6 (probability of fivefold defects, defined in Equation 3.14) versus the normalized chemical potential, μ�. (d) Total enthalpy. The transition, which is defined as the point in μ� above which the order parameter remains nearly constant, occurs at μ�tr ∼0.8. The discontinuity in H around μ�tr ∼ 0.8 indicates a first order transition with associated latent heat of the transition. The order parameter ρ5−6 after the transition is ∼ 0.017. Heat of the transition is ≈ 6.8 meV/atom. . . . . . . . . . . . 36

3.7 (color online). Computed adsorption isotherms for Ne, Xe, iNe(1), and dXe(1) on d-AlNiCo at T �=0.35. iNe(1) and dXe(1) are test noble gases having potential parameters described in the text and in Tables 3.1 and 3.2. The effect of varying the interaction strength of the adsorbates on the density increase ΔρN (while keeping the size constant) is negligible on large gases but significant on small gases. . . . . . . . . . . . . . . . . . . . . 39

3.8 (color online). Order parameters as a function of normalized chemical po- tential (as defined in the text) for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed on d-AlNiCo at T � = 0.35. A first-order fivefold to sixfold structural transition occurs in the last three systems, but not in dXe(2). . . 41

3.9 (color online). (a) The minimum of adsorption potential, Vmin(x, y), for Ne on a smoothed d-AlNiCo as described in the text. (b) The variations of the minimum adsorption potentials along the line at x = 0 shown in (a), for the modified and original interactions (solid and dotted curves). . . . . . 44

3.10 Xe on d-AlNiCo. Values of μ�tr for the fivefold to sixfold transition points from 40 K to 140 K (left axis). Transition points at μ�tr > 1 indicate that a transfer of atoms from the second layer to the first layer is required to complete the transition. Also shown is the defect probability as a function of T after the transition occurs (right axis), indicating an increase in defect probability with T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.11 Density plot of Xe on decagonal AlNiCo at 77 K, showing a superposition of the density slices for the 2nd and 4th layers. In the top right, 4th-layer atoms are located directly above the 2nd layer atoms indicating an hexagonal close- packed ABAB stacking, whereas in other regions, such as lower left, the two layers are offset due to stacking fault. . . . . . . . . . . . . . . . . . . . 46

x

3.12 Xe adsorption on d-AlNiCo. (a) Minimum potential energy surface of the adsorption potential with free boundary conditions. (b) Adsorption isotherms of the first layer from a set of 30 simulations at 77 K using the free cell described in the paper. Five density profiles and FTs at point p� of (b) are shown in (c) to (g), representing all possible orientations of hexagonal domains. (h) Schematic diagram illustrating the correspondence between the orientations of the hexagonal domains observed in the density profiles (c) to (g). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.13 Xe adsorption on d-AlNiCo. Pentagonal defects rotate the orientation of hexagons by (a) θ1 = 24◦ and (b) θ2 = 12◦. . . . . . . . . . . . . . . . . 51

3.14 (color online). Xe adsorption on d-AlNiCo. Locations in P , T of the vertical risers in the isotherms corresponding to the first (square), second (circle), and third (triangle) layer formation. The heats of adsorptions, qst, are 270, 129, and 125 meV/atom respectively, calculated as described in the text. The inset figure shows qst obtained from the simulations as well as from the experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 (color online). (a-e) Adsorption potential map, calculated by minimizing the adsorption potential of one molecule on a decagonal Al-Ni-Co along z direction and all rotational degrees of freedom at every coordinates (x, y). Red numbers represent the average value of the adsorption energies. (f) Top view of the decagonal Al-Ni-Co substrate 51.2x51.2 A˚2: Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red), Ni-toplayer (green), and Ni-otherlayers (blue). . . . . . . . . . . . . . . . 71

5.2 (color online). Isothermal adsorption densities of hydrocarbons on a decago- nal Al-Ni-Co: (a) methane (from left to right T = 68, 85, 136, 185 K), (b) propane (T = 80, 127, 245, 365 K), (c) hexane (T = 134, 170, 267, 450 K), (d) octane (T = 162, 210, 324, 450, 565 K), and (e) benzene (T = 209, 270, 418, 555 K). The inset in each figure is the density along z direction at pressure corresponding to point d. Xenon (red) is plotted in panel (a) for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 (a) and (b) Calculated density of methane adsorbed on a decagonal Al-Ni- Co at pressures corresponding to points ”a” and ”c” of the 68 K isotherm shown in Figure 5.2.a, respectively. (c) Fourier transform of the density plot shown in (b), consistent with 5-fold ordering of the methane near monolayer completion. (d) Order parameter (left axis, as calculated in Equation 3.14) as a function of pressure for the 68 K isotherm (right axis), indicating no sharp transition to 6-fold ordering. . . . . . . . . . . . . . . . . . . . . 74

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5.4 Calculated density of propane, hexane, and octane adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion. Propane forms a dis- ordered structures, whereas hexane and octane tend to form close-packed structures indicated by stripe features with increasing order for longer chain. 76

5.5 (Top row) Calculated density of benzene adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion for 209 K, 270 K, and 418 K. (middle row) Density profile of the geometrical center of density shown in the top row. (bottom row) Fourier transform of the density plot shown in the middle row, showing 5-fold ordering at 209 K, mixture of 5-fold and 6-fold structures at 270 K, and mostly 6-fold features at 418 K. . . . . . . 78

5.6 (left axis) Order parameter ρ5−6 = N5/(N5 +N6) (Nn denotes the number of molecules having n nearest neighbors) as a function of temperature at 0.01 atm of pressure. (right axis) Adsorption isobar showing the number of adsorbed molecules as a function of temperature. Verticel dashed lines correspond to T = 209, 270, and 418 K whose density profiles are plotted in Figure 5.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.1 (color online). Size-pressure approximation for Fe nanoparticles obtained by equating the deviation of average bond length from the bulk value due to curvature 1/R (in the case of particle) and due to pressure P (in the case of bulk). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.2 (color online). Ternary phase diagram for Fe-Mo-C nanoparticles of R ∼ ∞, 1.23, 0.62, 0.41 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.3 (color online). a) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x- Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”1”, represents [Fe4/5Mo1/5]1−x-Cx crossing the boundary phase Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed verti- cal lines, labeled as ”4” and ”5”, represent [Fe4/5Mo1/5]1−x-Cx crossing the boundary phases Fe←→Mo2C and τ3, as shown in figure 7.2(d). . . . . . . 97

xii

7.4 (color online). a) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x- Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”2” and ”3”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phase Fe2Mo↔Mo2C and Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”6”, ”7”, and ”8”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phases Fe2Mo↔Mo2C, Fe←→Mo2C, and τ3 ↔Mo2C, as shown in figure 7.2(d). . . . . . . . . . . 98

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List of Tables

3.1 Parameter values for the 12-6 Lennard-Jones interactions. TM is the label for Ni or Co. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Range, average (〈Vmin〉), and standard deviation (SD) of the interaction Vmin(x.y) on the d-AlNiCo. Effective parameters of the gas-substrate in- teractions (Dgs, σgs, D�gs, σ�gs), and, for comparison, the best estimated well depths DGrgs on graphite [106]. . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Results for Ne, Ar, Kr, and Xe adsorbed on d-AlNiCo. Tt is taken from reference [109]. The density increase (ΔρN ) in the first and second layers is calculated at T � = 0.35 from point (A) to (B) and (C) to (D) in Figure 3.2, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Summary of adsorbed noble gases on d-AlNiCo that undergo a first-order fivefold to sixfold structural transition and those that do not. . . . . . . . 42

4.1 List of structure prototypes used to fit the EAM potentials for hydrocarbon adsorption on Al-Co-Ni. For elemental Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure (DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed (HEX), simple cubic (SC), and simple hexagonal (SH). The Al-Co-Ni ternaries are taken from the database of alloys [122]. . . . . . . . . . . . . . . . . . . . . . . 53

4.2 (Top part) List of structures used to fit EAM potential for elemental alu- minum. ρi is charge density at atom-i, ΔE = EEAM−EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3 · re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 56

4.3 (Top part) List of structures used to fit EAM potential for elemental cobalt. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”HEX at a0 · c” indicates that the structure is HEX with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 57

xiv

4.4 (Top part) List of structures used to fit EAM potential for elemental nickel. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 58

4.5 (Top part) List of structures used to fit EAM potential for Al-Co-Ni systems. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV) and the first knot at (0,0) is assumed. The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . . . . . . . . . . . . . . . . . 60

4.6 (Top part) List of structures used to fit EAM potential for hydrocarbons. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . 61

4.7 (Top part) List of structures used to fit EAM potential for alkanes and benzene. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.8 Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. Fitting structures are given in Table 4.9 . . . 64

4.9 Fitted energies calculated using EAM parameters in Table 4.8. Methane up represents methane with one H below C and three H above C. Methane down is inverse of methane up. The unit for energy is eV/atom except for the adsorption energy which is in eV/molecule. . . . . . . . . . . . . . . . . 65

5.1 Parameter values for the adsorbate-adsorbate interactions used for hydro- carbon adsorption on a decagonal Al-Ni-Co. Intermolecular energies are calculated as a sum of pair interactions. For methane-methane, the C-H is taken as the geometrical mean for parameter A and as the arithmetic mean for parameters B and C. . . . . . . . . . . . . . . . . . . . . . . . . . . 69

xv

6.1 Fitted parameters for the charge density and pair interaction functionals of the EAM potentials for pure elements, continued in Table 6.2. The fitting structures are given in Tables 6.5 and 6.6. The parameters for the embedding functionals are given in Tables 6.3 and 6.4. . . . . . . . . . . 83

6.2 continuation of Table 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3 Fitted knots of cubic spline of the embedding functionals for the EAM potentials of pure elements, continued in Table 6.4. The first knot at (0,0) is assumed. The fitting structures are given in Tables 6.5 and 6.6. . . . . . 85

6.4 continuation of Table 6.3. . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.5 (left part) Structures used to fit EAM potentials for pure elements, con- tinued in Table 6.6. The EAM potentials are fitted to the ab initio ener- gies in body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close-packed (HEX), diamond structure (DIA), and groundstate structure at various pressures obtained by expanding/compressing the equillibrium lattice constant a by a factor from 0.9 to 1.16 corresponding to a range of charge density from ρmax to ρmin. (right part) Lattice constansts calculated using the fitted parameters. The literature values aLIT are taken from [146]. 87

6.6 continuation of Table 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . 88

xvi

Acknowledgements

I would like to thank my advisor, Prof. Stefano Curtarolo for all the academic and financial supports. I wish to thank all collaborators of this work, Prof. Renee D. Diehl, Prof. Milton W. Cole, Prof. L W. Bruch, Dr. Nicola N. Ferralis, and Dr. Andrea Trasca. I thank all my teachers for their mentoring and my committee members, especially Prof. Teh Y. Tan, Prof. Laurens E. Howle, and Prof. Xiaobai Sun for their dedications and insightful advice. I thank all my colleagues, in particular Dr. Neha Awasthi, Dr. Aiqin Jiang, Dr. Roman Chepulskyy for encouragements and valuable discussions, and Dr. Aleksey Kolmogorov for his mentoring and experienced advice on VASP. I also wish to thank all the staff in the department of Mechanical Engineering and Materials Science for the administrative helps.

I thank the National Science Foundation and Honda Research Institute for fund- ing this research. I also thank San Diego Super Computers (SDSC), Texas Ad- vanced Computing Center (TACC), National Center for Supercomputing Applica- tions (NCSA), and Pittsburgh Supercomputing Center (PSC) for all the computing time through Teragrid projects. Thanks also go to Duke Clusters for additional allocations.

I can never thank enough to my father whose talent in numbers has introduced me to math and engineering, my mother and my sisters for their endless prayers, love, and encouragements throughout my life. I am deeply grateful to be blessed with a brilliant, loving, and beautiful wife, Prof. Lisa M. Peloquin, who provides continuous prayers, love, and supports in so many ways. I also thank our friends especially Laura Heymann for her kind helps in many occassions. Thanks also go to Pepito, Clementine, Hamzah, John and other aquatic species for being excellent family members.

Finally, all praises are due to Allah, the Creator (al-Khaliq), the Loving (al- Rahman), and the All Knowing (al-Alim). I am thankful for all the opportunities, health, and blessings that enable me to finish this dissertation. I am deeply humbled by every piece of knowledge that I learned.

xvii

Chapter 1

Introduction

Quasicrystals (QCs) were discovered in 1982 by Dr. Shechtman during his X-ray mea-

surements on Al-Mn compounds. Similar to crystals, QCs consist of atoms arranged

in regular patterns having long-range order, i.e. the diffraction patterns show discrete

spots. However, they do not have any translational periodicities. The discrete spots

come from the rotational symmetries. A variety of stable and metastable QCs have

been successfully synthesized. Among the first high-quality samples are icosahedral

AlCuFe [1], decagonal AlNiCo [2], and icosahedral AlPdMn [3]. Today, hundreds of

quasicrystalline phases are known, tens of which are stable [4]. The majority of them

are derived from aluminum-transition metal family [5].

QCs have been shown to have lower coefficients of friction than most metals. For

example, the static friction μs between two clean surfaces of icosahedral AlPdMn is

≈ 0.6 [6], whereas μs for Ni(110) and Cu(111) is ≈ 4 [7]. Kinetic friction tests using pin-on-disk technique with diamond pin show that among materials with compara-

ble hardness included in the study, icosahedral AlPdMn exhibits the lowest friction

(μ = 0.05), compared to window glass (μ = 0.08), sintered Al2O3 (μ = 0.13), or

hard Cr-steel (μ = 0.13) [8]. Detailed measurements of friction as a function of struc-

tural perfection in icosahedral AlCuFe quasicrystal show that the minimum friction is

achieved for sample with the best quasilattice perfection [9]. Decagonal AlNiCoSi has

been verified to have lower friction than Cr2O3, which represents the most advanced

technology for use on piston rings in automotive engines [10]. Further evidence of

reduced friction is demonstrated in decagonal AlNiCo, in which the friction on the

2-fold periodic surface is eight times higher than that on the decagonal surface [11]

1

(Note that in decagonal quasicrystals, there is a direction along which the quasicrys-

talline surfaces are stacked periodically).

The reduced friction between two quasicrystalline surfaces can be understood by

considering the structure commensurability between them. For surfaces with enough

hardness, atoms can be regarded as fixed in their position in each material. Within

this approximation, it has been theoretically demonstrated that the total interaction

energy between the two surfaces is independent of their relative displacement parallel

to the interface [12]. Therefore, the frictional force which is the gradient of the en-

ergy with respect to the displacement is vanishingly small, a phenomenon known as

superlubricity [12]. Superlubricity has been observed experimentally between tung-

sten and graphite in which certain relative orientations result in nearly zero friction

beyond the limit of the instruments [13]. Since any two QCs do not have common

periodicities at any length scales, superlubricity is expected.

Another characteristic feature of QCs is their resistance to oxidation, which is

quite surprising given that their main constituent, aluminum, is readily oxidized in

ambient conditions [5]. The behavior is particularly spectacular in AlCuLi icosahedral

phase that resists oxidation very well in humid air [5]. The combination of high

hardness, oxidation resistance, and low friction has attracted interests in QCs as

coatings to reduce friction and wear in machine parts, e.g. at the piston-cylinder

interface and in gear boxes. In such environments, hydrocarbon-based oil lubricant

is typically used to overcome the friction caused by surface asperities. Therefore,

to yield a synergic performance of self-lubricating QC coating and lubricant, it is

important to understand the interactions between them. The lubricant must be

able to spread (wet) well on the QC. Furthermore, the structures of the thin film

of lubricant formed on the sliding surfaces which will affect the lubricity need to be

investigated.

2

In this work, we study the process of thin film growth of hydrocarbons and rare

gases adsorbed on a decagonal Al73Ni10Co17 quasicrystal (d-AlNiCo) via computer

simulations. A high-quality and large-size single grain d-AlNiCo has been routinely

grown, making it an excellent substrate to study adsorption [14, 15, 16]. Chapter 3 is

devoted to the adsorptions of rare gases. The absence of chemical reactivity of noble

gases will be utilized to elucidate the effects of quasicrytallinity of the substrate on the

growth and structure of the adsorbed film. In Chapter 5, the simulation results for

selected hydrocarbons are presented. We study the adsorption behaviors of propane,

hexane, and octane. From these molecules, we may extrapolate the results for longer

alkanes which constitute typical oil lubricants. In addition, methane and benzene are

also studied due to their interesting symmetries.

We develop classical many-body interatomic potentials of Al, Ni, Co, C, and H

which will be suitable for simulating hydrocarbons adsorptions on Al-Ni-Co involving

thousands of atoms. To our knowledge, such potentials have not existed. The poten-

tials are based on the embedded-atom method (EAM) [17] with parameters fitted to

electronic energies of various structures evaluated via first principle quantum calcu-

lations. The generation of the EAM potentials is presented in Chapter 4. In Chapter

6, we extend the procedures to develop a consistent set of EAM potentials for pure

elements in the periodic table. These potentials are also fitted to ab initio energies.

A consistent set of EAM potentials having the same parametrizations enables one to

use the same computer code for different potentials.

This manuscript contains a different reserach which is part of our doctoral work,

namely in the field of nanocatalysis for synthesis of single-walled carbon nanotubes

(SWCNTs). In Chapter 7, we present the calculation of phase diagram of Fe-Mo-

C system under pressure for studying the effects of Mo in the thermodynamics of

Fe:Mo nanoparticles as catalysts for growing SWCNTs. The decomposition of two

3

Fe:Mo particles namely FeMo and Fe4Mo into various stable phases are analyzed.

The implications of the formation of these phases to the growth of SWNCTs as well

as the estimated minimum size of nanotubes that can be produced are discussed.

4

Chapter 2

Methods

2.1 Density functional theory

2.1.1 Born-Oppenheimer approximation

In a calculation of electronic structure of materials, one solves an eigenvalue problem

of time-independent Schro¨dinger equation:

HΨ(R, r) = EΨ(R, r) (2.1)

where R ∈ {Rn} and r ∈ {ri} are the vector coordinates of the nuclei and electrons, respectively. The energy operator (Hamiltonian), H, is given by

H = − ∑ n

� 2

2Mn �2Rn−

∑ i

� 2

2mi �2ri+

∑ m<n

e2ZmZn Rn −Rm +

∑ i<j

e2

rj − ri − ∑ n,i

e2Zn ri −Rn (2.2)

The first two terms represent the kinetic energies of the nuclei and the electrons,

respectively. The third and fourth terms represent the nucleus-nucleus and electron-

electron potential energies. The last term is the nucleus-electron energy. It is compu-

tationally beyond the capability of current computers to solve Equation 2.1 using the

full Hamiltonian. An approximation, known as Born-Oppenheimer approximation,

is made by realizing that nuclei are significantly heavier than electrons so the nuclei

move much more slowly than the electrons. The electrons can adapt themselves to

the current configuration of nuclei. Using this approximation, we can decouple the

electronic and the ionic parts of the Hamiltonian. Consider the nuclei fixed at a given

5

configuration, α, and solve the following electronic Schro¨dinger equation:

[ − ∑

i

� 2

2mi �2ri +

∑ i<j

e2

rj − ri − ∑ n,i

e2Zn ri −Rαn

] ψα(r) = Eα(R)ψα(r) (2.3)

The total energy, E, is calculated by taking the electronic contribution, Eα(R), as a

potential energy operator in the ionic Schro¨dinger equation:

[ − ∑ n

� 2

2Mn �2Ri +

∑ m<n

e2ZmZn Rm −Rn + E

α(R)

] Φα(R) = EΦα(R) (2.4)

Mostly, one is interested only in the electronic part, i.e. Equation 2.3. Even though

Equation 2.3 contains only the electronic part of the system, the large number of

variables (e.g. coordinates of all electrons) makes it remain intractable. In addition,

it is the electron-electron interaction that makes the problem so difficult to solve. This

interaction is a result of correlation between electrons (the probability of finding an

electron depends on where the rest of the electrons are) and the fact that electrons

are fermions requiring antisymmetric wavefunctions (the many-electron wavefunction

gains a factor of -1 everytime two electrons exchange their coordinates). If this term

were absent, the Hamiltonian would be just a sum of many one-electron Hamiltonians,

known as independent electron approximation.

2.1.2 Slater determinant

An antisymmetric N -electron wavefunction can be constructed from N one-electron

wavefunctions using Slater determinant [18] defined as:

Ψ(x1,x2, . . . ,xN) = 1√ N !

ψ1(x1) ψ2(x1) . . . ψN(x1) ψ1(x2) ψ2(x2) . . . ψN(x2) . . . . . . . . . . . .

ψ1(xN) ψ2(xN) . . . ψN(xN)

(2.5)

6

where ψk(xi) denotes the k-th one-electron wavefunction being occupied by an elec-

tron with spin-orbital coordinate xi = (si, ri), with si being the spin state and ri the

spatial coordinate. For this reason, ψk is also called the spin-orbital k. The factor 1√ N !

arrives from the normalization of the total wavefunction and orthonormality among

the spin orbitals:

∫ ΨΨ∗dx1 . . . dxN = 1 (2.6)

∫ ψk(r)ψ

∗ k(r)dr = 1 (2.7)

∫ ψk(r)ψ

∗ l (r)dr = 0 (2.8)

Using the orthonormality of the spin-orbitals, it can be shown that the charge density

of the Slater determinant can be written as n(x) = ∑

k |ψk(x)|2.

2.1.3 Hartree-Fock equations

In the electronic Schro¨dinger equation (Equation 2.3), the Hamiltonian can be written

as follows:

H = ∑

i

h(i) + 1

2

∑ i�=j

g(i, j) (2.9)

h(i) ≡ −1 2 �2i −

∑ n

Zn |ri −Rn| (2.10)

g(i, j) ≡ 1|rj − rj| (2.11)

where h(i) depends only on ri and g(i, j) depends on ri and rj. The energy of

the system is calculated by taking the expectation value of the Hamiltonian in the

total wavefunction, E = 〈Ψ|H|Ψ〉. By employing the orthonormality of ψk as in the

7

calculation of charge density n(x), we have

〈Ψ| ∑

i

h(i)|Ψ〉 = ∑ k

〈ψk|h|ψk〉 = ∑ k

∫ dxψ∗k(x)h(r)ψk(x) (2.12)

Note that in the first equation, when |Ψ〉 is written as a Slater determinant, only i = k appears due to orthonormality of ψk, and the summation over electron index i inside

the many-electron wavefunction |Ψ〉 becomes a summation over k inside individual spin-orbital ψk, and we can drop the index h(i = k) for convenience. The integral∫

dx denotes integral over spatial coordinates and a sum over the spin-degrees of

freedom. Similarly, we have

〈Ψ| ∑ i,j

g(i, j)|Ψ〉 = ∑ k,l

〈ψkψl|g|ψkψl〉 − ∑ k,l

〈ψkψl|g|ψlψk〉 (2.13)

〈ψkψl|g|ψmψn〉 = ∫

dx1ψ ∗ k(x1)

[∫ dx2ψ

∗ l (x2)

1

|r1 − r2|ψn(x2) ] ψm(x1)(2.14)

The total energy is then

E = ∑ k

〈ψk|h|ψk〉+ 1 2

∑ k,l

[〈ψkψl|g|ψkψl〉 − 〈ψkψl|g|ψlψk〉] (2.15)

This expression shows that the electron-electron interaction, g(i, j), consists of two

terms, the first one is known as Coulomb energy (or Hartree term), and the second

one is the exchange energy term. To see how this derivation also reduces the many-

electron Schro¨dinger equation to a set of one-electron equations, we differentiate the

energy with respect to a particular spin-orbital, e.g. spin-orbital 〈ψi|. Note that i is not electron index, but a specific value of spin-orbital index k, after the derivation,

8

we may replace i by k to conform to the usual notation:

δE

δ〈ψi| = h|ψi〉+ 1

2

[∑ l

〈ψl| 1|r− r′| |ψl〉 ] |ψi〉+ 1

2

[∑ k

〈ψk| 1|r− r′| |ψk〉 ] |ψi〉

−1 2

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x)ψi(x ′)− 1

2

∑ k

∫ dxψ∗k(x

′) 1

|r− r′|ψi(x)ψk(x ′)

(2.16)

By changing indices (k → l) in the third term and (k → l,x→ x′) in the last term, we get

δE

δ〈ψi| = h|ψi〉+ [∑

l

〈ψl| 1|r− r′| |ψl〉 ] |ψi〉 −

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x)ψi(x ′)

(2.17)

This expression is known as Fock operator acting on |ψi〉. Replacing back i by k, and defining the eigenvalues of Fock operator as �k, we arrive at the Hartree-Fock

equation:

HHFψk = �kψk (2.18)

HHFψk =

[ −�

2

2 − ∑ n

Zn |r−Rn|

] ψk(x) +

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x ′)ψk(x)

− ∑

l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψk(x ′)ψl(x)

(2.19)

Note that the sum of the eigenvalues �k is not the total energy E. However, by

comparing Equation 2.15 and 2.19, E can be recovered using the following equation

E = 1

2

∑ k

[�k + 〈ψk|h|ψk〉] (2.20)

The last term (the exchange term) in Equation 2.19 is nonlocal because the Hamil-

tonian HHF operates on ψk(x) at a particular x, but the operator itself is a function

9

of ψk(x ′) at all possible x′. This nonlocality makes the Hartree-Fock Hamiltonian

difficult to evaluate in large systems involving many atoms and electrons, hence not

suitable for solids. Most electronic structure calculations for solids are based on

density functional theory (DFT) discussed in the following sections.

2.1.4 Kohn-Sham equations

In DFT, the nonlocal exchange term is given by an effective exchange potential that

depends on electronic charge density. Furthermore, all other potential operators are

expressed in charge density rather than in spin-orbitals. This approach reduces the

number of degrees of freedom significantly since all electron coordinates that enter in

the spin-orbitals are now replaced by charge density that is a function only on one

coordinate r. The DFT energy functional is given by

E(n) = T (n) +

∫ Vext(r)n(r)dr+

1

2

∫∫ n(r′)

1

|r− r′|n(r)dr ′dr+ Exc(n) (2.21)

where T being the electronic kinetic energy and Vext being the electron-nuclei electro-

static potential. We know that for given wavefunctions we can calculate the charge

density, n(r) = ∑

k |ψk(r)|2, however the reverse is not obvious. The formality that proves there exists one-to-one mapping between charge density and wavefunctions

was developed by Hohenberg-Kohn [19]. Hohenberg-Kohn theorem also proves that

there exists Exc(n) that will produce the exact ground state of the system. The DFT

Schro¨dinger equation can be derived by taking the variation of E(n) with respect to

n:

δE

δn =

δT

δn + Vext +

∫ n(r′)dr′

|r− r′| + δExc δn

(2.22)

As we have derived the Hartree-Fock equation (Equation 2.19), if we write the DFT

many-electron wavefunction as a Slater determinant of spin-orbitals and orthonor-

10

mality among the spin-orbitals, then each DFT spin-orbital equation satisfies

[ −1 2 �2 + Veff (r)

] ψk(r) = �kψk(r) (2.23)

Veff (r) ≡ Vext(r) + ∫

n(r′)dr′

|r− r′| + δExc δn

(2.24)

The total energy is related to the eigenvalues �k as follows

E = ∑ k

�k − 1 2

∫∫ n(r′)

1

|r− r′|n(r)dr ′dr+ Exc(n)−

∫ δExc(n)

δn n(r)dr (2.25)

Equations 2.24-2.25 are known as Kohn-Sham equations. Since the potential opera-

tors depend on charge density, Equation 2.24 must be solved self-consistently. One

starts from a trial electronic charge density to construct the potentials and solve for

the eigenvalues and spin-orbitals wavefunctions. A new charge density is then con-

structed from the spin-orbitals and the procedure is repeated until the charge and

the wavefunctions are self-consistent within a certain accuracy.

2.1.5 Local density and generalized gradient approximations

In Kohn-Sham equations, there are two terms involving exchange interaction, namely

Exc(n) and δExc/δn. For a given charge density, the first term (the exchange energy)

does not depend on the charge functional on r, whereas the second term (the ex-

change potential) will depend on r if the charge density does. This means that for

a nonhomogeneous systems, exchange potential can be expanded in charge density

and its derivatives:

Vxc ≡ δExc(n) δn(r)

= Vxc(n(r), |�n(r)|, |�(�(n(r)))|, ...) (2.26)

As a first approximation, one neglects all the gradients of charge density in the

exchange potential, this is known as local density approximation (LDA) since the

11

exchange potential depends on charge density only at a particular value of r. It

means that LDA gives an exact ground state for homogeneous electron gas since in

this case the gradients vanish. This allows one to write the LDA exchange energy as

a sum of exchange energy per electron in a homogeneous electron gas, �homxc (n):

ELDAxc =

∫ �homxc (n)ndr (2.27)

LDA also gives accurate results for systems where the charge density does not vary

too rapidly such as in metals. For nonhomogenous systems such as transition metals,

semiconductors, or slabs, LDA is known to underestimate the energy (hence the band

gap). A more accurate approximation, known as generalized gradient approximation

(GGA), includes the first gradient of charge and the exchange energy is given by

EGGAxc =

∫ �GGAxc (n(r), |∇n(r)|)n(r)dr (2.28)

Several techniques exist to parametrize �GGAxc : Perdew-Wang 1986 (PW86) [20, 21],

Perdew-Wang 1991 (PW91) [22], Becke [23], Lee-Yang-Parr (LYP) [24], and Perdew-

Burke-Enzerhof (PBE) [25, 26]. In this work, we use GGA-PBE functional for the

exchange energy.

2.1.6 Pseudopotentials

Kohn-Sham equations can be solved in different ways depending on the choice of

potentials and basis functions to expand the wavefunctions. Some considerations in

solving the Kohn-Sham equations are: (1) potentials becomes very strong near the

nuclei, whereas at far regions they are relatively weak, (2) wavefunctions fluctuate

more near nuclei than in the interstitial regions, (3) the symmetry of the potentials are

approximately spherical near the nuclei whereas at larger distances, the symmetry of

the crystal dominates. Some of the known methods are augmented plave wave (APW)

12

[27, 28], linearized augmented plane wave (LAPW) [29, 30], Orthogonalized plave-

wave (OPW) [31], and pseudopotential method [32, 33]. LAPW is the most accurate

method available. It uses spherical harmonics to expand the wavefunctions in the core

region near the nuclei and plane waves for the interstitial region. Pseudopotential

methods use only plane wave basis set. The number of plane waves can be kept

small by treating an atom as consisting of an effective core (nucleus + core electrons)

and valence electrons. Even though pseudopotentials are not as accurate as LAPW

(which as a contrast fall into the class of fullpotential methods), it provides good

results and becomes the most common choice for its lower computational costs than

LAPW.

2.2 Vienna Ab initio Simulation Package

All ab initio quantum calculations in this work are done using Vienna Ab initio Sim-

ulation Package (VASP) [34, 35]. The calculations are performed in the generalized

gradient approximation (GGA) [36, 37] with exchange correlation as parametrized

by Perdew, Burke, and Ernzerhof (PBE) [25]. To reduce the number of plane waves,

projector augmented-wave (PAW) pseudopotentials are used [38, 39].

VASP uses a plane wave basis set to expand the wave functions in solving the

Kohn-Sham equation in reciprocal space. The evaluation of total electronic energy per

unit cell is done by integrating the energy of all electronic states in the Brillouin zone.

In practice, the Brillouin zone is divided into grids and the integration is replaced by

a sum over k-points. In our work, the k-point grids are generated automatically using

Monkhorst-Pack scheme [40]. For hexagonal structures, an additional shift is used

so that the grids are centered at the Γ point (k = (0, 0, 0)). Typical setting of the

number of k-points as many as 3000/number-of-atoms is usually sufficient to achieve

a converged energy with an accuracy better than 10 meV/atom. Unless otherwise

13

stated, all structures are fully relaxed: atoms position as well as unit cell’s size and

shape are allowed to relax to find the equillibrium configurations.

2.3 Classical interatomic potentials

Due to QC’s lack of periodicity, a large enough simulation cell is necessary to cap-

ture the effect of substrate’s structure on the adsorbed films. A typical cell con-

tains thousands of atoms which makes the simulation unpractical to be performed

quantum-mechanically. Therefore, classical interatomic potentials are needed. For

simple systems, e.g. adsorption of noble gases, pair potentials such as Lennard-Jones

[41] or Morse [42] are sufficient. However, for complex systems, such as adsorption of

hydrocarbons, more accurate potentials are required to take into account many-body

effects in these systems, especially covalent bonds involving carbons which are higly

directional.

Several methods exist to incorporate many-body interations into classical poten-

tials, e.g. force field method (FF) [43, 44, 45, 46], Cluster Expansion method (CE)

[47, 48, 49], and embedded-atom method (EAM) [17, 50]. FF is based on energy-bond

order-bond length relationship [44] and is mostly used for biological systems [43, 51]

and chemical systems where bond formation and breaking are allowed [46]. In CE,

energy of a system is approximated by a converging sum over cluster contributions,

where contribution from large clusters are negligible. In nonperiodic systems, such as

amorphous and quasicrystalline phases, the number of clusters needed can be large

to reach the desired accuracy, therefore CE is mostly suitable to evaluate ground

state energy in periodic systems. EAM is based on the close relationship between

electronic charge density and energy of the system. In quantum physics, this relation-

ship becomes the basis for the density functional theory (DFT) [52, 19, 53], in which

it has been proven that there exists a one-to-one mapping between charge densities

14

and electronic wave functions, and hence energies [19]. Due to the universality of

charge density, in principle EAM can be used in any systems.

EAM was originally developed for examining metallic bulks and surfaces [17, 50].

Later on, charge screening methods have been proposed to modify EAM for use in

covalent systems such as Si [54] and Ge [55]. EAM has been successfully employed

to simulate surface relaxation/reconstructions [56, 57, 58, 59], film growth [60, 61],

and diffusion processes [62, 63]. In this study, we will develop EAM potentials to

simulate hydrocarbons adsorption on d-AlNiCo surface.

In EAM formalism [17, 50], each atom is viewed as being embedded in the material

consisting of all other atoms. The total energy of a system is defined by

Etot ≡ ∑

i

Fi(ρ¯i) + 1

2

∑ i

∑ j �=i

φij(rij), (2.29)

where Fi is the embedding energy of atom i, ρ¯i is the electron density at the vector

position ri, φij is the pair potential, and rij is the distance between atoms i and j.

If ρ¯i is approximated as a sum of individual contribution of the constituents [i.e.,

ρ¯i = ∑ j �=i

ρj(rij), where ρj is the atomic electron density of atom j, the energy is then

only a function of the position of atoms.

For an elemental potential, there are 3 functions needed: F (ρ¯), ρ(r), and φ(r).

For a binary system AB, we will need 7 functions: FA(ρ¯), ρA(r), φA(r), FB(ρ¯),

ρB(r), φB(r), and a cross-pair potential φAB(r). The cross-pair potential is needed

to calculate pair interaction of atoms of different types. In general, in a system

consisting of N different elements, we need N(N + 5)/2 functions.

15

We will use the following functional forms:

ρA(r) = ρAe e −βA(r/rAe −1), (2.30)

φAA(r) = DA[e−2α A(r−rA0 ) − 2e−αA(r−rA0 )], (2.31)

FA(ρ¯) = cubic− splines, (2.32)

φAB(r) = γAB

2 φAA(r) +

1

2γAB φBB(r). (2.33)

In the above equations, A and B denote atom type. ρe, re, and β will be taken from

the database of atomic electron density [64]. Equation 2.31 follows Morse potential

form [42]. The embedding function F will be taken as a natural cubic spline [65].

The expression for the cross-pair interaction φAB follows Haftel’s derivation [59] in

which γAB ≡ ZB/ZA (ZA is the effective charge of the core for atom type A).

2.4 Simplex method

The EAM potentials parameters need to be fit to quantum calculations. The fitting

will be performed using simplex method [66]. Simplex does not require evaluation

of function derivatives which makes it simple to implement. Simplex minimizes N -

dimensional function f by creating P > N simplex points. A simplex point is f

evaluated at a given coordinate. The simplex points will form P -polytope. A min-

imization move is made by replacing the maximum simplex point, hence the worst,

with a point reflected through the centroid of the remaining (P − 1)-polytope. Ex- pansion or shrinking of the polytope are allowed to overcome some local minima or

to converge, respectively.

16

2.5 Grand canonical Monte Carlo

The adsorption simulation will be performed using grand canonical Monte Carlo

(GCMC) method [67, 68, 69]. At constant temperature, T , and volume, V , the

GCMC method explores the configurational phase space using the Metropolis al-

gorithm and finds the equilibrium number of adsorbed atoms (adatoms), N , as a

function of the chemical potential, μ, of the gas, i.e. configuration(μ,N, V ). The

adsorbed atoms are in equilibrium with the coexisting gas: the chemical potential of

the gas is constant throughout the system. In addition, the coexisting gas is taken

to be ideal. With this method we determine adsorption isotherms, ρN , and density

profiles, ρ(x, y), as a function of the pressure, P (T, μ). For each data point in an

isotherm, we perform at least 18 million GCMC steps to reach equilibrium. Each

step is an attempted displacement, creation, or deletion of an atom with execution

probabilities equal to 0.2, 0.4, and 0.4, respectively [70, 71, 72]. At least 27 million

steps are performed in the subsequent data-gathering and -averaging phase.

17

Chapter 3

Noble gas adsorptions on d-AlNiCo

The observed unusual electronic [73, 74] and frictional [75, 76, 77] properties of qua-

sicrystal surfaces stimulate interesting fundamental questions about how these and

other physical properties are altered by quasiperiodicity. Recent progress in the char-

acterization and preparation of quasicrystal surfaces raises new possibilities for their

use as substrates in the growth of films having novel structural, electronic, dynamic

and mechanical properties [78, 79, 80]. The physical behavior of systems involving

competing interactions in adsorption is a subject of continuing interest and is par-

ticularly relevant to the growth of thin films [69]. Several different growth modes

have been observed for the growth of metal films on quasicrystals [81, 82, 83, 84]. A

form of competing interactions seen in adsorption involves either a length scale or a

symmetry mismatch between the adsorbate-adsorbate interaction and the adsorbate-

substrate interaction [85, 86]. Some consequences of such mismatches include den-

sity modulations [87, 88], domain walls [89], epitaxial rotation in the adsorbed layer

[90, 91, 92, 93, 94, 95], and a disruption of the normal periodicity and growth in the

film [96, 97, 98].

The wide range of behavior observed so far indicates that, even in the absence of

intermixing, film growth is strongly affected by chemical interactions between adsor-

bate and substrate. In order to separate these chemical effects from those specific to

quasiperiodic order, we have studied the adsorption of noble gases on a quasicrystal

surface, where both the gas-gas and gas-surface interactions are believed to be simple,

i.e., appreciable chemical interactions and adsorbate-induced surface reconstructions

are absent. In this chapter, we explore the implications of structural mismatch by

18

evaluating the nature of Ne, Ar, Kr, and Xe adsorption on a quasicrystal substrate,

namely the 10-fold surface of decagonal Al73Ni10Co17 quasicrystal (d-AlNiCo QC)

[15, 16].

3.1 Method

3.1.1 Simulation cell

The simulation cell is tetragonal. We take a square section of the surface, A, of side

5.12 nm, to be the (x, y) part of the unit cell in the simulation, for which we assume

periodic boundary conditions along the basal directions. Although this assumption

limits the accuracy of the long range QC structure, it is numerically necessary for

these simulations. To minimize the long range interaction corrections, a relatively

large cutoff (5σgg) is used. Since the size of the cell is relatively large compared to that

of the noble gases, the cell is accurately representative of order on short-to-moderate

length scales. The height of the cell, along the z (surface-normal) direction, is chosen

to be 10 nm (long enough to contain ∼20 layers of Xe). At the top of the cell, a hard-wall reflective potential is employed to confine the coexisting vapor phase. The

simulation results for Xe over d-AlNiCo, presented below, are consistent with both

our results from experiments [99] and virial calculations [100]. Hence, the calculations

may also be accurate for other systems.

3.1.2 Gas-gas and gas-substrate interactions

The gas-gas and gas-substrate interactions are modeled using Lennard-Jones (LJ) 12-

6 potentials, with the gas-gas parameter values �gg and σgg listed in Table 3.1. The

gas-substrate interactions are obtained by summing pair potentials for a gas atom

and all of the substrate atoms in an eigth-layer slab: Al, Ni and Co [15, 101, 100]. The

position of the atoms in the eight-layer slab are taken from the results of a low-energy

19

Table 3.1: Parameter values for the 12-6 Lennard-Jones interactions. TM is the label for Ni or Co.

�gg σgg �gas−Al σgas−Al �gas−TM σgas−TM (meV) (nm) (meV) (nm) (meV) (nm)

Ne 2.92 0.278 9.40 0.264 9.01 0.249 Ar 10.32 0.340 17.67 0.295 16.93 0.280 Kr 14.73 0.360 21.11 0.305 20.23 0.290 Xe 19.04 0.410 24.00 0.330 23.00 0.315

iNe(1) 2.92 0.410 5.45 0.330 5.22 0.315 dXe(1) 19.04 0.278 41.39 0.264 39.67 0.249 dXe(2) 19.04 0.390 25.88 0.320 24.80 0.305 iXe(1) 19.04 0.550 14.96 0.400 14.34 0.385 iXe(2) 19.04 0.675 10.52 0.462 10.08 0.447

electron diffraction LEED analysis of the surface structure of d-AlNiCo [15]. The

gas-substrate interaction parameters are derived using conventional combining rules,

σAB = (σA + σB)/2 and �AB = √

�A�B [102], and experimental heats of adsorption

[100, 99, 103]. The LJ gas-substrate parameters are �gas−Al and σgas−Al for Al, and

�gas−TM and σgas−TM for the two transition metals Ni and Co. All these values are

listed in the upper part of Table 3.1. In the calculation of the adsorption potential,

we assume a structure of the unrelaxed surface taken from the empirical fit to LEED

data [16].

3.1.3 Adsorption potentials

Figures 3.1(a), 3.1(c), 3.1(e), and 3.1(g) show the function Vmin(x, y) of Ne, Ar, Kr,

and Xe on the d-AlNiCo, respectively, which is calculated by minimizing the adsorp-

tion potentials, V (x, y, z), along the z direction at every value (x, y) coordinates:

Vmin(x, y) ≡ min {V (x, y, z)}|along z . (3.1)

20

The figures reveal the fivefold rotational symmetry of the substrate. Dark spots

correspond to the most attractive regions of the substrate. By choosing appropriate

sets of five dark spots, we can identify pentagons, whose sizes follow the inflationary

property of the d-AlNiCo. Note the pentagon at the center of each figure: it will be

used to extract the geometrical parameters λs and λc in Section 3.2.5.

To characterize the corrugation, not well-defined for aperiodic surfaces, we calcu-

late the distribution function f(Vmin), the average 〈Vmin〉 and standard deviation SD of Vmin(x, y) as:

f(Vmin)dVmin ≡ probability { Vmin ∈ [Vmin, Vmin + dVmin[

} (3.2)

〈Vmin〉 ≡ ∫ ∞ −∞

f(Vmin)Vmin dVmin, (3.3)

SD2 ≡ ∫ ∞ −∞

f(Vmin)(Vmin − 〈Vmin〉)2 dVmin. (3.4)

Figures 3.1(b), 3.1(d), 3.1(f), and 3.1(h) show f(Vmin) of the adsorption potential for

Ne, Ar, Kr, and Xe on the d-AlNiCo, respectively. Vmin(x, y) extends by more than

2·SD around its average, revealing the high corrugation of the gas-surface interaction in these four systems. The average and SD of Vmin(x, y) for these systems are listed

in the upper part of Table 3.2. In addition to highly corrugated, the potentials are

“deep” because the record maximum well-depth, e.g. for Xe, on a periodic surface is

about 160 meV, viz. on graphite [104]; and the record minimum well-depth is about

28 meV, on Cs [105].

3.1.4 Effective parameters

For every gas-substrate interaction we define two effective parameters σgs and Dgs.

σgs represents the averaged LJ size parameter of the interaction, calculated following

21

-70 -65 -60 -55 -50 -45 -40 -35 Vmin (meV)

f(V m

in )

(a)

(b)

-180 -160 -140 -120 -100 Vmin (meV)

f(V m

in )

(c)

(d)

-220 -200 -180 -160 -140 -120 Vmin (meV)

f(V m

in )

(e)

(f)

-280 -260 -240 -220 -200 -180 -160 Vmin (meV)

f(V m

in )

(g)

(h)

Ne/QC Ar/QC

Xe/QCKr/QC

Figure 3.1: (color online). Computed adsorption potentials for (a) Ne, (c) Ar, (e) Kr, and (g) Xe on the d-AlNiCo, obtained by minimizing V (x, y, z) with respect to z. The distribution of the minimum value of these potentials is plotted in (b, d, f, and h) respec- tively: the solid line marks the average value 〈Vmin〉, the dashed lines mark the values at 〈Vmin〉±SD.

22

Table 3.2: Range, average (〈Vmin〉), and standard deviation (SD) of the interaction Vmin(x.y) on the d-AlNiCo. Effective parameters of the gas-substrate interactions (Dgs, σgs, D�gs, σ�gs), and, for comparison, the best estimated well depths DGrgs on graphite [106].

Vmin range < Vmin > SD Dgs σgs D � gs σ

� gs D

Gr gs

(meV) (meV) (meV) (meV) (nm) (Dgs/�gg) (σgs/σgg) (meV) Ne -71 to -33 -47.43 6.63 43.89 0.260 15.03 0.935 33 Ar -181 to -85 -113.32 13.06 108.37 0.291 10.50 0.856 96 Kr -225 to -111 -145.71 15.68 140.18 0.301 9.52 0.836 125 Xe -283 to -155 -195.46 17.93 193.25 0.326 10.15 0.795 162

iNe(1) -65 to -36 -45.11 4.08 43.89 0.326 15.03 0.795

dXe(1) -305 to -150 -207.55 29.18 193.25 0.260 10.15 0.935

dXe(2) -295 to -155 -199.40 19.33 193.25 0.316 10.15 0.810

iXe(1) -248 to -170 -195.31 11.21 193.25 0.396 10.15 0.720

iXe(2) -230 to -180 -194.25 7.77 193.25 0.458 10.15 0.679

the traditional combining rules [102]:

σgs ≡ xAlσg−Al + xNiσg−Ni + xCoσg−Co, (3.5)

where xAl, xNi, and xCo are the concentrations of Al, Ni, and Co in the QC, respec-

tively. Dgs represents the well depth of the laterally averaged potential V (z):

Dgs ≡ −min {V (z)}|along z . (3.6)

In addition, we normalize the σgs and Dgs with respect to the gas-gas interactions:

σ�gs ≡ σgs/σgg, (3.7)

D�gs ≡ Dgs/�gg. (3.8)

The values of the effective parameters σgs, Dgs, σ � gs, and D

� gs for the four gas-surface

interactions are listed in the upper part of Table 3.2. We also include the well depth

for Ne, Ar, Kr, and Xe on graphite, as comparison [106].

3.1.5 Test rare gases

As shown in tables 3.1 and 3.2, Ne is the smallest atom and has the weakest gas-gas

and gas-surface interactions (minima of σgg, σgs, �gg and Dgs). In addition, Xe is the

23

largest atom and has the strongest gas-gas and gas-surface interactions (maxima of

σgg, σgs, �gg and Dgs). Therefore, for our analysis, it is useful to consider two test

gases, iNe(1) and dXe(1), which are combinations of Ne and Xe parameters.

iNe(1) represents an “inflated” version of Ne, having the same gas-gas and average

gas-substrate interactions of Ne but the geometrical dimensions of Xe:

{�gg, Dgs, D�gs}[iNe(1)] ≡ {�gg, Dgs, D�gs}[Ne], (3.9)

{σgg, σgs, σ�gs}[iNe(1)] ≡ {σgg, σgs, σ�gs}[Xe]. (3.10)

dXe(1) represents a “deflated” version of Xe, having the same gas-gas and average

gas-substrate interactions of Xe but the geometrical dimensions of Ne:

{�gg, Dgs, D�gs}[dXe(1)] ≡ {�gg, Dgs, D�gs}[Xe], (3.11)

{σgg, σgs, σ�gs}[dXe(1)] ≡ {σgg, σgs, σ�gs}[Ne]. (3.12)

The resulting LJ parameters for iNe(1) and dXe(1) are summarized in the central

parts of Tables 3.1 and 3.2. Furthermore, we also define three other test versions of

Xe: dXe(2), iXe(1), and iXe(2) which have the same gas-gas and average gas-substrate

interactions of Xe but deflated or inflated geometrical parameters. The last three test

gases will be used in Section 3.2.5. The LJ parameters for these gases are summarized

in the lower parts of Tables 3.1 and 3.2. In simulating test gases, we implicitly rescale

the substrate’s strengths so that the resulting adsorption potentials have the same

Dgs as the non-inflated or non-deflated ones (Equations 3.9 and 3.11).

24

3.1.6 Chemical potential, order parameter, and ordering tran-

sition

To conveniently characterize the evolution of the adsorption processes of the gases

we define a normalized chemical potential μ�, as:

μ� ≡ μ− μ1 μ2 − μ1 , (3.13)

where μ1 and μ2 are the chemical potentials at the onset of the first and second layer

formation, respectively. In addition, we introduce the order parameter ρ5−6, defined

as the probability of existence of fivefold defect [70, 71]:

ρ5−6 ≡ N5 N5 + N6

, (3.14)

where N5 and N6 are the numbers of atoms having 2D coordination equal to 5 and

6, respectively. The 2D coordination is the number of neighboring atoms within a

cutoff radius of aNN · 1.366 where aNN is the first nearest neighbor (NN) distance of the gas in the solid phase and 1.366 = cos(π/6) + 1/2 is the average factor of the

first and the second NN distances in a triangular lattice. Note that aNN does not

change appreciably with respect to temperature difference, e.g. aNN of Xe changes

from 0.440 nm at 77 K to 0.443 nm at 140 K.

In a fivefold ordering, most arrangements are hollow or filled pentagons with atoms

having mostly five neighbors. Hence, the particular choice of ρ5−6 is motivated by the

fact that such pentagons can become hexagons by gaining additional atoms with five

or six neighbors. Definition: the five to sixfold ordering transition is defined as a

decrease of the order parameter to a small or negligible final value. The phenomenon

can be abrupt (first-order) or continuous. Within this framework, ρ5−6 and (1−ρ5−6) can be considered as the fractions of pentagonal and triangular phases in the film,

respectively.

25

3.2 Results

3.2.1 Adsorption isotherms

Figure 3.2 shows the adsorption isotherms of Ne, Ar, Kr, and Xe on the d-AlNiCo.

The plotted quantity is the thermodynamic excess coverage (densities of adsorbed

atoms per unit area), ρN), defined as the difference between the total density of

atoms in the simulation cell and the density that would be present if the cell were filled

with uniform vapor at the specified values of P and T . The simulated ranges and the

experimental triple point temperatures (Tt) for Ne, Ar, Kr, and Xe are listed in Table

3.3. A layer-by-layer film growth is visible at low temperatures. Detailed inspection

of the isotherms reveals that there is a continuous film growth (i.e. complete wetting)

at temperatures above Tt (isotherms at T > Tt are shown as dotted curves). This

behavior, observed despite the high corrugation, is interesting as corrugation has

been shown to be capable of preventing wetting [107, 108].

Although vertical steps corresponding to layers’ formation are evident in the

isotherms, the slopes of the isotherms’ plateaus at the same normalized tempera-

tures (T � ≡ T/�gg = 0.35) differ between systems. To characterize this, we calculate the increase of each layer density, ΔρN , from the formation to the onset of the sub-

sequent layer. ΔρN is defined as ΔρN ≡ (ρB − ρA)/ρA and the values are reported in Table 3.3 (points (A) and (B) are specified in Figure 3.2). We observe that, as

the size of noble gas increases ΔρN become smaller, indicating that the substrate

corrugation has a more pronounced effect on smaller adsorbates, as expected since

they penetrate deeper into the corrugation pockets. However, Xe does not follow

this trend. This arises from the complex interplay between the corrugation energy

and length of the potential with respect to the parameters of the gas (σgg, �gg) in

determining the density of the adsorbed layers. In the case of Ne, Ar, and Kr, the

densities at points (A) are approximately the same (ρA = 5.4 atoms/nm 2), whereas

26

that of Xe is considerably smaller (ρA = 4.2 atoms/nm 2), because Xe dimension σgg

becomes comparable to the characteristic length (corrugation) of the potential. This

effect is clarified by the density profile of the films, ρ(x, y), shown in Figures 3.3 and

3.4. As can be seen at points (A), the density profiles of Ne, Ar, and Kr are the

same, i.e. the same set of dark spots appear in their plots. For Xe, some spots are

separated with distances smaller than its core radius (σgg), causing repulsive inter-

actions. Hence these spots will not likely appear in the density profile, resulting in

a lower ρA. More discussion on how interaction parameters affect the shape of the

isotherms is presented in Section 3.2.4. Note that the second layer in each system

has a smaller ΔρN than the first one. The explanation will be given when we discuss

the evolution of density profiles.

Table 3.3: Results for Ne, Ar, Kr, and Xe adsorbed on d-AlNiCo. Tt is taken from reference [109]. The density increase (ΔρN ) in the first and second layers is calculated at T � = 0.35 from point (A) to (B) and (C) to (D) in Figure 3.2, respectively.

simulated T T � ≡ T/�gg Tt ΔρN at T � = 0.35 θr (K) (K) for 1st layer for 2nd layer

Ne 11.8 → 46 0.35 → 1.36 24.55 (12.2-5.3)/5.3=1.30 (11.1-10.2)/10.2=0.09 6◦ Ar 41.7 → 155 0.35 → 1.29 83.81 (7.3-5.5)/5.5=0.33 (6.9-6.4)/6.4=0.08 30◦ Kr 59.6 → 225 0.35 → 1.32 115.76 (6.9-5.5)/5.5=0.25 (6.6-6.3)/6.3=0.05 42◦ Xe 77 → 280 0.35 → 1.27 161.39 (5.8-4.2)/4.2=0.38 (5.2-5.2)/5.2=0 54◦

3.2.2 Density profiles

Figures 3.3 and 3.4 show the density profiles ρ(x, y) at T � = 0.35 for the outer layers

of Ne, Ar, Kr, and Xe adsorbed on the d-AlNiCo at the pressures corresponding to

points (A) through (F) of the isotherms in Figure 3.2.

Ne/d-AlNiCo system. Figure 3.3(a) shows the evolution of adsorbed Ne. At

the formation of the first layer, adatoms are arranged in a pentagonal manner follow-

ing the order of the substrate, as shown by the discrete spots of the Fourier transform

27

10-25 10-20 10-15 10-10 10-5 1 0

10

20

30

40

50

P (atm)

ρ N (a

to m

s/ nm

2 )

Ne/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25

30

35 Ar/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25

30

35 Kr/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25 Xe/QC

P (atm)

ρ N (a

to m

s/ nm

2 )

P (atm)

ρ N (a

to m

s/ nm

2 )

P (atm)

ρ N (a

to m

s/ nm

2 )

(a) (b)

(d)(c)

A

B

C D

E

F

A B

C D

E

F

A B

C D

E

F

A B

C D

Figure 3.2: Computed adsorption isotherms for all the gas/d-AlNiCo systems. The ranges of temperatures under study are: Ne: T = 14 K to 46 K in 2 K steps, Ar: 45 K to 155 K in 5 K steps, Kr: 65 K to 225 K in 5 K steps, Xe: 80 K to 280 K in 10 K steps. Additional isotherms are shown with solid circles at T � = 0.35: T = 11.8 K (Ne), T = 41.7 K (Ar), T = 59.6 K (Kr), and T = 77 K (Xe). Isotherms above the triple point temperatures are shown as dotted curves.

(FT) having tenfold symmetry (point (A)). As the pressure increases, the arrange-

ment gradually loses its pentagonal character. In fact, at point (B) the adatoms are

arranged in patches of triangular lattices and the FT consists of uniformly-spaced

concentric rings with hexagonal resemblance. The absence of long-range ordering in

the density profile is indicated by the lack of discrete spots in the FT. This behavior

persists throughout the formation of the second layer (points (C) and (D)) until the

appearance of the third layer (point (E)). At this and higher pressures, the FT shows

patterns oriented as hexagons rotated by θr = 6 ◦, indicating the presence of short-

28

range triangular order on the outer layer (point (F)). In summary, between points (A)

and (F) the arrangement evolves from pentagonal fivefold to triangular sixfold with

considerable disorder, as the upper part of the density profile at point (F) shows. The

transformation of the density profile, from a lower-packing-density (pentagonal) to

a higher-packing-density structure (irregular triangular), occurs mostly in the mono-

layer from points (A) to (B), causing the largest density increase of the first layer with

respect to that of the other layers (see the end of Section 3.2.2 for more discussion).

Due to the considerable amount of disorder in the final state Ne/d-AlNiCo does not

satisfy the requirements for the transition as defined in Section 3.1.6.

Ar/d-AlNiCo and Kr/d-AlNiCo systems. Figures 3.3(b) and 3.4(a) show

the evolutions for Ar and Kr: they are similar to the Ne case. For Ar, the pentagonal

structure at the formation of the first layer is confirmed by the FT showing discrete

spots having tenfold symmetry (point (A)). The quasicrystal symmetry strongly af-

fects the overlayers’ structures up to the third layer by preventing the adatoms from

forming a triangular lattice (point (E)). This appears, finally, in the lower part of the

density profile at the formation of the fourth layer as confirmed by the FT showing

discrete spots with sixfold symmetry (point (F)). Similar to the Ne case, disorder does

not disappear but remain present in the middle of the density profile corresponding

to the highest coverage before saturation (point (F)). Similar situation occurs also

for the evolution of Kr as shown in Figure 3.4(a).

Xe/d-AlNiCo system. Figure 3.4(b) shows the evolution of adsorbed Xe. At

the formation of the first layer, adatoms are arranged in a fivefold ordering similar

to that of the substrate as shown by the discrete spots of the FT having tenfold

symmetry (point (A)). At point (B), the density profile shows a well-defined triangular

lattice not present in the other three systems: the FT shows discrete spots arranged

in regular and equally-spaced concentric hexagons with the smallest containing six

29

FTρ(x,y)(a)

θ

A

B

C

D

E

F

θ

(b)Ne/QC Ar/QC FTρ(x,y)

A

B

C

D

E

F

r r

Figure 3.3: Density profiles and Fourier transforms of the outer layer at T � = 0.35 for Ne/d-AlNiCo (T = 11.8 K) and Ar/d-AlNiCo (T = 41.7 K), corresponding to points (A) through (F) of Figure 3.2.

30

clear spots. Thus, at point (B) and at higher pressures, the Xe overlayers can be

considered to have regular closed-packed structure with negligible irregularities.

It is interesting to compare the orientation of the hexagons on FT for these four

adsorbed gases at the highest available pressures before saturation (point (F) for Ne,

Ar, and Kr, and point (D) for Xe). We define the orientation angles as the smallest

of the possible clockwise rotations to be applied to the hexagons to obtain one side

horizontal, as shown in Figures 3.3 and 3.4. Such angles are θr = 6 ◦, 30◦, 42◦, and

54◦, for adsorbed Ne, Ar, Kr, and Xe, respectively. These orientations, induced by

the fivefold symmetry of the d-AlNiCo, can differ only by multiples of n ·12◦ [70, 71]. Since hexagons have sixfold symmetry, our systems can access only five possible

orientations (6, 18, 30, 42, 54◦), and the final angles are determined by the interplay

between the adsorbate solid phase lattice spacing, the periodic simulation cell size,

and the potential corrugation. For systems without periodic boundary conditions,

the ground state has been found to be fivefold degenerate, as should be the case

[70, 71].

Xe adsorption on this surface was studied experimentally using LEED, in which the

isobar measurements indicate that the Xe film grows layer-by-layer in the temperature

range 65 K to 80 K [99], consistent with the simulations. Under similar conditions

to the simulation at 77 K, at the lowest coverage, the only discernible change in

the LEED pattern from that of the clean surface is an attenuation of the substrate

beams. After the adsorption of one layer, there are still no resolvable features that

would indicate an overlayer having order different from the substrate. At the onset of

the adsorption of the second layer, however, the LEED pattern shows new diffraction

spots that correspond to 5 rotational domains of a hexagonal structure. Within

each of these domains, the close-packed direction of the Xe is aligned with the 5-fold

directions of the substrate, as also observed in the simulation. In the experiments,

31

all possible alignments are observed owing to the presence of all possible rotational

alignments present within the width of the electron beam (0.25 mm). When the

second layer is complete, these spots are well-defined and their widths are the same as

the substrate spots, indicating a coherence length of at least 15 nm. The average Xe-

Xe spacing measured in the experiment is consistent with the bulk nearest-neighbor

spacing of 0.44 nm. A dynamical LEED analysis of the intensities indicates that the

structure of the multilayer film is consistent with face-centered cubic (FCC) Xe(111).

These structure parameters for the bilayer film are essentially identical to the results

obtained for Xe growth on Ag(111) [110, 111], a much weaker and less corrugated

substrate. This suggests that effect of the symmetry and corrugation of the substrate

potential on the Xe film structure is largely confined to the monolayer.

In every system, the increase of the density for each layer is strongly correlated

to the commensurability with its support: the more similar they are, the more flat

the adsorption isotherm will be (note that the support for the (N + 1)th-layer is the

N th-layer). For example, the Xe/d-AlNiCo system has an almost perfect hexagonal

structure at point (B) (due to its first-order five to sixfold ordering transition as

described in the next section). Hence, all the further overlayers growing on the top of

the monolayer will be at least “as regular” as the first layer, and have the negligible

density increase as listed in Table 3.3.

3.2.3 Order parameters (ρ5−6)

The evolution of the order parameter ρ5−6 is plotted in Figure 3.5 as a function of

the normalized chemical potential, μ�, at T �=0.35 for all the noble gas/d-AlNiCo

systems.

Ne/d-AlNiCo, Ar/d-AlNiCo, and Kr/d-AlNiCo systems. The ρ5−6 plots

for the first four layers observed before bulk condensation are shown in panels (a)−(c).

32

FTρ(x,y)(a)

A

B

C

D

E

F

(b)Kr/QC Xe/QC FTρ(x,y)

A

B

C

D

θr

θr

Figure 3.4: Density profiles and Fourier transforms of the outer layer at T � = 0.35 for of Kr/d-AlNiCo (T = 59.6 K) and Xe/d-AlNiCo (T = 77 K), corresponding to points (A) through (F) of Figure 3.2.

33

μ*

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

μ*

ρ 5 -6

layer 1

(a) (b)

(d)(c)

Ne/QC Ar/QC

Kr/QC Xe/QC

0 0.2 0.4 0.6 0.8 1 1.2 0

0.2

0.4

0.6

0.8

1

μ*

ρ 5 -6

1 1.05 1.1 1.15

0.4

0.5

0.6

layer 1

layer 2

layer 3 layer 4

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.06 1.12

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 μ

ρ 5 -6

layer 1

layer 2

layer 3 layer 4

0 0.2 0.4 0.6 0.8 1 1.2 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.06 1.12 0.3

0.5

0.7

μ*

ρ 5 -6

layer 1

layer 2 layer 3 layer 4

5- to 6- fold transition

Figure 3.5: (color online). Order parameters, ρ5−6, as a function of normalized chemical potential, μ�, (as defined in the text) at T � = 0.35 for the first four layers of (a) Ne, (b) Ar, (c) Kr, and for the first layer of Xe (d) adsorbed on d-AlNiCo. A sudden drop of the order parameter in Xe/QC to a constant value of ∼ 0.017 at μ� ∼ 0.8 indicates the existence of a first-order structural transition from fivefold to sixfold in the system.

As the chemical potential μ� increases, ρ5−6 decreases continuously reaching a con-

stant value only for Kr. At bulk condensation, the values of ρ5−6 are still high,

approximately 0.35 ∼ 0.45. Data at higher temperatures shows a similar behavior (up to T=24 K (T �=0.71) for Ne, T =70 K (T �=0.58) for Ar, and T=90 K (T �=0.53)

for Kr). Thus, we conclude that these systems do not undergo the ordering transition.

Xe/d-AlNiCo system. The ρ5−6 plot for the first layer is shown in panel (d).

In this system, as the chemical potential μ� increases, the order parameter gradually

34

decreases reaching a value of ∼ 0.3 at μ�tr ∼ 0.8. Suddenly it drops to 0.017 and remains constant until bulk condensation. Similar behavior is observed at higher

temperatures up to T=140 K (T �=0.63). This is a clear indication of a five to sixfold

ordering transition, as the first layer has undergone a transformation to an almost

perfect triangular lattice. Figure 3.6(d) shows the total enthalpy in the system, H.

At the transition point μ�tr, the enthalpy has a little step indicating a latent heat of

the transition. The discontinuity of the order parameter ρ5−6 and the presence of

latent heat indicate that the ransition is first-order. The latent heat of this transition

is estimated to be 1.021/151 ≈ 6.8 meV/atom. Despite the evidence for a first-order transition, the nearest neighbor distance, la-

beled rNN , e.g. Xe/d-AlNiCO in Figure 3.6(b), appears to change continuously. This

nearest neighbor distance is defined as the location of the first peak in the pair corre-

lation function because the latter property is more directly comparable to diffraction

measurements. We have also calculated the average spacing between neighbors, d¯NN ,

which is a thermodynamically meaningful quantity (related to the density). This has

a small discontinuity at the transition, providing additional evidence for the first-

order character of the transition. Both quantities, rNN and d¯NN , are shown in Figure

3.6(b). The NN Xe-Xe distance rNN decreases continuously as P increases, starting

from 0.45 nm and saturating at 0.44 nm. The Xe-Xe distance reaches saturation

value before the appearance of the second layer; therefore, the transition is complete

within the first layer. We note that a similar decrease in NN distance was measured

for Xe/Ag(111), but in that case, the NN spacing did not saturate before the onset

of the second layer adsorption [112, 110].

The observed transition from fivefold to sixfold order within the first layer of Xe

on d-AlNiCo can be viewed as a commensurate-incommensurate transition (CIT),

since at the lower coverage, the layer is commensurate with the substrate symmetry

35

0 0.2 0.4 0.6 0.8 1 μ*

ρ 5- 6

2 4 6 8

10

ρ N (a

to m

/n m

2 )

layer formation

st1

layer formation

nd2

b

a

0.2

0.4

0.6 5- to 6- fold transition

0.44

0.45

0.46

(n m

)

c

d

rNN

dNN

r NN d N

N ,

-45

-40

-35

-30

-25

H (e

V )

Δ Η

Figure 3.6: (color online). Xe on d-AlNiCo at T = 77 K. (a) Adsorption isotherm, ρN , versus the normalized chemical potential, μ�. (b) Nearest neighbor distance derived from the first peak of pair correlation function, rNN , (black line), and average spacing between neighbors at equilibrium, d¯NN , (red line). (c) Order parameter ρ5−6 (probability of fivefold defects, defined in Equation 3.14) versus the normalized chemical potential, μ�. (d) Total enthalpy. The transition, which is defined as the point in μ� above which the order parameter remains nearly constant, occurs at μ�tr ∼0.8. The discontinuity in H around μ�tr ∼ 0.8 indicates a first order transition with associated latent heat of the transition. The order parameter ρ5−6 after the transition is ∼ 0.017. Heat of the transition is ≈ 6.8 meV/atom.

36

and aperiodic, while at higher coverage, it is incommensurate with the substrate.

Such transitions within the first layer have been observed before for adsorbed gases,

perhaps most notably for Kr on graphite [113]. There, as here for Xe, the Kr forms

a commensurate structure at low coverage, which is compressed into an incommen-

surate structure at higher coverage. The opposite occurs for Xe on graphite, which

is incommensurate at low coverage and commensurate at high coverage [114]. Such

commensurate-incommensurate transitions have been studied theoretically in many

ways, but perhaps most simply as a harmonic system (balls and springs) having a

natural spacing that experiences a force field having a different spacing [115]. Such a

transition has been found to be first-order for strongly corrugated potentials (in 1D)

but continuous for more weakly corrugated potentials [116]. The transition observed

in our quasicrystal surface suggests that system is within the regime of “strong” cor-

rugation, which was not the case of Kr over graphite [113]. In fact, for the latter

system, both commensurate and incommensurate structures have sixfold symmetry.

A more relevant comparison may be the transition of Xe on Pt(111), from a rect-

angular symmetry incommensurate phase to a hexagonal symmetry commensurate

one, although in that case, the low-temperature phase was incommensurate. That

transition was also found to be continuous [117]. Therefore, while our simulations

indicate that Xe on d-AlNiCo undergoes a CIT, as observed for other adsorbed gases,

the observation of a first-order CIT is new, to our knowledge, and likely arises from

the large corrugation.

3.2.4 Effects of �gg and σgg on adsorption isotherms

In Section 3.2.1 we have briefly discussed how the density increase of each layer (ΔρN)

is affected by the size of the adsorbate (σgg). In addition, since the corrugation of the

potential depends also on the gas-gas interaction (�gg), the latter quantity could a

37

priori have an effect on the density increase. To decouple the effects of σgg and �gg on

ΔρN we calculate ΔρN while keeping one parameter constant, σgg or �gg, and varying

the other. For this purpose, we introduce two test gases iNe(1) and dXe(1), which

represent “inflated” or “deflated” versions of Ne and Xe, respectively (parameters

are defined in Equations 3.9-3.12 and listed in Tables 3.1 and 3.2). Then we perform

four tests summarized as the following:

(1) constant strength �gg, size σgg increases [Ne→iNe(1)]: ΔρN reduces,

(2) constant strength �gg, size σgg decreases [Xe→dXe(1)]: ΔρN increases,

(3) constant size σgg, strength �gg decreases [Xe→iNe(1)]: ΔρN ∼ constant,

(4) constant size σgg, strength �gg increases [Ne→dXe(1)]: enhanced agglomer- ation.

Figure 3.7 shows the adsorption isotherms at T � = 0.35 for Ne, iNe(1), Xe, and

dXe(1) on d-AlNiCo. By keeping the strength constant and varying the size of the

adsorbates, tests 1 and 2 ([Ne→iNe(1)] and [Xe→dXe(1)]), we find that we can reduce or increase the value of the density increase (when ΔρN decreases the continuous

growth tends to become stepwise and vice versa). These two tests indicate that the

larger the size, the smaller the ΔρN . By keeping the size constant and decreasing

the strength, test 3 ([Xe→iNe(1)]), we find that ΔρN does not change appreciably. An interesting phenomenon occurs in test 4 where we keep the size constant and

increase the strength ([Ne→dXe(1)]). In this test the growth of the film loses its step-like shape. We suspect that this is caused by an enhanced agglomeration effect

as follows. Ne and dXe(1) have the same size which is the smallest of the simulated

gases, allowing them to easily follow the substrate corrugation, in which case, the

corrugation helps to bring adatoms closer to each other [100] (agglomeration effect).

The stronger gas-gas self interaction of dXe(1) compared to Ne will further enhance

38

10-25 10-20 10-15 10-10 10-5 1 0

10

20

30

40

P (atm)

ρ (a

to m

s/ nm

2 )

Ne iNe Xe

dX e

Ν

(1 )

(1)

test 1: [Ne → iNe(1)]

test 2: [Xe → dXe(1)]

test 3: [Xe → iNe(1)]

test 4: [Ne → dXe(1)]

Figure 3.7: (color online). Computed adsorption isotherms for Ne, Xe, iNe(1), and dXe(1)

on d-AlNiCo at T �=0.35. iNe(1) and dXe(1) are test noble gases having potential parameters described in the text and in Tables 3.1 and 3.2. The effect of varying the interaction strength of the adsorbates on the density increase ΔρN (while keeping the size constant) is negligible on large gases but significant on small gases.

this agglomeration effect, resulting in a less stepwise film growth of dXe(1) than Ne.

As can be seen, dXe(1) grows continuously, suggesting a strong enhancement of the

agglomeration. In summary, the last two tests (3 and 4) indicate that the effect of

varying the interaction strength of the adsorbates (while keeping the size constant)

is negligible on large gases but significant on small gases.

3.2.5 Effects of �gg and σgg on 5- to 6-fold transition

Strength �gg and size σgg of the adsorbates also affect the existence of the first-order

transition (present in Xe/d-AlNiCo, but absent in Ne, Ar, and Kr on d-AlNiCo).

Hence we perform the same four tests described before and observe the evolution of

the order parameter. The results are the following:

(1) constant strength �gg, size σgg increases [Ne→iNe(1)]: transition appears

(2) constant strength �gg, size σgg decreases [Xe→dXe(1)]: transition disappears 39

(3) constant size σgg, strength �gg decreases [Xe→iNe(1)]: transition remains

(4) constant size σgg, strength �gg increases [Ne→dXe(1)]: remains no transition

The strength �gg has no effect on the existence of the transition (tests 3 and 4), which

instead is controlled by the size of the adsorbates (tests 1 and 2). To further charac-

terize such dependence, we add three additional test gases with the same strength �gg

of Xe but different sizes σgg. The three gases are denoted as dXe (2), iXe(1), and iXe(2)

(the prefixes d- and i- stand for deflated and inflated, respectively). The interaction

parameters, defined in the following equations, are listed in Tables 3.1 and 3.2:

{�gg, Dgs, σgg} [dXe(2)] ≡ {�gg, Dgs, 0.95σgg}[Xe], (3.15)

{�gg, Dgs, σgg}[iXe(1)] ≡ {�gg, Dgs, 1.34σgg}[Xe], (3.16)

{�gg, Dgs, σgg}[iXe(2)] ≡ {�gg, Dgs, 1.65σgg}[Xe]. (3.17)

Figure 3.8 shows the evolutions of the order parameter as a function of the normal-

ized chemical potential for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed

on d-AlNiCo at T �=0.35. All these systems undergo a transition, except dXe(2), i.e.

the transition occurs only in systems with σgg ≥ σgg[Xe] indicating the existence of a critical value for the appearance of the phenomenon. Furthermore, as σgg increases

(iNe(1) → iXe(1) → iXe(2)), the transition shifts towards smaller critical chemical potentials.

3.2.6 Prediction of 5- to 6-fold transition

The critical value of σgg associated with the transition can be related to the charac-

teristic length of the d-AlNiCo by introducing a gas-substrate mismatch parameter

defined as

δm ≡ k · σgg − λr λr

. (3.18)

40

0 0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

μ*

ρ 5 -6

iXe

dXe

iNe

(2)

(1) (1)iXe(2)

Figure 3.8: (color online). Order parameters as a function of normalized chemical poten- tial (as defined in the text) for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed on d-AlNiCo at T � = 0.35. A first-order fivefold to sixfold structural transition occurs in the last three systems, but not in dXe(2).

where k = 0.944 is the distance between rows in a close-packed plane of a bulk LJ

gas (calculated at T = 0 K with σ = 1 [118]), and λr is the characteristic spacing of

the d-AlNiCo, determined from the momentum transfer analysis of LEED patterns

[99] (our d-AlNiCo surface has λr=0.381 nm [99]). With such ad hoc definition, δm

measures the mismatch between an adsorbed FCC[111] plane of adatoms and the d-

AlNiCo surface. In Table 3.4 we show that δm perfectly correlates with the presence

of the transition in our test cases (transition exists ⇔ δm > 0). The definition of a gas-substrate mismatch parameter is not unique. For example,

one can substitute k · σgg with the first NN distance of the bulk gas, and λr with one of the following characteristic lengths: a) side length of the central pentagon in

the potential plots in Figure 3.1 (λs = 0.45nm), b) distance between the center of

the central pentagon and one of its vertices (λc = 0.40nm), c) L = τ · S = 0.45 nm, where τ = 1.618 is the golden ratio of the d-AlNiCo and S = 0.243 nm is the

41

Table 3.4: Summary of adsorbed noble gases on d-AlNiCo that undergo a first-order fivefold to sixfold structural transition and those that do not.

δm transition Ne -0.311 No Ar -0.158 No Kr -0.108 No Xe 0.016 Yes

iNe(1) 0.016 Yes dXe(1) -0.311 No dXe(2) -0.034 No iXe(1) 0.363 Yes iXe(2) 0.672 Yes

k = 0.944 [118] λr = 0.381 nm [99]

δm ≡ (k · σgg − λr)/λr

side length of the rhombic Penrose tiles [15]. Although there is no a priori reason to

choose one definition over the others, the one that we select (Equation 3.18) has the

convenience of being perfectly correlated with the presence of the transition, and of

using reference lengths commonly determined in experimental measurements (λr) or

quantities easy to extract (k · σgg).

3.2.7 Transitions on smoothed substrates

In Figure 3.1 we can observe that near the center of each potential there is a set of five

points with the highest binding interaction (the dark spots constituting the central

pentagons). A real QC surface contains an infinite number of these very attractive

positions which are located at regular distances and with five fold symmetry. Due to

the limited size and shape of the simulation cell, our surface contains only one set

of these points. Therefore, it is of our concern to check if the results regarding the

existence of the transition are real or artifacts of the method. We perform simulation

tests by mitigating the effect of the attractive spots through a Gaussian smoothing

function which reduces the corrugation of the original potential. The definitions are

42

the following:

G(x, y, z) ≡ AGe−(x2+y2+z2)/2σ2G , (3.19)

V (z) ≡ 〈V (x, y, z)〉(x,y) , (3.20)

Vmod(x, y, z) ≡ V (x, y, z) · [1−G(x, y, z)] + V (z) ·G(x, y, z). (3.21)

where G(x, y, z) is the Gaussian smoothing function (centered on the origin and with

parameters AG and σG), V (z) is the average over (x, y) of the original potential

V (x, y, z), and Vmod(x, y, z) is the final smoothed interaction. An example is shown

in Figure 3.9(a) where we plot the minimum of the adsorption potential for a Ne/d-

AlNiCo modified interaction (smoothed using AG = 0.5 and σG = 0.4 nm). In

addition, in panel (b) we show the variations of the minimum adsorption potentials

along line x = 0 for the modified and original interactions (solid and dotted curves,

respectively).

Using the modified interactions (with AG = 0.5 and σG = 0.4 nm) we simulate all

the noble gases of Table 3.4. The results regarding the phase transition on modified

surfaces do not differ from those on unmodified ones, confirming that the observed

transition behavior is a consequence of competing interactions between the adsorbate

and the whole QC substrate rather than just depinning of the monolayer epitaxially

nucleated. Therefore, the simple criterion for the existence of the transition (δm > 0)

might also be relevant for predicting such phenomena on other decagonal quasicrystal

substrates.

3.2.8 Temperature vs substrate effect

Using Xe/d-AlNiCo data, we observe that defects are present at all temperatures that

are simulated (20 to 286 K). The probability of defects increases with temperature,

implying that their origin is entropic, as is the case for a periodic crystal. Figure

43

(a) (b)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-75

-65

-55

-45

-35

Y (nm)

V m

in (y

) ( m

eV )

modified potential original potential

Figure 3.9: (color online). (a) The minimum of adsorption potential, Vmin(x, y), for Ne on a smoothed d-AlNiCo as described in the text. (b) The variations of the minimum adsorption potentials along the line at x = 0 shown in (a), for the modified and original interactions (solid and dotted curves).

3.10-(right axis) shows that the defect probability increases as T increases, while

Figure 3.10-(left axis) shows the trend of the transition point in function of T. At low

temperatures, the sixfold ordering occurs earlier (at lower μ�tr) as the temperature

is increased from 40K to 70K. This trend is expected because the ordering effect

imposed by the substrate corrugation becomes relatively smaller as the temperature

increases. However, this trend is not observed in the higher temperature region (from

70K to 140K). In fact, at higher temperatures the transition point shifts again to

higher μ�tr. This is most likely due to the monolayer becoming less two-dimensional,

allowing more structural freedom of the Xe atoms and thus decreasing the effect of

the repulsive Xe-Xe interaction that would stabilize the sixfold structure. Transitions

having critical μ�tr > 1 indicate that the onset of second-layer adsorption occurs

earlier than the transition to the sixfold structure. When the second layer adsorbs at

T >130K, the density of the monolayer increases by a few percent, thereby increasing

the effect of the repulsive interactions and driving the fivefold to sixfold transition.

44

o f 5

-6 fo

ld tr

an si

tio n

μ t r*

ρ 5-6

μtr*

ρ 5-6

Figure 3.10: Xe on d-AlNiCo. Values of μ�tr for the fivefold to sixfold transition points from 40 K to 140 K (left axis). Transition points at μ�tr > 1 indicate that a transfer of atoms from the second layer to the first layer is required to complete the transition. Also shown is the defect probability as a function of T after the transition occurs (right axis), indicating an increase in defect probability with T .

Interestingly, stacking faults are evident in the multilayer films. This is consistent

with x-ray diffraction studies of the growth of Xe on Ag(111), where stacking faults

were observed for Xe growth under various growth conditions [110, 111], although the

overall structure observed was FCC(111). Such a stacking fault is evident in Figure

3.11, which shows a superposition of Xe layers 2 and 4 at 77 K. The coincidence of

the atom locations in the top left part of this figure is consistent with an hexagonal

close-packed structure (HEX), ABAB stacking, whereas the offsets observed in the

lower part of the figure indicate the presence of stacking faults caused by dislocations

in the layers. We note that while bulk Xe has an FCC structure, and indeed an FCC

structure was found for the multilayer film in the LEED study, calculations of the

bulk structure using LJ pair potentials such as those employed here result in a more

stable HEX structure [119]. The energy difference between the two structures is very

small, and apparently arises from a neglect of d-orbital overlap interactions, which

45

Figure 3.11: Density plot of Xe on decagonal AlNiCo at 77 K, showing a superposition of the density slices for the 2nd and 4th layers. In the top right, 4th-layer atoms are located directly above the 2nd layer atoms indicating an hexagonal close-packed ABAB stacking, whereas in other regions, such as lower left, the two layers are offset due to stacking fault.

are more effective in FCC than in HEX structures [119, 120]. Although the simulated

film is HEX instead of FCC, the main conclusions concerning the growth mode of Xe

on the quasicrystal are not affected [70].

3.2.9 Orientational degeneracy of the ground state

In was mentioned in the previous section that after the ordering transition is complete,

the resulting sixfold structure is aligned parallel to one of the sides of the pentagons

in the Vmin map of the adsorption potential (there are five possible orientations). In

the experiments, all five orientations are observed, due to the presence of all possi-

ble alignments of hexagons along five sides of a pentagon in the QC sample within

the width of the electrons beam (∼0.25 mm). In an ideal infinite GCMC frame- work the ground state of the system would be degenerate and all five orientations

would have the same energy and be equally probable. However, the square periodic

boundary conditions of our GCMC break this orientational degeneracy, causing some

orientations to become more likely to appear.

To find all the possible orientations, we performed simulations with a cell having

free boundary conditions. The cell is a 5.12 x 5.12 nm2 quasicrystal surface sur-

rounded by vacuum. Figure 3.12(a) shows the Vmin map of the adsorption potential.

46

Thirty simulations at 77 K are performed with this cell. The isotherms from these

runs are plotted in Figure 3.12(b). Only the first layer is shown, and the finite size

of the surface makes the growth of the first layer continuous. The density profiles

ρ(x, y) of all the simulations are analyzed at point p� of Figure 3.12(b). In this cell,

all five orientations of hexagons are observed with equal frequency indicating the ori-

entational degeneracy of the ground state. To represent the five orientations, density

profiles of five calculations (c, d, e, f, and g) are shown in Figures 3.12(c) to 3.12(g)

with their FT plotted on the side. Figure 3.12(h) presents a schematic depiction of

which orientations of hexagons are exemplified in each simulation.

Figures 3.13(a) and 3.13(b) illustrate the effect of pentagonal defects on the ori-

entation of hexagons at point p� of Figure 3.12(b). In most of the density profiles

corresponding to this coverage, we find the behavior shown in Figure 3.13(a). Here,

the effect of the pentagonal defect, which is the center of a dislocation in the hexag-

onal structure, is to rotate the orientation of the hexagons above the pentagon by

2 · 60◦/5 = 24◦ with respect to the hexagons below the pentagon. The possible rota- tions are n · 12◦, where n = 1,2,3,4, or 5. The rotation by 12◦ is usually mediated by more than one equivalent pentagon, as is shown in Figure 3.13(b) (note the “up” pen-

tagon at the middle-bottom part and the “down” pentagon near the middle-top part

of the figure). The “up” pentagon (with one vertex on the top) is equivalent with the

“down” pentagon (with one vertex on the bottom) since they have five orientationally

equivalent sides). These pentagonal defects are induced by the fivefold symmetry of

the substrate, and their concentration decreases in the subsequent layers.

3.2.10 Isosteric heat of adsorption

Figure 3.14 shows a P -T diagram for three different coverages of Xe adsorbed on

d-AlNiCo constructed from the isotherms in the range 40 K < T < 110 K. In the

47

simulations, the layers grow step-wise; at 70 K the first step occurs between coverage

∼0.06 and ∼0.7, the second step occurs between coverage 1.0 and ∼1.9, and the third step occurs between coverage ∼1.9 and ∼2.8 (unit is in fractions of monolayer). Figure 3.14 shows the T , P location of these steps, denoted “cov 0.5”, “cov 1.5”,

and “cov 2.5” for the first, second, and third steps, respectively. The isosteric heat

of adsorption per atom at these steps can be calculated from the P -T diagram as

follows [112]:

qst ≡ −kB d(lnP ) d(1/T )n

. (3.22)

The inset of Figure 3.14 summarizes the values of qst obtained from simulations

and experiments. The agreement between experiment and the simulations for the

half monolayer heat of adsorption is good. The values obtained in the simulation for

the 1.5 and 2.5 layer heats are about 20% lower than the bulk value of 165 meV [99].

The lower values suggest that bulk formation should be preferred at coverages above

one layer. However, layer-by-layer growth is observed at all T for at least the first

few layers in these simulations. We therefore believe that the low heats of adsorption

arise from slight inaccuracies in the Xe-Xe LJ parameters used in this calculation, as

the heats of adsorption are very sensitive to the gas parameters.

3.2.11 Effect of vertical dimension

In a standard unit cell, only 2 steps, corresponding to the first and second layer ad-

sorption, are apparent in the isotherms [101]. Further simulations indicate that when

the cell is extended in the vertical direction, additional steps are observed. Therefore

the number of observable steps is related to the size of the cell. Nevertheless, layering

is clearly evident in the ρ(z) profile, and the main features of the film growth are not

altered. For Xe on d-AlNiCo, the average interlayer distance is calculated to be about

0.37 nm, compared to 0.358 nm for the interlayer distance in the < 111 > direction

48

of bulk Xe [121]. Our simulations of multilayer films show variable adsorption as

the simulation cell is expanded in the direction perpendicular to the surface. This

is a result of sensitivity to perturbations (here, cell size) close to the bulk chemical

potential, where the wetting film’s compressibility diverges. This dependence has

been seen previously in large scale simulations. See e.g. Figure 3 of reference [108].

The analog of this effect in real experiments is capillary condensation at pressures

just below saturated vapor pressure (svp), the difference varying as the inverse pore

radius.

3.3 Summary

The results of GCMC simulations of noble gas films on QC have been presented. Ne,

Ar, Kr, and Xe grow layer-by-layer at low temperatures up to several layers before

bulk condensation. We observe interesting phenomena that can only be attributed

to the quasicrystallinity and/or corrugation of the substrate, including structural

evolution of the overlayer films from commensurate pentagonal to incommensurate

triangular, substrate-induced alignment of the incommensurate films, and density

increase in each layer with the largest one observed in the first layer and in the smallest

gas. Two-dimensional quasicrystalline epitaxial structures of the overlayer form in

all the systems only in the monolayer regime and at low pressure. The final structure

of the films is a triangular lattice with a considerable amount of defects except in

Xe/QC. Here a first-order transition occurs in the monolayer regime resulting in an

almost perfect triangular lattice. The subsequent layers of Xe/QC have hexagonal

close-packed structures. By simulating test systems with various sizes and strengths,

we find that the dimension of the noble gas, σgg, is the most crucial parameter in

determining the existence of the phenomenon which is found only in systems with

σgg ≥ σgg[Xe].

49

a 10 -12

10 -10

10 -8

10 -60

2

4

6

8

10

p (atm)

ρ= N

/A

(a to

m s/

nm 2

)

p*

-5 5X(nm)

5

Y (n

m )

-5

0

-50

-100

-150

-200

-250

(meV)

b

c d

e

g

f

h c

d,g

f

g

e,g

Figure 3.12: Xe adsorption on d-AlNiCo. (a) Minimum potential energy surface of the adsorption potential with free boundary conditions. (b) Adsorption isotherms of the first layer from a set of 30 simulations at 77 K using the free cell described in the paper. Five density profiles and FTs at point p� of (b) are shown in (c) to (g), representing all possible orientations of hexagonal domains. (h) Schematic diagram illustrating the correspondence between the orientations of the hexagonal domains observed in the density profiles (c) to (g).

50

θ1 θ2

a b

Figure 3.13: Xe adsorption on d-AlNiCo. Pentagonal defects rotate the orientation of hexagons by (a) θ1 = 24◦ and (b) θ2 = 12◦.

Figure 3.14: (color online). Xe adsorption on d-AlNiCo. Locations in P , T of the vertical risers in the isotherms corresponding to the first (square), second (circle), and third (triangle) layer formation. The heats of adsorptions, qst, are 270, 129, and 125 meV/atom respectively, calculated as described in the text. The inset figure shows qst obtained from the simulations as well as from the experiments.

51

Chapter 4

Embedded-atom method potentials

The interatomic potentials to simulate hydrocarbon adsorptions on d-AlNiCo are

generated within the embedded-atom method (EAM) formalism. The parametriza-

tion of the potentials are given in Equations 2.30 - 2.33. These parameters are fitted

to the energy of various structures computed via ab initio quantum calculations using

VASP code. The structure prototypes are summarized in Table 4.1. For elemental

Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure

(DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed

(HEX), simple cubic (SC), and simple hexagonal (SH). The structures are first re-

laxed from the initial configurations to achieve the equilibrium ones. For the Al-Co-Ni

ternary phases, the initial configurations are taken from the database of alloys [122].

After the structures are relaxed, the ground state electronic energies are calculated

and are used as the fitting data. The technical details in relaxing the structures and

evaluating the energies with VASP are given in Section 2.2.

Fitting of the parameters of the EAM potentials are performed using SIMPLEX

method (Section 2.4). SIMPLEX is relatively slower than other methods such as

nonlinear least square or conjugate gradient. However, since the speed of the fitting

procedure is not a concern in this work (most time is spent in the ab initio calcula-

tions), SIMPLEX is advantageous because it does not require evaluation of function’s

derivatives or orthogonality. In this way, a new parametrization of EAM potentials

requires changes only in the function evaluation routines. A fitting code is developed

to be able to fit in a bulk or adsorption mode. In the bulk mode, the function to

52

Table 4.1: List of structure prototypes used to fit the EAM potentials for hydrocarbon ad- sorption on Al-Co-Ni. For elemental Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure (DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed (HEX), simple cubic (SC), and simple hexagonal (SH). The Al– Co-Ni ternaries are taken from the database of alloys [122].

group structure Al BCC, DIA, FCC, GRA, HEX, SC, SH Co BCC, DIA, FCC, GRA, HEX, SC, SH Ni BCC, DIA, FCC, GRA, HEX, SC, SH CHn CH4 C2Hn C2H2, C2H4, C2H6-isotactic, C2H6-syntactic C3Hn propane, C3H4 allene, propyne, propene,

cyclopropane, cyclopropene, cyclopropyne C4Hn isobutane, 1-butene, 1-butyne, cyclobutane, methylcyclopropane C5Hn pentane, cyclopentane Other alkanes hexane, heptane, nonane, decane, undecane, dodecane AlxCoyNiz Al32Co12Ni12 (cI112)

Al32Co16Ni12 (cI128) Al29Co4Ni8 (dB1) Al17Co5Ni3 (dH1) Al34Co4Ni12 (dH2) Al20Co7Ni1 (hP28) Al18Co4Ni4 (mC28) Al12Co2Ni2 (mC32) Al18Co2Ni2 (mP22) Al36Co8Ni4 (oI96) Al12Co1Ni3 (oP16) Al34Co12Ni4 (mC102)

53

minimize is the err = ΔEbulk defined as:

ΔEbulk = ∑

i

|EEAMbulk,i − EV ASPbulk,i | (4.1)

The energies are per atom and the summation is over all structures of the fit. In

the adsorption mode, the error function consists of the error in the bulk energies

of molecule structures (ΔEbulk,mol), substrate structures (ΔEbulk,sub), molecule-on-

substrate structure (ΔEbulk,mol+sub), as well as in the adsorption energies (ΔEads):

Eads = Ebulk,mol+sub − (Ebulk,mol + Ebulk,sub) (4.2)

ΔEads = ∑

i

|EEAMads,i − EV ASPads,i | (4.3)

err = c1ΔEbulk,mol + c2ΔEbulk,sub + c3ΔEbulk,mol+sub + c4ΔEads

c1 + c2 + c3 + c4 (4.4)

The coefficients c1, c2, c3, and c4 are introduced as weighing factors. To drive the

parameters toward physically meaningful convergence point, the fitting is performed

in multiple stages:

(1) fit potentials for elemental Al, Co, and Ni,

(2) fit potentials for Al-Co-Ni,

(3) fit potentials for hydrocarbons,

(4) fit final potentials for hydrocarbon on Al-Co-Ni.

The fitted parameters from stage (1) are used as initial conditions in stage (2). The

fitted parameters from stage (2) and (3) are used as initial values in stage (4).

4.1 Stage 1: Aluminum, cobalt, and nickel

In stage (1), the elemental potentials for Al, Co, and Ni are fitted to elemental

bulk energies (top part of Tables 4.2-4.4). The potentials are also trained at various

54

pressures to ensure stability under compression/expansion and to yield resonable lat-

tice constants. In the calculations, different pressures are achieved by expanding or

compressing the relaxed ground state structures (i.e. Al(FCC) Co(HEX) Ni(FCC)),

namely at lattice constants from a = 0.95a0 to a = 1.1a0, where a0 is the equilib-

rium lattice constant at zero pressure. Training at various pressures increases the

transferability of the potentials due to a wider range of charge density covered. The

fitted EAM potentials (bottom part of Tables 4.2-4.4) are able to find the ground

state structure as well as the relative stability of each structure. Note that for pure

elements, the bulk energy represents the cohesive energy. Going from the most stable

structure to the least stable one, the EAM potentials correctly predict the following:

FCC → HEX → BCC → SC → SH → GRA → DIA (for Al and Ni) and HEX → FCC → BCC → SC → SH → GRA → DIA (for Co). The potentials also give accurate equilibrium lattice constants (middle part of Tables 4.2-4.4).

4.2 Stage 2: Al-Co-Ni potentials

Results from stage (1) are used as initial conditions in stage (2). The potential

for systems containing Al, Ni, and Co (AlCoNi-pot) are fitted to energies in bulk

structures and in slab configurations. The latter is intended to tune the AlCoNi-

pot at low charge density having different atomic environments from bulk. Slab

configurations are created from dB1, dH1, and dH2. Note that dB1, dH1, and dH2

are decagonal AlNiCo quasicrystal approximants. They are crystals with a large unit

cell which represent the short-ranged order of the quasicrystal. Approximants exist in

the vicinity of region of chemical compositions of their quasicrystalline counterparts.

The bulk unit cell of dB1, dH1, and dH2 contains 41, 25, and 50 atoms, respectively.

The unit cells consist of two well-defined layers. There are two possible terminations

for the bulk. The top(bottom) layer of the unit cell is labeled A(B), respectively. A

55

Table 4.2: (Top part) List of structures used to fit EAM potential for elemental aluminum. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3 · re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.199 -3.375 -3.368 -0.007 DIA 0.135 -2.847 -2.722 -0.126 FCC at a0·0.95 0.312 -3.347 -3.347 0.000 FCC at a0·0.96 0.289 -3.396 -3.390 -0.006 FCC at a0·0.97 0.267 -3.425 -3.422 -0.003 FCC at a0·0.98 0.248 -3.440 -3.444 0.004 FCC at a0·0.99 0.229 -3.443 -3.457 0.014 FCC at a0·1.00 0.212 -3.439 -3.462 0.023 FCC at a0·1.05 0.144 -3.356 -3.390 0.034 FCC at a0·1.10 0.098 -3.221 -3.221 0.000 GRA 0.151 -2.882 -2.958 0.075 HEX 0.211 -3.436 -3.426 -0.009 SC 0.167 -3.108 -3.097 -0.011 SH 0.178 -3.241 -3.241 0.000

a0(FCC) vasp = 4.04 A˚ a0(FCC) EAM = 4.00 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.026 7.182 2.700 33.64 3.038 3.012

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.078 0.121 0.495 0.896 1.381 2.200 Fx (eV) 0.000 -2.810 -3.106 -3.980 -3.879 -3.665 -3.074

56

Table 4.3: (Top part) List of structures used to fit EAM potential for elemental cobalt. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”HEX at a0 · c” indicates that the structure is HEX with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.665 -5.380 -5.377 -0.003 DIA 0.602 -4.115 -4.141 0.027 FCC 0.693 -5.439 -5.450 0.011 GRA 0.566 -4.596 -4.567 -0.029 HEX at a0·0.95 0.973 -5.213 -5.214 0.001 HEX at a0·0.96 0.910 -5.317 -5.305 -0.012 HEX at a0·0.97 0.849 -5.386 -5.375 -0.011 HEX at a0·0.98 0.795 -5.425 -5.425 -0.000 HEX at a0·0.99 0.744 -5.442 -5.457 0.016 HEX at a0·1.00 0.697 -5.440 -5.473 0.033 HEX at a0·1.05 0.501 -5.295 -5.383 0.088 HEX at a0·1.10 0.358 -5.098 -5.098 0.000 SC 0.569 -4.984 -4.708 -0.276 SH 0.608 -5.155 -4.966 -0.189

a0(HEX) vasp = 2.47 A˚ a0(HEX) EAM = 2.45 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.090 5.945 2.280 36.70 2.832 2.701

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.146 0.380 0.821 1.306 1.683 2.200 Fx (eV) 0.000 -3.374 -4.894 -5.517 -5.509 -5.008 -3.952

57

Table 4.4: (Top part) List of structures used to fit EAM potential for elemental nickel. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 ·c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.699 -4.928 -4.924 -0.004 DIA 0.500 -3.839 -3.839 -0.000 FCC at a0·0.95 0.980 -4.754 -4.783 0.029 FCC at a0·0.96 0.918 -4.870 -4.868 -0.002 FCC at a0·0.97 0.861 -4.941 -4.932 -0.009 FCC at a0·0.98 0.807 -4.980 -4.980 0.000 FCC at a0·0.99 0.757 -4.995 -5.010 0.015 FCC at a0·1.00 0.709 -4.993 -5.024 0.031 FCC at a0·1.05 0.507 -4.871 -4.955 0.084 FCC at a0·1.10 0.369 -4.687 -4.687 -0.000 GRA 0.543 -4.081 -4.109 0.028 HEX 0.706 -4.993 -4.998 0.005 SC 0.591 -4.464 -4.342 -0.122 SH 0.637 -4.682 -4.630 -0.052

a0(FCC) vasp = 3.49 A˚ a0(FCC) EAM = 3.46 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.126 5.447 2.150 39.17 2.839 2.717

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.223 0.336 0.566 1.028 1.355 2.200 Fx (eV) 0.000 -3.899 -4.373 -4.731 -5.280 -5.028 -4.172

58

slab is labeled A if the top layer is layer A, and vice versa. The unit cell for slab

configurations contains vacuum space of 12 A˚ in the vertical dimension to minimize

the interaction between unit cells due to periodic boundary conditions. In relaxing

the slabs, cell’s size/shape and atoms are allowed to relax except the bottom atoms.

In this way, the bottom atoms will be at the same coordinates as they were in the

bulk which will be more appropriate (than if bottom atoms are relaxed) when the

slabs are used as the substrates in the adsorption configurations (stage (4) of fitting).

The bottom part of Table 4.5 shows the parameters of AlCoNi-pot fitted to structures

listed in the top part of Table 4.5.

4.3 Stage 3: Hydrocarbon potentials

The EAM potentials for hydrocarbons (CH-pot) are fitted to the energies of isolated

molecules. The ab initio calculations are performed in a fairly large cubic cell with

vacuum size of larger than 10 A˚ to minimize the interaction between molecules due to

periodic boundary conditions implemented in VASP. The structures are fully relaxed.

In the beginning, all molecules listed in Table 4.1 are included in the fit. The result-

ing fitted parameters and the fitted energies are reported in Table 4.6. The results

show that EAM is fairly accurate for hydrocarbons, especially for alkanes. This is

interesting because EAM formalism was introduced for metals where bondings are

due to nonlocal electrons. Nevertheless, not all hydrocarbons can be fit simultane-

ously, as significant errors are found in some molecules, e.g C3H4-allene, propylene,

cyclopropane, cyclopropyne, 1-butene, and benzene. For our simulations, presented

in Chapter 5, a more accurate fit for alkanes and benzene is needed. Table 4.7 shows

the EAM potentials fitted to alkanes and benzene.

.

59

Table 4.5: (Top part) List of structures used to fit EAM potential for Al-Co-Ni systems. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV) and the first knot at (0,0) is assumed. The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

cI112 bulk -4.828 -4.828 -0.000 cI128 bulk -4.916 -4.915 -0.000 dB1 bulk -4.398 -4.395 -0.003 dH1 bulk -4.483 -4.516 0.032 dH2 bulk -4.441 -4.480 0.038 hP28 bulk -4.484 -4.490 0.006 mC102 bulk -4.520 -4.517 -0.003 mC28 bulk -4.492 -4.509 0.017 mC32 bulk -4.323 -4.322 -0.001 mP22 bulk -4.121 -4.111 -0.010 oI96 bulk -4.334 -4.350 0.016 oP16 bulk -4.292 -4.300 0.007 dB1 slab3A -3.816 -3.853 0.037 dB1 slab3B -3.913 -3.921 0.008 dB1 slab4A -4.105 -4.096 -0.009 dB1 slab4B -4.015 -4.015 0.000 dH1 slab3A -4.058 -4.043 -0.015 dH1 slab3B -4.056 -4.028 -0.029 dH1 slab4A -4.153 -4.155 0.003 dH1 slab4B -4.158 -4.153 -0.005 dH2 slab4A -4.144 -4.143 -0.001 dH2 slab4B -3.980 -3.980 0.000

Al Co Ni ρe (A˚−3) 0.026 0.090 0.126 β 7.182 5.945 5.447 re (A˚) 2.70 2.28 2.15 D (eV) 0.034 0.037 0.039 α (A˚−1) 1.833 3.350 3.258 r0 (A˚) 3.009 2.744 2.735 Z/ZAl 1 0.862 0.521 Z/ZCo 1 0.975 knot2 0.086,-2.758 0.235,-3.249 0.285,-3.772 knot3 0.120,-2.974 0.383,-4.834 0.341,-4.449 knot4 0.517,-4.688 0.864,-5.579 0.604,-4.931 knot5 0.868,-3.777 1.321,-5.420 1.050,-5.279 knot6 1.365,-3.736 1.705,-5.061 1.355,-5.160 knot7 2.200,-3.151 2.200,-4.090 2.200,-4.196

60

Table 4.6: (Top part) List of structures used to fit EAM potential for hydrocarbons. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

methane -4.807 -4.806 -0.001 ethane syntactic -5.076 -5.061 -0.015 ethane isotactic -5.068 -5.047 -0.021 C2H2 -5.735 -5.736 0.001 C2H4 -5.382 -5.328 -0.054 propane -5.189 -5.183 -0.004 C3H4 allene -5.751 -5.342 -0.409 propylene -5.188 -5.002 -0.186 propyne -5.679 -5.687 0.008 cyclopropane -5.308 -4.850 -0.459 cyclopropene -5.366 -5.273 -0.094 cyclopropyne -5.340 -5.477 0.137 isobutane -5.249 -5.253 0.004 1-butene -5.250 -5.128 -0.122 1-butyne -5.648 -5.630 -0.018 cyclobutane -3.899 -3.899 0.000 methylcyclopropane -5.404 -5.410 0.006 pentane -5.291 -5.297 0.006 cyclopentane -5.024 -5.047 0.022 hexane -5.321 -5.329 0.008 benzene -6.136 -6.334 0.198 heptane -5.343 -5.352 0.009 octane -5.358 -5.370 0.012 nonane -5.368 -5.375 0.007 decane -5.383 -5.381 -0.002 undecane -5.392 -5.391 -0.001 dodecane -5.400 -5.413 0.013

ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

C 0.932 5.473 1.24 0.123 3.116 1.489 H 1.639 2.798 0.74 0.211 4.296 1.125

ZH/ZC = 0.880 knot1 knot2 knot3 knot4 knot5 knot6 knot7

C ρ 0.000 0.334 1.103 1.357 1.635 1.870 2.200 Fx 0.000 -3.851 -6.067 -8.011 -7.771 -8.17 -7.352

H ρ 0.000 1.071 1.528 1.773 2.008 2.200 - Fx 0.000 -2.997 -3.482 -4.189 -3.764 -3.409

61

Table 4.7: (Top part) List of structures used to fit EAM potential for alkanes and benzene. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

methane -4.806 -4.806 -0.000 ethane syntactic -5.060 -5.061 0.001 propane -5.182 -5.183 0.001 isobutane -5.254 -5.253 -0.001 pentane -5.298 -5.297 -0.001 hexane -5.329 -5.329 0.000 heptane -5.349 -5.352 0.003 octane -5.367 -5.370 0.003 nonane -5.380 -5.375 -0.005 decane -5.393 -5.381 -0.012 undecane -5.402 -5.391 -0.011 dodecane -5.411 -5.413 0.002 benzene -6.334 -6.334 0.000

ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

C 0.932 5.473 1.24 0.116 2.793 1.477 H 1.639 2.798 0.74 0.225 3.154 1.077

ZH/ZC = 0.533 C H

knot1 0,0 0,0 knot2 0.366,-3.793 1.040,-3.543 knot3 0.509,-5.454 1.534,-4.003 knot4 0.761,-6.130 1.801,-3.656 knot5 1.012,-7.611 1.911,-3.874 knot6 1.834,-8.037 2.200,-3.736 knot7 2.200,-7.288 -

62

4.4 Stage 4: Hydrocarbon on Al-Co-Ni

The final EAM potentials for hydrocarbon adsorption on Al-Co-Ni are fitted to the

adsorption energies of a hydrocarbon on a substrate. As the substrates, dB1-slab4A

and dH1-slab4A are used for methane, and dH2-slab4A are used for larger molecules.

The initial conditions are taken from Tables 4.5 and 4.7 and the fitting is performed

in the adsorption mode as explained previously. During the fitting, it is found that

the adsorption energies are more difficult to fit than those of molecules, substrates,

and molecule-on-substrate. This originates from the different nature of bonding. As

discussed later in Chapter 5, alkanes do not make strong chemical bonds with the Al-

Co-Ni. They are only physically adsorbed. Whereas the bondings within a molecule

and within the substrate are strong. Weighing factors of c1 = c2 = c3 = 1 and c4 > 1

are used to prioritize adsorption energies. Typically, values of c4 < 4 are used to get

the final fit. Increasing c4 beyond 4 does not considerably improve the fit, indicating

the limitations inherent with EAM formalism. The fitted structures and energies are

shown in Table 4.9 while the fitted parameters are tabulated in the Table 4.8.

63

Table 4.8: Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA+ZAφB/ZB)/2. Fitting structures are given in Table 4.9

C H Al Co Ni ρe (A˚−3) 0.932 1.639 0.026 0.09 0.126 β 5.473 2.798 7.182 5.945 5.447 re (A˚) 1.24 0.74 2.7 2.28 2.15 D (eV) 0.116 0.225 0.034 0.037 0.039 α (A˚−1) 2.786 3.156 1.740 3.636 3.198 r0 (A˚) 1.478 1.076 2.938 2.730 2.731 Z/ZC 1 0.532 0.989 1.115 1.258 Z/ZH 1 0.857 1.140 0.850 Z/ZAl 1 0.370 0.498 Z/ZCo 1 0.970 knot1 0,0 0,0 0,0 0,0 0,0 knot2 0.353,-3.860 1.037,-3.475 0.079,-2.816 0.217,-3.250 0.252,-3.932 knot3 0.519,-5.429 1.527,-3.974 0.108,-3.056 0.358,-5.176 0.327,-4.425 knot4 0.746,-6.254 1.797,-3.797 0.507,-4.214 0.774,-5.376 0.574,-4.849 knot5 1.033,-7.443 1.899,-3.744 0.893,-3.618 1.439,-5.139 1.032,-5.194 knot6 1.824,-7.988 2.200,-3.674 1.408,-3.681 1.635,-4.834 1.354,-5.065 knot7 2.200,-7.264 2.200,-2.911 2.200,-3.577 2.200,-4.130

64

Table 4.9: Fitted energies calculated using EAM parameters in Table 4.8. Methane up represents methane with one H below C and three H above C. Methane down is inverse of methane up. The unit for energy is eV/atom except for the adsorption energy which is in eV/molecule.

structure EEAM EV ASP ΔE Molecule energy (eV/atom): methane -4.805 -4.805 0.000 ethane syntactic -5.078 -5.061 -0.017 propane -5.189 -5.183 -0.006 butane -5.252 -5.252 0.000 pentane -5.293 -5.297 0.004 hexane -5.322 -5.328 0.007 benzene -6.294 -6.334 0.040 Substrate energy (eV/atom): dB1 slab4A -4.132 -4.096 -0.036 dH1 slab4A -4.141 -4.155 0.014 dH2 slab4A -4.135 -4.143 0.008 Molecule and substrate energy (eV/atom): methane on dB1 slab4A -4.173 -4.139 -0.034 methane up on dH1 slab4A -4.204 -4.219 0.015 methane down on dH1 slab4A -4.205 -4.219 0.014 ethane syntactic on dH2 slab4A -4.207 -4.214 0.006 propane on dH2 slab4A -4.242 -4.249 0.007 butane on dH2 slab4A -4.276 -4.283 0.008 pentane on dH2 slab4A -4.306 -4.315 0.009 hexane on dH2 slab4A -4.337 -4.346 0.009 benzene on dH2 slab4A -4.381 -4.385 0.004 Adsorption energy (eV/molecule): methane on dB1 slab4A -0.225 -0.238 0.013 methane up on dH1 slab4A -0.238 -0.243 0.005 methane down on dH1 slab4A -0.304 -0.257 -0.047 ethane syntactic on dH2 slab4A -0.348 -0.253 -0.095 propane on dH2 slab4A -0.458 -0.328 -0.131 butane on dH2 slab4A -0.439 -0.439 0.000 pentane on dH2 slab4A -0.465 -0.527 0.062 hexane on dH2 slab4A -0.520 -0.589 0.069 benzene on dH2 slab4A -0.790 -0.791 0.001

65

Chapter 5

Hydrocarbon adsorptions on d-AlNiCo

The low friction properties of quasicrystals in ambient conditions coupled with their

high hardness and oxidation resistance led to the development of applications of

quasicrystal coatings, for instance on machine parts, cutting blades, and non-stick

frying pans [5]. In machine parts, hydrocarbons are commonly used as a lubricant.

Superlubricity is the name given to the phenomenon in which two parallel single

crystal surfaces slide over each other with vanishingly small friction because their

structures are incommensurate. This phenomenon was proposed in the early 1990’s

[12] and experiment evidence for this effect has been seen in studies of mica sliding

on mica [123], W(110) on Si(100) [124], Ni(100) on Ni(100) [125], and tungsten on

graphite [13]. This effect is also expected in quasicrystals due to their aperiodic

structures at all length scales. Indeed, quasicrystal surfaces were observed to have

low friction not long after they were first discovered [126], but pinning down the exact

origin of the low friction has been elusive.

Recent experiments in ultra high vacuum (UHV) have demonstrated a frictional

dependence on aperiodicity for decagonal Al-Ni-Co quasicrystal (d-AlNiCo) against

thiol-passivated titanium-nitride tip [11]. In coating applications, it is expected that

even if superlubricity exists between moving parts, some additional lubricant would

still be needed to counter the macroscopic frictions due to grain boundaries, asperities,

and other defects in the surfaces of the moving parts. Some of the requirements of

the lubricant in such a situation are that it must wet the surface and that it must

not remove or reduce the superlubricity. Therefore it is desirable to have a good

understanding of how gases, hydrocarbons in particular, interact with quasicrystal

66

surfaces.

Very little is currently known about the interaction of hydrocarbons, their struc-

tures and growth on alloy or quasicrystalline surfaces. Some earlier experiments using

Fourier transform infrared spectroscopy (FTIR) and low energy electron diffraction

(LEED) suggest that on the 5-fold surface of Al-Pd-Mn, carbon monoxide (CO) does

not adsorb at 100 K, benzene adsorb at 100 K with possibly commensurate or dis-

ordered structure (LEED pattern unchanged) [127]. The same experiments on the

10-fold surface of d-AlNiCo show that CO bonds to the Ni sites at > 132 K, no struc-

ture reported, while there is no experiment available for benzene on Al-Ni-Co [127].

Later, scaning tunneling microscopy (STM) experiments on benzene adsorption on

Al-Pd-Mn show that the adsorbed benzene has a disordered structure [128]. In this

chapter, we report the simulation results of small hydrocarbons adsorb on the 10-fold

surface of decagonal Al73Ni10Co17 [15, 16], namely for methane, propane, hexane,

octane, and benzene.

5.1 Model

The simulations are performed within the framework of grand canonical ensemble

using Monte Carlo (GCMC) method as previously described in Section 2.5. The sim-

ulation cell is tetragonal. We take a square section of the surface, A, of side 5.12 nm,

to be the (x, y) part of the unit cell in the simulation, for which we assume periodic

boundary conditions along the basal directions. The coordinates of substrate atoms

are taken from Ref. [100]. The interaction potentials are modeled as the following.

The intermolecular interactions (adsorbate-adsorbate) are calculated as a sum of

pair interactions between atoms. For methane-methane [129, 130] Buckingham-type

potentials are used:

V (r) = Ae−Br − C/r6 (5.1)

67

Buckingham potentials also used to parametrize the benzene-benzene interactions

[131, 132]. For linear alkane-alkane, Morse-type potentials are used:

V (r) = −A(1− (1− e−B(r−C))2) (5.2)

The parameters for these potentials are summarized in Table 5.1. EAM potentials

generated in Chapter 4 are used for the rest of the interactions, namely the in-

tramolecular, adsorbate-substrate (C-Al, C-Co, C-Ni, H-Al, H-Co, H-Ni), and be-

tween substrate atoms (Al-Al, Al-Co, Al-Ni, Co-Ni). As previously mentioned in

Section 4.4, the ab initio calculations show that alkanes and benzene do not dissoci-

ate on dB1, dH1, and dH2 (these are decagonal Al-Ni-Co approximants). In these

systems, the surface of the substrate does not undergo any considerable relaxation

upon the adsorption of the molecules. Therefore, as a first approximation, in our

GCMC simulations, the substrate and the molecules are considered as rigid. How-

ever, molecules are allowed to explore all rotational degrees of freedom to achieve the

equilibrium configurations.

5.2 Adsorption potentials

Figure 5.1 displays the minima of the adsorption potential for methane (a), propane

(b), hexane (c), octane (d), and benzene (e), generated by minimizing the adsorption

potential of a molecule on d-AlNiCo with respect to z (Equation 3.1) and all rota-

tional degrees of freedom. The average adsorption energies are 221 (methane), 374

(propane), 620 (hexane), 794 (octane), and 931 (benzene), given in meV/molecule.

The figure shows the distribution of binding sites (dark spots) for the molecule.

Methane, propane, and benzene are small enough to follow the local atomic envi-

ronments of the substrate, whereas hexane and octane show considerable smearing

due to their large size. The location of dark spots in methane is similar to that

68

Table 5.1: Parameter values for the adsorbate-adsorbate interactions used for hydrocar- bon adsorption on a decagonal Al-Ni-Co. Intermolecular energies are calculated as a sum of pair interactions. For methane-methane, the C-H is taken as the geometrical mean for parameter A and as the arithmetic mean for parameters B and C.

A B C ref. (eV) (A˚−1) (A˚6)

methane C-C 82.132 2.693 449.53 [130] V (r) = Ae−Br − C/r6 C-H 66.217 2.892 167.51

H-H 53.381 3.105 62.42 [129] benzene C-C 11527.700 3.909 524 [131, 132] V (r) = Ae−Br − C/r6 C-H 348.518 3.703 75 [131, 132]

H-H 127.447 3.746 39 [131, 132] (meV) (A˚−1) (A˚)

alkane C-C 6.984 1.2655 4.1844 [133] V (r) = −A(1− (1− x)2) C-H 23.921 2.2744 2.544 [133] x = e−B(r−C) H-H 0.002 1.255 6.1543 [133]

[129] Tsuzuki S, et al 1993 J. Mol. Struct. 280 273 [130] Tsuzuki S, et al 1994 J. Phys. Chem. 98 1830 [131] Califano S, et al 1979 Chem. Phys. Lett. 64 491 [132] Chelli R, et al 2001 Phys. Chem. Chem. Phys. 3 2803 [133] Jalkanen J-P, et al 2002 J. Chem. Phys. 116 1303

69

of propane. However, it is interesting to see that the dark spots in methane and

propane become the bright ones in benzene. To study more, the topview of the

substrate atoms is depicted in panel (f) of Figure 5.1. The legend for the atoms is:

Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red),

Ni-toplayer (green), and Ni-otherlayers (blue). In methane and propane, dark spots

occur when the center of the molecule is located on top of black-Al (5 gray-Al on

toplayer forming a pentagon). In benzene, this location gives a bright spot (less bind-

ing). Except the binding site at the center of the figure, strong binding sites (dark

spots) in benzene occur when the center of benzene is located on top of gray-Al (on

the top layer, the pentagon consists of 2 Al and 3 Ni, or 3 Al and 2 Ni).

5.3 Molecule orientations

First we study the orientations of a molecule as it is adsorbed on d-AlNiCo. During

the calculation of minima of the adsorption potential (Section 5.2, the orientation of

the molecule is recorded when a minimum is achieved. It is found that for methane,

the rotational ground state is degenerate indicating its spherical nature. Ab initio

calculations of methane dimers indicate that the interaction energy within the dimer

depends on the relative orientations of the two [129, 130]. Therefore, the degeneracy

observed is due to the limitation of the EAM model. For linear alkanes, we can

define θ as the angle between the substrate’s xy-plane and the main axis of the

alkanes. Again, θ is recorded when an adsorption minimum is achieved. We find that

θ decreases with increasing alkane chain, e.g. propane (θ = 10◦), hexane (θ = 5◦),

and octane (θ ∼ 0◦). For benzene, θ would be the angle between the molecule’s plane and substrate’s xy-plane, and it is θ ∼ 0◦. Therefore, we conclude that alkanes and benzene prefer most contact with d-AlNiCo.

70

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-180

-200

-220

-240

-260

-300

1

-280

-221

-300

-350

-400

-450

-500

-550

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

1 -374

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-550

-600

-650

-700

-800

1

-750

-620

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-750

-800

-850

-900

1

-950

-700

-794

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-800

-900

-1100

1

-1000

-931

a) b)

c) d)

e) f)

Figure 5.1: (color online). (a-e) Adsorption potential map, calculated by minimizing the adsorption potential of one molecule on a decagonal Al-Ni-Co along z direction and all rotational degrees of freedom at every coordinates (x, y). Red numbers represent the average value of the adsorption energies. (f) Top view of the decagonal Al-Ni-Co substrate 51.2x51.2 A˚2: Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red), Ni-toplayer (green), and Ni-otherlayers (blue).

71

5.4 Adsorption isotherms

To use quasicrystals as low-friction coatings in conjunction with oil lubricants, the

lubricants must be able to spread well on the quasicrystals. Lubricants consist mostly

of alkanes. The wetting of d-AlNiCo by alkanes and benzene is demonstrated in

Figure 5.2. In the figure, the number of molecules adsorbed on the substrate per area

is plotted as a function of pressure at various temperatures. The results for methane,

propane, hexane, octane, and benzene are shown. The simulated temperatures are:

for methane T = 68, 85, 136, 185 K, for propane T = 80, 127, 245, 365 K, for hexane

T = 134, 170, 267, 450 K, for octane T = 162, 210, 324, 450, 565 K, for benzene T =

209, 270, 418, 555 K. Note that T = 450 K represents a typical temperature of the

inner wall of cylinder in regular car engines [134]. In the isotherms, even though

the step corresponding to the adsorption of the first layer is well observed, steps

corresponding to the second and further layers are not evident for methane, propane,

and benzene, and barely visible for hexane and octane. The second layer condensation

occurs near the bulk condensation and extends a short range of pressure, nevertheless

multilayer adsorptions can be seen at higher pressures (point d) as shown in the inset

of each panel in which the distribution of adsorbed molecules is plotted against the

z direction.

5.5 Density profiles

Methane on d-AlNiCo. Figure 5.3 shows the density plots for methane at two

different coverages, corresponding to points ”a” and ”c” on the 68 K isotherm in

Figure 5.2.a. At submonolayer regime, methanes occupy the strong binding sites

on the surface. At monolayer coverage, the ordering is 5-fold commensurate with

the substrate, also indicated by the Fourier transform (panel c), and there is no

transition to 6-fold. The evidence that there is no such transition is shown more

72

10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

14

16

18

20

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

a

b c

d

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5 Z (nm)

Xe CH4

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a

b c

d

ρ N (m

ol ec

ul es

/n m

2 )

P (atm)

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5 Z (nm)

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a b c

d

Z (nm)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a b c

d

Z (nm)

ρ Z (a

rb .)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

0 0.5 1 1.5 2 2.5

10-25 10-20 10-15 10-10 10-5 1 0

2

4

6

8

10

12

14

16

18

20

a

b c

d

0 1 2 3 Z (nm)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

ρ Z (a

rb .)

c) d)

e)

a) b)

Figure 5.2: (color online). Isothermal adsorption densities of hydrocarbons on a decago- nal Al-Ni-Co: (a) methane (from left to right T = 68, 85, 136, 185 K), (b) propane (T = 80, 127, 245, 365 K), (c) hexane (T = 134, 170, 267, 450 K), (d) octane (T = 162, 210, 324, 450, 565 K), and (e) benzene (T = 209, 270, 418, 555 K). The inset in each figure is the density along z direction at pressure corresponding to point d. Xenon (red) is plotted in panel (a) for comparison.

73

10 -15

10 -10

10 -5 1

0

5

10

15

ρN (m olecules/nm

2)

P (atm)

0

0.2

0.4

0.6

0.8

1

ρ 5 -6

a b

c d

Figure 5.3: (a) and (b) Calculated density of methane adsorbed on a decagonal Al-Ni-Co at pressures corresponding to points ”a” and ”c” of the 68 K isotherm shown in Figure 5.2.a, respectively. (c) Fourier transform of the density plot shown in (b), consistent with 5-fold ordering of the methane near monolayer completion. (d) Order parameter (left axis, as calculated in Equation 3.14) as a function of pressure for the 68 K isotherm (right axis), indicating no sharp transition to 6-fold ordering.

clearly in panel d, where the order parameter (left axis, as calculated in Equation

3.14) does not present any sharp transition as seen in the case of xenon adsorption on

the same substrate (Section 3.2.3). The lack of 5-fold to 6-fold transition is consistent

with our proposed rule based on rare gases (Section 3.2.6), that the size of the rare

gas must be at least as large as Xe on this quasicrystalline surface for the transition to

occur [135]. Note that the size of methane relative to xenon (ratio of Lennard-Jones

σ parameters) is 0.8 [69].

Propane, Hexane, and Octane on d-AlNiCo. At submonolayer coverages,

propane, hexane, and octane do not show any clear evidence of binding to specific sites

of the surface even though molecules tend to bind in the center of the surface with

74

the most attractive site. Figure 5.4 shows the density profiles for these hydrocarbons

at pressure corresponding to the first layer completion at various temperatures. The

simulated temperatures have been indicated in Section 5.4. At monolayer coverage,

propane adsorbs in a disordered fashion at all temperatures, whereas hexane and

octane tend to form close-packed structures as suggested by domains with stripe

feature. As comparisons, the crystal structures of solid propane at 30 K is monoclinic

(space group #11, P21/m) [136], hexane is triclinic (space group #2, P1¯) at 90 K

[136] and at 158 K [137], octane is triclinic (space group #2, P1¯) at 90 K [136]

and at 213 K [138]. The 2-dimensional structures of these hydrocarbons are close-

packed structures as charactereized by stripe structures with 1 and 2 molecules per

unit cell for even- and odd-alkanes, respectively [136]. In general, even-alkanes form

triclinic (6 ≤ NC ≤ 26), monoclinic (28 ≤ NC ≤ 36), or orthorombic (38 ≤ NC) with decreasing packing density [136, 137, 138, 139], where NC being the number of

carbon atoms. Whereas for odd-alkanes form triclinic (7 ≤ NC ≤ 9) or orthorombic (11 ≤ NC) [136, 139, 140] also with decreasing packing density for longer chain. Note that the even-alkanes have higher packing density than odd-alkanes.

Benzene on d-AlNiCo. As in the case of methane, at submonolayer coverage,

benzene preferentially adsorbs at sites offering the strongest binding at all simulated

temperatures. At pressure near to first layer completion, the density profiles show a

temperature-dependence as plotted in Figure 5.5 for 209 K, 270 K, and 418 K. The

structures are more clearly charaterized by the plotting the geometrical center of

density as shown in the middle row of the figure. At T = 209 K, pentagonal ordering

is observed. As the temperature is increased, a mixture of 5-fold and 6-fold ordering

is seen, e.g. at T = 270 K. At higher temperature, T = 418 K, 6-fold structure

dominates the ordering of the monolayer as confirmed by the Fourier analysis of the

density showing hexagonal spots characteristic of triangular lattice (bottom row, last

75

267 K170 K 450 K

324 K210 K 450 K

hexane

octane

80 K 127 K 245 Kpropane

Figure 5.4: Calculated density of propane, hexane, and octane adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion. Propane forms a disordered structures, whereas hexane and octane tend to form close-packed structures indicated by stripe features with increasing order for longer chain.

76

column in Figure 5.5). The crystal structure of bulk benzene has been determined

at 4.2 K and 270.15 K to be orthorombic (space group #61, Pbca) with 4 molecules

per unit cell [141].

The evolution of the density profile over temperature from being 5-fold to 6-fold

can be studied more clearly by plotting the order parameter, ρ5−6 = N5/(N5 + N6)

(Nn denotes the number of molecules having n nearest neighbors as defined in Section

3.1.6) as a function of T as shown in Figure 5.6 (left axis). The plot is taken at a

constant pressure of 0.01 atm corresponding to point c in Figure 5.2 panel e. The

adsorption isobar representing the number of adsorbed molecules N as a function of

T is plotted in right axis of Figure 5.6. Three dashed lines corresponding to T =209,

270, and 418 K, whose adsorption isotherms are plotted earlier in Figure 5.2 panel e,

are added for guidance in comparing the density profile to the order parameter value

at each of these temperatures. We observe the following trends:

• (region 1) T < 260 K, 0.5 < ρ5−6 → 5-fold ordering dominates, N = 44.

• (region 2) 260 ≤ T ≤ 280 K, 0.3 ≤ ρ5−6 ≤ 0.5 → 5-fold becomes mostly 6-fold, N = 44.

• (region 3) 280 ≤ T ≤ 370 K, ρ5−6 = 0.3 → 6-fold ordering dominates, N = 44.

• (region 4) 370 ≤ T ≤ 390 K, 0.1 ≤ ρ5−6 ≤ 0.3→ transition to 6-fold ordering, 44 ≤ N ≤ 46.

• (region 5) T > 390 K, ρ5−6 increases from 0.1 → 6-fold ordering weakens, N decreases from 46.

The highest 6-fold ordering occurs at T = 390 K which is mediated by a gain of 2

additional molecules adsorbed on the substrate. Beyong this temperature, thermal

77

209 K 270 K 418 K

Figure 5.5: (Top row) Calculated density of benzene adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion for 209 K, 270 K, and 418 K. (middle row) Density profile of the geometrical center of density shown in the top row. (bottom row) Fourier transform of the density plot shown in the middle row, showing 5-fold ordering at 209 K, mixture of 5-fold and 6-fold structures at 270 K, and mostly 6-fold features at 418 K.

78

200 250 300 350 400 450 500

0.1

0.2

0.3

0.4

0.5

0.6

T (K)

ρ 5 -6

38

40

42

44

46 N

m olecules

Figure 5.6: (left axis) Order parameter ρ5−6 = N5/(N5+N6) (Nn denotes the number of molecules having n nearest neighbors) as a function of temperature at 0.01 atm of pressure. (right axis) Adsorption isobar showing the number of adsorbed molecules as a function of temperature. Verticel dashed lines correspond to T = 209, 270, and 418 K whose density profiles are plotted in Figure 5.5.

energy causes reduction of N resulting in less ordered 6-fold structure as T increases.

Nevertheless, the 6-fold ordering are still well resolved up to T = 418 K as shown

before in Figure 5.5.

5.6 Summary

In this chapter, we have presented our studies on methane, propane, hexane, octane,

and benzene adsorption on decagonal Al-Ni-Co quasicrystal. All of these hydro-

carbons form very well defined step corresponding to the first layer condensation.

Eventhough multilayer formation is evident from the density distribution along ver-

tical dimension, the step corresponding to the condensation of the second and further

layers can not be observed clearly in the isotherm due narrow range of pressure for

these layers to form before bulk condensation occurs. Monolayer of methane has

79

been determined to be pentagonal commensurate with the substrate for 68 K ≤ T ≤ 136 K. Monolayer of propane shows disordered structures for 80 K ≤ T ≤ 245 K. Monolayers of hexane (134 K≤ T ≤ 450 K) and octane (162 K ≤ T ≤ 450 K) form 2- dimensional close-packed structures characterized by stripe ordering consistent with

their bulk crystal structures. Benzene monolayer is pentagonal at T ≤ 260 K, which transforms into 6-fold structure at T ≥ 280 K with the highest 6-fold ordering occurs at T = 390 K. Beyong 390 K, thermal energy causes fewer adsorbed benzene and the

6-fold ordering starts to deterioates. Nevertheless, the 6-fold ordering are still well

observed up to T = 418 K.

80

Chapter 6

Embedded-atom potentials for selected

pure elements in the periodic table

Embedded-atom method (EAM) formalism provides only the skeleton of calculat-

ing the total electronic energy for a given charge density. It does not specify the

parametrization of the three functionals: atomic charge density, embedding en-

ergy, and pair interaction. Many different parametrizations have been introduced

[54, 55, 58, 59, 59, 65, 142, 143, 144, 145]. This arises inconveniences when one needs

to use different potentials for different elements. Steps toward universal EAM poten-

tials are needed. Recently, a consistent set of atomic charge density for all elements

in the periodic table suitable for EAM potentials has been developed [64]. The set

is calculated by spherically averaging the atomic charge density of the solution of

Hartree-Fock (HF) equations (Equation 2.19) of an isolated atom. The exchange

term in the HF equation contains an atom-specific parameter which is adjusted to

reproduce the experimental first ionization energy, hence are semiempirical. In this

chapter we use these charge densities to generate a consistent set of EAM potentials

for pure elements in the periodic table.

The parametrizations are given in Section 2.3. For each element, the potential is

fitted to the ab initio cohesive energies in body-centered cubic (BCC), face-centered

cubic (FCC), hexagonal close-packed (HEX), and diamond structure (DIA). In ad-

dition, energies in the ground state structure at different pressures are included in

the fitting procedure to ensure the mechanical stability of the potentials. Energies at

various pressures are achieved by expanding or compressing the equillibrium lattice

81

constant a by a factor from 0.90 to 1.16. Within an accuracy of 30 meV, all the

EAM potentials successfully predict the correct ground state structure with respect

to those that are not included in the fitting step, namely graphite (GRA), simple

cubic (SC), and simple hexagonal (SH) structures. The fitted parameters are sum-

marized in Tables 6.1-6.4. Tables 6.5 and 6.6 list the ground state structures of

the elements, range of charge density covered in the fit, and the equillibrium lattice

constants calculated by the ab initio method as well as by using the fitted EAM

parameters. Literature value of the lattice constants is also included as comparisons.

Overall, the EAM potentials predict the lattice constants with within 0.5 A˚ of the

literature value except for tellurium. Indeed, elements in the V-A, VI-A, and VII-A

colums of periodic table tend to form complex structures with large unit cells and

more EAM parameters might be needed to fit these structures better. Manganese is

excluded also because of its complex cubic structure with 58 atoms/cell. Noble gases

are excluded because they are already well described by simple pair potentials. Ele-

ments in the Actinide series are excluded because they do not have a stable structure

due to their radioactivity.

82

Table 6.1: Fitted parameters for the charge density and pair interaction functionals of the EAM potentials for pure elements, continued in Table 6.2. The fitting structures are given in Tables 6.5 and 6.6. The parameters for the embedding functionals are given in Tables 6.3 and 6.4.

Z Struct. ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

3 Li 0.038 4.813 2.670 0.010 3.015 2.422 4 Be 0.031 7.174 2.450 0.016 3.317 2.216 6 C 0.932 5.473 1.240 0.115 3.152 1.441 11 Na 0.021 5.552 3.080 0.011 5.800 3.500 12 Mg 0.001 10.030 3.890 0.012 2.807 3.110 13 Al 0.026 7.182 2.700 0.034 3.038 3.012 14 Si 0.072 7.171 2.250 0.043 4.115 1.601 19 K 0.008 6.412 3.920 0.015 2.926 4.088 20 Ca 0.002 9.719 4.280 0.020 2.940 3.646 21 Sc 0.058 6.053 2.700 0.021 2.264 3.210 22 Ti 0.287 4.333 1.940 0.022 2.282 2.850 23 V 0.394 4.025 1.780 0.023 2.501 2.615 24 Cr 0.297 4.538 1.680 0.024 2.519 2.494 26 Fe 0.192 5.214 2.000 0.026 5.747 2.412 27 Co 0.090 5.945 2.280 0.037 2.832 2.701 28 Ni 0.126 5.447 2.150 0.039 2.839 2.717 29 Cu 0.052 5.864 2.220 0.029 2.589 2.553 30 Zn 0.0005 14.630 4.800 0.030 2.655 2.560 32 Ge 0.042 7.419 2.440 0.045 2.659 2.001 37 Rb 0.006 6.788 4.210 0.037 3.599 4.694 38 Sr 0.002 9.614 4.450 0.038 2.926 3.299 39 Y 0.037 6.593 3.000 0.039 2.123 3.300 40 Zr 0.158 5.287 2.300 0.040 2.132 3.230 41 Nb 0.145 5.556 2.080 0.041 2.347 2.858 42 Mo 0.201 5.621 1.940 0.042 2.153 2.728 43 Tc 0.365 4.968 1.880 0.043 2.559 2.640 44 Ru 0.047 6.360 2.400 0.044 2.702 2.700 45 Rh 0.020 6.981 2.680 0.045 2.609 2.687 46 Pd 0.017 8.389 2.500 0.036 2.719 2.791 47 Ag 0.028 6.619 2.530 0.027 2.982 3.055 48 Cd 0.0005 13.340 4.500 0.048 2.284 3.170 52 Te 0.047 8.157 2.560 0.052 2.477 2.245 56 Ba 0.001 10.020 4.900 0.056 2.815 3.847 57 La 0.009 8.106 3.890 0.057 2.518 3.350

83

Table 6.2: continuation of Table 6.1.

58 Ce 0.013 7.575 3.682 0.058 2.488 2.949 59 Pr 0.016 7.342 3.646 0.059 2.502 2.970 60 Nd 0.015 7.361 3.632 0.060 2.543 3.548 63 Eu 0.016 7.306 3.542 0.033 2.596 3.300 64 Gd 0.015 7.724 3.470 0.064 2.549 3.240 65 Tb 0.017 7.323 3.464 0.065 2.527 3.300 66 Dy 0.017 7.303 3.420 0.066 2.512 3.240 67 Ho 0.018 7.323 3.392 0.067 2.532 3.180 68 Er 0.019 7.271 3.346 0.068 2.560 3.260 69 Tm 0.020 7.284 3.320 0.069 2.597 3.140 70 Yb 0.020 7.250 3.274 0.070 2.542 3.522 71 Lu 0.023 6.972 3.342 0.071 2.596 3.210 72 Hf 0.138 5.581 2.350 0.072 2.454 2.900 73 Ta 0.142 6.016 2.300 0.073 2.026 2.766 74 W 0.137 6.176 2.270 0.074 2.350 2.630 75 Re 0.242 3.686 2.040 0.075 3.055 2.760 76 Os 0.029 7.771 2.760 0.076 2.696 2.540 77 Ir 0.244 6.055 1.960 0.077 3.142 2.715 78 Pt 0.021 7.539 2.500 0.078 2.919 2.772 79 Au 0.024 7.553 2.470 0.079 2.917 2.885 81 Tl 0.004 8.850 3.500 0.081 2.576 3.300 82 Pb 0.002 8.368 2.930 0.082 2.935 3.242

84

Table 6.3: Fitted knots of cubic spline of the embedding functionals for the EAM po- tentials of pure elements, continued in Table 6.4. The first knot at (0,0) is assumed. The fitting structures are given in Tables 6.5 and 6.6.

knot2 knot3 knot4 knot5 knot6 knot7 (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV)

Li 0.048,-1.058 0.119,-1.52850 0.190,-1.54750 0.246,-1.607 0.291,-1.554 0.469,-1.469 Be 0.178,-3.079 0.228,-3.241 0.423,-3.509 0.731,-3.647 1.057,-3.588 1.670,-3.324 C 0.301,-4.967 0.344,-5.905 0.476,-6.986 0.861,-7.490 1.419,-7.264 2.389,-5.970 Na 0.014,-0.725 0.029,-0.9756 0.062,-1.059 0.082,-1.057 0.135,-1.059 0.181,-1.424 Mg 0.015,-0.939 0.033,-1.292 0.062,-1.414 0.096,-1.413 0.133,-1.356 0.242,-1.175 Al 0.078,-2.811 0.121,-3.106 0.495,-3.980 0.897,-3.879 1.381,-3.665 2.200,-3.074 Si 0.057,-3.662 0.087,-4.153 0.157,-4.476 0.286,-4.409 0.349,-4.204 0.561,-3.168 K 0.005,-0.613 0.009,-0.757 0.025,-0.844 0.057,-0.710 0.089,-0.551 0.149,-0.415 Ca 0.009,-1.202 0.021,-1.692 0.062,-1.814 0.094,-1.719 0.135,-1.578 0.201,-1.434 Sc 0.055,-3.363 0.080,-3.787 0.140,-4.092 0.226,-4.140 0.315,-4.009 0.481,-3.612 Ti 0.133,-4.307 0.188,-4.850 0.314,-5.315 0.444,-5.410 0.712,-5.033 1.075,-4.406 V 0.245,-4.014 0.333,-4.545 0.606,-5.153 0.858,-5.111 1.058,-4.850 1.520,-4.005 Cr 0.168,-3.162 0.182,-3.277 0.300,-3.808 0.487,-3.758 0.634,-3.324 0.935,-1.915 Fe 0.171,-4.007 0.247,-4.554 0.505,-5.071 0.834,-4.962 1.072,-4.734 1.445,-4.613 Co 0.146,-3.374 0.380,-4.894 0.821,-5.517 1.306,-5.509 1.683,-5.008 2.200,-3.952 Ni 0.223,-3.899 0.336,-4.373 0.566,-4.731 1.028,-5.280 1.355,-5.028 2.200,-4.172 Cu 0.091,-2.811 0.127,-3.100 0.238,-3.280 0.334,-3.215 0.455,-2.935 0.604,-2.374 Zn 1.012,-0.752 1.604,-0.880 2.794,-0.940 3.544,-0.919 4.805,-0.822 8.006,-0.406 Ge 0.056,-3.285 0.072,-3.429 0.115,-3.592 0.188,-3.553 0.241,-3.359 0.364,-2.645 Rb 0.004,-0.552 0.009,-0.707 0.020,-0.663 0.029,-0.594 0.036,-0.633 0.047,-0.977 Sr 0.005,-1.128 0.008,-1.304 0.046,-1.558 0.063,-1.490 0.088,-1.331 0.108,-1.205 Y 0.012,-3.002 0.021,-3.548 0.098,-4.152 0.167,-4.060 0.260,-3.712 0.338,-3.459 Zr 0.052,-4.542 0.082,-5.367 0.186,-6.136 0.299,-6.147 0.489,-5.725 0.685,-5.428 Nb 0.034,-6.744 0.054,-7.933 0.139,-9.012 0.240,-8.889 0.347,-8.192 0.468,-7.278 Mo 0.059,-4.588 0.088,-5.395 0.192,-6.024 0.271,-5.941 0.363,-5.304 0.542,-4.467 Tc 0.164,-5.372 0.217,-5.913 0.317,-6.476 0.482,-6.730 0.720,-6.230 1.108,-4.437 Ru 0.088,-6.348 0.130,-7.166 0.241,-7.687 0.311,-7.756 0.457,-7.052 0.638,-6.282 Rh 0.074,-4.522 0.095,-4.900 0.176,-5.540 0.279,-5.529 0.381,-4.971 0.495,-3.965 Pd 0.017,-2.667 0.025,-3.029 0.078,-3.490 0.128,-3.256 0.166,-2.905 0.232,-2.126 Ag 0.027,-1.749 0.035,-1.894 0.117,-2.349 0.178,-2.390 0.239,-2.422 0.294,-2.488 Cd 0.061,-0.301 0.110,-0.379 0.245,-0.409 0.364,-0.446 0.690,-0.333 0.921,-0.278 Te 0.006,-1.550 0.011,-1.922 0.029,-2.088 0.038,-2.159 0.069,-1.969 0.117,-1.615 Ba 0.004,-1.228 0.010,-1.721 0.020,-1.822 0.027,-1.813 0.050,-1.658 0.081,-1.393 La 0.019,-2.564 0.051,-3.977 0.121,-4.240 0.210,-4.086 0.322,-3.777 0.416,-3.536 Ce 0.040,-2.826 0.106,-4.210 0.207,-4.481 0.329,-4.503 0.526,-4.234 0.677,-3.908

85

Table 6.4: continuation of Table 6.3.

Pr 0.035,-2.470 0.088,-3.600 0.166,-3.856 0.246,-3.891 0.357,-3.821 0.554,-3.605 Nd 0.021,-2.403 0.062,-3.730 0.157,-3.834 0.308,-3.858 0.456,-3.972 0.554,-3.776 Eu 0.010,-1.188 0.021,-1.599 0.055,-1.790 0.125,-1.734 0.191,-1.516 0.275,-1.357 Gd 0.017,-3.615 0.043,-5.278 0.085,-5.539 0.129,-5.542 0.219,-5.281 0.330,-4.894 Tb 0.027,-2.642 0.067,-3.837 0.142,-4.061 0.202,-3.963 0.319,-3.614 0.409,-3.340 Dy 0.027,-2.672 0.068,-3.683 0.158,-4.026 0.199,-3.866 0.279,-3.728 0.404,-3.139 Ho 0.028,-2.672 0.069,-3.851 0.162,-4.020 0.199,-3.927 0.283,-3.689 0.406,-3.248 Er 0.028,-2.681 0.069,-3.729 0.162,-3.947 0.196,-3.823 0.276,-3.654 0.406,-3.278 Tm 0.030,-2.762 0.069,-3.659 0.168,-3.992 0.217,-3.801 0.289,-3.658 0.397,-3.268 Yb 0.012,-0.958 0.022,-1.224 0.056,-1.366 0.122,-1.191 0.188,-1.044 0.264,-0.965 Lu 0.040,-2.683 0.087,-3.659 0.155,-3.911 0.210,-3.900 0.340,-3.630 0.482,-3.254 Hf 0.036,-4.599 0.057,-5.285 0.209,-6.329 0.279,-6.266 0.399,-5.886 0.680,-4.554 Ta 0.066,-5.540 0.109,-6.815 0.210,-7.729 0.409,-7.719 0.624,-6.877 0.903,-6.333 W 0.111,-6.179 0.149,-6.886 0.256,-7.726 0.465,-7.681 0.689,-6.553 0.953,-4.666 Re 0.307,-4.531 0.441,-5.847 0.680,-7.039 0.803,-7.237 1.102,-6.969 1.636,-5.140 Os 0.106,-6.174 0.206,-7.540 0.326,-7.866 0.394,-7.945 0.570,-7.555 0.908,-5.803 Ir 0.089,-5.498 0.106,-6.040 0.228,-7.004 0.342,-7.142 0.486,-6.271 0.667,-5.735 Pt 0.025,-3.842 0.035,-4.280 0.094,-5.002 0.149,-4.728 0.188,-4.327 0.250,-3.495 Au 0.015,-1.932 0.023,-2.231 0.075,-2.492 0.108,-2.276 0.134,-2.038 0.179,-1.590 Tl 0.007,-1.336 0.013,-1.611 0.029,-1.661 0.062,-1.476 0.091,-1.268 0.133,-1.001 Pb 0.0004,-1.813 0.0006,-2.159 0.0010,-2.454 0.0016,-2.537 0.0046,-2.491 0.0125,-1.686

86

Table 6.5: (left part) Structures used to fit EAM potentials for pure elements, continued in Table 6.6. The EAM potentials are fitted to the ab initio energies in body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close-packed (HEX), diamond structure (DIA), and groundstate structure at various pressures obtained by expanding/compressing the equillibrium lattice constant a by a factor from 0.9 to 1.16 corresponding to a range of charge density from ρmax to ρmin. (right part) Lattice constansts calculated using the fitted parameters. The literature values aLIT are taken from [146].

Z Name Struct. ρmin → ρmax aLIT aV ASP aEAM (A˚−3) (A˚) (A˚) (A˚)

3 Li BCC 0.104 → 0.441 3.49 3.44 3.44 4 Be HEX 0.268 → 1.437 2.29 2.27 2.24 6 C DIA 0.329 → 2.038 3.57 3.57 3.57 11 Na BCC 0.027 → 0.163 4.23 4.19 4.19 12 Mg HEX 0.014 → 0.174 3.21 3.19 3.16 13 Al FCC 0.098 → 0.312 4.05 4.04 4.00 14 Si DIA 0.060 → 0.443 5.43 5.46 5.46 19 K BCC 0.013 → 0.089 5.23 5.29 5.29 20 Ca FCC 0.014 → 0.186 5.58 5.54 5.54 21 Sc HEX 0.064 → 0.441 3.31 3.30 3.34 22 Ti HEX 0.150 → 0.845 2.95 2.92 2.92 23 V BCC 0.263 → 1.320 3.02 2.98 2.98 24 Cr BCC 0.126 → 0.765 2.88 2.84 2.84 26 Fe BCC 0.213 → 1.221 2.87 2.83 2.83 27 Co HEX 0.361 → 0.974 2.51 2.47 2.45 28 Ni FCC 0.374 → 0.983 3.52 3.49 3.46 29 Cu FCC 0.113 → 0.519 3.61 3.64 3.64 30 Zn HEX 0.777 → 7.050 2.66 2.71 2.71 32 Ge DIA 0.056 → 0.306 5.66 5.79 5.79 37 Rb BCC 0.004 → 0.048 5.59 5.65 5.65 38 Sr FCC 0.009 → 0.097 6.08 6.05 6.05 39 Y HEX 0.036 → 0.282 3.65 3.64 3.64 40 Zr HEX 0.073 → 0.511 3.23 3.22 3.22 41 Nb BCC 0.050 → 0.390 3.30 3.31 3.31 42 Mo BCC 0.055 → 0.463 3.15 3.15 3.12 43 Tc HEX 0.147 → 0.993 2.74 2.76 2.76 44 Ru HEX 0.084 → 0.558 2.70 2.73 2.70 45 Rh FCC 0.072 → 0.469 3.80 3.84 3.84 46 Pd FCC 0.017 → 0.195 3.89 3.96 3.96 47 Ag FCC 0.034 → 0.256 4.09 4.16 4.16 48 Cd HEX 0.076 → 0.855 2.98 3.09 3.03 52 Te HEX 0.007 → 0.108 4.45 4.06 4.02

87

Table 6.6: continuation of Table 6.5.

56 Ba BCC 0.007 → 0.074 5.02 5.02 5.02 57 La HEX 0.042 → 0.331 3.75 3.74 3.74 58 Ce FCC 0.096 → 0.610 5.16 4.74 4.74 59 Pr HEX 0.076 → 0.502 3.67 3.45 3.45 60 Nd HEX 0.068 → 0.457 3.66 3.56 3.56 63 Eu BCC 0.029 → 0.232 4.61 4.46 4.41 64 Gd HEX 0.037 → 0.303 3.64 3.62 3.62 65 Tb HEX 0.049 → 0.362 3.60 3.62 3.62 66 Dy HEX 0.047 → 0.345 3.59 3.61 3.61 67 Ho HEX 0.048 → 0.356 3.58 3.59 3.63 68 Er HEX 0.047 → 0.352 3.56 3.58 3.61 69 Tm HEX 0.049 → 0.367 3.54 3.55 3.55 70 Yb FCC 0.023 → 0.202 5.49 5.29 5.29 71 Lu HEX 0.061 → 0.425 3.51 3.48 3.48 72 Hf HEX 0.073 → 0.530 3.20 3.20 3.20 73 Ta BCC 0.091 → 0.692 3.31 3.32 3.32 74 W BCC 0.105 → 0.794 3.16 3.19 3.19 75 Re HEX 0.378 → 1.509 2.76 2.78 2.78 76 Os HEX 0.114 → 0.861 2.74 2.76 2.73 77 Ir FCC 0.068 → 0.629 3.84 3.88 3.80 78 Pt FCC 0.025 → 0.236 3.92 3.98 3.98 79 Au FCC 0.016 → 0.167 4.08 4.17 4.17 81 Tl HEX 0.011 → 0.112 3.46 3.54 3.54 82 Pb FCC 0.001 → 0.011 4.95 5.02 5.02

88

Chapter 7

Effects of Mo on the thermodynamics of

Fe:Mo:C nanocatalyst for single-walled

carbon nanotube growth

Among the established methods for single-walled carbon nanotubes (SWCNTs) syn-

thesis [147, 148, 149], catalytic chemical vapor decomposition (CCVD) technique is

preferred for growing nanotubes on a substrate at a target position due to its rel-

atively low synthesis temperature. Temperature as low as ∼ 450 oC was reported by using hydrocarbon feedstock with exothermic catalytic decomposition reaction

[150, 151]. Critical factors for the efficient growth via CCVD are the compositions

of the interacting species, the preparation of the catalysts, and the synthesis condi-

tions. Efficient catalysts must have long active lifetimes (with respect to feedstock

dissociation and nanotube growth), high selectivity and be less prone to contamina-

tion. Common factors that lead to reduction in catalytic activity are deactivation

(e.g. due to coating with carbon or nucleation of inactive phases) [152, 153, 154] and

thermal sintering (e.g. caused by highly exothermic reactions on the clusters surface

[150, 151, 155] with insufficient heat [156]).

Metal alloy catalysts, such as Fe:Co, Co:Mo, and Fe:Mo, improve the growth of

CNTs [157, 158, 159, 160, 162, 163], because the presence of more than one metal

species can significantly enhance the activity of a catalyst [159, 164], and can prevent

catalyst particle aggregation [163, 164]. In the case of Fe:Mo nanoparticles supported

on Al2O3 substrates, the enhanced catalyst activity has been shown to be larger than

the linear combination of the individual Fe/Al2O3 and Mo/Al2O3 activities [159, 160].

89

This is explained in terms of substantial intermetallic interaction between Mo, Fe and

C [159, 165, 166]. The addition of Mo in mechanical alloying of powder Fe and C

mixtures promotes solid state reactions even at low Mo concentrations by forming

ternary phases, such as the (Fe,Mo)23C6 and Fe2(MoO4)3 type carbides [167]. It has

been found that low Mo concentration in Fe:Mo is favored for growing SWCNTs

(on Al2O3 substrates) since the presence, after activation, of the phase Fe2(MoO4)3

can lead to the formation of small metallic clusters [168]. Considering the vapor-

liquid-solid model (VLS), which is the most probable mechanism for CNT growth

[159, 169]. The metallic nanoparticles are very efficient catalysts when they are in

the liquid or viscous states, probably due to considerable carbon bulk-diffusion in

this phase (compared to surface or sub-surface diffusion). Generally, unless stable

intermetallic compounds form, alloying metals reduce the melting point below those

of the constituents [165, 166]. Hence, to improve the yield and quality of nanotubes,

one can tailor the composition of the catalyst particle to move its liquidus line below

the synthesis temperature [159]. However, identifying the perfect alloy composition

is non trivial. In fact, the presence of more than two metallic species allows for the

possibility of different carbon pollution mechanisms by thermodynamic promotion of

ternary carbides. In this Chapter, we study the phase diagram of Fe:Mo:C system

and the possible roles of Mo in the catalytic properties of Fe:Mo.

7.1 Size-pressure approximation

Determining the thermodynamic stability of different phases in nanoparticles of dif-

ferent sizes with ab initio calculations is computationally expensive. We develop

a simple model, called the ”size-pressure approximation”, which allows one to es-

timate the phase diagram at the nanoscale starting from bulk calculations under

pressure [152]. Surface curvature and superficial dangling bonds on nanoparticles are

90

responsible for internal stress fields which modify the atomic bond lengths. As a

first approximation, where all surface effects that are not included in the curvature

are neglected, we can map the particle radius R to the pressure P by equating the

deviation from the bulk value of the average bond lengths due to surface curvature

(in the case of particle) and pressure (in the case of bulk). For spherical clusters,

the phenomenon can be modeled with the Young-Laplace equation P = 2γ/R where

the proportionality constant γ can be calculated with ab initio methods. In our

study, since the concentration of Mo in Fe:Mo is small, we use the ”size-pressure

approximation” of Fe particle.

Figure 7.1 shows the implementation of the ”size-pressure approximation” for Fe

nanoparticles. On the left hand side we show the ab initio calculations of the deviation

of the average bond length inside the cluster Δdnn ≡ d0nn − dnn (d0nn = 0.2455 nm is our bulk bond length), for body-centered cubic (BCC) particles of size N = 59, 113,

137, 169, 307 and ∞ (bulk) as a function of the inverse radius (1/R). The particles were created by intersecting a BCC lattice with different size spheres. The particle

radius is defined as 1/R ≡ 1/Nscp ∑

i 1/Ri where the sum is taken over the atoms

belonging to the surface convex polytope (Nscp vertices) and Ri are the distances to

the geometric center of the cluster. The left straight line is a linear interpolation

between 1/R and Δdnn calculated with the constraint of passing through 1/R = 0

and Δdnn = 0 (N = ∞, bulk). The right hand side shows the ab initio value of dnn in bulk BCC Fe as a function of hydrostatic pressure, P . The straight line is a linear

interpolation between P and Δdnn calculated with the constraint of passing through

P = 0 and Δdnn = 0 (bulk lattice). By following the colored dashed paths indicated

by the arrows we can map the analysis of nanoparticles stability as a function of R

onto bulk stability as a function of P , and obtain the relation between the radius of

particle/nanotube and the effective pressure P ·R = 2.46 GPa · nm. It is important

91

Figure 7.1: (color online). Size-pressure approximation for Fe nanoparticles obtained by equating the deviation of average bond length from the bulk value due to curvature 1/R (in the case of particle) and due to pressure P (in the case of bulk).

to mention that our γ = 1.23 J/m2 is not a real surface tension but an ab initio fitting

parameter describing size-induced stress in nanoparticles. With this γ, we deduce the

Fe-Mo-C phase diagram of nanoparticles of radius R from ab initio calculations of

the bulk material under pressure P .

92

7.2 Fe-Mo-C phase diagram under pressure

Simulations are performed with VASP as described in Section 2.2. The hydrostatic

pressure estimated from the pressure-size model is implemented as Pulay stress [170].

Ternary phase diagrams are calculated using BCC-Mo, BCC-Fe and SWCNTs as

references (pure-Fe phase is taken to be BCC because our simulations are aimed at

the low temperature regime of catalytic growth). The reference SWCNTs have the

same diameter of the particle to minimize the curvature-strain energy. In fact, CVD

experiments of SWCNT growth from small (∼ 0.6-2.1 nm) particles indicate that the diameter of the nanotube is similar to the diameter of the catalyst particle from

which it grows. In some experiments where the growth mechanism is thought to be

root-growth, the ratio of the catalyst particle diameter to SWCNT diameter is ∼ 1.0, whereas in experiments involving pre-made floating catalyst particles this ratio is ∼ 1.6 [171]. Formation energies are calculated with respect to decomposition into the

nearby stable phases, depending the position in the ternary phase diagram.

Binary and Ternary phases are included if they are stable in the temperature

range used in CVD growth of SWCNTs or if they have been reported experimentally

during or after the growth [165, 166, 172]. Thus, we include the binaries Mo2C,

Fe2Mo and Fe3C. In addition, since our Fe-rich Fe:Mo experiments were performed

with compositions close to Fe4Mo [159], we include a random phase Fe4Mo generated

with the special quasi-random structure formalism (SQS). Bulk ternary carbides,

which have been widely investigated due to their importance in alloys and steel, can

be considered as derivatives of binary structures with extra C atoms in the interstices

of the basic metal alloy structures. Three possible ternary phases have been reported

for bulk Fe-Mo-C [173] and they are referred as τ1 (M6C), τ2 (M3C) and τ3 (M23C6)

(M is the metal species). For simplicity, we follow the same nomenclature. τ1 is

the wellknown M6C phase, which has been observed experimentally as Fe4Mo2C and

93

Fe3Mo3C structures (η carbides) [174, 175]. Both of these structures are FCC but

have different lattice spacings. Our calculations show that the most stable variant

τ1 is Fe4Mo2C, and we denote it as τ1 henceforth. τ2 is the Fe2MoC phase, which

has an orthorhombic symmetry distinct from that of Fe3C [176, 177]. We consider

Fe21Mo2C6 as the third τ3 FCC phase [178]. We use the Cr23C6 as the prototype

structure [179] where Fe and Mo substitute for Cr. Although M23C6 type phases do

not appear in the stable C-Fe or C-Mo systems, they have been reported in ternary C-

Fe-Mo systems and also appear as transitional products in solid state reactions [173].

Time-temperature precipitation diagrams of low-C steels have identified τ2, τ3 and τ1

as low-temperature, metastable and stable carbides, respectively [180]. Furthermore,

τ2 carbides precipitate quickly due to carbon-diffusion controlled reaction while τ3

carbides precipitate due to substitutional-diffusion controlled reactions. The latter

phenomenon, requiring high temperature, longer times and producing metastable

phases is not expected to enhance the catalytic deactivation of the nanoparticle. In

summary, as long as the presence of carbon does not lead to excessive formation of

Fe3C and τ2), the catalyst should remain active for SWCNT growth.

Figure 7.2 shows the phase diagram at zero temperature of nanoparticles of radii

R ∼ ∞, 1.23, 0.62, 0.41 nm, calculated at P = 0, 2, 4, and 6 GPa, respectively. Stable and unstable phases are shown as black squares and red dots, respectively.

The solid green lines connect the stable phases. The numbers ”1”...”8” in panels (c)

and (d) indicate the intersections between the phases’ boundary and the dotted lines

representing the path of carbon pollution to the two test phases Fe4Mo and FeMo.

Fe4Mo has been reported to be an effective catalyst composition [159] while FeMo

represents a hypothetical Fe:Mo particle with a Mo content larger than 33%.

94

Figure 7.2: (color online). Ternary phase diagram for Fe-Mo-C nanoparticles of R ∼ ∞, 1.23, 0.62, 0.41 nm.

7.3 Fe4Mo particles

An advantage of an Fe4Mo particle has over a pure Fe particle is that the [Fe4/5Mo1/5]1−x-

Cx line does not intercept any carbide (Fe3C, τ3, τ2). This implies that, at least at low

temperatures, there is a surplus of unbounded metal (probably even at high temper-

atures since the line is far from all of the competing stable phases). This is illustrated

in figure 7.3, which shows the fractional evolution of species as one progresses along

the [Fe4/5Mo1/5]1−x-Cx line in figure 7.2.

For a large Fe4Mo particle (R ≥ 0.62 nm), the decomposition into stable phases is shown in figure 7.3(a)). At concentrations between 0 < xc < 0.09 there is available

free Fe for catalysis, however there is no carbon content in the particle. At higher

95

concentrations, the particle starts to contain free carbon while still providing free Fe.

Therefore, a steady state growth of SWCNTs is possible from large Fe4Mo particles.

Figure 7.3(b) shows the decomposition for a small Fe4Mo particle (R ≤ 0.41 nm)). In this case, the free Fe is consumed and transformed into τ3 carbide before the particle

has enough free carbon, hence, the SWCNT growth will not occur. We can estimate

the minimum size of the particle by calculting the pressure at which τ3 starts to form.

By linear interpolation, we obtain R = 0.52 nm. This size is smaller than that if one

uses pure Fe nanocatalyst (R = 0.56 nm) [152, 153].

7.4 FeMo particles

Figure 7.4 shows the decomposition analysis for FeMo particle. For a large particle of

size R ≥ 0.62 nm, The particle contains free Fe and carbon only after the concentra- tion xc is larger than 0.2 which then permits the growth of the nanotube. Comparing

to the Fe4Mo particle, since the fraction of Fe in the FeMo particle is considerably

smaller than in Fe4Mo, the expected yield is lower and the synthesis temperature

needs to be increased (to overcome the reduced fraction of catalytically active free

Fe). Small FeMo particles (R ≤ 0.41 nm) are similar to small Fe4Mo clusters. Nucle- ation of τ3 and the abscence of free Fe and excess carbons indicate that the particles

are catalytically inactive. The minimum size of FeMo and Fe4Mo particles able to

grow nanotube is the same since it is determined by the stabilization of the same τ3

carbide.

96

Figure 7.3: (color online). a) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”1”, represents [Fe4/5Mo1/5]1−x-Cx crossing the boundary phase Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”4” and ”5”, represent [Fe4/5Mo1/5]1−x-Cx crossing the boundary phases Fe←→Mo2C and τ3, as shown in figure 7.2(d).

97

Figure 7.4: (color online). a) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”2” and ”3”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phase Fe2Mo↔Mo2C and Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composi- tion [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”6”, ”7”, and ”8”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phases Fe2Mo↔Mo2C, Fe←→Mo2C, and τ3 ↔Mo2C, as shown in figure 7.2(d).

98

Chapter 8

Conclusions

We have presented the results of our computational studies on adsorptions of hy-

drocarbons (alkanes, benzene) and rare gases on a decagonal surface of Al-Ni-Co

quasicrystal (d-AlNiCo). Ab initio calculations show that upon the adsorptions, the

surface of the d-AlNiCo does not undergo relaxations and that there are no disso-

ciations of the adsorbates. The simulations of thin film growth of these adsorbates

have been performed in the grand canonical ensemble using Monte Carlo method.

We use semiempirical pair interactions for the rare gases and develop classical many-

body potentials based on embedded-atom method (EAM) for the hydrocarbons. All

of the simulated atoms/molecules wet the substrate as a consequence of compara-

ble strengths between the substrate-adsorbate and adsorbate-adsorbate. Another

consequence is the wide range of overlayer structures observed in these systems.

Methane monolayer has a quasicrystalline pentagonal order commensurate with the

substrate. Propane forms a disordered structure. Hexane and octane monolayers

show 2-dimensional close-packed features consistent with their bulk structure. Ben-

zene forms a pentagonal monolayer at low and moderate temperatures which trans-

forms into a triangular lattice at high temperatures. Similar structural transition

also occurs in xenon monolayer, however in this case, the transition is observed at all

temperatures below the triple-point temperature and as a function of pressure. We

have characterized that such transition is first-order with an associated latent heat.

Smaller noble gases such as Ne, Ar, Kr form monolayers with mixed pentagonal and

triangular patterns and do not show any structural transitions.

By systematically simulating test noble gases of various sizes and strengths, we

99

observe that the relative strengths between the competing interactions determine the

growth mode. Agglomeration occurs when the adsorbate-adsorbate interactions are

much stronger than the substrate-adsorbate ones. In the comparable strength regime,

a layer-by-layer film growth is observed. In this regime, the mismatch between the

size of the gas and the substrate’s characteristic length plays a major role in affecting

the structure of the adsorbed films. In general, on the d-AlNiCo substrate, we found

that structural transition from 5- to 6-fold occurs when the gas size is larger than

λ/0.944 (λ represents the average row-row spacing in the quasicrystalline plane of

the d-AlNiCo). Even though this rule is derived from rare gases, it is consistent with

methane, benzene, hexane and octane. Therefore, it might be useful as a guidance in

the search for suitable quasicrystalls for which alkanes (as the main constituent of oil

lubricants) will form quasiperiodic structures for the low-friction coating applications.

It is a natural extension of this work to investigate other quasicrystalls with

larger characterictic lengths than the d-AlNiCo used in this study. In fact, d-AlNiCo

is stable in many decorations depending on the concentration of Al, Ni, and Co.

Due to the stripe nature in the close-packed structure of linear alkanes, the film

structure of these molecules on one dimensional quasicrystals is interesting to study.

Two surfaces with stripe pattern will be commensurate only when the stripes are

perfectly alligned, therefore, one dimensional quasicrystalls might be good candidate

for low-friction coatings.

100

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Biography

The author was born in May 5th, 1978 in a Javanese village 8 miles southwest of

Solo, Central Java, Indonesia. He is the last(fourth) child of Ladiya (father) and

Sri Suratmi (mother). His father was an elementary school teacher and his mother

is a devoted house wife. Prior to attending college education, the author won the

third award in the International Physics Olympiad 1996 in Norway. The author

received a B.S. degree from Institut Teknologi Bandung, Indonesia, in Electrical

Engineering in 2000. In the Fall of 2000, he went to Florida State University on a

research assistantship from the Physics department in which he completed a M.S.

degree in 2004. In Summer 2004, he received a research assitantship to pursue a

doctoral degree in Mechanical Engineering and Materials Science department from

Duke University under the advisory of Dr. Stefano Curtarolo. He completed his PhD

work in computational materials science in Spring 2008. In addition, he received a

Graduate Certificate degree in Computational Science, Engineering, and Medicine,

in Spring 2008 also from Duke University.

List of publications:

• Diehl R D, Setyawan W and Curtarolo S 2008 J. Phys.: Cond. Mat. in press

• Harutyunyan A R, Awasthi N, Jiang A, Setyawan W, Mora E, Tokune T, Bolton K and Curtarolo S 2008 Phys. Rev. Lett. inpress

• Curtarolo S, Awasthi N, Setyawan W, Li N, Jiang A, Tan T Y, Mora E, Bolton K and Harutyunyan A T 2008 Proc. of Comp. Simul. Studies in Cond. Matt.

Phys. XXI Eds Landau D P, Lewis S P and Schuttler H-B (Springer, Berlin,

Heidelberg)

• Setyawan W, Diehl R D, Ferralis N, Cole M W and Curtarolo S 2007 J. Phys.:

111

Cond. Mat. 19 016007

• Diehl R D, Setyawan W, Ferralis N, Trasca R, Cole M W and Curtarolo S 2007 Phil. Mag. 87 2973

• Jiang A, Awasthi N, Kolmogorov A N, Setyawan W, Bo¨rjesson A, Bolton K, Harutyunyan A R and Curtarolo S 2007 Phys. Rev. B 75 205426

• Setyawan W, Ferralis N, Diehl R D, Cole M W and Curtarolo S 2006 Phys. Rev. B 74 125425

• Diehl R D, Ferralis N, Pussi K, Cole M W, Setyawan W and Curtarolo S 2006 Phil. Mag. 86 863

• Curtarolo S, Setyawan W, Diehl R D, Ferralis N and Cole M W 2005 Phys. Rev. Lett. 95 136104

• Rao S G, Huang L, Setyawan W and Hong S 2003 Nature 425 36

• Setyawan W, Rao S G and Hong S 2002 Mat. Res. Soc. Proc. NN Fall

• Mumtaz A, Setyawan W and Shaheen S A 2002 Phys. Rev. B 65 020503

112COMPUTATIONAL STUDY OF LOW-FRICTION

QUASICRYSTALLINE COATINGS VIA SIMULATIONS

OF THIN FILM GROWTH OF HYDROCARBONS AND

RARE GASES

by

Wahyu Setyawan

Department of Mechanical Engineering and Materials Science Duke University

Date: Approved:

Dr. Stefano Curtarolo, Supervisor

Dr. Teh Y. Tan

Dr. Laurens E. Howle

Dr. Xiaobai Sun

Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in the Department of Mechanical Engineering and Materials Science in the Graduate School of

Duke University

2008

ABSTRACT

COMPUTATIONAL STUDY OF LOW-FRICTION

QUASICRYSTALLINE COATINGS VIA SIMULATIONS

OF THIN FILM GROWTH OF HYDROCARBONS AND

RARE GASES

by

Wahyu Setyawan

Department of Mechanical Engineering and Materials Science Duke University

Date: Approved:

Dr. Stefano Curtarolo, Supervisor

Dr. Teh Y. Tan

Dr. Laurens E. Howle

Dr. Xiaobai Sun

An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in the Department of Mechanical Engineering and Materials Science in the Graduate School of

Duke University

2008

Copyright c© 2008 by Wahyu Setyawan All rights reserved

Abstract

Quasicrystalline compounds (QC) have been shown to have lower friction compared

to other structures of the same constituents. The abscence of structural interlocking

when two QC surfaces slide against one another yields the low friction. To use QC

as low-friction coatings in combustion engines where hydrocarbon-based oil lubri-

cant is commonly used, knowledge of how a film of lubricant forms on the coating is

required. Any adsorbed films having non-quasicrystalline structure will reduce the

self-lubricity of the coatings. In this manuscript, we report the results of simula-

tions on thin films growth of selected hydrocarbons and rare gases on a decagonal

Al73Ni10Co17 quasicrystal (d-AlNiCo). Grand canonical Monte Carlo method is used

to perform the simulations. We develop a set of classical interatomic many-body

potentials which are based on the embedded-atom method to study the adsorption

processes for hydrocarbons. Methane, propane, hexane, octane, and benzene are

simulated and show complete wetting and layered films. Methane monolayer forms

a pentagonal order commensurate with the d-AlNiCo. Propane forms disordered

monolayer. Hexane and octane adsorb in a close-packed manner consistent with

their bulk structure. The results of hexane and octane are expected to represent

those of longer alkanes which constitute typical lubricants. Benzene monolayer has

pentagonal order at low temperatures which transforms into triangular lattice at high

temperatures. The effects of size mismatch and relative strength of the competing

interactions (adsorbate-substrate and between adsorbates) on the film growth and

structure are systematically studied using rare gases with Lennard-Jones pair poten-

tials. It is found that the relative strength of the interactions determines the growth

mode, while the structure of the film is affected mostly by the size mismatch between

adsorbate and substrate’s characteristic length. On d-AlNiCo, xenon monolayer un-

iv

dergoes a first-order structural transition from quasiperiodic pentagonal to periodic

triangular. Smaller gases such as Ne, Ar, Kr do not show such transition. A simple

rule is proposed to predict the existence of the transition which will be useful in the

search of the appropriate quasicrystalline coatings for certain oil lubricants.

Another part of this thesis is the calculation of phase diagram of Fe-Mo-C sys-

tem under pressure for studying the effects of Mo on the thermodynamics of Fe:Mo

nanoparticles as catalysts for growing single-walled carbon nanotubes (SWCNTs).

Adding an appropriate amount of Mo to Fe particles avoids the formation of stable

binary Fe3C carbide that can terminate SWCNTs growth. Eventhough the formation

of ternary carbides in Fe-Mo-C system might also reduce the activity of the catalyst,

there are regions in the Fe:Mo which contain enough free Fe and excess carbon to

yield nanotubes. Furthermore, the ternary carbides become stable at a smaller size

of particle as compared to Fe3C indicating that Fe:Mo particles can be used to grow

smaller SWCNTs.

v

Contents

Abstract iv

List of Figures ix

List of Tables xiv

Acknowledgements xvii

1 Introduction 1

2 Methods 5

2.1 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . 5

2.1.2 Slater determinant . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 Hartree-Fock equations . . . . . . . . . . . . . . . . . . . . . . 7

2.1.4 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . 10

2.1.5 Local density and generalized gradient approximations . . . . 11

2.1.6 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Vienna Ab initio Simulation Package . . . . . . . . . . . . . . . . . . 13

2.3 Classical interatomic potentials . . . . . . . . . . . . . . . . . . . . . 14

2.4 Simplex method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Grand canonical Monte Carlo . . . . . . . . . . . . . . . . . . . . . . 17

3 Noble gas adsorptions on d-AlNiCo 18

3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Simulation cell . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.2 Gas-gas and gas-substrate interactions . . . . . . . . . . . . . 19

vi

3.1.3 Adsorption potentials . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.4 Effective parameters . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.5 Test rare gases . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.6 Chemical potential, order parameter, and ordering transition . 25

3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Adsorption isotherms . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.3 Order parameters (ρ5−6) . . . . . . . . . . . . . . . . . . . . . 32

3.2.4 Effects of �gg and σgg on adsorption isotherms . . . . . . . . . 37

3.2.5 Effects of �gg and σgg on 5- to 6-fold transition . . . . . . . . . 39

3.2.6 Prediction of 5- to 6-fold transition . . . . . . . . . . . . . . . 40

3.2.7 Transitions on smoothed substrates . . . . . . . . . . . . . . . 42

3.2.8 Temperature vs substrate effect . . . . . . . . . . . . . . . . . 43

3.2.9 Orientational degeneracy of the ground state . . . . . . . . . . 46

3.2.10 Isosteric heat of adsorption . . . . . . . . . . . . . . . . . . . . 47

3.2.11 Effect of vertical dimension . . . . . . . . . . . . . . . . . . . 48

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Embedded-atom method potentials 52

4.1 Stage 1: Aluminum, cobalt, and nickel . . . . . . . . . . . . . . . . . 54

4.2 Stage 2: Al-Co-Ni potentials . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Stage 3: Hydrocarbon potentials . . . . . . . . . . . . . . . . . . . . . 59

4.4 Stage 4: Hydrocarbon on Al-Co-Ni . . . . . . . . . . . . . . . . . . . 63

5 Hydrocarbon adsorptions on d-AlNiCo 66

5.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

vii

5.2 Adsorption potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Molecule orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4 Adsorption isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.5 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Embedded-atom potentials for selected pure elements in the pe- riodic table 81

7 Effects of Mo on the thermodynamics of Fe:Mo:C nanocatalyst for single-walled carbon nanotube growth 89

7.1 Size-pressure approximation . . . . . . . . . . . . . . . . . . . . . . . 90

7.2 Fe-Mo-C phase diagram under pressure . . . . . . . . . . . . . . . . . 93

7.3 Fe4Mo particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.4 FeMo particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8 Conclusions 99

Bibliography 101

Biography 111

viii

List of Figures

3.1 (color online). Computed adsorption potentials for (a) Ne, (c) Ar, (e) Kr, and (g) Xe on the d-AlNiCo, obtained by minimizing V (x, y, z) with respect to z. The distribution of the minimum value of these potentials is plotted in (b, d, f, and h) respectively: the solid line marks the average value 〈Vmin〉, the dashed lines mark the values at 〈Vmin〉±SD. . . . . . . . . . . . . . . 22

3.2 Computed adsorption isotherms for all the gas/d-AlNiCo systems. The ranges of temperatures under study are: Ne: T = 14 K to 46 K in 2 K steps, Ar: 45 K to 155 K in 5 K steps, Kr: 65 K to 225 K in 5 K steps, Xe: 80 K to 280 K in 10 K steps. Additional isotherms are shown with solid circles at T � = 0.35: T = 11.8 K (Ne), T = 41.7 K (Ar), T = 59.6 K (Kr), and T = 77 K (Xe). Isotherms above the triple point temperatures are shown as dotted curves. . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Density profiles and Fourier transforms of the outer layer at T � = 0.35 for Ne/d-AlNiCo (T = 11.8 K) and Ar/d-AlNiCo (T = 41.7 K), corresponding to points (A) through (F) of Figure 3.2. . . . . . . . . . . . . . . . . . . 30

3.4 Density profiles and Fourier transforms of the outer layer at T � = 0.35 for of Kr/d-AlNiCo (T = 59.6 K) and Xe/d-AlNiCo (T = 77 K), corresponding to points (A) through (F) of Figure 3.2. . . . . . . . . . . . . . . . . . . 33

3.5 (color online). Order parameters, ρ5−6, as a function of normalized chemical potential, μ�, (as defined in the text) at T � = 0.35 for the first four layers of (a) Ne, (b) Ar, (c) Kr, and for the first layer of Xe (d) adsorbed on d-AlNiCo. A sudden drop of the order parameter in Xe/QC to a constant value of ∼ 0.017 at μ� ∼ 0.8 indicates the existence of a first-order structural transition from fivefold to sixfold in the system. . . . . . . . . . . . . . . 34

ix

3.6 (color online). Xe on d-AlNiCo at T = 77 K. (a) Adsorption isotherm, ρN , versus the normalized chemical potential, μ�. (b) Nearest neighbor distance derived from the first peak of pair correlation function, rNN , (black line), and average spacing between neighbors at equilibrium, d¯NN , (red line). (c) Order parameter ρ5−6 (probability of fivefold defects, defined in Equation 3.14) versus the normalized chemical potential, μ�. (d) Total enthalpy. The transition, which is defined as the point in μ� above which the order parameter remains nearly constant, occurs at μ�tr ∼0.8. The discontinuity in H around μ�tr ∼ 0.8 indicates a first order transition with associated latent heat of the transition. The order parameter ρ5−6 after the transition is ∼ 0.017. Heat of the transition is ≈ 6.8 meV/atom. . . . . . . . . . . . 36

3.7 (color online). Computed adsorption isotherms for Ne, Xe, iNe(1), and dXe(1) on d-AlNiCo at T �=0.35. iNe(1) and dXe(1) are test noble gases having potential parameters described in the text and in Tables 3.1 and 3.2. The effect of varying the interaction strength of the adsorbates on the density increase ΔρN (while keeping the size constant) is negligible on large gases but significant on small gases. . . . . . . . . . . . . . . . . . . . . 39

3.8 (color online). Order parameters as a function of normalized chemical po- tential (as defined in the text) for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed on d-AlNiCo at T � = 0.35. A first-order fivefold to sixfold structural transition occurs in the last three systems, but not in dXe(2). . . 41

3.9 (color online). (a) The minimum of adsorption potential, Vmin(x, y), for Ne on a smoothed d-AlNiCo as described in the text. (b) The variations of the minimum adsorption potentials along the line at x = 0 shown in (a), for the modified and original interactions (solid and dotted curves). . . . . . 44

3.10 Xe on d-AlNiCo. Values of μ�tr for the fivefold to sixfold transition points from 40 K to 140 K (left axis). Transition points at μ�tr > 1 indicate that a transfer of atoms from the second layer to the first layer is required to complete the transition. Also shown is the defect probability as a function of T after the transition occurs (right axis), indicating an increase in defect probability with T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.11 Density plot of Xe on decagonal AlNiCo at 77 K, showing a superposition of the density slices for the 2nd and 4th layers. In the top right, 4th-layer atoms are located directly above the 2nd layer atoms indicating an hexagonal close- packed ABAB stacking, whereas in other regions, such as lower left, the two layers are offset due to stacking fault. . . . . . . . . . . . . . . . . . . . 46

x

3.12 Xe adsorption on d-AlNiCo. (a) Minimum potential energy surface of the adsorption potential with free boundary conditions. (b) Adsorption isotherms of the first layer from a set of 30 simulations at 77 K using the free cell described in the paper. Five density profiles and FTs at point p� of (b) are shown in (c) to (g), representing all possible orientations of hexagonal domains. (h) Schematic diagram illustrating the correspondence between the orientations of the hexagonal domains observed in the density profiles (c) to (g). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.13 Xe adsorption on d-AlNiCo. Pentagonal defects rotate the orientation of hexagons by (a) θ1 = 24◦ and (b) θ2 = 12◦. . . . . . . . . . . . . . . . . 51

3.14 (color online). Xe adsorption on d-AlNiCo. Locations in P , T of the vertical risers in the isotherms corresponding to the first (square), second (circle), and third (triangle) layer formation. The heats of adsorptions, qst, are 270, 129, and 125 meV/atom respectively, calculated as described in the text. The inset figure shows qst obtained from the simulations as well as from the experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 (color online). (a-e) Adsorption potential map, calculated by minimizing the adsorption potential of one molecule on a decagonal Al-Ni-Co along z direction and all rotational degrees of freedom at every coordinates (x, y). Red numbers represent the average value of the adsorption energies. (f) Top view of the decagonal Al-Ni-Co substrate 51.2x51.2 A˚2: Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red), Ni-toplayer (green), and Ni-otherlayers (blue). . . . . . . . . . . . . . . . 71

5.2 (color online). Isothermal adsorption densities of hydrocarbons on a decago- nal Al-Ni-Co: (a) methane (from left to right T = 68, 85, 136, 185 K), (b) propane (T = 80, 127, 245, 365 K), (c) hexane (T = 134, 170, 267, 450 K), (d) octane (T = 162, 210, 324, 450, 565 K), and (e) benzene (T = 209, 270, 418, 555 K). The inset in each figure is the density along z direction at pressure corresponding to point d. Xenon (red) is plotted in panel (a) for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 (a) and (b) Calculated density of methane adsorbed on a decagonal Al-Ni- Co at pressures corresponding to points ”a” and ”c” of the 68 K isotherm shown in Figure 5.2.a, respectively. (c) Fourier transform of the density plot shown in (b), consistent with 5-fold ordering of the methane near monolayer completion. (d) Order parameter (left axis, as calculated in Equation 3.14) as a function of pressure for the 68 K isotherm (right axis), indicating no sharp transition to 6-fold ordering. . . . . . . . . . . . . . . . . . . . . 74

xi

5.4 Calculated density of propane, hexane, and octane adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion. Propane forms a dis- ordered structures, whereas hexane and octane tend to form close-packed structures indicated by stripe features with increasing order for longer chain. 76

5.5 (Top row) Calculated density of benzene adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion for 209 K, 270 K, and 418 K. (middle row) Density profile of the geometrical center of density shown in the top row. (bottom row) Fourier transform of the density plot shown in the middle row, showing 5-fold ordering at 209 K, mixture of 5-fold and 6-fold structures at 270 K, and mostly 6-fold features at 418 K. . . . . . . 78

5.6 (left axis) Order parameter ρ5−6 = N5/(N5 +N6) (Nn denotes the number of molecules having n nearest neighbors) as a function of temperature at 0.01 atm of pressure. (right axis) Adsorption isobar showing the number of adsorbed molecules as a function of temperature. Verticel dashed lines correspond to T = 209, 270, and 418 K whose density profiles are plotted in Figure 5.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.1 (color online). Size-pressure approximation for Fe nanoparticles obtained by equating the deviation of average bond length from the bulk value due to curvature 1/R (in the case of particle) and due to pressure P (in the case of bulk). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.2 (color online). Ternary phase diagram for Fe-Mo-C nanoparticles of R ∼ ∞, 1.23, 0.62, 0.41 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.3 (color online). a) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x- Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”1”, represents [Fe4/5Mo1/5]1−x-Cx crossing the boundary phase Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed verti- cal lines, labeled as ”4” and ”5”, represent [Fe4/5Mo1/5]1−x-Cx crossing the boundary phases Fe←→Mo2C and τ3, as shown in figure 7.2(d). . . . . . . 97

xii

7.4 (color online). a) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x- Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”2” and ”3”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phase Fe2Mo↔Mo2C and Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”6”, ”7”, and ”8”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phases Fe2Mo↔Mo2C, Fe←→Mo2C, and τ3 ↔Mo2C, as shown in figure 7.2(d). . . . . . . . . . . 98

xiii

List of Tables

3.1 Parameter values for the 12-6 Lennard-Jones interactions. TM is the label for Ni or Co. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Range, average (〈Vmin〉), and standard deviation (SD) of the interaction Vmin(x.y) on the d-AlNiCo. Effective parameters of the gas-substrate in- teractions (Dgs, σgs, D�gs, σ�gs), and, for comparison, the best estimated well depths DGrgs on graphite [106]. . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Results for Ne, Ar, Kr, and Xe adsorbed on d-AlNiCo. Tt is taken from reference [109]. The density increase (ΔρN ) in the first and second layers is calculated at T � = 0.35 from point (A) to (B) and (C) to (D) in Figure 3.2, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Summary of adsorbed noble gases on d-AlNiCo that undergo a first-order fivefold to sixfold structural transition and those that do not. . . . . . . . 42

4.1 List of structure prototypes used to fit the EAM potentials for hydrocarbon adsorption on Al-Co-Ni. For elemental Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure (DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed (HEX), simple cubic (SC), and simple hexagonal (SH). The Al-Co-Ni ternaries are taken from the database of alloys [122]. . . . . . . . . . . . . . . . . . . . . . . 53

4.2 (Top part) List of structures used to fit EAM potential for elemental alu- minum. ρi is charge density at atom-i, ΔE = EEAM−EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3 · re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 56

4.3 (Top part) List of structures used to fit EAM potential for elemental cobalt. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”HEX at a0 · c” indicates that the structure is HEX with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 57

xiv

4.4 (Top part) List of structures used to fit EAM potential for elemental nickel. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 58

4.5 (Top part) List of structures used to fit EAM potential for Al-Co-Ni systems. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV) and the first knot at (0,0) is assumed. The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . . . . . . . . . . . . . . . . . 60

4.6 (Top part) List of structures used to fit EAM potential for hydrocarbons. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . 61

4.7 (Top part) List of structures used to fit EAM potential for alkanes and benzene. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.8 Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. Fitting structures are given in Table 4.9 . . . 64

4.9 Fitted energies calculated using EAM parameters in Table 4.8. Methane up represents methane with one H below C and three H above C. Methane down is inverse of methane up. The unit for energy is eV/atom except for the adsorption energy which is in eV/molecule. . . . . . . . . . . . . . . . . 65

5.1 Parameter values for the adsorbate-adsorbate interactions used for hydro- carbon adsorption on a decagonal Al-Ni-Co. Intermolecular energies are calculated as a sum of pair interactions. For methane-methane, the C-H is taken as the geometrical mean for parameter A and as the arithmetic mean for parameters B and C. . . . . . . . . . . . . . . . . . . . . . . . . . . 69

xv

6.1 Fitted parameters for the charge density and pair interaction functionals of the EAM potentials for pure elements, continued in Table 6.2. The fitting structures are given in Tables 6.5 and 6.6. The parameters for the embedding functionals are given in Tables 6.3 and 6.4. . . . . . . . . . . 83

6.2 continuation of Table 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3 Fitted knots of cubic spline of the embedding functionals for the EAM potentials of pure elements, continued in Table 6.4. The first knot at (0,0) is assumed. The fitting structures are given in Tables 6.5 and 6.6. . . . . . 85

6.4 continuation of Table 6.3. . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.5 (left part) Structures used to fit EAM potentials for pure elements, con- tinued in Table 6.6. The EAM potentials are fitted to the ab initio ener- gies in body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close-packed (HEX), diamond structure (DIA), and groundstate structure at various pressures obtained by expanding/compressing the equillibrium lattice constant a by a factor from 0.9 to 1.16 corresponding to a range of charge density from ρmax to ρmin. (right part) Lattice constansts calculated using the fitted parameters. The literature values aLIT are taken from [146]. 87

6.6 continuation of Table 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . 88

xvi

Acknowledgements

I would like to thank my advisor, Prof. Stefano Curtarolo for all the academic and financial supports. I wish to thank all collaborators of this work, Prof. Renee D. Diehl, Prof. Milton W. Cole, Prof. L W. Bruch, Dr. Nicola N. Ferralis, and Dr. Andrea Trasca. I thank all my teachers for their mentoring and my committee members, especially Prof. Teh Y. Tan, Prof. Laurens E. Howle, and Prof. Xiaobai Sun for their dedications and insightful advice. I thank all my colleagues, in particular Dr. Neha Awasthi, Dr. Aiqin Jiang, Dr. Roman Chepulskyy for encouragements and valuable discussions, and Dr. Aleksey Kolmogorov for his mentoring and experienced advice on VASP. I also wish to thank all the staff in the department of Mechanical Engineering and Materials Science for the administrative helps.

I thank the National Science Foundation and Honda Research Institute for fund- ing this research. I also thank San Diego Super Computers (SDSC), Texas Ad- vanced Computing Center (TACC), National Center for Supercomputing Applica- tions (NCSA), and Pittsburgh Supercomputing Center (PSC) for all the computing time through Teragrid projects. Thanks also go to Duke Clusters for additional allocations.

I can never thank enough to my father whose talent in numbers has introduced me to math and engineering, my mother and my sisters for their endless prayers, love, and encouragements throughout my life. I am deeply grateful to be blessed with a brilliant, loving, and beautiful wife, Prof. Lisa M. Peloquin, who provides continuous prayers, love, and supports in so many ways. I also thank our friends especially Laura Heymann for her kind helps in many occassions. Thanks also go to Pepito, Clementine, Hamzah, John and other aquatic species for being excellent family members.

Finally, all praises are due to Allah, the Creator (al-Khaliq), the Loving (al- Rahman), and the All Knowing (al-Alim). I am thankful for all the opportunities, health, and blessings that enable me to finish this dissertation. I am deeply humbled by every piece of knowledge that I learned.

xvii

Chapter 1

Introduction

Quasicrystals (QCs) were discovered in 1982 by Dr. Shechtman during his X-ray mea-

surements on Al-Mn compounds. Similar to crystals, QCs consist of atoms arranged

in regular patterns having long-range order, i.e. the diffraction patterns show discrete

spots. However, they do not have any translational periodicities. The discrete spots

come from the rotational symmetries. A variety of stable and metastable QCs have

been successfully synthesized. Among the first high-quality samples are icosahedral

AlCuFe [1], decagonal AlNiCo [2], and icosahedral AlPdMn [3]. Today, hundreds of

quasicrystalline phases are known, tens of which are stable [4]. The majority of them

are derived from aluminum-transition metal family [5].

QCs have been shown to have lower coefficients of friction than most metals. For

example, the static friction μs between two clean surfaces of icosahedral AlPdMn is

≈ 0.6 [6], whereas μs for Ni(110) and Cu(111) is ≈ 4 [7]. Kinetic friction tests using pin-on-disk technique with diamond pin show that among materials with compara-

ble hardness included in the study, icosahedral AlPdMn exhibits the lowest friction

(μ = 0.05), compared to window glass (μ = 0.08), sintered Al2O3 (μ = 0.13), or

hard Cr-steel (μ = 0.13) [8]. Detailed measurements of friction as a function of struc-

tural perfection in icosahedral AlCuFe quasicrystal show that the minimum friction is

achieved for sample with the best quasilattice perfection [9]. Decagonal AlNiCoSi has

been verified to have lower friction than Cr2O3, which represents the most advanced

technology for use on piston rings in automotive engines [10]. Further evidence of

reduced friction is demonstrated in decagonal AlNiCo, in which the friction on the

2-fold periodic surface is eight times higher than that on the decagonal surface [11]

1

(Note that in decagonal quasicrystals, there is a direction along which the quasicrys-

talline surfaces are stacked periodically).

The reduced friction between two quasicrystalline surfaces can be understood by

considering the structure commensurability between them. For surfaces with enough

hardness, atoms can be regarded as fixed in their position in each material. Within

this approximation, it has been theoretically demonstrated that the total interaction

energy between the two surfaces is independent of their relative displacement parallel

to the interface [12]. Therefore, the frictional force which is the gradient of the en-

ergy with respect to the displacement is vanishingly small, a phenomenon known as

superlubricity [12]. Superlubricity has been observed experimentally between tung-

sten and graphite in which certain relative orientations result in nearly zero friction

beyond the limit of the instruments [13]. Since any two QCs do not have common

periodicities at any length scales, superlubricity is expected.

Another characteristic feature of QCs is their resistance to oxidation, which is

quite surprising given that their main constituent, aluminum, is readily oxidized in

ambient conditions [5]. The behavior is particularly spectacular in AlCuLi icosahedral

phase that resists oxidation very well in humid air [5]. The combination of high

hardness, oxidation resistance, and low friction has attracted interests in QCs as

coatings to reduce friction and wear in machine parts, e.g. at the piston-cylinder

interface and in gear boxes. In such environments, hydrocarbon-based oil lubricant

is typically used to overcome the friction caused by surface asperities. Therefore,

to yield a synergic performance of self-lubricating QC coating and lubricant, it is

important to understand the interactions between them. The lubricant must be

able to spread (wet) well on the QC. Furthermore, the structures of the thin film

of lubricant formed on the sliding surfaces which will affect the lubricity need to be

investigated.

2

In this work, we study the process of thin film growth of hydrocarbons and rare

gases adsorbed on a decagonal Al73Ni10Co17 quasicrystal (d-AlNiCo) via computer

simulations. A high-quality and large-size single grain d-AlNiCo has been routinely

grown, making it an excellent substrate to study adsorption [14, 15, 16]. Chapter 3 is

devoted to the adsorptions of rare gases. The absence of chemical reactivity of noble

gases will be utilized to elucidate the effects of quasicrytallinity of the substrate on the

growth and structure of the adsorbed film. In Chapter 5, the simulation results for

selected hydrocarbons are presented. We study the adsorption behaviors of propane,

hexane, and octane. From these molecules, we may extrapolate the results for longer

alkanes which constitute typical oil lubricants. In addition, methane and benzene are

also studied due to their interesting symmetries.

We develop classical many-body interatomic potentials of Al, Ni, Co, C, and H

which will be suitable for simulating hydrocarbons adsorptions on Al-Ni-Co involving

thousands of atoms. To our knowledge, such potentials have not existed. The poten-

tials are based on the embedded-atom method (EAM) [17] with parameters fitted to

electronic energies of various structures evaluated via first principle quantum calcu-

lations. The generation of the EAM potentials is presented in Chapter 4. In Chapter

6, we extend the procedures to develop a consistent set of EAM potentials for pure

elements in the periodic table. These potentials are also fitted to ab initio energies.

A consistent set of EAM potentials having the same parametrizations enables one to

use the same computer code for different potentials.

This manuscript contains a different reserach which is part of our doctoral work,

namely in the field of nanocatalysis for synthesis of single-walled carbon nanotubes

(SWCNTs). In Chapter 7, we present the calculation of phase diagram of Fe-Mo-

C system under pressure for studying the effects of Mo in the thermodynamics of

Fe:Mo nanoparticles as catalysts for growing SWCNTs. The decomposition of two

3

Fe:Mo particles namely FeMo and Fe4Mo into various stable phases are analyzed.

The implications of the formation of these phases to the growth of SWNCTs as well

as the estimated minimum size of nanotubes that can be produced are discussed.

4

Chapter 2

Methods

2.1 Density functional theory

2.1.1 Born-Oppenheimer approximation

In a calculation of electronic structure of materials, one solves an eigenvalue problem

of time-independent Schro¨dinger equation:

HΨ(R, r) = EΨ(R, r) (2.1)

where R ∈ {Rn} and r ∈ {ri} are the vector coordinates of the nuclei and electrons, respectively. The energy operator (Hamiltonian), H, is given by

H = − ∑ n

� 2

2Mn �2Rn−

∑ i

� 2

2mi �2ri+

∑ m<n

e2ZmZn Rn −Rm +

∑ i<j

e2

rj − ri − ∑ n,i

e2Zn ri −Rn (2.2)

The first two terms represent the kinetic energies of the nuclei and the electrons,

respectively. The third and fourth terms represent the nucleus-nucleus and electron-

electron potential energies. The last term is the nucleus-electron energy. It is compu-

tationally beyond the capability of current computers to solve Equation 2.1 using the

full Hamiltonian. An approximation, known as Born-Oppenheimer approximation,

is made by realizing that nuclei are significantly heavier than electrons so the nuclei

move much more slowly than the electrons. The electrons can adapt themselves to

the current configuration of nuclei. Using this approximation, we can decouple the

electronic and the ionic parts of the Hamiltonian. Consider the nuclei fixed at a given

5

configuration, α, and solve the following electronic Schro¨dinger equation:

[ − ∑

i

� 2

2mi �2ri +

∑ i<j

e2

rj − ri − ∑ n,i

e2Zn ri −Rαn

] ψα(r) = Eα(R)ψα(r) (2.3)

The total energy, E, is calculated by taking the electronic contribution, Eα(R), as a

potential energy operator in the ionic Schro¨dinger equation:

[ − ∑ n

� 2

2Mn �2Ri +

∑ m<n

e2ZmZn Rm −Rn + E

α(R)

] Φα(R) = EΦα(R) (2.4)

Mostly, one is interested only in the electronic part, i.e. Equation 2.3. Even though

Equation 2.3 contains only the electronic part of the system, the large number of

variables (e.g. coordinates of all electrons) makes it remain intractable. In addition,

it is the electron-electron interaction that makes the problem so difficult to solve. This

interaction is a result of correlation between electrons (the probability of finding an

electron depends on where the rest of the electrons are) and the fact that electrons

are fermions requiring antisymmetric wavefunctions (the many-electron wavefunction

gains a factor of -1 everytime two electrons exchange their coordinates). If this term

were absent, the Hamiltonian would be just a sum of many one-electron Hamiltonians,

known as independent electron approximation.

2.1.2 Slater determinant

An antisymmetric N -electron wavefunction can be constructed from N one-electron

wavefunctions using Slater determinant [18] defined as:

Ψ(x1,x2, . . . ,xN) = 1√ N !

ψ1(x1) ψ2(x1) . . . ψN(x1) ψ1(x2) ψ2(x2) . . . ψN(x2) . . . . . . . . . . . .

ψ1(xN) ψ2(xN) . . . ψN(xN)

(2.5)

6

where ψk(xi) denotes the k-th one-electron wavefunction being occupied by an elec-

tron with spin-orbital coordinate xi = (si, ri), with si being the spin state and ri the

spatial coordinate. For this reason, ψk is also called the spin-orbital k. The factor 1√ N !

arrives from the normalization of the total wavefunction and orthonormality among

the spin orbitals:

∫ ΨΨ∗dx1 . . . dxN = 1 (2.6)

∫ ψk(r)ψ

∗ k(r)dr = 1 (2.7)

∫ ψk(r)ψ

∗ l (r)dr = 0 (2.8)

Using the orthonormality of the spin-orbitals, it can be shown that the charge density

of the Slater determinant can be written as n(x) = ∑

k |ψk(x)|2.

2.1.3 Hartree-Fock equations

In the electronic Schro¨dinger equation (Equation 2.3), the Hamiltonian can be written

as follows:

H = ∑

i

h(i) + 1

2

∑ i�=j

g(i, j) (2.9)

h(i) ≡ −1 2 �2i −

∑ n

Zn |ri −Rn| (2.10)

g(i, j) ≡ 1|rj − rj| (2.11)

where h(i) depends only on ri and g(i, j) depends on ri and rj. The energy of

the system is calculated by taking the expectation value of the Hamiltonian in the

total wavefunction, E = 〈Ψ|H|Ψ〉. By employing the orthonormality of ψk as in the

7

calculation of charge density n(x), we have

〈Ψ| ∑

i

h(i)|Ψ〉 = ∑ k

〈ψk|h|ψk〉 = ∑ k

∫ dxψ∗k(x)h(r)ψk(x) (2.12)

Note that in the first equation, when |Ψ〉 is written as a Slater determinant, only i = k appears due to orthonormality of ψk, and the summation over electron index i inside

the many-electron wavefunction |Ψ〉 becomes a summation over k inside individual spin-orbital ψk, and we can drop the index h(i = k) for convenience. The integral∫

dx denotes integral over spatial coordinates and a sum over the spin-degrees of

freedom. Similarly, we have

〈Ψ| ∑ i,j

g(i, j)|Ψ〉 = ∑ k,l

〈ψkψl|g|ψkψl〉 − ∑ k,l

〈ψkψl|g|ψlψk〉 (2.13)

〈ψkψl|g|ψmψn〉 = ∫

dx1ψ ∗ k(x1)

[∫ dx2ψ

∗ l (x2)

1

|r1 − r2|ψn(x2) ] ψm(x1)(2.14)

The total energy is then

E = ∑ k

〈ψk|h|ψk〉+ 1 2

∑ k,l

[〈ψkψl|g|ψkψl〉 − 〈ψkψl|g|ψlψk〉] (2.15)

This expression shows that the electron-electron interaction, g(i, j), consists of two

terms, the first one is known as Coulomb energy (or Hartree term), and the second

one is the exchange energy term. To see how this derivation also reduces the many-

electron Schro¨dinger equation to a set of one-electron equations, we differentiate the

energy with respect to a particular spin-orbital, e.g. spin-orbital 〈ψi|. Note that i is not electron index, but a specific value of spin-orbital index k, after the derivation,

8

we may replace i by k to conform to the usual notation:

δE

δ〈ψi| = h|ψi〉+ 1

2

[∑ l

〈ψl| 1|r− r′| |ψl〉 ] |ψi〉+ 1

2

[∑ k

〈ψk| 1|r− r′| |ψk〉 ] |ψi〉

−1 2

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x)ψi(x ′)− 1

2

∑ k

∫ dxψ∗k(x

′) 1

|r− r′|ψi(x)ψk(x ′)

(2.16)

By changing indices (k → l) in the third term and (k → l,x→ x′) in the last term, we get

δE

δ〈ψi| = h|ψi〉+ [∑

l

〈ψl| 1|r− r′| |ψl〉 ] |ψi〉 −

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x)ψi(x ′)

(2.17)

This expression is known as Fock operator acting on |ψi〉. Replacing back i by k, and defining the eigenvalues of Fock operator as �k, we arrive at the Hartree-Fock

equation:

HHFψk = �kψk (2.18)

HHFψk =

[ −�

2

2 − ∑ n

Zn |r−Rn|

] ψk(x) +

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x ′)ψk(x)

− ∑

l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψk(x ′)ψl(x)

(2.19)

Note that the sum of the eigenvalues �k is not the total energy E. However, by

comparing Equation 2.15 and 2.19, E can be recovered using the following equation

E = 1

2

∑ k

[�k + 〈ψk|h|ψk〉] (2.20)

The last term (the exchange term) in Equation 2.19 is nonlocal because the Hamil-

tonian HHF operates on ψk(x) at a particular x, but the operator itself is a function

9

of ψk(x ′) at all possible x′. This nonlocality makes the Hartree-Fock Hamiltonian

difficult to evaluate in large systems involving many atoms and electrons, hence not

suitable for solids. Most electronic structure calculations for solids are based on

density functional theory (DFT) discussed in the following sections.

2.1.4 Kohn-Sham equations

In DFT, the nonlocal exchange term is given by an effective exchange potential that

depends on electronic charge density. Furthermore, all other potential operators are

expressed in charge density rather than in spin-orbitals. This approach reduces the

number of degrees of freedom significantly since all electron coordinates that enter in

the spin-orbitals are now replaced by charge density that is a function only on one

coordinate r. The DFT energy functional is given by

E(n) = T (n) +

∫ Vext(r)n(r)dr+

1

2

∫∫ n(r′)

1

|r− r′|n(r)dr ′dr+ Exc(n) (2.21)

where T being the electronic kinetic energy and Vext being the electron-nuclei electro-

static potential. We know that for given wavefunctions we can calculate the charge

density, n(r) = ∑

k |ψk(r)|2, however the reverse is not obvious. The formality that proves there exists one-to-one mapping between charge density and wavefunctions

was developed by Hohenberg-Kohn [19]. Hohenberg-Kohn theorem also proves that

there exists Exc(n) that will produce the exact ground state of the system. The DFT

Schro¨dinger equation can be derived by taking the variation of E(n) with respect to

n:

δE

δn =

δT

δn + Vext +

∫ n(r′)dr′

|r− r′| + δExc δn

(2.22)

As we have derived the Hartree-Fock equation (Equation 2.19), if we write the DFT

many-electron wavefunction as a Slater determinant of spin-orbitals and orthonor-

10

mality among the spin-orbitals, then each DFT spin-orbital equation satisfies

[ −1 2 �2 + Veff (r)

] ψk(r) = �kψk(r) (2.23)

Veff (r) ≡ Vext(r) + ∫

n(r′)dr′

|r− r′| + δExc δn

(2.24)

The total energy is related to the eigenvalues �k as follows

E = ∑ k

�k − 1 2

∫∫ n(r′)

1

|r− r′|n(r)dr ′dr+ Exc(n)−

∫ δExc(n)

δn n(r)dr (2.25)

Equations 2.24-2.25 are known as Kohn-Sham equations. Since the potential opera-

tors depend on charge density, Equation 2.24 must be solved self-consistently. One

starts from a trial electronic charge density to construct the potentials and solve for

the eigenvalues and spin-orbitals wavefunctions. A new charge density is then con-

structed from the spin-orbitals and the procedure is repeated until the charge and

the wavefunctions are self-consistent within a certain accuracy.

2.1.5 Local density and generalized gradient approximations

In Kohn-Sham equations, there are two terms involving exchange interaction, namely

Exc(n) and δExc/δn. For a given charge density, the first term (the exchange energy)

does not depend on the charge functional on r, whereas the second term (the ex-

change potential) will depend on r if the charge density does. This means that for

a nonhomogeneous systems, exchange potential can be expanded in charge density

and its derivatives:

Vxc ≡ δExc(n) δn(r)

= Vxc(n(r), |�n(r)|, |�(�(n(r)))|, ...) (2.26)

As a first approximation, one neglects all the gradients of charge density in the

exchange potential, this is known as local density approximation (LDA) since the

11

exchange potential depends on charge density only at a particular value of r. It

means that LDA gives an exact ground state for homogeneous electron gas since in

this case the gradients vanish. This allows one to write the LDA exchange energy as

a sum of exchange energy per electron in a homogeneous electron gas, �homxc (n):

ELDAxc =

∫ �homxc (n)ndr (2.27)

LDA also gives accurate results for systems where the charge density does not vary

too rapidly such as in metals. For nonhomogenous systems such as transition metals,

semiconductors, or slabs, LDA is known to underestimate the energy (hence the band

gap). A more accurate approximation, known as generalized gradient approximation

(GGA), includes the first gradient of charge and the exchange energy is given by

EGGAxc =

∫ �GGAxc (n(r), |∇n(r)|)n(r)dr (2.28)

Several techniques exist to parametrize �GGAxc : Perdew-Wang 1986 (PW86) [20, 21],

Perdew-Wang 1991 (PW91) [22], Becke [23], Lee-Yang-Parr (LYP) [24], and Perdew-

Burke-Enzerhof (PBE) [25, 26]. In this work, we use GGA-PBE functional for the

exchange energy.

2.1.6 Pseudopotentials

Kohn-Sham equations can be solved in different ways depending on the choice of

potentials and basis functions to expand the wavefunctions. Some considerations in

solving the Kohn-Sham equations are: (1) potentials becomes very strong near the

nuclei, whereas at far regions they are relatively weak, (2) wavefunctions fluctuate

more near nuclei than in the interstitial regions, (3) the symmetry of the potentials are

approximately spherical near the nuclei whereas at larger distances, the symmetry of

the crystal dominates. Some of the known methods are augmented plave wave (APW)

12

[27, 28], linearized augmented plane wave (LAPW) [29, 30], Orthogonalized plave-

wave (OPW) [31], and pseudopotential method [32, 33]. LAPW is the most accurate

method available. It uses spherical harmonics to expand the wavefunctions in the core

region near the nuclei and plane waves for the interstitial region. Pseudopotential

methods use only plane wave basis set. The number of plane waves can be kept

small by treating an atom as consisting of an effective core (nucleus + core electrons)

and valence electrons. Even though pseudopotentials are not as accurate as LAPW

(which as a contrast fall into the class of fullpotential methods), it provides good

results and becomes the most common choice for its lower computational costs than

LAPW.

2.2 Vienna Ab initio Simulation Package

All ab initio quantum calculations in this work are done using Vienna Ab initio Sim-

ulation Package (VASP) [34, 35]. The calculations are performed in the generalized

gradient approximation (GGA) [36, 37] with exchange correlation as parametrized

by Perdew, Burke, and Ernzerhof (PBE) [25]. To reduce the number of plane waves,

projector augmented-wave (PAW) pseudopotentials are used [38, 39].

VASP uses a plane wave basis set to expand the wave functions in solving the

Kohn-Sham equation in reciprocal space. The evaluation of total electronic energy per

unit cell is done by integrating the energy of all electronic states in the Brillouin zone.

In practice, the Brillouin zone is divided into grids and the integration is replaced by

a sum over k-points. In our work, the k-point grids are generated automatically using

Monkhorst-Pack scheme [40]. For hexagonal structures, an additional shift is used

so that the grids are centered at the Γ point (k = (0, 0, 0)). Typical setting of the

number of k-points as many as 3000/number-of-atoms is usually sufficient to achieve

a converged energy with an accuracy better than 10 meV/atom. Unless otherwise

13

stated, all structures are fully relaxed: atoms position as well as unit cell’s size and

shape are allowed to relax to find the equillibrium configurations.

2.3 Classical interatomic potentials

Due to QC’s lack of periodicity, a large enough simulation cell is necessary to cap-

ture the effect of substrate’s structure on the adsorbed films. A typical cell con-

tains thousands of atoms which makes the simulation unpractical to be performed

quantum-mechanically. Therefore, classical interatomic potentials are needed. For

simple systems, e.g. adsorption of noble gases, pair potentials such as Lennard-Jones

[41] or Morse [42] are sufficient. However, for complex systems, such as adsorption of

hydrocarbons, more accurate potentials are required to take into account many-body

effects in these systems, especially covalent bonds involving carbons which are higly

directional.

Several methods exist to incorporate many-body interations into classical poten-

tials, e.g. force field method (FF) [43, 44, 45, 46], Cluster Expansion method (CE)

[47, 48, 49], and embedded-atom method (EAM) [17, 50]. FF is based on energy-bond

order-bond length relationship [44] and is mostly used for biological systems [43, 51]

and chemical systems where bond formation and breaking are allowed [46]. In CE,

energy of a system is approximated by a converging sum over cluster contributions,

where contribution from large clusters are negligible. In nonperiodic systems, such as

amorphous and quasicrystalline phases, the number of clusters needed can be large

to reach the desired accuracy, therefore CE is mostly suitable to evaluate ground

state energy in periodic systems. EAM is based on the close relationship between

electronic charge density and energy of the system. In quantum physics, this relation-

ship becomes the basis for the density functional theory (DFT) [52, 19, 53], in which

it has been proven that there exists a one-to-one mapping between charge densities

14

and electronic wave functions, and hence energies [19]. Due to the universality of

charge density, in principle EAM can be used in any systems.

EAM was originally developed for examining metallic bulks and surfaces [17, 50].

Later on, charge screening methods have been proposed to modify EAM for use in

covalent systems such as Si [54] and Ge [55]. EAM has been successfully employed

to simulate surface relaxation/reconstructions [56, 57, 58, 59], film growth [60, 61],

and diffusion processes [62, 63]. In this study, we will develop EAM potentials to

simulate hydrocarbons adsorption on d-AlNiCo surface.

In EAM formalism [17, 50], each atom is viewed as being embedded in the material

consisting of all other atoms. The total energy of a system is defined by

Etot ≡ ∑

i

Fi(ρ¯i) + 1

2

∑ i

∑ j �=i

φij(rij), (2.29)

where Fi is the embedding energy of atom i, ρ¯i is the electron density at the vector

position ri, φij is the pair potential, and rij is the distance between atoms i and j.

If ρ¯i is approximated as a sum of individual contribution of the constituents [i.e.,

ρ¯i = ∑ j �=i

ρj(rij), where ρj is the atomic electron density of atom j, the energy is then

only a function of the position of atoms.

For an elemental potential, there are 3 functions needed: F (ρ¯), ρ(r), and φ(r).

For a binary system AB, we will need 7 functions: FA(ρ¯), ρA(r), φA(r), FB(ρ¯),

ρB(r), φB(r), and a cross-pair potential φAB(r). The cross-pair potential is needed

to calculate pair interaction of atoms of different types. In general, in a system

consisting of N different elements, we need N(N + 5)/2 functions.

15

We will use the following functional forms:

ρA(r) = ρAe e −βA(r/rAe −1), (2.30)

φAA(r) = DA[e−2α A(r−rA0 ) − 2e−αA(r−rA0 )], (2.31)

FA(ρ¯) = cubic− splines, (2.32)

φAB(r) = γAB

2 φAA(r) +

1

2γAB φBB(r). (2.33)

In the above equations, A and B denote atom type. ρe, re, and β will be taken from

the database of atomic electron density [64]. Equation 2.31 follows Morse potential

form [42]. The embedding function F will be taken as a natural cubic spline [65].

The expression for the cross-pair interaction φAB follows Haftel’s derivation [59] in

which γAB ≡ ZB/ZA (ZA is the effective charge of the core for atom type A).

2.4 Simplex method

The EAM potentials parameters need to be fit to quantum calculations. The fitting

will be performed using simplex method [66]. Simplex does not require evaluation

of function derivatives which makes it simple to implement. Simplex minimizes N -

dimensional function f by creating P > N simplex points. A simplex point is f

evaluated at a given coordinate. The simplex points will form P -polytope. A min-

imization move is made by replacing the maximum simplex point, hence the worst,

with a point reflected through the centroid of the remaining (P − 1)-polytope. Ex- pansion or shrinking of the polytope are allowed to overcome some local minima or

to converge, respectively.

16

2.5 Grand canonical Monte Carlo

The adsorption simulation will be performed using grand canonical Monte Carlo

(GCMC) method [67, 68, 69]. At constant temperature, T , and volume, V , the

GCMC method explores the configurational phase space using the Metropolis al-

gorithm and finds the equilibrium number of adsorbed atoms (adatoms), N , as a

function of the chemical potential, μ, of the gas, i.e. configuration(μ,N, V ). The

adsorbed atoms are in equilibrium with the coexisting gas: the chemical potential of

the gas is constant throughout the system. In addition, the coexisting gas is taken

to be ideal. With this method we determine adsorption isotherms, ρN , and density

profiles, ρ(x, y), as a function of the pressure, P (T, μ). For each data point in an

isotherm, we perform at least 18 million GCMC steps to reach equilibrium. Each

step is an attempted displacement, creation, or deletion of an atom with execution

probabilities equal to 0.2, 0.4, and 0.4, respectively [70, 71, 72]. At least 27 million

steps are performed in the subsequent data-gathering and -averaging phase.

17

Chapter 3

Noble gas adsorptions on d-AlNiCo

The observed unusual electronic [73, 74] and frictional [75, 76, 77] properties of qua-

sicrystal surfaces stimulate interesting fundamental questions about how these and

other physical properties are altered by quasiperiodicity. Recent progress in the char-

acterization and preparation of quasicrystal surfaces raises new possibilities for their

use as substrates in the growth of films having novel structural, electronic, dynamic

and mechanical properties [78, 79, 80]. The physical behavior of systems involving

competing interactions in adsorption is a subject of continuing interest and is par-

ticularly relevant to the growth of thin films [69]. Several different growth modes

have been observed for the growth of metal films on quasicrystals [81, 82, 83, 84]. A

form of competing interactions seen in adsorption involves either a length scale or a

symmetry mismatch between the adsorbate-adsorbate interaction and the adsorbate-

substrate interaction [85, 86]. Some consequences of such mismatches include den-

sity modulations [87, 88], domain walls [89], epitaxial rotation in the adsorbed layer

[90, 91, 92, 93, 94, 95], and a disruption of the normal periodicity and growth in the

film [96, 97, 98].

The wide range of behavior observed so far indicates that, even in the absence of

intermixing, film growth is strongly affected by chemical interactions between adsor-

bate and substrate. In order to separate these chemical effects from those specific to

quasiperiodic order, we have studied the adsorption of noble gases on a quasicrystal

surface, where both the gas-gas and gas-surface interactions are believed to be simple,

i.e., appreciable chemical interactions and adsorbate-induced surface reconstructions

are absent. In this chapter, we explore the implications of structural mismatch by

18

evaluating the nature of Ne, Ar, Kr, and Xe adsorption on a quasicrystal substrate,

namely the 10-fold surface of decagonal Al73Ni10Co17 quasicrystal (d-AlNiCo QC)

[15, 16].

3.1 Method

3.1.1 Simulation cell

The simulation cell is tetragonal. We take a square section of the surface, A, of side

5.12 nm, to be the (x, y) part of the unit cell in the simulation, for which we assume

periodic boundary conditions along the basal directions. Although this assumption

limits the accuracy of the long range QC structure, it is numerically necessary for

these simulations. To minimize the long range interaction corrections, a relatively

large cutoff (5σgg) is used. Since the size of the cell is relatively large compared to that

of the noble gases, the cell is accurately representative of order on short-to-moderate

length scales. The height of the cell, along the z (surface-normal) direction, is chosen

to be 10 nm (long enough to contain ∼20 layers of Xe). At the top of the cell, a hard-wall reflective potential is employed to confine the coexisting vapor phase. The

simulation results for Xe over d-AlNiCo, presented below, are consistent with both

our results from experiments [99] and virial calculations [100]. Hence, the calculations

may also be accurate for other systems.

3.1.2 Gas-gas and gas-substrate interactions

The gas-gas and gas-substrate interactions are modeled using Lennard-Jones (LJ) 12-

6 potentials, with the gas-gas parameter values �gg and σgg listed in Table 3.1. The

gas-substrate interactions are obtained by summing pair potentials for a gas atom

and all of the substrate atoms in an eigth-layer slab: Al, Ni and Co [15, 101, 100]. The

position of the atoms in the eight-layer slab are taken from the results of a low-energy

19

Table 3.1: Parameter values for the 12-6 Lennard-Jones interactions. TM is the label for Ni or Co.

�gg σgg �gas−Al σgas−Al �gas−TM σgas−TM (meV) (nm) (meV) (nm) (meV) (nm)

Ne 2.92 0.278 9.40 0.264 9.01 0.249 Ar 10.32 0.340 17.67 0.295 16.93 0.280 Kr 14.73 0.360 21.11 0.305 20.23 0.290 Xe 19.04 0.410 24.00 0.330 23.00 0.315

iNe(1) 2.92 0.410 5.45 0.330 5.22 0.315 dXe(1) 19.04 0.278 41.39 0.264 39.67 0.249 dXe(2) 19.04 0.390 25.88 0.320 24.80 0.305 iXe(1) 19.04 0.550 14.96 0.400 14.34 0.385 iXe(2) 19.04 0.675 10.52 0.462 10.08 0.447

electron diffraction LEED analysis of the surface structure of d-AlNiCo [15]. The

gas-substrate interaction parameters are derived using conventional combining rules,

σAB = (σA + σB)/2 and �AB = √

�A�B [102], and experimental heats of adsorption

[100, 99, 103]. The LJ gas-substrate parameters are �gas−Al and σgas−Al for Al, and

�gas−TM and σgas−TM for the two transition metals Ni and Co. All these values are

listed in the upper part of Table 3.1. In the calculation of the adsorption potential,

we assume a structure of the unrelaxed surface taken from the empirical fit to LEED

data [16].

3.1.3 Adsorption potentials

Figures 3.1(a), 3.1(c), 3.1(e), and 3.1(g) show the function Vmin(x, y) of Ne, Ar, Kr,

and Xe on the d-AlNiCo, respectively, which is calculated by minimizing the adsorp-

tion potentials, V (x, y, z), along the z direction at every value (x, y) coordinates:

Vmin(x, y) ≡ min {V (x, y, z)}|along z . (3.1)

20

The figures reveal the fivefold rotational symmetry of the substrate. Dark spots

correspond to the most attractive regions of the substrate. By choosing appropriate

sets of five dark spots, we can identify pentagons, whose sizes follow the inflationary

property of the d-AlNiCo. Note the pentagon at the center of each figure: it will be

used to extract the geometrical parameters λs and λc in Section 3.2.5.

To characterize the corrugation, not well-defined for aperiodic surfaces, we calcu-

late the distribution function f(Vmin), the average 〈Vmin〉 and standard deviation SD of Vmin(x, y) as:

f(Vmin)dVmin ≡ probability { Vmin ∈ [Vmin, Vmin + dVmin[

} (3.2)

〈Vmin〉 ≡ ∫ ∞ −∞

f(Vmin)Vmin dVmin, (3.3)

SD2 ≡ ∫ ∞ −∞

f(Vmin)(Vmin − 〈Vmin〉)2 dVmin. (3.4)

Figures 3.1(b), 3.1(d), 3.1(f), and 3.1(h) show f(Vmin) of the adsorption potential for

Ne, Ar, Kr, and Xe on the d-AlNiCo, respectively. Vmin(x, y) extends by more than

2·SD around its average, revealing the high corrugation of the gas-surface interaction in these four systems. The average and SD of Vmin(x, y) for these systems are listed

in the upper part of Table 3.2. In addition to highly corrugated, the potentials are

“deep” because the record maximum well-depth, e.g. for Xe, on a periodic surface is

about 160 meV, viz. on graphite [104]; and the record minimum well-depth is about

28 meV, on Cs [105].

3.1.4 Effective parameters

For every gas-substrate interaction we define two effective parameters σgs and Dgs.

σgs represents the averaged LJ size parameter of the interaction, calculated following

21

-70 -65 -60 -55 -50 -45 -40 -35 Vmin (meV)

f(V m

in )

(a)

(b)

-180 -160 -140 -120 -100 Vmin (meV)

f(V m

in )

(c)

(d)

-220 -200 -180 -160 -140 -120 Vmin (meV)

f(V m

in )

(e)

(f)

-280 -260 -240 -220 -200 -180 -160 Vmin (meV)

f(V m

in )

(g)

(h)

Ne/QC Ar/QC

Xe/QCKr/QC

Figure 3.1: (color online). Computed adsorption potentials for (a) Ne, (c) Ar, (e) Kr, and (g) Xe on the d-AlNiCo, obtained by minimizing V (x, y, z) with respect to z. The distribution of the minimum value of these potentials is plotted in (b, d, f, and h) respec- tively: the solid line marks the average value 〈Vmin〉, the dashed lines mark the values at 〈Vmin〉±SD.

22

Table 3.2: Range, average (〈Vmin〉), and standard deviation (SD) of the interaction Vmin(x.y) on the d-AlNiCo. Effective parameters of the gas-substrate interactions (Dgs, σgs, D�gs, σ�gs), and, for comparison, the best estimated well depths DGrgs on graphite [106].

Vmin range < Vmin > SD Dgs σgs D � gs σ

� gs D

Gr gs

(meV) (meV) (meV) (meV) (nm) (Dgs/�gg) (σgs/σgg) (meV) Ne -71 to -33 -47.43 6.63 43.89 0.260 15.03 0.935 33 Ar -181 to -85 -113.32 13.06 108.37 0.291 10.50 0.856 96 Kr -225 to -111 -145.71 15.68 140.18 0.301 9.52 0.836 125 Xe -283 to -155 -195.46 17.93 193.25 0.326 10.15 0.795 162

iNe(1) -65 to -36 -45.11 4.08 43.89 0.326 15.03 0.795

dXe(1) -305 to -150 -207.55 29.18 193.25 0.260 10.15 0.935

dXe(2) -295 to -155 -199.40 19.33 193.25 0.316 10.15 0.810

iXe(1) -248 to -170 -195.31 11.21 193.25 0.396 10.15 0.720

iXe(2) -230 to -180 -194.25 7.77 193.25 0.458 10.15 0.679

the traditional combining rules [102]:

σgs ≡ xAlσg−Al + xNiσg−Ni + xCoσg−Co, (3.5)

where xAl, xNi, and xCo are the concentrations of Al, Ni, and Co in the QC, respec-

tively. Dgs represents the well depth of the laterally averaged potential V (z):

Dgs ≡ −min {V (z)}|along z . (3.6)

In addition, we normalize the σgs and Dgs with respect to the gas-gas interactions:

σ�gs ≡ σgs/σgg, (3.7)

D�gs ≡ Dgs/�gg. (3.8)

The values of the effective parameters σgs, Dgs, σ � gs, and D

� gs for the four gas-surface

interactions are listed in the upper part of Table 3.2. We also include the well depth

for Ne, Ar, Kr, and Xe on graphite, as comparison [106].

3.1.5 Test rare gases

As shown in tables 3.1 and 3.2, Ne is the smallest atom and has the weakest gas-gas

and gas-surface interactions (minima of σgg, σgs, �gg and Dgs). In addition, Xe is the

23

largest atom and has the strongest gas-gas and gas-surface interactions (maxima of

σgg, σgs, �gg and Dgs). Therefore, for our analysis, it is useful to consider two test

gases, iNe(1) and dXe(1), which are combinations of Ne and Xe parameters.

iNe(1) represents an “inflated” version of Ne, having the same gas-gas and average

gas-substrate interactions of Ne but the geometrical dimensions of Xe:

{�gg, Dgs, D�gs}[iNe(1)] ≡ {�gg, Dgs, D�gs}[Ne], (3.9)

{σgg, σgs, σ�gs}[iNe(1)] ≡ {σgg, σgs, σ�gs}[Xe]. (3.10)

dXe(1) represents a “deflated” version of Xe, having the same gas-gas and average

gas-substrate interactions of Xe but the geometrical dimensions of Ne:

{�gg, Dgs, D�gs}[dXe(1)] ≡ {�gg, Dgs, D�gs}[Xe], (3.11)

{σgg, σgs, σ�gs}[dXe(1)] ≡ {σgg, σgs, σ�gs}[Ne]. (3.12)

The resulting LJ parameters for iNe(1) and dXe(1) are summarized in the central

parts of Tables 3.1 and 3.2. Furthermore, we also define three other test versions of

Xe: dXe(2), iXe(1), and iXe(2) which have the same gas-gas and average gas-substrate

interactions of Xe but deflated or inflated geometrical parameters. The last three test

gases will be used in Section 3.2.5. The LJ parameters for these gases are summarized

in the lower parts of Tables 3.1 and 3.2. In simulating test gases, we implicitly rescale

the substrate’s strengths so that the resulting adsorption potentials have the same

Dgs as the non-inflated or non-deflated ones (Equations 3.9 and 3.11).

24

3.1.6 Chemical potential, order parameter, and ordering tran-

sition

To conveniently characterize the evolution of the adsorption processes of the gases

we define a normalized chemical potential μ�, as:

μ� ≡ μ− μ1 μ2 − μ1 , (3.13)

where μ1 and μ2 are the chemical potentials at the onset of the first and second layer

formation, respectively. In addition, we introduce the order parameter ρ5−6, defined

as the probability of existence of fivefold defect [70, 71]:

ρ5−6 ≡ N5 N5 + N6

, (3.14)

where N5 and N6 are the numbers of atoms having 2D coordination equal to 5 and

6, respectively. The 2D coordination is the number of neighboring atoms within a

cutoff radius of aNN · 1.366 where aNN is the first nearest neighbor (NN) distance of the gas in the solid phase and 1.366 = cos(π/6) + 1/2 is the average factor of the

first and the second NN distances in a triangular lattice. Note that aNN does not

change appreciably with respect to temperature difference, e.g. aNN of Xe changes

from 0.440 nm at 77 K to 0.443 nm at 140 K.

In a fivefold ordering, most arrangements are hollow or filled pentagons with atoms

having mostly five neighbors. Hence, the particular choice of ρ5−6 is motivated by the

fact that such pentagons can become hexagons by gaining additional atoms with five

or six neighbors. Definition: the five to sixfold ordering transition is defined as a

decrease of the order parameter to a small or negligible final value. The phenomenon

can be abrupt (first-order) or continuous. Within this framework, ρ5−6 and (1−ρ5−6) can be considered as the fractions of pentagonal and triangular phases in the film,

respectively.

25

3.2 Results

3.2.1 Adsorption isotherms

Figure 3.2 shows the adsorption isotherms of Ne, Ar, Kr, and Xe on the d-AlNiCo.

The plotted quantity is the thermodynamic excess coverage (densities of adsorbed

atoms per unit area), ρN), defined as the difference between the total density of

atoms in the simulation cell and the density that would be present if the cell were filled

with uniform vapor at the specified values of P and T . The simulated ranges and the

experimental triple point temperatures (Tt) for Ne, Ar, Kr, and Xe are listed in Table

3.3. A layer-by-layer film growth is visible at low temperatures. Detailed inspection

of the isotherms reveals that there is a continuous film growth (i.e. complete wetting)

at temperatures above Tt (isotherms at T > Tt are shown as dotted curves). This

behavior, observed despite the high corrugation, is interesting as corrugation has

been shown to be capable of preventing wetting [107, 108].

Although vertical steps corresponding to layers’ formation are evident in the

isotherms, the slopes of the isotherms’ plateaus at the same normalized tempera-

tures (T � ≡ T/�gg = 0.35) differ between systems. To characterize this, we calculate the increase of each layer density, ΔρN , from the formation to the onset of the sub-

sequent layer. ΔρN is defined as ΔρN ≡ (ρB − ρA)/ρA and the values are reported in Table 3.3 (points (A) and (B) are specified in Figure 3.2). We observe that, as

the size of noble gas increases ΔρN become smaller, indicating that the substrate

corrugation has a more pronounced effect on smaller adsorbates, as expected since

they penetrate deeper into the corrugation pockets. However, Xe does not follow

this trend. This arises from the complex interplay between the corrugation energy

and length of the potential with respect to the parameters of the gas (σgg, �gg) in

determining the density of the adsorbed layers. In the case of Ne, Ar, and Kr, the

densities at points (A) are approximately the same (ρA = 5.4 atoms/nm 2), whereas

26

that of Xe is considerably smaller (ρA = 4.2 atoms/nm 2), because Xe dimension σgg

becomes comparable to the characteristic length (corrugation) of the potential. This

effect is clarified by the density profile of the films, ρ(x, y), shown in Figures 3.3 and

3.4. As can be seen at points (A), the density profiles of Ne, Ar, and Kr are the

same, i.e. the same set of dark spots appear in their plots. For Xe, some spots are

separated with distances smaller than its core radius (σgg), causing repulsive inter-

actions. Hence these spots will not likely appear in the density profile, resulting in

a lower ρA. More discussion on how interaction parameters affect the shape of the

isotherms is presented in Section 3.2.4. Note that the second layer in each system

has a smaller ΔρN than the first one. The explanation will be given when we discuss

the evolution of density profiles.

Table 3.3: Results for Ne, Ar, Kr, and Xe adsorbed on d-AlNiCo. Tt is taken from reference [109]. The density increase (ΔρN ) in the first and second layers is calculated at T � = 0.35 from point (A) to (B) and (C) to (D) in Figure 3.2, respectively.

simulated T T � ≡ T/�gg Tt ΔρN at T � = 0.35 θr (K) (K) for 1st layer for 2nd layer

Ne 11.8 → 46 0.35 → 1.36 24.55 (12.2-5.3)/5.3=1.30 (11.1-10.2)/10.2=0.09 6◦ Ar 41.7 → 155 0.35 → 1.29 83.81 (7.3-5.5)/5.5=0.33 (6.9-6.4)/6.4=0.08 30◦ Kr 59.6 → 225 0.35 → 1.32 115.76 (6.9-5.5)/5.5=0.25 (6.6-6.3)/6.3=0.05 42◦ Xe 77 → 280 0.35 → 1.27 161.39 (5.8-4.2)/4.2=0.38 (5.2-5.2)/5.2=0 54◦

3.2.2 Density profiles

Figures 3.3 and 3.4 show the density profiles ρ(x, y) at T � = 0.35 for the outer layers

of Ne, Ar, Kr, and Xe adsorbed on the d-AlNiCo at the pressures corresponding to

points (A) through (F) of the isotherms in Figure 3.2.

Ne/d-AlNiCo system. Figure 3.3(a) shows the evolution of adsorbed Ne. At

the formation of the first layer, adatoms are arranged in a pentagonal manner follow-

ing the order of the substrate, as shown by the discrete spots of the Fourier transform

27

10-25 10-20 10-15 10-10 10-5 1 0

10

20

30

40

50

P (atm)

ρ N (a

to m

s/ nm

2 )

Ne/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25

30

35 Ar/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25

30

35 Kr/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25 Xe/QC

P (atm)

ρ N (a

to m

s/ nm

2 )

P (atm)

ρ N (a

to m

s/ nm

2 )

P (atm)

ρ N (a

to m

s/ nm

2 )

(a) (b)

(d)(c)

A

B

C D

E

F

A B

C D

E

F

A B

C D

E

F

A B

C D

Figure 3.2: Computed adsorption isotherms for all the gas/d-AlNiCo systems. The ranges of temperatures under study are: Ne: T = 14 K to 46 K in 2 K steps, Ar: 45 K to 155 K in 5 K steps, Kr: 65 K to 225 K in 5 K steps, Xe: 80 K to 280 K in 10 K steps. Additional isotherms are shown with solid circles at T � = 0.35: T = 11.8 K (Ne), T = 41.7 K (Ar), T = 59.6 K (Kr), and T = 77 K (Xe). Isotherms above the triple point temperatures are shown as dotted curves.

(FT) having tenfold symmetry (point (A)). As the pressure increases, the arrange-

ment gradually loses its pentagonal character. In fact, at point (B) the adatoms are

arranged in patches of triangular lattices and the FT consists of uniformly-spaced

concentric rings with hexagonal resemblance. The absence of long-range ordering in

the density profile is indicated by the lack of discrete spots in the FT. This behavior

persists throughout the formation of the second layer (points (C) and (D)) until the

appearance of the third layer (point (E)). At this and higher pressures, the FT shows

patterns oriented as hexagons rotated by θr = 6 ◦, indicating the presence of short-

28

range triangular order on the outer layer (point (F)). In summary, between points (A)

and (F) the arrangement evolves from pentagonal fivefold to triangular sixfold with

considerable disorder, as the upper part of the density profile at point (F) shows. The

transformation of the density profile, from a lower-packing-density (pentagonal) to

a higher-packing-density structure (irregular triangular), occurs mostly in the mono-

layer from points (A) to (B), causing the largest density increase of the first layer with

respect to that of the other layers (see the end of Section 3.2.2 for more discussion).

Due to the considerable amount of disorder in the final state Ne/d-AlNiCo does not

satisfy the requirements for the transition as defined in Section 3.1.6.

Ar/d-AlNiCo and Kr/d-AlNiCo systems. Figures 3.3(b) and 3.4(a) show

the evolutions for Ar and Kr: they are similar to the Ne case. For Ar, the pentagonal

structure at the formation of the first layer is confirmed by the FT showing discrete

spots having tenfold symmetry (point (A)). The quasicrystal symmetry strongly af-

fects the overlayers’ structures up to the third layer by preventing the adatoms from

forming a triangular lattice (point (E)). This appears, finally, in the lower part of the

density profile at the formation of the fourth layer as confirmed by the FT showing

discrete spots with sixfold symmetry (point (F)). Similar to the Ne case, disorder does

not disappear but remain present in the middle of the density profile corresponding

to the highest coverage before saturation (point (F)). Similar situation occurs also

for the evolution of Kr as shown in Figure 3.4(a).

Xe/d-AlNiCo system. Figure 3.4(b) shows the evolution of adsorbed Xe. At

the formation of the first layer, adatoms are arranged in a fivefold ordering similar

to that of the substrate as shown by the discrete spots of the FT having tenfold

symmetry (point (A)). At point (B), the density profile shows a well-defined triangular

lattice not present in the other three systems: the FT shows discrete spots arranged

in regular and equally-spaced concentric hexagons with the smallest containing six

29

FTρ(x,y)(a)

θ

A

B

C

D

E

F

θ

(b)Ne/QC Ar/QC FTρ(x,y)

A

B

C

D

E

F

r r

Figure 3.3: Density profiles and Fourier transforms of the outer layer at T � = 0.35 for Ne/d-AlNiCo (T = 11.8 K) and Ar/d-AlNiCo (T = 41.7 K), corresponding to points (A) through (F) of Figure 3.2.

30

clear spots. Thus, at point (B) and at higher pressures, the Xe overlayers can be

considered to have regular closed-packed structure with negligible irregularities.

It is interesting to compare the orientation of the hexagons on FT for these four

adsorbed gases at the highest available pressures before saturation (point (F) for Ne,

Ar, and Kr, and point (D) for Xe). We define the orientation angles as the smallest

of the possible clockwise rotations to be applied to the hexagons to obtain one side

horizontal, as shown in Figures 3.3 and 3.4. Such angles are θr = 6 ◦, 30◦, 42◦, and

54◦, for adsorbed Ne, Ar, Kr, and Xe, respectively. These orientations, induced by

the fivefold symmetry of the d-AlNiCo, can differ only by multiples of n ·12◦ [70, 71]. Since hexagons have sixfold symmetry, our systems can access only five possible

orientations (6, 18, 30, 42, 54◦), and the final angles are determined by the interplay

between the adsorbate solid phase lattice spacing, the periodic simulation cell size,

and the potential corrugation. For systems without periodic boundary conditions,

the ground state has been found to be fivefold degenerate, as should be the case

[70, 71].

Xe adsorption on this surface was studied experimentally using LEED, in which the

isobar measurements indicate that the Xe film grows layer-by-layer in the temperature

range 65 K to 80 K [99], consistent with the simulations. Under similar conditions

to the simulation at 77 K, at the lowest coverage, the only discernible change in

the LEED pattern from that of the clean surface is an attenuation of the substrate

beams. After the adsorption of one layer, there are still no resolvable features that

would indicate an overlayer having order different from the substrate. At the onset of

the adsorption of the second layer, however, the LEED pattern shows new diffraction

spots that correspond to 5 rotational domains of a hexagonal structure. Within

each of these domains, the close-packed direction of the Xe is aligned with the 5-fold

directions of the substrate, as also observed in the simulation. In the experiments,

31

all possible alignments are observed owing to the presence of all possible rotational

alignments present within the width of the electron beam (0.25 mm). When the

second layer is complete, these spots are well-defined and their widths are the same as

the substrate spots, indicating a coherence length of at least 15 nm. The average Xe-

Xe spacing measured in the experiment is consistent with the bulk nearest-neighbor

spacing of 0.44 nm. A dynamical LEED analysis of the intensities indicates that the

structure of the multilayer film is consistent with face-centered cubic (FCC) Xe(111).

These structure parameters for the bilayer film are essentially identical to the results

obtained for Xe growth on Ag(111) [110, 111], a much weaker and less corrugated

substrate. This suggests that effect of the symmetry and corrugation of the substrate

potential on the Xe film structure is largely confined to the monolayer.

In every system, the increase of the density for each layer is strongly correlated

to the commensurability with its support: the more similar they are, the more flat

the adsorption isotherm will be (note that the support for the (N + 1)th-layer is the

N th-layer). For example, the Xe/d-AlNiCo system has an almost perfect hexagonal

structure at point (B) (due to its first-order five to sixfold ordering transition as

described in the next section). Hence, all the further overlayers growing on the top of

the monolayer will be at least “as regular” as the first layer, and have the negligible

density increase as listed in Table 3.3.

3.2.3 Order parameters (ρ5−6)

The evolution of the order parameter ρ5−6 is plotted in Figure 3.5 as a function of

the normalized chemical potential, μ�, at T �=0.35 for all the noble gas/d-AlNiCo

systems.

Ne/d-AlNiCo, Ar/d-AlNiCo, and Kr/d-AlNiCo systems. The ρ5−6 plots

for the first four layers observed before bulk condensation are shown in panels (a)−(c).

32

FTρ(x,y)(a)

A

B

C

D

E

F

(b)Kr/QC Xe/QC FTρ(x,y)

A

B

C

D

θr

θr

Figure 3.4: Density profiles and Fourier transforms of the outer layer at T � = 0.35 for of Kr/d-AlNiCo (T = 59.6 K) and Xe/d-AlNiCo (T = 77 K), corresponding to points (A) through (F) of Figure 3.2.

33

μ*

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

μ*

ρ 5 -6

layer 1

(a) (b)

(d)(c)

Ne/QC Ar/QC

Kr/QC Xe/QC

0 0.2 0.4 0.6 0.8 1 1.2 0

0.2

0.4

0.6

0.8

1

μ*

ρ 5 -6

1 1.05 1.1 1.15

0.4

0.5

0.6

layer 1

layer 2

layer 3 layer 4

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.06 1.12

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 μ

ρ 5 -6

layer 1

layer 2

layer 3 layer 4

0 0.2 0.4 0.6 0.8 1 1.2 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.06 1.12 0.3

0.5

0.7

μ*

ρ 5 -6

layer 1

layer 2 layer 3 layer 4

5- to 6- fold transition

Figure 3.5: (color online). Order parameters, ρ5−6, as a function of normalized chemical potential, μ�, (as defined in the text) at T � = 0.35 for the first four layers of (a) Ne, (b) Ar, (c) Kr, and for the first layer of Xe (d) adsorbed on d-AlNiCo. A sudden drop of the order parameter in Xe/QC to a constant value of ∼ 0.017 at μ� ∼ 0.8 indicates the existence of a first-order structural transition from fivefold to sixfold in the system.

As the chemical potential μ� increases, ρ5−6 decreases continuously reaching a con-

stant value only for Kr. At bulk condensation, the values of ρ5−6 are still high,

approximately 0.35 ∼ 0.45. Data at higher temperatures shows a similar behavior (up to T=24 K (T �=0.71) for Ne, T =70 K (T �=0.58) for Ar, and T=90 K (T �=0.53)

for Kr). Thus, we conclude that these systems do not undergo the ordering transition.

Xe/d-AlNiCo system. The ρ5−6 plot for the first layer is shown in panel (d).

In this system, as the chemical potential μ� increases, the order parameter gradually

34

decreases reaching a value of ∼ 0.3 at μ�tr ∼ 0.8. Suddenly it drops to 0.017 and remains constant until bulk condensation. Similar behavior is observed at higher

temperatures up to T=140 K (T �=0.63). This is a clear indication of a five to sixfold

ordering transition, as the first layer has undergone a transformation to an almost

perfect triangular lattice. Figure 3.6(d) shows the total enthalpy in the system, H.

At the transition point μ�tr, the enthalpy has a little step indicating a latent heat of

the transition. The discontinuity of the order parameter ρ5−6 and the presence of

latent heat indicate that the ransition is first-order. The latent heat of this transition

is estimated to be 1.021/151 ≈ 6.8 meV/atom. Despite the evidence for a first-order transition, the nearest neighbor distance, la-

beled rNN , e.g. Xe/d-AlNiCO in Figure 3.6(b), appears to change continuously. This

nearest neighbor distance is defined as the location of the first peak in the pair corre-

lation function because the latter property is more directly comparable to diffraction

measurements. We have also calculated the average spacing between neighbors, d¯NN ,

which is a thermodynamically meaningful quantity (related to the density). This has

a small discontinuity at the transition, providing additional evidence for the first-

order character of the transition. Both quantities, rNN and d¯NN , are shown in Figure

3.6(b). The NN Xe-Xe distance rNN decreases continuously as P increases, starting

from 0.45 nm and saturating at 0.44 nm. The Xe-Xe distance reaches saturation

value before the appearance of the second layer; therefore, the transition is complete

within the first layer. We note that a similar decrease in NN distance was measured

for Xe/Ag(111), but in that case, the NN spacing did not saturate before the onset

of the second layer adsorption [112, 110].

The observed transition from fivefold to sixfold order within the first layer of Xe

on d-AlNiCo can be viewed as a commensurate-incommensurate transition (CIT),

since at the lower coverage, the layer is commensurate with the substrate symmetry

35

0 0.2 0.4 0.6 0.8 1 μ*

ρ 5- 6

2 4 6 8

10

ρ N (a

to m

/n m

2 )

layer formation

st1

layer formation

nd2

b

a

0.2

0.4

0.6 5- to 6- fold transition

0.44

0.45

0.46

(n m

)

c

d

rNN

dNN

r NN d N

N ,

-45

-40

-35

-30

-25

H (e

V )

Δ Η

Figure 3.6: (color online). Xe on d-AlNiCo at T = 77 K. (a) Adsorption isotherm, ρN , versus the normalized chemical potential, μ�. (b) Nearest neighbor distance derived from the first peak of pair correlation function, rNN , (black line), and average spacing between neighbors at equilibrium, d¯NN , (red line). (c) Order parameter ρ5−6 (probability of fivefold defects, defined in Equation 3.14) versus the normalized chemical potential, μ�. (d) Total enthalpy. The transition, which is defined as the point in μ� above which the order parameter remains nearly constant, occurs at μ�tr ∼0.8. The discontinuity in H around μ�tr ∼ 0.8 indicates a first order transition with associated latent heat of the transition. The order parameter ρ5−6 after the transition is ∼ 0.017. Heat of the transition is ≈ 6.8 meV/atom.

36

and aperiodic, while at higher coverage, it is incommensurate with the substrate.

Such transitions within the first layer have been observed before for adsorbed gases,

perhaps most notably for Kr on graphite [113]. There, as here for Xe, the Kr forms

a commensurate structure at low coverage, which is compressed into an incommen-

surate structure at higher coverage. The opposite occurs for Xe on graphite, which

is incommensurate at low coverage and commensurate at high coverage [114]. Such

commensurate-incommensurate transitions have been studied theoretically in many

ways, but perhaps most simply as a harmonic system (balls and springs) having a

natural spacing that experiences a force field having a different spacing [115]. Such a

transition has been found to be first-order for strongly corrugated potentials (in 1D)

but continuous for more weakly corrugated potentials [116]. The transition observed

in our quasicrystal surface suggests that system is within the regime of “strong” cor-

rugation, which was not the case of Kr over graphite [113]. In fact, for the latter

system, both commensurate and incommensurate structures have sixfold symmetry.

A more relevant comparison may be the transition of Xe on Pt(111), from a rect-

angular symmetry incommensurate phase to a hexagonal symmetry commensurate

one, although in that case, the low-temperature phase was incommensurate. That

transition was also found to be continuous [117]. Therefore, while our simulations

indicate that Xe on d-AlNiCo undergoes a CIT, as observed for other adsorbed gases,

the observation of a first-order CIT is new, to our knowledge, and likely arises from

the large corrugation.

3.2.4 Effects of �gg and σgg on adsorption isotherms

In Section 3.2.1 we have briefly discussed how the density increase of each layer (ΔρN)

is affected by the size of the adsorbate (σgg). In addition, since the corrugation of the

potential depends also on the gas-gas interaction (�gg), the latter quantity could a

37

priori have an effect on the density increase. To decouple the effects of σgg and �gg on

ΔρN we calculate ΔρN while keeping one parameter constant, σgg or �gg, and varying

the other. For this purpose, we introduce two test gases iNe(1) and dXe(1), which

represent “inflated” or “deflated” versions of Ne and Xe, respectively (parameters

are defined in Equations 3.9-3.12 and listed in Tables 3.1 and 3.2). Then we perform

four tests summarized as the following:

(1) constant strength �gg, size σgg increases [Ne→iNe(1)]: ΔρN reduces,

(2) constant strength �gg, size σgg decreases [Xe→dXe(1)]: ΔρN increases,

(3) constant size σgg, strength �gg decreases [Xe→iNe(1)]: ΔρN ∼ constant,

(4) constant size σgg, strength �gg increases [Ne→dXe(1)]: enhanced agglomer- ation.

Figure 3.7 shows the adsorption isotherms at T � = 0.35 for Ne, iNe(1), Xe, and

dXe(1) on d-AlNiCo. By keeping the strength constant and varying the size of the

adsorbates, tests 1 and 2 ([Ne→iNe(1)] and [Xe→dXe(1)]), we find that we can reduce or increase the value of the density increase (when ΔρN decreases the continuous

growth tends to become stepwise and vice versa). These two tests indicate that the

larger the size, the smaller the ΔρN . By keeping the size constant and decreasing

the strength, test 3 ([Xe→iNe(1)]), we find that ΔρN does not change appreciably. An interesting phenomenon occurs in test 4 where we keep the size constant and

increase the strength ([Ne→dXe(1)]). In this test the growth of the film loses its step-like shape. We suspect that this is caused by an enhanced agglomeration effect

as follows. Ne and dXe(1) have the same size which is the smallest of the simulated

gases, allowing them to easily follow the substrate corrugation, in which case, the

corrugation helps to bring adatoms closer to each other [100] (agglomeration effect).

The stronger gas-gas self interaction of dXe(1) compared to Ne will further enhance

38

10-25 10-20 10-15 10-10 10-5 1 0

10

20

30

40

P (atm)

ρ (a

to m

s/ nm

2 )

Ne iNe Xe

dX e

Ν

(1 )

(1)

test 1: [Ne → iNe(1)]

test 2: [Xe → dXe(1)]

test 3: [Xe → iNe(1)]

test 4: [Ne → dXe(1)]

Figure 3.7: (color online). Computed adsorption isotherms for Ne, Xe, iNe(1), and dXe(1)

on d-AlNiCo at T �=0.35. iNe(1) and dXe(1) are test noble gases having potential parameters described in the text and in Tables 3.1 and 3.2. The effect of varying the interaction strength of the adsorbates on the density increase ΔρN (while keeping the size constant) is negligible on large gases but significant on small gases.

this agglomeration effect, resulting in a less stepwise film growth of dXe(1) than Ne.

As can be seen, dXe(1) grows continuously, suggesting a strong enhancement of the

agglomeration. In summary, the last two tests (3 and 4) indicate that the effect of

varying the interaction strength of the adsorbates (while keeping the size constant)

is negligible on large gases but significant on small gases.

3.2.5 Effects of �gg and σgg on 5- to 6-fold transition

Strength �gg and size σgg of the adsorbates also affect the existence of the first-order

transition (present in Xe/d-AlNiCo, but absent in Ne, Ar, and Kr on d-AlNiCo).

Hence we perform the same four tests described before and observe the evolution of

the order parameter. The results are the following:

(1) constant strength �gg, size σgg increases [Ne→iNe(1)]: transition appears

(2) constant strength �gg, size σgg decreases [Xe→dXe(1)]: transition disappears 39

(3) constant size σgg, strength �gg decreases [Xe→iNe(1)]: transition remains

(4) constant size σgg, strength �gg increases [Ne→dXe(1)]: remains no transition

The strength �gg has no effect on the existence of the transition (tests 3 and 4), which

instead is controlled by the size of the adsorbates (tests 1 and 2). To further charac-

terize such dependence, we add three additional test gases with the same strength �gg

of Xe but different sizes σgg. The three gases are denoted as dXe (2), iXe(1), and iXe(2)

(the prefixes d- and i- stand for deflated and inflated, respectively). The interaction

parameters, defined in the following equations, are listed in Tables 3.1 and 3.2:

{�gg, Dgs, σgg} [dXe(2)] ≡ {�gg, Dgs, 0.95σgg}[Xe], (3.15)

{�gg, Dgs, σgg}[iXe(1)] ≡ {�gg, Dgs, 1.34σgg}[Xe], (3.16)

{�gg, Dgs, σgg}[iXe(2)] ≡ {�gg, Dgs, 1.65σgg}[Xe]. (3.17)

Figure 3.8 shows the evolutions of the order parameter as a function of the normal-

ized chemical potential for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed

on d-AlNiCo at T �=0.35. All these systems undergo a transition, except dXe(2), i.e.

the transition occurs only in systems with σgg ≥ σgg[Xe] indicating the existence of a critical value for the appearance of the phenomenon. Furthermore, as σgg increases

(iNe(1) → iXe(1) → iXe(2)), the transition shifts towards smaller critical chemical potentials.

3.2.6 Prediction of 5- to 6-fold transition

The critical value of σgg associated with the transition can be related to the charac-

teristic length of the d-AlNiCo by introducing a gas-substrate mismatch parameter

defined as

δm ≡ k · σgg − λr λr

. (3.18)

40

0 0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

μ*

ρ 5 -6

iXe

dXe

iNe

(2)

(1) (1)iXe(2)

Figure 3.8: (color online). Order parameters as a function of normalized chemical poten- tial (as defined in the text) for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed on d-AlNiCo at T � = 0.35. A first-order fivefold to sixfold structural transition occurs in the last three systems, but not in dXe(2).

where k = 0.944 is the distance between rows in a close-packed plane of a bulk LJ

gas (calculated at T = 0 K with σ = 1 [118]), and λr is the characteristic spacing of

the d-AlNiCo, determined from the momentum transfer analysis of LEED patterns

[99] (our d-AlNiCo surface has λr=0.381 nm [99]). With such ad hoc definition, δm

measures the mismatch between an adsorbed FCC[111] plane of adatoms and the d-

AlNiCo surface. In Table 3.4 we show that δm perfectly correlates with the presence

of the transition in our test cases (transition exists ⇔ δm > 0). The definition of a gas-substrate mismatch parameter is not unique. For example,

one can substitute k · σgg with the first NN distance of the bulk gas, and λr with one of the following characteristic lengths: a) side length of the central pentagon in

the potential plots in Figure 3.1 (λs = 0.45nm), b) distance between the center of

the central pentagon and one of its vertices (λc = 0.40nm), c) L = τ · S = 0.45 nm, where τ = 1.618 is the golden ratio of the d-AlNiCo and S = 0.243 nm is the

41

Table 3.4: Summary of adsorbed noble gases on d-AlNiCo that undergo a first-order fivefold to sixfold structural transition and those that do not.

δm transition Ne -0.311 No Ar -0.158 No Kr -0.108 No Xe 0.016 Yes

iNe(1) 0.016 Yes dXe(1) -0.311 No dXe(2) -0.034 No iXe(1) 0.363 Yes iXe(2) 0.672 Yes

k = 0.944 [118] λr = 0.381 nm [99]

δm ≡ (k · σgg − λr)/λr

side length of the rhombic Penrose tiles [15]. Although there is no a priori reason to

choose one definition over the others, the one that we select (Equation 3.18) has the

convenience of being perfectly correlated with the presence of the transition, and of

using reference lengths commonly determined in experimental measurements (λr) or

quantities easy to extract (k · σgg).

3.2.7 Transitions on smoothed substrates

In Figure 3.1 we can observe that near the center of each potential there is a set of five

points with the highest binding interaction (the dark spots constituting the central

pentagons). A real QC surface contains an infinite number of these very attractive

positions which are located at regular distances and with five fold symmetry. Due to

the limited size and shape of the simulation cell, our surface contains only one set

of these points. Therefore, it is of our concern to check if the results regarding the

existence of the transition are real or artifacts of the method. We perform simulation

tests by mitigating the effect of the attractive spots through a Gaussian smoothing

function which reduces the corrugation of the original potential. The definitions are

42

the following:

G(x, y, z) ≡ AGe−(x2+y2+z2)/2σ2G , (3.19)

V (z) ≡ 〈V (x, y, z)〉(x,y) , (3.20)

Vmod(x, y, z) ≡ V (x, y, z) · [1−G(x, y, z)] + V (z) ·G(x, y, z). (3.21)

where G(x, y, z) is the Gaussian smoothing function (centered on the origin and with

parameters AG and σG), V (z) is the average over (x, y) of the original potential

V (x, y, z), and Vmod(x, y, z) is the final smoothed interaction. An example is shown

in Figure 3.9(a) where we plot the minimum of the adsorption potential for a Ne/d-

AlNiCo modified interaction (smoothed using AG = 0.5 and σG = 0.4 nm). In

addition, in panel (b) we show the variations of the minimum adsorption potentials

along line x = 0 for the modified and original interactions (solid and dotted curves,

respectively).

Using the modified interactions (with AG = 0.5 and σG = 0.4 nm) we simulate all

the noble gases of Table 3.4. The results regarding the phase transition on modified

surfaces do not differ from those on unmodified ones, confirming that the observed

transition behavior is a consequence of competing interactions between the adsorbate

and the whole QC substrate rather than just depinning of the monolayer epitaxially

nucleated. Therefore, the simple criterion for the existence of the transition (δm > 0)

might also be relevant for predicting such phenomena on other decagonal quasicrystal

substrates.

3.2.8 Temperature vs substrate effect

Using Xe/d-AlNiCo data, we observe that defects are present at all temperatures that

are simulated (20 to 286 K). The probability of defects increases with temperature,

implying that their origin is entropic, as is the case for a periodic crystal. Figure

43

(a) (b)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-75

-65

-55

-45

-35

Y (nm)

V m

in (y

) ( m

eV )

modified potential original potential

Figure 3.9: (color online). (a) The minimum of adsorption potential, Vmin(x, y), for Ne on a smoothed d-AlNiCo as described in the text. (b) The variations of the minimum adsorption potentials along the line at x = 0 shown in (a), for the modified and original interactions (solid and dotted curves).

3.10-(right axis) shows that the defect probability increases as T increases, while

Figure 3.10-(left axis) shows the trend of the transition point in function of T. At low

temperatures, the sixfold ordering occurs earlier (at lower μ�tr) as the temperature

is increased from 40K to 70K. This trend is expected because the ordering effect

imposed by the substrate corrugation becomes relatively smaller as the temperature

increases. However, this trend is not observed in the higher temperature region (from

70K to 140K). In fact, at higher temperatures the transition point shifts again to

higher μ�tr. This is most likely due to the monolayer becoming less two-dimensional,

allowing more structural freedom of the Xe atoms and thus decreasing the effect of

the repulsive Xe-Xe interaction that would stabilize the sixfold structure. Transitions

having critical μ�tr > 1 indicate that the onset of second-layer adsorption occurs

earlier than the transition to the sixfold structure. When the second layer adsorbs at

T >130K, the density of the monolayer increases by a few percent, thereby increasing

the effect of the repulsive interactions and driving the fivefold to sixfold transition.

44

o f 5

-6 fo

ld tr

an si

tio n

μ t r*

ρ 5-6

μtr*

ρ 5-6

Figure 3.10: Xe on d-AlNiCo. Values of μ�tr for the fivefold to sixfold transition points from 40 K to 140 K (left axis). Transition points at μ�tr > 1 indicate that a transfer of atoms from the second layer to the first layer is required to complete the transition. Also shown is the defect probability as a function of T after the transition occurs (right axis), indicating an increase in defect probability with T .

Interestingly, stacking faults are evident in the multilayer films. This is consistent

with x-ray diffraction studies of the growth of Xe on Ag(111), where stacking faults

were observed for Xe growth under various growth conditions [110, 111], although the

overall structure observed was FCC(111). Such a stacking fault is evident in Figure

3.11, which shows a superposition of Xe layers 2 and 4 at 77 K. The coincidence of

the atom locations in the top left part of this figure is consistent with an hexagonal

close-packed structure (HEX), ABAB stacking, whereas the offsets observed in the

lower part of the figure indicate the presence of stacking faults caused by dislocations

in the layers. We note that while bulk Xe has an FCC structure, and indeed an FCC

structure was found for the multilayer film in the LEED study, calculations of the

bulk structure using LJ pair potentials such as those employed here result in a more

stable HEX structure [119]. The energy difference between the two structures is very

small, and apparently arises from a neglect of d-orbital overlap interactions, which

45

Figure 3.11: Density plot of Xe on decagonal AlNiCo at 77 K, showing a superposition of the density slices for the 2nd and 4th layers. In the top right, 4th-layer atoms are located directly above the 2nd layer atoms indicating an hexagonal close-packed ABAB stacking, whereas in other regions, such as lower left, the two layers are offset due to stacking fault.

are more effective in FCC than in HEX structures [119, 120]. Although the simulated

film is HEX instead of FCC, the main conclusions concerning the growth mode of Xe

on the quasicrystal are not affected [70].

3.2.9 Orientational degeneracy of the ground state

In was mentioned in the previous section that after the ordering transition is complete,

the resulting sixfold structure is aligned parallel to one of the sides of the pentagons

in the Vmin map of the adsorption potential (there are five possible orientations). In

the experiments, all five orientations are observed, due to the presence of all possi-

ble alignments of hexagons along five sides of a pentagon in the QC sample within

the width of the electrons beam (∼0.25 mm). In an ideal infinite GCMC frame- work the ground state of the system would be degenerate and all five orientations

would have the same energy and be equally probable. However, the square periodic

boundary conditions of our GCMC break this orientational degeneracy, causing some

orientations to become more likely to appear.

To find all the possible orientations, we performed simulations with a cell having

free boundary conditions. The cell is a 5.12 x 5.12 nm2 quasicrystal surface sur-

rounded by vacuum. Figure 3.12(a) shows the Vmin map of the adsorption potential.

46

Thirty simulations at 77 K are performed with this cell. The isotherms from these

runs are plotted in Figure 3.12(b). Only the first layer is shown, and the finite size

of the surface makes the growth of the first layer continuous. The density profiles

ρ(x, y) of all the simulations are analyzed at point p� of Figure 3.12(b). In this cell,

all five orientations of hexagons are observed with equal frequency indicating the ori-

entational degeneracy of the ground state. To represent the five orientations, density

profiles of five calculations (c, d, e, f, and g) are shown in Figures 3.12(c) to 3.12(g)

with their FT plotted on the side. Figure 3.12(h) presents a schematic depiction of

which orientations of hexagons are exemplified in each simulation.

Figures 3.13(a) and 3.13(b) illustrate the effect of pentagonal defects on the ori-

entation of hexagons at point p� of Figure 3.12(b). In most of the density profiles

corresponding to this coverage, we find the behavior shown in Figure 3.13(a). Here,

the effect of the pentagonal defect, which is the center of a dislocation in the hexag-

onal structure, is to rotate the orientation of the hexagons above the pentagon by

2 · 60◦/5 = 24◦ with respect to the hexagons below the pentagon. The possible rota- tions are n · 12◦, where n = 1,2,3,4, or 5. The rotation by 12◦ is usually mediated by more than one equivalent pentagon, as is shown in Figure 3.13(b) (note the “up” pen-

tagon at the middle-bottom part and the “down” pentagon near the middle-top part

of the figure). The “up” pentagon (with one vertex on the top) is equivalent with the

“down” pentagon (with one vertex on the bottom) since they have five orientationally

equivalent sides). These pentagonal defects are induced by the fivefold symmetry of

the substrate, and their concentration decreases in the subsequent layers.

3.2.10 Isosteric heat of adsorption

Figure 3.14 shows a P -T diagram for three different coverages of Xe adsorbed on

d-AlNiCo constructed from the isotherms in the range 40 K < T < 110 K. In the

47

simulations, the layers grow step-wise; at 70 K the first step occurs between coverage

∼0.06 and ∼0.7, the second step occurs between coverage 1.0 and ∼1.9, and the third step occurs between coverage ∼1.9 and ∼2.8 (unit is in fractions of monolayer). Figure 3.14 shows the T , P location of these steps, denoted “cov 0.5”, “cov 1.5”,

and “cov 2.5” for the first, second, and third steps, respectively. The isosteric heat

of adsorption per atom at these steps can be calculated from the P -T diagram as

follows [112]:

qst ≡ −kB d(lnP ) d(1/T )n

. (3.22)

The inset of Figure 3.14 summarizes the values of qst obtained from simulations

and experiments. The agreement between experiment and the simulations for the

half monolayer heat of adsorption is good. The values obtained in the simulation for

the 1.5 and 2.5 layer heats are about 20% lower than the bulk value of 165 meV [99].

The lower values suggest that bulk formation should be preferred at coverages above

one layer. However, layer-by-layer growth is observed at all T for at least the first

few layers in these simulations. We therefore believe that the low heats of adsorption

arise from slight inaccuracies in the Xe-Xe LJ parameters used in this calculation, as

the heats of adsorption are very sensitive to the gas parameters.

3.2.11 Effect of vertical dimension

In a standard unit cell, only 2 steps, corresponding to the first and second layer ad-

sorption, are apparent in the isotherms [101]. Further simulations indicate that when

the cell is extended in the vertical direction, additional steps are observed. Therefore

the number of observable steps is related to the size of the cell. Nevertheless, layering

is clearly evident in the ρ(z) profile, and the main features of the film growth are not

altered. For Xe on d-AlNiCo, the average interlayer distance is calculated to be about

0.37 nm, compared to 0.358 nm for the interlayer distance in the < 111 > direction

48

of bulk Xe [121]. Our simulations of multilayer films show variable adsorption as

the simulation cell is expanded in the direction perpendicular to the surface. This

is a result of sensitivity to perturbations (here, cell size) close to the bulk chemical

potential, where the wetting film’s compressibility diverges. This dependence has

been seen previously in large scale simulations. See e.g. Figure 3 of reference [108].

The analog of this effect in real experiments is capillary condensation at pressures

just below saturated vapor pressure (svp), the difference varying as the inverse pore

radius.

3.3 Summary

The results of GCMC simulations of noble gas films on QC have been presented. Ne,

Ar, Kr, and Xe grow layer-by-layer at low temperatures up to several layers before

bulk condensation. We observe interesting phenomena that can only be attributed

to the quasicrystallinity and/or corrugation of the substrate, including structural

evolution of the overlayer films from commensurate pentagonal to incommensurate

triangular, substrate-induced alignment of the incommensurate films, and density

increase in each layer with the largest one observed in the first layer and in the smallest

gas. Two-dimensional quasicrystalline epitaxial structures of the overlayer form in

all the systems only in the monolayer regime and at low pressure. The final structure

of the films is a triangular lattice with a considerable amount of defects except in

Xe/QC. Here a first-order transition occurs in the monolayer regime resulting in an

almost perfect triangular lattice. The subsequent layers of Xe/QC have hexagonal

close-packed structures. By simulating test systems with various sizes and strengths,

we find that the dimension of the noble gas, σgg, is the most crucial parameter in

determining the existence of the phenomenon which is found only in systems with

σgg ≥ σgg[Xe].

49

a 10 -12

10 -10

10 -8

10 -60

2

4

6

8

10

p (atm)

ρ= N

/A

(a to

m s/

nm 2

)

p*

-5 5X(nm)

5

Y (n

m )

-5

0

-50

-100

-150

-200

-250

(meV)

b

c d

e

g

f

h c

d,g

f

g

e,g

Figure 3.12: Xe adsorption on d-AlNiCo. (a) Minimum potential energy surface of the adsorption potential with free boundary conditions. (b) Adsorption isotherms of the first layer from a set of 30 simulations at 77 K using the free cell described in the paper. Five density profiles and FTs at point p� of (b) are shown in (c) to (g), representing all possible orientations of hexagonal domains. (h) Schematic diagram illustrating the correspondence between the orientations of the hexagonal domains observed in the density profiles (c) to (g).

50

θ1 θ2

a b

Figure 3.13: Xe adsorption on d-AlNiCo. Pentagonal defects rotate the orientation of hexagons by (a) θ1 = 24◦ and (b) θ2 = 12◦.

Figure 3.14: (color online). Xe adsorption on d-AlNiCo. Locations in P , T of the vertical risers in the isotherms corresponding to the first (square), second (circle), and third (triangle) layer formation. The heats of adsorptions, qst, are 270, 129, and 125 meV/atom respectively, calculated as described in the text. The inset figure shows qst obtained from the simulations as well as from the experiments.

51

Chapter 4

Embedded-atom method potentials

The interatomic potentials to simulate hydrocarbon adsorptions on d-AlNiCo are

generated within the embedded-atom method (EAM) formalism. The parametriza-

tion of the potentials are given in Equations 2.30 - 2.33. These parameters are fitted

to the energy of various structures computed via ab initio quantum calculations using

VASP code. The structure prototypes are summarized in Table 4.1. For elemental

Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure

(DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed

(HEX), simple cubic (SC), and simple hexagonal (SH). The structures are first re-

laxed from the initial configurations to achieve the equilibrium ones. For the Al-Co-Ni

ternary phases, the initial configurations are taken from the database of alloys [122].

After the structures are relaxed, the ground state electronic energies are calculated

and are used as the fitting data. The technical details in relaxing the structures and

evaluating the energies with VASP are given in Section 2.2.

Fitting of the parameters of the EAM potentials are performed using SIMPLEX

method (Section 2.4). SIMPLEX is relatively slower than other methods such as

nonlinear least square or conjugate gradient. However, since the speed of the fitting

procedure is not a concern in this work (most time is spent in the ab initio calcula-

tions), SIMPLEX is advantageous because it does not require evaluation of function’s

derivatives or orthogonality. In this way, a new parametrization of EAM potentials

requires changes only in the function evaluation routines. A fitting code is developed

to be able to fit in a bulk or adsorption mode. In the bulk mode, the function to

52

Table 4.1: List of structure prototypes used to fit the EAM potentials for hydrocarbon ad- sorption on Al-Co-Ni. For elemental Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure (DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed (HEX), simple cubic (SC), and simple hexagonal (SH). The Al– Co-Ni ternaries are taken from the database of alloys [122].

group structure Al BCC, DIA, FCC, GRA, HEX, SC, SH Co BCC, DIA, FCC, GRA, HEX, SC, SH Ni BCC, DIA, FCC, GRA, HEX, SC, SH CHn CH4 C2Hn C2H2, C2H4, C2H6-isotactic, C2H6-syntactic C3Hn propane, C3H4 allene, propyne, propene,

cyclopropane, cyclopropene, cyclopropyne C4Hn isobutane, 1-butene, 1-butyne, cyclobutane, methylcyclopropane C5Hn pentane, cyclopentane Other alkanes hexane, heptane, nonane, decane, undecane, dodecane AlxCoyNiz Al32Co12Ni12 (cI112)

Al32Co16Ni12 (cI128) Al29Co4Ni8 (dB1) Al17Co5Ni3 (dH1) Al34Co4Ni12 (dH2) Al20Co7Ni1 (hP28) Al18Co4Ni4 (mC28) Al12Co2Ni2 (mC32) Al18Co2Ni2 (mP22) Al36Co8Ni4 (oI96) Al12Co1Ni3 (oP16) Al34Co12Ni4 (mC102)

53

minimize is the err = ΔEbulk defined as:

ΔEbulk = ∑

i

|EEAMbulk,i − EV ASPbulk,i | (4.1)

The energies are per atom and the summation is over all structures of the fit. In

the adsorption mode, the error function consists of the error in the bulk energies

of molecule structures (ΔEbulk,mol), substrate structures (ΔEbulk,sub), molecule-on-

substrate structure (ΔEbulk,mol+sub), as well as in the adsorption energies (ΔEads):

Eads = Ebulk,mol+sub − (Ebulk,mol + Ebulk,sub) (4.2)

ΔEads = ∑

i

|EEAMads,i − EV ASPads,i | (4.3)

err = c1ΔEbulk,mol + c2ΔEbulk,sub + c3ΔEbulk,mol+sub + c4ΔEads

c1 + c2 + c3 + c4 (4.4)

The coefficients c1, c2, c3, and c4 are introduced as weighing factors. To drive the

parameters toward physically meaningful convergence point, the fitting is performed

in multiple stages:

(1) fit potentials for elemental Al, Co, and Ni,

(2) fit potentials for Al-Co-Ni,

(3) fit potentials for hydrocarbons,

(4) fit final potentials for hydrocarbon on Al-Co-Ni.

The fitted parameters from stage (1) are used as initial conditions in stage (2). The

fitted parameters from stage (2) and (3) are used as initial values in stage (4).

4.1 Stage 1: Aluminum, cobalt, and nickel

In stage (1), the elemental potentials for Al, Co, and Ni are fitted to elemental

bulk energies (top part of Tables 4.2-4.4). The potentials are also trained at various

54

pressures to ensure stability under compression/expansion and to yield resonable lat-

tice constants. In the calculations, different pressures are achieved by expanding or

compressing the relaxed ground state structures (i.e. Al(FCC) Co(HEX) Ni(FCC)),

namely at lattice constants from a = 0.95a0 to a = 1.1a0, where a0 is the equilib-

rium lattice constant at zero pressure. Training at various pressures increases the

transferability of the potentials due to a wider range of charge density covered. The

fitted EAM potentials (bottom part of Tables 4.2-4.4) are able to find the ground

state structure as well as the relative stability of each structure. Note that for pure

elements, the bulk energy represents the cohesive energy. Going from the most stable

structure to the least stable one, the EAM potentials correctly predict the following:

FCC → HEX → BCC → SC → SH → GRA → DIA (for Al and Ni) and HEX → FCC → BCC → SC → SH → GRA → DIA (for Co). The potentials also give accurate equilibrium lattice constants (middle part of Tables 4.2-4.4).

4.2 Stage 2: Al-Co-Ni potentials

Results from stage (1) are used as initial conditions in stage (2). The potential

for systems containing Al, Ni, and Co (AlCoNi-pot) are fitted to energies in bulk

structures and in slab configurations. The latter is intended to tune the AlCoNi-

pot at low charge density having different atomic environments from bulk. Slab

configurations are created from dB1, dH1, and dH2. Note that dB1, dH1, and dH2

are decagonal AlNiCo quasicrystal approximants. They are crystals with a large unit

cell which represent the short-ranged order of the quasicrystal. Approximants exist in

the vicinity of region of chemical compositions of their quasicrystalline counterparts.

The bulk unit cell of dB1, dH1, and dH2 contains 41, 25, and 50 atoms, respectively.

The unit cells consist of two well-defined layers. There are two possible terminations

for the bulk. The top(bottom) layer of the unit cell is labeled A(B), respectively. A

55

Table 4.2: (Top part) List of structures used to fit EAM potential for elemental aluminum. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3 · re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.199 -3.375 -3.368 -0.007 DIA 0.135 -2.847 -2.722 -0.126 FCC at a0·0.95 0.312 -3.347 -3.347 0.000 FCC at a0·0.96 0.289 -3.396 -3.390 -0.006 FCC at a0·0.97 0.267 -3.425 -3.422 -0.003 FCC at a0·0.98 0.248 -3.440 -3.444 0.004 FCC at a0·0.99 0.229 -3.443 -3.457 0.014 FCC at a0·1.00 0.212 -3.439 -3.462 0.023 FCC at a0·1.05 0.144 -3.356 -3.390 0.034 FCC at a0·1.10 0.098 -3.221 -3.221 0.000 GRA 0.151 -2.882 -2.958 0.075 HEX 0.211 -3.436 -3.426 -0.009 SC 0.167 -3.108 -3.097 -0.011 SH 0.178 -3.241 -3.241 0.000

a0(FCC) vasp = 4.04 A˚ a0(FCC) EAM = 4.00 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.026 7.182 2.700 33.64 3.038 3.012

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.078 0.121 0.495 0.896 1.381 2.200 Fx (eV) 0.000 -2.810 -3.106 -3.980 -3.879 -3.665 -3.074

56

Table 4.3: (Top part) List of structures used to fit EAM potential for elemental cobalt. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”HEX at a0 · c” indicates that the structure is HEX with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.665 -5.380 -5.377 -0.003 DIA 0.602 -4.115 -4.141 0.027 FCC 0.693 -5.439 -5.450 0.011 GRA 0.566 -4.596 -4.567 -0.029 HEX at a0·0.95 0.973 -5.213 -5.214 0.001 HEX at a0·0.96 0.910 -5.317 -5.305 -0.012 HEX at a0·0.97 0.849 -5.386 -5.375 -0.011 HEX at a0·0.98 0.795 -5.425 -5.425 -0.000 HEX at a0·0.99 0.744 -5.442 -5.457 0.016 HEX at a0·1.00 0.697 -5.440 -5.473 0.033 HEX at a0·1.05 0.501 -5.295 -5.383 0.088 HEX at a0·1.10 0.358 -5.098 -5.098 0.000 SC 0.569 -4.984 -4.708 -0.276 SH 0.608 -5.155 -4.966 -0.189

a0(HEX) vasp = 2.47 A˚ a0(HEX) EAM = 2.45 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.090 5.945 2.280 36.70 2.832 2.701

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.146 0.380 0.821 1.306 1.683 2.200 Fx (eV) 0.000 -3.374 -4.894 -5.517 -5.509 -5.008 -3.952

57

Table 4.4: (Top part) List of structures used to fit EAM potential for elemental nickel. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 ·c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.699 -4.928 -4.924 -0.004 DIA 0.500 -3.839 -3.839 -0.000 FCC at a0·0.95 0.980 -4.754 -4.783 0.029 FCC at a0·0.96 0.918 -4.870 -4.868 -0.002 FCC at a0·0.97 0.861 -4.941 -4.932 -0.009 FCC at a0·0.98 0.807 -4.980 -4.980 0.000 FCC at a0·0.99 0.757 -4.995 -5.010 0.015 FCC at a0·1.00 0.709 -4.993 -5.024 0.031 FCC at a0·1.05 0.507 -4.871 -4.955 0.084 FCC at a0·1.10 0.369 -4.687 -4.687 -0.000 GRA 0.543 -4.081 -4.109 0.028 HEX 0.706 -4.993 -4.998 0.005 SC 0.591 -4.464 -4.342 -0.122 SH 0.637 -4.682 -4.630 -0.052

a0(FCC) vasp = 3.49 A˚ a0(FCC) EAM = 3.46 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.126 5.447 2.150 39.17 2.839 2.717

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.223 0.336 0.566 1.028 1.355 2.200 Fx (eV) 0.000 -3.899 -4.373 -4.731 -5.280 -5.028 -4.172

58

slab is labeled A if the top layer is layer A, and vice versa. The unit cell for slab

configurations contains vacuum space of 12 A˚ in the vertical dimension to minimize

the interaction between unit cells due to periodic boundary conditions. In relaxing

the slabs, cell’s size/shape and atoms are allowed to relax except the bottom atoms.

In this way, the bottom atoms will be at the same coordinates as they were in the

bulk which will be more appropriate (than if bottom atoms are relaxed) when the

slabs are used as the substrates in the adsorption configurations (stage (4) of fitting).

The bottom part of Table 4.5 shows the parameters of AlCoNi-pot fitted to structures

listed in the top part of Table 4.5.

4.3 Stage 3: Hydrocarbon potentials

The EAM potentials for hydrocarbons (CH-pot) are fitted to the energies of isolated

molecules. The ab initio calculations are performed in a fairly large cubic cell with

vacuum size of larger than 10 A˚ to minimize the interaction between molecules due to

periodic boundary conditions implemented in VASP. The structures are fully relaxed.

In the beginning, all molecules listed in Table 4.1 are included in the fit. The result-

ing fitted parameters and the fitted energies are reported in Table 4.6. The results

show that EAM is fairly accurate for hydrocarbons, especially for alkanes. This is

interesting because EAM formalism was introduced for metals where bondings are

due to nonlocal electrons. Nevertheless, not all hydrocarbons can be fit simultane-

ously, as significant errors are found in some molecules, e.g C3H4-allene, propylene,

cyclopropane, cyclopropyne, 1-butene, and benzene. For our simulations, presented

in Chapter 5, a more accurate fit for alkanes and benzene is needed. Table 4.7 shows

the EAM potentials fitted to alkanes and benzene.

.

59

Table 4.5: (Top part) List of structures used to fit EAM potential for Al-Co-Ni systems. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV) and the first knot at (0,0) is assumed. The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

cI112 bulk -4.828 -4.828 -0.000 cI128 bulk -4.916 -4.915 -0.000 dB1 bulk -4.398 -4.395 -0.003 dH1 bulk -4.483 -4.516 0.032 dH2 bulk -4.441 -4.480 0.038 hP28 bulk -4.484 -4.490 0.006 mC102 bulk -4.520 -4.517 -0.003 mC28 bulk -4.492 -4.509 0.017 mC32 bulk -4.323 -4.322 -0.001 mP22 bulk -4.121 -4.111 -0.010 oI96 bulk -4.334 -4.350 0.016 oP16 bulk -4.292 -4.300 0.007 dB1 slab3A -3.816 -3.853 0.037 dB1 slab3B -3.913 -3.921 0.008 dB1 slab4A -4.105 -4.096 -0.009 dB1 slab4B -4.015 -4.015 0.000 dH1 slab3A -4.058 -4.043 -0.015 dH1 slab3B -4.056 -4.028 -0.029 dH1 slab4A -4.153 -4.155 0.003 dH1 slab4B -4.158 -4.153 -0.005 dH2 slab4A -4.144 -4.143 -0.001 dH2 slab4B -3.980 -3.980 0.000

Al Co Ni ρe (A˚−3) 0.026 0.090 0.126 β 7.182 5.945 5.447 re (A˚) 2.70 2.28 2.15 D (eV) 0.034 0.037 0.039 α (A˚−1) 1.833 3.350 3.258 r0 (A˚) 3.009 2.744 2.735 Z/ZAl 1 0.862 0.521 Z/ZCo 1 0.975 knot2 0.086,-2.758 0.235,-3.249 0.285,-3.772 knot3 0.120,-2.974 0.383,-4.834 0.341,-4.449 knot4 0.517,-4.688 0.864,-5.579 0.604,-4.931 knot5 0.868,-3.777 1.321,-5.420 1.050,-5.279 knot6 1.365,-3.736 1.705,-5.061 1.355,-5.160 knot7 2.200,-3.151 2.200,-4.090 2.200,-4.196

60

Table 4.6: (Top part) List of structures used to fit EAM potential for hydrocarbons. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

methane -4.807 -4.806 -0.001 ethane syntactic -5.076 -5.061 -0.015 ethane isotactic -5.068 -5.047 -0.021 C2H2 -5.735 -5.736 0.001 C2H4 -5.382 -5.328 -0.054 propane -5.189 -5.183 -0.004 C3H4 allene -5.751 -5.342 -0.409 propylene -5.188 -5.002 -0.186 propyne -5.679 -5.687 0.008 cyclopropane -5.308 -4.850 -0.459 cyclopropene -5.366 -5.273 -0.094 cyclopropyne -5.340 -5.477 0.137 isobutane -5.249 -5.253 0.004 1-butene -5.250 -5.128 -0.122 1-butyne -5.648 -5.630 -0.018 cyclobutane -3.899 -3.899 0.000 methylcyclopropane -5.404 -5.410 0.006 pentane -5.291 -5.297 0.006 cyclopentane -5.024 -5.047 0.022 hexane -5.321 -5.329 0.008 benzene -6.136 -6.334 0.198 heptane -5.343 -5.352 0.009 octane -5.358 -5.370 0.012 nonane -5.368 -5.375 0.007 decane -5.383 -5.381 -0.002 undecane -5.392 -5.391 -0.001 dodecane -5.400 -5.413 0.013

ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

C 0.932 5.473 1.24 0.123 3.116 1.489 H 1.639 2.798 0.74 0.211 4.296 1.125

ZH/ZC = 0.880 knot1 knot2 knot3 knot4 knot5 knot6 knot7

C ρ 0.000 0.334 1.103 1.357 1.635 1.870 2.200 Fx 0.000 -3.851 -6.067 -8.011 -7.771 -8.17 -7.352

H ρ 0.000 1.071 1.528 1.773 2.008 2.200 - Fx 0.000 -2.997 -3.482 -4.189 -3.764 -3.409

61

Table 4.7: (Top part) List of structures used to fit EAM potential for alkanes and benzene. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

methane -4.806 -4.806 -0.000 ethane syntactic -5.060 -5.061 0.001 propane -5.182 -5.183 0.001 isobutane -5.254 -5.253 -0.001 pentane -5.298 -5.297 -0.001 hexane -5.329 -5.329 0.000 heptane -5.349 -5.352 0.003 octane -5.367 -5.370 0.003 nonane -5.380 -5.375 -0.005 decane -5.393 -5.381 -0.012 undecane -5.402 -5.391 -0.011 dodecane -5.411 -5.413 0.002 benzene -6.334 -6.334 0.000

ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

C 0.932 5.473 1.24 0.116 2.793 1.477 H 1.639 2.798 0.74 0.225 3.154 1.077

ZH/ZC = 0.533 C H

knot1 0,0 0,0 knot2 0.366,-3.793 1.040,-3.543 knot3 0.509,-5.454 1.534,-4.003 knot4 0.761,-6.130 1.801,-3.656 knot5 1.012,-7.611 1.911,-3.874 knot6 1.834,-8.037 2.200,-3.736 knot7 2.200,-7.288 -

62

4.4 Stage 4: Hydrocarbon on Al-Co-Ni

The final EAM potentials for hydrocarbon adsorption on Al-Co-Ni are fitted to the

adsorption energies of a hydrocarbon on a substrate. As the substrates, dB1-slab4A

and dH1-slab4A are used for methane, and dH2-slab4A are used for larger molecules.

The initial conditions are taken from Tables 4.5 and 4.7 and the fitting is performed

in the adsorption mode as explained previously. During the fitting, it is found that

the adsorption energies are more difficult to fit than those of molecules, substrates,

and molecule-on-substrate. This originates from the different nature of bonding. As

discussed later in Chapter 5, alkanes do not make strong chemical bonds with the Al-

Co-Ni. They are only physically adsorbed. Whereas the bondings within a molecule

and within the substrate are strong. Weighing factors of c1 = c2 = c3 = 1 and c4 > 1

are used to prioritize adsorption energies. Typically, values of c4 < 4 are used to get

the final fit. Increasing c4 beyond 4 does not considerably improve the fit, indicating

the limitations inherent with EAM formalism. The fitted structures and energies are

shown in Table 4.9 while the fitted parameters are tabulated in the Table 4.8.

63

Table 4.8: Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA+ZAφB/ZB)/2. Fitting structures are given in Table 4.9

C H Al Co Ni ρe (A˚−3) 0.932 1.639 0.026 0.09 0.126 β 5.473 2.798 7.182 5.945 5.447 re (A˚) 1.24 0.74 2.7 2.28 2.15 D (eV) 0.116 0.225 0.034 0.037 0.039 α (A˚−1) 2.786 3.156 1.740 3.636 3.198 r0 (A˚) 1.478 1.076 2.938 2.730 2.731 Z/ZC 1 0.532 0.989 1.115 1.258 Z/ZH 1 0.857 1.140 0.850 Z/ZAl 1 0.370 0.498 Z/ZCo 1 0.970 knot1 0,0 0,0 0,0 0,0 0,0 knot2 0.353,-3.860 1.037,-3.475 0.079,-2.816 0.217,-3.250 0.252,-3.932 knot3 0.519,-5.429 1.527,-3.974 0.108,-3.056 0.358,-5.176 0.327,-4.425 knot4 0.746,-6.254 1.797,-3.797 0.507,-4.214 0.774,-5.376 0.574,-4.849 knot5 1.033,-7.443 1.899,-3.744 0.893,-3.618 1.439,-5.139 1.032,-5.194 knot6 1.824,-7.988 2.200,-3.674 1.408,-3.681 1.635,-4.834 1.354,-5.065 knot7 2.200,-7.264 2.200,-2.911 2.200,-3.577 2.200,-4.130

64

Table 4.9: Fitted energies calculated using EAM parameters in Table 4.8. Methane up represents methane with one H below C and three H above C. Methane down is inverse of methane up. The unit for energy is eV/atom except for the adsorption energy which is in eV/molecule.

structure EEAM EV ASP ΔE Molecule energy (eV/atom): methane -4.805 -4.805 0.000 ethane syntactic -5.078 -5.061 -0.017 propane -5.189 -5.183 -0.006 butane -5.252 -5.252 0.000 pentane -5.293 -5.297 0.004 hexane -5.322 -5.328 0.007 benzene -6.294 -6.334 0.040 Substrate energy (eV/atom): dB1 slab4A -4.132 -4.096 -0.036 dH1 slab4A -4.141 -4.155 0.014 dH2 slab4A -4.135 -4.143 0.008 Molecule and substrate energy (eV/atom): methane on dB1 slab4A -4.173 -4.139 -0.034 methane up on dH1 slab4A -4.204 -4.219 0.015 methane down on dH1 slab4A -4.205 -4.219 0.014 ethane syntactic on dH2 slab4A -4.207 -4.214 0.006 propane on dH2 slab4A -4.242 -4.249 0.007 butane on dH2 slab4A -4.276 -4.283 0.008 pentane on dH2 slab4A -4.306 -4.315 0.009 hexane on dH2 slab4A -4.337 -4.346 0.009 benzene on dH2 slab4A -4.381 -4.385 0.004 Adsorption energy (eV/molecule): methane on dB1 slab4A -0.225 -0.238 0.013 methane up on dH1 slab4A -0.238 -0.243 0.005 methane down on dH1 slab4A -0.304 -0.257 -0.047 ethane syntactic on dH2 slab4A -0.348 -0.253 -0.095 propane on dH2 slab4A -0.458 -0.328 -0.131 butane on dH2 slab4A -0.439 -0.439 0.000 pentane on dH2 slab4A -0.465 -0.527 0.062 hexane on dH2 slab4A -0.520 -0.589 0.069 benzene on dH2 slab4A -0.790 -0.791 0.001

65

Chapter 5

Hydrocarbon adsorptions on d-AlNiCo

The low friction properties of quasicrystals in ambient conditions coupled with their

high hardness and oxidation resistance led to the development of applications of

quasicrystal coatings, for instance on machine parts, cutting blades, and non-stick

frying pans [5]. In machine parts, hydrocarbons are commonly used as a lubricant.

Superlubricity is the name given to the phenomenon in which two parallel single

crystal surfaces slide over each other with vanishingly small friction because their

structures are incommensurate. This phenomenon was proposed in the early 1990’s

[12] and experiment evidence for this effect has been seen in studies of mica sliding

on mica [123], W(110) on Si(100) [124], Ni(100) on Ni(100) [125], and tungsten on

graphite [13]. This effect is also expected in quasicrystals due to their aperiodic

structures at all length scales. Indeed, quasicrystal surfaces were observed to have

low friction not long after they were first discovered [126], but pinning down the exact

origin of the low friction has been elusive.

Recent experiments in ultra high vacuum (UHV) have demonstrated a frictional

dependence on aperiodicity for decagonal Al-Ni-Co quasicrystal (d-AlNiCo) against

thiol-passivated titanium-nitride tip [11]. In coating applications, it is expected that

even if superlubricity exists between moving parts, some additional lubricant would

still be needed to counter the macroscopic frictions due to grain boundaries, asperities,

and other defects in the surfaces of the moving parts. Some of the requirements of

the lubricant in such a situation are that it must wet the surface and that it must

not remove or reduce the superlubricity. Therefore it is desirable to have a good

understanding of how gases, hydrocarbons in particular, interact with quasicrystal

66

surfaces.

Very little is currently known about the interaction of hydrocarbons, their struc-

tures and growth on alloy or quasicrystalline surfaces. Some earlier experiments using

Fourier transform infrared spectroscopy (FTIR) and low energy electron diffraction

(LEED) suggest that on the 5-fold surface of Al-Pd-Mn, carbon monoxide (CO) does

not adsorb at 100 K, benzene adsorb at 100 K with possibly commensurate or dis-

ordered structure (LEED pattern unchanged) [127]. The same experiments on the

10-fold surface of d-AlNiCo show that CO bonds to the Ni sites at > 132 K, no struc-

ture reported, while there is no experiment available for benzene on Al-Ni-Co [127].

Later, scaning tunneling microscopy (STM) experiments on benzene adsorption on

Al-Pd-Mn show that the adsorbed benzene has a disordered structure [128]. In this

chapter, we report the simulation results of small hydrocarbons adsorb on the 10-fold

surface of decagonal Al73Ni10Co17 [15, 16], namely for methane, propane, hexane,

octane, and benzene.

5.1 Model

The simulations are performed within the framework of grand canonical ensemble

using Monte Carlo (GCMC) method as previously described in Section 2.5. The sim-

ulation cell is tetragonal. We take a square section of the surface, A, of side 5.12 nm,

to be the (x, y) part of the unit cell in the simulation, for which we assume periodic

boundary conditions along the basal directions. The coordinates of substrate atoms

are taken from Ref. [100]. The interaction potentials are modeled as the following.

The intermolecular interactions (adsorbate-adsorbate) are calculated as a sum of

pair interactions between atoms. For methane-methane [129, 130] Buckingham-type

potentials are used:

V (r) = Ae−Br − C/r6 (5.1)

67

Buckingham potentials also used to parametrize the benzene-benzene interactions

[131, 132]. For linear alkane-alkane, Morse-type potentials are used:

V (r) = −A(1− (1− e−B(r−C))2) (5.2)

The parameters for these potentials are summarized in Table 5.1. EAM potentials

generated in Chapter 4 are used for the rest of the interactions, namely the in-

tramolecular, adsorbate-substrate (C-Al, C-Co, C-Ni, H-Al, H-Co, H-Ni), and be-

tween substrate atoms (Al-Al, Al-Co, Al-Ni, Co-Ni). As previously mentioned in

Section 4.4, the ab initio calculations show that alkanes and benzene do not dissoci-

ate on dB1, dH1, and dH2 (these are decagonal Al-Ni-Co approximants). In these

systems, the surface of the substrate does not undergo any considerable relaxation

upon the adsorption of the molecules. Therefore, as a first approximation, in our

GCMC simulations, the substrate and the molecules are considered as rigid. How-

ever, molecules are allowed to explore all rotational degrees of freedom to achieve the

equilibrium configurations.

5.2 Adsorption potentials

Figure 5.1 displays the minima of the adsorption potential for methane (a), propane

(b), hexane (c), octane (d), and benzene (e), generated by minimizing the adsorption

potential of a molecule on d-AlNiCo with respect to z (Equation 3.1) and all rota-

tional degrees of freedom. The average adsorption energies are 221 (methane), 374

(propane), 620 (hexane), 794 (octane), and 931 (benzene), given in meV/molecule.

The figure shows the distribution of binding sites (dark spots) for the molecule.

Methane, propane, and benzene are small enough to follow the local atomic envi-

ronments of the substrate, whereas hexane and octane show considerable smearing

due to their large size. The location of dark spots in methane is similar to that

68

Table 5.1: Parameter values for the adsorbate-adsorbate interactions used for hydrocar- bon adsorption on a decagonal Al-Ni-Co. Intermolecular energies are calculated as a sum of pair interactions. For methane-methane, the C-H is taken as the geometrical mean for parameter A and as the arithmetic mean for parameters B and C.

A B C ref. (eV) (A˚−1) (A˚6)

methane C-C 82.132 2.693 449.53 [130] V (r) = Ae−Br − C/r6 C-H 66.217 2.892 167.51

H-H 53.381 3.105 62.42 [129] benzene C-C 11527.700 3.909 524 [131, 132] V (r) = Ae−Br − C/r6 C-H 348.518 3.703 75 [131, 132]

H-H 127.447 3.746 39 [131, 132] (meV) (A˚−1) (A˚)

alkane C-C 6.984 1.2655 4.1844 [133] V (r) = −A(1− (1− x)2) C-H 23.921 2.2744 2.544 [133] x = e−B(r−C) H-H 0.002 1.255 6.1543 [133]

[129] Tsuzuki S, et al 1993 J. Mol. Struct. 280 273 [130] Tsuzuki S, et al 1994 J. Phys. Chem. 98 1830 [131] Califano S, et al 1979 Chem. Phys. Lett. 64 491 [132] Chelli R, et al 2001 Phys. Chem. Chem. Phys. 3 2803 [133] Jalkanen J-P, et al 2002 J. Chem. Phys. 116 1303

69

of propane. However, it is interesting to see that the dark spots in methane and

propane become the bright ones in benzene. To study more, the topview of the

substrate atoms is depicted in panel (f) of Figure 5.1. The legend for the atoms is:

Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red),

Ni-toplayer (green), and Ni-otherlayers (blue). In methane and propane, dark spots

occur when the center of the molecule is located on top of black-Al (5 gray-Al on

toplayer forming a pentagon). In benzene, this location gives a bright spot (less bind-

ing). Except the binding site at the center of the figure, strong binding sites (dark

spots) in benzene occur when the center of benzene is located on top of gray-Al (on

the top layer, the pentagon consists of 2 Al and 3 Ni, or 3 Al and 2 Ni).

5.3 Molecule orientations

First we study the orientations of a molecule as it is adsorbed on d-AlNiCo. During

the calculation of minima of the adsorption potential (Section 5.2, the orientation of

the molecule is recorded when a minimum is achieved. It is found that for methane,

the rotational ground state is degenerate indicating its spherical nature. Ab initio

calculations of methane dimers indicate that the interaction energy within the dimer

depends on the relative orientations of the two [129, 130]. Therefore, the degeneracy

observed is due to the limitation of the EAM model. For linear alkanes, we can

define θ as the angle between the substrate’s xy-plane and the main axis of the

alkanes. Again, θ is recorded when an adsorption minimum is achieved. We find that

θ decreases with increasing alkane chain, e.g. propane (θ = 10◦), hexane (θ = 5◦),

and octane (θ ∼ 0◦). For benzene, θ would be the angle between the molecule’s plane and substrate’s xy-plane, and it is θ ∼ 0◦. Therefore, we conclude that alkanes and benzene prefer most contact with d-AlNiCo.

70

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-180

-200

-220

-240

-260

-300

1

-280

-221

-300

-350

-400

-450

-500

-550

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

1 -374

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-550

-600

-650

-700

-800

1

-750

-620

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-750

-800

-850

-900

1

-950

-700

-794

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-800

-900

-1100

1

-1000

-931

a) b)

c) d)

e) f)

Figure 5.1: (color online). (a-e) Adsorption potential map, calculated by minimizing the adsorption potential of one molecule on a decagonal Al-Ni-Co along z direction and all rotational degrees of freedom at every coordinates (x, y). Red numbers represent the average value of the adsorption energies. (f) Top view of the decagonal Al-Ni-Co substrate 51.2x51.2 A˚2: Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red), Ni-toplayer (green), and Ni-otherlayers (blue).

71

5.4 Adsorption isotherms

To use quasicrystals as low-friction coatings in conjunction with oil lubricants, the

lubricants must be able to spread well on the quasicrystals. Lubricants consist mostly

of alkanes. The wetting of d-AlNiCo by alkanes and benzene is demonstrated in

Figure 5.2. In the figure, the number of molecules adsorbed on the substrate per area

is plotted as a function of pressure at various temperatures. The results for methane,

propane, hexane, octane, and benzene are shown. The simulated temperatures are:

for methane T = 68, 85, 136, 185 K, for propane T = 80, 127, 245, 365 K, for hexane

T = 134, 170, 267, 450 K, for octane T = 162, 210, 324, 450, 565 K, for benzene T =

209, 270, 418, 555 K. Note that T = 450 K represents a typical temperature of the

inner wall of cylinder in regular car engines [134]. In the isotherms, even though

the step corresponding to the adsorption of the first layer is well observed, steps

corresponding to the second and further layers are not evident for methane, propane,

and benzene, and barely visible for hexane and octane. The second layer condensation

occurs near the bulk condensation and extends a short range of pressure, nevertheless

multilayer adsorptions can be seen at higher pressures (point d) as shown in the inset

of each panel in which the distribution of adsorbed molecules is plotted against the

z direction.

5.5 Density profiles

Methane on d-AlNiCo. Figure 5.3 shows the density plots for methane at two

different coverages, corresponding to points ”a” and ”c” on the 68 K isotherm in

Figure 5.2.a. At submonolayer regime, methanes occupy the strong binding sites

on the surface. At monolayer coverage, the ordering is 5-fold commensurate with

the substrate, also indicated by the Fourier transform (panel c), and there is no

transition to 6-fold. The evidence that there is no such transition is shown more

72

10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

14

16

18

20

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

a

b c

d

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5 Z (nm)

Xe CH4

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a

b c

d

ρ N (m

ol ec

ul es

/n m

2 )

P (atm)

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5 Z (nm)

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a b c

d

Z (nm)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a b c

d

Z (nm)

ρ Z (a

rb .)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

0 0.5 1 1.5 2 2.5

10-25 10-20 10-15 10-10 10-5 1 0

2

4

6

8

10

12

14

16

18

20

a

b c

d

0 1 2 3 Z (nm)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

ρ Z (a

rb .)

c) d)

e)

a) b)

Figure 5.2: (color online). Isothermal adsorption densities of hydrocarbons on a decago- nal Al-Ni-Co: (a) methane (from left to right T = 68, 85, 136, 185 K), (b) propane (T = 80, 127, 245, 365 K), (c) hexane (T = 134, 170, 267, 450 K), (d) octane (T = 162, 210, 324, 450, 565 K), and (e) benzene (T = 209, 270, 418, 555 K). The inset in each figure is the density along z direction at pressure corresponding to point d. Xenon (red) is plotted in panel (a) for comparison.

73

10 -15

10 -10

10 -5 1

0

5

10

15

ρN (m olecules/nm

2)

P (atm)

0

0.2

0.4

0.6

0.8

1

ρ 5 -6

a b

c d

Figure 5.3: (a) and (b) Calculated density of methane adsorbed on a decagonal Al-Ni-Co at pressures corresponding to points ”a” and ”c” of the 68 K isotherm shown in Figure 5.2.a, respectively. (c) Fourier transform of the density plot shown in (b), consistent with 5-fold ordering of the methane near monolayer completion. (d) Order parameter (left axis, as calculated in Equation 3.14) as a function of pressure for the 68 K isotherm (right axis), indicating no sharp transition to 6-fold ordering.

clearly in panel d, where the order parameter (left axis, as calculated in Equation

3.14) does not present any sharp transition as seen in the case of xenon adsorption on

the same substrate (Section 3.2.3). The lack of 5-fold to 6-fold transition is consistent

with our proposed rule based on rare gases (Section 3.2.6), that the size of the rare

gas must be at least as large as Xe on this quasicrystalline surface for the transition to

occur [135]. Note that the size of methane relative to xenon (ratio of Lennard-Jones

σ parameters) is 0.8 [69].

Propane, Hexane, and Octane on d-AlNiCo. At submonolayer coverages,

propane, hexane, and octane do not show any clear evidence of binding to specific sites

of the surface even though molecules tend to bind in the center of the surface with

74

the most attractive site. Figure 5.4 shows the density profiles for these hydrocarbons

at pressure corresponding to the first layer completion at various temperatures. The

simulated temperatures have been indicated in Section 5.4. At monolayer coverage,

propane adsorbs in a disordered fashion at all temperatures, whereas hexane and

octane tend to form close-packed structures as suggested by domains with stripe

feature. As comparisons, the crystal structures of solid propane at 30 K is monoclinic

(space group #11, P21/m) [136], hexane is triclinic (space group #2, P1¯) at 90 K

[136] and at 158 K [137], octane is triclinic (space group #2, P1¯) at 90 K [136]

and at 213 K [138]. The 2-dimensional structures of these hydrocarbons are close-

packed structures as charactereized by stripe structures with 1 and 2 molecules per

unit cell for even- and odd-alkanes, respectively [136]. In general, even-alkanes form

triclinic (6 ≤ NC ≤ 26), monoclinic (28 ≤ NC ≤ 36), or orthorombic (38 ≤ NC) with decreasing packing density [136, 137, 138, 139], where NC being the number of

carbon atoms. Whereas for odd-alkanes form triclinic (7 ≤ NC ≤ 9) or orthorombic (11 ≤ NC) [136, 139, 140] also with decreasing packing density for longer chain. Note that the even-alkanes have higher packing density than odd-alkanes.

Benzene on d-AlNiCo. As in the case of methane, at submonolayer coverage,

benzene preferentially adsorbs at sites offering the strongest binding at all simulated

temperatures. At pressure near to first layer completion, the density profiles show a

temperature-dependence as plotted in Figure 5.5 for 209 K, 270 K, and 418 K. The

structures are more clearly charaterized by the plotting the geometrical center of

density as shown in the middle row of the figure. At T = 209 K, pentagonal ordering

is observed. As the temperature is increased, a mixture of 5-fold and 6-fold ordering

is seen, e.g. at T = 270 K. At higher temperature, T = 418 K, 6-fold structure

dominates the ordering of the monolayer as confirmed by the Fourier analysis of the

density showing hexagonal spots characteristic of triangular lattice (bottom row, last

75

267 K170 K 450 K

324 K210 K 450 K

hexane

octane

80 K 127 K 245 Kpropane

Figure 5.4: Calculated density of propane, hexane, and octane adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion. Propane forms a disordered structures, whereas hexane and octane tend to form close-packed structures indicated by stripe features with increasing order for longer chain.

76

column in Figure 5.5). The crystal structure of bulk benzene has been determined

at 4.2 K and 270.15 K to be orthorombic (space group #61, Pbca) with 4 molecules

per unit cell [141].

The evolution of the density profile over temperature from being 5-fold to 6-fold

can be studied more clearly by plotting the order parameter, ρ5−6 = N5/(N5 + N6)

(Nn denotes the number of molecules having n nearest neighbors as defined in Section

3.1.6) as a function of T as shown in Figure 5.6 (left axis). The plot is taken at a

constant pressure of 0.01 atm corresponding to point c in Figure 5.2 panel e. The

adsorption isobar representing the number of adsorbed molecules N as a function of

T is plotted in right axis of Figure 5.6. Three dashed lines corresponding to T =209,

270, and 418 K, whose adsorption isotherms are plotted earlier in Figure 5.2 panel e,

are added for guidance in comparing the density profile to the order parameter value

at each of these temperatures. We observe the following trends:

• (region 1) T < 260 K, 0.5 < ρ5−6 → 5-fold ordering dominates, N = 44.

• (region 2) 260 ≤ T ≤ 280 K, 0.3 ≤ ρ5−6 ≤ 0.5 → 5-fold becomes mostly 6-fold, N = 44.

• (region 3) 280 ≤ T ≤ 370 K, ρ5−6 = 0.3 → 6-fold ordering dominates, N = 44.

• (region 4) 370 ≤ T ≤ 390 K, 0.1 ≤ ρ5−6 ≤ 0.3→ transition to 6-fold ordering, 44 ≤ N ≤ 46.

• (region 5) T > 390 K, ρ5−6 increases from 0.1 → 6-fold ordering weakens, N decreases from 46.

The highest 6-fold ordering occurs at T = 390 K which is mediated by a gain of 2

additional molecules adsorbed on the substrate. Beyong this temperature, thermal

77

209 K 270 K 418 K

Figure 5.5: (Top row) Calculated density of benzene adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion for 209 K, 270 K, and 418 K. (middle row) Density profile of the geometrical center of density shown in the top row. (bottom row) Fourier transform of the density plot shown in the middle row, showing 5-fold ordering at 209 K, mixture of 5-fold and 6-fold structures at 270 K, and mostly 6-fold features at 418 K.

78

200 250 300 350 400 450 500

0.1

0.2

0.3

0.4

0.5

0.6

T (K)

ρ 5 -6

38

40

42

44

46 N

m olecules

Figure 5.6: (left axis) Order parameter ρ5−6 = N5/(N5+N6) (Nn denotes the number of molecules having n nearest neighbors) as a function of temperature at 0.01 atm of pressure. (right axis) Adsorption isobar showing the number of adsorbed molecules as a function of temperature. Verticel dashed lines correspond to T = 209, 270, and 418 K whose density profiles are plotted in Figure 5.5.

energy causes reduction of N resulting in less ordered 6-fold structure as T increases.

Nevertheless, the 6-fold ordering are still well resolved up to T = 418 K as shown

before in Figure 5.5.

5.6 Summary

In this chapter, we have presented our studies on methane, propane, hexane, octane,

and benzene adsorption on decagonal Al-Ni-Co quasicrystal. All of these hydro-

carbons form very well defined step corresponding to the first layer condensation.

Eventhough multilayer formation is evident from the density distribution along ver-

tical dimension, the step corresponding to the condensation of the second and further

layers can not be observed clearly in the isotherm due narrow range of pressure for

these layers to form before bulk condensation occurs. Monolayer of methane has

79

been determined to be pentagonal commensurate with the substrate for 68 K ≤ T ≤ 136 K. Monolayer of propane shows disordered structures for 80 K ≤ T ≤ 245 K. Monolayers of hexane (134 K≤ T ≤ 450 K) and octane (162 K ≤ T ≤ 450 K) form 2- dimensional close-packed structures characterized by stripe ordering consistent with

their bulk crystal structures. Benzene monolayer is pentagonal at T ≤ 260 K, which transforms into 6-fold structure at T ≥ 280 K with the highest 6-fold ordering occurs at T = 390 K. Beyong 390 K, thermal energy causes fewer adsorbed benzene and the

6-fold ordering starts to deterioates. Nevertheless, the 6-fold ordering are still well

observed up to T = 418 K.

80

Chapter 6

Embedded-atom potentials for selected

pure elements in the periodic table

Embedded-atom method (EAM) formalism provides only the skeleton of calculat-

ing the total electronic energy for a given charge density. It does not specify the

parametrization of the three functionals: atomic charge density, embedding en-

ergy, and pair interaction. Many different parametrizations have been introduced

[54, 55, 58, 59, 59, 65, 142, 143, 144, 145]. This arises inconveniences when one needs

to use different potentials for different elements. Steps toward universal EAM poten-

tials are needed. Recently, a consistent set of atomic charge density for all elements

in the periodic table suitable for EAM potentials has been developed [64]. The set

is calculated by spherically averaging the atomic charge density of the solution of

Hartree-Fock (HF) equations (Equation 2.19) of an isolated atom. The exchange

term in the HF equation contains an atom-specific parameter which is adjusted to

reproduce the experimental first ionization energy, hence are semiempirical. In this

chapter we use these charge densities to generate a consistent set of EAM potentials

for pure elements in the periodic table.

The parametrizations are given in Section 2.3. For each element, the potential is

fitted to the ab initio cohesive energies in body-centered cubic (BCC), face-centered

cubic (FCC), hexagonal close-packed (HEX), and diamond structure (DIA). In ad-

dition, energies in the ground state structure at different pressures are included in

the fitting procedure to ensure the mechanical stability of the potentials. Energies at

various pressures are achieved by expanding or compressing the equillibrium lattice

81

constant a by a factor from 0.90 to 1.16. Within an accuracy of 30 meV, all the

EAM potentials successfully predict the correct ground state structure with respect

to those that are not included in the fitting step, namely graphite (GRA), simple

cubic (SC), and simple hexagonal (SH) structures. The fitted parameters are sum-

marized in Tables 6.1-6.4. Tables 6.5 and 6.6 list the ground state structures of

the elements, range of charge density covered in the fit, and the equillibrium lattice

constants calculated by the ab initio method as well as by using the fitted EAM

parameters. Literature value of the lattice constants is also included as comparisons.

Overall, the EAM potentials predict the lattice constants with within 0.5 A˚ of the

literature value except for tellurium. Indeed, elements in the V-A, VI-A, and VII-A

colums of periodic table tend to form complex structures with large unit cells and

more EAM parameters might be needed to fit these structures better. Manganese is

excluded also because of its complex cubic structure with 58 atoms/cell. Noble gases

are excluded because they are already well described by simple pair potentials. Ele-

ments in the Actinide series are excluded because they do not have a stable structure

due to their radioactivity.

82

Table 6.1: Fitted parameters for the charge density and pair interaction functionals of the EAM potentials for pure elements, continued in Table 6.2. The fitting structures are given in Tables 6.5 and 6.6. The parameters for the embedding functionals are given in Tables 6.3 and 6.4.

Z Struct. ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

3 Li 0.038 4.813 2.670 0.010 3.015 2.422 4 Be 0.031 7.174 2.450 0.016 3.317 2.216 6 C 0.932 5.473 1.240 0.115 3.152 1.441 11 Na 0.021 5.552 3.080 0.011 5.800 3.500 12 Mg 0.001 10.030 3.890 0.012 2.807 3.110 13 Al 0.026 7.182 2.700 0.034 3.038 3.012 14 Si 0.072 7.171 2.250 0.043 4.115 1.601 19 K 0.008 6.412 3.920 0.015 2.926 4.088 20 Ca 0.002 9.719 4.280 0.020 2.940 3.646 21 Sc 0.058 6.053 2.700 0.021 2.264 3.210 22 Ti 0.287 4.333 1.940 0.022 2.282 2.850 23 V 0.394 4.025 1.780 0.023 2.501 2.615 24 Cr 0.297 4.538 1.680 0.024 2.519 2.494 26 Fe 0.192 5.214 2.000 0.026 5.747 2.412 27 Co 0.090 5.945 2.280 0.037 2.832 2.701 28 Ni 0.126 5.447 2.150 0.039 2.839 2.717 29 Cu 0.052 5.864 2.220 0.029 2.589 2.553 30 Zn 0.0005 14.630 4.800 0.030 2.655 2.560 32 Ge 0.042 7.419 2.440 0.045 2.659 2.001 37 Rb 0.006 6.788 4.210 0.037 3.599 4.694 38 Sr 0.002 9.614 4.450 0.038 2.926 3.299 39 Y 0.037 6.593 3.000 0.039 2.123 3.300 40 Zr 0.158 5.287 2.300 0.040 2.132 3.230 41 Nb 0.145 5.556 2.080 0.041 2.347 2.858 42 Mo 0.201 5.621 1.940 0.042 2.153 2.728 43 Tc 0.365 4.968 1.880 0.043 2.559 2.640 44 Ru 0.047 6.360 2.400 0.044 2.702 2.700 45 Rh 0.020 6.981 2.680 0.045 2.609 2.687 46 Pd 0.017 8.389 2.500 0.036 2.719 2.791 47 Ag 0.028 6.619 2.530 0.027 2.982 3.055 48 Cd 0.0005 13.340 4.500 0.048 2.284 3.170 52 Te 0.047 8.157 2.560 0.052 2.477 2.245 56 Ba 0.001 10.020 4.900 0.056 2.815 3.847 57 La 0.009 8.106 3.890 0.057 2.518 3.350

83

Table 6.2: continuation of Table 6.1.

58 Ce 0.013 7.575 3.682 0.058 2.488 2.949 59 Pr 0.016 7.342 3.646 0.059 2.502 2.970 60 Nd 0.015 7.361 3.632 0.060 2.543 3.548 63 Eu 0.016 7.306 3.542 0.033 2.596 3.300 64 Gd 0.015 7.724 3.470 0.064 2.549 3.240 65 Tb 0.017 7.323 3.464 0.065 2.527 3.300 66 Dy 0.017 7.303 3.420 0.066 2.512 3.240 67 Ho 0.018 7.323 3.392 0.067 2.532 3.180 68 Er 0.019 7.271 3.346 0.068 2.560 3.260 69 Tm 0.020 7.284 3.320 0.069 2.597 3.140 70 Yb 0.020 7.250 3.274 0.070 2.542 3.522 71 Lu 0.023 6.972 3.342 0.071 2.596 3.210 72 Hf 0.138 5.581 2.350 0.072 2.454 2.900 73 Ta 0.142 6.016 2.300 0.073 2.026 2.766 74 W 0.137 6.176 2.270 0.074 2.350 2.630 75 Re 0.242 3.686 2.040 0.075 3.055 2.760 76 Os 0.029 7.771 2.760 0.076 2.696 2.540 77 Ir 0.244 6.055 1.960 0.077 3.142 2.715 78 Pt 0.021 7.539 2.500 0.078 2.919 2.772 79 Au 0.024 7.553 2.470 0.079 2.917 2.885 81 Tl 0.004 8.850 3.500 0.081 2.576 3.300 82 Pb 0.002 8.368 2.930 0.082 2.935 3.242

84

Table 6.3: Fitted knots of cubic spline of the embedding functionals for the EAM po- tentials of pure elements, continued in Table 6.4. The first knot at (0,0) is assumed. The fitting structures are given in Tables 6.5 and 6.6.

knot2 knot3 knot4 knot5 knot6 knot7 (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV)

Li 0.048,-1.058 0.119,-1.52850 0.190,-1.54750 0.246,-1.607 0.291,-1.554 0.469,-1.469 Be 0.178,-3.079 0.228,-3.241 0.423,-3.509 0.731,-3.647 1.057,-3.588 1.670,-3.324 C 0.301,-4.967 0.344,-5.905 0.476,-6.986 0.861,-7.490 1.419,-7.264 2.389,-5.970 Na 0.014,-0.725 0.029,-0.9756 0.062,-1.059 0.082,-1.057 0.135,-1.059 0.181,-1.424 Mg 0.015,-0.939 0.033,-1.292 0.062,-1.414 0.096,-1.413 0.133,-1.356 0.242,-1.175 Al 0.078,-2.811 0.121,-3.106 0.495,-3.980 0.897,-3.879 1.381,-3.665 2.200,-3.074 Si 0.057,-3.662 0.087,-4.153 0.157,-4.476 0.286,-4.409 0.349,-4.204 0.561,-3.168 K 0.005,-0.613 0.009,-0.757 0.025,-0.844 0.057,-0.710 0.089,-0.551 0.149,-0.415 Ca 0.009,-1.202 0.021,-1.692 0.062,-1.814 0.094,-1.719 0.135,-1.578 0.201,-1.434 Sc 0.055,-3.363 0.080,-3.787 0.140,-4.092 0.226,-4.140 0.315,-4.009 0.481,-3.612 Ti 0.133,-4.307 0.188,-4.850 0.314,-5.315 0.444,-5.410 0.712,-5.033 1.075,-4.406 V 0.245,-4.014 0.333,-4.545 0.606,-5.153 0.858,-5.111 1.058,-4.850 1.520,-4.005 Cr 0.168,-3.162 0.182,-3.277 0.300,-3.808 0.487,-3.758 0.634,-3.324 0.935,-1.915 Fe 0.171,-4.007 0.247,-4.554 0.505,-5.071 0.834,-4.962 1.072,-4.734 1.445,-4.613 Co 0.146,-3.374 0.380,-4.894 0.821,-5.517 1.306,-5.509 1.683,-5.008 2.200,-3.952 Ni 0.223,-3.899 0.336,-4.373 0.566,-4.731 1.028,-5.280 1.355,-5.028 2.200,-4.172 Cu 0.091,-2.811 0.127,-3.100 0.238,-3.280 0.334,-3.215 0.455,-2.935 0.604,-2.374 Zn 1.012,-0.752 1.604,-0.880 2.794,-0.940 3.544,-0.919 4.805,-0.822 8.006,-0.406 Ge 0.056,-3.285 0.072,-3.429 0.115,-3.592 0.188,-3.553 0.241,-3.359 0.364,-2.645 Rb 0.004,-0.552 0.009,-0.707 0.020,-0.663 0.029,-0.594 0.036,-0.633 0.047,-0.977 Sr 0.005,-1.128 0.008,-1.304 0.046,-1.558 0.063,-1.490 0.088,-1.331 0.108,-1.205 Y 0.012,-3.002 0.021,-3.548 0.098,-4.152 0.167,-4.060 0.260,-3.712 0.338,-3.459 Zr 0.052,-4.542 0.082,-5.367 0.186,-6.136 0.299,-6.147 0.489,-5.725 0.685,-5.428 Nb 0.034,-6.744 0.054,-7.933 0.139,-9.012 0.240,-8.889 0.347,-8.192 0.468,-7.278 Mo 0.059,-4.588 0.088,-5.395 0.192,-6.024 0.271,-5.941 0.363,-5.304 0.542,-4.467 Tc 0.164,-5.372 0.217,-5.913 0.317,-6.476 0.482,-6.730 0.720,-6.230 1.108,-4.437 Ru 0.088,-6.348 0.130,-7.166 0.241,-7.687 0.311,-7.756 0.457,-7.052 0.638,-6.282 Rh 0.074,-4.522 0.095,-4.900 0.176,-5.540 0.279,-5.529 0.381,-4.971 0.495,-3.965 Pd 0.017,-2.667 0.025,-3.029 0.078,-3.490 0.128,-3.256 0.166,-2.905 0.232,-2.126 Ag 0.027,-1.749 0.035,-1.894 0.117,-2.349 0.178,-2.390 0.239,-2.422 0.294,-2.488 Cd 0.061,-0.301 0.110,-0.379 0.245,-0.409 0.364,-0.446 0.690,-0.333 0.921,-0.278 Te 0.006,-1.550 0.011,-1.922 0.029,-2.088 0.038,-2.159 0.069,-1.969 0.117,-1.615 Ba 0.004,-1.228 0.010,-1.721 0.020,-1.822 0.027,-1.813 0.050,-1.658 0.081,-1.393 La 0.019,-2.564 0.051,-3.977 0.121,-4.240 0.210,-4.086 0.322,-3.777 0.416,-3.536 Ce 0.040,-2.826 0.106,-4.210 0.207,-4.481 0.329,-4.503 0.526,-4.234 0.677,-3.908

85

Table 6.4: continuation of Table 6.3.

Pr 0.035,-2.470 0.088,-3.600 0.166,-3.856 0.246,-3.891 0.357,-3.821 0.554,-3.605 Nd 0.021,-2.403 0.062,-3.730 0.157,-3.834 0.308,-3.858 0.456,-3.972 0.554,-3.776 Eu 0.010,-1.188 0.021,-1.599 0.055,-1.790 0.125,-1.734 0.191,-1.516 0.275,-1.357 Gd 0.017,-3.615 0.043,-5.278 0.085,-5.539 0.129,-5.542 0.219,-5.281 0.330,-4.894 Tb 0.027,-2.642 0.067,-3.837 0.142,-4.061 0.202,-3.963 0.319,-3.614 0.409,-3.340 Dy 0.027,-2.672 0.068,-3.683 0.158,-4.026 0.199,-3.866 0.279,-3.728 0.404,-3.139 Ho 0.028,-2.672 0.069,-3.851 0.162,-4.020 0.199,-3.927 0.283,-3.689 0.406,-3.248 Er 0.028,-2.681 0.069,-3.729 0.162,-3.947 0.196,-3.823 0.276,-3.654 0.406,-3.278 Tm 0.030,-2.762 0.069,-3.659 0.168,-3.992 0.217,-3.801 0.289,-3.658 0.397,-3.268 Yb 0.012,-0.958 0.022,-1.224 0.056,-1.366 0.122,-1.191 0.188,-1.044 0.264,-0.965 Lu 0.040,-2.683 0.087,-3.659 0.155,-3.911 0.210,-3.900 0.340,-3.630 0.482,-3.254 Hf 0.036,-4.599 0.057,-5.285 0.209,-6.329 0.279,-6.266 0.399,-5.886 0.680,-4.554 Ta 0.066,-5.540 0.109,-6.815 0.210,-7.729 0.409,-7.719 0.624,-6.877 0.903,-6.333 W 0.111,-6.179 0.149,-6.886 0.256,-7.726 0.465,-7.681 0.689,-6.553 0.953,-4.666 Re 0.307,-4.531 0.441,-5.847 0.680,-7.039 0.803,-7.237 1.102,-6.969 1.636,-5.140 Os 0.106,-6.174 0.206,-7.540 0.326,-7.866 0.394,-7.945 0.570,-7.555 0.908,-5.803 Ir 0.089,-5.498 0.106,-6.040 0.228,-7.004 0.342,-7.142 0.486,-6.271 0.667,-5.735 Pt 0.025,-3.842 0.035,-4.280 0.094,-5.002 0.149,-4.728 0.188,-4.327 0.250,-3.495 Au 0.015,-1.932 0.023,-2.231 0.075,-2.492 0.108,-2.276 0.134,-2.038 0.179,-1.590 Tl 0.007,-1.336 0.013,-1.611 0.029,-1.661 0.062,-1.476 0.091,-1.268 0.133,-1.001 Pb 0.0004,-1.813 0.0006,-2.159 0.0010,-2.454 0.0016,-2.537 0.0046,-2.491 0.0125,-1.686

86

Table 6.5: (left part) Structures used to fit EAM potentials for pure elements, continued in Table 6.6. The EAM potentials are fitted to the ab initio energies in body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close-packed (HEX), diamond structure (DIA), and groundstate structure at various pressures obtained by expanding/compressing the equillibrium lattice constant a by a factor from 0.9 to 1.16 corresponding to a range of charge density from ρmax to ρmin. (right part) Lattice constansts calculated using the fitted parameters. The literature values aLIT are taken from [146].

Z Name Struct. ρmin → ρmax aLIT aV ASP aEAM (A˚−3) (A˚) (A˚) (A˚)

3 Li BCC 0.104 → 0.441 3.49 3.44 3.44 4 Be HEX 0.268 → 1.437 2.29 2.27 2.24 6 C DIA 0.329 → 2.038 3.57 3.57 3.57 11 Na BCC 0.027 → 0.163 4.23 4.19 4.19 12 Mg HEX 0.014 → 0.174 3.21 3.19 3.16 13 Al FCC 0.098 → 0.312 4.05 4.04 4.00 14 Si DIA 0.060 → 0.443 5.43 5.46 5.46 19 K BCC 0.013 → 0.089 5.23 5.29 5.29 20 Ca FCC 0.014 → 0.186 5.58 5.54 5.54 21 Sc HEX 0.064 → 0.441 3.31 3.30 3.34 22 Ti HEX 0.150 → 0.845 2.95 2.92 2.92 23 V BCC 0.263 → 1.320 3.02 2.98 2.98 24 Cr BCC 0.126 → 0.765 2.88 2.84 2.84 26 Fe BCC 0.213 → 1.221 2.87 2.83 2.83 27 Co HEX 0.361 → 0.974 2.51 2.47 2.45 28 Ni FCC 0.374 → 0.983 3.52 3.49 3.46 29 Cu FCC 0.113 → 0.519 3.61 3.64 3.64 30 Zn HEX 0.777 → 7.050 2.66 2.71 2.71 32 Ge DIA 0.056 → 0.306 5.66 5.79 5.79 37 Rb BCC 0.004 → 0.048 5.59 5.65 5.65 38 Sr FCC 0.009 → 0.097 6.08 6.05 6.05 39 Y HEX 0.036 → 0.282 3.65 3.64 3.64 40 Zr HEX 0.073 → 0.511 3.23 3.22 3.22 41 Nb BCC 0.050 → 0.390 3.30 3.31 3.31 42 Mo BCC 0.055 → 0.463 3.15 3.15 3.12 43 Tc HEX 0.147 → 0.993 2.74 2.76 2.76 44 Ru HEX 0.084 → 0.558 2.70 2.73 2.70 45 Rh FCC 0.072 → 0.469 3.80 3.84 3.84 46 Pd FCC 0.017 → 0.195 3.89 3.96 3.96 47 Ag FCC 0.034 → 0.256 4.09 4.16 4.16 48 Cd HEX 0.076 → 0.855 2.98 3.09 3.03 52 Te HEX 0.007 → 0.108 4.45 4.06 4.02

87

Table 6.6: continuation of Table 6.5.

56 Ba BCC 0.007 → 0.074 5.02 5.02 5.02 57 La HEX 0.042 → 0.331 3.75 3.74 3.74 58 Ce FCC 0.096 → 0.610 5.16 4.74 4.74 59 Pr HEX 0.076 → 0.502 3.67 3.45 3.45 60 Nd HEX 0.068 → 0.457 3.66 3.56 3.56 63 Eu BCC 0.029 → 0.232 4.61 4.46 4.41 64 Gd HEX 0.037 → 0.303 3.64 3.62 3.62 65 Tb HEX 0.049 → 0.362 3.60 3.62 3.62 66 Dy HEX 0.047 → 0.345 3.59 3.61 3.61 67 Ho HEX 0.048 → 0.356 3.58 3.59 3.63 68 Er HEX 0.047 → 0.352 3.56 3.58 3.61 69 Tm HEX 0.049 → 0.367 3.54 3.55 3.55 70 Yb FCC 0.023 → 0.202 5.49 5.29 5.29 71 Lu HEX 0.061 → 0.425 3.51 3.48 3.48 72 Hf HEX 0.073 → 0.530 3.20 3.20 3.20 73 Ta BCC 0.091 → 0.692 3.31 3.32 3.32 74 W BCC 0.105 → 0.794 3.16 3.19 3.19 75 Re HEX 0.378 → 1.509 2.76 2.78 2.78 76 Os HEX 0.114 → 0.861 2.74 2.76 2.73 77 Ir FCC 0.068 → 0.629 3.84 3.88 3.80 78 Pt FCC 0.025 → 0.236 3.92 3.98 3.98 79 Au FCC 0.016 → 0.167 4.08 4.17 4.17 81 Tl HEX 0.011 → 0.112 3.46 3.54 3.54 82 Pb FCC 0.001 → 0.011 4.95 5.02 5.02

88

Chapter 7

Effects of Mo on the thermodynamics of

Fe:Mo:C nanocatalyst for single-walled

carbon nanotube growth

Among the established methods for single-walled carbon nanotubes (SWCNTs) syn-

thesis [147, 148, 149], catalytic chemical vapor decomposition (CCVD) technique is

preferred for growing nanotubes on a substrate at a target position due to its rel-

atively low synthesis temperature. Temperature as low as ∼ 450 oC was reported by using hydrocarbon feedstock with exothermic catalytic decomposition reaction

[150, 151]. Critical factors for the efficient growth via CCVD are the compositions

of the interacting species, the preparation of the catalysts, and the synthesis condi-

tions. Efficient catalysts must have long active lifetimes (with respect to feedstock

dissociation and nanotube growth), high selectivity and be less prone to contamina-

tion. Common factors that lead to reduction in catalytic activity are deactivation

(e.g. due to coating with carbon or nucleation of inactive phases) [152, 153, 154] and

thermal sintering (e.g. caused by highly exothermic reactions on the clusters surface

[150, 151, 155] with insufficient heat [156]).

Metal alloy catalysts, such as Fe:Co, Co:Mo, and Fe:Mo, improve the growth of

CNTs [157, 158, 159, 160, 162, 163], because the presence of more than one metal

species can significantly enhance the activity of a catalyst [159, 164], and can prevent

catalyst particle aggregation [163, 164]. In the case of Fe:Mo nanoparticles supported

on Al2O3 substrates, the enhanced catalyst activity has been shown to be larger than

the linear combination of the individual Fe/Al2O3 and Mo/Al2O3 activities [159, 160].

89

This is explained in terms of substantial intermetallic interaction between Mo, Fe and

C [159, 165, 166]. The addition of Mo in mechanical alloying of powder Fe and C

mixtures promotes solid state reactions even at low Mo concentrations by forming

ternary phases, such as the (Fe,Mo)23C6 and Fe2(MoO4)3 type carbides [167]. It has

been found that low Mo concentration in Fe:Mo is favored for growing SWCNTs

(on Al2O3 substrates) since the presence, after activation, of the phase Fe2(MoO4)3

can lead to the formation of small metallic clusters [168]. Considering the vapor-

liquid-solid model (VLS), which is the most probable mechanism for CNT growth

[159, 169]. The metallic nanoparticles are very efficient catalysts when they are in

the liquid or viscous states, probably due to considerable carbon bulk-diffusion in

this phase (compared to surface or sub-surface diffusion). Generally, unless stable

intermetallic compounds form, alloying metals reduce the melting point below those

of the constituents [165, 166]. Hence, to improve the yield and quality of nanotubes,

one can tailor the composition of the catalyst particle to move its liquidus line below

the synthesis temperature [159]. However, identifying the perfect alloy composition

is non trivial. In fact, the presence of more than two metallic species allows for the

possibility of different carbon pollution mechanisms by thermodynamic promotion of

ternary carbides. In this Chapter, we study the phase diagram of Fe:Mo:C system

and the possible roles of Mo in the catalytic properties of Fe:Mo.

7.1 Size-pressure approximation

Determining the thermodynamic stability of different phases in nanoparticles of dif-

ferent sizes with ab initio calculations is computationally expensive. We develop

a simple model, called the ”size-pressure approximation”, which allows one to es-

timate the phase diagram at the nanoscale starting from bulk calculations under

pressure [152]. Surface curvature and superficial dangling bonds on nanoparticles are

90

responsible for internal stress fields which modify the atomic bond lengths. As a

first approximation, where all surface effects that are not included in the curvature

are neglected, we can map the particle radius R to the pressure P by equating the

deviation from the bulk value of the average bond lengths due to surface curvature

(in the case of particle) and pressure (in the case of bulk). For spherical clusters,

the phenomenon can be modeled with the Young-Laplace equation P = 2γ/R where

the proportionality constant γ can be calculated with ab initio methods. In our

study, since the concentration of Mo in Fe:Mo is small, we use the ”size-pressure

approximation” of Fe particle.

Figure 7.1 shows the implementation of the ”size-pressure approximation” for Fe

nanoparticles. On the left hand side we show the ab initio calculations of the deviation

of the average bond length inside the cluster Δdnn ≡ d0nn − dnn (d0nn = 0.2455 nm is our bulk bond length), for body-centered cubic (BCC) particles of size N = 59, 113,

137, 169, 307 and ∞ (bulk) as a function of the inverse radius (1/R). The particles were created by intersecting a BCC lattice with different size spheres. The particle

radius is defined as 1/R ≡ 1/Nscp ∑

i 1/Ri where the sum is taken over the atoms

belonging to the surface convex polytope (Nscp vertices) and Ri are the distances to

the geometric center of the cluster. The left straight line is a linear interpolation

between 1/R and Δdnn calculated with the constraint of passing through 1/R = 0

and Δdnn = 0 (N = ∞, bulk). The right hand side shows the ab initio value of dnn in bulk BCC Fe as a function of hydrostatic pressure, P . The straight line is a linear

interpolation between P and Δdnn calculated with the constraint of passing through

P = 0 and Δdnn = 0 (bulk lattice). By following the colored dashed paths indicated

by the arrows we can map the analysis of nanoparticles stability as a function of R

onto bulk stability as a function of P , and obtain the relation between the radius of

particle/nanotube and the effective pressure P ·R = 2.46 GPa · nm. It is important

91

Figure 7.1: (color online). Size-pressure approximation for Fe nanoparticles obtained by equating the deviation of average bond length from the bulk value due to curvature 1/R (in the case of particle) and due to pressure P (in the case of bulk).

to mention that our γ = 1.23 J/m2 is not a real surface tension but an ab initio fitting

parameter describing size-induced stress in nanoparticles. With this γ, we deduce the

Fe-Mo-C phase diagram of nanoparticles of radius R from ab initio calculations of

the bulk material under pressure P .

92

7.2 Fe-Mo-C phase diagram under pressure

Simulations are performed with VASP as described in Section 2.2. The hydrostatic

pressure estimated from the pressure-size model is implemented as Pulay stress [170].

Ternary phase diagrams are calculated using BCC-Mo, BCC-Fe and SWCNTs as

references (pure-Fe phase is taken to be BCC because our simulations are aimed at

the low temperature regime of catalytic growth). The reference SWCNTs have the

same diameter of the particle to minimize the curvature-strain energy. In fact, CVD

experiments of SWCNT growth from small (∼ 0.6-2.1 nm) particles indicate that the diameter of the nanotube is similar to the diameter of the catalyst particle from

which it grows. In some experiments where the growth mechanism is thought to be

root-growth, the ratio of the catalyst particle diameter to SWCNT diameter is ∼ 1.0, whereas in experiments involving pre-made floating catalyst particles this ratio is ∼ 1.6 [171]. Formation energies are calculated with respect to decomposition into the

nearby stable phases, depending the position in the ternary phase diagram.

Binary and Ternary phases are included if they are stable in the temperature

range used in CVD growth of SWCNTs or if they have been reported experimentally

during or after the growth [165, 166, 172]. Thus, we include the binaries Mo2C,

Fe2Mo and Fe3C. In addition, since our Fe-rich Fe:Mo experiments were performed

with compositions close to Fe4Mo [159], we include a random phase Fe4Mo generated

with the special quasi-random structure formalism (SQS). Bulk ternary carbides,

which have been widely investigated due to their importance in alloys and steel, can

be considered as derivatives of binary structures with extra C atoms in the interstices

of the basic metal alloy structures. Three possible ternary phases have been reported

for bulk Fe-Mo-C [173] and they are referred as τ1 (M6C), τ2 (M3C) and τ3 (M23C6)

(M is the metal species). For simplicity, we follow the same nomenclature. τ1 is

the wellknown M6C phase, which has been observed experimentally as Fe4Mo2C and

93

Fe3Mo3C structures (η carbides) [174, 175]. Both of these structures are FCC but

have different lattice spacings. Our calculations show that the most stable variant

τ1 is Fe4Mo2C, and we denote it as τ1 henceforth. τ2 is the Fe2MoC phase, which

has an orthorhombic symmetry distinct from that of Fe3C [176, 177]. We consider

Fe21Mo2C6 as the third τ3 FCC phase [178]. We use the Cr23C6 as the prototype

structure [179] where Fe and Mo substitute for Cr. Although M23C6 type phases do

not appear in the stable C-Fe or C-Mo systems, they have been reported in ternary C-

Fe-Mo systems and also appear as transitional products in solid state reactions [173].

Time-temperature precipitation diagrams of low-C steels have identified τ2, τ3 and τ1

as low-temperature, metastable and stable carbides, respectively [180]. Furthermore,

τ2 carbides precipitate quickly due to carbon-diffusion controlled reaction while τ3

carbides precipitate due to substitutional-diffusion controlled reactions. The latter

phenomenon, requiring high temperature, longer times and producing metastable

phases is not expected to enhance the catalytic deactivation of the nanoparticle. In

summary, as long as the presence of carbon does not lead to excessive formation of

Fe3C and τ2), the catalyst should remain active for SWCNT growth.

Figure 7.2 shows the phase diagram at zero temperature of nanoparticles of radii

R ∼ ∞, 1.23, 0.62, 0.41 nm, calculated at P = 0, 2, 4, and 6 GPa, respectively. Stable and unstable phases are shown as black squares and red dots, respectively.

The solid green lines connect the stable phases. The numbers ”1”...”8” in panels (c)

and (d) indicate the intersections between the phases’ boundary and the dotted lines

representing the path of carbon pollution to the two test phases Fe4Mo and FeMo.

Fe4Mo has been reported to be an effective catalyst composition [159] while FeMo

represents a hypothetical Fe:Mo particle with a Mo content larger than 33%.

94

Figure 7.2: (color online). Ternary phase diagram for Fe-Mo-C nanoparticles of R ∼ ∞, 1.23, 0.62, 0.41 nm.

7.3 Fe4Mo particles

An advantage of an Fe4Mo particle has over a pure Fe particle is that the [Fe4/5Mo1/5]1−x-

Cx line does not intercept any carbide (Fe3C, τ3, τ2). This implies that, at least at low

temperatures, there is a surplus of unbounded metal (probably even at high temper-

atures since the line is far from all of the competing stable phases). This is illustrated

in figure 7.3, which shows the fractional evolution of species as one progresses along

the [Fe4/5Mo1/5]1−x-Cx line in figure 7.2.

For a large Fe4Mo particle (R ≥ 0.62 nm), the decomposition into stable phases is shown in figure 7.3(a)). At concentrations between 0 < xc < 0.09 there is available

free Fe for catalysis, however there is no carbon content in the particle. At higher

95

concentrations, the particle starts to contain free carbon while still providing free Fe.

Therefore, a steady state growth of SWCNTs is possible from large Fe4Mo particles.

Figure 7.3(b) shows the decomposition for a small Fe4Mo particle (R ≤ 0.41 nm)). In this case, the free Fe is consumed and transformed into τ3 carbide before the particle

has enough free carbon, hence, the SWCNT growth will not occur. We can estimate

the minimum size of the particle by calculting the pressure at which τ3 starts to form.

By linear interpolation, we obtain R = 0.52 nm. This size is smaller than that if one

uses pure Fe nanocatalyst (R = 0.56 nm) [152, 153].

7.4 FeMo particles

Figure 7.4 shows the decomposition analysis for FeMo particle. For a large particle of

size R ≥ 0.62 nm, The particle contains free Fe and carbon only after the concentra- tion xc is larger than 0.2 which then permits the growth of the nanotube. Comparing

to the Fe4Mo particle, since the fraction of Fe in the FeMo particle is considerably

smaller than in Fe4Mo, the expected yield is lower and the synthesis temperature

needs to be increased (to overcome the reduced fraction of catalytically active free

Fe). Small FeMo particles (R ≤ 0.41 nm) are similar to small Fe4Mo clusters. Nucle- ation of τ3 and the abscence of free Fe and excess carbons indicate that the particles

are catalytically inactive. The minimum size of FeMo and Fe4Mo particles able to

grow nanotube is the same since it is determined by the stabilization of the same τ3

carbide.

96

Figure 7.3: (color online). a) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”1”, represents [Fe4/5Mo1/5]1−x-Cx crossing the boundary phase Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”4” and ”5”, represent [Fe4/5Mo1/5]1−x-Cx crossing the boundary phases Fe←→Mo2C and τ3, as shown in figure 7.2(d).

97

Figure 7.4: (color online). a) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”2” and ”3”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phase Fe2Mo↔Mo2C and Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composi- tion [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”6”, ”7”, and ”8”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phases Fe2Mo↔Mo2C, Fe←→Mo2C, and τ3 ↔Mo2C, as shown in figure 7.2(d).

98

Chapter 8

Conclusions

We have presented the results of our computational studies on adsorptions of hy-

drocarbons (alkanes, benzene) and rare gases on a decagonal surface of Al-Ni-Co

quasicrystal (d-AlNiCo). Ab initio calculations show that upon the adsorptions, the

surface of the d-AlNiCo does not undergo relaxations and that there are no disso-

ciations of the adsorbates. The simulations of thin film growth of these adsorbates

have been performed in the grand canonical ensemble using Monte Carlo method.

We use semiempirical pair interactions for the rare gases and develop classical many-

body potentials based on embedded-atom method (EAM) for the hydrocarbons. All

of the simulated atoms/molecules wet the substrate as a consequence of compara-

ble strengths between the substrate-adsorbate and adsorbate-adsorbate. Another

consequence is the wide range of overlayer structures observed in these systems.

Methane monolayer has a quasicrystalline pentagonal order commensurate with the

substrate. Propane forms a disordered structure. Hexane and octane monolayers

show 2-dimensional close-packed features consistent with their bulk structure. Ben-

zene forms a pentagonal monolayer at low and moderate temperatures which trans-

forms into a triangular lattice at high temperatures. Similar structural transition

also occurs in xenon monolayer, however in this case, the transition is observed at all

temperatures below the triple-point temperature and as a function of pressure. We

have characterized that such transition is first-order with an associated latent heat.

Smaller noble gases such as Ne, Ar, Kr form monolayers with mixed pentagonal and

triangular patterns and do not show any structural transitions.

By systematically simulating test noble gases of various sizes and strengths, we

99

observe that the relative strengths between the competing interactions determine the

growth mode. Agglomeration occurs when the adsorbate-adsorbate interactions are

much stronger than the substrate-adsorbate ones. In the comparable strength regime,

a layer-by-layer film growth is observed. In this regime, the mismatch between the

size of the gas and the substrate’s characteristic length plays a major role in affecting

the structure of the adsorbed films. In general, on the d-AlNiCo substrate, we found

that structural transition from 5- to 6-fold occurs when the gas size is larger than

λ/0.944 (λ represents the average row-row spacing in the quasicrystalline plane of

the d-AlNiCo). Even though this rule is derived from rare gases, it is consistent with

methane, benzene, hexane and octane. Therefore, it might be useful as a guidance in

the search for suitable quasicrystalls for which alkanes (as the main constituent of oil

lubricants) will form quasiperiodic structures for the low-friction coating applications.

It is a natural extension of this work to investigate other quasicrystalls with

larger characterictic lengths than the d-AlNiCo used in this study. In fact, d-AlNiCo

is stable in many decorations depending on the concentration of Al, Ni, and Co.

Due to the stripe nature in the close-packed structure of linear alkanes, the film

structure of these molecules on one dimensional quasicrystals is interesting to study.

Two surfaces with stripe pattern will be commensurate only when the stripes are

perfectly alligned, therefore, one dimensional quasicrystalls might be good candidate

for low-friction coatings.

100

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110

Biography

The author was born in May 5th, 1978 in a Javanese village 8 miles southwest of

Solo, Central Java, Indonesia. He is the last(fourth) child of Ladiya (father) and

Sri Suratmi (mother). His father was an elementary school teacher and his mother

is a devoted house wife. Prior to attending college education, the author won the

third award in the International Physics Olympiad 1996 in Norway. The author

received a B.S. degree from Institut Teknologi Bandung, Indonesia, in Electrical

Engineering in 2000. In the Fall of 2000, he went to Florida State University on a

research assistantship from the Physics department in which he completed a M.S.

degree in 2004. In Summer 2004, he received a research assitantship to pursue a

doctoral degree in Mechanical Engineering and Materials Science department from

Duke University under the advisory of Dr. Stefano Curtarolo. He completed his PhD

work in computational materials science in Spring 2008. In addition, he received a

Graduate Certificate degree in Computational Science, Engineering, and Medicine,

in Spring 2008 also from Duke University.

List of publications:

• Diehl R D, Setyawan W and Curtarolo S 2008 J. Phys.: Cond. Mat. in press

• Harutyunyan A R, Awasthi N, Jiang A, Setyawan W, Mora E, Tokune T, Bolton K and Curtarolo S 2008 Phys. Rev. Lett. inpress

• Curtarolo S, Awasthi N, Setyawan W, Li N, Jiang A, Tan T Y, Mora E, Bolton K and Harutyunyan A T 2008 Proc. of Comp. Simul. Studies in Cond. Matt.

Phys. XXI Eds Landau D P, Lewis S P and Schuttler H-B (Springer, Berlin,

Heidelberg)

• Setyawan W, Diehl R D, Ferralis N, Cole M W and Curtarolo S 2007 J. Phys.:

111

Cond. Mat. 19 016007

• Diehl R D, Setyawan W, Ferralis N, Trasca R, Cole M W and Curtarolo S 2007 Phil. Mag. 87 2973

• Jiang A, Awasthi N, Kolmogorov A N, Setyawan W, Bo¨rjesson A, Bolton K, Harutyunyan A R and Curtarolo S 2007 Phys. Rev. B 75 205426

• Setyawan W, Ferralis N, Diehl R D, Cole M W and Curtarolo S 2006 Phys. Rev. B 74 125425

• Diehl R D, Ferralis N, Pussi K, Cole M W, Setyawan W and Curtarolo S 2006 Phil. Mag. 86 863

• Curtarolo S, Setyawan W, Diehl R D, Ferralis N and Cole M W 2005 Phys. Rev. Lett. 95 136104

• Rao S G, Huang L, Setyawan W and Hong S 2003 Nature 425 36

• Setyawan W, Rao S G and Hong S 2002 Mat. Res. Soc. Proc. NN Fall

• Mumtaz A, Setyawan W and Shaheen S A 2002 Phys. Rev. B 65 020503

112

QUASICRYSTALLINE COATINGS VIA SIMULATIONS

OF THIN FILM GROWTH OF HYDROCARBONS AND

RARE GASES

by

Wahyu Setyawan

Department of Mechanical Engineering and Materials Science Duke University

Date: Approved:

Dr. Stefano Curtarolo, Supervisor

Dr. Teh Y. Tan

Dr. Laurens E. Howle

Dr. Xiaobai Sun

Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in the Department of Mechanical Engineering and Materials Science in the Graduate School of

Duke University

2008

ABSTRACT

COMPUTATIONAL STUDY OF LOW-FRICTION

QUASICRYSTALLINE COATINGS VIA SIMULATIONS

OF THIN FILM GROWTH OF HYDROCARBONS AND

RARE GASES

by

Wahyu Setyawan

Department of Mechanical Engineering and Materials Science Duke University

Date: Approved:

Dr. Stefano Curtarolo, Supervisor

Dr. Teh Y. Tan

Dr. Laurens E. Howle

Dr. Xiaobai Sun

An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in the Department of Mechanical Engineering and Materials Science in the Graduate School of

Duke University

2008

Copyright c© 2008 by Wahyu Setyawan All rights reserved

Abstract

Quasicrystalline compounds (QC) have been shown to have lower friction compared

to other structures of the same constituents. The abscence of structural interlocking

when two QC surfaces slide against one another yields the low friction. To use QC

as low-friction coatings in combustion engines where hydrocarbon-based oil lubri-

cant is commonly used, knowledge of how a film of lubricant forms on the coating is

required. Any adsorbed films having non-quasicrystalline structure will reduce the

self-lubricity of the coatings. In this manuscript, we report the results of simula-

tions on thin films growth of selected hydrocarbons and rare gases on a decagonal

Al73Ni10Co17 quasicrystal (d-AlNiCo). Grand canonical Monte Carlo method is used

to perform the simulations. We develop a set of classical interatomic many-body

potentials which are based on the embedded-atom method to study the adsorption

processes for hydrocarbons. Methane, propane, hexane, octane, and benzene are

simulated and show complete wetting and layered films. Methane monolayer forms

a pentagonal order commensurate with the d-AlNiCo. Propane forms disordered

monolayer. Hexane and octane adsorb in a close-packed manner consistent with

their bulk structure. The results of hexane and octane are expected to represent

those of longer alkanes which constitute typical lubricants. Benzene monolayer has

pentagonal order at low temperatures which transforms into triangular lattice at high

temperatures. The effects of size mismatch and relative strength of the competing

interactions (adsorbate-substrate and between adsorbates) on the film growth and

structure are systematically studied using rare gases with Lennard-Jones pair poten-

tials. It is found that the relative strength of the interactions determines the growth

mode, while the structure of the film is affected mostly by the size mismatch between

adsorbate and substrate’s characteristic length. On d-AlNiCo, xenon monolayer un-

iv

dergoes a first-order structural transition from quasiperiodic pentagonal to periodic

triangular. Smaller gases such as Ne, Ar, Kr do not show such transition. A simple

rule is proposed to predict the existence of the transition which will be useful in the

search of the appropriate quasicrystalline coatings for certain oil lubricants.

Another part of this thesis is the calculation of phase diagram of Fe-Mo-C sys-

tem under pressure for studying the effects of Mo on the thermodynamics of Fe:Mo

nanoparticles as catalysts for growing single-walled carbon nanotubes (SWCNTs).

Adding an appropriate amount of Mo to Fe particles avoids the formation of stable

binary Fe3C carbide that can terminate SWCNTs growth. Eventhough the formation

of ternary carbides in Fe-Mo-C system might also reduce the activity of the catalyst,

there are regions in the Fe:Mo which contain enough free Fe and excess carbon to

yield nanotubes. Furthermore, the ternary carbides become stable at a smaller size

of particle as compared to Fe3C indicating that Fe:Mo particles can be used to grow

smaller SWCNTs.

v

Contents

Abstract iv

List of Figures ix

List of Tables xiv

Acknowledgements xvii

1 Introduction 1

2 Methods 5

2.1 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . 5

2.1.2 Slater determinant . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 Hartree-Fock equations . . . . . . . . . . . . . . . . . . . . . . 7

2.1.4 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . 10

2.1.5 Local density and generalized gradient approximations . . . . 11

2.1.6 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Vienna Ab initio Simulation Package . . . . . . . . . . . . . . . . . . 13

2.3 Classical interatomic potentials . . . . . . . . . . . . . . . . . . . . . 14

2.4 Simplex method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Grand canonical Monte Carlo . . . . . . . . . . . . . . . . . . . . . . 17

3 Noble gas adsorptions on d-AlNiCo 18

3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Simulation cell . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.2 Gas-gas and gas-substrate interactions . . . . . . . . . . . . . 19

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3.1.3 Adsorption potentials . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.4 Effective parameters . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.5 Test rare gases . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.6 Chemical potential, order parameter, and ordering transition . 25

3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Adsorption isotherms . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.3 Order parameters (ρ5−6) . . . . . . . . . . . . . . . . . . . . . 32

3.2.4 Effects of �gg and σgg on adsorption isotherms . . . . . . . . . 37

3.2.5 Effects of �gg and σgg on 5- to 6-fold transition . . . . . . . . . 39

3.2.6 Prediction of 5- to 6-fold transition . . . . . . . . . . . . . . . 40

3.2.7 Transitions on smoothed substrates . . . . . . . . . . . . . . . 42

3.2.8 Temperature vs substrate effect . . . . . . . . . . . . . . . . . 43

3.2.9 Orientational degeneracy of the ground state . . . . . . . . . . 46

3.2.10 Isosteric heat of adsorption . . . . . . . . . . . . . . . . . . . . 47

3.2.11 Effect of vertical dimension . . . . . . . . . . . . . . . . . . . 48

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Embedded-atom method potentials 52

4.1 Stage 1: Aluminum, cobalt, and nickel . . . . . . . . . . . . . . . . . 54

4.2 Stage 2: Al-Co-Ni potentials . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Stage 3: Hydrocarbon potentials . . . . . . . . . . . . . . . . . . . . . 59

4.4 Stage 4: Hydrocarbon on Al-Co-Ni . . . . . . . . . . . . . . . . . . . 63

5 Hydrocarbon adsorptions on d-AlNiCo 66

5.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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5.2 Adsorption potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Molecule orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4 Adsorption isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.5 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Embedded-atom potentials for selected pure elements in the pe- riodic table 81

7 Effects of Mo on the thermodynamics of Fe:Mo:C nanocatalyst for single-walled carbon nanotube growth 89

7.1 Size-pressure approximation . . . . . . . . . . . . . . . . . . . . . . . 90

7.2 Fe-Mo-C phase diagram under pressure . . . . . . . . . . . . . . . . . 93

7.3 Fe4Mo particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.4 FeMo particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8 Conclusions 99

Bibliography 101

Biography 111

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List of Figures

3.1 (color online). Computed adsorption potentials for (a) Ne, (c) Ar, (e) Kr, and (g) Xe on the d-AlNiCo, obtained by minimizing V (x, y, z) with respect to z. The distribution of the minimum value of these potentials is plotted in (b, d, f, and h) respectively: the solid line marks the average value 〈Vmin〉, the dashed lines mark the values at 〈Vmin〉±SD. . . . . . . . . . . . . . . 22

3.2 Computed adsorption isotherms for all the gas/d-AlNiCo systems. The ranges of temperatures under study are: Ne: T = 14 K to 46 K in 2 K steps, Ar: 45 K to 155 K in 5 K steps, Kr: 65 K to 225 K in 5 K steps, Xe: 80 K to 280 K in 10 K steps. Additional isotherms are shown with solid circles at T � = 0.35: T = 11.8 K (Ne), T = 41.7 K (Ar), T = 59.6 K (Kr), and T = 77 K (Xe). Isotherms above the triple point temperatures are shown as dotted curves. . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Density profiles and Fourier transforms of the outer layer at T � = 0.35 for Ne/d-AlNiCo (T = 11.8 K) and Ar/d-AlNiCo (T = 41.7 K), corresponding to points (A) through (F) of Figure 3.2. . . . . . . . . . . . . . . . . . . 30

3.4 Density profiles and Fourier transforms of the outer layer at T � = 0.35 for of Kr/d-AlNiCo (T = 59.6 K) and Xe/d-AlNiCo (T = 77 K), corresponding to points (A) through (F) of Figure 3.2. . . . . . . . . . . . . . . . . . . 33

3.5 (color online). Order parameters, ρ5−6, as a function of normalized chemical potential, μ�, (as defined in the text) at T � = 0.35 for the first four layers of (a) Ne, (b) Ar, (c) Kr, and for the first layer of Xe (d) adsorbed on d-AlNiCo. A sudden drop of the order parameter in Xe/QC to a constant value of ∼ 0.017 at μ� ∼ 0.8 indicates the existence of a first-order structural transition from fivefold to sixfold in the system. . . . . . . . . . . . . . . 34

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3.6 (color online). Xe on d-AlNiCo at T = 77 K. (a) Adsorption isotherm, ρN , versus the normalized chemical potential, μ�. (b) Nearest neighbor distance derived from the first peak of pair correlation function, rNN , (black line), and average spacing between neighbors at equilibrium, d¯NN , (red line). (c) Order parameter ρ5−6 (probability of fivefold defects, defined in Equation 3.14) versus the normalized chemical potential, μ�. (d) Total enthalpy. The transition, which is defined as the point in μ� above which the order parameter remains nearly constant, occurs at μ�tr ∼0.8. The discontinuity in H around μ�tr ∼ 0.8 indicates a first order transition with associated latent heat of the transition. The order parameter ρ5−6 after the transition is ∼ 0.017. Heat of the transition is ≈ 6.8 meV/atom. . . . . . . . . . . . 36

3.7 (color online). Computed adsorption isotherms for Ne, Xe, iNe(1), and dXe(1) on d-AlNiCo at T �=0.35. iNe(1) and dXe(1) are test noble gases having potential parameters described in the text and in Tables 3.1 and 3.2. The effect of varying the interaction strength of the adsorbates on the density increase ΔρN (while keeping the size constant) is negligible on large gases but significant on small gases. . . . . . . . . . . . . . . . . . . . . 39

3.8 (color online). Order parameters as a function of normalized chemical po- tential (as defined in the text) for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed on d-AlNiCo at T � = 0.35. A first-order fivefold to sixfold structural transition occurs in the last three systems, but not in dXe(2). . . 41

3.9 (color online). (a) The minimum of adsorption potential, Vmin(x, y), for Ne on a smoothed d-AlNiCo as described in the text. (b) The variations of the minimum adsorption potentials along the line at x = 0 shown in (a), for the modified and original interactions (solid and dotted curves). . . . . . 44

3.10 Xe on d-AlNiCo. Values of μ�tr for the fivefold to sixfold transition points from 40 K to 140 K (left axis). Transition points at μ�tr > 1 indicate that a transfer of atoms from the second layer to the first layer is required to complete the transition. Also shown is the defect probability as a function of T after the transition occurs (right axis), indicating an increase in defect probability with T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.11 Density plot of Xe on decagonal AlNiCo at 77 K, showing a superposition of the density slices for the 2nd and 4th layers. In the top right, 4th-layer atoms are located directly above the 2nd layer atoms indicating an hexagonal close- packed ABAB stacking, whereas in other regions, such as lower left, the two layers are offset due to stacking fault. . . . . . . . . . . . . . . . . . . . 46

x

3.12 Xe adsorption on d-AlNiCo. (a) Minimum potential energy surface of the adsorption potential with free boundary conditions. (b) Adsorption isotherms of the first layer from a set of 30 simulations at 77 K using the free cell described in the paper. Five density profiles and FTs at point p� of (b) are shown in (c) to (g), representing all possible orientations of hexagonal domains. (h) Schematic diagram illustrating the correspondence between the orientations of the hexagonal domains observed in the density profiles (c) to (g). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.13 Xe adsorption on d-AlNiCo. Pentagonal defects rotate the orientation of hexagons by (a) θ1 = 24◦ and (b) θ2 = 12◦. . . . . . . . . . . . . . . . . 51

3.14 (color online). Xe adsorption on d-AlNiCo. Locations in P , T of the vertical risers in the isotherms corresponding to the first (square), second (circle), and third (triangle) layer formation. The heats of adsorptions, qst, are 270, 129, and 125 meV/atom respectively, calculated as described in the text. The inset figure shows qst obtained from the simulations as well as from the experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 (color online). (a-e) Adsorption potential map, calculated by minimizing the adsorption potential of one molecule on a decagonal Al-Ni-Co along z direction and all rotational degrees of freedom at every coordinates (x, y). Red numbers represent the average value of the adsorption energies. (f) Top view of the decagonal Al-Ni-Co substrate 51.2x51.2 A˚2: Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red), Ni-toplayer (green), and Ni-otherlayers (blue). . . . . . . . . . . . . . . . 71

5.2 (color online). Isothermal adsorption densities of hydrocarbons on a decago- nal Al-Ni-Co: (a) methane (from left to right T = 68, 85, 136, 185 K), (b) propane (T = 80, 127, 245, 365 K), (c) hexane (T = 134, 170, 267, 450 K), (d) octane (T = 162, 210, 324, 450, 565 K), and (e) benzene (T = 209, 270, 418, 555 K). The inset in each figure is the density along z direction at pressure corresponding to point d. Xenon (red) is plotted in panel (a) for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 (a) and (b) Calculated density of methane adsorbed on a decagonal Al-Ni- Co at pressures corresponding to points ”a” and ”c” of the 68 K isotherm shown in Figure 5.2.a, respectively. (c) Fourier transform of the density plot shown in (b), consistent with 5-fold ordering of the methane near monolayer completion. (d) Order parameter (left axis, as calculated in Equation 3.14) as a function of pressure for the 68 K isotherm (right axis), indicating no sharp transition to 6-fold ordering. . . . . . . . . . . . . . . . . . . . . 74

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5.4 Calculated density of propane, hexane, and octane adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion. Propane forms a dis- ordered structures, whereas hexane and octane tend to form close-packed structures indicated by stripe features with increasing order for longer chain. 76

5.5 (Top row) Calculated density of benzene adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion for 209 K, 270 K, and 418 K. (middle row) Density profile of the geometrical center of density shown in the top row. (bottom row) Fourier transform of the density plot shown in the middle row, showing 5-fold ordering at 209 K, mixture of 5-fold and 6-fold structures at 270 K, and mostly 6-fold features at 418 K. . . . . . . 78

5.6 (left axis) Order parameter ρ5−6 = N5/(N5 +N6) (Nn denotes the number of molecules having n nearest neighbors) as a function of temperature at 0.01 atm of pressure. (right axis) Adsorption isobar showing the number of adsorbed molecules as a function of temperature. Verticel dashed lines correspond to T = 209, 270, and 418 K whose density profiles are plotted in Figure 5.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.1 (color online). Size-pressure approximation for Fe nanoparticles obtained by equating the deviation of average bond length from the bulk value due to curvature 1/R (in the case of particle) and due to pressure P (in the case of bulk). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.2 (color online). Ternary phase diagram for Fe-Mo-C nanoparticles of R ∼ ∞, 1.23, 0.62, 0.41 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.3 (color online). a) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x- Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”1”, represents [Fe4/5Mo1/5]1−x-Cx crossing the boundary phase Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed verti- cal lines, labeled as ”4” and ”5”, represent [Fe4/5Mo1/5]1−x-Cx crossing the boundary phases Fe←→Mo2C and τ3, as shown in figure 7.2(d). . . . . . . 97

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7.4 (color online). a) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x- Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”2” and ”3”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phase Fe2Mo↔Mo2C and Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”6”, ”7”, and ”8”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phases Fe2Mo↔Mo2C, Fe←→Mo2C, and τ3 ↔Mo2C, as shown in figure 7.2(d). . . . . . . . . . . 98

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List of Tables

3.1 Parameter values for the 12-6 Lennard-Jones interactions. TM is the label for Ni or Co. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Range, average (〈Vmin〉), and standard deviation (SD) of the interaction Vmin(x.y) on the d-AlNiCo. Effective parameters of the gas-substrate in- teractions (Dgs, σgs, D�gs, σ�gs), and, for comparison, the best estimated well depths DGrgs on graphite [106]. . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Results for Ne, Ar, Kr, and Xe adsorbed on d-AlNiCo. Tt is taken from reference [109]. The density increase (ΔρN ) in the first and second layers is calculated at T � = 0.35 from point (A) to (B) and (C) to (D) in Figure 3.2, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Summary of adsorbed noble gases on d-AlNiCo that undergo a first-order fivefold to sixfold structural transition and those that do not. . . . . . . . 42

4.1 List of structure prototypes used to fit the EAM potentials for hydrocarbon adsorption on Al-Co-Ni. For elemental Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure (DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed (HEX), simple cubic (SC), and simple hexagonal (SH). The Al-Co-Ni ternaries are taken from the database of alloys [122]. . . . . . . . . . . . . . . . . . . . . . . 53

4.2 (Top part) List of structures used to fit EAM potential for elemental alu- minum. ρi is charge density at atom-i, ΔE = EEAM−EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3 · re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 56

4.3 (Top part) List of structures used to fit EAM potential for elemental cobalt. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”HEX at a0 · c” indicates that the structure is HEX with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 57

xiv

4.4 (Top part) List of structures used to fit EAM potential for elemental nickel. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 58

4.5 (Top part) List of structures used to fit EAM potential for Al-Co-Ni systems. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV) and the first knot at (0,0) is assumed. The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . . . . . . . . . . . . . . . . . 60

4.6 (Top part) List of structures used to fit EAM potential for hydrocarbons. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . 61

4.7 (Top part) List of structures used to fit EAM potential for alkanes and benzene. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.8 Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. Fitting structures are given in Table 4.9 . . . 64

4.9 Fitted energies calculated using EAM parameters in Table 4.8. Methane up represents methane with one H below C and three H above C. Methane down is inverse of methane up. The unit for energy is eV/atom except for the adsorption energy which is in eV/molecule. . . . . . . . . . . . . . . . . 65

5.1 Parameter values for the adsorbate-adsorbate interactions used for hydro- carbon adsorption on a decagonal Al-Ni-Co. Intermolecular energies are calculated as a sum of pair interactions. For methane-methane, the C-H is taken as the geometrical mean for parameter A and as the arithmetic mean for parameters B and C. . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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6.1 Fitted parameters for the charge density and pair interaction functionals of the EAM potentials for pure elements, continued in Table 6.2. The fitting structures are given in Tables 6.5 and 6.6. The parameters for the embedding functionals are given in Tables 6.3 and 6.4. . . . . . . . . . . 83

6.2 continuation of Table 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3 Fitted knots of cubic spline of the embedding functionals for the EAM potentials of pure elements, continued in Table 6.4. The first knot at (0,0) is assumed. The fitting structures are given in Tables 6.5 and 6.6. . . . . . 85

6.4 continuation of Table 6.3. . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.5 (left part) Structures used to fit EAM potentials for pure elements, con- tinued in Table 6.6. The EAM potentials are fitted to the ab initio ener- gies in body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close-packed (HEX), diamond structure (DIA), and groundstate structure at various pressures obtained by expanding/compressing the equillibrium lattice constant a by a factor from 0.9 to 1.16 corresponding to a range of charge density from ρmax to ρmin. (right part) Lattice constansts calculated using the fitted parameters. The literature values aLIT are taken from [146]. 87

6.6 continuation of Table 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . 88

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Acknowledgements

I would like to thank my advisor, Prof. Stefano Curtarolo for all the academic and financial supports. I wish to thank all collaborators of this work, Prof. Renee D. Diehl, Prof. Milton W. Cole, Prof. L W. Bruch, Dr. Nicola N. Ferralis, and Dr. Andrea Trasca. I thank all my teachers for their mentoring and my committee members, especially Prof. Teh Y. Tan, Prof. Laurens E. Howle, and Prof. Xiaobai Sun for their dedications and insightful advice. I thank all my colleagues, in particular Dr. Neha Awasthi, Dr. Aiqin Jiang, Dr. Roman Chepulskyy for encouragements and valuable discussions, and Dr. Aleksey Kolmogorov for his mentoring and experienced advice on VASP. I also wish to thank all the staff in the department of Mechanical Engineering and Materials Science for the administrative helps.

I thank the National Science Foundation and Honda Research Institute for fund- ing this research. I also thank San Diego Super Computers (SDSC), Texas Ad- vanced Computing Center (TACC), National Center for Supercomputing Applica- tions (NCSA), and Pittsburgh Supercomputing Center (PSC) for all the computing time through Teragrid projects. Thanks also go to Duke Clusters for additional allocations.

I can never thank enough to my father whose talent in numbers has introduced me to math and engineering, my mother and my sisters for their endless prayers, love, and encouragements throughout my life. I am deeply grateful to be blessed with a brilliant, loving, and beautiful wife, Prof. Lisa M. Peloquin, who provides continuous prayers, love, and supports in so many ways. I also thank our friends especially Laura Heymann for her kind helps in many occassions. Thanks also go to Pepito, Clementine, Hamzah, John and other aquatic species for being excellent family members.

Finally, all praises are due to Allah, the Creator (al-Khaliq), the Loving (al- Rahman), and the All Knowing (al-Alim). I am thankful for all the opportunities, health, and blessings that enable me to finish this dissertation. I am deeply humbled by every piece of knowledge that I learned.

xvii

Chapter 1

Introduction

Quasicrystals (QCs) were discovered in 1982 by Dr. Shechtman during his X-ray mea-

surements on Al-Mn compounds. Similar to crystals, QCs consist of atoms arranged

in regular patterns having long-range order, i.e. the diffraction patterns show discrete

spots. However, they do not have any translational periodicities. The discrete spots

come from the rotational symmetries. A variety of stable and metastable QCs have

been successfully synthesized. Among the first high-quality samples are icosahedral

AlCuFe [1], decagonal AlNiCo [2], and icosahedral AlPdMn [3]. Today, hundreds of

quasicrystalline phases are known, tens of which are stable [4]. The majority of them

are derived from aluminum-transition metal family [5].

QCs have been shown to have lower coefficients of friction than most metals. For

example, the static friction μs between two clean surfaces of icosahedral AlPdMn is

≈ 0.6 [6], whereas μs for Ni(110) and Cu(111) is ≈ 4 [7]. Kinetic friction tests using pin-on-disk technique with diamond pin show that among materials with compara-

ble hardness included in the study, icosahedral AlPdMn exhibits the lowest friction

(μ = 0.05), compared to window glass (μ = 0.08), sintered Al2O3 (μ = 0.13), or

hard Cr-steel (μ = 0.13) [8]. Detailed measurements of friction as a function of struc-

tural perfection in icosahedral AlCuFe quasicrystal show that the minimum friction is

achieved for sample with the best quasilattice perfection [9]. Decagonal AlNiCoSi has

been verified to have lower friction than Cr2O3, which represents the most advanced

technology for use on piston rings in automotive engines [10]. Further evidence of

reduced friction is demonstrated in decagonal AlNiCo, in which the friction on the

2-fold periodic surface is eight times higher than that on the decagonal surface [11]

1

(Note that in decagonal quasicrystals, there is a direction along which the quasicrys-

talline surfaces are stacked periodically).

The reduced friction between two quasicrystalline surfaces can be understood by

considering the structure commensurability between them. For surfaces with enough

hardness, atoms can be regarded as fixed in their position in each material. Within

this approximation, it has been theoretically demonstrated that the total interaction

energy between the two surfaces is independent of their relative displacement parallel

to the interface [12]. Therefore, the frictional force which is the gradient of the en-

ergy with respect to the displacement is vanishingly small, a phenomenon known as

superlubricity [12]. Superlubricity has been observed experimentally between tung-

sten and graphite in which certain relative orientations result in nearly zero friction

beyond the limit of the instruments [13]. Since any two QCs do not have common

periodicities at any length scales, superlubricity is expected.

Another characteristic feature of QCs is their resistance to oxidation, which is

quite surprising given that their main constituent, aluminum, is readily oxidized in

ambient conditions [5]. The behavior is particularly spectacular in AlCuLi icosahedral

phase that resists oxidation very well in humid air [5]. The combination of high

hardness, oxidation resistance, and low friction has attracted interests in QCs as

coatings to reduce friction and wear in machine parts, e.g. at the piston-cylinder

interface and in gear boxes. In such environments, hydrocarbon-based oil lubricant

is typically used to overcome the friction caused by surface asperities. Therefore,

to yield a synergic performance of self-lubricating QC coating and lubricant, it is

important to understand the interactions between them. The lubricant must be

able to spread (wet) well on the QC. Furthermore, the structures of the thin film

of lubricant formed on the sliding surfaces which will affect the lubricity need to be

investigated.

2

In this work, we study the process of thin film growth of hydrocarbons and rare

gases adsorbed on a decagonal Al73Ni10Co17 quasicrystal (d-AlNiCo) via computer

simulations. A high-quality and large-size single grain d-AlNiCo has been routinely

grown, making it an excellent substrate to study adsorption [14, 15, 16]. Chapter 3 is

devoted to the adsorptions of rare gases. The absence of chemical reactivity of noble

gases will be utilized to elucidate the effects of quasicrytallinity of the substrate on the

growth and structure of the adsorbed film. In Chapter 5, the simulation results for

selected hydrocarbons are presented. We study the adsorption behaviors of propane,

hexane, and octane. From these molecules, we may extrapolate the results for longer

alkanes which constitute typical oil lubricants. In addition, methane and benzene are

also studied due to their interesting symmetries.

We develop classical many-body interatomic potentials of Al, Ni, Co, C, and H

which will be suitable for simulating hydrocarbons adsorptions on Al-Ni-Co involving

thousands of atoms. To our knowledge, such potentials have not existed. The poten-

tials are based on the embedded-atom method (EAM) [17] with parameters fitted to

electronic energies of various structures evaluated via first principle quantum calcu-

lations. The generation of the EAM potentials is presented in Chapter 4. In Chapter

6, we extend the procedures to develop a consistent set of EAM potentials for pure

elements in the periodic table. These potentials are also fitted to ab initio energies.

A consistent set of EAM potentials having the same parametrizations enables one to

use the same computer code for different potentials.

This manuscript contains a different reserach which is part of our doctoral work,

namely in the field of nanocatalysis for synthesis of single-walled carbon nanotubes

(SWCNTs). In Chapter 7, we present the calculation of phase diagram of Fe-Mo-

C system under pressure for studying the effects of Mo in the thermodynamics of

Fe:Mo nanoparticles as catalysts for growing SWCNTs. The decomposition of two

3

Fe:Mo particles namely FeMo and Fe4Mo into various stable phases are analyzed.

The implications of the formation of these phases to the growth of SWNCTs as well

as the estimated minimum size of nanotubes that can be produced are discussed.

4

Chapter 2

Methods

2.1 Density functional theory

2.1.1 Born-Oppenheimer approximation

In a calculation of electronic structure of materials, one solves an eigenvalue problem

of time-independent Schro¨dinger equation:

HΨ(R, r) = EΨ(R, r) (2.1)

where R ∈ {Rn} and r ∈ {ri} are the vector coordinates of the nuclei and electrons, respectively. The energy operator (Hamiltonian), H, is given by

H = − ∑ n

� 2

2Mn �2Rn−

∑ i

� 2

2mi �2ri+

∑ m<n

e2ZmZn Rn −Rm +

∑ i<j

e2

rj − ri − ∑ n,i

e2Zn ri −Rn (2.2)

The first two terms represent the kinetic energies of the nuclei and the electrons,

respectively. The third and fourth terms represent the nucleus-nucleus and electron-

electron potential energies. The last term is the nucleus-electron energy. It is compu-

tationally beyond the capability of current computers to solve Equation 2.1 using the

full Hamiltonian. An approximation, known as Born-Oppenheimer approximation,

is made by realizing that nuclei are significantly heavier than electrons so the nuclei

move much more slowly than the electrons. The electrons can adapt themselves to

the current configuration of nuclei. Using this approximation, we can decouple the

electronic and the ionic parts of the Hamiltonian. Consider the nuclei fixed at a given

5

configuration, α, and solve the following electronic Schro¨dinger equation:

[ − ∑

i

� 2

2mi �2ri +

∑ i<j

e2

rj − ri − ∑ n,i

e2Zn ri −Rαn

] ψα(r) = Eα(R)ψα(r) (2.3)

The total energy, E, is calculated by taking the electronic contribution, Eα(R), as a

potential energy operator in the ionic Schro¨dinger equation:

[ − ∑ n

� 2

2Mn �2Ri +

∑ m<n

e2ZmZn Rm −Rn + E

α(R)

] Φα(R) = EΦα(R) (2.4)

Mostly, one is interested only in the electronic part, i.e. Equation 2.3. Even though

Equation 2.3 contains only the electronic part of the system, the large number of

variables (e.g. coordinates of all electrons) makes it remain intractable. In addition,

it is the electron-electron interaction that makes the problem so difficult to solve. This

interaction is a result of correlation between electrons (the probability of finding an

electron depends on where the rest of the electrons are) and the fact that electrons

are fermions requiring antisymmetric wavefunctions (the many-electron wavefunction

gains a factor of -1 everytime two electrons exchange their coordinates). If this term

were absent, the Hamiltonian would be just a sum of many one-electron Hamiltonians,

known as independent electron approximation.

2.1.2 Slater determinant

An antisymmetric N -electron wavefunction can be constructed from N one-electron

wavefunctions using Slater determinant [18] defined as:

Ψ(x1,x2, . . . ,xN) = 1√ N !

ψ1(x1) ψ2(x1) . . . ψN(x1) ψ1(x2) ψ2(x2) . . . ψN(x2) . . . . . . . . . . . .

ψ1(xN) ψ2(xN) . . . ψN(xN)

(2.5)

6

where ψk(xi) denotes the k-th one-electron wavefunction being occupied by an elec-

tron with spin-orbital coordinate xi = (si, ri), with si being the spin state and ri the

spatial coordinate. For this reason, ψk is also called the spin-orbital k. The factor 1√ N !

arrives from the normalization of the total wavefunction and orthonormality among

the spin orbitals:

∫ ΨΨ∗dx1 . . . dxN = 1 (2.6)

∫ ψk(r)ψ

∗ k(r)dr = 1 (2.7)

∫ ψk(r)ψ

∗ l (r)dr = 0 (2.8)

Using the orthonormality of the spin-orbitals, it can be shown that the charge density

of the Slater determinant can be written as n(x) = ∑

k |ψk(x)|2.

2.1.3 Hartree-Fock equations

In the electronic Schro¨dinger equation (Equation 2.3), the Hamiltonian can be written

as follows:

H = ∑

i

h(i) + 1

2

∑ i�=j

g(i, j) (2.9)

h(i) ≡ −1 2 �2i −

∑ n

Zn |ri −Rn| (2.10)

g(i, j) ≡ 1|rj − rj| (2.11)

where h(i) depends only on ri and g(i, j) depends on ri and rj. The energy of

the system is calculated by taking the expectation value of the Hamiltonian in the

total wavefunction, E = 〈Ψ|H|Ψ〉. By employing the orthonormality of ψk as in the

7

calculation of charge density n(x), we have

〈Ψ| ∑

i

h(i)|Ψ〉 = ∑ k

〈ψk|h|ψk〉 = ∑ k

∫ dxψ∗k(x)h(r)ψk(x) (2.12)

Note that in the first equation, when |Ψ〉 is written as a Slater determinant, only i = k appears due to orthonormality of ψk, and the summation over electron index i inside

the many-electron wavefunction |Ψ〉 becomes a summation over k inside individual spin-orbital ψk, and we can drop the index h(i = k) for convenience. The integral∫

dx denotes integral over spatial coordinates and a sum over the spin-degrees of

freedom. Similarly, we have

〈Ψ| ∑ i,j

g(i, j)|Ψ〉 = ∑ k,l

〈ψkψl|g|ψkψl〉 − ∑ k,l

〈ψkψl|g|ψlψk〉 (2.13)

〈ψkψl|g|ψmψn〉 = ∫

dx1ψ ∗ k(x1)

[∫ dx2ψ

∗ l (x2)

1

|r1 − r2|ψn(x2) ] ψm(x1)(2.14)

The total energy is then

E = ∑ k

〈ψk|h|ψk〉+ 1 2

∑ k,l

[〈ψkψl|g|ψkψl〉 − 〈ψkψl|g|ψlψk〉] (2.15)

This expression shows that the electron-electron interaction, g(i, j), consists of two

terms, the first one is known as Coulomb energy (or Hartree term), and the second

one is the exchange energy term. To see how this derivation also reduces the many-

electron Schro¨dinger equation to a set of one-electron equations, we differentiate the

energy with respect to a particular spin-orbital, e.g. spin-orbital 〈ψi|. Note that i is not electron index, but a specific value of spin-orbital index k, after the derivation,

8

we may replace i by k to conform to the usual notation:

δE

δ〈ψi| = h|ψi〉+ 1

2

[∑ l

〈ψl| 1|r− r′| |ψl〉 ] |ψi〉+ 1

2

[∑ k

〈ψk| 1|r− r′| |ψk〉 ] |ψi〉

−1 2

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x)ψi(x ′)− 1

2

∑ k

∫ dxψ∗k(x

′) 1

|r− r′|ψi(x)ψk(x ′)

(2.16)

By changing indices (k → l) in the third term and (k → l,x→ x′) in the last term, we get

δE

δ〈ψi| = h|ψi〉+ [∑

l

〈ψl| 1|r− r′| |ψl〉 ] |ψi〉 −

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x)ψi(x ′)

(2.17)

This expression is known as Fock operator acting on |ψi〉. Replacing back i by k, and defining the eigenvalues of Fock operator as �k, we arrive at the Hartree-Fock

equation:

HHFψk = �kψk (2.18)

HHFψk =

[ −�

2

2 − ∑ n

Zn |r−Rn|

] ψk(x) +

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x ′)ψk(x)

− ∑

l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψk(x ′)ψl(x)

(2.19)

Note that the sum of the eigenvalues �k is not the total energy E. However, by

comparing Equation 2.15 and 2.19, E can be recovered using the following equation

E = 1

2

∑ k

[�k + 〈ψk|h|ψk〉] (2.20)

The last term (the exchange term) in Equation 2.19 is nonlocal because the Hamil-

tonian HHF operates on ψk(x) at a particular x, but the operator itself is a function

9

of ψk(x ′) at all possible x′. This nonlocality makes the Hartree-Fock Hamiltonian

difficult to evaluate in large systems involving many atoms and electrons, hence not

suitable for solids. Most electronic structure calculations for solids are based on

density functional theory (DFT) discussed in the following sections.

2.1.4 Kohn-Sham equations

In DFT, the nonlocal exchange term is given by an effective exchange potential that

depends on electronic charge density. Furthermore, all other potential operators are

expressed in charge density rather than in spin-orbitals. This approach reduces the

number of degrees of freedom significantly since all electron coordinates that enter in

the spin-orbitals are now replaced by charge density that is a function only on one

coordinate r. The DFT energy functional is given by

E(n) = T (n) +

∫ Vext(r)n(r)dr+

1

2

∫∫ n(r′)

1

|r− r′|n(r)dr ′dr+ Exc(n) (2.21)

where T being the electronic kinetic energy and Vext being the electron-nuclei electro-

static potential. We know that for given wavefunctions we can calculate the charge

density, n(r) = ∑

k |ψk(r)|2, however the reverse is not obvious. The formality that proves there exists one-to-one mapping between charge density and wavefunctions

was developed by Hohenberg-Kohn [19]. Hohenberg-Kohn theorem also proves that

there exists Exc(n) that will produce the exact ground state of the system. The DFT

Schro¨dinger equation can be derived by taking the variation of E(n) with respect to

n:

δE

δn =

δT

δn + Vext +

∫ n(r′)dr′

|r− r′| + δExc δn

(2.22)

As we have derived the Hartree-Fock equation (Equation 2.19), if we write the DFT

many-electron wavefunction as a Slater determinant of spin-orbitals and orthonor-

10

mality among the spin-orbitals, then each DFT spin-orbital equation satisfies

[ −1 2 �2 + Veff (r)

] ψk(r) = �kψk(r) (2.23)

Veff (r) ≡ Vext(r) + ∫

n(r′)dr′

|r− r′| + δExc δn

(2.24)

The total energy is related to the eigenvalues �k as follows

E = ∑ k

�k − 1 2

∫∫ n(r′)

1

|r− r′|n(r)dr ′dr+ Exc(n)−

∫ δExc(n)

δn n(r)dr (2.25)

Equations 2.24-2.25 are known as Kohn-Sham equations. Since the potential opera-

tors depend on charge density, Equation 2.24 must be solved self-consistently. One

starts from a trial electronic charge density to construct the potentials and solve for

the eigenvalues and spin-orbitals wavefunctions. A new charge density is then con-

structed from the spin-orbitals and the procedure is repeated until the charge and

the wavefunctions are self-consistent within a certain accuracy.

2.1.5 Local density and generalized gradient approximations

In Kohn-Sham equations, there are two terms involving exchange interaction, namely

Exc(n) and δExc/δn. For a given charge density, the first term (the exchange energy)

does not depend on the charge functional on r, whereas the second term (the ex-

change potential) will depend on r if the charge density does. This means that for

a nonhomogeneous systems, exchange potential can be expanded in charge density

and its derivatives:

Vxc ≡ δExc(n) δn(r)

= Vxc(n(r), |�n(r)|, |�(�(n(r)))|, ...) (2.26)

As a first approximation, one neglects all the gradients of charge density in the

exchange potential, this is known as local density approximation (LDA) since the

11

exchange potential depends on charge density only at a particular value of r. It

means that LDA gives an exact ground state for homogeneous electron gas since in

this case the gradients vanish. This allows one to write the LDA exchange energy as

a sum of exchange energy per electron in a homogeneous electron gas, �homxc (n):

ELDAxc =

∫ �homxc (n)ndr (2.27)

LDA also gives accurate results for systems where the charge density does not vary

too rapidly such as in metals. For nonhomogenous systems such as transition metals,

semiconductors, or slabs, LDA is known to underestimate the energy (hence the band

gap). A more accurate approximation, known as generalized gradient approximation

(GGA), includes the first gradient of charge and the exchange energy is given by

EGGAxc =

∫ �GGAxc (n(r), |∇n(r)|)n(r)dr (2.28)

Several techniques exist to parametrize �GGAxc : Perdew-Wang 1986 (PW86) [20, 21],

Perdew-Wang 1991 (PW91) [22], Becke [23], Lee-Yang-Parr (LYP) [24], and Perdew-

Burke-Enzerhof (PBE) [25, 26]. In this work, we use GGA-PBE functional for the

exchange energy.

2.1.6 Pseudopotentials

Kohn-Sham equations can be solved in different ways depending on the choice of

potentials and basis functions to expand the wavefunctions. Some considerations in

solving the Kohn-Sham equations are: (1) potentials becomes very strong near the

nuclei, whereas at far regions they are relatively weak, (2) wavefunctions fluctuate

more near nuclei than in the interstitial regions, (3) the symmetry of the potentials are

approximately spherical near the nuclei whereas at larger distances, the symmetry of

the crystal dominates. Some of the known methods are augmented plave wave (APW)

12

[27, 28], linearized augmented plane wave (LAPW) [29, 30], Orthogonalized plave-

wave (OPW) [31], and pseudopotential method [32, 33]. LAPW is the most accurate

method available. It uses spherical harmonics to expand the wavefunctions in the core

region near the nuclei and plane waves for the interstitial region. Pseudopotential

methods use only plane wave basis set. The number of plane waves can be kept

small by treating an atom as consisting of an effective core (nucleus + core electrons)

and valence electrons. Even though pseudopotentials are not as accurate as LAPW

(which as a contrast fall into the class of fullpotential methods), it provides good

results and becomes the most common choice for its lower computational costs than

LAPW.

2.2 Vienna Ab initio Simulation Package

All ab initio quantum calculations in this work are done using Vienna Ab initio Sim-

ulation Package (VASP) [34, 35]. The calculations are performed in the generalized

gradient approximation (GGA) [36, 37] with exchange correlation as parametrized

by Perdew, Burke, and Ernzerhof (PBE) [25]. To reduce the number of plane waves,

projector augmented-wave (PAW) pseudopotentials are used [38, 39].

VASP uses a plane wave basis set to expand the wave functions in solving the

Kohn-Sham equation in reciprocal space. The evaluation of total electronic energy per

unit cell is done by integrating the energy of all electronic states in the Brillouin zone.

In practice, the Brillouin zone is divided into grids and the integration is replaced by

a sum over k-points. In our work, the k-point grids are generated automatically using

Monkhorst-Pack scheme [40]. For hexagonal structures, an additional shift is used

so that the grids are centered at the Γ point (k = (0, 0, 0)). Typical setting of the

number of k-points as many as 3000/number-of-atoms is usually sufficient to achieve

a converged energy with an accuracy better than 10 meV/atom. Unless otherwise

13

stated, all structures are fully relaxed: atoms position as well as unit cell’s size and

shape are allowed to relax to find the equillibrium configurations.

2.3 Classical interatomic potentials

Due to QC’s lack of periodicity, a large enough simulation cell is necessary to cap-

ture the effect of substrate’s structure on the adsorbed films. A typical cell con-

tains thousands of atoms which makes the simulation unpractical to be performed

quantum-mechanically. Therefore, classical interatomic potentials are needed. For

simple systems, e.g. adsorption of noble gases, pair potentials such as Lennard-Jones

[41] or Morse [42] are sufficient. However, for complex systems, such as adsorption of

hydrocarbons, more accurate potentials are required to take into account many-body

effects in these systems, especially covalent bonds involving carbons which are higly

directional.

Several methods exist to incorporate many-body interations into classical poten-

tials, e.g. force field method (FF) [43, 44, 45, 46], Cluster Expansion method (CE)

[47, 48, 49], and embedded-atom method (EAM) [17, 50]. FF is based on energy-bond

order-bond length relationship [44] and is mostly used for biological systems [43, 51]

and chemical systems where bond formation and breaking are allowed [46]. In CE,

energy of a system is approximated by a converging sum over cluster contributions,

where contribution from large clusters are negligible. In nonperiodic systems, such as

amorphous and quasicrystalline phases, the number of clusters needed can be large

to reach the desired accuracy, therefore CE is mostly suitable to evaluate ground

state energy in periodic systems. EAM is based on the close relationship between

electronic charge density and energy of the system. In quantum physics, this relation-

ship becomes the basis for the density functional theory (DFT) [52, 19, 53], in which

it has been proven that there exists a one-to-one mapping between charge densities

14

and electronic wave functions, and hence energies [19]. Due to the universality of

charge density, in principle EAM can be used in any systems.

EAM was originally developed for examining metallic bulks and surfaces [17, 50].

Later on, charge screening methods have been proposed to modify EAM for use in

covalent systems such as Si [54] and Ge [55]. EAM has been successfully employed

to simulate surface relaxation/reconstructions [56, 57, 58, 59], film growth [60, 61],

and diffusion processes [62, 63]. In this study, we will develop EAM potentials to

simulate hydrocarbons adsorption on d-AlNiCo surface.

In EAM formalism [17, 50], each atom is viewed as being embedded in the material

consisting of all other atoms. The total energy of a system is defined by

Etot ≡ ∑

i

Fi(ρ¯i) + 1

2

∑ i

∑ j �=i

φij(rij), (2.29)

where Fi is the embedding energy of atom i, ρ¯i is the electron density at the vector

position ri, φij is the pair potential, and rij is the distance between atoms i and j.

If ρ¯i is approximated as a sum of individual contribution of the constituents [i.e.,

ρ¯i = ∑ j �=i

ρj(rij), where ρj is the atomic electron density of atom j, the energy is then

only a function of the position of atoms.

For an elemental potential, there are 3 functions needed: F (ρ¯), ρ(r), and φ(r).

For a binary system AB, we will need 7 functions: FA(ρ¯), ρA(r), φA(r), FB(ρ¯),

ρB(r), φB(r), and a cross-pair potential φAB(r). The cross-pair potential is needed

to calculate pair interaction of atoms of different types. In general, in a system

consisting of N different elements, we need N(N + 5)/2 functions.

15

We will use the following functional forms:

ρA(r) = ρAe e −βA(r/rAe −1), (2.30)

φAA(r) = DA[e−2α A(r−rA0 ) − 2e−αA(r−rA0 )], (2.31)

FA(ρ¯) = cubic− splines, (2.32)

φAB(r) = γAB

2 φAA(r) +

1

2γAB φBB(r). (2.33)

In the above equations, A and B denote atom type. ρe, re, and β will be taken from

the database of atomic electron density [64]. Equation 2.31 follows Morse potential

form [42]. The embedding function F will be taken as a natural cubic spline [65].

The expression for the cross-pair interaction φAB follows Haftel’s derivation [59] in

which γAB ≡ ZB/ZA (ZA is the effective charge of the core for atom type A).

2.4 Simplex method

The EAM potentials parameters need to be fit to quantum calculations. The fitting

will be performed using simplex method [66]. Simplex does not require evaluation

of function derivatives which makes it simple to implement. Simplex minimizes N -

dimensional function f by creating P > N simplex points. A simplex point is f

evaluated at a given coordinate. The simplex points will form P -polytope. A min-

imization move is made by replacing the maximum simplex point, hence the worst,

with a point reflected through the centroid of the remaining (P − 1)-polytope. Ex- pansion or shrinking of the polytope are allowed to overcome some local minima or

to converge, respectively.

16

2.5 Grand canonical Monte Carlo

The adsorption simulation will be performed using grand canonical Monte Carlo

(GCMC) method [67, 68, 69]. At constant temperature, T , and volume, V , the

GCMC method explores the configurational phase space using the Metropolis al-

gorithm and finds the equilibrium number of adsorbed atoms (adatoms), N , as a

function of the chemical potential, μ, of the gas, i.e. configuration(μ,N, V ). The

adsorbed atoms are in equilibrium with the coexisting gas: the chemical potential of

the gas is constant throughout the system. In addition, the coexisting gas is taken

to be ideal. With this method we determine adsorption isotherms, ρN , and density

profiles, ρ(x, y), as a function of the pressure, P (T, μ). For each data point in an

isotherm, we perform at least 18 million GCMC steps to reach equilibrium. Each

step is an attempted displacement, creation, or deletion of an atom with execution

probabilities equal to 0.2, 0.4, and 0.4, respectively [70, 71, 72]. At least 27 million

steps are performed in the subsequent data-gathering and -averaging phase.

17

Chapter 3

Noble gas adsorptions on d-AlNiCo

The observed unusual electronic [73, 74] and frictional [75, 76, 77] properties of qua-

sicrystal surfaces stimulate interesting fundamental questions about how these and

other physical properties are altered by quasiperiodicity. Recent progress in the char-

acterization and preparation of quasicrystal surfaces raises new possibilities for their

use as substrates in the growth of films having novel structural, electronic, dynamic

and mechanical properties [78, 79, 80]. The physical behavior of systems involving

competing interactions in adsorption is a subject of continuing interest and is par-

ticularly relevant to the growth of thin films [69]. Several different growth modes

have been observed for the growth of metal films on quasicrystals [81, 82, 83, 84]. A

form of competing interactions seen in adsorption involves either a length scale or a

symmetry mismatch between the adsorbate-adsorbate interaction and the adsorbate-

substrate interaction [85, 86]. Some consequences of such mismatches include den-

sity modulations [87, 88], domain walls [89], epitaxial rotation in the adsorbed layer

[90, 91, 92, 93, 94, 95], and a disruption of the normal periodicity and growth in the

film [96, 97, 98].

The wide range of behavior observed so far indicates that, even in the absence of

intermixing, film growth is strongly affected by chemical interactions between adsor-

bate and substrate. In order to separate these chemical effects from those specific to

quasiperiodic order, we have studied the adsorption of noble gases on a quasicrystal

surface, where both the gas-gas and gas-surface interactions are believed to be simple,

i.e., appreciable chemical interactions and adsorbate-induced surface reconstructions

are absent. In this chapter, we explore the implications of structural mismatch by

18

evaluating the nature of Ne, Ar, Kr, and Xe adsorption on a quasicrystal substrate,

namely the 10-fold surface of decagonal Al73Ni10Co17 quasicrystal (d-AlNiCo QC)

[15, 16].

3.1 Method

3.1.1 Simulation cell

The simulation cell is tetragonal. We take a square section of the surface, A, of side

5.12 nm, to be the (x, y) part of the unit cell in the simulation, for which we assume

periodic boundary conditions along the basal directions. Although this assumption

limits the accuracy of the long range QC structure, it is numerically necessary for

these simulations. To minimize the long range interaction corrections, a relatively

large cutoff (5σgg) is used. Since the size of the cell is relatively large compared to that

of the noble gases, the cell is accurately representative of order on short-to-moderate

length scales. The height of the cell, along the z (surface-normal) direction, is chosen

to be 10 nm (long enough to contain ∼20 layers of Xe). At the top of the cell, a hard-wall reflective potential is employed to confine the coexisting vapor phase. The

simulation results for Xe over d-AlNiCo, presented below, are consistent with both

our results from experiments [99] and virial calculations [100]. Hence, the calculations

may also be accurate for other systems.

3.1.2 Gas-gas and gas-substrate interactions

The gas-gas and gas-substrate interactions are modeled using Lennard-Jones (LJ) 12-

6 potentials, with the gas-gas parameter values �gg and σgg listed in Table 3.1. The

gas-substrate interactions are obtained by summing pair potentials for a gas atom

and all of the substrate atoms in an eigth-layer slab: Al, Ni and Co [15, 101, 100]. The

position of the atoms in the eight-layer slab are taken from the results of a low-energy

19

Table 3.1: Parameter values for the 12-6 Lennard-Jones interactions. TM is the label for Ni or Co.

�gg σgg �gas−Al σgas−Al �gas−TM σgas−TM (meV) (nm) (meV) (nm) (meV) (nm)

Ne 2.92 0.278 9.40 0.264 9.01 0.249 Ar 10.32 0.340 17.67 0.295 16.93 0.280 Kr 14.73 0.360 21.11 0.305 20.23 0.290 Xe 19.04 0.410 24.00 0.330 23.00 0.315

iNe(1) 2.92 0.410 5.45 0.330 5.22 0.315 dXe(1) 19.04 0.278 41.39 0.264 39.67 0.249 dXe(2) 19.04 0.390 25.88 0.320 24.80 0.305 iXe(1) 19.04 0.550 14.96 0.400 14.34 0.385 iXe(2) 19.04 0.675 10.52 0.462 10.08 0.447

electron diffraction LEED analysis of the surface structure of d-AlNiCo [15]. The

gas-substrate interaction parameters are derived using conventional combining rules,

σAB = (σA + σB)/2 and �AB = √

�A�B [102], and experimental heats of adsorption

[100, 99, 103]. The LJ gas-substrate parameters are �gas−Al and σgas−Al for Al, and

�gas−TM and σgas−TM for the two transition metals Ni and Co. All these values are

listed in the upper part of Table 3.1. In the calculation of the adsorption potential,

we assume a structure of the unrelaxed surface taken from the empirical fit to LEED

data [16].

3.1.3 Adsorption potentials

Figures 3.1(a), 3.1(c), 3.1(e), and 3.1(g) show the function Vmin(x, y) of Ne, Ar, Kr,

and Xe on the d-AlNiCo, respectively, which is calculated by minimizing the adsorp-

tion potentials, V (x, y, z), along the z direction at every value (x, y) coordinates:

Vmin(x, y) ≡ min {V (x, y, z)}|along z . (3.1)

20

The figures reveal the fivefold rotational symmetry of the substrate. Dark spots

correspond to the most attractive regions of the substrate. By choosing appropriate

sets of five dark spots, we can identify pentagons, whose sizes follow the inflationary

property of the d-AlNiCo. Note the pentagon at the center of each figure: it will be

used to extract the geometrical parameters λs and λc in Section 3.2.5.

To characterize the corrugation, not well-defined for aperiodic surfaces, we calcu-

late the distribution function f(Vmin), the average 〈Vmin〉 and standard deviation SD of Vmin(x, y) as:

f(Vmin)dVmin ≡ probability { Vmin ∈ [Vmin, Vmin + dVmin[

} (3.2)

〈Vmin〉 ≡ ∫ ∞ −∞

f(Vmin)Vmin dVmin, (3.3)

SD2 ≡ ∫ ∞ −∞

f(Vmin)(Vmin − 〈Vmin〉)2 dVmin. (3.4)

Figures 3.1(b), 3.1(d), 3.1(f), and 3.1(h) show f(Vmin) of the adsorption potential for

Ne, Ar, Kr, and Xe on the d-AlNiCo, respectively. Vmin(x, y) extends by more than

2·SD around its average, revealing the high corrugation of the gas-surface interaction in these four systems. The average and SD of Vmin(x, y) for these systems are listed

in the upper part of Table 3.2. In addition to highly corrugated, the potentials are

“deep” because the record maximum well-depth, e.g. for Xe, on a periodic surface is

about 160 meV, viz. on graphite [104]; and the record minimum well-depth is about

28 meV, on Cs [105].

3.1.4 Effective parameters

For every gas-substrate interaction we define two effective parameters σgs and Dgs.

σgs represents the averaged LJ size parameter of the interaction, calculated following

21

-70 -65 -60 -55 -50 -45 -40 -35 Vmin (meV)

f(V m

in )

(a)

(b)

-180 -160 -140 -120 -100 Vmin (meV)

f(V m

in )

(c)

(d)

-220 -200 -180 -160 -140 -120 Vmin (meV)

f(V m

in )

(e)

(f)

-280 -260 -240 -220 -200 -180 -160 Vmin (meV)

f(V m

in )

(g)

(h)

Ne/QC Ar/QC

Xe/QCKr/QC

Figure 3.1: (color online). Computed adsorption potentials for (a) Ne, (c) Ar, (e) Kr, and (g) Xe on the d-AlNiCo, obtained by minimizing V (x, y, z) with respect to z. The distribution of the minimum value of these potentials is plotted in (b, d, f, and h) respec- tively: the solid line marks the average value 〈Vmin〉, the dashed lines mark the values at 〈Vmin〉±SD.

22

Table 3.2: Range, average (〈Vmin〉), and standard deviation (SD) of the interaction Vmin(x.y) on the d-AlNiCo. Effective parameters of the gas-substrate interactions (Dgs, σgs, D�gs, σ�gs), and, for comparison, the best estimated well depths DGrgs on graphite [106].

Vmin range < Vmin > SD Dgs σgs D � gs σ

� gs D

Gr gs

(meV) (meV) (meV) (meV) (nm) (Dgs/�gg) (σgs/σgg) (meV) Ne -71 to -33 -47.43 6.63 43.89 0.260 15.03 0.935 33 Ar -181 to -85 -113.32 13.06 108.37 0.291 10.50 0.856 96 Kr -225 to -111 -145.71 15.68 140.18 0.301 9.52 0.836 125 Xe -283 to -155 -195.46 17.93 193.25 0.326 10.15 0.795 162

iNe(1) -65 to -36 -45.11 4.08 43.89 0.326 15.03 0.795

dXe(1) -305 to -150 -207.55 29.18 193.25 0.260 10.15 0.935

dXe(2) -295 to -155 -199.40 19.33 193.25 0.316 10.15 0.810

iXe(1) -248 to -170 -195.31 11.21 193.25 0.396 10.15 0.720

iXe(2) -230 to -180 -194.25 7.77 193.25 0.458 10.15 0.679

the traditional combining rules [102]:

σgs ≡ xAlσg−Al + xNiσg−Ni + xCoσg−Co, (3.5)

where xAl, xNi, and xCo are the concentrations of Al, Ni, and Co in the QC, respec-

tively. Dgs represents the well depth of the laterally averaged potential V (z):

Dgs ≡ −min {V (z)}|along z . (3.6)

In addition, we normalize the σgs and Dgs with respect to the gas-gas interactions:

σ�gs ≡ σgs/σgg, (3.7)

D�gs ≡ Dgs/�gg. (3.8)

The values of the effective parameters σgs, Dgs, σ � gs, and D

� gs for the four gas-surface

interactions are listed in the upper part of Table 3.2. We also include the well depth

for Ne, Ar, Kr, and Xe on graphite, as comparison [106].

3.1.5 Test rare gases

As shown in tables 3.1 and 3.2, Ne is the smallest atom and has the weakest gas-gas

and gas-surface interactions (minima of σgg, σgs, �gg and Dgs). In addition, Xe is the

23

largest atom and has the strongest gas-gas and gas-surface interactions (maxima of

σgg, σgs, �gg and Dgs). Therefore, for our analysis, it is useful to consider two test

gases, iNe(1) and dXe(1), which are combinations of Ne and Xe parameters.

iNe(1) represents an “inflated” version of Ne, having the same gas-gas and average

gas-substrate interactions of Ne but the geometrical dimensions of Xe:

{�gg, Dgs, D�gs}[iNe(1)] ≡ {�gg, Dgs, D�gs}[Ne], (3.9)

{σgg, σgs, σ�gs}[iNe(1)] ≡ {σgg, σgs, σ�gs}[Xe]. (3.10)

dXe(1) represents a “deflated” version of Xe, having the same gas-gas and average

gas-substrate interactions of Xe but the geometrical dimensions of Ne:

{�gg, Dgs, D�gs}[dXe(1)] ≡ {�gg, Dgs, D�gs}[Xe], (3.11)

{σgg, σgs, σ�gs}[dXe(1)] ≡ {σgg, σgs, σ�gs}[Ne]. (3.12)

The resulting LJ parameters for iNe(1) and dXe(1) are summarized in the central

parts of Tables 3.1 and 3.2. Furthermore, we also define three other test versions of

Xe: dXe(2), iXe(1), and iXe(2) which have the same gas-gas and average gas-substrate

interactions of Xe but deflated or inflated geometrical parameters. The last three test

gases will be used in Section 3.2.5. The LJ parameters for these gases are summarized

in the lower parts of Tables 3.1 and 3.2. In simulating test gases, we implicitly rescale

the substrate’s strengths so that the resulting adsorption potentials have the same

Dgs as the non-inflated or non-deflated ones (Equations 3.9 and 3.11).

24

3.1.6 Chemical potential, order parameter, and ordering tran-

sition

To conveniently characterize the evolution of the adsorption processes of the gases

we define a normalized chemical potential μ�, as:

μ� ≡ μ− μ1 μ2 − μ1 , (3.13)

where μ1 and μ2 are the chemical potentials at the onset of the first and second layer

formation, respectively. In addition, we introduce the order parameter ρ5−6, defined

as the probability of existence of fivefold defect [70, 71]:

ρ5−6 ≡ N5 N5 + N6

, (3.14)

where N5 and N6 are the numbers of atoms having 2D coordination equal to 5 and

6, respectively. The 2D coordination is the number of neighboring atoms within a

cutoff radius of aNN · 1.366 where aNN is the first nearest neighbor (NN) distance of the gas in the solid phase and 1.366 = cos(π/6) + 1/2 is the average factor of the

first and the second NN distances in a triangular lattice. Note that aNN does not

change appreciably with respect to temperature difference, e.g. aNN of Xe changes

from 0.440 nm at 77 K to 0.443 nm at 140 K.

In a fivefold ordering, most arrangements are hollow or filled pentagons with atoms

having mostly five neighbors. Hence, the particular choice of ρ5−6 is motivated by the

fact that such pentagons can become hexagons by gaining additional atoms with five

or six neighbors. Definition: the five to sixfold ordering transition is defined as a

decrease of the order parameter to a small or negligible final value. The phenomenon

can be abrupt (first-order) or continuous. Within this framework, ρ5−6 and (1−ρ5−6) can be considered as the fractions of pentagonal and triangular phases in the film,

respectively.

25

3.2 Results

3.2.1 Adsorption isotherms

Figure 3.2 shows the adsorption isotherms of Ne, Ar, Kr, and Xe on the d-AlNiCo.

The plotted quantity is the thermodynamic excess coverage (densities of adsorbed

atoms per unit area), ρN), defined as the difference between the total density of

atoms in the simulation cell and the density that would be present if the cell were filled

with uniform vapor at the specified values of P and T . The simulated ranges and the

experimental triple point temperatures (Tt) for Ne, Ar, Kr, and Xe are listed in Table

3.3. A layer-by-layer film growth is visible at low temperatures. Detailed inspection

of the isotherms reveals that there is a continuous film growth (i.e. complete wetting)

at temperatures above Tt (isotherms at T > Tt are shown as dotted curves). This

behavior, observed despite the high corrugation, is interesting as corrugation has

been shown to be capable of preventing wetting [107, 108].

Although vertical steps corresponding to layers’ formation are evident in the

isotherms, the slopes of the isotherms’ plateaus at the same normalized tempera-

tures (T � ≡ T/�gg = 0.35) differ between systems. To characterize this, we calculate the increase of each layer density, ΔρN , from the formation to the onset of the sub-

sequent layer. ΔρN is defined as ΔρN ≡ (ρB − ρA)/ρA and the values are reported in Table 3.3 (points (A) and (B) are specified in Figure 3.2). We observe that, as

the size of noble gas increases ΔρN become smaller, indicating that the substrate

corrugation has a more pronounced effect on smaller adsorbates, as expected since

they penetrate deeper into the corrugation pockets. However, Xe does not follow

this trend. This arises from the complex interplay between the corrugation energy

and length of the potential with respect to the parameters of the gas (σgg, �gg) in

determining the density of the adsorbed layers. In the case of Ne, Ar, and Kr, the

densities at points (A) are approximately the same (ρA = 5.4 atoms/nm 2), whereas

26

that of Xe is considerably smaller (ρA = 4.2 atoms/nm 2), because Xe dimension σgg

becomes comparable to the characteristic length (corrugation) of the potential. This

effect is clarified by the density profile of the films, ρ(x, y), shown in Figures 3.3 and

3.4. As can be seen at points (A), the density profiles of Ne, Ar, and Kr are the

same, i.e. the same set of dark spots appear in their plots. For Xe, some spots are

separated with distances smaller than its core radius (σgg), causing repulsive inter-

actions. Hence these spots will not likely appear in the density profile, resulting in

a lower ρA. More discussion on how interaction parameters affect the shape of the

isotherms is presented in Section 3.2.4. Note that the second layer in each system

has a smaller ΔρN than the first one. The explanation will be given when we discuss

the evolution of density profiles.

Table 3.3: Results for Ne, Ar, Kr, and Xe adsorbed on d-AlNiCo. Tt is taken from reference [109]. The density increase (ΔρN ) in the first and second layers is calculated at T � = 0.35 from point (A) to (B) and (C) to (D) in Figure 3.2, respectively.

simulated T T � ≡ T/�gg Tt ΔρN at T � = 0.35 θr (K) (K) for 1st layer for 2nd layer

Ne 11.8 → 46 0.35 → 1.36 24.55 (12.2-5.3)/5.3=1.30 (11.1-10.2)/10.2=0.09 6◦ Ar 41.7 → 155 0.35 → 1.29 83.81 (7.3-5.5)/5.5=0.33 (6.9-6.4)/6.4=0.08 30◦ Kr 59.6 → 225 0.35 → 1.32 115.76 (6.9-5.5)/5.5=0.25 (6.6-6.3)/6.3=0.05 42◦ Xe 77 → 280 0.35 → 1.27 161.39 (5.8-4.2)/4.2=0.38 (5.2-5.2)/5.2=0 54◦

3.2.2 Density profiles

Figures 3.3 and 3.4 show the density profiles ρ(x, y) at T � = 0.35 for the outer layers

of Ne, Ar, Kr, and Xe adsorbed on the d-AlNiCo at the pressures corresponding to

points (A) through (F) of the isotherms in Figure 3.2.

Ne/d-AlNiCo system. Figure 3.3(a) shows the evolution of adsorbed Ne. At

the formation of the first layer, adatoms are arranged in a pentagonal manner follow-

ing the order of the substrate, as shown by the discrete spots of the Fourier transform

27

10-25 10-20 10-15 10-10 10-5 1 0

10

20

30

40

50

P (atm)

ρ N (a

to m

s/ nm

2 )

Ne/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25

30

35 Ar/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25

30

35 Kr/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25 Xe/QC

P (atm)

ρ N (a

to m

s/ nm

2 )

P (atm)

ρ N (a

to m

s/ nm

2 )

P (atm)

ρ N (a

to m

s/ nm

2 )

(a) (b)

(d)(c)

A

B

C D

E

F

A B

C D

E

F

A B

C D

E

F

A B

C D

Figure 3.2: Computed adsorption isotherms for all the gas/d-AlNiCo systems. The ranges of temperatures under study are: Ne: T = 14 K to 46 K in 2 K steps, Ar: 45 K to 155 K in 5 K steps, Kr: 65 K to 225 K in 5 K steps, Xe: 80 K to 280 K in 10 K steps. Additional isotherms are shown with solid circles at T � = 0.35: T = 11.8 K (Ne), T = 41.7 K (Ar), T = 59.6 K (Kr), and T = 77 K (Xe). Isotherms above the triple point temperatures are shown as dotted curves.

(FT) having tenfold symmetry (point (A)). As the pressure increases, the arrange-

ment gradually loses its pentagonal character. In fact, at point (B) the adatoms are

arranged in patches of triangular lattices and the FT consists of uniformly-spaced

concentric rings with hexagonal resemblance. The absence of long-range ordering in

the density profile is indicated by the lack of discrete spots in the FT. This behavior

persists throughout the formation of the second layer (points (C) and (D)) until the

appearance of the third layer (point (E)). At this and higher pressures, the FT shows

patterns oriented as hexagons rotated by θr = 6 ◦, indicating the presence of short-

28

range triangular order on the outer layer (point (F)). In summary, between points (A)

and (F) the arrangement evolves from pentagonal fivefold to triangular sixfold with

considerable disorder, as the upper part of the density profile at point (F) shows. The

transformation of the density profile, from a lower-packing-density (pentagonal) to

a higher-packing-density structure (irregular triangular), occurs mostly in the mono-

layer from points (A) to (B), causing the largest density increase of the first layer with

respect to that of the other layers (see the end of Section 3.2.2 for more discussion).

Due to the considerable amount of disorder in the final state Ne/d-AlNiCo does not

satisfy the requirements for the transition as defined in Section 3.1.6.

Ar/d-AlNiCo and Kr/d-AlNiCo systems. Figures 3.3(b) and 3.4(a) show

the evolutions for Ar and Kr: they are similar to the Ne case. For Ar, the pentagonal

structure at the formation of the first layer is confirmed by the FT showing discrete

spots having tenfold symmetry (point (A)). The quasicrystal symmetry strongly af-

fects the overlayers’ structures up to the third layer by preventing the adatoms from

forming a triangular lattice (point (E)). This appears, finally, in the lower part of the

density profile at the formation of the fourth layer as confirmed by the FT showing

discrete spots with sixfold symmetry (point (F)). Similar to the Ne case, disorder does

not disappear but remain present in the middle of the density profile corresponding

to the highest coverage before saturation (point (F)). Similar situation occurs also

for the evolution of Kr as shown in Figure 3.4(a).

Xe/d-AlNiCo system. Figure 3.4(b) shows the evolution of adsorbed Xe. At

the formation of the first layer, adatoms are arranged in a fivefold ordering similar

to that of the substrate as shown by the discrete spots of the FT having tenfold

symmetry (point (A)). At point (B), the density profile shows a well-defined triangular

lattice not present in the other three systems: the FT shows discrete spots arranged

in regular and equally-spaced concentric hexagons with the smallest containing six

29

FTρ(x,y)(a)

θ

A

B

C

D

E

F

θ

(b)Ne/QC Ar/QC FTρ(x,y)

A

B

C

D

E

F

r r

Figure 3.3: Density profiles and Fourier transforms of the outer layer at T � = 0.35 for Ne/d-AlNiCo (T = 11.8 K) and Ar/d-AlNiCo (T = 41.7 K), corresponding to points (A) through (F) of Figure 3.2.

30

clear spots. Thus, at point (B) and at higher pressures, the Xe overlayers can be

considered to have regular closed-packed structure with negligible irregularities.

It is interesting to compare the orientation of the hexagons on FT for these four

adsorbed gases at the highest available pressures before saturation (point (F) for Ne,

Ar, and Kr, and point (D) for Xe). We define the orientation angles as the smallest

of the possible clockwise rotations to be applied to the hexagons to obtain one side

horizontal, as shown in Figures 3.3 and 3.4. Such angles are θr = 6 ◦, 30◦, 42◦, and

54◦, for adsorbed Ne, Ar, Kr, and Xe, respectively. These orientations, induced by

the fivefold symmetry of the d-AlNiCo, can differ only by multiples of n ·12◦ [70, 71]. Since hexagons have sixfold symmetry, our systems can access only five possible

orientations (6, 18, 30, 42, 54◦), and the final angles are determined by the interplay

between the adsorbate solid phase lattice spacing, the periodic simulation cell size,

and the potential corrugation. For systems without periodic boundary conditions,

the ground state has been found to be fivefold degenerate, as should be the case

[70, 71].

Xe adsorption on this surface was studied experimentally using LEED, in which the

isobar measurements indicate that the Xe film grows layer-by-layer in the temperature

range 65 K to 80 K [99], consistent with the simulations. Under similar conditions

to the simulation at 77 K, at the lowest coverage, the only discernible change in

the LEED pattern from that of the clean surface is an attenuation of the substrate

beams. After the adsorption of one layer, there are still no resolvable features that

would indicate an overlayer having order different from the substrate. At the onset of

the adsorption of the second layer, however, the LEED pattern shows new diffraction

spots that correspond to 5 rotational domains of a hexagonal structure. Within

each of these domains, the close-packed direction of the Xe is aligned with the 5-fold

directions of the substrate, as also observed in the simulation. In the experiments,

31

all possible alignments are observed owing to the presence of all possible rotational

alignments present within the width of the electron beam (0.25 mm). When the

second layer is complete, these spots are well-defined and their widths are the same as

the substrate spots, indicating a coherence length of at least 15 nm. The average Xe-

Xe spacing measured in the experiment is consistent with the bulk nearest-neighbor

spacing of 0.44 nm. A dynamical LEED analysis of the intensities indicates that the

structure of the multilayer film is consistent with face-centered cubic (FCC) Xe(111).

These structure parameters for the bilayer film are essentially identical to the results

obtained for Xe growth on Ag(111) [110, 111], a much weaker and less corrugated

substrate. This suggests that effect of the symmetry and corrugation of the substrate

potential on the Xe film structure is largely confined to the monolayer.

In every system, the increase of the density for each layer is strongly correlated

to the commensurability with its support: the more similar they are, the more flat

the adsorption isotherm will be (note that the support for the (N + 1)th-layer is the

N th-layer). For example, the Xe/d-AlNiCo system has an almost perfect hexagonal

structure at point (B) (due to its first-order five to sixfold ordering transition as

described in the next section). Hence, all the further overlayers growing on the top of

the monolayer will be at least “as regular” as the first layer, and have the negligible

density increase as listed in Table 3.3.

3.2.3 Order parameters (ρ5−6)

The evolution of the order parameter ρ5−6 is plotted in Figure 3.5 as a function of

the normalized chemical potential, μ�, at T �=0.35 for all the noble gas/d-AlNiCo

systems.

Ne/d-AlNiCo, Ar/d-AlNiCo, and Kr/d-AlNiCo systems. The ρ5−6 plots

for the first four layers observed before bulk condensation are shown in panels (a)−(c).

32

FTρ(x,y)(a)

A

B

C

D

E

F

(b)Kr/QC Xe/QC FTρ(x,y)

A

B

C

D

θr

θr

Figure 3.4: Density profiles and Fourier transforms of the outer layer at T � = 0.35 for of Kr/d-AlNiCo (T = 59.6 K) and Xe/d-AlNiCo (T = 77 K), corresponding to points (A) through (F) of Figure 3.2.

33

μ*

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

μ*

ρ 5 -6

layer 1

(a) (b)

(d)(c)

Ne/QC Ar/QC

Kr/QC Xe/QC

0 0.2 0.4 0.6 0.8 1 1.2 0

0.2

0.4

0.6

0.8

1

μ*

ρ 5 -6

1 1.05 1.1 1.15

0.4

0.5

0.6

layer 1

layer 2

layer 3 layer 4

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.06 1.12

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 μ

ρ 5 -6

layer 1

layer 2

layer 3 layer 4

0 0.2 0.4 0.6 0.8 1 1.2 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.06 1.12 0.3

0.5

0.7

μ*

ρ 5 -6

layer 1

layer 2 layer 3 layer 4

5- to 6- fold transition

Figure 3.5: (color online). Order parameters, ρ5−6, as a function of normalized chemical potential, μ�, (as defined in the text) at T � = 0.35 for the first four layers of (a) Ne, (b) Ar, (c) Kr, and for the first layer of Xe (d) adsorbed on d-AlNiCo. A sudden drop of the order parameter in Xe/QC to a constant value of ∼ 0.017 at μ� ∼ 0.8 indicates the existence of a first-order structural transition from fivefold to sixfold in the system.

As the chemical potential μ� increases, ρ5−6 decreases continuously reaching a con-

stant value only for Kr. At bulk condensation, the values of ρ5−6 are still high,

approximately 0.35 ∼ 0.45. Data at higher temperatures shows a similar behavior (up to T=24 K (T �=0.71) for Ne, T =70 K (T �=0.58) for Ar, and T=90 K (T �=0.53)

for Kr). Thus, we conclude that these systems do not undergo the ordering transition.

Xe/d-AlNiCo system. The ρ5−6 plot for the first layer is shown in panel (d).

In this system, as the chemical potential μ� increases, the order parameter gradually

34

decreases reaching a value of ∼ 0.3 at μ�tr ∼ 0.8. Suddenly it drops to 0.017 and remains constant until bulk condensation. Similar behavior is observed at higher

temperatures up to T=140 K (T �=0.63). This is a clear indication of a five to sixfold

ordering transition, as the first layer has undergone a transformation to an almost

perfect triangular lattice. Figure 3.6(d) shows the total enthalpy in the system, H.

At the transition point μ�tr, the enthalpy has a little step indicating a latent heat of

the transition. The discontinuity of the order parameter ρ5−6 and the presence of

latent heat indicate that the ransition is first-order. The latent heat of this transition

is estimated to be 1.021/151 ≈ 6.8 meV/atom. Despite the evidence for a first-order transition, the nearest neighbor distance, la-

beled rNN , e.g. Xe/d-AlNiCO in Figure 3.6(b), appears to change continuously. This

nearest neighbor distance is defined as the location of the first peak in the pair corre-

lation function because the latter property is more directly comparable to diffraction

measurements. We have also calculated the average spacing between neighbors, d¯NN ,

which is a thermodynamically meaningful quantity (related to the density). This has

a small discontinuity at the transition, providing additional evidence for the first-

order character of the transition. Both quantities, rNN and d¯NN , are shown in Figure

3.6(b). The NN Xe-Xe distance rNN decreases continuously as P increases, starting

from 0.45 nm and saturating at 0.44 nm. The Xe-Xe distance reaches saturation

value before the appearance of the second layer; therefore, the transition is complete

within the first layer. We note that a similar decrease in NN distance was measured

for Xe/Ag(111), but in that case, the NN spacing did not saturate before the onset

of the second layer adsorption [112, 110].

The observed transition from fivefold to sixfold order within the first layer of Xe

on d-AlNiCo can be viewed as a commensurate-incommensurate transition (CIT),

since at the lower coverage, the layer is commensurate with the substrate symmetry

35

0 0.2 0.4 0.6 0.8 1 μ*

ρ 5- 6

2 4 6 8

10

ρ N (a

to m

/n m

2 )

layer formation

st1

layer formation

nd2

b

a

0.2

0.4

0.6 5- to 6- fold transition

0.44

0.45

0.46

(n m

)

c

d

rNN

dNN

r NN d N

N ,

-45

-40

-35

-30

-25

H (e

V )

Δ Η

Figure 3.6: (color online). Xe on d-AlNiCo at T = 77 K. (a) Adsorption isotherm, ρN , versus the normalized chemical potential, μ�. (b) Nearest neighbor distance derived from the first peak of pair correlation function, rNN , (black line), and average spacing between neighbors at equilibrium, d¯NN , (red line). (c) Order parameter ρ5−6 (probability of fivefold defects, defined in Equation 3.14) versus the normalized chemical potential, μ�. (d) Total enthalpy. The transition, which is defined as the point in μ� above which the order parameter remains nearly constant, occurs at μ�tr ∼0.8. The discontinuity in H around μ�tr ∼ 0.8 indicates a first order transition with associated latent heat of the transition. The order parameter ρ5−6 after the transition is ∼ 0.017. Heat of the transition is ≈ 6.8 meV/atom.

36

and aperiodic, while at higher coverage, it is incommensurate with the substrate.

Such transitions within the first layer have been observed before for adsorbed gases,

perhaps most notably for Kr on graphite [113]. There, as here for Xe, the Kr forms

a commensurate structure at low coverage, which is compressed into an incommen-

surate structure at higher coverage. The opposite occurs for Xe on graphite, which

is incommensurate at low coverage and commensurate at high coverage [114]. Such

commensurate-incommensurate transitions have been studied theoretically in many

ways, but perhaps most simply as a harmonic system (balls and springs) having a

natural spacing that experiences a force field having a different spacing [115]. Such a

transition has been found to be first-order for strongly corrugated potentials (in 1D)

but continuous for more weakly corrugated potentials [116]. The transition observed

in our quasicrystal surface suggests that system is within the regime of “strong” cor-

rugation, which was not the case of Kr over graphite [113]. In fact, for the latter

system, both commensurate and incommensurate structures have sixfold symmetry.

A more relevant comparison may be the transition of Xe on Pt(111), from a rect-

angular symmetry incommensurate phase to a hexagonal symmetry commensurate

one, although in that case, the low-temperature phase was incommensurate. That

transition was also found to be continuous [117]. Therefore, while our simulations

indicate that Xe on d-AlNiCo undergoes a CIT, as observed for other adsorbed gases,

the observation of a first-order CIT is new, to our knowledge, and likely arises from

the large corrugation.

3.2.4 Effects of �gg and σgg on adsorption isotherms

In Section 3.2.1 we have briefly discussed how the density increase of each layer (ΔρN)

is affected by the size of the adsorbate (σgg). In addition, since the corrugation of the

potential depends also on the gas-gas interaction (�gg), the latter quantity could a

37

priori have an effect on the density increase. To decouple the effects of σgg and �gg on

ΔρN we calculate ΔρN while keeping one parameter constant, σgg or �gg, and varying

the other. For this purpose, we introduce two test gases iNe(1) and dXe(1), which

represent “inflated” or “deflated” versions of Ne and Xe, respectively (parameters

are defined in Equations 3.9-3.12 and listed in Tables 3.1 and 3.2). Then we perform

four tests summarized as the following:

(1) constant strength �gg, size σgg increases [Ne→iNe(1)]: ΔρN reduces,

(2) constant strength �gg, size σgg decreases [Xe→dXe(1)]: ΔρN increases,

(3) constant size σgg, strength �gg decreases [Xe→iNe(1)]: ΔρN ∼ constant,

(4) constant size σgg, strength �gg increases [Ne→dXe(1)]: enhanced agglomer- ation.

Figure 3.7 shows the adsorption isotherms at T � = 0.35 for Ne, iNe(1), Xe, and

dXe(1) on d-AlNiCo. By keeping the strength constant and varying the size of the

adsorbates, tests 1 and 2 ([Ne→iNe(1)] and [Xe→dXe(1)]), we find that we can reduce or increase the value of the density increase (when ΔρN decreases the continuous

growth tends to become stepwise and vice versa). These two tests indicate that the

larger the size, the smaller the ΔρN . By keeping the size constant and decreasing

the strength, test 3 ([Xe→iNe(1)]), we find that ΔρN does not change appreciably. An interesting phenomenon occurs in test 4 where we keep the size constant and

increase the strength ([Ne→dXe(1)]). In this test the growth of the film loses its step-like shape. We suspect that this is caused by an enhanced agglomeration effect

as follows. Ne and dXe(1) have the same size which is the smallest of the simulated

gases, allowing them to easily follow the substrate corrugation, in which case, the

corrugation helps to bring adatoms closer to each other [100] (agglomeration effect).

The stronger gas-gas self interaction of dXe(1) compared to Ne will further enhance

38

10-25 10-20 10-15 10-10 10-5 1 0

10

20

30

40

P (atm)

ρ (a

to m

s/ nm

2 )

Ne iNe Xe

dX e

Ν

(1 )

(1)

test 1: [Ne → iNe(1)]

test 2: [Xe → dXe(1)]

test 3: [Xe → iNe(1)]

test 4: [Ne → dXe(1)]

Figure 3.7: (color online). Computed adsorption isotherms for Ne, Xe, iNe(1), and dXe(1)

on d-AlNiCo at T �=0.35. iNe(1) and dXe(1) are test noble gases having potential parameters described in the text and in Tables 3.1 and 3.2. The effect of varying the interaction strength of the adsorbates on the density increase ΔρN (while keeping the size constant) is negligible on large gases but significant on small gases.

this agglomeration effect, resulting in a less stepwise film growth of dXe(1) than Ne.

As can be seen, dXe(1) grows continuously, suggesting a strong enhancement of the

agglomeration. In summary, the last two tests (3 and 4) indicate that the effect of

varying the interaction strength of the adsorbates (while keeping the size constant)

is negligible on large gases but significant on small gases.

3.2.5 Effects of �gg and σgg on 5- to 6-fold transition

Strength �gg and size σgg of the adsorbates also affect the existence of the first-order

transition (present in Xe/d-AlNiCo, but absent in Ne, Ar, and Kr on d-AlNiCo).

Hence we perform the same four tests described before and observe the evolution of

the order parameter. The results are the following:

(1) constant strength �gg, size σgg increases [Ne→iNe(1)]: transition appears

(2) constant strength �gg, size σgg decreases [Xe→dXe(1)]: transition disappears 39

(3) constant size σgg, strength �gg decreases [Xe→iNe(1)]: transition remains

(4) constant size σgg, strength �gg increases [Ne→dXe(1)]: remains no transition

The strength �gg has no effect on the existence of the transition (tests 3 and 4), which

instead is controlled by the size of the adsorbates (tests 1 and 2). To further charac-

terize such dependence, we add three additional test gases with the same strength �gg

of Xe but different sizes σgg. The three gases are denoted as dXe (2), iXe(1), and iXe(2)

(the prefixes d- and i- stand for deflated and inflated, respectively). The interaction

parameters, defined in the following equations, are listed in Tables 3.1 and 3.2:

{�gg, Dgs, σgg} [dXe(2)] ≡ {�gg, Dgs, 0.95σgg}[Xe], (3.15)

{�gg, Dgs, σgg}[iXe(1)] ≡ {�gg, Dgs, 1.34σgg}[Xe], (3.16)

{�gg, Dgs, σgg}[iXe(2)] ≡ {�gg, Dgs, 1.65σgg}[Xe]. (3.17)

Figure 3.8 shows the evolutions of the order parameter as a function of the normal-

ized chemical potential for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed

on d-AlNiCo at T �=0.35. All these systems undergo a transition, except dXe(2), i.e.

the transition occurs only in systems with σgg ≥ σgg[Xe] indicating the existence of a critical value for the appearance of the phenomenon. Furthermore, as σgg increases

(iNe(1) → iXe(1) → iXe(2)), the transition shifts towards smaller critical chemical potentials.

3.2.6 Prediction of 5- to 6-fold transition

The critical value of σgg associated with the transition can be related to the charac-

teristic length of the d-AlNiCo by introducing a gas-substrate mismatch parameter

defined as

δm ≡ k · σgg − λr λr

. (3.18)

40

0 0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

μ*

ρ 5 -6

iXe

dXe

iNe

(2)

(1) (1)iXe(2)

Figure 3.8: (color online). Order parameters as a function of normalized chemical poten- tial (as defined in the text) for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed on d-AlNiCo at T � = 0.35. A first-order fivefold to sixfold structural transition occurs in the last three systems, but not in dXe(2).

where k = 0.944 is the distance between rows in a close-packed plane of a bulk LJ

gas (calculated at T = 0 K with σ = 1 [118]), and λr is the characteristic spacing of

the d-AlNiCo, determined from the momentum transfer analysis of LEED patterns

[99] (our d-AlNiCo surface has λr=0.381 nm [99]). With such ad hoc definition, δm

measures the mismatch between an adsorbed FCC[111] plane of adatoms and the d-

AlNiCo surface. In Table 3.4 we show that δm perfectly correlates with the presence

of the transition in our test cases (transition exists ⇔ δm > 0). The definition of a gas-substrate mismatch parameter is not unique. For example,

one can substitute k · σgg with the first NN distance of the bulk gas, and λr with one of the following characteristic lengths: a) side length of the central pentagon in

the potential plots in Figure 3.1 (λs = 0.45nm), b) distance between the center of

the central pentagon and one of its vertices (λc = 0.40nm), c) L = τ · S = 0.45 nm, where τ = 1.618 is the golden ratio of the d-AlNiCo and S = 0.243 nm is the

41

Table 3.4: Summary of adsorbed noble gases on d-AlNiCo that undergo a first-order fivefold to sixfold structural transition and those that do not.

δm transition Ne -0.311 No Ar -0.158 No Kr -0.108 No Xe 0.016 Yes

iNe(1) 0.016 Yes dXe(1) -0.311 No dXe(2) -0.034 No iXe(1) 0.363 Yes iXe(2) 0.672 Yes

k = 0.944 [118] λr = 0.381 nm [99]

δm ≡ (k · σgg − λr)/λr

side length of the rhombic Penrose tiles [15]. Although there is no a priori reason to

choose one definition over the others, the one that we select (Equation 3.18) has the

convenience of being perfectly correlated with the presence of the transition, and of

using reference lengths commonly determined in experimental measurements (λr) or

quantities easy to extract (k · σgg).

3.2.7 Transitions on smoothed substrates

In Figure 3.1 we can observe that near the center of each potential there is a set of five

points with the highest binding interaction (the dark spots constituting the central

pentagons). A real QC surface contains an infinite number of these very attractive

positions which are located at regular distances and with five fold symmetry. Due to

the limited size and shape of the simulation cell, our surface contains only one set

of these points. Therefore, it is of our concern to check if the results regarding the

existence of the transition are real or artifacts of the method. We perform simulation

tests by mitigating the effect of the attractive spots through a Gaussian smoothing

function which reduces the corrugation of the original potential. The definitions are

42

the following:

G(x, y, z) ≡ AGe−(x2+y2+z2)/2σ2G , (3.19)

V (z) ≡ 〈V (x, y, z)〉(x,y) , (3.20)

Vmod(x, y, z) ≡ V (x, y, z) · [1−G(x, y, z)] + V (z) ·G(x, y, z). (3.21)

where G(x, y, z) is the Gaussian smoothing function (centered on the origin and with

parameters AG and σG), V (z) is the average over (x, y) of the original potential

V (x, y, z), and Vmod(x, y, z) is the final smoothed interaction. An example is shown

in Figure 3.9(a) where we plot the minimum of the adsorption potential for a Ne/d-

AlNiCo modified interaction (smoothed using AG = 0.5 and σG = 0.4 nm). In

addition, in panel (b) we show the variations of the minimum adsorption potentials

along line x = 0 for the modified and original interactions (solid and dotted curves,

respectively).

Using the modified interactions (with AG = 0.5 and σG = 0.4 nm) we simulate all

the noble gases of Table 3.4. The results regarding the phase transition on modified

surfaces do not differ from those on unmodified ones, confirming that the observed

transition behavior is a consequence of competing interactions between the adsorbate

and the whole QC substrate rather than just depinning of the monolayer epitaxially

nucleated. Therefore, the simple criterion for the existence of the transition (δm > 0)

might also be relevant for predicting such phenomena on other decagonal quasicrystal

substrates.

3.2.8 Temperature vs substrate effect

Using Xe/d-AlNiCo data, we observe that defects are present at all temperatures that

are simulated (20 to 286 K). The probability of defects increases with temperature,

implying that their origin is entropic, as is the case for a periodic crystal. Figure

43

(a) (b)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-75

-65

-55

-45

-35

Y (nm)

V m

in (y

) ( m

eV )

modified potential original potential

Figure 3.9: (color online). (a) The minimum of adsorption potential, Vmin(x, y), for Ne on a smoothed d-AlNiCo as described in the text. (b) The variations of the minimum adsorption potentials along the line at x = 0 shown in (a), for the modified and original interactions (solid and dotted curves).

3.10-(right axis) shows that the defect probability increases as T increases, while

Figure 3.10-(left axis) shows the trend of the transition point in function of T. At low

temperatures, the sixfold ordering occurs earlier (at lower μ�tr) as the temperature

is increased from 40K to 70K. This trend is expected because the ordering effect

imposed by the substrate corrugation becomes relatively smaller as the temperature

increases. However, this trend is not observed in the higher temperature region (from

70K to 140K). In fact, at higher temperatures the transition point shifts again to

higher μ�tr. This is most likely due to the monolayer becoming less two-dimensional,

allowing more structural freedom of the Xe atoms and thus decreasing the effect of

the repulsive Xe-Xe interaction that would stabilize the sixfold structure. Transitions

having critical μ�tr > 1 indicate that the onset of second-layer adsorption occurs

earlier than the transition to the sixfold structure. When the second layer adsorbs at

T >130K, the density of the monolayer increases by a few percent, thereby increasing

the effect of the repulsive interactions and driving the fivefold to sixfold transition.

44

o f 5

-6 fo

ld tr

an si

tio n

μ t r*

ρ 5-6

μtr*

ρ 5-6

Figure 3.10: Xe on d-AlNiCo. Values of μ�tr for the fivefold to sixfold transition points from 40 K to 140 K (left axis). Transition points at μ�tr > 1 indicate that a transfer of atoms from the second layer to the first layer is required to complete the transition. Also shown is the defect probability as a function of T after the transition occurs (right axis), indicating an increase in defect probability with T .

Interestingly, stacking faults are evident in the multilayer films. This is consistent

with x-ray diffraction studies of the growth of Xe on Ag(111), where stacking faults

were observed for Xe growth under various growth conditions [110, 111], although the

overall structure observed was FCC(111). Such a stacking fault is evident in Figure

3.11, which shows a superposition of Xe layers 2 and 4 at 77 K. The coincidence of

the atom locations in the top left part of this figure is consistent with an hexagonal

close-packed structure (HEX), ABAB stacking, whereas the offsets observed in the

lower part of the figure indicate the presence of stacking faults caused by dislocations

in the layers. We note that while bulk Xe has an FCC structure, and indeed an FCC

structure was found for the multilayer film in the LEED study, calculations of the

bulk structure using LJ pair potentials such as those employed here result in a more

stable HEX structure [119]. The energy difference between the two structures is very

small, and apparently arises from a neglect of d-orbital overlap interactions, which

45

Figure 3.11: Density plot of Xe on decagonal AlNiCo at 77 K, showing a superposition of the density slices for the 2nd and 4th layers. In the top right, 4th-layer atoms are located directly above the 2nd layer atoms indicating an hexagonal close-packed ABAB stacking, whereas in other regions, such as lower left, the two layers are offset due to stacking fault.

are more effective in FCC than in HEX structures [119, 120]. Although the simulated

film is HEX instead of FCC, the main conclusions concerning the growth mode of Xe

on the quasicrystal are not affected [70].

3.2.9 Orientational degeneracy of the ground state

In was mentioned in the previous section that after the ordering transition is complete,

the resulting sixfold structure is aligned parallel to one of the sides of the pentagons

in the Vmin map of the adsorption potential (there are five possible orientations). In

the experiments, all five orientations are observed, due to the presence of all possi-

ble alignments of hexagons along five sides of a pentagon in the QC sample within

the width of the electrons beam (∼0.25 mm). In an ideal infinite GCMC frame- work the ground state of the system would be degenerate and all five orientations

would have the same energy and be equally probable. However, the square periodic

boundary conditions of our GCMC break this orientational degeneracy, causing some

orientations to become more likely to appear.

To find all the possible orientations, we performed simulations with a cell having

free boundary conditions. The cell is a 5.12 x 5.12 nm2 quasicrystal surface sur-

rounded by vacuum. Figure 3.12(a) shows the Vmin map of the adsorption potential.

46

Thirty simulations at 77 K are performed with this cell. The isotherms from these

runs are plotted in Figure 3.12(b). Only the first layer is shown, and the finite size

of the surface makes the growth of the first layer continuous. The density profiles

ρ(x, y) of all the simulations are analyzed at point p� of Figure 3.12(b). In this cell,

all five orientations of hexagons are observed with equal frequency indicating the ori-

entational degeneracy of the ground state. To represent the five orientations, density

profiles of five calculations (c, d, e, f, and g) are shown in Figures 3.12(c) to 3.12(g)

with their FT plotted on the side. Figure 3.12(h) presents a schematic depiction of

which orientations of hexagons are exemplified in each simulation.

Figures 3.13(a) and 3.13(b) illustrate the effect of pentagonal defects on the ori-

entation of hexagons at point p� of Figure 3.12(b). In most of the density profiles

corresponding to this coverage, we find the behavior shown in Figure 3.13(a). Here,

the effect of the pentagonal defect, which is the center of a dislocation in the hexag-

onal structure, is to rotate the orientation of the hexagons above the pentagon by

2 · 60◦/5 = 24◦ with respect to the hexagons below the pentagon. The possible rota- tions are n · 12◦, where n = 1,2,3,4, or 5. The rotation by 12◦ is usually mediated by more than one equivalent pentagon, as is shown in Figure 3.13(b) (note the “up” pen-

tagon at the middle-bottom part and the “down” pentagon near the middle-top part

of the figure). The “up” pentagon (with one vertex on the top) is equivalent with the

“down” pentagon (with one vertex on the bottom) since they have five orientationally

equivalent sides). These pentagonal defects are induced by the fivefold symmetry of

the substrate, and their concentration decreases in the subsequent layers.

3.2.10 Isosteric heat of adsorption

Figure 3.14 shows a P -T diagram for three different coverages of Xe adsorbed on

d-AlNiCo constructed from the isotherms in the range 40 K < T < 110 K. In the

47

simulations, the layers grow step-wise; at 70 K the first step occurs between coverage

∼0.06 and ∼0.7, the second step occurs between coverage 1.0 and ∼1.9, and the third step occurs between coverage ∼1.9 and ∼2.8 (unit is in fractions of monolayer). Figure 3.14 shows the T , P location of these steps, denoted “cov 0.5”, “cov 1.5”,

and “cov 2.5” for the first, second, and third steps, respectively. The isosteric heat

of adsorption per atom at these steps can be calculated from the P -T diagram as

follows [112]:

qst ≡ −kB d(lnP ) d(1/T )n

. (3.22)

The inset of Figure 3.14 summarizes the values of qst obtained from simulations

and experiments. The agreement between experiment and the simulations for the

half monolayer heat of adsorption is good. The values obtained in the simulation for

the 1.5 and 2.5 layer heats are about 20% lower than the bulk value of 165 meV [99].

The lower values suggest that bulk formation should be preferred at coverages above

one layer. However, layer-by-layer growth is observed at all T for at least the first

few layers in these simulations. We therefore believe that the low heats of adsorption

arise from slight inaccuracies in the Xe-Xe LJ parameters used in this calculation, as

the heats of adsorption are very sensitive to the gas parameters.

3.2.11 Effect of vertical dimension

In a standard unit cell, only 2 steps, corresponding to the first and second layer ad-

sorption, are apparent in the isotherms [101]. Further simulations indicate that when

the cell is extended in the vertical direction, additional steps are observed. Therefore

the number of observable steps is related to the size of the cell. Nevertheless, layering

is clearly evident in the ρ(z) profile, and the main features of the film growth are not

altered. For Xe on d-AlNiCo, the average interlayer distance is calculated to be about

0.37 nm, compared to 0.358 nm for the interlayer distance in the < 111 > direction

48

of bulk Xe [121]. Our simulations of multilayer films show variable adsorption as

the simulation cell is expanded in the direction perpendicular to the surface. This

is a result of sensitivity to perturbations (here, cell size) close to the bulk chemical

potential, where the wetting film’s compressibility diverges. This dependence has

been seen previously in large scale simulations. See e.g. Figure 3 of reference [108].

The analog of this effect in real experiments is capillary condensation at pressures

just below saturated vapor pressure (svp), the difference varying as the inverse pore

radius.

3.3 Summary

The results of GCMC simulations of noble gas films on QC have been presented. Ne,

Ar, Kr, and Xe grow layer-by-layer at low temperatures up to several layers before

bulk condensation. We observe interesting phenomena that can only be attributed

to the quasicrystallinity and/or corrugation of the substrate, including structural

evolution of the overlayer films from commensurate pentagonal to incommensurate

triangular, substrate-induced alignment of the incommensurate films, and density

increase in each layer with the largest one observed in the first layer and in the smallest

gas. Two-dimensional quasicrystalline epitaxial structures of the overlayer form in

all the systems only in the monolayer regime and at low pressure. The final structure

of the films is a triangular lattice with a considerable amount of defects except in

Xe/QC. Here a first-order transition occurs in the monolayer regime resulting in an

almost perfect triangular lattice. The subsequent layers of Xe/QC have hexagonal

close-packed structures. By simulating test systems with various sizes and strengths,

we find that the dimension of the noble gas, σgg, is the most crucial parameter in

determining the existence of the phenomenon which is found only in systems with

σgg ≥ σgg[Xe].

49

a 10 -12

10 -10

10 -8

10 -60

2

4

6

8

10

p (atm)

ρ= N

/A

(a to

m s/

nm 2

)

p*

-5 5X(nm)

5

Y (n

m )

-5

0

-50

-100

-150

-200

-250

(meV)

b

c d

e

g

f

h c

d,g

f

g

e,g

Figure 3.12: Xe adsorption on d-AlNiCo. (a) Minimum potential energy surface of the adsorption potential with free boundary conditions. (b) Adsorption isotherms of the first layer from a set of 30 simulations at 77 K using the free cell described in the paper. Five density profiles and FTs at point p� of (b) are shown in (c) to (g), representing all possible orientations of hexagonal domains. (h) Schematic diagram illustrating the correspondence between the orientations of the hexagonal domains observed in the density profiles (c) to (g).

50

θ1 θ2

a b

Figure 3.13: Xe adsorption on d-AlNiCo. Pentagonal defects rotate the orientation of hexagons by (a) θ1 = 24◦ and (b) θ2 = 12◦.

Figure 3.14: (color online). Xe adsorption on d-AlNiCo. Locations in P , T of the vertical risers in the isotherms corresponding to the first (square), second (circle), and third (triangle) layer formation. The heats of adsorptions, qst, are 270, 129, and 125 meV/atom respectively, calculated as described in the text. The inset figure shows qst obtained from the simulations as well as from the experiments.

51

Chapter 4

Embedded-atom method potentials

The interatomic potentials to simulate hydrocarbon adsorptions on d-AlNiCo are

generated within the embedded-atom method (EAM) formalism. The parametriza-

tion of the potentials are given in Equations 2.30 - 2.33. These parameters are fitted

to the energy of various structures computed via ab initio quantum calculations using

VASP code. The structure prototypes are summarized in Table 4.1. For elemental

Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure

(DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed

(HEX), simple cubic (SC), and simple hexagonal (SH). The structures are first re-

laxed from the initial configurations to achieve the equilibrium ones. For the Al-Co-Ni

ternary phases, the initial configurations are taken from the database of alloys [122].

After the structures are relaxed, the ground state electronic energies are calculated

and are used as the fitting data. The technical details in relaxing the structures and

evaluating the energies with VASP are given in Section 2.2.

Fitting of the parameters of the EAM potentials are performed using SIMPLEX

method (Section 2.4). SIMPLEX is relatively slower than other methods such as

nonlinear least square or conjugate gradient. However, since the speed of the fitting

procedure is not a concern in this work (most time is spent in the ab initio calcula-

tions), SIMPLEX is advantageous because it does not require evaluation of function’s

derivatives or orthogonality. In this way, a new parametrization of EAM potentials

requires changes only in the function evaluation routines. A fitting code is developed

to be able to fit in a bulk or adsorption mode. In the bulk mode, the function to

52

Table 4.1: List of structure prototypes used to fit the EAM potentials for hydrocarbon ad- sorption on Al-Co-Ni. For elemental Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure (DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed (HEX), simple cubic (SC), and simple hexagonal (SH). The Al– Co-Ni ternaries are taken from the database of alloys [122].

group structure Al BCC, DIA, FCC, GRA, HEX, SC, SH Co BCC, DIA, FCC, GRA, HEX, SC, SH Ni BCC, DIA, FCC, GRA, HEX, SC, SH CHn CH4 C2Hn C2H2, C2H4, C2H6-isotactic, C2H6-syntactic C3Hn propane, C3H4 allene, propyne, propene,

cyclopropane, cyclopropene, cyclopropyne C4Hn isobutane, 1-butene, 1-butyne, cyclobutane, methylcyclopropane C5Hn pentane, cyclopentane Other alkanes hexane, heptane, nonane, decane, undecane, dodecane AlxCoyNiz Al32Co12Ni12 (cI112)

Al32Co16Ni12 (cI128) Al29Co4Ni8 (dB1) Al17Co5Ni3 (dH1) Al34Co4Ni12 (dH2) Al20Co7Ni1 (hP28) Al18Co4Ni4 (mC28) Al12Co2Ni2 (mC32) Al18Co2Ni2 (mP22) Al36Co8Ni4 (oI96) Al12Co1Ni3 (oP16) Al34Co12Ni4 (mC102)

53

minimize is the err = ΔEbulk defined as:

ΔEbulk = ∑

i

|EEAMbulk,i − EV ASPbulk,i | (4.1)

The energies are per atom and the summation is over all structures of the fit. In

the adsorption mode, the error function consists of the error in the bulk energies

of molecule structures (ΔEbulk,mol), substrate structures (ΔEbulk,sub), molecule-on-

substrate structure (ΔEbulk,mol+sub), as well as in the adsorption energies (ΔEads):

Eads = Ebulk,mol+sub − (Ebulk,mol + Ebulk,sub) (4.2)

ΔEads = ∑

i

|EEAMads,i − EV ASPads,i | (4.3)

err = c1ΔEbulk,mol + c2ΔEbulk,sub + c3ΔEbulk,mol+sub + c4ΔEads

c1 + c2 + c3 + c4 (4.4)

The coefficients c1, c2, c3, and c4 are introduced as weighing factors. To drive the

parameters toward physically meaningful convergence point, the fitting is performed

in multiple stages:

(1) fit potentials for elemental Al, Co, and Ni,

(2) fit potentials for Al-Co-Ni,

(3) fit potentials for hydrocarbons,

(4) fit final potentials for hydrocarbon on Al-Co-Ni.

The fitted parameters from stage (1) are used as initial conditions in stage (2). The

fitted parameters from stage (2) and (3) are used as initial values in stage (4).

4.1 Stage 1: Aluminum, cobalt, and nickel

In stage (1), the elemental potentials for Al, Co, and Ni are fitted to elemental

bulk energies (top part of Tables 4.2-4.4). The potentials are also trained at various

54

pressures to ensure stability under compression/expansion and to yield resonable lat-

tice constants. In the calculations, different pressures are achieved by expanding or

compressing the relaxed ground state structures (i.e. Al(FCC) Co(HEX) Ni(FCC)),

namely at lattice constants from a = 0.95a0 to a = 1.1a0, where a0 is the equilib-

rium lattice constant at zero pressure. Training at various pressures increases the

transferability of the potentials due to a wider range of charge density covered. The

fitted EAM potentials (bottom part of Tables 4.2-4.4) are able to find the ground

state structure as well as the relative stability of each structure. Note that for pure

elements, the bulk energy represents the cohesive energy. Going from the most stable

structure to the least stable one, the EAM potentials correctly predict the following:

FCC → HEX → BCC → SC → SH → GRA → DIA (for Al and Ni) and HEX → FCC → BCC → SC → SH → GRA → DIA (for Co). The potentials also give accurate equilibrium lattice constants (middle part of Tables 4.2-4.4).

4.2 Stage 2: Al-Co-Ni potentials

Results from stage (1) are used as initial conditions in stage (2). The potential

for systems containing Al, Ni, and Co (AlCoNi-pot) are fitted to energies in bulk

structures and in slab configurations. The latter is intended to tune the AlCoNi-

pot at low charge density having different atomic environments from bulk. Slab

configurations are created from dB1, dH1, and dH2. Note that dB1, dH1, and dH2

are decagonal AlNiCo quasicrystal approximants. They are crystals with a large unit

cell which represent the short-ranged order of the quasicrystal. Approximants exist in

the vicinity of region of chemical compositions of their quasicrystalline counterparts.

The bulk unit cell of dB1, dH1, and dH2 contains 41, 25, and 50 atoms, respectively.

The unit cells consist of two well-defined layers. There are two possible terminations

for the bulk. The top(bottom) layer of the unit cell is labeled A(B), respectively. A

55

Table 4.2: (Top part) List of structures used to fit EAM potential for elemental aluminum. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3 · re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.199 -3.375 -3.368 -0.007 DIA 0.135 -2.847 -2.722 -0.126 FCC at a0·0.95 0.312 -3.347 -3.347 0.000 FCC at a0·0.96 0.289 -3.396 -3.390 -0.006 FCC at a0·0.97 0.267 -3.425 -3.422 -0.003 FCC at a0·0.98 0.248 -3.440 -3.444 0.004 FCC at a0·0.99 0.229 -3.443 -3.457 0.014 FCC at a0·1.00 0.212 -3.439 -3.462 0.023 FCC at a0·1.05 0.144 -3.356 -3.390 0.034 FCC at a0·1.10 0.098 -3.221 -3.221 0.000 GRA 0.151 -2.882 -2.958 0.075 HEX 0.211 -3.436 -3.426 -0.009 SC 0.167 -3.108 -3.097 -0.011 SH 0.178 -3.241 -3.241 0.000

a0(FCC) vasp = 4.04 A˚ a0(FCC) EAM = 4.00 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.026 7.182 2.700 33.64 3.038 3.012

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.078 0.121 0.495 0.896 1.381 2.200 Fx (eV) 0.000 -2.810 -3.106 -3.980 -3.879 -3.665 -3.074

56

Table 4.3: (Top part) List of structures used to fit EAM potential for elemental cobalt. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”HEX at a0 · c” indicates that the structure is HEX with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.665 -5.380 -5.377 -0.003 DIA 0.602 -4.115 -4.141 0.027 FCC 0.693 -5.439 -5.450 0.011 GRA 0.566 -4.596 -4.567 -0.029 HEX at a0·0.95 0.973 -5.213 -5.214 0.001 HEX at a0·0.96 0.910 -5.317 -5.305 -0.012 HEX at a0·0.97 0.849 -5.386 -5.375 -0.011 HEX at a0·0.98 0.795 -5.425 -5.425 -0.000 HEX at a0·0.99 0.744 -5.442 -5.457 0.016 HEX at a0·1.00 0.697 -5.440 -5.473 0.033 HEX at a0·1.05 0.501 -5.295 -5.383 0.088 HEX at a0·1.10 0.358 -5.098 -5.098 0.000 SC 0.569 -4.984 -4.708 -0.276 SH 0.608 -5.155 -4.966 -0.189

a0(HEX) vasp = 2.47 A˚ a0(HEX) EAM = 2.45 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.090 5.945 2.280 36.70 2.832 2.701

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.146 0.380 0.821 1.306 1.683 2.200 Fx (eV) 0.000 -3.374 -4.894 -5.517 -5.509 -5.008 -3.952

57

Table 4.4: (Top part) List of structures used to fit EAM potential for elemental nickel. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 ·c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.699 -4.928 -4.924 -0.004 DIA 0.500 -3.839 -3.839 -0.000 FCC at a0·0.95 0.980 -4.754 -4.783 0.029 FCC at a0·0.96 0.918 -4.870 -4.868 -0.002 FCC at a0·0.97 0.861 -4.941 -4.932 -0.009 FCC at a0·0.98 0.807 -4.980 -4.980 0.000 FCC at a0·0.99 0.757 -4.995 -5.010 0.015 FCC at a0·1.00 0.709 -4.993 -5.024 0.031 FCC at a0·1.05 0.507 -4.871 -4.955 0.084 FCC at a0·1.10 0.369 -4.687 -4.687 -0.000 GRA 0.543 -4.081 -4.109 0.028 HEX 0.706 -4.993 -4.998 0.005 SC 0.591 -4.464 -4.342 -0.122 SH 0.637 -4.682 -4.630 -0.052

a0(FCC) vasp = 3.49 A˚ a0(FCC) EAM = 3.46 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.126 5.447 2.150 39.17 2.839 2.717

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.223 0.336 0.566 1.028 1.355 2.200 Fx (eV) 0.000 -3.899 -4.373 -4.731 -5.280 -5.028 -4.172

58

slab is labeled A if the top layer is layer A, and vice versa. The unit cell for slab

configurations contains vacuum space of 12 A˚ in the vertical dimension to minimize

the interaction between unit cells due to periodic boundary conditions. In relaxing

the slabs, cell’s size/shape and atoms are allowed to relax except the bottom atoms.

In this way, the bottom atoms will be at the same coordinates as they were in the

bulk which will be more appropriate (than if bottom atoms are relaxed) when the

slabs are used as the substrates in the adsorption configurations (stage (4) of fitting).

The bottom part of Table 4.5 shows the parameters of AlCoNi-pot fitted to structures

listed in the top part of Table 4.5.

4.3 Stage 3: Hydrocarbon potentials

The EAM potentials for hydrocarbons (CH-pot) are fitted to the energies of isolated

molecules. The ab initio calculations are performed in a fairly large cubic cell with

vacuum size of larger than 10 A˚ to minimize the interaction between molecules due to

periodic boundary conditions implemented in VASP. The structures are fully relaxed.

In the beginning, all molecules listed in Table 4.1 are included in the fit. The result-

ing fitted parameters and the fitted energies are reported in Table 4.6. The results

show that EAM is fairly accurate for hydrocarbons, especially for alkanes. This is

interesting because EAM formalism was introduced for metals where bondings are

due to nonlocal electrons. Nevertheless, not all hydrocarbons can be fit simultane-

ously, as significant errors are found in some molecules, e.g C3H4-allene, propylene,

cyclopropane, cyclopropyne, 1-butene, and benzene. For our simulations, presented

in Chapter 5, a more accurate fit for alkanes and benzene is needed. Table 4.7 shows

the EAM potentials fitted to alkanes and benzene.

.

59

Table 4.5: (Top part) List of structures used to fit EAM potential for Al-Co-Ni systems. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV) and the first knot at (0,0) is assumed. The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

cI112 bulk -4.828 -4.828 -0.000 cI128 bulk -4.916 -4.915 -0.000 dB1 bulk -4.398 -4.395 -0.003 dH1 bulk -4.483 -4.516 0.032 dH2 bulk -4.441 -4.480 0.038 hP28 bulk -4.484 -4.490 0.006 mC102 bulk -4.520 -4.517 -0.003 mC28 bulk -4.492 -4.509 0.017 mC32 bulk -4.323 -4.322 -0.001 mP22 bulk -4.121 -4.111 -0.010 oI96 bulk -4.334 -4.350 0.016 oP16 bulk -4.292 -4.300 0.007 dB1 slab3A -3.816 -3.853 0.037 dB1 slab3B -3.913 -3.921 0.008 dB1 slab4A -4.105 -4.096 -0.009 dB1 slab4B -4.015 -4.015 0.000 dH1 slab3A -4.058 -4.043 -0.015 dH1 slab3B -4.056 -4.028 -0.029 dH1 slab4A -4.153 -4.155 0.003 dH1 slab4B -4.158 -4.153 -0.005 dH2 slab4A -4.144 -4.143 -0.001 dH2 slab4B -3.980 -3.980 0.000

Al Co Ni ρe (A˚−3) 0.026 0.090 0.126 β 7.182 5.945 5.447 re (A˚) 2.70 2.28 2.15 D (eV) 0.034 0.037 0.039 α (A˚−1) 1.833 3.350 3.258 r0 (A˚) 3.009 2.744 2.735 Z/ZAl 1 0.862 0.521 Z/ZCo 1 0.975 knot2 0.086,-2.758 0.235,-3.249 0.285,-3.772 knot3 0.120,-2.974 0.383,-4.834 0.341,-4.449 knot4 0.517,-4.688 0.864,-5.579 0.604,-4.931 knot5 0.868,-3.777 1.321,-5.420 1.050,-5.279 knot6 1.365,-3.736 1.705,-5.061 1.355,-5.160 knot7 2.200,-3.151 2.200,-4.090 2.200,-4.196

60

Table 4.6: (Top part) List of structures used to fit EAM potential for hydrocarbons. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

methane -4.807 -4.806 -0.001 ethane syntactic -5.076 -5.061 -0.015 ethane isotactic -5.068 -5.047 -0.021 C2H2 -5.735 -5.736 0.001 C2H4 -5.382 -5.328 -0.054 propane -5.189 -5.183 -0.004 C3H4 allene -5.751 -5.342 -0.409 propylene -5.188 -5.002 -0.186 propyne -5.679 -5.687 0.008 cyclopropane -5.308 -4.850 -0.459 cyclopropene -5.366 -5.273 -0.094 cyclopropyne -5.340 -5.477 0.137 isobutane -5.249 -5.253 0.004 1-butene -5.250 -5.128 -0.122 1-butyne -5.648 -5.630 -0.018 cyclobutane -3.899 -3.899 0.000 methylcyclopropane -5.404 -5.410 0.006 pentane -5.291 -5.297 0.006 cyclopentane -5.024 -5.047 0.022 hexane -5.321 -5.329 0.008 benzene -6.136 -6.334 0.198 heptane -5.343 -5.352 0.009 octane -5.358 -5.370 0.012 nonane -5.368 -5.375 0.007 decane -5.383 -5.381 -0.002 undecane -5.392 -5.391 -0.001 dodecane -5.400 -5.413 0.013

ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

C 0.932 5.473 1.24 0.123 3.116 1.489 H 1.639 2.798 0.74 0.211 4.296 1.125

ZH/ZC = 0.880 knot1 knot2 knot3 knot4 knot5 knot6 knot7

C ρ 0.000 0.334 1.103 1.357 1.635 1.870 2.200 Fx 0.000 -3.851 -6.067 -8.011 -7.771 -8.17 -7.352

H ρ 0.000 1.071 1.528 1.773 2.008 2.200 - Fx 0.000 -2.997 -3.482 -4.189 -3.764 -3.409

61

Table 4.7: (Top part) List of structures used to fit EAM potential for alkanes and benzene. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

methane -4.806 -4.806 -0.000 ethane syntactic -5.060 -5.061 0.001 propane -5.182 -5.183 0.001 isobutane -5.254 -5.253 -0.001 pentane -5.298 -5.297 -0.001 hexane -5.329 -5.329 0.000 heptane -5.349 -5.352 0.003 octane -5.367 -5.370 0.003 nonane -5.380 -5.375 -0.005 decane -5.393 -5.381 -0.012 undecane -5.402 -5.391 -0.011 dodecane -5.411 -5.413 0.002 benzene -6.334 -6.334 0.000

ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

C 0.932 5.473 1.24 0.116 2.793 1.477 H 1.639 2.798 0.74 0.225 3.154 1.077

ZH/ZC = 0.533 C H

knot1 0,0 0,0 knot2 0.366,-3.793 1.040,-3.543 knot3 0.509,-5.454 1.534,-4.003 knot4 0.761,-6.130 1.801,-3.656 knot5 1.012,-7.611 1.911,-3.874 knot6 1.834,-8.037 2.200,-3.736 knot7 2.200,-7.288 -

62

4.4 Stage 4: Hydrocarbon on Al-Co-Ni

The final EAM potentials for hydrocarbon adsorption on Al-Co-Ni are fitted to the

adsorption energies of a hydrocarbon on a substrate. As the substrates, dB1-slab4A

and dH1-slab4A are used for methane, and dH2-slab4A are used for larger molecules.

The initial conditions are taken from Tables 4.5 and 4.7 and the fitting is performed

in the adsorption mode as explained previously. During the fitting, it is found that

the adsorption energies are more difficult to fit than those of molecules, substrates,

and molecule-on-substrate. This originates from the different nature of bonding. As

discussed later in Chapter 5, alkanes do not make strong chemical bonds with the Al-

Co-Ni. They are only physically adsorbed. Whereas the bondings within a molecule

and within the substrate are strong. Weighing factors of c1 = c2 = c3 = 1 and c4 > 1

are used to prioritize adsorption energies. Typically, values of c4 < 4 are used to get

the final fit. Increasing c4 beyond 4 does not considerably improve the fit, indicating

the limitations inherent with EAM formalism. The fitted structures and energies are

shown in Table 4.9 while the fitted parameters are tabulated in the Table 4.8.

63

Table 4.8: Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA+ZAφB/ZB)/2. Fitting structures are given in Table 4.9

C H Al Co Ni ρe (A˚−3) 0.932 1.639 0.026 0.09 0.126 β 5.473 2.798 7.182 5.945 5.447 re (A˚) 1.24 0.74 2.7 2.28 2.15 D (eV) 0.116 0.225 0.034 0.037 0.039 α (A˚−1) 2.786 3.156 1.740 3.636 3.198 r0 (A˚) 1.478 1.076 2.938 2.730 2.731 Z/ZC 1 0.532 0.989 1.115 1.258 Z/ZH 1 0.857 1.140 0.850 Z/ZAl 1 0.370 0.498 Z/ZCo 1 0.970 knot1 0,0 0,0 0,0 0,0 0,0 knot2 0.353,-3.860 1.037,-3.475 0.079,-2.816 0.217,-3.250 0.252,-3.932 knot3 0.519,-5.429 1.527,-3.974 0.108,-3.056 0.358,-5.176 0.327,-4.425 knot4 0.746,-6.254 1.797,-3.797 0.507,-4.214 0.774,-5.376 0.574,-4.849 knot5 1.033,-7.443 1.899,-3.744 0.893,-3.618 1.439,-5.139 1.032,-5.194 knot6 1.824,-7.988 2.200,-3.674 1.408,-3.681 1.635,-4.834 1.354,-5.065 knot7 2.200,-7.264 2.200,-2.911 2.200,-3.577 2.200,-4.130

64

Table 4.9: Fitted energies calculated using EAM parameters in Table 4.8. Methane up represents methane with one H below C and three H above C. Methane down is inverse of methane up. The unit for energy is eV/atom except for the adsorption energy which is in eV/molecule.

structure EEAM EV ASP ΔE Molecule energy (eV/atom): methane -4.805 -4.805 0.000 ethane syntactic -5.078 -5.061 -0.017 propane -5.189 -5.183 -0.006 butane -5.252 -5.252 0.000 pentane -5.293 -5.297 0.004 hexane -5.322 -5.328 0.007 benzene -6.294 -6.334 0.040 Substrate energy (eV/atom): dB1 slab4A -4.132 -4.096 -0.036 dH1 slab4A -4.141 -4.155 0.014 dH2 slab4A -4.135 -4.143 0.008 Molecule and substrate energy (eV/atom): methane on dB1 slab4A -4.173 -4.139 -0.034 methane up on dH1 slab4A -4.204 -4.219 0.015 methane down on dH1 slab4A -4.205 -4.219 0.014 ethane syntactic on dH2 slab4A -4.207 -4.214 0.006 propane on dH2 slab4A -4.242 -4.249 0.007 butane on dH2 slab4A -4.276 -4.283 0.008 pentane on dH2 slab4A -4.306 -4.315 0.009 hexane on dH2 slab4A -4.337 -4.346 0.009 benzene on dH2 slab4A -4.381 -4.385 0.004 Adsorption energy (eV/molecule): methane on dB1 slab4A -0.225 -0.238 0.013 methane up on dH1 slab4A -0.238 -0.243 0.005 methane down on dH1 slab4A -0.304 -0.257 -0.047 ethane syntactic on dH2 slab4A -0.348 -0.253 -0.095 propane on dH2 slab4A -0.458 -0.328 -0.131 butane on dH2 slab4A -0.439 -0.439 0.000 pentane on dH2 slab4A -0.465 -0.527 0.062 hexane on dH2 slab4A -0.520 -0.589 0.069 benzene on dH2 slab4A -0.790 -0.791 0.001

65

Chapter 5

Hydrocarbon adsorptions on d-AlNiCo

The low friction properties of quasicrystals in ambient conditions coupled with their

high hardness and oxidation resistance led to the development of applications of

quasicrystal coatings, for instance on machine parts, cutting blades, and non-stick

frying pans [5]. In machine parts, hydrocarbons are commonly used as a lubricant.

Superlubricity is the name given to the phenomenon in which two parallel single

crystal surfaces slide over each other with vanishingly small friction because their

structures are incommensurate. This phenomenon was proposed in the early 1990’s

[12] and experiment evidence for this effect has been seen in studies of mica sliding

on mica [123], W(110) on Si(100) [124], Ni(100) on Ni(100) [125], and tungsten on

graphite [13]. This effect is also expected in quasicrystals due to their aperiodic

structures at all length scales. Indeed, quasicrystal surfaces were observed to have

low friction not long after they were first discovered [126], but pinning down the exact

origin of the low friction has been elusive.

Recent experiments in ultra high vacuum (UHV) have demonstrated a frictional

dependence on aperiodicity for decagonal Al-Ni-Co quasicrystal (d-AlNiCo) against

thiol-passivated titanium-nitride tip [11]. In coating applications, it is expected that

even if superlubricity exists between moving parts, some additional lubricant would

still be needed to counter the macroscopic frictions due to grain boundaries, asperities,

and other defects in the surfaces of the moving parts. Some of the requirements of

the lubricant in such a situation are that it must wet the surface and that it must

not remove or reduce the superlubricity. Therefore it is desirable to have a good

understanding of how gases, hydrocarbons in particular, interact with quasicrystal

66

surfaces.

Very little is currently known about the interaction of hydrocarbons, their struc-

tures and growth on alloy or quasicrystalline surfaces. Some earlier experiments using

Fourier transform infrared spectroscopy (FTIR) and low energy electron diffraction

(LEED) suggest that on the 5-fold surface of Al-Pd-Mn, carbon monoxide (CO) does

not adsorb at 100 K, benzene adsorb at 100 K with possibly commensurate or dis-

ordered structure (LEED pattern unchanged) [127]. The same experiments on the

10-fold surface of d-AlNiCo show that CO bonds to the Ni sites at > 132 K, no struc-

ture reported, while there is no experiment available for benzene on Al-Ni-Co [127].

Later, scaning tunneling microscopy (STM) experiments on benzene adsorption on

Al-Pd-Mn show that the adsorbed benzene has a disordered structure [128]. In this

chapter, we report the simulation results of small hydrocarbons adsorb on the 10-fold

surface of decagonal Al73Ni10Co17 [15, 16], namely for methane, propane, hexane,

octane, and benzene.

5.1 Model

The simulations are performed within the framework of grand canonical ensemble

using Monte Carlo (GCMC) method as previously described in Section 2.5. The sim-

ulation cell is tetragonal. We take a square section of the surface, A, of side 5.12 nm,

to be the (x, y) part of the unit cell in the simulation, for which we assume periodic

boundary conditions along the basal directions. The coordinates of substrate atoms

are taken from Ref. [100]. The interaction potentials are modeled as the following.

The intermolecular interactions (adsorbate-adsorbate) are calculated as a sum of

pair interactions between atoms. For methane-methane [129, 130] Buckingham-type

potentials are used:

V (r) = Ae−Br − C/r6 (5.1)

67

Buckingham potentials also used to parametrize the benzene-benzene interactions

[131, 132]. For linear alkane-alkane, Morse-type potentials are used:

V (r) = −A(1− (1− e−B(r−C))2) (5.2)

The parameters for these potentials are summarized in Table 5.1. EAM potentials

generated in Chapter 4 are used for the rest of the interactions, namely the in-

tramolecular, adsorbate-substrate (C-Al, C-Co, C-Ni, H-Al, H-Co, H-Ni), and be-

tween substrate atoms (Al-Al, Al-Co, Al-Ni, Co-Ni). As previously mentioned in

Section 4.4, the ab initio calculations show that alkanes and benzene do not dissoci-

ate on dB1, dH1, and dH2 (these are decagonal Al-Ni-Co approximants). In these

systems, the surface of the substrate does not undergo any considerable relaxation

upon the adsorption of the molecules. Therefore, as a first approximation, in our

GCMC simulations, the substrate and the molecules are considered as rigid. How-

ever, molecules are allowed to explore all rotational degrees of freedom to achieve the

equilibrium configurations.

5.2 Adsorption potentials

Figure 5.1 displays the minima of the adsorption potential for methane (a), propane

(b), hexane (c), octane (d), and benzene (e), generated by minimizing the adsorption

potential of a molecule on d-AlNiCo with respect to z (Equation 3.1) and all rota-

tional degrees of freedom. The average adsorption energies are 221 (methane), 374

(propane), 620 (hexane), 794 (octane), and 931 (benzene), given in meV/molecule.

The figure shows the distribution of binding sites (dark spots) for the molecule.

Methane, propane, and benzene are small enough to follow the local atomic envi-

ronments of the substrate, whereas hexane and octane show considerable smearing

due to their large size. The location of dark spots in methane is similar to that

68

Table 5.1: Parameter values for the adsorbate-adsorbate interactions used for hydrocar- bon adsorption on a decagonal Al-Ni-Co. Intermolecular energies are calculated as a sum of pair interactions. For methane-methane, the C-H is taken as the geometrical mean for parameter A and as the arithmetic mean for parameters B and C.

A B C ref. (eV) (A˚−1) (A˚6)

methane C-C 82.132 2.693 449.53 [130] V (r) = Ae−Br − C/r6 C-H 66.217 2.892 167.51

H-H 53.381 3.105 62.42 [129] benzene C-C 11527.700 3.909 524 [131, 132] V (r) = Ae−Br − C/r6 C-H 348.518 3.703 75 [131, 132]

H-H 127.447 3.746 39 [131, 132] (meV) (A˚−1) (A˚)

alkane C-C 6.984 1.2655 4.1844 [133] V (r) = −A(1− (1− x)2) C-H 23.921 2.2744 2.544 [133] x = e−B(r−C) H-H 0.002 1.255 6.1543 [133]

[129] Tsuzuki S, et al 1993 J. Mol. Struct. 280 273 [130] Tsuzuki S, et al 1994 J. Phys. Chem. 98 1830 [131] Califano S, et al 1979 Chem. Phys. Lett. 64 491 [132] Chelli R, et al 2001 Phys. Chem. Chem. Phys. 3 2803 [133] Jalkanen J-P, et al 2002 J. Chem. Phys. 116 1303

69

of propane. However, it is interesting to see that the dark spots in methane and

propane become the bright ones in benzene. To study more, the topview of the

substrate atoms is depicted in panel (f) of Figure 5.1. The legend for the atoms is:

Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red),

Ni-toplayer (green), and Ni-otherlayers (blue). In methane and propane, dark spots

occur when the center of the molecule is located on top of black-Al (5 gray-Al on

toplayer forming a pentagon). In benzene, this location gives a bright spot (less bind-

ing). Except the binding site at the center of the figure, strong binding sites (dark

spots) in benzene occur when the center of benzene is located on top of gray-Al (on

the top layer, the pentagon consists of 2 Al and 3 Ni, or 3 Al and 2 Ni).

5.3 Molecule orientations

First we study the orientations of a molecule as it is adsorbed on d-AlNiCo. During

the calculation of minima of the adsorption potential (Section 5.2, the orientation of

the molecule is recorded when a minimum is achieved. It is found that for methane,

the rotational ground state is degenerate indicating its spherical nature. Ab initio

calculations of methane dimers indicate that the interaction energy within the dimer

depends on the relative orientations of the two [129, 130]. Therefore, the degeneracy

observed is due to the limitation of the EAM model. For linear alkanes, we can

define θ as the angle between the substrate’s xy-plane and the main axis of the

alkanes. Again, θ is recorded when an adsorption minimum is achieved. We find that

θ decreases with increasing alkane chain, e.g. propane (θ = 10◦), hexane (θ = 5◦),

and octane (θ ∼ 0◦). For benzene, θ would be the angle between the molecule’s plane and substrate’s xy-plane, and it is θ ∼ 0◦. Therefore, we conclude that alkanes and benzene prefer most contact with d-AlNiCo.

70

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-180

-200

-220

-240

-260

-300

1

-280

-221

-300

-350

-400

-450

-500

-550

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

1 -374

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-550

-600

-650

-700

-800

1

-750

-620

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-750

-800

-850

-900

1

-950

-700

-794

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-800

-900

-1100

1

-1000

-931

a) b)

c) d)

e) f)

Figure 5.1: (color online). (a-e) Adsorption potential map, calculated by minimizing the adsorption potential of one molecule on a decagonal Al-Ni-Co along z direction and all rotational degrees of freedom at every coordinates (x, y). Red numbers represent the average value of the adsorption energies. (f) Top view of the decagonal Al-Ni-Co substrate 51.2x51.2 A˚2: Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red), Ni-toplayer (green), and Ni-otherlayers (blue).

71

5.4 Adsorption isotherms

To use quasicrystals as low-friction coatings in conjunction with oil lubricants, the

lubricants must be able to spread well on the quasicrystals. Lubricants consist mostly

of alkanes. The wetting of d-AlNiCo by alkanes and benzene is demonstrated in

Figure 5.2. In the figure, the number of molecules adsorbed on the substrate per area

is plotted as a function of pressure at various temperatures. The results for methane,

propane, hexane, octane, and benzene are shown. The simulated temperatures are:

for methane T = 68, 85, 136, 185 K, for propane T = 80, 127, 245, 365 K, for hexane

T = 134, 170, 267, 450 K, for octane T = 162, 210, 324, 450, 565 K, for benzene T =

209, 270, 418, 555 K. Note that T = 450 K represents a typical temperature of the

inner wall of cylinder in regular car engines [134]. In the isotherms, even though

the step corresponding to the adsorption of the first layer is well observed, steps

corresponding to the second and further layers are not evident for methane, propane,

and benzene, and barely visible for hexane and octane. The second layer condensation

occurs near the bulk condensation and extends a short range of pressure, nevertheless

multilayer adsorptions can be seen at higher pressures (point d) as shown in the inset

of each panel in which the distribution of adsorbed molecules is plotted against the

z direction.

5.5 Density profiles

Methane on d-AlNiCo. Figure 5.3 shows the density plots for methane at two

different coverages, corresponding to points ”a” and ”c” on the 68 K isotherm in

Figure 5.2.a. At submonolayer regime, methanes occupy the strong binding sites

on the surface. At monolayer coverage, the ordering is 5-fold commensurate with

the substrate, also indicated by the Fourier transform (panel c), and there is no

transition to 6-fold. The evidence that there is no such transition is shown more

72

10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

14

16

18

20

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

a

b c

d

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5 Z (nm)

Xe CH4

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a

b c

d

ρ N (m

ol ec

ul es

/n m

2 )

P (atm)

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5 Z (nm)

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a b c

d

Z (nm)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a b c

d

Z (nm)

ρ Z (a

rb .)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

0 0.5 1 1.5 2 2.5

10-25 10-20 10-15 10-10 10-5 1 0

2

4

6

8

10

12

14

16

18

20

a

b c

d

0 1 2 3 Z (nm)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

ρ Z (a

rb .)

c) d)

e)

a) b)

Figure 5.2: (color online). Isothermal adsorption densities of hydrocarbons on a decago- nal Al-Ni-Co: (a) methane (from left to right T = 68, 85, 136, 185 K), (b) propane (T = 80, 127, 245, 365 K), (c) hexane (T = 134, 170, 267, 450 K), (d) octane (T = 162, 210, 324, 450, 565 K), and (e) benzene (T = 209, 270, 418, 555 K). The inset in each figure is the density along z direction at pressure corresponding to point d. Xenon (red) is plotted in panel (a) for comparison.

73

10 -15

10 -10

10 -5 1

0

5

10

15

ρN (m olecules/nm

2)

P (atm)

0

0.2

0.4

0.6

0.8

1

ρ 5 -6

a b

c d

Figure 5.3: (a) and (b) Calculated density of methane adsorbed on a decagonal Al-Ni-Co at pressures corresponding to points ”a” and ”c” of the 68 K isotherm shown in Figure 5.2.a, respectively. (c) Fourier transform of the density plot shown in (b), consistent with 5-fold ordering of the methane near monolayer completion. (d) Order parameter (left axis, as calculated in Equation 3.14) as a function of pressure for the 68 K isotherm (right axis), indicating no sharp transition to 6-fold ordering.

clearly in panel d, where the order parameter (left axis, as calculated in Equation

3.14) does not present any sharp transition as seen in the case of xenon adsorption on

the same substrate (Section 3.2.3). The lack of 5-fold to 6-fold transition is consistent

with our proposed rule based on rare gases (Section 3.2.6), that the size of the rare

gas must be at least as large as Xe on this quasicrystalline surface for the transition to

occur [135]. Note that the size of methane relative to xenon (ratio of Lennard-Jones

σ parameters) is 0.8 [69].

Propane, Hexane, and Octane on d-AlNiCo. At submonolayer coverages,

propane, hexane, and octane do not show any clear evidence of binding to specific sites

of the surface even though molecules tend to bind in the center of the surface with

74

the most attractive site. Figure 5.4 shows the density profiles for these hydrocarbons

at pressure corresponding to the first layer completion at various temperatures. The

simulated temperatures have been indicated in Section 5.4. At monolayer coverage,

propane adsorbs in a disordered fashion at all temperatures, whereas hexane and

octane tend to form close-packed structures as suggested by domains with stripe

feature. As comparisons, the crystal structures of solid propane at 30 K is monoclinic

(space group #11, P21/m) [136], hexane is triclinic (space group #2, P1¯) at 90 K

[136] and at 158 K [137], octane is triclinic (space group #2, P1¯) at 90 K [136]

and at 213 K [138]. The 2-dimensional structures of these hydrocarbons are close-

packed structures as charactereized by stripe structures with 1 and 2 molecules per

unit cell for even- and odd-alkanes, respectively [136]. In general, even-alkanes form

triclinic (6 ≤ NC ≤ 26), monoclinic (28 ≤ NC ≤ 36), or orthorombic (38 ≤ NC) with decreasing packing density [136, 137, 138, 139], where NC being the number of

carbon atoms. Whereas for odd-alkanes form triclinic (7 ≤ NC ≤ 9) or orthorombic (11 ≤ NC) [136, 139, 140] also with decreasing packing density for longer chain. Note that the even-alkanes have higher packing density than odd-alkanes.

Benzene on d-AlNiCo. As in the case of methane, at submonolayer coverage,

benzene preferentially adsorbs at sites offering the strongest binding at all simulated

temperatures. At pressure near to first layer completion, the density profiles show a

temperature-dependence as plotted in Figure 5.5 for 209 K, 270 K, and 418 K. The

structures are more clearly charaterized by the plotting the geometrical center of

density as shown in the middle row of the figure. At T = 209 K, pentagonal ordering

is observed. As the temperature is increased, a mixture of 5-fold and 6-fold ordering

is seen, e.g. at T = 270 K. At higher temperature, T = 418 K, 6-fold structure

dominates the ordering of the monolayer as confirmed by the Fourier analysis of the

density showing hexagonal spots characteristic of triangular lattice (bottom row, last

75

267 K170 K 450 K

324 K210 K 450 K

hexane

octane

80 K 127 K 245 Kpropane

Figure 5.4: Calculated density of propane, hexane, and octane adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion. Propane forms a disordered structures, whereas hexane and octane tend to form close-packed structures indicated by stripe features with increasing order for longer chain.

76

column in Figure 5.5). The crystal structure of bulk benzene has been determined

at 4.2 K and 270.15 K to be orthorombic (space group #61, Pbca) with 4 molecules

per unit cell [141].

The evolution of the density profile over temperature from being 5-fold to 6-fold

can be studied more clearly by plotting the order parameter, ρ5−6 = N5/(N5 + N6)

(Nn denotes the number of molecules having n nearest neighbors as defined in Section

3.1.6) as a function of T as shown in Figure 5.6 (left axis). The plot is taken at a

constant pressure of 0.01 atm corresponding to point c in Figure 5.2 panel e. The

adsorption isobar representing the number of adsorbed molecules N as a function of

T is plotted in right axis of Figure 5.6. Three dashed lines corresponding to T =209,

270, and 418 K, whose adsorption isotherms are plotted earlier in Figure 5.2 panel e,

are added for guidance in comparing the density profile to the order parameter value

at each of these temperatures. We observe the following trends:

• (region 1) T < 260 K, 0.5 < ρ5−6 → 5-fold ordering dominates, N = 44.

• (region 2) 260 ≤ T ≤ 280 K, 0.3 ≤ ρ5−6 ≤ 0.5 → 5-fold becomes mostly 6-fold, N = 44.

• (region 3) 280 ≤ T ≤ 370 K, ρ5−6 = 0.3 → 6-fold ordering dominates, N = 44.

• (region 4) 370 ≤ T ≤ 390 K, 0.1 ≤ ρ5−6 ≤ 0.3→ transition to 6-fold ordering, 44 ≤ N ≤ 46.

• (region 5) T > 390 K, ρ5−6 increases from 0.1 → 6-fold ordering weakens, N decreases from 46.

The highest 6-fold ordering occurs at T = 390 K which is mediated by a gain of 2

additional molecules adsorbed on the substrate. Beyong this temperature, thermal

77

209 K 270 K 418 K

Figure 5.5: (Top row) Calculated density of benzene adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion for 209 K, 270 K, and 418 K. (middle row) Density profile of the geometrical center of density shown in the top row. (bottom row) Fourier transform of the density plot shown in the middle row, showing 5-fold ordering at 209 K, mixture of 5-fold and 6-fold structures at 270 K, and mostly 6-fold features at 418 K.

78

200 250 300 350 400 450 500

0.1

0.2

0.3

0.4

0.5

0.6

T (K)

ρ 5 -6

38

40

42

44

46 N

m olecules

Figure 5.6: (left axis) Order parameter ρ5−6 = N5/(N5+N6) (Nn denotes the number of molecules having n nearest neighbors) as a function of temperature at 0.01 atm of pressure. (right axis) Adsorption isobar showing the number of adsorbed molecules as a function of temperature. Verticel dashed lines correspond to T = 209, 270, and 418 K whose density profiles are plotted in Figure 5.5.

energy causes reduction of N resulting in less ordered 6-fold structure as T increases.

Nevertheless, the 6-fold ordering are still well resolved up to T = 418 K as shown

before in Figure 5.5.

5.6 Summary

In this chapter, we have presented our studies on methane, propane, hexane, octane,

and benzene adsorption on decagonal Al-Ni-Co quasicrystal. All of these hydro-

carbons form very well defined step corresponding to the first layer condensation.

Eventhough multilayer formation is evident from the density distribution along ver-

tical dimension, the step corresponding to the condensation of the second and further

layers can not be observed clearly in the isotherm due narrow range of pressure for

these layers to form before bulk condensation occurs. Monolayer of methane has

79

been determined to be pentagonal commensurate with the substrate for 68 K ≤ T ≤ 136 K. Monolayer of propane shows disordered structures for 80 K ≤ T ≤ 245 K. Monolayers of hexane (134 K≤ T ≤ 450 K) and octane (162 K ≤ T ≤ 450 K) form 2- dimensional close-packed structures characterized by stripe ordering consistent with

their bulk crystal structures. Benzene monolayer is pentagonal at T ≤ 260 K, which transforms into 6-fold structure at T ≥ 280 K with the highest 6-fold ordering occurs at T = 390 K. Beyong 390 K, thermal energy causes fewer adsorbed benzene and the

6-fold ordering starts to deterioates. Nevertheless, the 6-fold ordering are still well

observed up to T = 418 K.

80

Chapter 6

Embedded-atom potentials for selected

pure elements in the periodic table

Embedded-atom method (EAM) formalism provides only the skeleton of calculat-

ing the total electronic energy for a given charge density. It does not specify the

parametrization of the three functionals: atomic charge density, embedding en-

ergy, and pair interaction. Many different parametrizations have been introduced

[54, 55, 58, 59, 59, 65, 142, 143, 144, 145]. This arises inconveniences when one needs

to use different potentials for different elements. Steps toward universal EAM poten-

tials are needed. Recently, a consistent set of atomic charge density for all elements

in the periodic table suitable for EAM potentials has been developed [64]. The set

is calculated by spherically averaging the atomic charge density of the solution of

Hartree-Fock (HF) equations (Equation 2.19) of an isolated atom. The exchange

term in the HF equation contains an atom-specific parameter which is adjusted to

reproduce the experimental first ionization energy, hence are semiempirical. In this

chapter we use these charge densities to generate a consistent set of EAM potentials

for pure elements in the periodic table.

The parametrizations are given in Section 2.3. For each element, the potential is

fitted to the ab initio cohesive energies in body-centered cubic (BCC), face-centered

cubic (FCC), hexagonal close-packed (HEX), and diamond structure (DIA). In ad-

dition, energies in the ground state structure at different pressures are included in

the fitting procedure to ensure the mechanical stability of the potentials. Energies at

various pressures are achieved by expanding or compressing the equillibrium lattice

81

constant a by a factor from 0.90 to 1.16. Within an accuracy of 30 meV, all the

EAM potentials successfully predict the correct ground state structure with respect

to those that are not included in the fitting step, namely graphite (GRA), simple

cubic (SC), and simple hexagonal (SH) structures. The fitted parameters are sum-

marized in Tables 6.1-6.4. Tables 6.5 and 6.6 list the ground state structures of

the elements, range of charge density covered in the fit, and the equillibrium lattice

constants calculated by the ab initio method as well as by using the fitted EAM

parameters. Literature value of the lattice constants is also included as comparisons.

Overall, the EAM potentials predict the lattice constants with within 0.5 A˚ of the

literature value except for tellurium. Indeed, elements in the V-A, VI-A, and VII-A

colums of periodic table tend to form complex structures with large unit cells and

more EAM parameters might be needed to fit these structures better. Manganese is

excluded also because of its complex cubic structure with 58 atoms/cell. Noble gases

are excluded because they are already well described by simple pair potentials. Ele-

ments in the Actinide series are excluded because they do not have a stable structure

due to their radioactivity.

82

Table 6.1: Fitted parameters for the charge density and pair interaction functionals of the EAM potentials for pure elements, continued in Table 6.2. The fitting structures are given in Tables 6.5 and 6.6. The parameters for the embedding functionals are given in Tables 6.3 and 6.4.

Z Struct. ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

3 Li 0.038 4.813 2.670 0.010 3.015 2.422 4 Be 0.031 7.174 2.450 0.016 3.317 2.216 6 C 0.932 5.473 1.240 0.115 3.152 1.441 11 Na 0.021 5.552 3.080 0.011 5.800 3.500 12 Mg 0.001 10.030 3.890 0.012 2.807 3.110 13 Al 0.026 7.182 2.700 0.034 3.038 3.012 14 Si 0.072 7.171 2.250 0.043 4.115 1.601 19 K 0.008 6.412 3.920 0.015 2.926 4.088 20 Ca 0.002 9.719 4.280 0.020 2.940 3.646 21 Sc 0.058 6.053 2.700 0.021 2.264 3.210 22 Ti 0.287 4.333 1.940 0.022 2.282 2.850 23 V 0.394 4.025 1.780 0.023 2.501 2.615 24 Cr 0.297 4.538 1.680 0.024 2.519 2.494 26 Fe 0.192 5.214 2.000 0.026 5.747 2.412 27 Co 0.090 5.945 2.280 0.037 2.832 2.701 28 Ni 0.126 5.447 2.150 0.039 2.839 2.717 29 Cu 0.052 5.864 2.220 0.029 2.589 2.553 30 Zn 0.0005 14.630 4.800 0.030 2.655 2.560 32 Ge 0.042 7.419 2.440 0.045 2.659 2.001 37 Rb 0.006 6.788 4.210 0.037 3.599 4.694 38 Sr 0.002 9.614 4.450 0.038 2.926 3.299 39 Y 0.037 6.593 3.000 0.039 2.123 3.300 40 Zr 0.158 5.287 2.300 0.040 2.132 3.230 41 Nb 0.145 5.556 2.080 0.041 2.347 2.858 42 Mo 0.201 5.621 1.940 0.042 2.153 2.728 43 Tc 0.365 4.968 1.880 0.043 2.559 2.640 44 Ru 0.047 6.360 2.400 0.044 2.702 2.700 45 Rh 0.020 6.981 2.680 0.045 2.609 2.687 46 Pd 0.017 8.389 2.500 0.036 2.719 2.791 47 Ag 0.028 6.619 2.530 0.027 2.982 3.055 48 Cd 0.0005 13.340 4.500 0.048 2.284 3.170 52 Te 0.047 8.157 2.560 0.052 2.477 2.245 56 Ba 0.001 10.020 4.900 0.056 2.815 3.847 57 La 0.009 8.106 3.890 0.057 2.518 3.350

83

Table 6.2: continuation of Table 6.1.

58 Ce 0.013 7.575 3.682 0.058 2.488 2.949 59 Pr 0.016 7.342 3.646 0.059 2.502 2.970 60 Nd 0.015 7.361 3.632 0.060 2.543 3.548 63 Eu 0.016 7.306 3.542 0.033 2.596 3.300 64 Gd 0.015 7.724 3.470 0.064 2.549 3.240 65 Tb 0.017 7.323 3.464 0.065 2.527 3.300 66 Dy 0.017 7.303 3.420 0.066 2.512 3.240 67 Ho 0.018 7.323 3.392 0.067 2.532 3.180 68 Er 0.019 7.271 3.346 0.068 2.560 3.260 69 Tm 0.020 7.284 3.320 0.069 2.597 3.140 70 Yb 0.020 7.250 3.274 0.070 2.542 3.522 71 Lu 0.023 6.972 3.342 0.071 2.596 3.210 72 Hf 0.138 5.581 2.350 0.072 2.454 2.900 73 Ta 0.142 6.016 2.300 0.073 2.026 2.766 74 W 0.137 6.176 2.270 0.074 2.350 2.630 75 Re 0.242 3.686 2.040 0.075 3.055 2.760 76 Os 0.029 7.771 2.760 0.076 2.696 2.540 77 Ir 0.244 6.055 1.960 0.077 3.142 2.715 78 Pt 0.021 7.539 2.500 0.078 2.919 2.772 79 Au 0.024 7.553 2.470 0.079 2.917 2.885 81 Tl 0.004 8.850 3.500 0.081 2.576 3.300 82 Pb 0.002 8.368 2.930 0.082 2.935 3.242

84

Table 6.3: Fitted knots of cubic spline of the embedding functionals for the EAM po- tentials of pure elements, continued in Table 6.4. The first knot at (0,0) is assumed. The fitting structures are given in Tables 6.5 and 6.6.

knot2 knot3 knot4 knot5 knot6 knot7 (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV)

Li 0.048,-1.058 0.119,-1.52850 0.190,-1.54750 0.246,-1.607 0.291,-1.554 0.469,-1.469 Be 0.178,-3.079 0.228,-3.241 0.423,-3.509 0.731,-3.647 1.057,-3.588 1.670,-3.324 C 0.301,-4.967 0.344,-5.905 0.476,-6.986 0.861,-7.490 1.419,-7.264 2.389,-5.970 Na 0.014,-0.725 0.029,-0.9756 0.062,-1.059 0.082,-1.057 0.135,-1.059 0.181,-1.424 Mg 0.015,-0.939 0.033,-1.292 0.062,-1.414 0.096,-1.413 0.133,-1.356 0.242,-1.175 Al 0.078,-2.811 0.121,-3.106 0.495,-3.980 0.897,-3.879 1.381,-3.665 2.200,-3.074 Si 0.057,-3.662 0.087,-4.153 0.157,-4.476 0.286,-4.409 0.349,-4.204 0.561,-3.168 K 0.005,-0.613 0.009,-0.757 0.025,-0.844 0.057,-0.710 0.089,-0.551 0.149,-0.415 Ca 0.009,-1.202 0.021,-1.692 0.062,-1.814 0.094,-1.719 0.135,-1.578 0.201,-1.434 Sc 0.055,-3.363 0.080,-3.787 0.140,-4.092 0.226,-4.140 0.315,-4.009 0.481,-3.612 Ti 0.133,-4.307 0.188,-4.850 0.314,-5.315 0.444,-5.410 0.712,-5.033 1.075,-4.406 V 0.245,-4.014 0.333,-4.545 0.606,-5.153 0.858,-5.111 1.058,-4.850 1.520,-4.005 Cr 0.168,-3.162 0.182,-3.277 0.300,-3.808 0.487,-3.758 0.634,-3.324 0.935,-1.915 Fe 0.171,-4.007 0.247,-4.554 0.505,-5.071 0.834,-4.962 1.072,-4.734 1.445,-4.613 Co 0.146,-3.374 0.380,-4.894 0.821,-5.517 1.306,-5.509 1.683,-5.008 2.200,-3.952 Ni 0.223,-3.899 0.336,-4.373 0.566,-4.731 1.028,-5.280 1.355,-5.028 2.200,-4.172 Cu 0.091,-2.811 0.127,-3.100 0.238,-3.280 0.334,-3.215 0.455,-2.935 0.604,-2.374 Zn 1.012,-0.752 1.604,-0.880 2.794,-0.940 3.544,-0.919 4.805,-0.822 8.006,-0.406 Ge 0.056,-3.285 0.072,-3.429 0.115,-3.592 0.188,-3.553 0.241,-3.359 0.364,-2.645 Rb 0.004,-0.552 0.009,-0.707 0.020,-0.663 0.029,-0.594 0.036,-0.633 0.047,-0.977 Sr 0.005,-1.128 0.008,-1.304 0.046,-1.558 0.063,-1.490 0.088,-1.331 0.108,-1.205 Y 0.012,-3.002 0.021,-3.548 0.098,-4.152 0.167,-4.060 0.260,-3.712 0.338,-3.459 Zr 0.052,-4.542 0.082,-5.367 0.186,-6.136 0.299,-6.147 0.489,-5.725 0.685,-5.428 Nb 0.034,-6.744 0.054,-7.933 0.139,-9.012 0.240,-8.889 0.347,-8.192 0.468,-7.278 Mo 0.059,-4.588 0.088,-5.395 0.192,-6.024 0.271,-5.941 0.363,-5.304 0.542,-4.467 Tc 0.164,-5.372 0.217,-5.913 0.317,-6.476 0.482,-6.730 0.720,-6.230 1.108,-4.437 Ru 0.088,-6.348 0.130,-7.166 0.241,-7.687 0.311,-7.756 0.457,-7.052 0.638,-6.282 Rh 0.074,-4.522 0.095,-4.900 0.176,-5.540 0.279,-5.529 0.381,-4.971 0.495,-3.965 Pd 0.017,-2.667 0.025,-3.029 0.078,-3.490 0.128,-3.256 0.166,-2.905 0.232,-2.126 Ag 0.027,-1.749 0.035,-1.894 0.117,-2.349 0.178,-2.390 0.239,-2.422 0.294,-2.488 Cd 0.061,-0.301 0.110,-0.379 0.245,-0.409 0.364,-0.446 0.690,-0.333 0.921,-0.278 Te 0.006,-1.550 0.011,-1.922 0.029,-2.088 0.038,-2.159 0.069,-1.969 0.117,-1.615 Ba 0.004,-1.228 0.010,-1.721 0.020,-1.822 0.027,-1.813 0.050,-1.658 0.081,-1.393 La 0.019,-2.564 0.051,-3.977 0.121,-4.240 0.210,-4.086 0.322,-3.777 0.416,-3.536 Ce 0.040,-2.826 0.106,-4.210 0.207,-4.481 0.329,-4.503 0.526,-4.234 0.677,-3.908

85

Table 6.4: continuation of Table 6.3.

Pr 0.035,-2.470 0.088,-3.600 0.166,-3.856 0.246,-3.891 0.357,-3.821 0.554,-3.605 Nd 0.021,-2.403 0.062,-3.730 0.157,-3.834 0.308,-3.858 0.456,-3.972 0.554,-3.776 Eu 0.010,-1.188 0.021,-1.599 0.055,-1.790 0.125,-1.734 0.191,-1.516 0.275,-1.357 Gd 0.017,-3.615 0.043,-5.278 0.085,-5.539 0.129,-5.542 0.219,-5.281 0.330,-4.894 Tb 0.027,-2.642 0.067,-3.837 0.142,-4.061 0.202,-3.963 0.319,-3.614 0.409,-3.340 Dy 0.027,-2.672 0.068,-3.683 0.158,-4.026 0.199,-3.866 0.279,-3.728 0.404,-3.139 Ho 0.028,-2.672 0.069,-3.851 0.162,-4.020 0.199,-3.927 0.283,-3.689 0.406,-3.248 Er 0.028,-2.681 0.069,-3.729 0.162,-3.947 0.196,-3.823 0.276,-3.654 0.406,-3.278 Tm 0.030,-2.762 0.069,-3.659 0.168,-3.992 0.217,-3.801 0.289,-3.658 0.397,-3.268 Yb 0.012,-0.958 0.022,-1.224 0.056,-1.366 0.122,-1.191 0.188,-1.044 0.264,-0.965 Lu 0.040,-2.683 0.087,-3.659 0.155,-3.911 0.210,-3.900 0.340,-3.630 0.482,-3.254 Hf 0.036,-4.599 0.057,-5.285 0.209,-6.329 0.279,-6.266 0.399,-5.886 0.680,-4.554 Ta 0.066,-5.540 0.109,-6.815 0.210,-7.729 0.409,-7.719 0.624,-6.877 0.903,-6.333 W 0.111,-6.179 0.149,-6.886 0.256,-7.726 0.465,-7.681 0.689,-6.553 0.953,-4.666 Re 0.307,-4.531 0.441,-5.847 0.680,-7.039 0.803,-7.237 1.102,-6.969 1.636,-5.140 Os 0.106,-6.174 0.206,-7.540 0.326,-7.866 0.394,-7.945 0.570,-7.555 0.908,-5.803 Ir 0.089,-5.498 0.106,-6.040 0.228,-7.004 0.342,-7.142 0.486,-6.271 0.667,-5.735 Pt 0.025,-3.842 0.035,-4.280 0.094,-5.002 0.149,-4.728 0.188,-4.327 0.250,-3.495 Au 0.015,-1.932 0.023,-2.231 0.075,-2.492 0.108,-2.276 0.134,-2.038 0.179,-1.590 Tl 0.007,-1.336 0.013,-1.611 0.029,-1.661 0.062,-1.476 0.091,-1.268 0.133,-1.001 Pb 0.0004,-1.813 0.0006,-2.159 0.0010,-2.454 0.0016,-2.537 0.0046,-2.491 0.0125,-1.686

86

Table 6.5: (left part) Structures used to fit EAM potentials for pure elements, continued in Table 6.6. The EAM potentials are fitted to the ab initio energies in body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close-packed (HEX), diamond structure (DIA), and groundstate structure at various pressures obtained by expanding/compressing the equillibrium lattice constant a by a factor from 0.9 to 1.16 corresponding to a range of charge density from ρmax to ρmin. (right part) Lattice constansts calculated using the fitted parameters. The literature values aLIT are taken from [146].

Z Name Struct. ρmin → ρmax aLIT aV ASP aEAM (A˚−3) (A˚) (A˚) (A˚)

3 Li BCC 0.104 → 0.441 3.49 3.44 3.44 4 Be HEX 0.268 → 1.437 2.29 2.27 2.24 6 C DIA 0.329 → 2.038 3.57 3.57 3.57 11 Na BCC 0.027 → 0.163 4.23 4.19 4.19 12 Mg HEX 0.014 → 0.174 3.21 3.19 3.16 13 Al FCC 0.098 → 0.312 4.05 4.04 4.00 14 Si DIA 0.060 → 0.443 5.43 5.46 5.46 19 K BCC 0.013 → 0.089 5.23 5.29 5.29 20 Ca FCC 0.014 → 0.186 5.58 5.54 5.54 21 Sc HEX 0.064 → 0.441 3.31 3.30 3.34 22 Ti HEX 0.150 → 0.845 2.95 2.92 2.92 23 V BCC 0.263 → 1.320 3.02 2.98 2.98 24 Cr BCC 0.126 → 0.765 2.88 2.84 2.84 26 Fe BCC 0.213 → 1.221 2.87 2.83 2.83 27 Co HEX 0.361 → 0.974 2.51 2.47 2.45 28 Ni FCC 0.374 → 0.983 3.52 3.49 3.46 29 Cu FCC 0.113 → 0.519 3.61 3.64 3.64 30 Zn HEX 0.777 → 7.050 2.66 2.71 2.71 32 Ge DIA 0.056 → 0.306 5.66 5.79 5.79 37 Rb BCC 0.004 → 0.048 5.59 5.65 5.65 38 Sr FCC 0.009 → 0.097 6.08 6.05 6.05 39 Y HEX 0.036 → 0.282 3.65 3.64 3.64 40 Zr HEX 0.073 → 0.511 3.23 3.22 3.22 41 Nb BCC 0.050 → 0.390 3.30 3.31 3.31 42 Mo BCC 0.055 → 0.463 3.15 3.15 3.12 43 Tc HEX 0.147 → 0.993 2.74 2.76 2.76 44 Ru HEX 0.084 → 0.558 2.70 2.73 2.70 45 Rh FCC 0.072 → 0.469 3.80 3.84 3.84 46 Pd FCC 0.017 → 0.195 3.89 3.96 3.96 47 Ag FCC 0.034 → 0.256 4.09 4.16 4.16 48 Cd HEX 0.076 → 0.855 2.98 3.09 3.03 52 Te HEX 0.007 → 0.108 4.45 4.06 4.02

87

Table 6.6: continuation of Table 6.5.

56 Ba BCC 0.007 → 0.074 5.02 5.02 5.02 57 La HEX 0.042 → 0.331 3.75 3.74 3.74 58 Ce FCC 0.096 → 0.610 5.16 4.74 4.74 59 Pr HEX 0.076 → 0.502 3.67 3.45 3.45 60 Nd HEX 0.068 → 0.457 3.66 3.56 3.56 63 Eu BCC 0.029 → 0.232 4.61 4.46 4.41 64 Gd HEX 0.037 → 0.303 3.64 3.62 3.62 65 Tb HEX 0.049 → 0.362 3.60 3.62 3.62 66 Dy HEX 0.047 → 0.345 3.59 3.61 3.61 67 Ho HEX 0.048 → 0.356 3.58 3.59 3.63 68 Er HEX 0.047 → 0.352 3.56 3.58 3.61 69 Tm HEX 0.049 → 0.367 3.54 3.55 3.55 70 Yb FCC 0.023 → 0.202 5.49 5.29 5.29 71 Lu HEX 0.061 → 0.425 3.51 3.48 3.48 72 Hf HEX 0.073 → 0.530 3.20 3.20 3.20 73 Ta BCC 0.091 → 0.692 3.31 3.32 3.32 74 W BCC 0.105 → 0.794 3.16 3.19 3.19 75 Re HEX 0.378 → 1.509 2.76 2.78 2.78 76 Os HEX 0.114 → 0.861 2.74 2.76 2.73 77 Ir FCC 0.068 → 0.629 3.84 3.88 3.80 78 Pt FCC 0.025 → 0.236 3.92 3.98 3.98 79 Au FCC 0.016 → 0.167 4.08 4.17 4.17 81 Tl HEX 0.011 → 0.112 3.46 3.54 3.54 82 Pb FCC 0.001 → 0.011 4.95 5.02 5.02

88

Chapter 7

Effects of Mo on the thermodynamics of

Fe:Mo:C nanocatalyst for single-walled

carbon nanotube growth

Among the established methods for single-walled carbon nanotubes (SWCNTs) syn-

thesis [147, 148, 149], catalytic chemical vapor decomposition (CCVD) technique is

preferred for growing nanotubes on a substrate at a target position due to its rel-

atively low synthesis temperature. Temperature as low as ∼ 450 oC was reported by using hydrocarbon feedstock with exothermic catalytic decomposition reaction

[150, 151]. Critical factors for the efficient growth via CCVD are the compositions

of the interacting species, the preparation of the catalysts, and the synthesis condi-

tions. Efficient catalysts must have long active lifetimes (with respect to feedstock

dissociation and nanotube growth), high selectivity and be less prone to contamina-

tion. Common factors that lead to reduction in catalytic activity are deactivation

(e.g. due to coating with carbon or nucleation of inactive phases) [152, 153, 154] and

thermal sintering (e.g. caused by highly exothermic reactions on the clusters surface

[150, 151, 155] with insufficient heat [156]).

Metal alloy catalysts, such as Fe:Co, Co:Mo, and Fe:Mo, improve the growth of

CNTs [157, 158, 159, 160, 162, 163], because the presence of more than one metal

species can significantly enhance the activity of a catalyst [159, 164], and can prevent

catalyst particle aggregation [163, 164]. In the case of Fe:Mo nanoparticles supported

on Al2O3 substrates, the enhanced catalyst activity has been shown to be larger than

the linear combination of the individual Fe/Al2O3 and Mo/Al2O3 activities [159, 160].

89

This is explained in terms of substantial intermetallic interaction between Mo, Fe and

C [159, 165, 166]. The addition of Mo in mechanical alloying of powder Fe and C

mixtures promotes solid state reactions even at low Mo concentrations by forming

ternary phases, such as the (Fe,Mo)23C6 and Fe2(MoO4)3 type carbides [167]. It has

been found that low Mo concentration in Fe:Mo is favored for growing SWCNTs

(on Al2O3 substrates) since the presence, after activation, of the phase Fe2(MoO4)3

can lead to the formation of small metallic clusters [168]. Considering the vapor-

liquid-solid model (VLS), which is the most probable mechanism for CNT growth

[159, 169]. The metallic nanoparticles are very efficient catalysts when they are in

the liquid or viscous states, probably due to considerable carbon bulk-diffusion in

this phase (compared to surface or sub-surface diffusion). Generally, unless stable

intermetallic compounds form, alloying metals reduce the melting point below those

of the constituents [165, 166]. Hence, to improve the yield and quality of nanotubes,

one can tailor the composition of the catalyst particle to move its liquidus line below

the synthesis temperature [159]. However, identifying the perfect alloy composition

is non trivial. In fact, the presence of more than two metallic species allows for the

possibility of different carbon pollution mechanisms by thermodynamic promotion of

ternary carbides. In this Chapter, we study the phase diagram of Fe:Mo:C system

and the possible roles of Mo in the catalytic properties of Fe:Mo.

7.1 Size-pressure approximation

Determining the thermodynamic stability of different phases in nanoparticles of dif-

ferent sizes with ab initio calculations is computationally expensive. We develop

a simple model, called the ”size-pressure approximation”, which allows one to es-

timate the phase diagram at the nanoscale starting from bulk calculations under

pressure [152]. Surface curvature and superficial dangling bonds on nanoparticles are

90

responsible for internal stress fields which modify the atomic bond lengths. As a

first approximation, where all surface effects that are not included in the curvature

are neglected, we can map the particle radius R to the pressure P by equating the

deviation from the bulk value of the average bond lengths due to surface curvature

(in the case of particle) and pressure (in the case of bulk). For spherical clusters,

the phenomenon can be modeled with the Young-Laplace equation P = 2γ/R where

the proportionality constant γ can be calculated with ab initio methods. In our

study, since the concentration of Mo in Fe:Mo is small, we use the ”size-pressure

approximation” of Fe particle.

Figure 7.1 shows the implementation of the ”size-pressure approximation” for Fe

nanoparticles. On the left hand side we show the ab initio calculations of the deviation

of the average bond length inside the cluster Δdnn ≡ d0nn − dnn (d0nn = 0.2455 nm is our bulk bond length), for body-centered cubic (BCC) particles of size N = 59, 113,

137, 169, 307 and ∞ (bulk) as a function of the inverse radius (1/R). The particles were created by intersecting a BCC lattice with different size spheres. The particle

radius is defined as 1/R ≡ 1/Nscp ∑

i 1/Ri where the sum is taken over the atoms

belonging to the surface convex polytope (Nscp vertices) and Ri are the distances to

the geometric center of the cluster. The left straight line is a linear interpolation

between 1/R and Δdnn calculated with the constraint of passing through 1/R = 0

and Δdnn = 0 (N = ∞, bulk). The right hand side shows the ab initio value of dnn in bulk BCC Fe as a function of hydrostatic pressure, P . The straight line is a linear

interpolation between P and Δdnn calculated with the constraint of passing through

P = 0 and Δdnn = 0 (bulk lattice). By following the colored dashed paths indicated

by the arrows we can map the analysis of nanoparticles stability as a function of R

onto bulk stability as a function of P , and obtain the relation between the radius of

particle/nanotube and the effective pressure P ·R = 2.46 GPa · nm. It is important

91

Figure 7.1: (color online). Size-pressure approximation for Fe nanoparticles obtained by equating the deviation of average bond length from the bulk value due to curvature 1/R (in the case of particle) and due to pressure P (in the case of bulk).

to mention that our γ = 1.23 J/m2 is not a real surface tension but an ab initio fitting

parameter describing size-induced stress in nanoparticles. With this γ, we deduce the

Fe-Mo-C phase diagram of nanoparticles of radius R from ab initio calculations of

the bulk material under pressure P .

92

7.2 Fe-Mo-C phase diagram under pressure

Simulations are performed with VASP as described in Section 2.2. The hydrostatic

pressure estimated from the pressure-size model is implemented as Pulay stress [170].

Ternary phase diagrams are calculated using BCC-Mo, BCC-Fe and SWCNTs as

references (pure-Fe phase is taken to be BCC because our simulations are aimed at

the low temperature regime of catalytic growth). The reference SWCNTs have the

same diameter of the particle to minimize the curvature-strain energy. In fact, CVD

experiments of SWCNT growth from small (∼ 0.6-2.1 nm) particles indicate that the diameter of the nanotube is similar to the diameter of the catalyst particle from

which it grows. In some experiments where the growth mechanism is thought to be

root-growth, the ratio of the catalyst particle diameter to SWCNT diameter is ∼ 1.0, whereas in experiments involving pre-made floating catalyst particles this ratio is ∼ 1.6 [171]. Formation energies are calculated with respect to decomposition into the

nearby stable phases, depending the position in the ternary phase diagram.

Binary and Ternary phases are included if they are stable in the temperature

range used in CVD growth of SWCNTs or if they have been reported experimentally

during or after the growth [165, 166, 172]. Thus, we include the binaries Mo2C,

Fe2Mo and Fe3C. In addition, since our Fe-rich Fe:Mo experiments were performed

with compositions close to Fe4Mo [159], we include a random phase Fe4Mo generated

with the special quasi-random structure formalism (SQS). Bulk ternary carbides,

which have been widely investigated due to their importance in alloys and steel, can

be considered as derivatives of binary structures with extra C atoms in the interstices

of the basic metal alloy structures. Three possible ternary phases have been reported

for bulk Fe-Mo-C [173] and they are referred as τ1 (M6C), τ2 (M3C) and τ3 (M23C6)

(M is the metal species). For simplicity, we follow the same nomenclature. τ1 is

the wellknown M6C phase, which has been observed experimentally as Fe4Mo2C and

93

Fe3Mo3C structures (η carbides) [174, 175]. Both of these structures are FCC but

have different lattice spacings. Our calculations show that the most stable variant

τ1 is Fe4Mo2C, and we denote it as τ1 henceforth. τ2 is the Fe2MoC phase, which

has an orthorhombic symmetry distinct from that of Fe3C [176, 177]. We consider

Fe21Mo2C6 as the third τ3 FCC phase [178]. We use the Cr23C6 as the prototype

structure [179] where Fe and Mo substitute for Cr. Although M23C6 type phases do

not appear in the stable C-Fe or C-Mo systems, they have been reported in ternary C-

Fe-Mo systems and also appear as transitional products in solid state reactions [173].

Time-temperature precipitation diagrams of low-C steels have identified τ2, τ3 and τ1

as low-temperature, metastable and stable carbides, respectively [180]. Furthermore,

τ2 carbides precipitate quickly due to carbon-diffusion controlled reaction while τ3

carbides precipitate due to substitutional-diffusion controlled reactions. The latter

phenomenon, requiring high temperature, longer times and producing metastable

phases is not expected to enhance the catalytic deactivation of the nanoparticle. In

summary, as long as the presence of carbon does not lead to excessive formation of

Fe3C and τ2), the catalyst should remain active for SWCNT growth.

Figure 7.2 shows the phase diagram at zero temperature of nanoparticles of radii

R ∼ ∞, 1.23, 0.62, 0.41 nm, calculated at P = 0, 2, 4, and 6 GPa, respectively. Stable and unstable phases are shown as black squares and red dots, respectively.

The solid green lines connect the stable phases. The numbers ”1”...”8” in panels (c)

and (d) indicate the intersections between the phases’ boundary and the dotted lines

representing the path of carbon pollution to the two test phases Fe4Mo and FeMo.

Fe4Mo has been reported to be an effective catalyst composition [159] while FeMo

represents a hypothetical Fe:Mo particle with a Mo content larger than 33%.

94

Figure 7.2: (color online). Ternary phase diagram for Fe-Mo-C nanoparticles of R ∼ ∞, 1.23, 0.62, 0.41 nm.

7.3 Fe4Mo particles

An advantage of an Fe4Mo particle has over a pure Fe particle is that the [Fe4/5Mo1/5]1−x-

Cx line does not intercept any carbide (Fe3C, τ3, τ2). This implies that, at least at low

temperatures, there is a surplus of unbounded metal (probably even at high temper-

atures since the line is far from all of the competing stable phases). This is illustrated

in figure 7.3, which shows the fractional evolution of species as one progresses along

the [Fe4/5Mo1/5]1−x-Cx line in figure 7.2.

For a large Fe4Mo particle (R ≥ 0.62 nm), the decomposition into stable phases is shown in figure 7.3(a)). At concentrations between 0 < xc < 0.09 there is available

free Fe for catalysis, however there is no carbon content in the particle. At higher

95

concentrations, the particle starts to contain free carbon while still providing free Fe.

Therefore, a steady state growth of SWCNTs is possible from large Fe4Mo particles.

Figure 7.3(b) shows the decomposition for a small Fe4Mo particle (R ≤ 0.41 nm)). In this case, the free Fe is consumed and transformed into τ3 carbide before the particle

has enough free carbon, hence, the SWCNT growth will not occur. We can estimate

the minimum size of the particle by calculting the pressure at which τ3 starts to form.

By linear interpolation, we obtain R = 0.52 nm. This size is smaller than that if one

uses pure Fe nanocatalyst (R = 0.56 nm) [152, 153].

7.4 FeMo particles

Figure 7.4 shows the decomposition analysis for FeMo particle. For a large particle of

size R ≥ 0.62 nm, The particle contains free Fe and carbon only after the concentra- tion xc is larger than 0.2 which then permits the growth of the nanotube. Comparing

to the Fe4Mo particle, since the fraction of Fe in the FeMo particle is considerably

smaller than in Fe4Mo, the expected yield is lower and the synthesis temperature

needs to be increased (to overcome the reduced fraction of catalytically active free

Fe). Small FeMo particles (R ≤ 0.41 nm) are similar to small Fe4Mo clusters. Nucle- ation of τ3 and the abscence of free Fe and excess carbons indicate that the particles

are catalytically inactive. The minimum size of FeMo and Fe4Mo particles able to

grow nanotube is the same since it is determined by the stabilization of the same τ3

carbide.

96

Figure 7.3: (color online). a) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”1”, represents [Fe4/5Mo1/5]1−x-Cx crossing the boundary phase Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”4” and ”5”, represent [Fe4/5Mo1/5]1−x-Cx crossing the boundary phases Fe←→Mo2C and τ3, as shown in figure 7.2(d).

97

Figure 7.4: (color online). a) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”2” and ”3”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phase Fe2Mo↔Mo2C and Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composi- tion [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”6”, ”7”, and ”8”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phases Fe2Mo↔Mo2C, Fe←→Mo2C, and τ3 ↔Mo2C, as shown in figure 7.2(d).

98

Chapter 8

Conclusions

We have presented the results of our computational studies on adsorptions of hy-

drocarbons (alkanes, benzene) and rare gases on a decagonal surface of Al-Ni-Co

quasicrystal (d-AlNiCo). Ab initio calculations show that upon the adsorptions, the

surface of the d-AlNiCo does not undergo relaxations and that there are no disso-

ciations of the adsorbates. The simulations of thin film growth of these adsorbates

have been performed in the grand canonical ensemble using Monte Carlo method.

We use semiempirical pair interactions for the rare gases and develop classical many-

body potentials based on embedded-atom method (EAM) for the hydrocarbons. All

of the simulated atoms/molecules wet the substrate as a consequence of compara-

ble strengths between the substrate-adsorbate and adsorbate-adsorbate. Another

consequence is the wide range of overlayer structures observed in these systems.

Methane monolayer has a quasicrystalline pentagonal order commensurate with the

substrate. Propane forms a disordered structure. Hexane and octane monolayers

show 2-dimensional close-packed features consistent with their bulk structure. Ben-

zene forms a pentagonal monolayer at low and moderate temperatures which trans-

forms into a triangular lattice at high temperatures. Similar structural transition

also occurs in xenon monolayer, however in this case, the transition is observed at all

temperatures below the triple-point temperature and as a function of pressure. We

have characterized that such transition is first-order with an associated latent heat.

Smaller noble gases such as Ne, Ar, Kr form monolayers with mixed pentagonal and

triangular patterns and do not show any structural transitions.

By systematically simulating test noble gases of various sizes and strengths, we

99

observe that the relative strengths between the competing interactions determine the

growth mode. Agglomeration occurs when the adsorbate-adsorbate interactions are

much stronger than the substrate-adsorbate ones. In the comparable strength regime,

a layer-by-layer film growth is observed. In this regime, the mismatch between the

size of the gas and the substrate’s characteristic length plays a major role in affecting

the structure of the adsorbed films. In general, on the d-AlNiCo substrate, we found

that structural transition from 5- to 6-fold occurs when the gas size is larger than

λ/0.944 (λ represents the average row-row spacing in the quasicrystalline plane of

the d-AlNiCo). Even though this rule is derived from rare gases, it is consistent with

methane, benzene, hexane and octane. Therefore, it might be useful as a guidance in

the search for suitable quasicrystalls for which alkanes (as the main constituent of oil

lubricants) will form quasiperiodic structures for the low-friction coating applications.

It is a natural extension of this work to investigate other quasicrystalls with

larger characterictic lengths than the d-AlNiCo used in this study. In fact, d-AlNiCo

is stable in many decorations depending on the concentration of Al, Ni, and Co.

Due to the stripe nature in the close-packed structure of linear alkanes, the film

structure of these molecules on one dimensional quasicrystals is interesting to study.

Two surfaces with stripe pattern will be commensurate only when the stripes are

perfectly alligned, therefore, one dimensional quasicrystalls might be good candidate

for low-friction coatings.

100

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Biography

The author was born in May 5th, 1978 in a Javanese village 8 miles southwest of

Solo, Central Java, Indonesia. He is the last(fourth) child of Ladiya (father) and

Sri Suratmi (mother). His father was an elementary school teacher and his mother

is a devoted house wife. Prior to attending college education, the author won the

third award in the International Physics Olympiad 1996 in Norway. The author

received a B.S. degree from Institut Teknologi Bandung, Indonesia, in Electrical

Engineering in 2000. In the Fall of 2000, he went to Florida State University on a

research assistantship from the Physics department in which he completed a M.S.

degree in 2004. In Summer 2004, he received a research assitantship to pursue a

doctoral degree in Mechanical Engineering and Materials Science department from

Duke University under the advisory of Dr. Stefano Curtarolo. He completed his PhD

work in computational materials science in Spring 2008. In addition, he received a

Graduate Certificate degree in Computational Science, Engineering, and Medicine,

in Spring 2008 also from Duke University.

List of publications:

• Diehl R D, Setyawan W and Curtarolo S 2008 J. Phys.: Cond. Mat. in press

• Harutyunyan A R, Awasthi N, Jiang A, Setyawan W, Mora E, Tokune T, Bolton K and Curtarolo S 2008 Phys. Rev. Lett. inpress

• Curtarolo S, Awasthi N, Setyawan W, Li N, Jiang A, Tan T Y, Mora E, Bolton K and Harutyunyan A T 2008 Proc. of Comp. Simul. Studies in Cond. Matt.

Phys. XXI Eds Landau D P, Lewis S P and Schuttler H-B (Springer, Berlin,

Heidelberg)

• Setyawan W, Diehl R D, Ferralis N, Cole M W and Curtarolo S 2007 J. Phys.:

111

Cond. Mat. 19 016007

• Diehl R D, Setyawan W, Ferralis N, Trasca R, Cole M W and Curtarolo S 2007 Phil. Mag. 87 2973

• Jiang A, Awasthi N, Kolmogorov A N, Setyawan W, Bo¨rjesson A, Bolton K, Harutyunyan A R and Curtarolo S 2007 Phys. Rev. B 75 205426

• Setyawan W, Ferralis N, Diehl R D, Cole M W and Curtarolo S 2006 Phys. Rev. B 74 125425

• Diehl R D, Ferralis N, Pussi K, Cole M W, Setyawan W and Curtarolo S 2006 Phil. Mag. 86 863

• Curtarolo S, Setyawan W, Diehl R D, Ferralis N and Cole M W 2005 Phys. Rev. Lett. 95 136104

• Rao S G, Huang L, Setyawan W and Hong S 2003 Nature 425 36

• Setyawan W, Rao S G and Hong S 2002 Mat. Res. Soc. Proc. NN Fall

• Mumtaz A, Setyawan W and Shaheen S A 2002 Phys. Rev. B 65 020503

112COMPUTATIONAL STUDY OF LOW-FRICTION

QUASICRYSTALLINE COATINGS VIA SIMULATIONS

OF THIN FILM GROWTH OF HYDROCARBONS AND

RARE GASES

by

Wahyu Setyawan

Department of Mechanical Engineering and Materials Science Duke University

Date: Approved:

Dr. Stefano Curtarolo, Supervisor

Dr. Teh Y. Tan

Dr. Laurens E. Howle

Dr. Xiaobai Sun

Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in the Department of Mechanical Engineering and Materials Science in the Graduate School of

Duke University

2008

ABSTRACT

COMPUTATIONAL STUDY OF LOW-FRICTION

QUASICRYSTALLINE COATINGS VIA SIMULATIONS

OF THIN FILM GROWTH OF HYDROCARBONS AND

RARE GASES

by

Wahyu Setyawan

Department of Mechanical Engineering and Materials Science Duke University

Date: Approved:

Dr. Stefano Curtarolo, Supervisor

Dr. Teh Y. Tan

Dr. Laurens E. Howle

Dr. Xiaobai Sun

An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in the Department of Mechanical Engineering and Materials Science in the Graduate School of

Duke University

2008

Copyright c© 2008 by Wahyu Setyawan All rights reserved

Abstract

Quasicrystalline compounds (QC) have been shown to have lower friction compared

to other structures of the same constituents. The abscence of structural interlocking

when two QC surfaces slide against one another yields the low friction. To use QC

as low-friction coatings in combustion engines where hydrocarbon-based oil lubri-

cant is commonly used, knowledge of how a film of lubricant forms on the coating is

required. Any adsorbed films having non-quasicrystalline structure will reduce the

self-lubricity of the coatings. In this manuscript, we report the results of simula-

tions on thin films growth of selected hydrocarbons and rare gases on a decagonal

Al73Ni10Co17 quasicrystal (d-AlNiCo). Grand canonical Monte Carlo method is used

to perform the simulations. We develop a set of classical interatomic many-body

potentials which are based on the embedded-atom method to study the adsorption

processes for hydrocarbons. Methane, propane, hexane, octane, and benzene are

simulated and show complete wetting and layered films. Methane monolayer forms

a pentagonal order commensurate with the d-AlNiCo. Propane forms disordered

monolayer. Hexane and octane adsorb in a close-packed manner consistent with

their bulk structure. The results of hexane and octane are expected to represent

those of longer alkanes which constitute typical lubricants. Benzene monolayer has

pentagonal order at low temperatures which transforms into triangular lattice at high

temperatures. The effects of size mismatch and relative strength of the competing

interactions (adsorbate-substrate and between adsorbates) on the film growth and

structure are systematically studied using rare gases with Lennard-Jones pair poten-

tials. It is found that the relative strength of the interactions determines the growth

mode, while the structure of the film is affected mostly by the size mismatch between

adsorbate and substrate’s characteristic length. On d-AlNiCo, xenon monolayer un-

iv

dergoes a first-order structural transition from quasiperiodic pentagonal to periodic

triangular. Smaller gases such as Ne, Ar, Kr do not show such transition. A simple

rule is proposed to predict the existence of the transition which will be useful in the

search of the appropriate quasicrystalline coatings for certain oil lubricants.

Another part of this thesis is the calculation of phase diagram of Fe-Mo-C sys-

tem under pressure for studying the effects of Mo on the thermodynamics of Fe:Mo

nanoparticles as catalysts for growing single-walled carbon nanotubes (SWCNTs).

Adding an appropriate amount of Mo to Fe particles avoids the formation of stable

binary Fe3C carbide that can terminate SWCNTs growth. Eventhough the formation

of ternary carbides in Fe-Mo-C system might also reduce the activity of the catalyst,

there are regions in the Fe:Mo which contain enough free Fe and excess carbon to

yield nanotubes. Furthermore, the ternary carbides become stable at a smaller size

of particle as compared to Fe3C indicating that Fe:Mo particles can be used to grow

smaller SWCNTs.

v

Contents

Abstract iv

List of Figures ix

List of Tables xiv

Acknowledgements xvii

1 Introduction 1

2 Methods 5

2.1 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . 5

2.1.2 Slater determinant . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 Hartree-Fock equations . . . . . . . . . . . . . . . . . . . . . . 7

2.1.4 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . 10

2.1.5 Local density and generalized gradient approximations . . . . 11

2.1.6 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Vienna Ab initio Simulation Package . . . . . . . . . . . . . . . . . . 13

2.3 Classical interatomic potentials . . . . . . . . . . . . . . . . . . . . . 14

2.4 Simplex method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Grand canonical Monte Carlo . . . . . . . . . . . . . . . . . . . . . . 17

3 Noble gas adsorptions on d-AlNiCo 18

3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Simulation cell . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.2 Gas-gas and gas-substrate interactions . . . . . . . . . . . . . 19

vi

3.1.3 Adsorption potentials . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.4 Effective parameters . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.5 Test rare gases . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.6 Chemical potential, order parameter, and ordering transition . 25

3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Adsorption isotherms . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.3 Order parameters (ρ5−6) . . . . . . . . . . . . . . . . . . . . . 32

3.2.4 Effects of �gg and σgg on adsorption isotherms . . . . . . . . . 37

3.2.5 Effects of �gg and σgg on 5- to 6-fold transition . . . . . . . . . 39

3.2.6 Prediction of 5- to 6-fold transition . . . . . . . . . . . . . . . 40

3.2.7 Transitions on smoothed substrates . . . . . . . . . . . . . . . 42

3.2.8 Temperature vs substrate effect . . . . . . . . . . . . . . . . . 43

3.2.9 Orientational degeneracy of the ground state . . . . . . . . . . 46

3.2.10 Isosteric heat of adsorption . . . . . . . . . . . . . . . . . . . . 47

3.2.11 Effect of vertical dimension . . . . . . . . . . . . . . . . . . . 48

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Embedded-atom method potentials 52

4.1 Stage 1: Aluminum, cobalt, and nickel . . . . . . . . . . . . . . . . . 54

4.2 Stage 2: Al-Co-Ni potentials . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Stage 3: Hydrocarbon potentials . . . . . . . . . . . . . . . . . . . . . 59

4.4 Stage 4: Hydrocarbon on Al-Co-Ni . . . . . . . . . . . . . . . . . . . 63

5 Hydrocarbon adsorptions on d-AlNiCo 66

5.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

vii

5.2 Adsorption potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Molecule orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4 Adsorption isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.5 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Embedded-atom potentials for selected pure elements in the pe- riodic table 81

7 Effects of Mo on the thermodynamics of Fe:Mo:C nanocatalyst for single-walled carbon nanotube growth 89

7.1 Size-pressure approximation . . . . . . . . . . . . . . . . . . . . . . . 90

7.2 Fe-Mo-C phase diagram under pressure . . . . . . . . . . . . . . . . . 93

7.3 Fe4Mo particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.4 FeMo particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8 Conclusions 99

Bibliography 101

Biography 111

viii

List of Figures

3.1 (color online). Computed adsorption potentials for (a) Ne, (c) Ar, (e) Kr, and (g) Xe on the d-AlNiCo, obtained by minimizing V (x, y, z) with respect to z. The distribution of the minimum value of these potentials is plotted in (b, d, f, and h) respectively: the solid line marks the average value 〈Vmin〉, the dashed lines mark the values at 〈Vmin〉±SD. . . . . . . . . . . . . . . 22

3.2 Computed adsorption isotherms for all the gas/d-AlNiCo systems. The ranges of temperatures under study are: Ne: T = 14 K to 46 K in 2 K steps, Ar: 45 K to 155 K in 5 K steps, Kr: 65 K to 225 K in 5 K steps, Xe: 80 K to 280 K in 10 K steps. Additional isotherms are shown with solid circles at T � = 0.35: T = 11.8 K (Ne), T = 41.7 K (Ar), T = 59.6 K (Kr), and T = 77 K (Xe). Isotherms above the triple point temperatures are shown as dotted curves. . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Density profiles and Fourier transforms of the outer layer at T � = 0.35 for Ne/d-AlNiCo (T = 11.8 K) and Ar/d-AlNiCo (T = 41.7 K), corresponding to points (A) through (F) of Figure 3.2. . . . . . . . . . . . . . . . . . . 30

3.4 Density profiles and Fourier transforms of the outer layer at T � = 0.35 for of Kr/d-AlNiCo (T = 59.6 K) and Xe/d-AlNiCo (T = 77 K), corresponding to points (A) through (F) of Figure 3.2. . . . . . . . . . . . . . . . . . . 33

3.5 (color online). Order parameters, ρ5−6, as a function of normalized chemical potential, μ�, (as defined in the text) at T � = 0.35 for the first four layers of (a) Ne, (b) Ar, (c) Kr, and for the first layer of Xe (d) adsorbed on d-AlNiCo. A sudden drop of the order parameter in Xe/QC to a constant value of ∼ 0.017 at μ� ∼ 0.8 indicates the existence of a first-order structural transition from fivefold to sixfold in the system. . . . . . . . . . . . . . . 34

ix

3.6 (color online). Xe on d-AlNiCo at T = 77 K. (a) Adsorption isotherm, ρN , versus the normalized chemical potential, μ�. (b) Nearest neighbor distance derived from the first peak of pair correlation function, rNN , (black line), and average spacing between neighbors at equilibrium, d¯NN , (red line). (c) Order parameter ρ5−6 (probability of fivefold defects, defined in Equation 3.14) versus the normalized chemical potential, μ�. (d) Total enthalpy. The transition, which is defined as the point in μ� above which the order parameter remains nearly constant, occurs at μ�tr ∼0.8. The discontinuity in H around μ�tr ∼ 0.8 indicates a first order transition with associated latent heat of the transition. The order parameter ρ5−6 after the transition is ∼ 0.017. Heat of the transition is ≈ 6.8 meV/atom. . . . . . . . . . . . 36

3.7 (color online). Computed adsorption isotherms for Ne, Xe, iNe(1), and dXe(1) on d-AlNiCo at T �=0.35. iNe(1) and dXe(1) are test noble gases having potential parameters described in the text and in Tables 3.1 and 3.2. The effect of varying the interaction strength of the adsorbates on the density increase ΔρN (while keeping the size constant) is negligible on large gases but significant on small gases. . . . . . . . . . . . . . . . . . . . . 39

3.8 (color online). Order parameters as a function of normalized chemical po- tential (as defined in the text) for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed on d-AlNiCo at T � = 0.35. A first-order fivefold to sixfold structural transition occurs in the last three systems, but not in dXe(2). . . 41

3.9 (color online). (a) The minimum of adsorption potential, Vmin(x, y), for Ne on a smoothed d-AlNiCo as described in the text. (b) The variations of the minimum adsorption potentials along the line at x = 0 shown in (a), for the modified and original interactions (solid and dotted curves). . . . . . 44

3.10 Xe on d-AlNiCo. Values of μ�tr for the fivefold to sixfold transition points from 40 K to 140 K (left axis). Transition points at μ�tr > 1 indicate that a transfer of atoms from the second layer to the first layer is required to complete the transition. Also shown is the defect probability as a function of T after the transition occurs (right axis), indicating an increase in defect probability with T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.11 Density plot of Xe on decagonal AlNiCo at 77 K, showing a superposition of the density slices for the 2nd and 4th layers. In the top right, 4th-layer atoms are located directly above the 2nd layer atoms indicating an hexagonal close- packed ABAB stacking, whereas in other regions, such as lower left, the two layers are offset due to stacking fault. . . . . . . . . . . . . . . . . . . . 46

x

3.12 Xe adsorption on d-AlNiCo. (a) Minimum potential energy surface of the adsorption potential with free boundary conditions. (b) Adsorption isotherms of the first layer from a set of 30 simulations at 77 K using the free cell described in the paper. Five density profiles and FTs at point p� of (b) are shown in (c) to (g), representing all possible orientations of hexagonal domains. (h) Schematic diagram illustrating the correspondence between the orientations of the hexagonal domains observed in the density profiles (c) to (g). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.13 Xe adsorption on d-AlNiCo. Pentagonal defects rotate the orientation of hexagons by (a) θ1 = 24◦ and (b) θ2 = 12◦. . . . . . . . . . . . . . . . . 51

3.14 (color online). Xe adsorption on d-AlNiCo. Locations in P , T of the vertical risers in the isotherms corresponding to the first (square), second (circle), and third (triangle) layer formation. The heats of adsorptions, qst, are 270, 129, and 125 meV/atom respectively, calculated as described in the text. The inset figure shows qst obtained from the simulations as well as from the experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 (color online). (a-e) Adsorption potential map, calculated by minimizing the adsorption potential of one molecule on a decagonal Al-Ni-Co along z direction and all rotational degrees of freedom at every coordinates (x, y). Red numbers represent the average value of the adsorption energies. (f) Top view of the decagonal Al-Ni-Co substrate 51.2x51.2 A˚2: Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red), Ni-toplayer (green), and Ni-otherlayers (blue). . . . . . . . . . . . . . . . 71

5.2 (color online). Isothermal adsorption densities of hydrocarbons on a decago- nal Al-Ni-Co: (a) methane (from left to right T = 68, 85, 136, 185 K), (b) propane (T = 80, 127, 245, 365 K), (c) hexane (T = 134, 170, 267, 450 K), (d) octane (T = 162, 210, 324, 450, 565 K), and (e) benzene (T = 209, 270, 418, 555 K). The inset in each figure is the density along z direction at pressure corresponding to point d. Xenon (red) is plotted in panel (a) for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 (a) and (b) Calculated density of methane adsorbed on a decagonal Al-Ni- Co at pressures corresponding to points ”a” and ”c” of the 68 K isotherm shown in Figure 5.2.a, respectively. (c) Fourier transform of the density plot shown in (b), consistent with 5-fold ordering of the methane near monolayer completion. (d) Order parameter (left axis, as calculated in Equation 3.14) as a function of pressure for the 68 K isotherm (right axis), indicating no sharp transition to 6-fold ordering. . . . . . . . . . . . . . . . . . . . . 74

xi

5.4 Calculated density of propane, hexane, and octane adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion. Propane forms a dis- ordered structures, whereas hexane and octane tend to form close-packed structures indicated by stripe features with increasing order for longer chain. 76

5.5 (Top row) Calculated density of benzene adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion for 209 K, 270 K, and 418 K. (middle row) Density profile of the geometrical center of density shown in the top row. (bottom row) Fourier transform of the density plot shown in the middle row, showing 5-fold ordering at 209 K, mixture of 5-fold and 6-fold structures at 270 K, and mostly 6-fold features at 418 K. . . . . . . 78

5.6 (left axis) Order parameter ρ5−6 = N5/(N5 +N6) (Nn denotes the number of molecules having n nearest neighbors) as a function of temperature at 0.01 atm of pressure. (right axis) Adsorption isobar showing the number of adsorbed molecules as a function of temperature. Verticel dashed lines correspond to T = 209, 270, and 418 K whose density profiles are plotted in Figure 5.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.1 (color online). Size-pressure approximation for Fe nanoparticles obtained by equating the deviation of average bond length from the bulk value due to curvature 1/R (in the case of particle) and due to pressure P (in the case of bulk). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.2 (color online). Ternary phase diagram for Fe-Mo-C nanoparticles of R ∼ ∞, 1.23, 0.62, 0.41 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.3 (color online). a) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x- Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”1”, represents [Fe4/5Mo1/5]1−x-Cx crossing the boundary phase Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed verti- cal lines, labeled as ”4” and ”5”, represent [Fe4/5Mo1/5]1−x-Cx crossing the boundary phases Fe←→Mo2C and τ3, as shown in figure 7.2(d). . . . . . . 97

xii

7.4 (color online). a) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x- Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”2” and ”3”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phase Fe2Mo↔Mo2C and Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”6”, ”7”, and ”8”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phases Fe2Mo↔Mo2C, Fe←→Mo2C, and τ3 ↔Mo2C, as shown in figure 7.2(d). . . . . . . . . . . 98

xiii

List of Tables

3.1 Parameter values for the 12-6 Lennard-Jones interactions. TM is the label for Ni or Co. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Range, average (〈Vmin〉), and standard deviation (SD) of the interaction Vmin(x.y) on the d-AlNiCo. Effective parameters of the gas-substrate in- teractions (Dgs, σgs, D�gs, σ�gs), and, for comparison, the best estimated well depths DGrgs on graphite [106]. . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Results for Ne, Ar, Kr, and Xe adsorbed on d-AlNiCo. Tt is taken from reference [109]. The density increase (ΔρN ) in the first and second layers is calculated at T � = 0.35 from point (A) to (B) and (C) to (D) in Figure 3.2, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Summary of adsorbed noble gases on d-AlNiCo that undergo a first-order fivefold to sixfold structural transition and those that do not. . . . . . . . 42

4.1 List of structure prototypes used to fit the EAM potentials for hydrocarbon adsorption on Al-Co-Ni. For elemental Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure (DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed (HEX), simple cubic (SC), and simple hexagonal (SH). The Al-Co-Ni ternaries are taken from the database of alloys [122]. . . . . . . . . . . . . . . . . . . . . . . 53

4.2 (Top part) List of structures used to fit EAM potential for elemental alu- minum. ρi is charge density at atom-i, ΔE = EEAM−EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3 · re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 56

4.3 (Top part) List of structures used to fit EAM potential for elemental cobalt. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”HEX at a0 · c” indicates that the structure is HEX with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 57

xiv

4.4 (Top part) List of structures used to fit EAM potential for elemental nickel. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx). . . . . . . 58

4.5 (Top part) List of structures used to fit EAM potential for Al-Co-Ni systems. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV) and the first knot at (0,0) is assumed. The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . . . . . . . . . . . . . . . . . 60

4.6 (Top part) List of structures used to fit EAM potential for hydrocarbons. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . 61

4.7 (Top part) List of structures used to fit EAM potential for alkanes and benzene. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.8 Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2. Fitting structures are given in Table 4.9 . . . 64

4.9 Fitted energies calculated using EAM parameters in Table 4.8. Methane up represents methane with one H below C and three H above C. Methane down is inverse of methane up. The unit for energy is eV/atom except for the adsorption energy which is in eV/molecule. . . . . . . . . . . . . . . . . 65

5.1 Parameter values for the adsorbate-adsorbate interactions used for hydro- carbon adsorption on a decagonal Al-Ni-Co. Intermolecular energies are calculated as a sum of pair interactions. For methane-methane, the C-H is taken as the geometrical mean for parameter A and as the arithmetic mean for parameters B and C. . . . . . . . . . . . . . . . . . . . . . . . . . . 69

xv

6.1 Fitted parameters for the charge density and pair interaction functionals of the EAM potentials for pure elements, continued in Table 6.2. The fitting structures are given in Tables 6.5 and 6.6. The parameters for the embedding functionals are given in Tables 6.3 and 6.4. . . . . . . . . . . 83

6.2 continuation of Table 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3 Fitted knots of cubic spline of the embedding functionals for the EAM potentials of pure elements, continued in Table 6.4. The first knot at (0,0) is assumed. The fitting structures are given in Tables 6.5 and 6.6. . . . . . 85

6.4 continuation of Table 6.3. . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.5 (left part) Structures used to fit EAM potentials for pure elements, con- tinued in Table 6.6. The EAM potentials are fitted to the ab initio ener- gies in body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close-packed (HEX), diamond structure (DIA), and groundstate structure at various pressures obtained by expanding/compressing the equillibrium lattice constant a by a factor from 0.9 to 1.16 corresponding to a range of charge density from ρmax to ρmin. (right part) Lattice constansts calculated using the fitted parameters. The literature values aLIT are taken from [146]. 87

6.6 continuation of Table 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . 88

xvi

Acknowledgements

I would like to thank my advisor, Prof. Stefano Curtarolo for all the academic and financial supports. I wish to thank all collaborators of this work, Prof. Renee D. Diehl, Prof. Milton W. Cole, Prof. L W. Bruch, Dr. Nicola N. Ferralis, and Dr. Andrea Trasca. I thank all my teachers for their mentoring and my committee members, especially Prof. Teh Y. Tan, Prof. Laurens E. Howle, and Prof. Xiaobai Sun for their dedications and insightful advice. I thank all my colleagues, in particular Dr. Neha Awasthi, Dr. Aiqin Jiang, Dr. Roman Chepulskyy for encouragements and valuable discussions, and Dr. Aleksey Kolmogorov for his mentoring and experienced advice on VASP. I also wish to thank all the staff in the department of Mechanical Engineering and Materials Science for the administrative helps.

I thank the National Science Foundation and Honda Research Institute for fund- ing this research. I also thank San Diego Super Computers (SDSC), Texas Ad- vanced Computing Center (TACC), National Center for Supercomputing Applica- tions (NCSA), and Pittsburgh Supercomputing Center (PSC) for all the computing time through Teragrid projects. Thanks also go to Duke Clusters for additional allocations.

I can never thank enough to my father whose talent in numbers has introduced me to math and engineering, my mother and my sisters for their endless prayers, love, and encouragements throughout my life. I am deeply grateful to be blessed with a brilliant, loving, and beautiful wife, Prof. Lisa M. Peloquin, who provides continuous prayers, love, and supports in so many ways. I also thank our friends especially Laura Heymann for her kind helps in many occassions. Thanks also go to Pepito, Clementine, Hamzah, John and other aquatic species for being excellent family members.

Finally, all praises are due to Allah, the Creator (al-Khaliq), the Loving (al- Rahman), and the All Knowing (al-Alim). I am thankful for all the opportunities, health, and blessings that enable me to finish this dissertation. I am deeply humbled by every piece of knowledge that I learned.

xvii

Chapter 1

Introduction

Quasicrystals (QCs) were discovered in 1982 by Dr. Shechtman during his X-ray mea-

surements on Al-Mn compounds. Similar to crystals, QCs consist of atoms arranged

in regular patterns having long-range order, i.e. the diffraction patterns show discrete

spots. However, they do not have any translational periodicities. The discrete spots

come from the rotational symmetries. A variety of stable and metastable QCs have

been successfully synthesized. Among the first high-quality samples are icosahedral

AlCuFe [1], decagonal AlNiCo [2], and icosahedral AlPdMn [3]. Today, hundreds of

quasicrystalline phases are known, tens of which are stable [4]. The majority of them

are derived from aluminum-transition metal family [5].

QCs have been shown to have lower coefficients of friction than most metals. For

example, the static friction μs between two clean surfaces of icosahedral AlPdMn is

≈ 0.6 [6], whereas μs for Ni(110) and Cu(111) is ≈ 4 [7]. Kinetic friction tests using pin-on-disk technique with diamond pin show that among materials with compara-

ble hardness included in the study, icosahedral AlPdMn exhibits the lowest friction

(μ = 0.05), compared to window glass (μ = 0.08), sintered Al2O3 (μ = 0.13), or

hard Cr-steel (μ = 0.13) [8]. Detailed measurements of friction as a function of struc-

tural perfection in icosahedral AlCuFe quasicrystal show that the minimum friction is

achieved for sample with the best quasilattice perfection [9]. Decagonal AlNiCoSi has

been verified to have lower friction than Cr2O3, which represents the most advanced

technology for use on piston rings in automotive engines [10]. Further evidence of

reduced friction is demonstrated in decagonal AlNiCo, in which the friction on the

2-fold periodic surface is eight times higher than that on the decagonal surface [11]

1

(Note that in decagonal quasicrystals, there is a direction along which the quasicrys-

talline surfaces are stacked periodically).

The reduced friction between two quasicrystalline surfaces can be understood by

considering the structure commensurability between them. For surfaces with enough

hardness, atoms can be regarded as fixed in their position in each material. Within

this approximation, it has been theoretically demonstrated that the total interaction

energy between the two surfaces is independent of their relative displacement parallel

to the interface [12]. Therefore, the frictional force which is the gradient of the en-

ergy with respect to the displacement is vanishingly small, a phenomenon known as

superlubricity [12]. Superlubricity has been observed experimentally between tung-

sten and graphite in which certain relative orientations result in nearly zero friction

beyond the limit of the instruments [13]. Since any two QCs do not have common

periodicities at any length scales, superlubricity is expected.

Another characteristic feature of QCs is their resistance to oxidation, which is

quite surprising given that their main constituent, aluminum, is readily oxidized in

ambient conditions [5]. The behavior is particularly spectacular in AlCuLi icosahedral

phase that resists oxidation very well in humid air [5]. The combination of high

hardness, oxidation resistance, and low friction has attracted interests in QCs as

coatings to reduce friction and wear in machine parts, e.g. at the piston-cylinder

interface and in gear boxes. In such environments, hydrocarbon-based oil lubricant

is typically used to overcome the friction caused by surface asperities. Therefore,

to yield a synergic performance of self-lubricating QC coating and lubricant, it is

important to understand the interactions between them. The lubricant must be

able to spread (wet) well on the QC. Furthermore, the structures of the thin film

of lubricant formed on the sliding surfaces which will affect the lubricity need to be

investigated.

2

In this work, we study the process of thin film growth of hydrocarbons and rare

gases adsorbed on a decagonal Al73Ni10Co17 quasicrystal (d-AlNiCo) via computer

simulations. A high-quality and large-size single grain d-AlNiCo has been routinely

grown, making it an excellent substrate to study adsorption [14, 15, 16]. Chapter 3 is

devoted to the adsorptions of rare gases. The absence of chemical reactivity of noble

gases will be utilized to elucidate the effects of quasicrytallinity of the substrate on the

growth and structure of the adsorbed film. In Chapter 5, the simulation results for

selected hydrocarbons are presented. We study the adsorption behaviors of propane,

hexane, and octane. From these molecules, we may extrapolate the results for longer

alkanes which constitute typical oil lubricants. In addition, methane and benzene are

also studied due to their interesting symmetries.

We develop classical many-body interatomic potentials of Al, Ni, Co, C, and H

which will be suitable for simulating hydrocarbons adsorptions on Al-Ni-Co involving

thousands of atoms. To our knowledge, such potentials have not existed. The poten-

tials are based on the embedded-atom method (EAM) [17] with parameters fitted to

electronic energies of various structures evaluated via first principle quantum calcu-

lations. The generation of the EAM potentials is presented in Chapter 4. In Chapter

6, we extend the procedures to develop a consistent set of EAM potentials for pure

elements in the periodic table. These potentials are also fitted to ab initio energies.

A consistent set of EAM potentials having the same parametrizations enables one to

use the same computer code for different potentials.

This manuscript contains a different reserach which is part of our doctoral work,

namely in the field of nanocatalysis for synthesis of single-walled carbon nanotubes

(SWCNTs). In Chapter 7, we present the calculation of phase diagram of Fe-Mo-

C system under pressure for studying the effects of Mo in the thermodynamics of

Fe:Mo nanoparticles as catalysts for growing SWCNTs. The decomposition of two

3

Fe:Mo particles namely FeMo and Fe4Mo into various stable phases are analyzed.

The implications of the formation of these phases to the growth of SWNCTs as well

as the estimated minimum size of nanotubes that can be produced are discussed.

4

Chapter 2

Methods

2.1 Density functional theory

2.1.1 Born-Oppenheimer approximation

In a calculation of electronic structure of materials, one solves an eigenvalue problem

of time-independent Schro¨dinger equation:

HΨ(R, r) = EΨ(R, r) (2.1)

where R ∈ {Rn} and r ∈ {ri} are the vector coordinates of the nuclei and electrons, respectively. The energy operator (Hamiltonian), H, is given by

H = − ∑ n

� 2

2Mn �2Rn−

∑ i

� 2

2mi �2ri+

∑ m<n

e2ZmZn Rn −Rm +

∑ i<j

e2

rj − ri − ∑ n,i

e2Zn ri −Rn (2.2)

The first two terms represent the kinetic energies of the nuclei and the electrons,

respectively. The third and fourth terms represent the nucleus-nucleus and electron-

electron potential energies. The last term is the nucleus-electron energy. It is compu-

tationally beyond the capability of current computers to solve Equation 2.1 using the

full Hamiltonian. An approximation, known as Born-Oppenheimer approximation,

is made by realizing that nuclei are significantly heavier than electrons so the nuclei

move much more slowly than the electrons. The electrons can adapt themselves to

the current configuration of nuclei. Using this approximation, we can decouple the

electronic and the ionic parts of the Hamiltonian. Consider the nuclei fixed at a given

5

configuration, α, and solve the following electronic Schro¨dinger equation:

[ − ∑

i

� 2

2mi �2ri +

∑ i<j

e2

rj − ri − ∑ n,i

e2Zn ri −Rαn

] ψα(r) = Eα(R)ψα(r) (2.3)

The total energy, E, is calculated by taking the electronic contribution, Eα(R), as a

potential energy operator in the ionic Schro¨dinger equation:

[ − ∑ n

� 2

2Mn �2Ri +

∑ m<n

e2ZmZn Rm −Rn + E

α(R)

] Φα(R) = EΦα(R) (2.4)

Mostly, one is interested only in the electronic part, i.e. Equation 2.3. Even though

Equation 2.3 contains only the electronic part of the system, the large number of

variables (e.g. coordinates of all electrons) makes it remain intractable. In addition,

it is the electron-electron interaction that makes the problem so difficult to solve. This

interaction is a result of correlation between electrons (the probability of finding an

electron depends on where the rest of the electrons are) and the fact that electrons

are fermions requiring antisymmetric wavefunctions (the many-electron wavefunction

gains a factor of -1 everytime two electrons exchange their coordinates). If this term

were absent, the Hamiltonian would be just a sum of many one-electron Hamiltonians,

known as independent electron approximation.

2.1.2 Slater determinant

An antisymmetric N -electron wavefunction can be constructed from N one-electron

wavefunctions using Slater determinant [18] defined as:

Ψ(x1,x2, . . . ,xN) = 1√ N !

ψ1(x1) ψ2(x1) . . . ψN(x1) ψ1(x2) ψ2(x2) . . . ψN(x2) . . . . . . . . . . . .

ψ1(xN) ψ2(xN) . . . ψN(xN)

(2.5)

6

where ψk(xi) denotes the k-th one-electron wavefunction being occupied by an elec-

tron with spin-orbital coordinate xi = (si, ri), with si being the spin state and ri the

spatial coordinate. For this reason, ψk is also called the spin-orbital k. The factor 1√ N !

arrives from the normalization of the total wavefunction and orthonormality among

the spin orbitals:

∫ ΨΨ∗dx1 . . . dxN = 1 (2.6)

∫ ψk(r)ψ

∗ k(r)dr = 1 (2.7)

∫ ψk(r)ψ

∗ l (r)dr = 0 (2.8)

Using the orthonormality of the spin-orbitals, it can be shown that the charge density

of the Slater determinant can be written as n(x) = ∑

k |ψk(x)|2.

2.1.3 Hartree-Fock equations

In the electronic Schro¨dinger equation (Equation 2.3), the Hamiltonian can be written

as follows:

H = ∑

i

h(i) + 1

2

∑ i�=j

g(i, j) (2.9)

h(i) ≡ −1 2 �2i −

∑ n

Zn |ri −Rn| (2.10)

g(i, j) ≡ 1|rj − rj| (2.11)

where h(i) depends only on ri and g(i, j) depends on ri and rj. The energy of

the system is calculated by taking the expectation value of the Hamiltonian in the

total wavefunction, E = 〈Ψ|H|Ψ〉. By employing the orthonormality of ψk as in the

7

calculation of charge density n(x), we have

〈Ψ| ∑

i

h(i)|Ψ〉 = ∑ k

〈ψk|h|ψk〉 = ∑ k

∫ dxψ∗k(x)h(r)ψk(x) (2.12)

Note that in the first equation, when |Ψ〉 is written as a Slater determinant, only i = k appears due to orthonormality of ψk, and the summation over electron index i inside

the many-electron wavefunction |Ψ〉 becomes a summation over k inside individual spin-orbital ψk, and we can drop the index h(i = k) for convenience. The integral∫

dx denotes integral over spatial coordinates and a sum over the spin-degrees of

freedom. Similarly, we have

〈Ψ| ∑ i,j

g(i, j)|Ψ〉 = ∑ k,l

〈ψkψl|g|ψkψl〉 − ∑ k,l

〈ψkψl|g|ψlψk〉 (2.13)

〈ψkψl|g|ψmψn〉 = ∫

dx1ψ ∗ k(x1)

[∫ dx2ψ

∗ l (x2)

1

|r1 − r2|ψn(x2) ] ψm(x1)(2.14)

The total energy is then

E = ∑ k

〈ψk|h|ψk〉+ 1 2

∑ k,l

[〈ψkψl|g|ψkψl〉 − 〈ψkψl|g|ψlψk〉] (2.15)

This expression shows that the electron-electron interaction, g(i, j), consists of two

terms, the first one is known as Coulomb energy (or Hartree term), and the second

one is the exchange energy term. To see how this derivation also reduces the many-

electron Schro¨dinger equation to a set of one-electron equations, we differentiate the

energy with respect to a particular spin-orbital, e.g. spin-orbital 〈ψi|. Note that i is not electron index, but a specific value of spin-orbital index k, after the derivation,

8

we may replace i by k to conform to the usual notation:

δE

δ〈ψi| = h|ψi〉+ 1

2

[∑ l

〈ψl| 1|r− r′| |ψl〉 ] |ψi〉+ 1

2

[∑ k

〈ψk| 1|r− r′| |ψk〉 ] |ψi〉

−1 2

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x)ψi(x ′)− 1

2

∑ k

∫ dxψ∗k(x

′) 1

|r− r′|ψi(x)ψk(x ′)

(2.16)

By changing indices (k → l) in the third term and (k → l,x→ x′) in the last term, we get

δE

δ〈ψi| = h|ψi〉+ [∑

l

〈ψl| 1|r− r′| |ψl〉 ] |ψi〉 −

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x)ψi(x ′)

(2.17)

This expression is known as Fock operator acting on |ψi〉. Replacing back i by k, and defining the eigenvalues of Fock operator as �k, we arrive at the Hartree-Fock

equation:

HHFψk = �kψk (2.18)

HHFψk =

[ −�

2

2 − ∑ n

Zn |r−Rn|

] ψk(x) +

∑ l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψl(x ′)ψk(x)

− ∑

l

∫ dx′ψ∗l (x

′) 1

|r− r′|ψk(x ′)ψl(x)

(2.19)

Note that the sum of the eigenvalues �k is not the total energy E. However, by

comparing Equation 2.15 and 2.19, E can be recovered using the following equation

E = 1

2

∑ k

[�k + 〈ψk|h|ψk〉] (2.20)

The last term (the exchange term) in Equation 2.19 is nonlocal because the Hamil-

tonian HHF operates on ψk(x) at a particular x, but the operator itself is a function

9

of ψk(x ′) at all possible x′. This nonlocality makes the Hartree-Fock Hamiltonian

difficult to evaluate in large systems involving many atoms and electrons, hence not

suitable for solids. Most electronic structure calculations for solids are based on

density functional theory (DFT) discussed in the following sections.

2.1.4 Kohn-Sham equations

In DFT, the nonlocal exchange term is given by an effective exchange potential that

depends on electronic charge density. Furthermore, all other potential operators are

expressed in charge density rather than in spin-orbitals. This approach reduces the

number of degrees of freedom significantly since all electron coordinates that enter in

the spin-orbitals are now replaced by charge density that is a function only on one

coordinate r. The DFT energy functional is given by

E(n) = T (n) +

∫ Vext(r)n(r)dr+

1

2

∫∫ n(r′)

1

|r− r′|n(r)dr ′dr+ Exc(n) (2.21)

where T being the electronic kinetic energy and Vext being the electron-nuclei electro-

static potential. We know that for given wavefunctions we can calculate the charge

density, n(r) = ∑

k |ψk(r)|2, however the reverse is not obvious. The formality that proves there exists one-to-one mapping between charge density and wavefunctions

was developed by Hohenberg-Kohn [19]. Hohenberg-Kohn theorem also proves that

there exists Exc(n) that will produce the exact ground state of the system. The DFT

Schro¨dinger equation can be derived by taking the variation of E(n) with respect to

n:

δE

δn =

δT

δn + Vext +

∫ n(r′)dr′

|r− r′| + δExc δn

(2.22)

As we have derived the Hartree-Fock equation (Equation 2.19), if we write the DFT

many-electron wavefunction as a Slater determinant of spin-orbitals and orthonor-

10

mality among the spin-orbitals, then each DFT spin-orbital equation satisfies

[ −1 2 �2 + Veff (r)

] ψk(r) = �kψk(r) (2.23)

Veff (r) ≡ Vext(r) + ∫

n(r′)dr′

|r− r′| + δExc δn

(2.24)

The total energy is related to the eigenvalues �k as follows

E = ∑ k

�k − 1 2

∫∫ n(r′)

1

|r− r′|n(r)dr ′dr+ Exc(n)−

∫ δExc(n)

δn n(r)dr (2.25)

Equations 2.24-2.25 are known as Kohn-Sham equations. Since the potential opera-

tors depend on charge density, Equation 2.24 must be solved self-consistently. One

starts from a trial electronic charge density to construct the potentials and solve for

the eigenvalues and spin-orbitals wavefunctions. A new charge density is then con-

structed from the spin-orbitals and the procedure is repeated until the charge and

the wavefunctions are self-consistent within a certain accuracy.

2.1.5 Local density and generalized gradient approximations

In Kohn-Sham equations, there are two terms involving exchange interaction, namely

Exc(n) and δExc/δn. For a given charge density, the first term (the exchange energy)

does not depend on the charge functional on r, whereas the second term (the ex-

change potential) will depend on r if the charge density does. This means that for

a nonhomogeneous systems, exchange potential can be expanded in charge density

and its derivatives:

Vxc ≡ δExc(n) δn(r)

= Vxc(n(r), |�n(r)|, |�(�(n(r)))|, ...) (2.26)

As a first approximation, one neglects all the gradients of charge density in the

exchange potential, this is known as local density approximation (LDA) since the

11

exchange potential depends on charge density only at a particular value of r. It

means that LDA gives an exact ground state for homogeneous electron gas since in

this case the gradients vanish. This allows one to write the LDA exchange energy as

a sum of exchange energy per electron in a homogeneous electron gas, �homxc (n):

ELDAxc =

∫ �homxc (n)ndr (2.27)

LDA also gives accurate results for systems where the charge density does not vary

too rapidly such as in metals. For nonhomogenous systems such as transition metals,

semiconductors, or slabs, LDA is known to underestimate the energy (hence the band

gap). A more accurate approximation, known as generalized gradient approximation

(GGA), includes the first gradient of charge and the exchange energy is given by

EGGAxc =

∫ �GGAxc (n(r), |∇n(r)|)n(r)dr (2.28)

Several techniques exist to parametrize �GGAxc : Perdew-Wang 1986 (PW86) [20, 21],

Perdew-Wang 1991 (PW91) [22], Becke [23], Lee-Yang-Parr (LYP) [24], and Perdew-

Burke-Enzerhof (PBE) [25, 26]. In this work, we use GGA-PBE functional for the

exchange energy.

2.1.6 Pseudopotentials

Kohn-Sham equations can be solved in different ways depending on the choice of

potentials and basis functions to expand the wavefunctions. Some considerations in

solving the Kohn-Sham equations are: (1) potentials becomes very strong near the

nuclei, whereas at far regions they are relatively weak, (2) wavefunctions fluctuate

more near nuclei than in the interstitial regions, (3) the symmetry of the potentials are

approximately spherical near the nuclei whereas at larger distances, the symmetry of

the crystal dominates. Some of the known methods are augmented plave wave (APW)

12

[27, 28], linearized augmented plane wave (LAPW) [29, 30], Orthogonalized plave-

wave (OPW) [31], and pseudopotential method [32, 33]. LAPW is the most accurate

method available. It uses spherical harmonics to expand the wavefunctions in the core

region near the nuclei and plane waves for the interstitial region. Pseudopotential

methods use only plane wave basis set. The number of plane waves can be kept

small by treating an atom as consisting of an effective core (nucleus + core electrons)

and valence electrons. Even though pseudopotentials are not as accurate as LAPW

(which as a contrast fall into the class of fullpotential methods), it provides good

results and becomes the most common choice for its lower computational costs than

LAPW.

2.2 Vienna Ab initio Simulation Package

All ab initio quantum calculations in this work are done using Vienna Ab initio Sim-

ulation Package (VASP) [34, 35]. The calculations are performed in the generalized

gradient approximation (GGA) [36, 37] with exchange correlation as parametrized

by Perdew, Burke, and Ernzerhof (PBE) [25]. To reduce the number of plane waves,

projector augmented-wave (PAW) pseudopotentials are used [38, 39].

VASP uses a plane wave basis set to expand the wave functions in solving the

Kohn-Sham equation in reciprocal space. The evaluation of total electronic energy per

unit cell is done by integrating the energy of all electronic states in the Brillouin zone.

In practice, the Brillouin zone is divided into grids and the integration is replaced by

a sum over k-points. In our work, the k-point grids are generated automatically using

Monkhorst-Pack scheme [40]. For hexagonal structures, an additional shift is used

so that the grids are centered at the Γ point (k = (0, 0, 0)). Typical setting of the

number of k-points as many as 3000/number-of-atoms is usually sufficient to achieve

a converged energy with an accuracy better than 10 meV/atom. Unless otherwise

13

stated, all structures are fully relaxed: atoms position as well as unit cell’s size and

shape are allowed to relax to find the equillibrium configurations.

2.3 Classical interatomic potentials

Due to QC’s lack of periodicity, a large enough simulation cell is necessary to cap-

ture the effect of substrate’s structure on the adsorbed films. A typical cell con-

tains thousands of atoms which makes the simulation unpractical to be performed

quantum-mechanically. Therefore, classical interatomic potentials are needed. For

simple systems, e.g. adsorption of noble gases, pair potentials such as Lennard-Jones

[41] or Morse [42] are sufficient. However, for complex systems, such as adsorption of

hydrocarbons, more accurate potentials are required to take into account many-body

effects in these systems, especially covalent bonds involving carbons which are higly

directional.

Several methods exist to incorporate many-body interations into classical poten-

tials, e.g. force field method (FF) [43, 44, 45, 46], Cluster Expansion method (CE)

[47, 48, 49], and embedded-atom method (EAM) [17, 50]. FF is based on energy-bond

order-bond length relationship [44] and is mostly used for biological systems [43, 51]

and chemical systems where bond formation and breaking are allowed [46]. In CE,

energy of a system is approximated by a converging sum over cluster contributions,

where contribution from large clusters are negligible. In nonperiodic systems, such as

amorphous and quasicrystalline phases, the number of clusters needed can be large

to reach the desired accuracy, therefore CE is mostly suitable to evaluate ground

state energy in periodic systems. EAM is based on the close relationship between

electronic charge density and energy of the system. In quantum physics, this relation-

ship becomes the basis for the density functional theory (DFT) [52, 19, 53], in which

it has been proven that there exists a one-to-one mapping between charge densities

14

and electronic wave functions, and hence energies [19]. Due to the universality of

charge density, in principle EAM can be used in any systems.

EAM was originally developed for examining metallic bulks and surfaces [17, 50].

Later on, charge screening methods have been proposed to modify EAM for use in

covalent systems such as Si [54] and Ge [55]. EAM has been successfully employed

to simulate surface relaxation/reconstructions [56, 57, 58, 59], film growth [60, 61],

and diffusion processes [62, 63]. In this study, we will develop EAM potentials to

simulate hydrocarbons adsorption on d-AlNiCo surface.

In EAM formalism [17, 50], each atom is viewed as being embedded in the material

consisting of all other atoms. The total energy of a system is defined by

Etot ≡ ∑

i

Fi(ρ¯i) + 1

2

∑ i

∑ j �=i

φij(rij), (2.29)

where Fi is the embedding energy of atom i, ρ¯i is the electron density at the vector

position ri, φij is the pair potential, and rij is the distance between atoms i and j.

If ρ¯i is approximated as a sum of individual contribution of the constituents [i.e.,

ρ¯i = ∑ j �=i

ρj(rij), where ρj is the atomic electron density of atom j, the energy is then

only a function of the position of atoms.

For an elemental potential, there are 3 functions needed: F (ρ¯), ρ(r), and φ(r).

For a binary system AB, we will need 7 functions: FA(ρ¯), ρA(r), φA(r), FB(ρ¯),

ρB(r), φB(r), and a cross-pair potential φAB(r). The cross-pair potential is needed

to calculate pair interaction of atoms of different types. In general, in a system

consisting of N different elements, we need N(N + 5)/2 functions.

15

We will use the following functional forms:

ρA(r) = ρAe e −βA(r/rAe −1), (2.30)

φAA(r) = DA[e−2α A(r−rA0 ) − 2e−αA(r−rA0 )], (2.31)

FA(ρ¯) = cubic− splines, (2.32)

φAB(r) = γAB

2 φAA(r) +

1

2γAB φBB(r). (2.33)

In the above equations, A and B denote atom type. ρe, re, and β will be taken from

the database of atomic electron density [64]. Equation 2.31 follows Morse potential

form [42]. The embedding function F will be taken as a natural cubic spline [65].

The expression for the cross-pair interaction φAB follows Haftel’s derivation [59] in

which γAB ≡ ZB/ZA (ZA is the effective charge of the core for atom type A).

2.4 Simplex method

The EAM potentials parameters need to be fit to quantum calculations. The fitting

will be performed using simplex method [66]. Simplex does not require evaluation

of function derivatives which makes it simple to implement. Simplex minimizes N -

dimensional function f by creating P > N simplex points. A simplex point is f

evaluated at a given coordinate. The simplex points will form P -polytope. A min-

imization move is made by replacing the maximum simplex point, hence the worst,

with a point reflected through the centroid of the remaining (P − 1)-polytope. Ex- pansion or shrinking of the polytope are allowed to overcome some local minima or

to converge, respectively.

16

2.5 Grand canonical Monte Carlo

The adsorption simulation will be performed using grand canonical Monte Carlo

(GCMC) method [67, 68, 69]. At constant temperature, T , and volume, V , the

GCMC method explores the configurational phase space using the Metropolis al-

gorithm and finds the equilibrium number of adsorbed atoms (adatoms), N , as a

function of the chemical potential, μ, of the gas, i.e. configuration(μ,N, V ). The

adsorbed atoms are in equilibrium with the coexisting gas: the chemical potential of

the gas is constant throughout the system. In addition, the coexisting gas is taken

to be ideal. With this method we determine adsorption isotherms, ρN , and density

profiles, ρ(x, y), as a function of the pressure, P (T, μ). For each data point in an

isotherm, we perform at least 18 million GCMC steps to reach equilibrium. Each

step is an attempted displacement, creation, or deletion of an atom with execution

probabilities equal to 0.2, 0.4, and 0.4, respectively [70, 71, 72]. At least 27 million

steps are performed in the subsequent data-gathering and -averaging phase.

17

Chapter 3

Noble gas adsorptions on d-AlNiCo

The observed unusual electronic [73, 74] and frictional [75, 76, 77] properties of qua-

sicrystal surfaces stimulate interesting fundamental questions about how these and

other physical properties are altered by quasiperiodicity. Recent progress in the char-

acterization and preparation of quasicrystal surfaces raises new possibilities for their

use as substrates in the growth of films having novel structural, electronic, dynamic

and mechanical properties [78, 79, 80]. The physical behavior of systems involving

competing interactions in adsorption is a subject of continuing interest and is par-

ticularly relevant to the growth of thin films [69]. Several different growth modes

have been observed for the growth of metal films on quasicrystals [81, 82, 83, 84]. A

form of competing interactions seen in adsorption involves either a length scale or a

symmetry mismatch between the adsorbate-adsorbate interaction and the adsorbate-

substrate interaction [85, 86]. Some consequences of such mismatches include den-

sity modulations [87, 88], domain walls [89], epitaxial rotation in the adsorbed layer

[90, 91, 92, 93, 94, 95], and a disruption of the normal periodicity and growth in the

film [96, 97, 98].

The wide range of behavior observed so far indicates that, even in the absence of

intermixing, film growth is strongly affected by chemical interactions between adsor-

bate and substrate. In order to separate these chemical effects from those specific to

quasiperiodic order, we have studied the adsorption of noble gases on a quasicrystal

surface, where both the gas-gas and gas-surface interactions are believed to be simple,

i.e., appreciable chemical interactions and adsorbate-induced surface reconstructions

are absent. In this chapter, we explore the implications of structural mismatch by

18

evaluating the nature of Ne, Ar, Kr, and Xe adsorption on a quasicrystal substrate,

namely the 10-fold surface of decagonal Al73Ni10Co17 quasicrystal (d-AlNiCo QC)

[15, 16].

3.1 Method

3.1.1 Simulation cell

The simulation cell is tetragonal. We take a square section of the surface, A, of side

5.12 nm, to be the (x, y) part of the unit cell in the simulation, for which we assume

periodic boundary conditions along the basal directions. Although this assumption

limits the accuracy of the long range QC structure, it is numerically necessary for

these simulations. To minimize the long range interaction corrections, a relatively

large cutoff (5σgg) is used. Since the size of the cell is relatively large compared to that

of the noble gases, the cell is accurately representative of order on short-to-moderate

length scales. The height of the cell, along the z (surface-normal) direction, is chosen

to be 10 nm (long enough to contain ∼20 layers of Xe). At the top of the cell, a hard-wall reflective potential is employed to confine the coexisting vapor phase. The

simulation results for Xe over d-AlNiCo, presented below, are consistent with both

our results from experiments [99] and virial calculations [100]. Hence, the calculations

may also be accurate for other systems.

3.1.2 Gas-gas and gas-substrate interactions

The gas-gas and gas-substrate interactions are modeled using Lennard-Jones (LJ) 12-

6 potentials, with the gas-gas parameter values �gg and σgg listed in Table 3.1. The

gas-substrate interactions are obtained by summing pair potentials for a gas atom

and all of the substrate atoms in an eigth-layer slab: Al, Ni and Co [15, 101, 100]. The

position of the atoms in the eight-layer slab are taken from the results of a low-energy

19

Table 3.1: Parameter values for the 12-6 Lennard-Jones interactions. TM is the label for Ni or Co.

�gg σgg �gas−Al σgas−Al �gas−TM σgas−TM (meV) (nm) (meV) (nm) (meV) (nm)

Ne 2.92 0.278 9.40 0.264 9.01 0.249 Ar 10.32 0.340 17.67 0.295 16.93 0.280 Kr 14.73 0.360 21.11 0.305 20.23 0.290 Xe 19.04 0.410 24.00 0.330 23.00 0.315

iNe(1) 2.92 0.410 5.45 0.330 5.22 0.315 dXe(1) 19.04 0.278 41.39 0.264 39.67 0.249 dXe(2) 19.04 0.390 25.88 0.320 24.80 0.305 iXe(1) 19.04 0.550 14.96 0.400 14.34 0.385 iXe(2) 19.04 0.675 10.52 0.462 10.08 0.447

electron diffraction LEED analysis of the surface structure of d-AlNiCo [15]. The

gas-substrate interaction parameters are derived using conventional combining rules,

σAB = (σA + σB)/2 and �AB = √

�A�B [102], and experimental heats of adsorption

[100, 99, 103]. The LJ gas-substrate parameters are �gas−Al and σgas−Al for Al, and

�gas−TM and σgas−TM for the two transition metals Ni and Co. All these values are

listed in the upper part of Table 3.1. In the calculation of the adsorption potential,

we assume a structure of the unrelaxed surface taken from the empirical fit to LEED

data [16].

3.1.3 Adsorption potentials

Figures 3.1(a), 3.1(c), 3.1(e), and 3.1(g) show the function Vmin(x, y) of Ne, Ar, Kr,

and Xe on the d-AlNiCo, respectively, which is calculated by minimizing the adsorp-

tion potentials, V (x, y, z), along the z direction at every value (x, y) coordinates:

Vmin(x, y) ≡ min {V (x, y, z)}|along z . (3.1)

20

The figures reveal the fivefold rotational symmetry of the substrate. Dark spots

correspond to the most attractive regions of the substrate. By choosing appropriate

sets of five dark spots, we can identify pentagons, whose sizes follow the inflationary

property of the d-AlNiCo. Note the pentagon at the center of each figure: it will be

used to extract the geometrical parameters λs and λc in Section 3.2.5.

To characterize the corrugation, not well-defined for aperiodic surfaces, we calcu-

late the distribution function f(Vmin), the average 〈Vmin〉 and standard deviation SD of Vmin(x, y) as:

f(Vmin)dVmin ≡ probability { Vmin ∈ [Vmin, Vmin + dVmin[

} (3.2)

〈Vmin〉 ≡ ∫ ∞ −∞

f(Vmin)Vmin dVmin, (3.3)

SD2 ≡ ∫ ∞ −∞

f(Vmin)(Vmin − 〈Vmin〉)2 dVmin. (3.4)

Figures 3.1(b), 3.1(d), 3.1(f), and 3.1(h) show f(Vmin) of the adsorption potential for

Ne, Ar, Kr, and Xe on the d-AlNiCo, respectively. Vmin(x, y) extends by more than

2·SD around its average, revealing the high corrugation of the gas-surface interaction in these four systems. The average and SD of Vmin(x, y) for these systems are listed

in the upper part of Table 3.2. In addition to highly corrugated, the potentials are

“deep” because the record maximum well-depth, e.g. for Xe, on a periodic surface is

about 160 meV, viz. on graphite [104]; and the record minimum well-depth is about

28 meV, on Cs [105].

3.1.4 Effective parameters

For every gas-substrate interaction we define two effective parameters σgs and Dgs.

σgs represents the averaged LJ size parameter of the interaction, calculated following

21

-70 -65 -60 -55 -50 -45 -40 -35 Vmin (meV)

f(V m

in )

(a)

(b)

-180 -160 -140 -120 -100 Vmin (meV)

f(V m

in )

(c)

(d)

-220 -200 -180 -160 -140 -120 Vmin (meV)

f(V m

in )

(e)

(f)

-280 -260 -240 -220 -200 -180 -160 Vmin (meV)

f(V m

in )

(g)

(h)

Ne/QC Ar/QC

Xe/QCKr/QC

Figure 3.1: (color online). Computed adsorption potentials for (a) Ne, (c) Ar, (e) Kr, and (g) Xe on the d-AlNiCo, obtained by minimizing V (x, y, z) with respect to z. The distribution of the minimum value of these potentials is plotted in (b, d, f, and h) respec- tively: the solid line marks the average value 〈Vmin〉, the dashed lines mark the values at 〈Vmin〉±SD.

22

Table 3.2: Range, average (〈Vmin〉), and standard deviation (SD) of the interaction Vmin(x.y) on the d-AlNiCo. Effective parameters of the gas-substrate interactions (Dgs, σgs, D�gs, σ�gs), and, for comparison, the best estimated well depths DGrgs on graphite [106].

Vmin range < Vmin > SD Dgs σgs D � gs σ

� gs D

Gr gs

(meV) (meV) (meV) (meV) (nm) (Dgs/�gg) (σgs/σgg) (meV) Ne -71 to -33 -47.43 6.63 43.89 0.260 15.03 0.935 33 Ar -181 to -85 -113.32 13.06 108.37 0.291 10.50 0.856 96 Kr -225 to -111 -145.71 15.68 140.18 0.301 9.52 0.836 125 Xe -283 to -155 -195.46 17.93 193.25 0.326 10.15 0.795 162

iNe(1) -65 to -36 -45.11 4.08 43.89 0.326 15.03 0.795

dXe(1) -305 to -150 -207.55 29.18 193.25 0.260 10.15 0.935

dXe(2) -295 to -155 -199.40 19.33 193.25 0.316 10.15 0.810

iXe(1) -248 to -170 -195.31 11.21 193.25 0.396 10.15 0.720

iXe(2) -230 to -180 -194.25 7.77 193.25 0.458 10.15 0.679

the traditional combining rules [102]:

σgs ≡ xAlσg−Al + xNiσg−Ni + xCoσg−Co, (3.5)

where xAl, xNi, and xCo are the concentrations of Al, Ni, and Co in the QC, respec-

tively. Dgs represents the well depth of the laterally averaged potential V (z):

Dgs ≡ −min {V (z)}|along z . (3.6)

In addition, we normalize the σgs and Dgs with respect to the gas-gas interactions:

σ�gs ≡ σgs/σgg, (3.7)

D�gs ≡ Dgs/�gg. (3.8)

The values of the effective parameters σgs, Dgs, σ � gs, and D

� gs for the four gas-surface

interactions are listed in the upper part of Table 3.2. We also include the well depth

for Ne, Ar, Kr, and Xe on graphite, as comparison [106].

3.1.5 Test rare gases

As shown in tables 3.1 and 3.2, Ne is the smallest atom and has the weakest gas-gas

and gas-surface interactions (minima of σgg, σgs, �gg and Dgs). In addition, Xe is the

23

largest atom and has the strongest gas-gas and gas-surface interactions (maxima of

σgg, σgs, �gg and Dgs). Therefore, for our analysis, it is useful to consider two test

gases, iNe(1) and dXe(1), which are combinations of Ne and Xe parameters.

iNe(1) represents an “inflated” version of Ne, having the same gas-gas and average

gas-substrate interactions of Ne but the geometrical dimensions of Xe:

{�gg, Dgs, D�gs}[iNe(1)] ≡ {�gg, Dgs, D�gs}[Ne], (3.9)

{σgg, σgs, σ�gs}[iNe(1)] ≡ {σgg, σgs, σ�gs}[Xe]. (3.10)

dXe(1) represents a “deflated” version of Xe, having the same gas-gas and average

gas-substrate interactions of Xe but the geometrical dimensions of Ne:

{�gg, Dgs, D�gs}[dXe(1)] ≡ {�gg, Dgs, D�gs}[Xe], (3.11)

{σgg, σgs, σ�gs}[dXe(1)] ≡ {σgg, σgs, σ�gs}[Ne]. (3.12)

The resulting LJ parameters for iNe(1) and dXe(1) are summarized in the central

parts of Tables 3.1 and 3.2. Furthermore, we also define three other test versions of

Xe: dXe(2), iXe(1), and iXe(2) which have the same gas-gas and average gas-substrate

interactions of Xe but deflated or inflated geometrical parameters. The last three test

gases will be used in Section 3.2.5. The LJ parameters for these gases are summarized

in the lower parts of Tables 3.1 and 3.2. In simulating test gases, we implicitly rescale

the substrate’s strengths so that the resulting adsorption potentials have the same

Dgs as the non-inflated or non-deflated ones (Equations 3.9 and 3.11).

24

3.1.6 Chemical potential, order parameter, and ordering tran-

sition

To conveniently characterize the evolution of the adsorption processes of the gases

we define a normalized chemical potential μ�, as:

μ� ≡ μ− μ1 μ2 − μ1 , (3.13)

where μ1 and μ2 are the chemical potentials at the onset of the first and second layer

formation, respectively. In addition, we introduce the order parameter ρ5−6, defined

as the probability of existence of fivefold defect [70, 71]:

ρ5−6 ≡ N5 N5 + N6

, (3.14)

where N5 and N6 are the numbers of atoms having 2D coordination equal to 5 and

6, respectively. The 2D coordination is the number of neighboring atoms within a

cutoff radius of aNN · 1.366 where aNN is the first nearest neighbor (NN) distance of the gas in the solid phase and 1.366 = cos(π/6) + 1/2 is the average factor of the

first and the second NN distances in a triangular lattice. Note that aNN does not

change appreciably with respect to temperature difference, e.g. aNN of Xe changes

from 0.440 nm at 77 K to 0.443 nm at 140 K.

In a fivefold ordering, most arrangements are hollow or filled pentagons with atoms

having mostly five neighbors. Hence, the particular choice of ρ5−6 is motivated by the

fact that such pentagons can become hexagons by gaining additional atoms with five

or six neighbors. Definition: the five to sixfold ordering transition is defined as a

decrease of the order parameter to a small or negligible final value. The phenomenon

can be abrupt (first-order) or continuous. Within this framework, ρ5−6 and (1−ρ5−6) can be considered as the fractions of pentagonal and triangular phases in the film,

respectively.

25

3.2 Results

3.2.1 Adsorption isotherms

Figure 3.2 shows the adsorption isotherms of Ne, Ar, Kr, and Xe on the d-AlNiCo.

The plotted quantity is the thermodynamic excess coverage (densities of adsorbed

atoms per unit area), ρN), defined as the difference between the total density of

atoms in the simulation cell and the density that would be present if the cell were filled

with uniform vapor at the specified values of P and T . The simulated ranges and the

experimental triple point temperatures (Tt) for Ne, Ar, Kr, and Xe are listed in Table

3.3. A layer-by-layer film growth is visible at low temperatures. Detailed inspection

of the isotherms reveals that there is a continuous film growth (i.e. complete wetting)

at temperatures above Tt (isotherms at T > Tt are shown as dotted curves). This

behavior, observed despite the high corrugation, is interesting as corrugation has

been shown to be capable of preventing wetting [107, 108].

Although vertical steps corresponding to layers’ formation are evident in the

isotherms, the slopes of the isotherms’ plateaus at the same normalized tempera-

tures (T � ≡ T/�gg = 0.35) differ between systems. To characterize this, we calculate the increase of each layer density, ΔρN , from the formation to the onset of the sub-

sequent layer. ΔρN is defined as ΔρN ≡ (ρB − ρA)/ρA and the values are reported in Table 3.3 (points (A) and (B) are specified in Figure 3.2). We observe that, as

the size of noble gas increases ΔρN become smaller, indicating that the substrate

corrugation has a more pronounced effect on smaller adsorbates, as expected since

they penetrate deeper into the corrugation pockets. However, Xe does not follow

this trend. This arises from the complex interplay between the corrugation energy

and length of the potential with respect to the parameters of the gas (σgg, �gg) in

determining the density of the adsorbed layers. In the case of Ne, Ar, and Kr, the

densities at points (A) are approximately the same (ρA = 5.4 atoms/nm 2), whereas

26

that of Xe is considerably smaller (ρA = 4.2 atoms/nm 2), because Xe dimension σgg

becomes comparable to the characteristic length (corrugation) of the potential. This

effect is clarified by the density profile of the films, ρ(x, y), shown in Figures 3.3 and

3.4. As can be seen at points (A), the density profiles of Ne, Ar, and Kr are the

same, i.e. the same set of dark spots appear in their plots. For Xe, some spots are

separated with distances smaller than its core radius (σgg), causing repulsive inter-

actions. Hence these spots will not likely appear in the density profile, resulting in

a lower ρA. More discussion on how interaction parameters affect the shape of the

isotherms is presented in Section 3.2.4. Note that the second layer in each system

has a smaller ΔρN than the first one. The explanation will be given when we discuss

the evolution of density profiles.

Table 3.3: Results for Ne, Ar, Kr, and Xe adsorbed on d-AlNiCo. Tt is taken from reference [109]. The density increase (ΔρN ) in the first and second layers is calculated at T � = 0.35 from point (A) to (B) and (C) to (D) in Figure 3.2, respectively.

simulated T T � ≡ T/�gg Tt ΔρN at T � = 0.35 θr (K) (K) for 1st layer for 2nd layer

Ne 11.8 → 46 0.35 → 1.36 24.55 (12.2-5.3)/5.3=1.30 (11.1-10.2)/10.2=0.09 6◦ Ar 41.7 → 155 0.35 → 1.29 83.81 (7.3-5.5)/5.5=0.33 (6.9-6.4)/6.4=0.08 30◦ Kr 59.6 → 225 0.35 → 1.32 115.76 (6.9-5.5)/5.5=0.25 (6.6-6.3)/6.3=0.05 42◦ Xe 77 → 280 0.35 → 1.27 161.39 (5.8-4.2)/4.2=0.38 (5.2-5.2)/5.2=0 54◦

3.2.2 Density profiles

Figures 3.3 and 3.4 show the density profiles ρ(x, y) at T � = 0.35 for the outer layers

of Ne, Ar, Kr, and Xe adsorbed on the d-AlNiCo at the pressures corresponding to

points (A) through (F) of the isotherms in Figure 3.2.

Ne/d-AlNiCo system. Figure 3.3(a) shows the evolution of adsorbed Ne. At

the formation of the first layer, adatoms are arranged in a pentagonal manner follow-

ing the order of the substrate, as shown by the discrete spots of the Fourier transform

27

10-25 10-20 10-15 10-10 10-5 1 0

10

20

30

40

50

P (atm)

ρ N (a

to m

s/ nm

2 )

Ne/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25

30

35 Ar/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25

30

35 Kr/QC

10-15 10-12 10-9 10-6 10-3 1 0

5

10

15

20

25 Xe/QC

P (atm)

ρ N (a

to m

s/ nm

2 )

P (atm)

ρ N (a

to m

s/ nm

2 )

P (atm)

ρ N (a

to m

s/ nm

2 )

(a) (b)

(d)(c)

A

B

C D

E

F

A B

C D

E

F

A B

C D

E

F

A B

C D

Figure 3.2: Computed adsorption isotherms for all the gas/d-AlNiCo systems. The ranges of temperatures under study are: Ne: T = 14 K to 46 K in 2 K steps, Ar: 45 K to 155 K in 5 K steps, Kr: 65 K to 225 K in 5 K steps, Xe: 80 K to 280 K in 10 K steps. Additional isotherms are shown with solid circles at T � = 0.35: T = 11.8 K (Ne), T = 41.7 K (Ar), T = 59.6 K (Kr), and T = 77 K (Xe). Isotherms above the triple point temperatures are shown as dotted curves.

(FT) having tenfold symmetry (point (A)). As the pressure increases, the arrange-

ment gradually loses its pentagonal character. In fact, at point (B) the adatoms are

arranged in patches of triangular lattices and the FT consists of uniformly-spaced

concentric rings with hexagonal resemblance. The absence of long-range ordering in

the density profile is indicated by the lack of discrete spots in the FT. This behavior

persists throughout the formation of the second layer (points (C) and (D)) until the

appearance of the third layer (point (E)). At this and higher pressures, the FT shows

patterns oriented as hexagons rotated by θr = 6 ◦, indicating the presence of short-

28

range triangular order on the outer layer (point (F)). In summary, between points (A)

and (F) the arrangement evolves from pentagonal fivefold to triangular sixfold with

considerable disorder, as the upper part of the density profile at point (F) shows. The

transformation of the density profile, from a lower-packing-density (pentagonal) to

a higher-packing-density structure (irregular triangular), occurs mostly in the mono-

layer from points (A) to (B), causing the largest density increase of the first layer with

respect to that of the other layers (see the end of Section 3.2.2 for more discussion).

Due to the considerable amount of disorder in the final state Ne/d-AlNiCo does not

satisfy the requirements for the transition as defined in Section 3.1.6.

Ar/d-AlNiCo and Kr/d-AlNiCo systems. Figures 3.3(b) and 3.4(a) show

the evolutions for Ar and Kr: they are similar to the Ne case. For Ar, the pentagonal

structure at the formation of the first layer is confirmed by the FT showing discrete

spots having tenfold symmetry (point (A)). The quasicrystal symmetry strongly af-

fects the overlayers’ structures up to the third layer by preventing the adatoms from

forming a triangular lattice (point (E)). This appears, finally, in the lower part of the

density profile at the formation of the fourth layer as confirmed by the FT showing

discrete spots with sixfold symmetry (point (F)). Similar to the Ne case, disorder does

not disappear but remain present in the middle of the density profile corresponding

to the highest coverage before saturation (point (F)). Similar situation occurs also

for the evolution of Kr as shown in Figure 3.4(a).

Xe/d-AlNiCo system. Figure 3.4(b) shows the evolution of adsorbed Xe. At

the formation of the first layer, adatoms are arranged in a fivefold ordering similar

to that of the substrate as shown by the discrete spots of the FT having tenfold

symmetry (point (A)). At point (B), the density profile shows a well-defined triangular

lattice not present in the other three systems: the FT shows discrete spots arranged

in regular and equally-spaced concentric hexagons with the smallest containing six

29

FTρ(x,y)(a)

θ

A

B

C

D

E

F

θ

(b)Ne/QC Ar/QC FTρ(x,y)

A

B

C

D

E

F

r r

Figure 3.3: Density profiles and Fourier transforms of the outer layer at T � = 0.35 for Ne/d-AlNiCo (T = 11.8 K) and Ar/d-AlNiCo (T = 41.7 K), corresponding to points (A) through (F) of Figure 3.2.

30

clear spots. Thus, at point (B) and at higher pressures, the Xe overlayers can be

considered to have regular closed-packed structure with negligible irregularities.

It is interesting to compare the orientation of the hexagons on FT for these four

adsorbed gases at the highest available pressures before saturation (point (F) for Ne,

Ar, and Kr, and point (D) for Xe). We define the orientation angles as the smallest

of the possible clockwise rotations to be applied to the hexagons to obtain one side

horizontal, as shown in Figures 3.3 and 3.4. Such angles are θr = 6 ◦, 30◦, 42◦, and

54◦, for adsorbed Ne, Ar, Kr, and Xe, respectively. These orientations, induced by

the fivefold symmetry of the d-AlNiCo, can differ only by multiples of n ·12◦ [70, 71]. Since hexagons have sixfold symmetry, our systems can access only five possible

orientations (6, 18, 30, 42, 54◦), and the final angles are determined by the interplay

between the adsorbate solid phase lattice spacing, the periodic simulation cell size,

and the potential corrugation. For systems without periodic boundary conditions,

the ground state has been found to be fivefold degenerate, as should be the case

[70, 71].

Xe adsorption on this surface was studied experimentally using LEED, in which the

isobar measurements indicate that the Xe film grows layer-by-layer in the temperature

range 65 K to 80 K [99], consistent with the simulations. Under similar conditions

to the simulation at 77 K, at the lowest coverage, the only discernible change in

the LEED pattern from that of the clean surface is an attenuation of the substrate

beams. After the adsorption of one layer, there are still no resolvable features that

would indicate an overlayer having order different from the substrate. At the onset of

the adsorption of the second layer, however, the LEED pattern shows new diffraction

spots that correspond to 5 rotational domains of a hexagonal structure. Within

each of these domains, the close-packed direction of the Xe is aligned with the 5-fold

directions of the substrate, as also observed in the simulation. In the experiments,

31

all possible alignments are observed owing to the presence of all possible rotational

alignments present within the width of the electron beam (0.25 mm). When the

second layer is complete, these spots are well-defined and their widths are the same as

the substrate spots, indicating a coherence length of at least 15 nm. The average Xe-

Xe spacing measured in the experiment is consistent with the bulk nearest-neighbor

spacing of 0.44 nm. A dynamical LEED analysis of the intensities indicates that the

structure of the multilayer film is consistent with face-centered cubic (FCC) Xe(111).

These structure parameters for the bilayer film are essentially identical to the results

obtained for Xe growth on Ag(111) [110, 111], a much weaker and less corrugated

substrate. This suggests that effect of the symmetry and corrugation of the substrate

potential on the Xe film structure is largely confined to the monolayer.

In every system, the increase of the density for each layer is strongly correlated

to the commensurability with its support: the more similar they are, the more flat

the adsorption isotherm will be (note that the support for the (N + 1)th-layer is the

N th-layer). For example, the Xe/d-AlNiCo system has an almost perfect hexagonal

structure at point (B) (due to its first-order five to sixfold ordering transition as

described in the next section). Hence, all the further overlayers growing on the top of

the monolayer will be at least “as regular” as the first layer, and have the negligible

density increase as listed in Table 3.3.

3.2.3 Order parameters (ρ5−6)

The evolution of the order parameter ρ5−6 is plotted in Figure 3.5 as a function of

the normalized chemical potential, μ�, at T �=0.35 for all the noble gas/d-AlNiCo

systems.

Ne/d-AlNiCo, Ar/d-AlNiCo, and Kr/d-AlNiCo systems. The ρ5−6 plots

for the first four layers observed before bulk condensation are shown in panels (a)−(c).

32

FTρ(x,y)(a)

A

B

C

D

E

F

(b)Kr/QC Xe/QC FTρ(x,y)

A

B

C

D

θr

θr

Figure 3.4: Density profiles and Fourier transforms of the outer layer at T � = 0.35 for of Kr/d-AlNiCo (T = 59.6 K) and Xe/d-AlNiCo (T = 77 K), corresponding to points (A) through (F) of Figure 3.2.

33

μ*

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

μ*

ρ 5 -6

layer 1

(a) (b)

(d)(c)

Ne/QC Ar/QC

Kr/QC Xe/QC

0 0.2 0.4 0.6 0.8 1 1.2 0

0.2

0.4

0.6

0.8

1

μ*

ρ 5 -6

1 1.05 1.1 1.15

0.4

0.5

0.6

layer 1

layer 2

layer 3 layer 4

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.06 1.12

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 μ

ρ 5 -6

layer 1

layer 2

layer 3 layer 4

0 0.2 0.4 0.6 0.8 1 1.2 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.06 1.12 0.3

0.5

0.7

μ*

ρ 5 -6

layer 1

layer 2 layer 3 layer 4

5- to 6- fold transition

Figure 3.5: (color online). Order parameters, ρ5−6, as a function of normalized chemical potential, μ�, (as defined in the text) at T � = 0.35 for the first four layers of (a) Ne, (b) Ar, (c) Kr, and for the first layer of Xe (d) adsorbed on d-AlNiCo. A sudden drop of the order parameter in Xe/QC to a constant value of ∼ 0.017 at μ� ∼ 0.8 indicates the existence of a first-order structural transition from fivefold to sixfold in the system.

As the chemical potential μ� increases, ρ5−6 decreases continuously reaching a con-

stant value only for Kr. At bulk condensation, the values of ρ5−6 are still high,

approximately 0.35 ∼ 0.45. Data at higher temperatures shows a similar behavior (up to T=24 K (T �=0.71) for Ne, T =70 K (T �=0.58) for Ar, and T=90 K (T �=0.53)

for Kr). Thus, we conclude that these systems do not undergo the ordering transition.

Xe/d-AlNiCo system. The ρ5−6 plot for the first layer is shown in panel (d).

In this system, as the chemical potential μ� increases, the order parameter gradually

34

decreases reaching a value of ∼ 0.3 at μ�tr ∼ 0.8. Suddenly it drops to 0.017 and remains constant until bulk condensation. Similar behavior is observed at higher

temperatures up to T=140 K (T �=0.63). This is a clear indication of a five to sixfold

ordering transition, as the first layer has undergone a transformation to an almost

perfect triangular lattice. Figure 3.6(d) shows the total enthalpy in the system, H.

At the transition point μ�tr, the enthalpy has a little step indicating a latent heat of

the transition. The discontinuity of the order parameter ρ5−6 and the presence of

latent heat indicate that the ransition is first-order. The latent heat of this transition

is estimated to be 1.021/151 ≈ 6.8 meV/atom. Despite the evidence for a first-order transition, the nearest neighbor distance, la-

beled rNN , e.g. Xe/d-AlNiCO in Figure 3.6(b), appears to change continuously. This

nearest neighbor distance is defined as the location of the first peak in the pair corre-

lation function because the latter property is more directly comparable to diffraction

measurements. We have also calculated the average spacing between neighbors, d¯NN ,

which is a thermodynamically meaningful quantity (related to the density). This has

a small discontinuity at the transition, providing additional evidence for the first-

order character of the transition. Both quantities, rNN and d¯NN , are shown in Figure

3.6(b). The NN Xe-Xe distance rNN decreases continuously as P increases, starting

from 0.45 nm and saturating at 0.44 nm. The Xe-Xe distance reaches saturation

value before the appearance of the second layer; therefore, the transition is complete

within the first layer. We note that a similar decrease in NN distance was measured

for Xe/Ag(111), but in that case, the NN spacing did not saturate before the onset

of the second layer adsorption [112, 110].

The observed transition from fivefold to sixfold order within the first layer of Xe

on d-AlNiCo can be viewed as a commensurate-incommensurate transition (CIT),

since at the lower coverage, the layer is commensurate with the substrate symmetry

35

0 0.2 0.4 0.6 0.8 1 μ*

ρ 5- 6

2 4 6 8

10

ρ N (a

to m

/n m

2 )

layer formation

st1

layer formation

nd2

b

a

0.2

0.4

0.6 5- to 6- fold transition

0.44

0.45

0.46

(n m

)

c

d

rNN

dNN

r NN d N

N ,

-45

-40

-35

-30

-25

H (e

V )

Δ Η

Figure 3.6: (color online). Xe on d-AlNiCo at T = 77 K. (a) Adsorption isotherm, ρN , versus the normalized chemical potential, μ�. (b) Nearest neighbor distance derived from the first peak of pair correlation function, rNN , (black line), and average spacing between neighbors at equilibrium, d¯NN , (red line). (c) Order parameter ρ5−6 (probability of fivefold defects, defined in Equation 3.14) versus the normalized chemical potential, μ�. (d) Total enthalpy. The transition, which is defined as the point in μ� above which the order parameter remains nearly constant, occurs at μ�tr ∼0.8. The discontinuity in H around μ�tr ∼ 0.8 indicates a first order transition with associated latent heat of the transition. The order parameter ρ5−6 after the transition is ∼ 0.017. Heat of the transition is ≈ 6.8 meV/atom.

36

and aperiodic, while at higher coverage, it is incommensurate with the substrate.

Such transitions within the first layer have been observed before for adsorbed gases,

perhaps most notably for Kr on graphite [113]. There, as here for Xe, the Kr forms

a commensurate structure at low coverage, which is compressed into an incommen-

surate structure at higher coverage. The opposite occurs for Xe on graphite, which

is incommensurate at low coverage and commensurate at high coverage [114]. Such

commensurate-incommensurate transitions have been studied theoretically in many

ways, but perhaps most simply as a harmonic system (balls and springs) having a

natural spacing that experiences a force field having a different spacing [115]. Such a

transition has been found to be first-order for strongly corrugated potentials (in 1D)

but continuous for more weakly corrugated potentials [116]. The transition observed

in our quasicrystal surface suggests that system is within the regime of “strong” cor-

rugation, which was not the case of Kr over graphite [113]. In fact, for the latter

system, both commensurate and incommensurate structures have sixfold symmetry.

A more relevant comparison may be the transition of Xe on Pt(111), from a rect-

angular symmetry incommensurate phase to a hexagonal symmetry commensurate

one, although in that case, the low-temperature phase was incommensurate. That

transition was also found to be continuous [117]. Therefore, while our simulations

indicate that Xe on d-AlNiCo undergoes a CIT, as observed for other adsorbed gases,

the observation of a first-order CIT is new, to our knowledge, and likely arises from

the large corrugation.

3.2.4 Effects of �gg and σgg on adsorption isotherms

In Section 3.2.1 we have briefly discussed how the density increase of each layer (ΔρN)

is affected by the size of the adsorbate (σgg). In addition, since the corrugation of the

potential depends also on the gas-gas interaction (�gg), the latter quantity could a

37

priori have an effect on the density increase. To decouple the effects of σgg and �gg on

ΔρN we calculate ΔρN while keeping one parameter constant, σgg or �gg, and varying

the other. For this purpose, we introduce two test gases iNe(1) and dXe(1), which

represent “inflated” or “deflated” versions of Ne and Xe, respectively (parameters

are defined in Equations 3.9-3.12 and listed in Tables 3.1 and 3.2). Then we perform

four tests summarized as the following:

(1) constant strength �gg, size σgg increases [Ne→iNe(1)]: ΔρN reduces,

(2) constant strength �gg, size σgg decreases [Xe→dXe(1)]: ΔρN increases,

(3) constant size σgg, strength �gg decreases [Xe→iNe(1)]: ΔρN ∼ constant,

(4) constant size σgg, strength �gg increases [Ne→dXe(1)]: enhanced agglomer- ation.

Figure 3.7 shows the adsorption isotherms at T � = 0.35 for Ne, iNe(1), Xe, and

dXe(1) on d-AlNiCo. By keeping the strength constant and varying the size of the

adsorbates, tests 1 and 2 ([Ne→iNe(1)] and [Xe→dXe(1)]), we find that we can reduce or increase the value of the density increase (when ΔρN decreases the continuous

growth tends to become stepwise and vice versa). These two tests indicate that the

larger the size, the smaller the ΔρN . By keeping the size constant and decreasing

the strength, test 3 ([Xe→iNe(1)]), we find that ΔρN does not change appreciably. An interesting phenomenon occurs in test 4 where we keep the size constant and

increase the strength ([Ne→dXe(1)]). In this test the growth of the film loses its step-like shape. We suspect that this is caused by an enhanced agglomeration effect

as follows. Ne and dXe(1) have the same size which is the smallest of the simulated

gases, allowing them to easily follow the substrate corrugation, in which case, the

corrugation helps to bring adatoms closer to each other [100] (agglomeration effect).

The stronger gas-gas self interaction of dXe(1) compared to Ne will further enhance

38

10-25 10-20 10-15 10-10 10-5 1 0

10

20

30

40

P (atm)

ρ (a

to m

s/ nm

2 )

Ne iNe Xe

dX e

Ν

(1 )

(1)

test 1: [Ne → iNe(1)]

test 2: [Xe → dXe(1)]

test 3: [Xe → iNe(1)]

test 4: [Ne → dXe(1)]

Figure 3.7: (color online). Computed adsorption isotherms for Ne, Xe, iNe(1), and dXe(1)

on d-AlNiCo at T �=0.35. iNe(1) and dXe(1) are test noble gases having potential parameters described in the text and in Tables 3.1 and 3.2. The effect of varying the interaction strength of the adsorbates on the density increase ΔρN (while keeping the size constant) is negligible on large gases but significant on small gases.

this agglomeration effect, resulting in a less stepwise film growth of dXe(1) than Ne.

As can be seen, dXe(1) grows continuously, suggesting a strong enhancement of the

agglomeration. In summary, the last two tests (3 and 4) indicate that the effect of

varying the interaction strength of the adsorbates (while keeping the size constant)

is negligible on large gases but significant on small gases.

3.2.5 Effects of �gg and σgg on 5- to 6-fold transition

Strength �gg and size σgg of the adsorbates also affect the existence of the first-order

transition (present in Xe/d-AlNiCo, but absent in Ne, Ar, and Kr on d-AlNiCo).

Hence we perform the same four tests described before and observe the evolution of

the order parameter. The results are the following:

(1) constant strength �gg, size σgg increases [Ne→iNe(1)]: transition appears

(2) constant strength �gg, size σgg decreases [Xe→dXe(1)]: transition disappears 39

(3) constant size σgg, strength �gg decreases [Xe→iNe(1)]: transition remains

(4) constant size σgg, strength �gg increases [Ne→dXe(1)]: remains no transition

The strength �gg has no effect on the existence of the transition (tests 3 and 4), which

instead is controlled by the size of the adsorbates (tests 1 and 2). To further charac-

terize such dependence, we add three additional test gases with the same strength �gg

of Xe but different sizes σgg. The three gases are denoted as dXe (2), iXe(1), and iXe(2)

(the prefixes d- and i- stand for deflated and inflated, respectively). The interaction

parameters, defined in the following equations, are listed in Tables 3.1 and 3.2:

{�gg, Dgs, σgg} [dXe(2)] ≡ {�gg, Dgs, 0.95σgg}[Xe], (3.15)

{�gg, Dgs, σgg}[iXe(1)] ≡ {�gg, Dgs, 1.34σgg}[Xe], (3.16)

{�gg, Dgs, σgg}[iXe(2)] ≡ {�gg, Dgs, 1.65σgg}[Xe]. (3.17)

Figure 3.8 shows the evolutions of the order parameter as a function of the normal-

ized chemical potential for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed

on d-AlNiCo at T �=0.35. All these systems undergo a transition, except dXe(2), i.e.

the transition occurs only in systems with σgg ≥ σgg[Xe] indicating the existence of a critical value for the appearance of the phenomenon. Furthermore, as σgg increases

(iNe(1) → iXe(1) → iXe(2)), the transition shifts towards smaller critical chemical potentials.

3.2.6 Prediction of 5- to 6-fold transition

The critical value of σgg associated with the transition can be related to the charac-

teristic length of the d-AlNiCo by introducing a gas-substrate mismatch parameter

defined as

δm ≡ k · σgg − λr λr

. (3.18)

40

0 0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

μ*

ρ 5 -6

iXe

dXe

iNe

(2)

(1) (1)iXe(2)

Figure 3.8: (color online). Order parameters as a function of normalized chemical poten- tial (as defined in the text) for the first layer of dXe(2), iNe(1), iXe(1), and iXe(2) adsorbed on d-AlNiCo at T � = 0.35. A first-order fivefold to sixfold structural transition occurs in the last three systems, but not in dXe(2).

where k = 0.944 is the distance between rows in a close-packed plane of a bulk LJ

gas (calculated at T = 0 K with σ = 1 [118]), and λr is the characteristic spacing of

the d-AlNiCo, determined from the momentum transfer analysis of LEED patterns

[99] (our d-AlNiCo surface has λr=0.381 nm [99]). With such ad hoc definition, δm

measures the mismatch between an adsorbed FCC[111] plane of adatoms and the d-

AlNiCo surface. In Table 3.4 we show that δm perfectly correlates with the presence

of the transition in our test cases (transition exists ⇔ δm > 0). The definition of a gas-substrate mismatch parameter is not unique. For example,

one can substitute k · σgg with the first NN distance of the bulk gas, and λr with one of the following characteristic lengths: a) side length of the central pentagon in

the potential plots in Figure 3.1 (λs = 0.45nm), b) distance between the center of

the central pentagon and one of its vertices (λc = 0.40nm), c) L = τ · S = 0.45 nm, where τ = 1.618 is the golden ratio of the d-AlNiCo and S = 0.243 nm is the

41

Table 3.4: Summary of adsorbed noble gases on d-AlNiCo that undergo a first-order fivefold to sixfold structural transition and those that do not.

δm transition Ne -0.311 No Ar -0.158 No Kr -0.108 No Xe 0.016 Yes

iNe(1) 0.016 Yes dXe(1) -0.311 No dXe(2) -0.034 No iXe(1) 0.363 Yes iXe(2) 0.672 Yes

k = 0.944 [118] λr = 0.381 nm [99]

δm ≡ (k · σgg − λr)/λr

side length of the rhombic Penrose tiles [15]. Although there is no a priori reason to

choose one definition over the others, the one that we select (Equation 3.18) has the

convenience of being perfectly correlated with the presence of the transition, and of

using reference lengths commonly determined in experimental measurements (λr) or

quantities easy to extract (k · σgg).

3.2.7 Transitions on smoothed substrates

In Figure 3.1 we can observe that near the center of each potential there is a set of five

points with the highest binding interaction (the dark spots constituting the central

pentagons). A real QC surface contains an infinite number of these very attractive

positions which are located at regular distances and with five fold symmetry. Due to

the limited size and shape of the simulation cell, our surface contains only one set

of these points. Therefore, it is of our concern to check if the results regarding the

existence of the transition are real or artifacts of the method. We perform simulation

tests by mitigating the effect of the attractive spots through a Gaussian smoothing

function which reduces the corrugation of the original potential. The definitions are

42

the following:

G(x, y, z) ≡ AGe−(x2+y2+z2)/2σ2G , (3.19)

V (z) ≡ 〈V (x, y, z)〉(x,y) , (3.20)

Vmod(x, y, z) ≡ V (x, y, z) · [1−G(x, y, z)] + V (z) ·G(x, y, z). (3.21)

where G(x, y, z) is the Gaussian smoothing function (centered on the origin and with

parameters AG and σG), V (z) is the average over (x, y) of the original potential

V (x, y, z), and Vmod(x, y, z) is the final smoothed interaction. An example is shown

in Figure 3.9(a) where we plot the minimum of the adsorption potential for a Ne/d-

AlNiCo modified interaction (smoothed using AG = 0.5 and σG = 0.4 nm). In

addition, in panel (b) we show the variations of the minimum adsorption potentials

along line x = 0 for the modified and original interactions (solid and dotted curves,

respectively).

Using the modified interactions (with AG = 0.5 and σG = 0.4 nm) we simulate all

the noble gases of Table 3.4. The results regarding the phase transition on modified

surfaces do not differ from those on unmodified ones, confirming that the observed

transition behavior is a consequence of competing interactions between the adsorbate

and the whole QC substrate rather than just depinning of the monolayer epitaxially

nucleated. Therefore, the simple criterion for the existence of the transition (δm > 0)

might also be relevant for predicting such phenomena on other decagonal quasicrystal

substrates.

3.2.8 Temperature vs substrate effect

Using Xe/d-AlNiCo data, we observe that defects are present at all temperatures that

are simulated (20 to 286 K). The probability of defects increases with temperature,

implying that their origin is entropic, as is the case for a periodic crystal. Figure

43

(a) (b)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-75

-65

-55

-45

-35

Y (nm)

V m

in (y

) ( m

eV )

modified potential original potential

Figure 3.9: (color online). (a) The minimum of adsorption potential, Vmin(x, y), for Ne on a smoothed d-AlNiCo as described in the text. (b) The variations of the minimum adsorption potentials along the line at x = 0 shown in (a), for the modified and original interactions (solid and dotted curves).

3.10-(right axis) shows that the defect probability increases as T increases, while

Figure 3.10-(left axis) shows the trend of the transition point in function of T. At low

temperatures, the sixfold ordering occurs earlier (at lower μ�tr) as the temperature

is increased from 40K to 70K. This trend is expected because the ordering effect

imposed by the substrate corrugation becomes relatively smaller as the temperature

increases. However, this trend is not observed in the higher temperature region (from

70K to 140K). In fact, at higher temperatures the transition point shifts again to

higher μ�tr. This is most likely due to the monolayer becoming less two-dimensional,

allowing more structural freedom of the Xe atoms and thus decreasing the effect of

the repulsive Xe-Xe interaction that would stabilize the sixfold structure. Transitions

having critical μ�tr > 1 indicate that the onset of second-layer adsorption occurs

earlier than the transition to the sixfold structure. When the second layer adsorbs at

T >130K, the density of the monolayer increases by a few percent, thereby increasing

the effect of the repulsive interactions and driving the fivefold to sixfold transition.

44

o f 5

-6 fo

ld tr

an si

tio n

μ t r*

ρ 5-6

μtr*

ρ 5-6

Figure 3.10: Xe on d-AlNiCo. Values of μ�tr for the fivefold to sixfold transition points from 40 K to 140 K (left axis). Transition points at μ�tr > 1 indicate that a transfer of atoms from the second layer to the first layer is required to complete the transition. Also shown is the defect probability as a function of T after the transition occurs (right axis), indicating an increase in defect probability with T .

Interestingly, stacking faults are evident in the multilayer films. This is consistent

with x-ray diffraction studies of the growth of Xe on Ag(111), where stacking faults

were observed for Xe growth under various growth conditions [110, 111], although the

overall structure observed was FCC(111). Such a stacking fault is evident in Figure

3.11, which shows a superposition of Xe layers 2 and 4 at 77 K. The coincidence of

the atom locations in the top left part of this figure is consistent with an hexagonal

close-packed structure (HEX), ABAB stacking, whereas the offsets observed in the

lower part of the figure indicate the presence of stacking faults caused by dislocations

in the layers. We note that while bulk Xe has an FCC structure, and indeed an FCC

structure was found for the multilayer film in the LEED study, calculations of the

bulk structure using LJ pair potentials such as those employed here result in a more

stable HEX structure [119]. The energy difference between the two structures is very

small, and apparently arises from a neglect of d-orbital overlap interactions, which

45

Figure 3.11: Density plot of Xe on decagonal AlNiCo at 77 K, showing a superposition of the density slices for the 2nd and 4th layers. In the top right, 4th-layer atoms are located directly above the 2nd layer atoms indicating an hexagonal close-packed ABAB stacking, whereas in other regions, such as lower left, the two layers are offset due to stacking fault.

are more effective in FCC than in HEX structures [119, 120]. Although the simulated

film is HEX instead of FCC, the main conclusions concerning the growth mode of Xe

on the quasicrystal are not affected [70].

3.2.9 Orientational degeneracy of the ground state

In was mentioned in the previous section that after the ordering transition is complete,

the resulting sixfold structure is aligned parallel to one of the sides of the pentagons

in the Vmin map of the adsorption potential (there are five possible orientations). In

the experiments, all five orientations are observed, due to the presence of all possi-

ble alignments of hexagons along five sides of a pentagon in the QC sample within

the width of the electrons beam (∼0.25 mm). In an ideal infinite GCMC frame- work the ground state of the system would be degenerate and all five orientations

would have the same energy and be equally probable. However, the square periodic

boundary conditions of our GCMC break this orientational degeneracy, causing some

orientations to become more likely to appear.

To find all the possible orientations, we performed simulations with a cell having

free boundary conditions. The cell is a 5.12 x 5.12 nm2 quasicrystal surface sur-

rounded by vacuum. Figure 3.12(a) shows the Vmin map of the adsorption potential.

46

Thirty simulations at 77 K are performed with this cell. The isotherms from these

runs are plotted in Figure 3.12(b). Only the first layer is shown, and the finite size

of the surface makes the growth of the first layer continuous. The density profiles

ρ(x, y) of all the simulations are analyzed at point p� of Figure 3.12(b). In this cell,

all five orientations of hexagons are observed with equal frequency indicating the ori-

entational degeneracy of the ground state. To represent the five orientations, density

profiles of five calculations (c, d, e, f, and g) are shown in Figures 3.12(c) to 3.12(g)

with their FT plotted on the side. Figure 3.12(h) presents a schematic depiction of

which orientations of hexagons are exemplified in each simulation.

Figures 3.13(a) and 3.13(b) illustrate the effect of pentagonal defects on the ori-

entation of hexagons at point p� of Figure 3.12(b). In most of the density profiles

corresponding to this coverage, we find the behavior shown in Figure 3.13(a). Here,

the effect of the pentagonal defect, which is the center of a dislocation in the hexag-

onal structure, is to rotate the orientation of the hexagons above the pentagon by

2 · 60◦/5 = 24◦ with respect to the hexagons below the pentagon. The possible rota- tions are n · 12◦, where n = 1,2,3,4, or 5. The rotation by 12◦ is usually mediated by more than one equivalent pentagon, as is shown in Figure 3.13(b) (note the “up” pen-

tagon at the middle-bottom part and the “down” pentagon near the middle-top part

of the figure). The “up” pentagon (with one vertex on the top) is equivalent with the

“down” pentagon (with one vertex on the bottom) since they have five orientationally

equivalent sides). These pentagonal defects are induced by the fivefold symmetry of

the substrate, and their concentration decreases in the subsequent layers.

3.2.10 Isosteric heat of adsorption

Figure 3.14 shows a P -T diagram for three different coverages of Xe adsorbed on

d-AlNiCo constructed from the isotherms in the range 40 K < T < 110 K. In the

47

simulations, the layers grow step-wise; at 70 K the first step occurs between coverage

∼0.06 and ∼0.7, the second step occurs between coverage 1.0 and ∼1.9, and the third step occurs between coverage ∼1.9 and ∼2.8 (unit is in fractions of monolayer). Figure 3.14 shows the T , P location of these steps, denoted “cov 0.5”, “cov 1.5”,

and “cov 2.5” for the first, second, and third steps, respectively. The isosteric heat

of adsorption per atom at these steps can be calculated from the P -T diagram as

follows [112]:

qst ≡ −kB d(lnP ) d(1/T )n

. (3.22)

The inset of Figure 3.14 summarizes the values of qst obtained from simulations

and experiments. The agreement between experiment and the simulations for the

half monolayer heat of adsorption is good. The values obtained in the simulation for

the 1.5 and 2.5 layer heats are about 20% lower than the bulk value of 165 meV [99].

The lower values suggest that bulk formation should be preferred at coverages above

one layer. However, layer-by-layer growth is observed at all T for at least the first

few layers in these simulations. We therefore believe that the low heats of adsorption

arise from slight inaccuracies in the Xe-Xe LJ parameters used in this calculation, as

the heats of adsorption are very sensitive to the gas parameters.

3.2.11 Effect of vertical dimension

In a standard unit cell, only 2 steps, corresponding to the first and second layer ad-

sorption, are apparent in the isotherms [101]. Further simulations indicate that when

the cell is extended in the vertical direction, additional steps are observed. Therefore

the number of observable steps is related to the size of the cell. Nevertheless, layering

is clearly evident in the ρ(z) profile, and the main features of the film growth are not

altered. For Xe on d-AlNiCo, the average interlayer distance is calculated to be about

0.37 nm, compared to 0.358 nm for the interlayer distance in the < 111 > direction

48

of bulk Xe [121]. Our simulations of multilayer films show variable adsorption as

the simulation cell is expanded in the direction perpendicular to the surface. This

is a result of sensitivity to perturbations (here, cell size) close to the bulk chemical

potential, where the wetting film’s compressibility diverges. This dependence has

been seen previously in large scale simulations. See e.g. Figure 3 of reference [108].

The analog of this effect in real experiments is capillary condensation at pressures

just below saturated vapor pressure (svp), the difference varying as the inverse pore

radius.

3.3 Summary

The results of GCMC simulations of noble gas films on QC have been presented. Ne,

Ar, Kr, and Xe grow layer-by-layer at low temperatures up to several layers before

bulk condensation. We observe interesting phenomena that can only be attributed

to the quasicrystallinity and/or corrugation of the substrate, including structural

evolution of the overlayer films from commensurate pentagonal to incommensurate

triangular, substrate-induced alignment of the incommensurate films, and density

increase in each layer with the largest one observed in the first layer and in the smallest

gas. Two-dimensional quasicrystalline epitaxial structures of the overlayer form in

all the systems only in the monolayer regime and at low pressure. The final structure

of the films is a triangular lattice with a considerable amount of defects except in

Xe/QC. Here a first-order transition occurs in the monolayer regime resulting in an

almost perfect triangular lattice. The subsequent layers of Xe/QC have hexagonal

close-packed structures. By simulating test systems with various sizes and strengths,

we find that the dimension of the noble gas, σgg, is the most crucial parameter in

determining the existence of the phenomenon which is found only in systems with

σgg ≥ σgg[Xe].

49

a 10 -12

10 -10

10 -8

10 -60

2

4

6

8

10

p (atm)

ρ= N

/A

(a to

m s/

nm 2

)

p*

-5 5X(nm)

5

Y (n

m )

-5

0

-50

-100

-150

-200

-250

(meV)

b

c d

e

g

f

h c

d,g

f

g

e,g

Figure 3.12: Xe adsorption on d-AlNiCo. (a) Minimum potential energy surface of the adsorption potential with free boundary conditions. (b) Adsorption isotherms of the first layer from a set of 30 simulations at 77 K using the free cell described in the paper. Five density profiles and FTs at point p� of (b) are shown in (c) to (g), representing all possible orientations of hexagonal domains. (h) Schematic diagram illustrating the correspondence between the orientations of the hexagonal domains observed in the density profiles (c) to (g).

50

θ1 θ2

a b

Figure 3.13: Xe adsorption on d-AlNiCo. Pentagonal defects rotate the orientation of hexagons by (a) θ1 = 24◦ and (b) θ2 = 12◦.

Figure 3.14: (color online). Xe adsorption on d-AlNiCo. Locations in P , T of the vertical risers in the isotherms corresponding to the first (square), second (circle), and third (triangle) layer formation. The heats of adsorptions, qst, are 270, 129, and 125 meV/atom respectively, calculated as described in the text. The inset figure shows qst obtained from the simulations as well as from the experiments.

51

Chapter 4

Embedded-atom method potentials

The interatomic potentials to simulate hydrocarbon adsorptions on d-AlNiCo are

generated within the embedded-atom method (EAM) formalism. The parametriza-

tion of the potentials are given in Equations 2.30 - 2.33. These parameters are fitted

to the energy of various structures computed via ab initio quantum calculations using

VASP code. The structure prototypes are summarized in Table 4.1. For elemental

Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure

(DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed

(HEX), simple cubic (SC), and simple hexagonal (SH). The structures are first re-

laxed from the initial configurations to achieve the equilibrium ones. For the Al-Co-Ni

ternary phases, the initial configurations are taken from the database of alloys [122].

After the structures are relaxed, the ground state electronic energies are calculated

and are used as the fitting data. The technical details in relaxing the structures and

evaluating the energies with VASP are given in Section 2.2.

Fitting of the parameters of the EAM potentials are performed using SIMPLEX

method (Section 2.4). SIMPLEX is relatively slower than other methods such as

nonlinear least square or conjugate gradient. However, since the speed of the fitting

procedure is not a concern in this work (most time is spent in the ab initio calcula-

tions), SIMPLEX is advantageous because it does not require evaluation of function’s

derivatives or orthogonality. In this way, a new parametrization of EAM potentials

requires changes only in the function evaluation routines. A fitting code is developed

to be able to fit in a bulk or adsorption mode. In the bulk mode, the function to

52

Table 4.1: List of structure prototypes used to fit the EAM potentials for hydrocarbon ad- sorption on Al-Co-Ni. For elemental Al, Co, and Ni, the prototypes are body-centered cubic (BCC), diamond structure (DIA), face-centered cubic (FCC), graphite structure (GRA), hexagonal close-packed (HEX), simple cubic (SC), and simple hexagonal (SH). The Al– Co-Ni ternaries are taken from the database of alloys [122].

group structure Al BCC, DIA, FCC, GRA, HEX, SC, SH Co BCC, DIA, FCC, GRA, HEX, SC, SH Ni BCC, DIA, FCC, GRA, HEX, SC, SH CHn CH4 C2Hn C2H2, C2H4, C2H6-isotactic, C2H6-syntactic C3Hn propane, C3H4 allene, propyne, propene,

cyclopropane, cyclopropene, cyclopropyne C4Hn isobutane, 1-butene, 1-butyne, cyclobutane, methylcyclopropane C5Hn pentane, cyclopentane Other alkanes hexane, heptane, nonane, decane, undecane, dodecane AlxCoyNiz Al32Co12Ni12 (cI112)

Al32Co16Ni12 (cI128) Al29Co4Ni8 (dB1) Al17Co5Ni3 (dH1) Al34Co4Ni12 (dH2) Al20Co7Ni1 (hP28) Al18Co4Ni4 (mC28) Al12Co2Ni2 (mC32) Al18Co2Ni2 (mP22) Al36Co8Ni4 (oI96) Al12Co1Ni3 (oP16) Al34Co12Ni4 (mC102)

53

minimize is the err = ΔEbulk defined as:

ΔEbulk = ∑

i

|EEAMbulk,i − EV ASPbulk,i | (4.1)

The energies are per atom and the summation is over all structures of the fit. In

the adsorption mode, the error function consists of the error in the bulk energies

of molecule structures (ΔEbulk,mol), substrate structures (ΔEbulk,sub), molecule-on-

substrate structure (ΔEbulk,mol+sub), as well as in the adsorption energies (ΔEads):

Eads = Ebulk,mol+sub − (Ebulk,mol + Ebulk,sub) (4.2)

ΔEads = ∑

i

|EEAMads,i − EV ASPads,i | (4.3)

err = c1ΔEbulk,mol + c2ΔEbulk,sub + c3ΔEbulk,mol+sub + c4ΔEads

c1 + c2 + c3 + c4 (4.4)

The coefficients c1, c2, c3, and c4 are introduced as weighing factors. To drive the

parameters toward physically meaningful convergence point, the fitting is performed

in multiple stages:

(1) fit potentials for elemental Al, Co, and Ni,

(2) fit potentials for Al-Co-Ni,

(3) fit potentials for hydrocarbons,

(4) fit final potentials for hydrocarbon on Al-Co-Ni.

The fitted parameters from stage (1) are used as initial conditions in stage (2). The

fitted parameters from stage (2) and (3) are used as initial values in stage (4).

4.1 Stage 1: Aluminum, cobalt, and nickel

In stage (1), the elemental potentials for Al, Co, and Ni are fitted to elemental

bulk energies (top part of Tables 4.2-4.4). The potentials are also trained at various

54

pressures to ensure stability under compression/expansion and to yield resonable lat-

tice constants. In the calculations, different pressures are achieved by expanding or

compressing the relaxed ground state structures (i.e. Al(FCC) Co(HEX) Ni(FCC)),

namely at lattice constants from a = 0.95a0 to a = 1.1a0, where a0 is the equilib-

rium lattice constant at zero pressure. Training at various pressures increases the

transferability of the potentials due to a wider range of charge density covered. The

fitted EAM potentials (bottom part of Tables 4.2-4.4) are able to find the ground

state structure as well as the relative stability of each structure. Note that for pure

elements, the bulk energy represents the cohesive energy. Going from the most stable

structure to the least stable one, the EAM potentials correctly predict the following:

FCC → HEX → BCC → SC → SH → GRA → DIA (for Al and Ni) and HEX → FCC → BCC → SC → SH → GRA → DIA (for Co). The potentials also give accurate equilibrium lattice constants (middle part of Tables 4.2-4.4).

4.2 Stage 2: Al-Co-Ni potentials

Results from stage (1) are used as initial conditions in stage (2). The potential

for systems containing Al, Ni, and Co (AlCoNi-pot) are fitted to energies in bulk

structures and in slab configurations. The latter is intended to tune the AlCoNi-

pot at low charge density having different atomic environments from bulk. Slab

configurations are created from dB1, dH1, and dH2. Note that dB1, dH1, and dH2

are decagonal AlNiCo quasicrystal approximants. They are crystals with a large unit

cell which represent the short-ranged order of the quasicrystal. Approximants exist in

the vicinity of region of chemical compositions of their quasicrystalline counterparts.

The bulk unit cell of dB1, dH1, and dH2 contains 41, 25, and 50 atoms, respectively.

The unit cells consist of two well-defined layers. There are two possible terminations

for the bulk. The top(bottom) layer of the unit cell is labeled A(B), respectively. A

55

Table 4.2: (Top part) List of structures used to fit EAM potential for elemental aluminum. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 · c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3 · re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.199 -3.375 -3.368 -0.007 DIA 0.135 -2.847 -2.722 -0.126 FCC at a0·0.95 0.312 -3.347 -3.347 0.000 FCC at a0·0.96 0.289 -3.396 -3.390 -0.006 FCC at a0·0.97 0.267 -3.425 -3.422 -0.003 FCC at a0·0.98 0.248 -3.440 -3.444 0.004 FCC at a0·0.99 0.229 -3.443 -3.457 0.014 FCC at a0·1.00 0.212 -3.439 -3.462 0.023 FCC at a0·1.05 0.144 -3.356 -3.390 0.034 FCC at a0·1.10 0.098 -3.221 -3.221 0.000 GRA 0.151 -2.882 -2.958 0.075 HEX 0.211 -3.436 -3.426 -0.009 SC 0.167 -3.108 -3.097 -0.011 SH 0.178 -3.241 -3.241 0.000

a0(FCC) vasp = 4.04 A˚ a0(FCC) EAM = 4.00 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.026 7.182 2.700 33.64 3.038 3.012

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.078 0.121 0.495 0.896 1.381 2.200 Fx (eV) 0.000 -2.810 -3.106 -3.980 -3.879 -3.665 -3.074

56

Table 4.3: (Top part) List of structures used to fit EAM potential for elemental cobalt. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”HEX at a0 · c” indicates that the structure is HEX with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.665 -5.380 -5.377 -0.003 DIA 0.602 -4.115 -4.141 0.027 FCC 0.693 -5.439 -5.450 0.011 GRA 0.566 -4.596 -4.567 -0.029 HEX at a0·0.95 0.973 -5.213 -5.214 0.001 HEX at a0·0.96 0.910 -5.317 -5.305 -0.012 HEX at a0·0.97 0.849 -5.386 -5.375 -0.011 HEX at a0·0.98 0.795 -5.425 -5.425 -0.000 HEX at a0·0.99 0.744 -5.442 -5.457 0.016 HEX at a0·1.00 0.697 -5.440 -5.473 0.033 HEX at a0·1.05 0.501 -5.295 -5.383 0.088 HEX at a0·1.10 0.358 -5.098 -5.098 0.000 SC 0.569 -4.984 -4.708 -0.276 SH 0.608 -5.155 -4.966 -0.189

a0(HEX) vasp = 2.47 A˚ a0(HEX) EAM = 2.45 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.090 5.945 2.280 36.70 2.832 2.701

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.146 0.380 0.821 1.306 1.683 2.200 Fx (eV) 0.000 -3.374 -4.894 -5.517 -5.509 -5.008 -3.952

57

Table 4.4: (Top part) List of structures used to fit EAM potential for elemental nickel. ρi is charge density at atom-i, ΔE = EEAM − EV ASP . The energies are per atom. The label ”FCC at a0 ·c” indicates that the structure is FCC with modified lattice contant by a factor of c from the equilibrium one (a0). (Bottom part) Fitted parameters using cutoff radius = 3·re. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx).

structure ρi EEAM EV ASP ΔE (A˚−3) (eV/at) (eV/at) (eV/at)

BCC 0.699 -4.928 -4.924 -0.004 DIA 0.500 -3.839 -3.839 -0.000 FCC at a0·0.95 0.980 -4.754 -4.783 0.029 FCC at a0·0.96 0.918 -4.870 -4.868 -0.002 FCC at a0·0.97 0.861 -4.941 -4.932 -0.009 FCC at a0·0.98 0.807 -4.980 -4.980 0.000 FCC at a0·0.99 0.757 -4.995 -5.010 0.015 FCC at a0·1.00 0.709 -4.993 -5.024 0.031 FCC at a0·1.05 0.507 -4.871 -4.955 0.084 FCC at a0·1.10 0.369 -4.687 -4.687 -0.000 GRA 0.543 -4.081 -4.109 0.028 HEX 0.706 -4.993 -4.998 0.005 SC 0.591 -4.464 -4.342 -0.122 SH 0.637 -4.682 -4.630 -0.052

a0(FCC) vasp = 3.49 A˚ a0(FCC) EAM = 3.46 A˚

ρe β re D α r0 (A˚−3) (A˚) (meV) (A˚−1) (A˚) 0.126 5.447 2.150 39.17 2.839 2.717

knot1 knot2 knot3 knot4 knot5 knot6 knot7 ρ (A˚−3) 0.000 0.223 0.336 0.566 1.028 1.355 2.200 Fx (eV) 0.000 -3.899 -4.373 -4.731 -5.280 -5.028 -4.172

58

slab is labeled A if the top layer is layer A, and vice versa. The unit cell for slab

configurations contains vacuum space of 12 A˚ in the vertical dimension to minimize

the interaction between unit cells due to periodic boundary conditions. In relaxing

the slabs, cell’s size/shape and atoms are allowed to relax except the bottom atoms.

In this way, the bottom atoms will be at the same coordinates as they were in the

bulk which will be more appropriate (than if bottom atoms are relaxed) when the

slabs are used as the substrates in the adsorption configurations (stage (4) of fitting).

The bottom part of Table 4.5 shows the parameters of AlCoNi-pot fitted to structures

listed in the top part of Table 4.5.

4.3 Stage 3: Hydrocarbon potentials

The EAM potentials for hydrocarbons (CH-pot) are fitted to the energies of isolated

molecules. The ab initio calculations are performed in a fairly large cubic cell with

vacuum size of larger than 10 A˚ to minimize the interaction between molecules due to

periodic boundary conditions implemented in VASP. The structures are fully relaxed.

In the beginning, all molecules listed in Table 4.1 are included in the fit. The result-

ing fitted parameters and the fitted energies are reported in Table 4.6. The results

show that EAM is fairly accurate for hydrocarbons, especially for alkanes. This is

interesting because EAM formalism was introduced for metals where bondings are

due to nonlocal electrons. Nevertheless, not all hydrocarbons can be fit simultane-

ously, as significant errors are found in some molecules, e.g C3H4-allene, propylene,

cyclopropane, cyclopropyne, 1-butene, and benzene. For our simulations, presented

in Chapter 5, a more accurate fit for alkanes and benzene is needed. Table 4.7 shows

the EAM potentials fitted to alkanes and benzene.

.

59

Table 4.5: (Top part) List of structures used to fit EAM potential for Al-Co-Ni systems. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV) and the first knot at (0,0) is assumed. The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

cI112 bulk -4.828 -4.828 -0.000 cI128 bulk -4.916 -4.915 -0.000 dB1 bulk -4.398 -4.395 -0.003 dH1 bulk -4.483 -4.516 0.032 dH2 bulk -4.441 -4.480 0.038 hP28 bulk -4.484 -4.490 0.006 mC102 bulk -4.520 -4.517 -0.003 mC28 bulk -4.492 -4.509 0.017 mC32 bulk -4.323 -4.322 -0.001 mP22 bulk -4.121 -4.111 -0.010 oI96 bulk -4.334 -4.350 0.016 oP16 bulk -4.292 -4.300 0.007 dB1 slab3A -3.816 -3.853 0.037 dB1 slab3B -3.913 -3.921 0.008 dB1 slab4A -4.105 -4.096 -0.009 dB1 slab4B -4.015 -4.015 0.000 dH1 slab3A -4.058 -4.043 -0.015 dH1 slab3B -4.056 -4.028 -0.029 dH1 slab4A -4.153 -4.155 0.003 dH1 slab4B -4.158 -4.153 -0.005 dH2 slab4A -4.144 -4.143 -0.001 dH2 slab4B -3.980 -3.980 0.000

Al Co Ni ρe (A˚−3) 0.026 0.090 0.126 β 7.182 5.945 5.447 re (A˚) 2.70 2.28 2.15 D (eV) 0.034 0.037 0.039 α (A˚−1) 1.833 3.350 3.258 r0 (A˚) 3.009 2.744 2.735 Z/ZAl 1 0.862 0.521 Z/ZCo 1 0.975 knot2 0.086,-2.758 0.235,-3.249 0.285,-3.772 knot3 0.120,-2.974 0.383,-4.834 0.341,-4.449 knot4 0.517,-4.688 0.864,-5.579 0.604,-4.931 knot5 0.868,-3.777 1.321,-5.420 1.050,-5.279 knot6 1.365,-3.736 1.705,-5.061 1.355,-5.160 knot7 2.200,-3.151 2.200,-4.090 2.200,-4.196

60

Table 4.6: (Top part) List of structures used to fit EAM potential for hydrocarbons. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

methane -4.807 -4.806 -0.001 ethane syntactic -5.076 -5.061 -0.015 ethane isotactic -5.068 -5.047 -0.021 C2H2 -5.735 -5.736 0.001 C2H4 -5.382 -5.328 -0.054 propane -5.189 -5.183 -0.004 C3H4 allene -5.751 -5.342 -0.409 propylene -5.188 -5.002 -0.186 propyne -5.679 -5.687 0.008 cyclopropane -5.308 -4.850 -0.459 cyclopropene -5.366 -5.273 -0.094 cyclopropyne -5.340 -5.477 0.137 isobutane -5.249 -5.253 0.004 1-butene -5.250 -5.128 -0.122 1-butyne -5.648 -5.630 -0.018 cyclobutane -3.899 -3.899 0.000 methylcyclopropane -5.404 -5.410 0.006 pentane -5.291 -5.297 0.006 cyclopentane -5.024 -5.047 0.022 hexane -5.321 -5.329 0.008 benzene -6.136 -6.334 0.198 heptane -5.343 -5.352 0.009 octane -5.358 -5.370 0.012 nonane -5.368 -5.375 0.007 decane -5.383 -5.381 -0.002 undecane -5.392 -5.391 -0.001 dodecane -5.400 -5.413 0.013

ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

C 0.932 5.473 1.24 0.123 3.116 1.489 H 1.639 2.798 0.74 0.211 4.296 1.125

ZH/ZC = 0.880 knot1 knot2 knot3 knot4 knot5 knot6 knot7

C ρ 0.000 0.334 1.103 1.357 1.635 1.870 2.200 Fx 0.000 -3.851 -6.067 -8.011 -7.771 -8.17 -7.352

H ρ 0.000 1.071 1.528 1.773 2.008 2.200 - Fx 0.000 -2.997 -3.482 -4.189 -3.764 -3.409

61

Table 4.7: (Top part) List of structures used to fit EAM potential for alkanes and benzene. The energies are in eV/atom. (Bottom part) Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA + ZAφB/ZB)/2.

structure EEAM EV ASP ΔE (eV/at) (eV/at) (eV/at)

methane -4.806 -4.806 -0.000 ethane syntactic -5.060 -5.061 0.001 propane -5.182 -5.183 0.001 isobutane -5.254 -5.253 -0.001 pentane -5.298 -5.297 -0.001 hexane -5.329 -5.329 0.000 heptane -5.349 -5.352 0.003 octane -5.367 -5.370 0.003 nonane -5.380 -5.375 -0.005 decane -5.393 -5.381 -0.012 undecane -5.402 -5.391 -0.011 dodecane -5.411 -5.413 0.002 benzene -6.334 -6.334 0.000

ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

C 0.932 5.473 1.24 0.116 2.793 1.477 H 1.639 2.798 0.74 0.225 3.154 1.077

ZH/ZC = 0.533 C H

knot1 0,0 0,0 knot2 0.366,-3.793 1.040,-3.543 knot3 0.509,-5.454 1.534,-4.003 knot4 0.761,-6.130 1.801,-3.656 knot5 1.012,-7.611 1.911,-3.874 knot6 1.834,-8.037 2.200,-3.736 knot7 2.200,-7.288 -

62

4.4 Stage 4: Hydrocarbon on Al-Co-Ni

The final EAM potentials for hydrocarbon adsorption on Al-Co-Ni are fitted to the

adsorption energies of a hydrocarbon on a substrate. As the substrates, dB1-slab4A

and dH1-slab4A are used for methane, and dH2-slab4A are used for larger molecules.

The initial conditions are taken from Tables 4.5 and 4.7 and the fitting is performed

in the adsorption mode as explained previously. During the fitting, it is found that

the adsorption energies are more difficult to fit than those of molecules, substrates,

and molecule-on-substrate. This originates from the different nature of bonding. As

discussed later in Chapter 5, alkanes do not make strong chemical bonds with the Al-

Co-Ni. They are only physically adsorbed. Whereas the bondings within a molecule

and within the substrate are strong. Weighing factors of c1 = c2 = c3 = 1 and c4 > 1

are used to prioritize adsorption energies. Typically, values of c4 < 4 are used to get

the final fit. Increasing c4 beyond 4 does not considerably improve the fit, indicating

the limitations inherent with EAM formalism. The fitted structures and energies are

shown in Table 4.9 while the fitted parameters are tabulated in the Table 4.8.

63

Table 4.8: Fitted parameters using cutoff radius = 3· max[re]. The knots for the cubic spline of the embedding energy F are given in (ρ, Fx) with unit (A˚−3,eV). The cross pair interaction between species A and B is defined as φAB ≡ (ZBφA/ZA+ZAφB/ZB)/2. Fitting structures are given in Table 4.9

C H Al Co Ni ρe (A˚−3) 0.932 1.639 0.026 0.09 0.126 β 5.473 2.798 7.182 5.945 5.447 re (A˚) 1.24 0.74 2.7 2.28 2.15 D (eV) 0.116 0.225 0.034 0.037 0.039 α (A˚−1) 2.786 3.156 1.740 3.636 3.198 r0 (A˚) 1.478 1.076 2.938 2.730 2.731 Z/ZC 1 0.532 0.989 1.115 1.258 Z/ZH 1 0.857 1.140 0.850 Z/ZAl 1 0.370 0.498 Z/ZCo 1 0.970 knot1 0,0 0,0 0,0 0,0 0,0 knot2 0.353,-3.860 1.037,-3.475 0.079,-2.816 0.217,-3.250 0.252,-3.932 knot3 0.519,-5.429 1.527,-3.974 0.108,-3.056 0.358,-5.176 0.327,-4.425 knot4 0.746,-6.254 1.797,-3.797 0.507,-4.214 0.774,-5.376 0.574,-4.849 knot5 1.033,-7.443 1.899,-3.744 0.893,-3.618 1.439,-5.139 1.032,-5.194 knot6 1.824,-7.988 2.200,-3.674 1.408,-3.681 1.635,-4.834 1.354,-5.065 knot7 2.200,-7.264 2.200,-2.911 2.200,-3.577 2.200,-4.130

64

Table 4.9: Fitted energies calculated using EAM parameters in Table 4.8. Methane up represents methane with one H below C and three H above C. Methane down is inverse of methane up. The unit for energy is eV/atom except for the adsorption energy which is in eV/molecule.

structure EEAM EV ASP ΔE Molecule energy (eV/atom): methane -4.805 -4.805 0.000 ethane syntactic -5.078 -5.061 -0.017 propane -5.189 -5.183 -0.006 butane -5.252 -5.252 0.000 pentane -5.293 -5.297 0.004 hexane -5.322 -5.328 0.007 benzene -6.294 -6.334 0.040 Substrate energy (eV/atom): dB1 slab4A -4.132 -4.096 -0.036 dH1 slab4A -4.141 -4.155 0.014 dH2 slab4A -4.135 -4.143 0.008 Molecule and substrate energy (eV/atom): methane on dB1 slab4A -4.173 -4.139 -0.034 methane up on dH1 slab4A -4.204 -4.219 0.015 methane down on dH1 slab4A -4.205 -4.219 0.014 ethane syntactic on dH2 slab4A -4.207 -4.214 0.006 propane on dH2 slab4A -4.242 -4.249 0.007 butane on dH2 slab4A -4.276 -4.283 0.008 pentane on dH2 slab4A -4.306 -4.315 0.009 hexane on dH2 slab4A -4.337 -4.346 0.009 benzene on dH2 slab4A -4.381 -4.385 0.004 Adsorption energy (eV/molecule): methane on dB1 slab4A -0.225 -0.238 0.013 methane up on dH1 slab4A -0.238 -0.243 0.005 methane down on dH1 slab4A -0.304 -0.257 -0.047 ethane syntactic on dH2 slab4A -0.348 -0.253 -0.095 propane on dH2 slab4A -0.458 -0.328 -0.131 butane on dH2 slab4A -0.439 -0.439 0.000 pentane on dH2 slab4A -0.465 -0.527 0.062 hexane on dH2 slab4A -0.520 -0.589 0.069 benzene on dH2 slab4A -0.790 -0.791 0.001

65

Chapter 5

Hydrocarbon adsorptions on d-AlNiCo

The low friction properties of quasicrystals in ambient conditions coupled with their

high hardness and oxidation resistance led to the development of applications of

quasicrystal coatings, for instance on machine parts, cutting blades, and non-stick

frying pans [5]. In machine parts, hydrocarbons are commonly used as a lubricant.

Superlubricity is the name given to the phenomenon in which two parallel single

crystal surfaces slide over each other with vanishingly small friction because their

structures are incommensurate. This phenomenon was proposed in the early 1990’s

[12] and experiment evidence for this effect has been seen in studies of mica sliding

on mica [123], W(110) on Si(100) [124], Ni(100) on Ni(100) [125], and tungsten on

graphite [13]. This effect is also expected in quasicrystals due to their aperiodic

structures at all length scales. Indeed, quasicrystal surfaces were observed to have

low friction not long after they were first discovered [126], but pinning down the exact

origin of the low friction has been elusive.

Recent experiments in ultra high vacuum (UHV) have demonstrated a frictional

dependence on aperiodicity for decagonal Al-Ni-Co quasicrystal (d-AlNiCo) against

thiol-passivated titanium-nitride tip [11]. In coating applications, it is expected that

even if superlubricity exists between moving parts, some additional lubricant would

still be needed to counter the macroscopic frictions due to grain boundaries, asperities,

and other defects in the surfaces of the moving parts. Some of the requirements of

the lubricant in such a situation are that it must wet the surface and that it must

not remove or reduce the superlubricity. Therefore it is desirable to have a good

understanding of how gases, hydrocarbons in particular, interact with quasicrystal

66

surfaces.

Very little is currently known about the interaction of hydrocarbons, their struc-

tures and growth on alloy or quasicrystalline surfaces. Some earlier experiments using

Fourier transform infrared spectroscopy (FTIR) and low energy electron diffraction

(LEED) suggest that on the 5-fold surface of Al-Pd-Mn, carbon monoxide (CO) does

not adsorb at 100 K, benzene adsorb at 100 K with possibly commensurate or dis-

ordered structure (LEED pattern unchanged) [127]. The same experiments on the

10-fold surface of d-AlNiCo show that CO bonds to the Ni sites at > 132 K, no struc-

ture reported, while there is no experiment available for benzene on Al-Ni-Co [127].

Later, scaning tunneling microscopy (STM) experiments on benzene adsorption on

Al-Pd-Mn show that the adsorbed benzene has a disordered structure [128]. In this

chapter, we report the simulation results of small hydrocarbons adsorb on the 10-fold

surface of decagonal Al73Ni10Co17 [15, 16], namely for methane, propane, hexane,

octane, and benzene.

5.1 Model

The simulations are performed within the framework of grand canonical ensemble

using Monte Carlo (GCMC) method as previously described in Section 2.5. The sim-

ulation cell is tetragonal. We take a square section of the surface, A, of side 5.12 nm,

to be the (x, y) part of the unit cell in the simulation, for which we assume periodic

boundary conditions along the basal directions. The coordinates of substrate atoms

are taken from Ref. [100]. The interaction potentials are modeled as the following.

The intermolecular interactions (adsorbate-adsorbate) are calculated as a sum of

pair interactions between atoms. For methane-methane [129, 130] Buckingham-type

potentials are used:

V (r) = Ae−Br − C/r6 (5.1)

67

Buckingham potentials also used to parametrize the benzene-benzene interactions

[131, 132]. For linear alkane-alkane, Morse-type potentials are used:

V (r) = −A(1− (1− e−B(r−C))2) (5.2)

The parameters for these potentials are summarized in Table 5.1. EAM potentials

generated in Chapter 4 are used for the rest of the interactions, namely the in-

tramolecular, adsorbate-substrate (C-Al, C-Co, C-Ni, H-Al, H-Co, H-Ni), and be-

tween substrate atoms (Al-Al, Al-Co, Al-Ni, Co-Ni). As previously mentioned in

Section 4.4, the ab initio calculations show that alkanes and benzene do not dissoci-

ate on dB1, dH1, and dH2 (these are decagonal Al-Ni-Co approximants). In these

systems, the surface of the substrate does not undergo any considerable relaxation

upon the adsorption of the molecules. Therefore, as a first approximation, in our

GCMC simulations, the substrate and the molecules are considered as rigid. How-

ever, molecules are allowed to explore all rotational degrees of freedom to achieve the

equilibrium configurations.

5.2 Adsorption potentials

Figure 5.1 displays the minima of the adsorption potential for methane (a), propane

(b), hexane (c), octane (d), and benzene (e), generated by minimizing the adsorption

potential of a molecule on d-AlNiCo with respect to z (Equation 3.1) and all rota-

tional degrees of freedom. The average adsorption energies are 221 (methane), 374

(propane), 620 (hexane), 794 (octane), and 931 (benzene), given in meV/molecule.

The figure shows the distribution of binding sites (dark spots) for the molecule.

Methane, propane, and benzene are small enough to follow the local atomic envi-

ronments of the substrate, whereas hexane and octane show considerable smearing

due to their large size. The location of dark spots in methane is similar to that

68

Table 5.1: Parameter values for the adsorbate-adsorbate interactions used for hydrocar- bon adsorption on a decagonal Al-Ni-Co. Intermolecular energies are calculated as a sum of pair interactions. For methane-methane, the C-H is taken as the geometrical mean for parameter A and as the arithmetic mean for parameters B and C.

A B C ref. (eV) (A˚−1) (A˚6)

methane C-C 82.132 2.693 449.53 [130] V (r) = Ae−Br − C/r6 C-H 66.217 2.892 167.51

H-H 53.381 3.105 62.42 [129] benzene C-C 11527.700 3.909 524 [131, 132] V (r) = Ae−Br − C/r6 C-H 348.518 3.703 75 [131, 132]

H-H 127.447 3.746 39 [131, 132] (meV) (A˚−1) (A˚)

alkane C-C 6.984 1.2655 4.1844 [133] V (r) = −A(1− (1− x)2) C-H 23.921 2.2744 2.544 [133] x = e−B(r−C) H-H 0.002 1.255 6.1543 [133]

[129] Tsuzuki S, et al 1993 J. Mol. Struct. 280 273 [130] Tsuzuki S, et al 1994 J. Phys. Chem. 98 1830 [131] Califano S, et al 1979 Chem. Phys. Lett. 64 491 [132] Chelli R, et al 2001 Phys. Chem. Chem. Phys. 3 2803 [133] Jalkanen J-P, et al 2002 J. Chem. Phys. 116 1303

69

of propane. However, it is interesting to see that the dark spots in methane and

propane become the bright ones in benzene. To study more, the topview of the

substrate atoms is depicted in panel (f) of Figure 5.1. The legend for the atoms is:

Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red),

Ni-toplayer (green), and Ni-otherlayers (blue). In methane and propane, dark spots

occur when the center of the molecule is located on top of black-Al (5 gray-Al on

toplayer forming a pentagon). In benzene, this location gives a bright spot (less bind-

ing). Except the binding site at the center of the figure, strong binding sites (dark

spots) in benzene occur when the center of benzene is located on top of gray-Al (on

the top layer, the pentagon consists of 2 Al and 3 Ni, or 3 Al and 2 Ni).

5.3 Molecule orientations

First we study the orientations of a molecule as it is adsorbed on d-AlNiCo. During

the calculation of minima of the adsorption potential (Section 5.2, the orientation of

the molecule is recorded when a minimum is achieved. It is found that for methane,

the rotational ground state is degenerate indicating its spherical nature. Ab initio

calculations of methane dimers indicate that the interaction energy within the dimer

depends on the relative orientations of the two [129, 130]. Therefore, the degeneracy

observed is due to the limitation of the EAM model. For linear alkanes, we can

define θ as the angle between the substrate’s xy-plane and the main axis of the

alkanes. Again, θ is recorded when an adsorption minimum is achieved. We find that

θ decreases with increasing alkane chain, e.g. propane (θ = 10◦), hexane (θ = 5◦),

and octane (θ ∼ 0◦). For benzene, θ would be the angle between the molecule’s plane and substrate’s xy-plane, and it is θ ∼ 0◦. Therefore, we conclude that alkanes and benzene prefer most contact with d-AlNiCo.

70

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-180

-200

-220

-240

-260

-300

1

-280

-221

-300

-350

-400

-450

-500

-550

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

1 -374

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-550

-600

-650

-700

-800

1

-750

-620

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-750

-800

-850

-900

1

-950

-700

-794

X (nm) -2.5 2.50 0.5 1 1.5 2-2 -1.5 -1 -0.5

-2.5

2.5

0

0.5

1.5

2

-2

-1.5

-1

-0.5

Y (n

m )

(meV)

-800

-900

-1100

1

-1000

-931

a) b)

c) d)

e) f)

Figure 5.1: (color online). (a-e) Adsorption potential map, calculated by minimizing the adsorption potential of one molecule on a decagonal Al-Ni-Co along z direction and all rotational degrees of freedom at every coordinates (x, y). Red numbers represent the average value of the adsorption energies. (f) Top view of the decagonal Al-Ni-Co substrate 51.2x51.2 A˚2: Al-toplayer (gray), Al-otherlayers (black), Co-toplayer (yellow), Co-otherlayers (red), Ni-toplayer (green), and Ni-otherlayers (blue).

71

5.4 Adsorption isotherms

To use quasicrystals as low-friction coatings in conjunction with oil lubricants, the

lubricants must be able to spread well on the quasicrystals. Lubricants consist mostly

of alkanes. The wetting of d-AlNiCo by alkanes and benzene is demonstrated in

Figure 5.2. In the figure, the number of molecules adsorbed on the substrate per area

is plotted as a function of pressure at various temperatures. The results for methane,

propane, hexane, octane, and benzene are shown. The simulated temperatures are:

for methane T = 68, 85, 136, 185 K, for propane T = 80, 127, 245, 365 K, for hexane

T = 134, 170, 267, 450 K, for octane T = 162, 210, 324, 450, 565 K, for benzene T =

209, 270, 418, 555 K. Note that T = 450 K represents a typical temperature of the

inner wall of cylinder in regular car engines [134]. In the isotherms, even though

the step corresponding to the adsorption of the first layer is well observed, steps

corresponding to the second and further layers are not evident for methane, propane,

and benzene, and barely visible for hexane and octane. The second layer condensation

occurs near the bulk condensation and extends a short range of pressure, nevertheless

multilayer adsorptions can be seen at higher pressures (point d) as shown in the inset

of each panel in which the distribution of adsorbed molecules is plotted against the

z direction.

5.5 Density profiles

Methane on d-AlNiCo. Figure 5.3 shows the density plots for methane at two

different coverages, corresponding to points ”a” and ”c” on the 68 K isotherm in

Figure 5.2.a. At submonolayer regime, methanes occupy the strong binding sites

on the surface. At monolayer coverage, the ordering is 5-fold commensurate with

the substrate, also indicated by the Fourier transform (panel c), and there is no

transition to 6-fold. The evidence that there is no such transition is shown more

72

10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

14

16

18

20

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

a

b c

d

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5 Z (nm)

Xe CH4

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a

b c

d

ρ N (m

ol ec

ul es

/n m

2 )

P (atm)

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5 Z (nm)

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a b c

d

Z (nm)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

ρ Z (a

rb .)

0 0.5 1 1.5 2 2.5

10-20 10-16 10-12 10-8 10-4 1 0

2

4

6

8

10

12

a b c

d

Z (nm)

ρ Z (a

rb .)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

0 0.5 1 1.5 2 2.5

10-25 10-20 10-15 10-10 10-5 1 0

2

4

6

8

10

12

14

16

18

20

a

b c

d

0 1 2 3 Z (nm)

P (atm)

ρ N (m

ol ec

ul es

/n m

2 )

ρ Z (a

rb .)

c) d)

e)

a) b)

Figure 5.2: (color online). Isothermal adsorption densities of hydrocarbons on a decago- nal Al-Ni-Co: (a) methane (from left to right T = 68, 85, 136, 185 K), (b) propane (T = 80, 127, 245, 365 K), (c) hexane (T = 134, 170, 267, 450 K), (d) octane (T = 162, 210, 324, 450, 565 K), and (e) benzene (T = 209, 270, 418, 555 K). The inset in each figure is the density along z direction at pressure corresponding to point d. Xenon (red) is plotted in panel (a) for comparison.

73

10 -15

10 -10

10 -5 1

0

5

10

15

ρN (m olecules/nm

2)

P (atm)

0

0.2

0.4

0.6

0.8

1

ρ 5 -6

a b

c d

Figure 5.3: (a) and (b) Calculated density of methane adsorbed on a decagonal Al-Ni-Co at pressures corresponding to points ”a” and ”c” of the 68 K isotherm shown in Figure 5.2.a, respectively. (c) Fourier transform of the density plot shown in (b), consistent with 5-fold ordering of the methane near monolayer completion. (d) Order parameter (left axis, as calculated in Equation 3.14) as a function of pressure for the 68 K isotherm (right axis), indicating no sharp transition to 6-fold ordering.

clearly in panel d, where the order parameter (left axis, as calculated in Equation

3.14) does not present any sharp transition as seen in the case of xenon adsorption on

the same substrate (Section 3.2.3). The lack of 5-fold to 6-fold transition is consistent

with our proposed rule based on rare gases (Section 3.2.6), that the size of the rare

gas must be at least as large as Xe on this quasicrystalline surface for the transition to

occur [135]. Note that the size of methane relative to xenon (ratio of Lennard-Jones

σ parameters) is 0.8 [69].

Propane, Hexane, and Octane on d-AlNiCo. At submonolayer coverages,

propane, hexane, and octane do not show any clear evidence of binding to specific sites

of the surface even though molecules tend to bind in the center of the surface with

74

the most attractive site. Figure 5.4 shows the density profiles for these hydrocarbons

at pressure corresponding to the first layer completion at various temperatures. The

simulated temperatures have been indicated in Section 5.4. At monolayer coverage,

propane adsorbs in a disordered fashion at all temperatures, whereas hexane and

octane tend to form close-packed structures as suggested by domains with stripe

feature. As comparisons, the crystal structures of solid propane at 30 K is monoclinic

(space group #11, P21/m) [136], hexane is triclinic (space group #2, P1¯) at 90 K

[136] and at 158 K [137], octane is triclinic (space group #2, P1¯) at 90 K [136]

and at 213 K [138]. The 2-dimensional structures of these hydrocarbons are close-

packed structures as charactereized by stripe structures with 1 and 2 molecules per

unit cell for even- and odd-alkanes, respectively [136]. In general, even-alkanes form

triclinic (6 ≤ NC ≤ 26), monoclinic (28 ≤ NC ≤ 36), or orthorombic (38 ≤ NC) with decreasing packing density [136, 137, 138, 139], where NC being the number of

carbon atoms. Whereas for odd-alkanes form triclinic (7 ≤ NC ≤ 9) or orthorombic (11 ≤ NC) [136, 139, 140] also with decreasing packing density for longer chain. Note that the even-alkanes have higher packing density than odd-alkanes.

Benzene on d-AlNiCo. As in the case of methane, at submonolayer coverage,

benzene preferentially adsorbs at sites offering the strongest binding at all simulated

temperatures. At pressure near to first layer completion, the density profiles show a

temperature-dependence as plotted in Figure 5.5 for 209 K, 270 K, and 418 K. The

structures are more clearly charaterized by the plotting the geometrical center of

density as shown in the middle row of the figure. At T = 209 K, pentagonal ordering

is observed. As the temperature is increased, a mixture of 5-fold and 6-fold ordering

is seen, e.g. at T = 270 K. At higher temperature, T = 418 K, 6-fold structure

dominates the ordering of the monolayer as confirmed by the Fourier analysis of the

density showing hexagonal spots characteristic of triangular lattice (bottom row, last

75

267 K170 K 450 K

324 K210 K 450 K

hexane

octane

80 K 127 K 245 Kpropane

Figure 5.4: Calculated density of propane, hexane, and octane adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion. Propane forms a disordered structures, whereas hexane and octane tend to form close-packed structures indicated by stripe features with increasing order for longer chain.

76

column in Figure 5.5). The crystal structure of bulk benzene has been determined

at 4.2 K and 270.15 K to be orthorombic (space group #61, Pbca) with 4 molecules

per unit cell [141].

The evolution of the density profile over temperature from being 5-fold to 6-fold

can be studied more clearly by plotting the order parameter, ρ5−6 = N5/(N5 + N6)

(Nn denotes the number of molecules having n nearest neighbors as defined in Section

3.1.6) as a function of T as shown in Figure 5.6 (left axis). The plot is taken at a

constant pressure of 0.01 atm corresponding to point c in Figure 5.2 panel e. The

adsorption isobar representing the number of adsorbed molecules N as a function of

T is plotted in right axis of Figure 5.6. Three dashed lines corresponding to T =209,

270, and 418 K, whose adsorption isotherms are plotted earlier in Figure 5.2 panel e,

are added for guidance in comparing the density profile to the order parameter value

at each of these temperatures. We observe the following trends:

• (region 1) T < 260 K, 0.5 < ρ5−6 → 5-fold ordering dominates, N = 44.

• (region 2) 260 ≤ T ≤ 280 K, 0.3 ≤ ρ5−6 ≤ 0.5 → 5-fold becomes mostly 6-fold, N = 44.

• (region 3) 280 ≤ T ≤ 370 K, ρ5−6 = 0.3 → 6-fold ordering dominates, N = 44.

• (region 4) 370 ≤ T ≤ 390 K, 0.1 ≤ ρ5−6 ≤ 0.3→ transition to 6-fold ordering, 44 ≤ N ≤ 46.

• (region 5) T > 390 K, ρ5−6 increases from 0.1 → 6-fold ordering weakens, N decreases from 46.

The highest 6-fold ordering occurs at T = 390 K which is mediated by a gain of 2

additional molecules adsorbed on the substrate. Beyong this temperature, thermal

77

209 K 270 K 418 K

Figure 5.5: (Top row) Calculated density of benzene adsorbed on a decagonal Al-Ni-Co at pressure near to first layer completion for 209 K, 270 K, and 418 K. (middle row) Density profile of the geometrical center of density shown in the top row. (bottom row) Fourier transform of the density plot shown in the middle row, showing 5-fold ordering at 209 K, mixture of 5-fold and 6-fold structures at 270 K, and mostly 6-fold features at 418 K.

78

200 250 300 350 400 450 500

0.1

0.2

0.3

0.4

0.5

0.6

T (K)

ρ 5 -6

38

40

42

44

46 N

m olecules

Figure 5.6: (left axis) Order parameter ρ5−6 = N5/(N5+N6) (Nn denotes the number of molecules having n nearest neighbors) as a function of temperature at 0.01 atm of pressure. (right axis) Adsorption isobar showing the number of adsorbed molecules as a function of temperature. Verticel dashed lines correspond to T = 209, 270, and 418 K whose density profiles are plotted in Figure 5.5.

energy causes reduction of N resulting in less ordered 6-fold structure as T increases.

Nevertheless, the 6-fold ordering are still well resolved up to T = 418 K as shown

before in Figure 5.5.

5.6 Summary

In this chapter, we have presented our studies on methane, propane, hexane, octane,

and benzene adsorption on decagonal Al-Ni-Co quasicrystal. All of these hydro-

carbons form very well defined step corresponding to the first layer condensation.

Eventhough multilayer formation is evident from the density distribution along ver-

tical dimension, the step corresponding to the condensation of the second and further

layers can not be observed clearly in the isotherm due narrow range of pressure for

these layers to form before bulk condensation occurs. Monolayer of methane has

79

been determined to be pentagonal commensurate with the substrate for 68 K ≤ T ≤ 136 K. Monolayer of propane shows disordered structures for 80 K ≤ T ≤ 245 K. Monolayers of hexane (134 K≤ T ≤ 450 K) and octane (162 K ≤ T ≤ 450 K) form 2- dimensional close-packed structures characterized by stripe ordering consistent with

their bulk crystal structures. Benzene monolayer is pentagonal at T ≤ 260 K, which transforms into 6-fold structure at T ≥ 280 K with the highest 6-fold ordering occurs at T = 390 K. Beyong 390 K, thermal energy causes fewer adsorbed benzene and the

6-fold ordering starts to deterioates. Nevertheless, the 6-fold ordering are still well

observed up to T = 418 K.

80

Chapter 6

Embedded-atom potentials for selected

pure elements in the periodic table

Embedded-atom method (EAM) formalism provides only the skeleton of calculat-

ing the total electronic energy for a given charge density. It does not specify the

parametrization of the three functionals: atomic charge density, embedding en-

ergy, and pair interaction. Many different parametrizations have been introduced

[54, 55, 58, 59, 59, 65, 142, 143, 144, 145]. This arises inconveniences when one needs

to use different potentials for different elements. Steps toward universal EAM poten-

tials are needed. Recently, a consistent set of atomic charge density for all elements

in the periodic table suitable for EAM potentials has been developed [64]. The set

is calculated by spherically averaging the atomic charge density of the solution of

Hartree-Fock (HF) equations (Equation 2.19) of an isolated atom. The exchange

term in the HF equation contains an atom-specific parameter which is adjusted to

reproduce the experimental first ionization energy, hence are semiempirical. In this

chapter we use these charge densities to generate a consistent set of EAM potentials

for pure elements in the periodic table.

The parametrizations are given in Section 2.3. For each element, the potential is

fitted to the ab initio cohesive energies in body-centered cubic (BCC), face-centered

cubic (FCC), hexagonal close-packed (HEX), and diamond structure (DIA). In ad-

dition, energies in the ground state structure at different pressures are included in

the fitting procedure to ensure the mechanical stability of the potentials. Energies at

various pressures are achieved by expanding or compressing the equillibrium lattice

81

constant a by a factor from 0.90 to 1.16. Within an accuracy of 30 meV, all the

EAM potentials successfully predict the correct ground state structure with respect

to those that are not included in the fitting step, namely graphite (GRA), simple

cubic (SC), and simple hexagonal (SH) structures. The fitted parameters are sum-

marized in Tables 6.1-6.4. Tables 6.5 and 6.6 list the ground state structures of

the elements, range of charge density covered in the fit, and the equillibrium lattice

constants calculated by the ab initio method as well as by using the fitted EAM

parameters. Literature value of the lattice constants is also included as comparisons.

Overall, the EAM potentials predict the lattice constants with within 0.5 A˚ of the

literature value except for tellurium. Indeed, elements in the V-A, VI-A, and VII-A

colums of periodic table tend to form complex structures with large unit cells and

more EAM parameters might be needed to fit these structures better. Manganese is

excluded also because of its complex cubic structure with 58 atoms/cell. Noble gases

are excluded because they are already well described by simple pair potentials. Ele-

ments in the Actinide series are excluded because they do not have a stable structure

due to their radioactivity.

82

Table 6.1: Fitted parameters for the charge density and pair interaction functionals of the EAM potentials for pure elements, continued in Table 6.2. The fitting structures are given in Tables 6.5 and 6.6. The parameters for the embedding functionals are given in Tables 6.3 and 6.4.

Z Struct. ρe β re D α r0 (A˚−3) (A˚) (eV) (A˚−1) (A˚)

3 Li 0.038 4.813 2.670 0.010 3.015 2.422 4 Be 0.031 7.174 2.450 0.016 3.317 2.216 6 C 0.932 5.473 1.240 0.115 3.152 1.441 11 Na 0.021 5.552 3.080 0.011 5.800 3.500 12 Mg 0.001 10.030 3.890 0.012 2.807 3.110 13 Al 0.026 7.182 2.700 0.034 3.038 3.012 14 Si 0.072 7.171 2.250 0.043 4.115 1.601 19 K 0.008 6.412 3.920 0.015 2.926 4.088 20 Ca 0.002 9.719 4.280 0.020 2.940 3.646 21 Sc 0.058 6.053 2.700 0.021 2.264 3.210 22 Ti 0.287 4.333 1.940 0.022 2.282 2.850 23 V 0.394 4.025 1.780 0.023 2.501 2.615 24 Cr 0.297 4.538 1.680 0.024 2.519 2.494 26 Fe 0.192 5.214 2.000 0.026 5.747 2.412 27 Co 0.090 5.945 2.280 0.037 2.832 2.701 28 Ni 0.126 5.447 2.150 0.039 2.839 2.717 29 Cu 0.052 5.864 2.220 0.029 2.589 2.553 30 Zn 0.0005 14.630 4.800 0.030 2.655 2.560 32 Ge 0.042 7.419 2.440 0.045 2.659 2.001 37 Rb 0.006 6.788 4.210 0.037 3.599 4.694 38 Sr 0.002 9.614 4.450 0.038 2.926 3.299 39 Y 0.037 6.593 3.000 0.039 2.123 3.300 40 Zr 0.158 5.287 2.300 0.040 2.132 3.230 41 Nb 0.145 5.556 2.080 0.041 2.347 2.858 42 Mo 0.201 5.621 1.940 0.042 2.153 2.728 43 Tc 0.365 4.968 1.880 0.043 2.559 2.640 44 Ru 0.047 6.360 2.400 0.044 2.702 2.700 45 Rh 0.020 6.981 2.680 0.045 2.609 2.687 46 Pd 0.017 8.389 2.500 0.036 2.719 2.791 47 Ag 0.028 6.619 2.530 0.027 2.982 3.055 48 Cd 0.0005 13.340 4.500 0.048 2.284 3.170 52 Te 0.047 8.157 2.560 0.052 2.477 2.245 56 Ba 0.001 10.020 4.900 0.056 2.815 3.847 57 La 0.009 8.106 3.890 0.057 2.518 3.350

83

Table 6.2: continuation of Table 6.1.

58 Ce 0.013 7.575 3.682 0.058 2.488 2.949 59 Pr 0.016 7.342 3.646 0.059 2.502 2.970 60 Nd 0.015 7.361 3.632 0.060 2.543 3.548 63 Eu 0.016 7.306 3.542 0.033 2.596 3.300 64 Gd 0.015 7.724 3.470 0.064 2.549 3.240 65 Tb 0.017 7.323 3.464 0.065 2.527 3.300 66 Dy 0.017 7.303 3.420 0.066 2.512 3.240 67 Ho 0.018 7.323 3.392 0.067 2.532 3.180 68 Er 0.019 7.271 3.346 0.068 2.560 3.260 69 Tm 0.020 7.284 3.320 0.069 2.597 3.140 70 Yb 0.020 7.250 3.274 0.070 2.542 3.522 71 Lu 0.023 6.972 3.342 0.071 2.596 3.210 72 Hf 0.138 5.581 2.350 0.072 2.454 2.900 73 Ta 0.142 6.016 2.300 0.073 2.026 2.766 74 W 0.137 6.176 2.270 0.074 2.350 2.630 75 Re 0.242 3.686 2.040 0.075 3.055 2.760 76 Os 0.029 7.771 2.760 0.076 2.696 2.540 77 Ir 0.244 6.055 1.960 0.077 3.142 2.715 78 Pt 0.021 7.539 2.500 0.078 2.919 2.772 79 Au 0.024 7.553 2.470 0.079 2.917 2.885 81 Tl 0.004 8.850 3.500 0.081 2.576 3.300 82 Pb 0.002 8.368 2.930 0.082 2.935 3.242

84

Table 6.3: Fitted knots of cubic spline of the embedding functionals for the EAM po- tentials of pure elements, continued in Table 6.4. The first knot at (0,0) is assumed. The fitting structures are given in Tables 6.5 and 6.6.

knot2 knot3 knot4 knot5 knot6 knot7 (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV) (A˚−3,eV)

Li 0.048,-1.058 0.119,-1.52850 0.190,-1.54750 0.246,-1.607 0.291,-1.554 0.469,-1.469 Be 0.178,-3.079 0.228,-3.241 0.423,-3.509 0.731,-3.647 1.057,-3.588 1.670,-3.324 C 0.301,-4.967 0.344,-5.905 0.476,-6.986 0.861,-7.490 1.419,-7.264 2.389,-5.970 Na 0.014,-0.725 0.029,-0.9756 0.062,-1.059 0.082,-1.057 0.135,-1.059 0.181,-1.424 Mg 0.015,-0.939 0.033,-1.292 0.062,-1.414 0.096,-1.413 0.133,-1.356 0.242,-1.175 Al 0.078,-2.811 0.121,-3.106 0.495,-3.980 0.897,-3.879 1.381,-3.665 2.200,-3.074 Si 0.057,-3.662 0.087,-4.153 0.157,-4.476 0.286,-4.409 0.349,-4.204 0.561,-3.168 K 0.005,-0.613 0.009,-0.757 0.025,-0.844 0.057,-0.710 0.089,-0.551 0.149,-0.415 Ca 0.009,-1.202 0.021,-1.692 0.062,-1.814 0.094,-1.719 0.135,-1.578 0.201,-1.434 Sc 0.055,-3.363 0.080,-3.787 0.140,-4.092 0.226,-4.140 0.315,-4.009 0.481,-3.612 Ti 0.133,-4.307 0.188,-4.850 0.314,-5.315 0.444,-5.410 0.712,-5.033 1.075,-4.406 V 0.245,-4.014 0.333,-4.545 0.606,-5.153 0.858,-5.111 1.058,-4.850 1.520,-4.005 Cr 0.168,-3.162 0.182,-3.277 0.300,-3.808 0.487,-3.758 0.634,-3.324 0.935,-1.915 Fe 0.171,-4.007 0.247,-4.554 0.505,-5.071 0.834,-4.962 1.072,-4.734 1.445,-4.613 Co 0.146,-3.374 0.380,-4.894 0.821,-5.517 1.306,-5.509 1.683,-5.008 2.200,-3.952 Ni 0.223,-3.899 0.336,-4.373 0.566,-4.731 1.028,-5.280 1.355,-5.028 2.200,-4.172 Cu 0.091,-2.811 0.127,-3.100 0.238,-3.280 0.334,-3.215 0.455,-2.935 0.604,-2.374 Zn 1.012,-0.752 1.604,-0.880 2.794,-0.940 3.544,-0.919 4.805,-0.822 8.006,-0.406 Ge 0.056,-3.285 0.072,-3.429 0.115,-3.592 0.188,-3.553 0.241,-3.359 0.364,-2.645 Rb 0.004,-0.552 0.009,-0.707 0.020,-0.663 0.029,-0.594 0.036,-0.633 0.047,-0.977 Sr 0.005,-1.128 0.008,-1.304 0.046,-1.558 0.063,-1.490 0.088,-1.331 0.108,-1.205 Y 0.012,-3.002 0.021,-3.548 0.098,-4.152 0.167,-4.060 0.260,-3.712 0.338,-3.459 Zr 0.052,-4.542 0.082,-5.367 0.186,-6.136 0.299,-6.147 0.489,-5.725 0.685,-5.428 Nb 0.034,-6.744 0.054,-7.933 0.139,-9.012 0.240,-8.889 0.347,-8.192 0.468,-7.278 Mo 0.059,-4.588 0.088,-5.395 0.192,-6.024 0.271,-5.941 0.363,-5.304 0.542,-4.467 Tc 0.164,-5.372 0.217,-5.913 0.317,-6.476 0.482,-6.730 0.720,-6.230 1.108,-4.437 Ru 0.088,-6.348 0.130,-7.166 0.241,-7.687 0.311,-7.756 0.457,-7.052 0.638,-6.282 Rh 0.074,-4.522 0.095,-4.900 0.176,-5.540 0.279,-5.529 0.381,-4.971 0.495,-3.965 Pd 0.017,-2.667 0.025,-3.029 0.078,-3.490 0.128,-3.256 0.166,-2.905 0.232,-2.126 Ag 0.027,-1.749 0.035,-1.894 0.117,-2.349 0.178,-2.390 0.239,-2.422 0.294,-2.488 Cd 0.061,-0.301 0.110,-0.379 0.245,-0.409 0.364,-0.446 0.690,-0.333 0.921,-0.278 Te 0.006,-1.550 0.011,-1.922 0.029,-2.088 0.038,-2.159 0.069,-1.969 0.117,-1.615 Ba 0.004,-1.228 0.010,-1.721 0.020,-1.822 0.027,-1.813 0.050,-1.658 0.081,-1.393 La 0.019,-2.564 0.051,-3.977 0.121,-4.240 0.210,-4.086 0.322,-3.777 0.416,-3.536 Ce 0.040,-2.826 0.106,-4.210 0.207,-4.481 0.329,-4.503 0.526,-4.234 0.677,-3.908

85

Table 6.4: continuation of Table 6.3.

Pr 0.035,-2.470 0.088,-3.600 0.166,-3.856 0.246,-3.891 0.357,-3.821 0.554,-3.605 Nd 0.021,-2.403 0.062,-3.730 0.157,-3.834 0.308,-3.858 0.456,-3.972 0.554,-3.776 Eu 0.010,-1.188 0.021,-1.599 0.055,-1.790 0.125,-1.734 0.191,-1.516 0.275,-1.357 Gd 0.017,-3.615 0.043,-5.278 0.085,-5.539 0.129,-5.542 0.219,-5.281 0.330,-4.894 Tb 0.027,-2.642 0.067,-3.837 0.142,-4.061 0.202,-3.963 0.319,-3.614 0.409,-3.340 Dy 0.027,-2.672 0.068,-3.683 0.158,-4.026 0.199,-3.866 0.279,-3.728 0.404,-3.139 Ho 0.028,-2.672 0.069,-3.851 0.162,-4.020 0.199,-3.927 0.283,-3.689 0.406,-3.248 Er 0.028,-2.681 0.069,-3.729 0.162,-3.947 0.196,-3.823 0.276,-3.654 0.406,-3.278 Tm 0.030,-2.762 0.069,-3.659 0.168,-3.992 0.217,-3.801 0.289,-3.658 0.397,-3.268 Yb 0.012,-0.958 0.022,-1.224 0.056,-1.366 0.122,-1.191 0.188,-1.044 0.264,-0.965 Lu 0.040,-2.683 0.087,-3.659 0.155,-3.911 0.210,-3.900 0.340,-3.630 0.482,-3.254 Hf 0.036,-4.599 0.057,-5.285 0.209,-6.329 0.279,-6.266 0.399,-5.886 0.680,-4.554 Ta 0.066,-5.540 0.109,-6.815 0.210,-7.729 0.409,-7.719 0.624,-6.877 0.903,-6.333 W 0.111,-6.179 0.149,-6.886 0.256,-7.726 0.465,-7.681 0.689,-6.553 0.953,-4.666 Re 0.307,-4.531 0.441,-5.847 0.680,-7.039 0.803,-7.237 1.102,-6.969 1.636,-5.140 Os 0.106,-6.174 0.206,-7.540 0.326,-7.866 0.394,-7.945 0.570,-7.555 0.908,-5.803 Ir 0.089,-5.498 0.106,-6.040 0.228,-7.004 0.342,-7.142 0.486,-6.271 0.667,-5.735 Pt 0.025,-3.842 0.035,-4.280 0.094,-5.002 0.149,-4.728 0.188,-4.327 0.250,-3.495 Au 0.015,-1.932 0.023,-2.231 0.075,-2.492 0.108,-2.276 0.134,-2.038 0.179,-1.590 Tl 0.007,-1.336 0.013,-1.611 0.029,-1.661 0.062,-1.476 0.091,-1.268 0.133,-1.001 Pb 0.0004,-1.813 0.0006,-2.159 0.0010,-2.454 0.0016,-2.537 0.0046,-2.491 0.0125,-1.686

86

Table 6.5: (left part) Structures used to fit EAM potentials for pure elements, continued in Table 6.6. The EAM potentials are fitted to the ab initio energies in body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close-packed (HEX), diamond structure (DIA), and groundstate structure at various pressures obtained by expanding/compressing the equillibrium lattice constant a by a factor from 0.9 to 1.16 corresponding to a range of charge density from ρmax to ρmin. (right part) Lattice constansts calculated using the fitted parameters. The literature values aLIT are taken from [146].

Z Name Struct. ρmin → ρmax aLIT aV ASP aEAM (A˚−3) (A˚) (A˚) (A˚)

3 Li BCC 0.104 → 0.441 3.49 3.44 3.44 4 Be HEX 0.268 → 1.437 2.29 2.27 2.24 6 C DIA 0.329 → 2.038 3.57 3.57 3.57 11 Na BCC 0.027 → 0.163 4.23 4.19 4.19 12 Mg HEX 0.014 → 0.174 3.21 3.19 3.16 13 Al FCC 0.098 → 0.312 4.05 4.04 4.00 14 Si DIA 0.060 → 0.443 5.43 5.46 5.46 19 K BCC 0.013 → 0.089 5.23 5.29 5.29 20 Ca FCC 0.014 → 0.186 5.58 5.54 5.54 21 Sc HEX 0.064 → 0.441 3.31 3.30 3.34 22 Ti HEX 0.150 → 0.845 2.95 2.92 2.92 23 V BCC 0.263 → 1.320 3.02 2.98 2.98 24 Cr BCC 0.126 → 0.765 2.88 2.84 2.84 26 Fe BCC 0.213 → 1.221 2.87 2.83 2.83 27 Co HEX 0.361 → 0.974 2.51 2.47 2.45 28 Ni FCC 0.374 → 0.983 3.52 3.49 3.46 29 Cu FCC 0.113 → 0.519 3.61 3.64 3.64 30 Zn HEX 0.777 → 7.050 2.66 2.71 2.71 32 Ge DIA 0.056 → 0.306 5.66 5.79 5.79 37 Rb BCC 0.004 → 0.048 5.59 5.65 5.65 38 Sr FCC 0.009 → 0.097 6.08 6.05 6.05 39 Y HEX 0.036 → 0.282 3.65 3.64 3.64 40 Zr HEX 0.073 → 0.511 3.23 3.22 3.22 41 Nb BCC 0.050 → 0.390 3.30 3.31 3.31 42 Mo BCC 0.055 → 0.463 3.15 3.15 3.12 43 Tc HEX 0.147 → 0.993 2.74 2.76 2.76 44 Ru HEX 0.084 → 0.558 2.70 2.73 2.70 45 Rh FCC 0.072 → 0.469 3.80 3.84 3.84 46 Pd FCC 0.017 → 0.195 3.89 3.96 3.96 47 Ag FCC 0.034 → 0.256 4.09 4.16 4.16 48 Cd HEX 0.076 → 0.855 2.98 3.09 3.03 52 Te HEX 0.007 → 0.108 4.45 4.06 4.02

87

Table 6.6: continuation of Table 6.5.

56 Ba BCC 0.007 → 0.074 5.02 5.02 5.02 57 La HEX 0.042 → 0.331 3.75 3.74 3.74 58 Ce FCC 0.096 → 0.610 5.16 4.74 4.74 59 Pr HEX 0.076 → 0.502 3.67 3.45 3.45 60 Nd HEX 0.068 → 0.457 3.66 3.56 3.56 63 Eu BCC 0.029 → 0.232 4.61 4.46 4.41 64 Gd HEX 0.037 → 0.303 3.64 3.62 3.62 65 Tb HEX 0.049 → 0.362 3.60 3.62 3.62 66 Dy HEX 0.047 → 0.345 3.59 3.61 3.61 67 Ho HEX 0.048 → 0.356 3.58 3.59 3.63 68 Er HEX 0.047 → 0.352 3.56 3.58 3.61 69 Tm HEX 0.049 → 0.367 3.54 3.55 3.55 70 Yb FCC 0.023 → 0.202 5.49 5.29 5.29 71 Lu HEX 0.061 → 0.425 3.51 3.48 3.48 72 Hf HEX 0.073 → 0.530 3.20 3.20 3.20 73 Ta BCC 0.091 → 0.692 3.31 3.32 3.32 74 W BCC 0.105 → 0.794 3.16 3.19 3.19 75 Re HEX 0.378 → 1.509 2.76 2.78 2.78 76 Os HEX 0.114 → 0.861 2.74 2.76 2.73 77 Ir FCC 0.068 → 0.629 3.84 3.88 3.80 78 Pt FCC 0.025 → 0.236 3.92 3.98 3.98 79 Au FCC 0.016 → 0.167 4.08 4.17 4.17 81 Tl HEX 0.011 → 0.112 3.46 3.54 3.54 82 Pb FCC 0.001 → 0.011 4.95 5.02 5.02

88

Chapter 7

Effects of Mo on the thermodynamics of

Fe:Mo:C nanocatalyst for single-walled

carbon nanotube growth

Among the established methods for single-walled carbon nanotubes (SWCNTs) syn-

thesis [147, 148, 149], catalytic chemical vapor decomposition (CCVD) technique is

preferred for growing nanotubes on a substrate at a target position due to its rel-

atively low synthesis temperature. Temperature as low as ∼ 450 oC was reported by using hydrocarbon feedstock with exothermic catalytic decomposition reaction

[150, 151]. Critical factors for the efficient growth via CCVD are the compositions

of the interacting species, the preparation of the catalysts, and the synthesis condi-

tions. Efficient catalysts must have long active lifetimes (with respect to feedstock

dissociation and nanotube growth), high selectivity and be less prone to contamina-

tion. Common factors that lead to reduction in catalytic activity are deactivation

(e.g. due to coating with carbon or nucleation of inactive phases) [152, 153, 154] and

thermal sintering (e.g. caused by highly exothermic reactions on the clusters surface

[150, 151, 155] with insufficient heat [156]).

Metal alloy catalysts, such as Fe:Co, Co:Mo, and Fe:Mo, improve the growth of

CNTs [157, 158, 159, 160, 162, 163], because the presence of more than one metal

species can significantly enhance the activity of a catalyst [159, 164], and can prevent

catalyst particle aggregation [163, 164]. In the case of Fe:Mo nanoparticles supported

on Al2O3 substrates, the enhanced catalyst activity has been shown to be larger than

the linear combination of the individual Fe/Al2O3 and Mo/Al2O3 activities [159, 160].

89

This is explained in terms of substantial intermetallic interaction between Mo, Fe and

C [159, 165, 166]. The addition of Mo in mechanical alloying of powder Fe and C

mixtures promotes solid state reactions even at low Mo concentrations by forming

ternary phases, such as the (Fe,Mo)23C6 and Fe2(MoO4)3 type carbides [167]. It has

been found that low Mo concentration in Fe:Mo is favored for growing SWCNTs

(on Al2O3 substrates) since the presence, after activation, of the phase Fe2(MoO4)3

can lead to the formation of small metallic clusters [168]. Considering the vapor-

liquid-solid model (VLS), which is the most probable mechanism for CNT growth

[159, 169]. The metallic nanoparticles are very efficient catalysts when they are in

the liquid or viscous states, probably due to considerable carbon bulk-diffusion in

this phase (compared to surface or sub-surface diffusion). Generally, unless stable

intermetallic compounds form, alloying metals reduce the melting point below those

of the constituents [165, 166]. Hence, to improve the yield and quality of nanotubes,

one can tailor the composition of the catalyst particle to move its liquidus line below

the synthesis temperature [159]. However, identifying the perfect alloy composition

is non trivial. In fact, the presence of more than two metallic species allows for the

possibility of different carbon pollution mechanisms by thermodynamic promotion of

ternary carbides. In this Chapter, we study the phase diagram of Fe:Mo:C system

and the possible roles of Mo in the catalytic properties of Fe:Mo.

7.1 Size-pressure approximation

Determining the thermodynamic stability of different phases in nanoparticles of dif-

ferent sizes with ab initio calculations is computationally expensive. We develop

a simple model, called the ”size-pressure approximation”, which allows one to es-

timate the phase diagram at the nanoscale starting from bulk calculations under

pressure [152]. Surface curvature and superficial dangling bonds on nanoparticles are

90

responsible for internal stress fields which modify the atomic bond lengths. As a

first approximation, where all surface effects that are not included in the curvature

are neglected, we can map the particle radius R to the pressure P by equating the

deviation from the bulk value of the average bond lengths due to surface curvature

(in the case of particle) and pressure (in the case of bulk). For spherical clusters,

the phenomenon can be modeled with the Young-Laplace equation P = 2γ/R where

the proportionality constant γ can be calculated with ab initio methods. In our

study, since the concentration of Mo in Fe:Mo is small, we use the ”size-pressure

approximation” of Fe particle.

Figure 7.1 shows the implementation of the ”size-pressure approximation” for Fe

nanoparticles. On the left hand side we show the ab initio calculations of the deviation

of the average bond length inside the cluster Δdnn ≡ d0nn − dnn (d0nn = 0.2455 nm is our bulk bond length), for body-centered cubic (BCC) particles of size N = 59, 113,

137, 169, 307 and ∞ (bulk) as a function of the inverse radius (1/R). The particles were created by intersecting a BCC lattice with different size spheres. The particle

radius is defined as 1/R ≡ 1/Nscp ∑

i 1/Ri where the sum is taken over the atoms

belonging to the surface convex polytope (Nscp vertices) and Ri are the distances to

the geometric center of the cluster. The left straight line is a linear interpolation

between 1/R and Δdnn calculated with the constraint of passing through 1/R = 0

and Δdnn = 0 (N = ∞, bulk). The right hand side shows the ab initio value of dnn in bulk BCC Fe as a function of hydrostatic pressure, P . The straight line is a linear

interpolation between P and Δdnn calculated with the constraint of passing through

P = 0 and Δdnn = 0 (bulk lattice). By following the colored dashed paths indicated

by the arrows we can map the analysis of nanoparticles stability as a function of R

onto bulk stability as a function of P , and obtain the relation between the radius of

particle/nanotube and the effective pressure P ·R = 2.46 GPa · nm. It is important

91

Figure 7.1: (color online). Size-pressure approximation for Fe nanoparticles obtained by equating the deviation of average bond length from the bulk value due to curvature 1/R (in the case of particle) and due to pressure P (in the case of bulk).

to mention that our γ = 1.23 J/m2 is not a real surface tension but an ab initio fitting

parameter describing size-induced stress in nanoparticles. With this γ, we deduce the

Fe-Mo-C phase diagram of nanoparticles of radius R from ab initio calculations of

the bulk material under pressure P .

92

7.2 Fe-Mo-C phase diagram under pressure

Simulations are performed with VASP as described in Section 2.2. The hydrostatic

pressure estimated from the pressure-size model is implemented as Pulay stress [170].

Ternary phase diagrams are calculated using BCC-Mo, BCC-Fe and SWCNTs as

references (pure-Fe phase is taken to be BCC because our simulations are aimed at

the low temperature regime of catalytic growth). The reference SWCNTs have the

same diameter of the particle to minimize the curvature-strain energy. In fact, CVD

experiments of SWCNT growth from small (∼ 0.6-2.1 nm) particles indicate that the diameter of the nanotube is similar to the diameter of the catalyst particle from

which it grows. In some experiments where the growth mechanism is thought to be

root-growth, the ratio of the catalyst particle diameter to SWCNT diameter is ∼ 1.0, whereas in experiments involving pre-made floating catalyst particles this ratio is ∼ 1.6 [171]. Formation energies are calculated with respect to decomposition into the

nearby stable phases, depending the position in the ternary phase diagram.

Binary and Ternary phases are included if they are stable in the temperature

range used in CVD growth of SWCNTs or if they have been reported experimentally

during or after the growth [165, 166, 172]. Thus, we include the binaries Mo2C,

Fe2Mo and Fe3C. In addition, since our Fe-rich Fe:Mo experiments were performed

with compositions close to Fe4Mo [159], we include a random phase Fe4Mo generated

with the special quasi-random structure formalism (SQS). Bulk ternary carbides,

which have been widely investigated due to their importance in alloys and steel, can

be considered as derivatives of binary structures with extra C atoms in the interstices

of the basic metal alloy structures. Three possible ternary phases have been reported

for bulk Fe-Mo-C [173] and they are referred as τ1 (M6C), τ2 (M3C) and τ3 (M23C6)

(M is the metal species). For simplicity, we follow the same nomenclature. τ1 is

the wellknown M6C phase, which has been observed experimentally as Fe4Mo2C and

93

Fe3Mo3C structures (η carbides) [174, 175]. Both of these structures are FCC but

have different lattice spacings. Our calculations show that the most stable variant

τ1 is Fe4Mo2C, and we denote it as τ1 henceforth. τ2 is the Fe2MoC phase, which

has an orthorhombic symmetry distinct from that of Fe3C [176, 177]. We consider

Fe21Mo2C6 as the third τ3 FCC phase [178]. We use the Cr23C6 as the prototype

structure [179] where Fe and Mo substitute for Cr. Although M23C6 type phases do

not appear in the stable C-Fe or C-Mo systems, they have been reported in ternary C-

Fe-Mo systems and also appear as transitional products in solid state reactions [173].

Time-temperature precipitation diagrams of low-C steels have identified τ2, τ3 and τ1

as low-temperature, metastable and stable carbides, respectively [180]. Furthermore,

τ2 carbides precipitate quickly due to carbon-diffusion controlled reaction while τ3

carbides precipitate due to substitutional-diffusion controlled reactions. The latter

phenomenon, requiring high temperature, longer times and producing metastable

phases is not expected to enhance the catalytic deactivation of the nanoparticle. In

summary, as long as the presence of carbon does not lead to excessive formation of

Fe3C and τ2), the catalyst should remain active for SWCNT growth.

Figure 7.2 shows the phase diagram at zero temperature of nanoparticles of radii

R ∼ ∞, 1.23, 0.62, 0.41 nm, calculated at P = 0, 2, 4, and 6 GPa, respectively. Stable and unstable phases are shown as black squares and red dots, respectively.

The solid green lines connect the stable phases. The numbers ”1”...”8” in panels (c)

and (d) indicate the intersections between the phases’ boundary and the dotted lines

representing the path of carbon pollution to the two test phases Fe4Mo and FeMo.

Fe4Mo has been reported to be an effective catalyst composition [159] while FeMo

represents a hypothetical Fe:Mo particle with a Mo content larger than 33%.

94

Figure 7.2: (color online). Ternary phase diagram for Fe-Mo-C nanoparticles of R ∼ ∞, 1.23, 0.62, 0.41 nm.

7.3 Fe4Mo particles

An advantage of an Fe4Mo particle has over a pure Fe particle is that the [Fe4/5Mo1/5]1−x-

Cx line does not intercept any carbide (Fe3C, τ3, τ2). This implies that, at least at low

temperatures, there is a surplus of unbounded metal (probably even at high temper-

atures since the line is far from all of the competing stable phases). This is illustrated

in figure 7.3, which shows the fractional evolution of species as one progresses along

the [Fe4/5Mo1/5]1−x-Cx line in figure 7.2.

For a large Fe4Mo particle (R ≥ 0.62 nm), the decomposition into stable phases is shown in figure 7.3(a)). At concentrations between 0 < xc < 0.09 there is available

free Fe for catalysis, however there is no carbon content in the particle. At higher

95

concentrations, the particle starts to contain free carbon while still providing free Fe.

Therefore, a steady state growth of SWCNTs is possible from large Fe4Mo particles.

Figure 7.3(b) shows the decomposition for a small Fe4Mo particle (R ≤ 0.41 nm)). In this case, the free Fe is consumed and transformed into τ3 carbide before the particle

has enough free carbon, hence, the SWCNT growth will not occur. We can estimate

the minimum size of the particle by calculting the pressure at which τ3 starts to form.

By linear interpolation, we obtain R = 0.52 nm. This size is smaller than that if one

uses pure Fe nanocatalyst (R = 0.56 nm) [152, 153].

7.4 FeMo particles

Figure 7.4 shows the decomposition analysis for FeMo particle. For a large particle of

size R ≥ 0.62 nm, The particle contains free Fe and carbon only after the concentra- tion xc is larger than 0.2 which then permits the growth of the nanotube. Comparing

to the Fe4Mo particle, since the fraction of Fe in the FeMo particle is considerably

smaller than in Fe4Mo, the expected yield is lower and the synthesis temperature

needs to be increased (to overcome the reduced fraction of catalytically active free

Fe). Small FeMo particles (R ≤ 0.41 nm) are similar to small Fe4Mo clusters. Nucle- ation of τ3 and the abscence of free Fe and excess carbons indicate that the particles

are catalytically inactive. The minimum size of FeMo and Fe4Mo particles able to

grow nanotube is the same since it is determined by the stabilization of the same τ3

carbide.

96

Figure 7.3: (color online). a) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”1”, represents [Fe4/5Mo1/5]1−x-Cx crossing the boundary phase Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composition [Fe4/5Mo1/5]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”4” and ”5”, represent [Fe4/5Mo1/5]1−x-Cx crossing the boundary phases Fe←→Mo2C and τ3, as shown in figure 7.2(d).

97

Figure 7.4: (color online). a) Fraction of species for catalyst composition [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≥ 0.62 nm. The dashed vertical line, labeled as ”2” and ”3”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phase Fe2Mo↔Mo2C and Fe↔Mo2C, as shown in figure 7.2(c). b) Fraction of species for catalyst composi- tion [Fe1/2Mo1/2]1−x-Cx and nanocatalyst of R ≤ 0.41 nm. The dashed vertical lines, labeled as ”6”, ”7”, and ”8”, represent [Fe1/2Mo1/2]1−x-Cx crossing the boundary phases Fe2Mo↔Mo2C, Fe←→Mo2C, and τ3 ↔Mo2C, as shown in figure 7.2(d).

98

Chapter 8

Conclusions

We have presented the results of our computational studies on adsorptions of hy-

drocarbons (alkanes, benzene) and rare gases on a decagonal surface of Al-Ni-Co

quasicrystal (d-AlNiCo). Ab initio calculations show that upon the adsorptions, the

surface of the d-AlNiCo does not undergo relaxations and that there are no disso-

ciations of the adsorbates. The simulations of thin film growth of these adsorbates

have been performed in the grand canonical ensemble using Monte Carlo method.

We use semiempirical pair interactions for the rare gases and develop classical many-

body potentials based on embedded-atom method (EAM) for the hydrocarbons. All

of the simulated atoms/molecules wet the substrate as a consequence of compara-

ble strengths between the substrate-adsorbate and adsorbate-adsorbate. Another

consequence is the wide range of overlayer structures observed in these systems.

Methane monolayer has a quasicrystalline pentagonal order commensurate with the

substrate. Propane forms a disordered structure. Hexane and octane monolayers

show 2-dimensional close-packed features consistent with their bulk structure. Ben-

zene forms a pentagonal monolayer at low and moderate temperatures which trans-

forms into a triangular lattice at high temperatures. Similar structural transition

also occurs in xenon monolayer, however in this case, the transition is observed at all

temperatures below the triple-point temperature and as a function of pressure. We

have characterized that such transition is first-order with an associated latent heat.

Smaller noble gases such as Ne, Ar, Kr form monolayers with mixed pentagonal and

triangular patterns and do not show any structural transitions.

By systematically simulating test noble gases of various sizes and strengths, we

99

observe that the relative strengths between the competing interactions determine the

growth mode. Agglomeration occurs when the adsorbate-adsorbate interactions are

much stronger than the substrate-adsorbate ones. In the comparable strength regime,

a layer-by-layer film growth is observed. In this regime, the mismatch between the

size of the gas and the substrate’s characteristic length plays a major role in affecting

the structure of the adsorbed films. In general, on the d-AlNiCo substrate, we found

that structural transition from 5- to 6-fold occurs when the gas size is larger than

λ/0.944 (λ represents the average row-row spacing in the quasicrystalline plane of

the d-AlNiCo). Even though this rule is derived from rare gases, it is consistent with

methane, benzene, hexane and octane. Therefore, it might be useful as a guidance in

the search for suitable quasicrystalls for which alkanes (as the main constituent of oil

lubricants) will form quasiperiodic structures for the low-friction coating applications.

It is a natural extension of this work to investigate other quasicrystalls with

larger characterictic lengths than the d-AlNiCo used in this study. In fact, d-AlNiCo

is stable in many decorations depending on the concentration of Al, Ni, and Co.

Due to the stripe nature in the close-packed structure of linear alkanes, the film

structure of these molecules on one dimensional quasicrystals is interesting to study.

Two surfaces with stripe pattern will be commensurate only when the stripes are

perfectly alligned, therefore, one dimensional quasicrystalls might be good candidate

for low-friction coatings.

100

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110

Biography

The author was born in May 5th, 1978 in a Javanese village 8 miles southwest of

Solo, Central Java, Indonesia. He is the last(fourth) child of Ladiya (father) and

Sri Suratmi (mother). His father was an elementary school teacher and his mother

is a devoted house wife. Prior to attending college education, the author won the

third award in the International Physics Olympiad 1996 in Norway. The author

received a B.S. degree from Institut Teknologi Bandung, Indonesia, in Electrical

Engineering in 2000. In the Fall of 2000, he went to Florida State University on a

research assistantship from the Physics department in which he completed a M.S.

degree in 2004. In Summer 2004, he received a research assitantship to pursue a

doctoral degree in Mechanical Engineering and Materials Science department from

Duke University under the advisory of Dr. Stefano Curtarolo. He completed his PhD

work in computational materials science in Spring 2008. In addition, he received a

Graduate Certificate degree in Computational Science, Engineering, and Medicine,

in Spring 2008 also from Duke University.

List of publications:

• Diehl R D, Setyawan W and Curtarolo S 2008 J. Phys.: Cond. Mat. in press

• Harutyunyan A R, Awasthi N, Jiang A, Setyawan W, Mora E, Tokune T, Bolton K and Curtarolo S 2008 Phys. Rev. Lett. inpress

• Curtarolo S, Awasthi N, Setyawan W, Li N, Jiang A, Tan T Y, Mora E, Bolton K and Harutyunyan A T 2008 Proc. of Comp. Simul. Studies in Cond. Matt.

Phys. XXI Eds Landau D P, Lewis S P and Schuttler H-B (Springer, Berlin,

Heidelberg)

• Setyawan W, Diehl R D, Ferralis N, Cole M W and Curtarolo S 2007 J. Phys.:

111

Cond. Mat. 19 016007

• Diehl R D, Setyawan W, Ferralis N, Trasca R, Cole M W and Curtarolo S 2007 Phil. Mag. 87 2973

• Jiang A, Awasthi N, Kolmogorov A N, Setyawan W, Bo¨rjesson A, Bolton K, Harutyunyan A R and Curtarolo S 2007 Phys. Rev. B 75 205426

• Setyawan W, Ferralis N, Diehl R D, Cole M W and Curtarolo S 2006 Phys. Rev. B 74 125425

• Diehl R D, Ferralis N, Pussi K, Cole M W, Setyawan W and Curtarolo S 2006 Phil. Mag. 86 863

• Curtarolo S, Setyawan W, Diehl R D, Ferralis N and Cole M W 2005 Phys. Rev. Lett. 95 136104

• Rao S G, Huang L, Setyawan W and Hong S 2003 Nature 425 36

• Setyawan W, Rao S G and Hong S 2002 Mat. Res. Soc. Proc. NN Fall

• Mumtaz A, Setyawan W and Shaheen S A 2002 Phys. Rev. B 65 020503

112

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