THE VALIDITY OF CLASSICAL NUCLEATION THEORY …rx036ms4124/SeunghwaRyu... · ITS APPLICATION TO DISLOCATION NUCLEATION A DISSERTATION ... changing my research ﬁeld so smoothly

THE VALIDITY OF CLASSICAL NUCLEATION THEORY AND

ITS APPLICATION TO DISLOCATION NUCLEATION

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Seunghwa Ryu

August 2011

http://creativecommons.org/licenses/by/3.0/us/

This dissertation is online at: http://purl.stanford.edu/rx036ms4124

© 2011 by Seunghwa Ryu. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-3.0 United States License.

ii



http://purl.stanford.edu/rx036ms4124

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Wei Cai, Primary Adviser


Douglas Osheroff, Co-Adviser


Paul McIntyre


William Nix

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

Nucleation has been the subject of intense research because it plays an important role

in the dynamics of most first-order phase transitions. The standard theory to describe

the nucleation phenomena is the classical nucleation theory (CNT) because it cor-

rectly captures the qualitative features of the nucleation process. However potential

problems with CNT have been suggested by previous studies. We systematically test

the individual components of CNT by computer simulations of the Ising model and

find that it accurately predicts the nucleation rate if the correct droplet free energy

computed by umbrella sampling is provided as input. This validates the fundamental

assumption of CNT that the system can be coarse grained into a one dimensional

Markov chain with the largest droplet size as the reaction coordinate.

Employing similar simulation techniques, we study the dislocation nucleation

which is essential to our understanding of plastic deformation, ductility, and me-

chanical strength of crystalline materials. We show that dislocation nucleation rates

can be accurately predicted over a wide range of conditions using CNT with the ac-

tivation free energy determined by umbrella sampling. Our data reveal very large

activation entropies, which contribute a multiplicative factor of many orders of mag-

nitude to the nucleation rate. The activation entropy at constant strain is caused by

thermal expansion, with negligible contribution from the vibrational entropy. The ac-

tivation entropy at constant stress is significantly larger than that at constant strain,

as a result of thermal softening. The large activation entropies are caused by anhar-

monic effects, showing the limitations of the harmonic approximation widely used for

rate estimation in solids. Similar behaviors are expected to occur in other nucleation

processes in solids.

iv

Acknowledgements

First of all, I am very much indebted to my principal adviser, Professor Wei Cai. It

is of great fortune for me to work with such a bright and gentle person who I want to

follow as a role model both as a scientist and as a gentleman. I have learned how to

approach a scientific problem and how to tackle it: under his guidance, my random

ideas transformed into a well defined research project, and a seemingly formidable

problem into a series of small problems that can be handled systematically. Pro-

fessor Cai has also helped me every single step that I need to grow as a scientist,

such as writing a concise paper and delivering an insightful presentation. In addition

to academic advices, I have also learned the virtues of a gentleman: he has consis-

tently shown positive attitude on life, humility in the quest of knowledge, respect on

other people, and dedication to his family. I am especially grateful for having many

discussions with him on non-academic subjects regarding various aspects of life and

being able to listen to his advices. Past four years that I worked with Professor Cai

have been one of the most important periods in my life in which I have grown both

intellectually and mentally.

I owe very special thanks to my co-adviser, Professor Douglas Osheroff, under

whom I had worked during the first three years of my graduate study. I had no

experience on the experimental physics when I arrived at Stanford, and joined his

group in the hope that I could learn a completely new subject under the guidance of

a famous Nobel Laureate. Indeed, I have learned a lot from his wizardly expertise

and rich experiences on the low temperature physics experiment. I still remember

the first moment when I saw the signature of He-3 superfluid transition with full of

joy, after several months of struggles to fix the dilution fridge. My experience in

v

Professor Osheroff’s group has aided and will continue to aid me to communicate

and cooperate with experimentalists. I deeply appreciate the generosity he showed

me when I decided to leave his group after I realized my natural preference toward

theoretical studies. Since then, gratefully, he has been my co-adviser and given me

precious advices on research, career, and life in general. His dedication to science

education for general public and humble attitude have shown me the way I want to

follow when becoming a senior scientist in the far future.

I would like to thank Professor William Nix and Paul McIntyre to serve on my

thesis committee. The dislocation course that I took from Professor Nix and the

kinetic process course from Professor McIntyre have provided me the theoretical basis

for the dissertation project. I appreciate Professor Nix for the discussion and valuable

assessments on the dislocation nucleation study. He exemplifies the ideal life as a

senior professor: he still actively works and enjoys the life at the same time, and

willingly shares his valuable time for helping students and young faculties. I would

like to thank Professor McIntyre for the invitation to his nanowire group meeting

and insightful advices on the nanowire growth simulation project, another branch

of my doctoral study. He exemplifies the quality of a real professional: he gives

critical assessments on students working in various subjects with his deep and broad

understanding in materials science research and organizes collaborations with several

groups very efficiently. I also want to thank Professor Evan Reed for serving as the

chair for my thesis defense meeting.

I am happy to thank two special seniors in our group, Dr. Keonwook Kang and

Dr. Eunseok Lee. I was going through a difficult time when I moved to Cai group

due to the anxiety from starting a completely new field in the midst of graduate

study and the ignorance in computational work. Without their moral support and

help on the technical skills on computer simulations, I would have not succeeded in

changing my research field so smoothly. I would like to thank all Cai group mem-

bers who shared valuable discussions on my project. And many thanks to former

lab mates in Osheroff group for the training on the low temperature physics exper-

iments and to McIntyre group members for sharing interesting experimental results

on semiconductor nanowire growth.

vi

Besides spending time in the lab doing research, I have been nourished by having

good friends and sharing unforgettable memories with them. I would like to thank my

fellow KAIST alumni at Stanford, friends in Cornerstone Community Church (special

thanks to Dr. Jungjoon Lee), Professor Lew group members, fellow Korean students

in physics and mechanical engineering departments, and all other close friends not

included in these groups. I have been so comfortable and relaxed with my friends

having many trips and parties, playing sports and games, tasting delicious foods and

liquors, and going museums and concerts together. Their encouragements and advices

on many aspects of life are also priceless.

I want to thank my home university, Korea Advanced Institute of Science and

Technology (KAIST) and the department of physics where I acquired a solid foun-

dation in physics as an undergraduate. Special thanks to my undergraduate adviser,

Professor Mahn Won Kim, and Professor Hawoong Jeong for their encouragement

and invaluable advices in my career.

I appreciate the financial supports from the Stanford Graduate Fellowship and the

Korea Science and Engineering Foundation Fellowship, which allowed me to choose

research projects more freely.

Lastly, I would like to thank my family for all their love, support, and encourage-

ment. I do not know how I can repay what I have received from my parents for the

rest of my life. The devotion, patience, and responsibility that they have shown in

their life have been and will be the source of my strength. I thank my younger brother

who kept encouraging me for the past years. I would like to thank my grandmother

in heaven who had dedicated her life to her family and showed me what the true love

is.

During seven years of graduate study, I have gradually realized that I owe every-

thing I accomplished to people around me and how important it is to interact well

with others. This precious lesson on the relationship, as well as the expertise I gained

in my doctoral research, is the true gem that I will cherish for my life.

vii

Preface

During my doctoral study with Professor Wei Cai, I have worked on a diverse spec-

trum of research projects in computational physics and computational materials sci-

ence. Since it was impossible to assemble all the published works in a coherent

manner, I have opted for presenting a tightly knit story with the theme of the com-

putational investigation of nucleation phenomena, out of some portion of them. For

that purpose, I have written a long introduction to well establish a niche, and a care-

ful review on existing theories and experiments, and computational methods, which

will help readers better understand the materials presented in this dissertation.

As noted in the abstract, the main text of this dissertation addresses two corre-

lated projects sharing common theory and numerical algorithms: (1) the test of the

classical nucleation theory using the Ising model and (2) the prediction of disloca-

tion nucleation rate from the classical nucleation theory using atomistic simulations.

Readers who prefer a more distilled presentation are refered to following journal ar-

ticles:

for the Ising model,

• Seunghwa Ryu and Wei Cai,“The Validity of Classical Nucleation Theory for

Ising Models”, Phys. Rev. E (Rapid Communications) 81, 030601 (R) (2010).

• Seunghwa Ryu and Wei Cai, “Numerical Tests of Nucleation Theories for the

Ising Models”, Phys. Rev. E 82, 011603 (2010).

The first one is a concise letter containing the crux of the work and the second one

is a full paper including more extensive discussions which contains most contents of

Chapter 4.

viii

for dislocation nucleation,

• Seunghwa Ryu, Keonwook Kang and Wei Cai, “Entropic Effect on the Rate of

Dislocation Nucleation”, Proc. Natl. Acad. Sci. USA 108, 5174 (2011).

• Seunghwa Ryu, Keonwook Kang, and Wei Cai, “Predicting the Dislocation Nu-

cleation Rate as a Function of Temperature and Stress”, J. Mater. Res. (2011),

in press.

• Sylvie Aubry, Keonwook Kang, Seunghwa Ryu and Wei Cai, “Energy Barrier

for Homogeneous Dislocation Nucleation: Comparing Atomistic and Continuum

Models”, Scripta Mater. 64, 1043 (2011).

The first one is a condensed article highlighting the entropic effect and the second

one is a full paper including more in-depth discussions which contains most contents

of Chapter 5. Third one considers the comparison between atomistic and contin-

uum model. This work is not included since I am not a main contributor, but it

is an interesting work that may attracts readers engrossed in dislocation nucleation

research.

Another major branch of my doctoral study is the simulation of the gold-catalyzed

growth of silicon nanowires via vapor-liquid-solid (VLS) mechanism. For this project,

thermal properties obtained from existing atomistic models are investigated for gold,

silicon, and various other materials. We have devised efficient methods for computing

the free energies of solid and liquid alloys, to improve and develop a gold-silicon poten-

tial that is fitted to the experimental binary phase diagram. These works are included

as appendices that are referred in Chapter 3 where various interatomic potential mod-

els and simulation methods are reviewed. Because the series of studies constitute a

complete set of story by themselves, readers interested solely in the nanowire growth

mechanism can skip the main text of the dissertation and read through Appendices

D, E, and F. Each of Appendices D, E, F contains the contents of following three

journal articles, with one-to-one correspondence:

• Seunghwa Ryu and Wei Cai, “Comparison of Thermal Properties Predicted by

Interatomic Potential Models”, Modell. Simul. Mater. Sci. Eng. 16, 085005

ix

(2008).

• Seunghwa Ryu, Christopher R. Weinberger, Michael I. Baskes, and Wei Cai,

“Improved Modified Embedded-Atom Method Potentials for Gold and Silicon”,

Modell. Simul. Mater. Sci. Eng. 17, 075008 (2009).

• Seunghwa Ryu and Wei Cai, “A Gold-Silicon Potential Fitted to the Binary

Phase Diagram”, J. Phys.: Condens. Matter. 22, 055401 (2010).

We have published a pioneering simulation of the silicon nanowire growth using the

potential developed in above studies, which is not included in the dissertation due

to weaker link with the main text. We investigated the origins of the orientation

dependence of nanowire growth rate and the shift of phase diagram in nanoscale.

Readers interested in this study are invited to read the following journal article.

• Seunghwa Ryu andWei Cai, “Molecular Dynamics Simulations of Gold-Catalyzed

Growth of Silicon Bulk Crystals and Nanowires”, J. Mater. Res. (2011), in

press.

We have also done an interesting piece of work on quantum entanglement for

intellectual amusement. A fast algorithm to calculate the entanglement of formation

of a mixed state was developed with which we obtain the statistics of the entanglement

of formation on ensembles of random density matrices of higher dimensions than

possible before. The correlations between the entanglement of formation and other

quantities that are easier to compute, such as participation ratio and negativity are

studied. The details of this work can be found in the following journal article.

• Seunghwa Ryu, Wei Cai and Alfredo Caro, “Quantum Entanglement of Forma-

tion between Qudits”, Phys. Rev. A 77, 052312 (2008).

x

Contents

Abstract iv

Acknowledgements v

Preface viii

1 Introduction 1

1.1 Computational Investigation of Nucleation . . . . . . . . . . . . . . . 1

1.2 Scope of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background and Motivation 7

2.1 Basic Thermodynamics of Nucleation . . . . . . . . . . . . . . . . . . 7

2.2 Nucleation Rate Predictions From Nucleation Theories . . . . . . . . 11

2.3 Nucleation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Dislocation Nucleation and Materials Strength at Small Scale . . . . 20

3 Computational Methods 25

3.1 Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Interatomic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 List of Empirical Potentials . . . . . . . . . . . . . . . . . . . 30

3.2.2 The Benchmarks of EAM Copper Potential . . . . . . . . . . 34

3.3 Molecular Dynamics Simulation . . . . . . . . . . . . . . . . . . . . . 38

3.4 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 Advanced Sampling Methods . . . . . . . . . . . . . . . . . . . . . . 41

xi

3.5.1 Forward Flux Sampling . . . . . . . . . . . . . . . . . . . . . . 42

3.5.2 Umbrella Sampling . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5.3 Computing Rate from Becker-Doring Theory . . . . . . . . . . 48

4 Numerical Tests of Nucleation Theories 50

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Nucleation Theories Applied to the Ising Model . . . . . . . . . . . . 53

4.2.1 Becker-Doring Theory . . . . . . . . . . . . . . . . . . . . . . 53

4.2.2 Langer’s Field Theory . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4.1 Nucleation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4.2 Critical Droplet Size and Shape . . . . . . . . . . . . . . . . . 61

4.4.3 Droplet Free Energy of 2D Ising Model . . . . . . . . . . . . . 63

4.4.4 Droplet Free Energy of 3D Ising model . . . . . . . . . . . . . 66

4.4.5 Effective Entropy of Nucleation . . . . . . . . . . . . . . . . . 70

4.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Predicting the Dislocation Nucleation Rate 75

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Thermodynamics of Nucleation . . . . . . . . . . . . . . . . . . . . . 78

5.2.1 Activation Free Energies . . . . . . . . . . . . . . . . . . . . . 78

5.2.2 Activation Entropies . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.3 Difference between the Two Activation Entropies . . . . . . . 83

5.2.4 Previous Estimates of Activation Entropy . . . . . . . . . . . 86

5.3 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3.1 Simulation Cell . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3.2 Nucleation Rate Calculation . . . . . . . . . . . . . . . . . . 92

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4.1 Benchmark with MD Simulations . . . . . . . . . . . . . . . 95

5.4.2 Homogeneous Dislocation Nucleation in Bulk Cu . . . . . . . 96

5.4.3 Heterogeneous Dislocation Nucleation in Cu Nano-Rod . . . 99

xii

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.5.1 Testing the “Thermodynamic Compensation Law” . . . . . . 100

5.5.2 Entropic Effect on Nucleation Rate and Yield Strength . . . . 103

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Summary and Outlook 107

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A Derivations 111

A.1 Nucleation Rate Prediction from Classical Nucleation Theory . . . . . 112

A.2 Nucleation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

B More Data on the Nucleation in the Ising Model 121

B.1 Attachment Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

B.2 Droplet Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

B.3 The Constant Term in Droplet Free Energy . . . . . . . . . . . . . . 123

B.4 Free Energy Curves F (n) for the Ising Model . . . . . . . . . . . . . . 126

C More Discussion on the Dislocation Nucleation 131

C.1 Equality of Critical Sizes nσc and nγ

c . . . . . . . . . . . . . . . . . . 131

C.2 Equality of Activation Gibbs and Helmholtz Free Energies . . . . . . 132

C.3 Physical Interpretation of Activation Entropy Difference ∆Sc . . . . . 133

C.4 Approximation of Sc(σ) . . . . . . . . . . . . . . . . . . . . . . . . . 135

C.5 Activation Free Energy Data . . . . . . . . . . . . . . . . . . . . . . . 136

C.6 Activation Volume and Critical Loop Size . . . . . . . . . . . . . . . 139

D Thermal Properties from Interatomic Potentials 142

D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

D.2 Comparison between Model Predictions and Experiments . . . . . . . 144

D.2.1 Semiconductors: Si and Ge . . . . . . . . . . . . . . . . . . . . 145

D.2.2 FCC Metals: Au, Cu, Ag and Pb . . . . . . . . . . . . . . . . 147

D.2.3 BCC Metals: Mo, Ta and W . . . . . . . . . . . . . . . . . . . 147

xiii

D.3 Free Energy Method for Melting Point Calculation . . . . . . . . . . 148

D.3.1 Solid Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . 150

D.3.2 Liquid Free Energy . . . . . . . . . . . . . . . . . . . . . . . . 153

D.3.3 Melting Point and Error Estimate . . . . . . . . . . . . . . . . 155

D.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

D.5 Error Estimates in Free Energy Calculations . . . . . . . . . . . . . . 158

E MEAM Potentials for Pure Au and Pure Si 160

E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

E.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

E.2.1 Limitations of the MEAM Gold Potential . . . . . . . . . . . . 163

E.2.2 Limitations of MEAM Silicon Potentials . . . . . . . . . . . . 165

E.3 Methods and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

E.3.1 Multi-body Screening Function . . . . . . . . . . . . . . . . . 168

E.3.2 Pair Potential and Equation of State . . . . . . . . . . . . . . 170

E.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

E.5 Multi-body Screening Function . . . . . . . . . . . . . . . . . . . . . 174

E.6 Further Benchmarks of the MEAM† Potentials . . . . . . . . . . . . . 177

F Gold-Silicon Binary Potential 180

F.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

F.2 MEAM Model for Gold and Silicon . . . . . . . . . . . . . . . . . . . 181

F.2.1 Functional Form . . . . . . . . . . . . . . . . . . . . . . . . . 181

F.2.2 Determining the Parameters . . . . . . . . . . . . . . . . . . . 182

F.3 Construction of Binary Phase Diagram . . . . . . . . . . . . . . . . . 185

F.3.1 Free Energy of Solid with Impurities . . . . . . . . . . . . . . 186

F.3.2 Free Energy of Liquid Alloy . . . . . . . . . . . . . . . . . . . 188

F.3.3 Construction of Binary Phase Diagram . . . . . . . . . . . . 191

F.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

F.5 Further Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Bibliography 196

xiv

List of Tables

3.1 Lattice properties of Cu predicted by the EAM potential [111]. . . . 34

3.2 The intrinsic stacking fault energy γSF and the unstable stacking fault

energy γUSF of Cu from experiments, EAM potential [111], and ab ini-

tio calculation [114]. ab initio data depend on the exchange-correlation

functionals used in the study. . . . . . . . . . . . . . . . . . . . . . . 36

C.1 Data for homogeneous nucleation: σxy in GPa, Ec, Ec and Fc in eV,

f+c in 1014 s−1. γxy and Γ are dimensionless. The error in Ec is about

0.003 eV, due to the small errors in equilibrating the simulation cell to

achieve the pure shear stress state. The error in Fc is about 0.5 kBT ,

i.e. approximately 0.01 eV, due to the statistical error in umbrella sam-

pling. The error in Zeldovich factor Γ is within ±0.01. The attachment

rate f+c has relative error of ±50%. . . . . . . . . . . . . . . . . . . . 138

C.2 Data for heterogeneous nucleation: σzz in GPa, Ec, and Fc in eV, f+c

in 1014 s−1. γxy and Γ are dimensionless. The error in Fc is about

0.5 kBT , i.e. approximately 0.01 eV, due to the statistical error in

umbrella sampling. The error in Zeldovich factor Γ is within ±0.01.

The attachment rate f+c has relative error of ±50%. Notice that, due

to the existence of thermal strain, the elastic strain values are slightly

different at different temperatures. . . . . . . . . . . . . . . . . . . . 139

xv

D.1 Thermal properties of various elements as predicted by several empir-

ical potentials and compared with experiments [183, 207, 208]. The

properties include the melting point Tm (in K), latent heat of fusion

L (in J/g), solid and liquid entropy at melting point, SS and SL (in

J/mol K), and thermal expansion coefficient α (in 10−6K−1) at 300 K.

The MEAM∗-Au and MEAM∗-Cu entries correspond to a modifica-

tion of the original MEAM model by changing cmin from 2.0 to 0.8.

The MEAM† entries of BCC metals are computed by the new MEAM

model that includes second nearest neighbor interactions [209, 210]. . 146

D.2 The estimated free energy difference ∆Fi and its standard deviation in

the 5 different adiabatic switching steps for the melting point calcula-

tions of SW-Si and MEAM-Si models. . . . . . . . . . . . . . . . . . 159

E.1 Model predictions and experimental data [207, 230, 231] on thermal

and mechanical properties of gold. The computation models include

original MEAM [206], EAM [204], 2nn-MEAM [227] and two modifi-

cations made in this study (2nn-MEAM∗ and 2nn-MEAM†), and first-

principles calculation with DFT/LDA [232]. The properties include the

melting point Tm (in K), latent heat of fusion L (in kJ mol−1), solid and

liquid entropy at the melting point, SS and SL (in J mol−1K−1), diffu-

sion constant of the liquid D at Tm (in 10−9m2s−1), thermal expansion

coefficient α of the solid (in 10−6K−1) at 300 K, ideal shear strength

τc (in GPa) of the solid at zero temperature, and volume change on

melting ∆Vm/Vsolid (in %). Statistical errors are on the order of the

last digit of the presented values. . . . . . . . . . . . . . . . . . . . . 163

xvi

E.2 Model predictions and experimental data [207, 230, 196] of thermal

and mechanical properties of silicon. The ideal shear strength [233]

obtained from first principle calculation is also presented for compari-

son. The computational models include the original MEAM [206] and

MEAM second nearest-neighbor (2nn) [228]. Superscripts ∗ and † rep-

resent modifications to these models introduced in this work. SW [105]

and Tersoff [106] models are also included for comparison. The vari-

ables and their units are identical to those in Table E.1. . . . . . . . . 166

E.3 The elastic constants, C11, C12 and C44 (in GPa), vacancy formation en-

ergy Ev (in eV) and surface energies E100, E110, and E111 (in ergs/cm2)

from experimental measurements [183, 206, 241, 242], first principle

calculations [241, 243, 244, 245] (except silicon elastic constants com-

puted here using DFT/LDA) and various MEAM models considered in

this study. The unrelaxed energies are given in parenthesis. The DFT

results are from several different pseudopotentials. Surface reconstruc-

tion such as dimer structure is not considered in calculation of relaxed

surface energy of silicon. . . . . . . . . . . . . . . . . . . . . . . . . . 178

F.1 Equilibrium lattice constant a, bulk modulus B, cohesive energy Ec,

and cubic elastic constants C11 and C44 for DC structure of Si, FCC

structure of Au, and B1 structure of Au-Si. For pure Si and Au, the

differences between the experimental and ab initio values are listed in

the column labelled “offset”. Their average is the expected “offset”

value for the hypothetical B1 structure. The values marked with ∗are the ab initio values plus the correction terms given in the “offset”

column. The last column is what the MEAM model is fitted to or

predicts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

F.2 MEAM and ab initio (DFT/LDA) predictions of impurity energies. E1

is the energy needed to substitute an atom in an FCC Au crystal by

a Si atom. E2 is the energy needed to substitute an atom in a DC Si

crystal with by a Au atom. . . . . . . . . . . . . . . . . . . . . . . . 184

xvii

F.3 Parameters for the Au-Si MEAM cross-potential using B1 as the ref-

erence structure. Cmin(i, j, k) are cut-off parameters in the multi-body

screening function. They describe the screening effect on the interac-

tion between atoms of type i and j by their common neighbor of type

k, where i, j, k = 1 (Au) or 2 (Si). The same Cmax(i, j, k) is used for

every combination of i, j, k . . . . . . . . . . . . . . . . . . . . . . . 185

F.4 Comparison between MEAM and ab initio predictions on the energy

and elastic properties of B1, B2 and L12 structures of Au-Si. a is the

equilibrium lattice constant, B is bulk modulus, Ec is the cohesive en-

ergy, and C11 and C44 are cubic elastic constants. MEAM data should

be compared with ab initio (DFT/LDA) results that have been ad-

justed for known differences from experimental values for pure elements.195

xviii

List of Figures

2.1 ∆G versus n showing volume and surface contributions resulting in a

peak at n = nc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Schematic of a solid spherical cap with radius of curvature r and con-

tact angle θ forming a wall. . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Comparison of experimental nucleation rates of water from vapor (cir-

cles) with the predictions of the classical theory [72]. The full lines

belong to the classical Becker-Doring (BD) theory. Logarithmic scales

are used for both axes. . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Schematic of dislocations [80]. (a) An edge dislocation is a defect where

an extra half-plane of atoms is introduced mid way through the crystal,

distorting nearby plane of atoms. (b) A screw dislocation is a shear

ripple extending from side to side. . . . . . . . . . . . . . . . . . . . . 20

2.5 Schematic of edge dislocation motion that induces plastic strain [80]. 21

2.6 (a) and (b) Two consecutive in situ TEM compression tests on a

FIB microfacbricated 160-nm-top-diameter Ni pillar with 〈111〉 orien-tation [89]. (a) Dark-field TEM image of the pillar before the tests;

note the high initial dislocation density. (b) Dark-field TEM image of

the sam pillar after the first test; the pillar is now free of dislocations.

(c) MD simulation of nanoindentation process [95]. Snapshot of dis-

location nucleation at the first plastic yield point on Au(111), Lower

figure shows a two-layer-thick cross section of a (111) plane containing

the partial dislocation loop on the right in the upper figure. . . . . . 22

xix

3.1 (a) Magnetization M = 1N

∑

si of the 2D square lattice Ising model

as function of temperature in the absence of external field, i.e. h = 0.

At T < Tc = 2/ ln(1 +√2)J/kB ∼ 2.269J/kB, majority of spins in

the system are spontaneously aligned in either up or down. However,

T > Tc, spontaneous ordering disappears because of thermal fluctua-

tion. (b) schematic of ferromagnetic state below Tc. (c) schematic of

paramagnetic state above Tc. . . . . . . . . . . . . . . . . . . . . . . 27

3.2 (a) Analogy between the Ising model demagnetization and nucleation

phenomena. (b) The snapshot of Monte Carlo simulation of demagne-

tization in the presence of an external field h > 0. The transition from

the down spin dominated meta-stable state to the up spin dominated

stable state occurs via nucleation of small island of up spins. . . . . 28

3.3 (a) The stacking pattern of the 111 planes in FCC crystals. (b) Perfect

Burgers vectors b1, b2, b3 and partial Burgers vectors bp1, bp2, bp3 on

the 111 plane. Figures are taken from the literature [113]. . . . . . . 35

3.4 Generalized stacking fault energy on (111) plane along a/6[211] direc-

tion calculated with EAM potential [111]. . . . . . . . . . . . . . . . 36

3.5 Linear thermal expansion of Cu calculated in the quasi-harmonic ap-

proximation (QHA) and by the Monte Carlo method using the EAM

potential [111]. The melting point of Cu (Tm) is indicated. . . . . . . 37

xx

3.6 Schematic of forward flux sampling. (a) To compute the nucleation

rate of small droplet with size n0, we count the number q that droplets

larger than n0 forms for time duration t. Then, nucleation rate I0

becomes q/t. Here, it is not desirable to count small fluctuation in n

around n0 as separate events. The counter reset only after n comes

back to the original basin n < nA and becomes ready to count another

event. nA can be considered as an error margin in the digital signal

processing. (b) At each interface i, we have an ensemble of configu-

rations having largest droplet size ni. N independent MC simulations

are performed, starting from a randomly chosen configuration from the

ensemble. Then, number M of reaching next interface i+ 1 before re-

turning back to nA is counted as successful forward flux. Then, the

probability of reaching next interface P (ni+1|ni) becomes N/M . . . . 42

3.7 (color online) The probability P (λi|λ0) (solid line) of reaching interface

λi from λ0 and average committor probability PB(λi) (circles) over

interface λi at (kBT, h) = (1.5, 0.05) for the 2D Ising model. The 50%

committor point is marked by *. . . . . . . . . . . . . . . . . . . . . 44

3.8 (a) For a given free energy landscape, we can sample a limited region

within the kBT range. (b) We can sample other regions with higher

free energy with umbrella sampling that limits the Monte Carlo move

within the range of bias function. . . . . . . . . . . . . . . . . . . . . 45

3.9 (From the top left cornet, in clock wide direction). Processing the raw

histogram taken from umbrella sampling simulations. (1) We first ob-

tain the biased relative distribution P′′

(n) at each window from series of

umbrella sampling. (2) Unbiased relative distribution P′

(n) can be ob-

tained by multiplying the inverse of weighing function, exp[12k(n−ni)

2

kBT],

to the data of each window i. (3) From the overlapping histogram

methods [125], we can merge the distributions into a single curve. Af-

ter normalization, we obtain the absolute probability P (n). (4) The

free energy curve ∆G(n) can be obtained from −kBT ln(∆G(n)). . . 46

xxi

3.10 (color online) (a) Droplet free energy F (n) obtained by US at kBT =

1.5 and h = 0.05 in the 2D Ising model. (b) Fluctuation of droplet size

〈∆n2(t)〉 as a function of time. . . . . . . . . . . . . . . . . . . . . . 48

4.1 (color online) Effective surface free energy σeff as a function of temper-

ature for the 2D Ising model from analytic expression [47]. The free

energy of the surface parallel to the sides of the squares, σ(10), is also

plotted for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Schematic of numerical test on the CNT nucleation rate. . . . . . . . 58

4.3 The nucleation rate I computed by FFS (open symbols) and Becker-

Doring theory with US free energies (filled symbols) in the (a) 2D and

(c) 3D Ising models. The ratio between nucleation rates obtained by

FFS and Becker-Doring theory at different temperatures in the (b) 2D

and (d) 3D Ising models. The symbols in (b) and (d) match those

defined in (a) and (c), respectively. . . . . . . . . . . . . . . . . . . . 60

4.4 (a) For the 2D Ising model, the critical droplet size n obtained from

FFS (filled symbols) and umbrella sampling (open symbols). nc pre-

dicted by Becker-Doring theory (dotted line) and by field theoretic

equation (solid line) are plotted for comparison. (b) For the 3D Ising

model, the critical droplet size n obtained from FFS (filled symbols)

and umbrella sampling (open symbols). . . . . . . . . . . . . . . . . 61

4.5 (a) Histogram of committor probability in an ensemble of spin configu-

rations with n = 496 for the 2D Ising model at kBT = 1.5 and h = 0.05.

Representative droplets are also shown, with black and white squares

corresponding to +1 and −1 spins, respectively. (b) Histogram of com-

mittor probability in an ensemble of spin configurations with n = 524

for the 3D Ising model at kBT = 2.20 and h = 0.40. . . . . . . . . . 62

xxii

4.6 (a) Droplet free energy curve F (n) of the 2D Ising model at kBT = 1.5

and h = 0.05 obtained by US (circles) is compared with Eq. (4.10)

(solid line) and Eq. (4.3) (dashed line). Logarithmic correction term54kBT lnn (dot-dashed line) and the constant term d (dotted line) are

also drawn for comparison. (b) Magnified view of (a) near n = 0,

together with the results from analytic expressions (squares) available

for n ≤ 17 (see Appendix B.3). . . . . . . . . . . . . . . . . . . . . . 64

4.7 (a) Droplet free energy F (n) of the 3D Ising model at kBT = 2.40

and h = 0 obtained by US (circles) is compared with Eq. (4.22) (solid

line) and Eq. (4.11) (dots). Logarithmic term τkBT lnn is also plotted

(dot-dashed line). The difference in predictions by classical expres-

sion Eq. (4.11) and field theory Eq. (4.22) are very small compared

to F (n) itself and cannot be observed at this scale. (b) Magnified

view of (a) near n = 0, together with the analytic solution of small

droplets (squares, see Appendix B.3) and the exponential correction

term (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.8 (a) Surface free energies of the 3D Ising model as functions of tempera-

ture. Circles are fitted values of σeff from Eq. (4.22), dashed line is the

expected behavior of σeff over a wider range of temperature, and solid

line is the free energy of the (100) surface [141]. Numerically fitted

values of σeff from Heermann et al. [137] are plotted as +. (b) τ values

that give the best fit to the free energy data from US. τ can be roughly

described by a linear function of T shown as a straight line. No abrupt

change is observed near the roughening temperature TR. . . . . . . . 69

xxiii

4.9 (a) Droplet free energy as a function of droplet size n at h = 0.1 and

different kBT for the 2D Ising model. The critical droplet free energy

is marked by circles. (b) Critical droplet free energy (circles) from (a)

as a function of kBT for the 2D Ising model. The solid line is a linear

fit of the data, and the dashed line is the prediction of Eq. (4.5). (c)

Droplet free energy as a function of droplet size n at h = 0.45 and

different kBT for the 3D Ising model. (d) Critical droplet free energy

from (c) as a function of kBT for the 3D Ising model. The solid line is a

linear fit of the data, and the dashed line is the prediction of Eq. (4.13). 71

5.1 Homogeneous dislocation nucleation rate per lattice site in Cu under

pure shear stress σ = 2.0 GPa on the (111) plane along the [112]

direction as a function of T−1, predicted by Becker-Doring theory using

free energy barrier computed from umbrella sampling (See Section 5.3).

The solid line is a fit to the predicted data (in circles). The slope of the

line is Hc/kB, while the intersection point of the extrapolated line with

the vertical axis is ν0 exp(Sc/kB). Dashed line presents the nucleation

rate predicted by ν0 exp(−Hc/kBT ), in which the activation entropy is

completely ignored, leading to an underestimate of the nucleation rate

by ∼ 20 orders of magnitude. . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Schematics of simulation cells designed for studying (a) homogeneous

and (c) heterogeneous nucleation. In (a), the spheres represent atoms

enclosed by the critical nucleus of a Shockley partial dislocation loop.

In (c), atoms on the surface are colored by gray and atoms enclosed by

the dislocation loop are colored by magenta. Shear stress-strain curves

of the Cu perfect crystal (before dislocation nucleation) at different

temperatures for (b) homogeneous and (d) heterogeneous nucleation

simulation cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

xxiv

5.3 (a) The Helmholtz free energy of the dislocation loop as a function of its

size n during homogeneous nucleation at T = 300 K, σxy = 2.16 GPa

(γxy = 0.135) obtained from umbrella sampling. (b) Size fluctuation

of critical nuclei from MD simulations. . . . . . . . . . . . . . . . . . 92

5.4 Atomistic configurations of dislocation loops at (a) 0 K and (b) 300 K. 93

5.5 The fraction of 192 MD simulations in which dislocation nucleation

has not occurred at time t, Ps(t), at T = 300 K and σxy = 2.16 GPa

(γxy = 0.135). Dotted curve presents the fitted curve exp(−IMDt) with

IMD = 2.5× 108s−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.6 Activation Helmholtz free energy for homogeneous dislocation nucle-

ation in Cu. (a) Fc as a function of shear strain γ at different T . (b)

Gc as a function of shear strain σ at different T . Squares represent um-

brella sampling data and dots represent zero temperature MEP search

results using simulation cells equilibrated at different temperatures.

(c) Fc as a function of T at γ = 0.092. (d) Gc as a function of T at

σ = 2.0 GPa. Circles represent umbrella sampling data and dashed

lines represent a polynomial fit. . . . . . . . . . . . . . . . . . . . . . 97

5.7 Activation free energy for heterogeneous dislocation nucleation from

the surface of a Cu nanorod. (a) Fc as a function of compressive

strain ǫzz at different T . (b) Gc as a function of compressive stress

σzz at different T . Squares represent umbrella sampling data and dots

represent zero temperature MEP search results using simulation cells

equilibrated at different temperatures. . . . . . . . . . . . . . . . . . 99

5.8 The relation between Ec and Sc in the temperature range of zero to

300 K for (a) homogeneous and (b) heterogeneous nucleation. The

relation between Hc and Sc for (c) homogeneous and (d) heterogeneous

nucleation. The solid lines represent simulation data and the dashed

lines are empirical fits of the form Sc = Ec/T∗ or Sc = Hc/T

∗. . . . . 102

xxv

5.9 Contour lines of (a) homogeneous and (b) heterogeneous dislocation

nucleation rate per site I as a function of T and σ. The predictions with

and without accounting for the activation entropy Sc(σ) are plotted in

thick and thin lines, respectively. The nucleation rate of I ∼ 106 s−1

per site is accessible in typical MD timescales whereas the nucleation

rate of I ∼ 10−4−10−9 is accessible in typical experimental timescales,

depending on the number of nucleation sites. . . . . . . . . . . . . . 103

5.10 (a) Nucleation stress of our bulk sample (containing 14,976 atoms) un-

der constant shear strain loading rate γ = 10−3 and (b) nucleation

stress of the nanorod under constant compressive strain loading rate

ǫ = 10−3. The strain rate 10−3 is experimentally accessible loading

rate. The solid lines are the prediction based on the activation free

energy computed by umbrella sampling. The dashed lines are the nu-

cleation stress prediction when the activation entropy is neglected. The

dotted line in (b) is the prediction based on the approximation by Zhu

et al. [153]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.1 (color online) (a) The pre-exponential factor f+c Γ in 2D computed from

Monte Carlo and US. (b) The ratio between the attachment rate f+c

in 2D computed by Monte Carlo and that predicted by Eq.(4.7). (a)

The pre-exponential factor f+c Γ in 3D computed from Monte Carlo and

US. (b) The ratio between the attachment rate f+c in 3D computed by

Monte Carlo and that predicted by Eq.(4.15). . . . . . . . . . . . . . 122

B.2 Droplets in (a) 2D and (b) 3D Ising models randomly chosen from FFS

simulations at different (T, h) conditions. n is the size of the droplet. 124

B.3 The free energy curve F (n) of 2D Ising system at kBT = 1.0 and (a)

h = 0.05, (b) h = 0.06, (c) h = 0.07, (d) h = 0.08, (e) h = 0.09,

(f) h = 0.10 obtained by US (circles) is compared with Eq. (6) (solid

line) and Eq. (8) (dashed line). Logarithmic correction term 54kBT lnn

(dot-dashed line) and the constant term d (dotted line) are also drawn

for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

xxvi


h = 0.04, (b) h = 0.05, (c) h = 0.06, (d) h = 0.07, (e) h = 0.08, (f)

h = 0.09, (g) h = 0.10, (h) h = 0.11, (i) h = 0.12, (j) h = 0.13 obtained

by US (circles) is compared with Eq. (6) (solid line) and Eq. (8) (dashed

line). Logarithmic correction term 54kBT lnn (dot-dashed line) and the

constant term d (dotted line) are also drawn for comparison. . . . . 129


h = 0.015, (b) h = 0.017, (c) h = 0.022, (d) h = 0.025, (e) h = 0.03,

(f) h = 0.035 obtained by US (circles) is compared with Eq. (6) (solid

line) and Eq. (8) (dashed line). Logarithmic correction term 54kBT lnn

(dot-dashed line) and the constant term d (dotted line) are also drawn

for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

C.1 The relation between critical dislocation size nc and the activation

volume Ωc ≡ −∂Gc

∂σ.(a) homogeneous nucleation (b) heterogeneous nu-

cleation. Circles represent the activation volume obtained from the

derivative of Gc with respect to σ. Squares represent the activation

volume data multiplied by 1/S where S is the Schmid factor. Dashed

lines are linear fits to the data. . . . . . . . . . . . . . . . . . . . . . 140

D.1 Gibb’s free energy per atom for both the solid phase (solid line) and

liquid phase (dashed line). The symbols represent data points in

Broughton and Li [198] with squares for the solid phase and circles

for the liquid phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

E.1 (Color Online) Generalized stacking fault energy of different poten-

tial models for gold: (a) 2nn-MEAM [227] (dashed line), 2nn-MEAM∗

(dotted line), and 2nn-MEAM† (solid line); (b) EAM [204] (dashed

line), MEAM [206] (dotted line) and DFT/LDA (solid line). . . . . . 165

xxvii

E.2 (Color Online) Generalized stacking fault energy of different potential

models for silicon. (a) MEAM [206](dashed curve), MEAM∗ (dotted

curve), MEAM† (solid curve). (b) 2nn-MEAM [228](dashed curve). (c)

Tersoff [106](solid curve), SW [105] (dashed curve) (d) DFT/LDA [233]

(solid curve) ,DFT/GGA [233] (dashed curve) . . . . . . . . . . . . . 167

E.3 (Color Online) (a) Pair-correlation functions of the solid (solid line) and

liquid (dotted line) phases of gold described by the 2nn-MEAM [227]

potential at its melting point. (b) The equation of state function in the

2nn-MEAM (dotted line) potential and the new 2nn-MEAM† potential

(solid line). (c) The Gibbs free energy of the 2nn-MEAM (thick lines)

and 2nn-MEAM† (thin lines) potentials for gold. Solid lines for the

solid phase and dashed lines for the liquid phase. . . . . . . . . . . . 171

E.4 (Color Online) (a) Pair-correlation functions of the solid (solid line)

and liquid (dotted line) phases of silicon described by the MEAM [206]

potential at its melting point. (b) The equation of state function in the

original MEAM (dotted line) potential and the new MEAM† potential

(solid line). (c) The Gibbs free energy of the MEAM (thick lines)

and MEAM† (thin lines) potentials for silicon. Solid lines for the solid

phase and dashed lines for the liquid phase. . . . . . . . . . . . . . . 173

E.5 (Color Online) Ellipses defined in Eq. (E.7) for different values of

C (0.8,2.0,2.8). The line segments represent nearest-neighbor bonds

i-j and j-k in BCC (square), FCC (circle), and Diamond-Cubic (aster-

isk) crystal structures, scaled by the second nearest-neighbor distance

rik. In these three crystal structures, the atom j lies on the ellipses

(not shown) corresponding to C = 0.5, 1.0 and 2.0, respectively. . . . 176

E.6 (Color Online) Bond angle distribution functions of liquid phase of sili-

con described by the MEAM† (solid line) ,DFT/LDA (dashed line) [246]

and SW (dotted line) [246]. . . . . . . . . . . . . . . . . . . . . . . . 179

xxviii

F.1 Binary phase diagram of Au-Si. MEAM prediction is plotted in thick

line and experimental phase diagram is plotted in thin line. L corre-

sponds to the liquid phase. Au(s) and Si(s) correspond to the Au-rich

and Si-rich solid phases, respectively. . . . . . . . . . . . . . . . . . . 186

F.2 Gibbs free energy ∆gimp(T ) and enthalpy ∆himp(T ) (a) a Si impurity

within Au crystal. (b) a Au impurity within Si crystal. . . . . . . . . 189

F.3 (a) Liquid free energy Gliq(x, T ) at T = 1250 K. Circles are simula-

tion results, which are fitted to a spline (solid line). A straight line

connecting the liquid free energy of pure Au and pure Si is drawn for

comparison. (b) The free energy of mixing Gmix(x, T ) for the liquid

phase at T = 1250 K. Predictions from the MEAM potential is plotted

in thick line, which is the difference between Gliq(x, T ) and the straight

line shown in (a). Free energy obtained from CALPHAD method [256]

are plotted in thin line. . . . . . . . . . . . . . . . . . . . . . . . . . 190

F.4 (a) Enthalpy of mixing ∆Hliq(x, T ) at T = 1373 K from experiments

(circles), MEAM (thick line), and CALPHAD (thin line). (b) Excess

free energy of mixing ∆GXSliq (x, T ) at T = 1685 K from experiments

(circles), MEAM (thick line), and CALPHAD (thin line). . . . . . . 192

F.5 Common tangent method to construct binary phase diagram from free

energy curves. (a) Gibbs free energy of the three phases, GFCC(x, T ),

GDC(x, T ) and Gliq(x, T ), as a function of composition x at T = 700 K.

All of them are referenced to free energies of pure Au liquid and pure

Si liquid. Common tangent lines are drawn between GFCC(x, T ) and

Gliq(x, T ) (from x1 = 0.011 to x2 = 0.225), and between Gliq(x, T ) and

GDC(x, T ) (from x3 = 0.251 to x4 ≈ 1). (b) Binary phase diagram of

the MEAM Au-Si potential. The phase boundaries at T = 700 K are

determined from the data in (a). . . . . . . . . . . . . . . . . . . . . 193

xxix

Chapter 1

Introduction

1.1 Computational Investigation of Nucleation

Nucleation refers to the formation of small region of a new phase in the background

of a meta-stable phase during a first order phase transition such as melting and

freezing. Such phase transitions are ubiquitous in the nature and hence have been

investigated in a wide range of scientific disciplines, including physics, materials sci-

ence, meteorology, medical science, biology, and nano technology. Phase transitions

that proceed via nucleation include supercooled fluids [1, 2, 3, 4], cloud formation [5],

kidney stone formation [6], polymerization [7], electro-weak phase transitions [8], and

nano materials [9].

To understand and have control over the nucleation processes, it is important to

predict the nucleation rate I as a function of ∆µ and T . Here, ∆µ refers to the

chemical potential difference between the meta-stable phase and the new phase, and

T is the absolute temperature. It is found that the nucleation rate is exponentially

sensitive to ∆µ and T , which makes the prediction of nucleation rate very difficult.

While ∆µ depends solely on T in most single component systems (ignoring stress

effects), it depends on both T and composition in multi-component systems, making

the understanding and prediction of the nucleation process even more challenging.

The nucleation phenomena has been investigated by three different approaches:

1

CHAPTER 1. INTRODUCTION 2

experiment, theory, and computer simulation [10, 11]. If appropriate tools are avail-

able, direct experiments on the system of interest would be an ideal way to understand

the nucleation. However, it is likely that nucleation experiments are affected by the

existence of impurities and surfaces that are hard to eliminate. Even if impurities are

controlled and nucleation rate can be measured with a reasonable accuracy, it is still

difficult to reveal the microscopic detail of the nucleation process from experiments.

Deeper understanding of the microscopic process is crucial to apply the insight gained

from an experiment on a simple system to predict phase transitions of more complex

systems where experimental investigation is very difficult.

Theoretical study can complement experiments by revealing the microscopic detail

based on known physics, and provides rate predictions that can be compared with the

experimental results. In many cases, the qualitative trend obtained from the theory

can be well compared with experiments, but the quantitative matching is difficult

to achieve [10, 11, 12, 13]. Because most theories rely on multiple assumptions that

are difficult to be validated, it is hard to pinpoint which part of the theory causes

the discrepancy. Besides, the theoretical expression on the nucleation rate requires

several pre-determined paramters as input, such as surface energy σ, characteristic

vibration frequency ν, and the chemical potential difference ∆µ at given temperature,

composition, external stress and so on. While the rate prediction is sensitive to these

parameters, it is difficult to accurately measure them in many cases.

As an alternative approach, computer simulations have also played an important

role in probing nucleation processes [14, 15]. The advantage of simulations lies in

creating a model system which is difficult to prepare in a laboratory. For example,

we can model the solidification of liquid that has no impurity, which is an ideal testbed

of homogeneous nucleation theories. We can trace positions and velocities of all atoms

in the model system, which allows a very accurate prediction of the nucleation rate as

well as the detailed microscopic processes. Of course, the interactions among particles

cannot be perfectly modeled and it is impossible to perform a virtual experiment

reproducing what happens in the nature exactly. Still, if we have model systems that

mimick real systems reasonably well, i.e. when the computational models capture

important characteristics of chemical bonding and predict phase diagrams close to


experiments, computer simulations often leads to new discoveries that have not been

predicted from existing theories. For instance, it has been found that multiple order

parameters other than the size of nucleus affect nucleation rates of crystallization in

many circumstances [16, 17]. It was a surprising discovery that the critical nuclei in

a binary suspension of oppositely charged colloid is not the one with the lowest free

energy barrier for nucleation, but the one with the fastest growth rate [18].

While computer simulations have been proved as a powerful tool to study the

nucleation phenomena, it is important to devise smart algorithms that can overcome

the limitations arising from the limited computing power, length scale limitation and

time scale limitation. Length scale limitation refers to the limited number of atoms

that can be modeled by computer simulations. For example, while a drop (about 0.3

cc) of water consists of ∼ 1022 molecules, it takes a few hours of modern CPU (central

processor unit) time to simulate the dynamics of 10, 000 water molecules for a few

hundreds picoseconds even though computationally cheap empirical potential is used

to model water molecules. Fortunately, the length scale problem is relatively easy to

solve and is not a critical obstacle. Because atoms involved in nucleation event can

be as small as a few hundred atoms in many substances, computer simulations of a

few tens of thousands particles are often big enough to capture the formation of the

critical nuclei and to describe the nucleation pathway. Even larger critical nucleus can

be handled by using multiple processors simultaneously. For example, a molecular

dynamics simulation has recently reached the 1011 atoms simulating a 1.5µm cubed

box, employing state-of-the-art parallel computing algorithms [19].

The major limiting factor in the simulation of nucleation process is the time scale

problem. The time step of molecular dynamics simulations must be on the order of

femtoseconds to stablize numerical integrators used to trace the motion of particles,

because the characteristic vibration frequency is around 1013s−1 in most condensed

matter systems. It takes a few days to proceed a few million time steps of simulations

which correspond to a few nano-seconds of simulations time. However, typical time

scale of nucleation events is a few millisecond to a few seconds which is many orders of

magnitude larger than the time scale of conventional molecular dynamics simulations.

It is recognized that a system spends most of time fluctuating around a meta-stable


phase, while a successful nucleation event is extremely rare. When it occurs, the

formation of a stable nucleus can happen within picoseconds. Because it is such

a rare event that controls the onset of phase transitions, many versions of advanced

sampling methods have been developed that captures such rare events selectively [20],

which allows us to estimate the nucleation rate.

This dissertation is devoted mainly to the study of nucleation processes via com-

puter simulations. Using the Ising model [21], the simplest and well-investigated

model of phase transitions (as well as of ferromagnetisms), we systematically test the

validity of the classical nucleation theory (CNT) [22, 23] which is a standard theory

that has been used to describe the nucleation phenomena for almost a century [24].

The validated part of the classical nucleation theory, in combination with computer

simulations, has been applied to predict the rate of dislocation nucleation [25] which

is essential to our understanding of plastic deformation, ductility, and mechanical

strength of crystalline materials. We have employed advanced sampling methods to

overcome the time scale problems when studying both the Ising model and dislocation

nucleation.

1.2 Scope of the Dissertation

The dissertation is organized as follows.

Chapter 2 introduces nucleation theories and experiments relevant to this work.

We begin with a short description of the basic thermodynamics of nucleation such

as the chemical potential difference, nucleation barrier, and population of droplets

of new phase. Nucleation rate predictions from three different nucleation theories

will be presented with special focus on the classical nucleation theory (CNT) and

its two fundamental assumptions. With brief historic review of nucleation experi-

ments, discrepancies between the CNT prediction and experimental results will be

highlighted. In the last part of the chapter, we briefly introduce the concepts of dislo-

cation and explain why dislocation nucleation plays an important role in determining

the mechanical behavior of materials at small scale.


Chapter 3 summarizes the computational methods used in this work. We in-

troduce the Ising model whose demagnetization process has a close analogy to the

nucleation dynamics. We review interatomic potentials that were developed to de-

scribe different bonding mechanisms, with a special focus on the Cu embedded-atom-

method (EAM) potential which is used in dislocation nucleation study. An overview

of molecular dynamics (MD) and Monte Carlo (MC) methods is presented with pros

and cons of each method. We also describe two advanced sampling methods that can

overcome the timescale limit of conventional MD and MC simulations.

In chapter 4, we test the validity of the classical nucleation theory (CNT) by

calculating the individual components of CNT via computer simulations of the Ising

models. We open this chapter with a brief description on how nucleation theories are

applied to the Ising model. Using two independent simulation techniques, we confirm

the fundamental assumption that nucleation process can be described by 1D Markov

chain, under a wide range of conditions in both 2D and 3D Ising models. However,

it is found that the free energy predicted by CNT does not match with numerical

results, unless appropriate correction term is added. Our analysis confirms that the

nucleation rate by CNT can be predicted accurately if a correct free energy barrier

obtained by umbrella sampling is used as an input.

Chapter 5 provides an in-depth description on the prediction of dislocation nucle-

ation rate based on the classical nucleation theory in combination with the umbrella

sampling technique. The results reveal very large activation entropies, originated

from the anharmonic effects, which can alter the nucleation rate by many orders of

magnitude. Here we discuss the thermodynamics and algorithms underlying these

calculations in great detail. In particular, we prove that the activation Helmholtz

free energy equals the activation Gibbs free energy in the thermodynamic limit, and

explain the large difference in the activation entropies in the constant stress and con-

stant strain ensembles. We also discuss the origin of the large activation entropies

for dislocation nucleation, along with previous theoretical estimates of the activation

entropy.

Finally, chapter 6 reviews the results presented in the dissertation and discuss

future research opportunities. One possibility is the investigation of the gold catalyzed


growth of silicon nanowire via vapor-liquid-solid (VLS) mechanism which involves

silicon crystal nucleation inside gold-silicon eutectic liquid alloy. As a preparation for

the project, we have developed a Au-Si potential that is fitted to the binary phase

diagram and efficient free energy calculation method for solid and liquid alloy. These

contributions are presented in Appendices D, E, and F.

Chapter 2

Background and Motivation

This chapter reviews theoretical and experimental studies of nucleation phenomena.

We begin with a brief explanation on the thermodynamic origin of the nucleation

barrier in the first order transitions. We present nucleation rate predictions from

three different nucleation theories, which is followed by a concise review of experiments

for testing the nucleation theories. The relation between dislocation nucleation and

materials strength at small scale will be discussed in the last section of the chapter.

2.1 Basic Thermodynamics of Nucleation

Most first order phase transitions require the appearance of small nuclei of the new

phase as a prerequisite. Nucleation refers to such localized budding of new phases

in the background of the ambient phases, i.e. meta-stable phases [10, 11]. Some

examples of the emergent phases include gaseous bubbles, small crystallites, and

liquid droplets in the volume of supersaturated solution of gas, undercooled liquid,

and undercooled vapor, respectively. It is statistical fluctuations that create nuclei

that undergo the transient appearance and disappearance. Only when a “critical”

size is exceeded, the dissolution probability of nuclei becomes small enough and the

new phase evolves into a macroscopic size. The work of formation of the critical

nucleus, so called “nucleation barrier”, is supplied by thermal fluctuation.

To quantitatively describe the process in terms of physics, we consider a volume

7

CHAPTER 2. BACKGROUND AND MOTIVATION 8

containing a original phase with chemical potential µ1 (i.e. the Gibbs free energy

per particle) which is a function of temperature T . We will consider only a single

component system and ignore stress effects for simplicity. Formation of a droplet

of new phase with chemical potential µ2 costs the surface free energy S σ where S

is the surface area of the droplet and σ is the interface free energy between two

phases. Hence, the change of the Gibbs free energy upon the formation of the droplet

containing n particle is

∆G(n) = −n(µ1 − µ2) + S(n)σ. (2.1)

The chemical potential difference ∆µ = µ1 − µ2 is the thermodynamics driving force

that induces the phase transition and becomes positive at conditions where the new

phase is thermodynamically favored, i.e. µ1 > µ2. For example, the chemical potential

µs of solid is lower than the chemical potential µl of liquid below the melting point, Tm,

and the undercooled liquid can lower the Gibbs free energy of system by transforming

into the solid phase. The surface area S(n) has sublinear power dependence S =

α n1−1/d where α is a geometrical constant and d is the dimension of the system. When

the droplet has the equilibrium shape determined by the Wulff construction [33], the

Gibbs free energy is minimized for a given n, which determines the value of α for each

system. Combining the volume contribution −n ∆µ and the surface contribution

S(n) σ, we find that ∆G displays a maximum at some critical size nc as shown in

Fig 2.1 and nc is given by

nc =

[

(1− 1/d)ασ

∆µ

]d

(2.2)

By plugging nc into the Eq. (2.1), we find the maximum value of ∆G, or the nucleation

barrier Gc to be

Gc = nc∆µ

d− 1=

[(1− 1/d)ασ]d

(d− 1)∆µd−1(2.3)

which is also called as the “activation Gibbs free energy” in some contexts such as

solid state rate processes.

When the size of droplet is smaller than critical size, i.e. n < nc, the evolution

is likely to lead to the dissolution of the droplet. In the opposite case of n > nc, the


Fre

e E

nerg

y ∆

G

n

volumesurfacetotal

nc

Gc

Figure 2.1: ∆G versus n showing volume and surface contributions resulting in apeak at n = nc.

droplet size is likely to increase in order to lower the Gibbs free energy of the system.

The presence of the critical nucleus size and the associated nucleation barrier is the

reason that the water can be supercooled and the solution can be supersaturated.

The probability of forming the critical size droplet is given by the Boltzmann factor

exp(−Gc/kBT ). Under small undercooling (or small supersaturation) where the driv-

ing force ∆µ is small, the nucleation barrier Gc becomes large, because Gc is inversely

proportional to ∆µd−1 as in the Eq. (2.3). The large nucleation barrier makes the

probability of forming the critical droplet extremely small. However, the probability

exp(−Gc/kBT ) raises very quickly for deeper undercooling (or supersaturation), due

to the increase of the driving force ∆µ.

In practice, nucleation of the new phase rarely takes place homogeneously in the

bulk of the current phase. Much smaller undercooling or supersaturation is usually

achieved in experiments compared to that predicted from the homogeneous nucleation

barrier. In most situation, nucleation initiates on the walls of containment vessel or

on an impurity particles. Homogeneous nucleation can be achieved only by dispersing


the liquid into droplets small enough so that there is an appreciable chance of not

having any heterogeneous nucleation sites in a droplet [10, 11].

Figure 2.2: Schematic of a solid spherical cap with radius of curvature r and contactangle θ forming a wall.

To examine how heterogeneity affects the nucleation barrier, we consider formation

of spherical cap of new phase on the wall, as shown in Fig. 2.2. We have three different

interface energies to consider: σnc is the interface energy between the new phase and

the current phase. σnw is the interface energy between the new phase and the wall.

σcw is the interface energy between the current phase and the wall. When σnc is larger

than the difference between σcw and σnw, i.e. σnc > |σcw−σnw|, the new phase can wet

on the wall with the angle θ determined by the Young’s equation σcw = σnw+σnc cos θ.

Because of the mechanical equilibrium, the wetting angle θ does not depend on the

size of nucleus size1. It is straightforward to show that the critical radius rc, more

precisely the critical radius of curvature, is identical to the homogeneous nucleation,

independent of the wetting angle [33]. Then, the work of forming the critical nucleus,

Gheteroc is written as

Gheteroc = Ghomo

c

Vc

V0(2.4)

where Vc is the volume of the spherical cap of critical size and V0 is the volume

1For simplicity, we ignores the line tension at the trijunction


of the complete sphere with radius rc. Ghomoc refers to the homogeneous nucleation

barrier in Eq. (2.3). The ratio Vc/V0 is a function of the wetting angle θ, given

by (1 − cos θ)2(2 + cos θ)/4 which is always less than the unity, and becomes half

at θ = π/2. The reduction of nucleation barrier at identical ∆µ explains why the

heterogeneous nucleation preferentially occurs in most circumstances.

2.2 Nucleation Rate Predictions From Nucleation

Theories

Having established the concept of nucleation barrier and critical droplet, we turn

our attention to the kinetic model for droplet formation and the nucleation rate

prediction. Here, we will briefly review three different versions of nucleation theories

that are relevant to the studies in the present dissertation.

In 1926, Volmer and Weber [34] first introduced the concept of critical droplet and

estimated the nucleation rate in a supersaturated vapor by the following equation,

I ≈ N f+c exp

(

− Gc

kBT

)

(2.5)

where N is the equivalent nucleation site and Gc is the formation free energy of

the critical droplet. f+c is the attachment rate of molecules to the critical droplet.

N exp(−Gc/kBT ) is the “equilibrium” population of the critical droplet. The Volmer-

Weber theory also gives the droplet free energy function in the form of Eq. (2.1). This

work was the first attempt to predict the nucleation rate with the concepts of critical

droplet, its free energy, and the attachment rate of molecules are developed. Other

dynamical factors, such as multiple recrossing of the free energy barrier, originally

ignored in the Volmer-Weber theory, was recognized in later studies.

The concepts recognized by Volmer-Weber have served as a basis for further

development of so called “classical nucleation theory” by Farkas [35], Becker and

Doring [36], Zeldovich [37], and Frenkel [38], and remain important to date for our

understanding of the nucleation process. They assumed that clusters consisting of


n particles (an atom or a molecule), Λn, grow or shrink by the addition or loss of a

single particle Λ1, following a series of bimolecular reactions:

Λn−1 + Λ1

f+n−1−−−−−−f−

n

Λn

Λn + Λ1

f+n−−−−−−

f−

n+1

Λn+1 (2.6)

Here, f+n is the rate of single-particle attachment to a cluster of size n and f−

n is the

rate of loss. It is implicitly assumed that reactions of clusters with dimers, trimers,

etc., are too infrequent to be comparable with single particle attachment. In short,

the nucleation process is modeled by the time evolution of the droplet population as

an one-dimensional Markov chain.

In 1935, Becker and Doring [36] obtained a steady-state solution for the nucleation

rate. Since then, the term “Becker-Doring theory” and the “classical nucleation

theory” are used interchangeably in literatures. While Volmer and Weber considered

the “equilibrium” population of critical droplet, Becker and Doring considered the

droplet population during steady-state nucleation process. Detailed derivation of the

nucleation rate from the classical nucleation theory can be found in Appendix A.1.

This solution finally pinpoints the kinetic prefactor2 in the nucleation rate, which is

expressed as

I = N f+c Γ exp

(

− Gc

kBT

)

(2.7)

where Γ is known as the Zeldovich factor [37, 38] defined by

Γ ≡(

η

2πkBT

)1/2

, η = − ∂2G(n)

∂ n2

∣

∣

∣

∣

n=nc

(2.8)

The flatter is the free energy curve near the critical size nc, the smaller is the Zeldovich

factor [37]. For two systems having the same free energy barriers, the system with the

flatter free energy landscape near the barrier has more diffusive nucleation dynamics

2Here the word “prefactor” means the factor in front of the exponential term.


and its nucleation rate is lower. Hence the Zeldovich factor captures the multiple re-

crossing of the free energy barrier. A systematic investigation of the relation between

the Zeldovich factor and recrossing can be found in Pan and Chandler [39].

There are two fundamental assumptions in CNT that are independent of each

other. First, the time evolution of the droplet population can be described by a 1D

Markov chain model as in Eq. (2.6). Second, the free energy of a droplet can be

written as Eq. (2.1), where σ is the surface tension of macroscopic interfaces. We

can test the first assumption if we can compute the nucleation rate using a numerical

method that does not rely on the Markovian assumption and compare it to Eq. (2.7).

We can test the second assumption by computing the free energy function by umbrella

sampling. Our numerical results using the Ising model shows that the nucleation rate

can be predicted from CNT accurately if correct free energy barrier obtained from

computer simulation is used as input, which confirms the first assumption. However,

it is found that additional correction factors to Eq. (2.1) are required to describe the

free energy of droplet.

In 1967, Langer [40] developed a field theoretical approach to take into account

all degrees of freedom of a droplet when calculating the steady-state solution for the

nucleation rate. This is a generalization of the Becker-Doring theory to incorporate

microscopic (fluctuation) degrees of freedom of the droplet. Langer’s field theory was

later used to derive a correction term to the nucleation rate in the droplet model [41,

42, 43, 44]. In the literature, the field theory correction is usually expressed as an

extra term in the pre-exponential factor in Eq. (2.7). But it can also be expressed as

a modification to the free energy function in Eq. (2.1), changing it to

G(n) = −∆µn + S σ + τkBT lnn (2.9)

While both approaches can give rise to similar predictions to the nucleation rate, we

will show later that it is more self-consistent to include the correction term in the free

energy.

The field theory predicts that, for an isotropic medium, the coefficient of logarith-

mic correction term is τ = 54in 2D [45] and τ = −1

9in 3D [46]. However, it was later


predicted that the shape fluctuation of a 3D droplet should be suppressed below the

roughening temperature [44], which leads to τ = −23. Our numerical results confirm

the τ = 54prediction for the 2D Ising model, under a wide range of temperatures.

This contradicts an earlier numerical study [47] which suggests that τ is close to zero

at low temperatures and only rise to 54at high temperatures. On the other hand,

our numerical results are not consistent with any of the above theoretical predictions

of τ for the 3D Ising model. This problem may be related to the finding by Zia

and Wallace [48], that the excitation spectrum around a 3D droplet is affected by the

anisotropy of the medium, but that around a 2D droplet is not affected by anisotropy.

Because the Ising model is fundamentally anisotropic (e.g. with cubic symmetry), the

field theoretic prediction based on isotropic medium may not apply to the 3D Ising

model.

All the nucleation theories mentioned above share several fundamental assump-

tions: (1) only isolated droplets are considered and the interaction between droplets

is neglected; (2) a droplet is assumed to be compact with a well-defined surface; (3)

the surface energy expression derived from a macroscopically planar surface can be

applied to the surface of a very small droplet. The first two assumptions are valid at

temperatures much lower than the critical temperature and at small chemical poten-

tial difference ∆µ. Under these conditions, the density of droplets is very small and

each droplet tends to be compact. We will not test these two assumptions in this

study. In other words, our numerical simulations will be limited to the low temper-

ature and small chemical potential difference ∆µ where these assumptions should be

valid. Models that account for droplet interactions exist in the literature [49] but will

not be discussed in this dissertation.

2.3 Nucleation Experiments

In 1724, Fahrenheit published the results of a systematic set of experiments that shows

the evidence of undercooling [50]. Surprisingly, he found that the water remained a

fluid, even when left outside overnight in an air temperature of 15F, while the freezing

point of water is 32F. When small ice particles were introduced into the supercooled


water, however, crystallization followed immediately, with the temperature of the ice-

water mixture rising to 32F 3. It took enormous numbers of studies [51, 52, 53, 54]

to establish that undercooling is a common properties of all liquids, regardless of

nature of their chemical bonding. This initiates the development of various versions

of nucleation theories [34, 35, 36, 37, 38, 40, 55, 56, 57] and numerous experimental

studies [58, 59, 60, 61, 62, 63].

In this section, we will present several findings from nucleation experiments to

demonstrate the applicability of nucleation theories. Out of many nucleation theories,

we will focus on the classical nucleation theory, a standard theory of nucleation,

that will be extensively tested and used in the remaining part of this dissertation.

Classical nucleation theory predicts that in 3D, the nucleation barrier is proportional

to the third-power of surface free energy σ between two phases as shown in Eq. (2.3),

which means that exact surface energy is crucial for the testing nucleation rate. We

will mainly discuss the gas-liquid transition experiments that is considered to be

a stringent test of nucleation theories because the surface tension between gas and

liquid phases is easily measurable to high accuracy as a function of temperature. On

the other hand, the liquid-solid surface energy is difficult to measure, and is available

only for certain special cases [64]. Moreover, while the nucleation process take place

at a temperature much lower (possibly more than a few hundred degrees lower) than

the melting point, the surface energy data is measured at the melting point.

In the nucleation experiment of gas-liquid transition, the thermodynamic driving

force is adjusted by the pressure of the gas. The vapor pressure Peq is the pressure

of the gas in the thermodynamic equilibrium with its condensed phases in a closed

system. The vapor pressure of any substance increases non-linearly with temperature

according to the well known Clausius-Clapeyron relation

dPeq

dT=

L

T∆V(2.10)

where dPeq/dT is the slope of the coexistence curve, L is the latent heat, and ∆V is

3In a modern point of view, the undercooling of water can be explained by the nucleation barrierthat is significantly larger than the thermal fluctuation kBT at 15F. The immediate crystallizationfrom the ice particle can be understood by the fast growth of a nucleus larger than the critical size.


the volume change of the phase transition. The chemical potential difference between

liquid and gas can be written as

∆µlg = kBT ln(P/Peq) (2.11)

where P is the pressure of gas, T is the absolute temperature, and the ratio P/Peq

is called the supersaturation S. The condensation of gas occurs via the nucleation of

small liquid droplets when S > 0.

The traditional method of studying gas-liquid nucleation uses a cloud chamber.

At a given temperature, the supersaturation S is adjusted until droplet formation is

observed. Because the nucleation rate changes so rapidly with supersaturation, but a

critical value Sc where I passes through a magnitude of order 1 (per second per cubic

micron) can be approximately determined instead of the absolute rate I. The critical

supersaturation Sc can then be compared with the prediction of classical nucleation

theory. It is found that the classical nucleation theory predicts values of Sc that are

typically accurate within 10% for most substances [59]. It should be underscored

that a variation of 10% in S can leads to changes in I by many orders of magnitude,

because I is exponentially sensitive to the nucleation barrier which is proportional to

∆µ−2, i.e. (lnS)−2.

Development of the upward thermal diffusion cloud chamber [65] and the fast-

expansion piston cloud chamber [66, 67] made possible the measurement of actual

rate I, instead of critical supersaturation Sc. The consensus of many works on the

nucleation rate is that the variation of nucleation with supersaturation predicted by

CNT

ln I ≈ A(T )−B(T )/(lnS)2 (2.12)

is approximately correct, but that the temperature dependence is not. As a example,

a experimental data on the water nucleation rate from vapor is presented in the

Fig. 2.3. For many substances, predicted nucleation rates from CNT are significantly

more sensitive to the temperature than experiments, underestimating the rate at low

temperature, and overestimating it at high temperature [68, 69, 70, 71, 72].


Figure 2.3: Comparison of experimental nucleation rates of water from vapor (circles)with the predictions of the classical theory [72]. The full lines belong to the classicalBecker-Doring (BD) theory. Logarithmic scales are used for both axes.

To test the validity of the classical nucleation theory, we can compare the exper-

imental results with two nucleation theorems whose applicability is not restricted to

the CNT assumption that droplet free energy has only surface and volume contribu-

tions as in the Eq. (2.1) (See Appendix A.2). Nucleation theorems states that (1) the

supersaturation dependence of the nucleation rate I is determined by the size nc of

critical droplet, i.e.∂ ln I

∂ lnS

∣

∣

∣

∣

T

= nc (2.13)

and (2) the temperature dependence of the nucleation rate is determined by the

formation energy ∆U 4 of a droplet with size nc in the absence of supersaturation,

i.e.∂ ln I

∂T

∣

∣

∣

∣

S

=∆U

kBT 2. (2.14)

4For the definition of ∆U , refer to Appendix A.2.


What can be inferred from the experimental tests on the classical nucleation the-

ory? Correct S dependence, but wrong temperature dependence suggests that the

classical nucleation theory predicts the size of the critical droplet with relatively small

error, but fails in describing the absolute value of the formation energy of the clus-

ter. In other words, the functional form Eq. (2.1) lead to reasonable prediction on

the size of critical droplet but overestimates the nucleation barrier. This states that

the nucleation barrier can be correctly predicted if an appropriate constant term or

slowlying varying function of n is added in the free energy as a function of size n,

the number of particles in the cluster. There are three possible corrections that can

improve the free energy function.

First, note that the free energy of forming a droplet of size n at the equilibrium

vapor pressure Peq is written as ∆G(n) = S(n)σ in CNT. However, it does not hold

down to size n = 1. The distribution of n-mer is predicted by N exp(−∆G(n)/kBT )

where N is the number of monomers. Because the ∆G must be zero for n = 1 by

definition, the contribution at n = 1 must be subtracted to make the expression ‘self-

consistent’, and use ∆G(n) = σ(T )S(n)−S(1) [73]. Due to this inconsistency, the

nucleation barrier would be overestimated by the classical nucleation theory.

Second, while the surface tension σ(T ) is measured at planar interface using a bulk

sample, σ(T ) decreases with decreasing size of droplet [74, 75, 76] in general, as it

loses long-range interactions. If the size dependence of σ(T ) is ignored and bulk value

is used to compute ∆G, the free energy barrier would be overestimated. Still, the

size of critical droplet nc can be predicted correctly, if the critical size is big enough

that bulk value of σ is applicable to compute the droplet formation free energy of the

critical droplet.

Third, it has been shown that a logarithmic correction term τkBT lnn may be

required to include the microscopic (fluctuation) degrees of freedom of droplet from

Langer’s field theory [40] and other non-classical theories. Because the logarithmic

function increases very quickly near the origin, but varies slowly at large n, this term

would mostly change the free energy barrier and would not affect the critical size nc

as much. Considering that negative τ has been predicted for the droplets in three

dimension [40], the nucleation barrier of real system would be smaller than the CNT


prediction where lnn term is ignored. These issues will be revisited later in the

Chapter 4 where we test the validity of the classical nucleation theory using the Ising

model.

There are many cases where even the measured critical supersaturation is largely

deviated from the prediction of classical nucleation theory. It has been shown that

the classical nucleation theory underestimates the nucleation barrier for highly polar

fluids such as acetonitrile, benzonitrile, nitromethane, and nitrobenzen [77, 78]. Com-

puter simulation study [79] found that the small clusters, that initiate the nucleation

process, are not compact spherical objects, but are chains, in which the dipoles align

head-to-tail. The relative large interface area induces a larger nucleation barrier than

the CNT prediction which is based on the compact droplet assumption. Another case

where substantial deviation from classical theory are seen is in the condensation of

liquid metals such as mercury from the vapor [63]. A reasonable explanation for the

effect is that the small clusters of mercury atoms are insulators, with much smaller

effective surface tensions than those measured for bulk metallic mercury. The basic

assumptions of classical nucleation theory do not hold in these substances, and the

investigation of these system is beyond the scope of the present dissertation.

We focus on the discrepancy between the nucleation rate and CNT prediction in

the substance where the two basic assumptions 5 on nuclei hold. The nucleation in the

Ising model at low temperature and low magnetic field condition can be considered

as the ideal model system where the CNT can be tested. While experiments have

enlightened many details of the nucleation process, there are still a significant room to

improve. Other than the surface tension σ(T ), it is practically impossible to directly

measure the attachment rate f+c , Zeldovich factor Γ, and the formation free energy

curve ∆G(n) from experiments. Using computer simulations, we calculate every

component of theory independently and provide a more stringent test of nucleation

theory (See Chapter 4).

5(1) Density of droplets is small enough that the interaction between them is negligible. (2) Adroplet is compact with a well-defined surface.


2.4 Dislocation Nucleation and Materials Strength

at Small Scale

Figure 2.4: Schematic of dislocations [80]. (a) An edge dislocation is a defect wherean extra half-plane of atoms is introduced mid way through the crystal, distortingnearby plane of atoms. (b) A screw dislocation is a shear ripple extending from sideto side.

Nucleation has mostly been studied as an initial step of first order phase transi-

tions. Here, we will introduce the deformation of crystalline materials at small scale

as another physical process where nucleation plays an important role. Instead of

nuclei of new phases, we will consider nucleation of a dislocation loop. Dislocations

are line defects that exist in crystalline materials and schematics of dislocations are

presented in Fig. 2.4. The Burgers vector represents the magnitude and direction

of the lattice distorsion of a dislocation. In an edge dislocation, the Burgers vector

and dislocation line are at right angles to one another. In a screw dislocation, they

are parallel. Different crystal structures have different magnitude and direction of

Burgers vector.

It is well known that the mechanical properties of crystals are controlled by defects

such as dislocations. In 1926, Frenkel estimated the ideal strength of a crystal to be


Figure 2.5: Schematic of edge dislocation motion that induces plastic strain [80].

around one-tenth of its modulus [81]. Frenkel suggests the simple argument that

plastic deformation would occur when the applied shear stress was sufficient to make

adjacent planes of atoms glide rigidly over one another. However, most metallic

materials we exploit today deform at around one-thousandth of their moduli and it is

clear that the Frenkel model does not apply. This is because the plastic deformation

could be initiated by the motion of dislocation at stresses well below the ideal shear

strength as shown in Fig. 2.5. Frenkel envisioned the direct deformation from (a) to

(f) to define ideal shear strength, which requires all atomic bondings in the shear plane

break at once. However, the sequential deformation from (a),(b),. . . ,(f) associated

with dislocation motion requires only atomic bondings along the dislocation line to

break. This explains the orders of magnitude discrepancy between the Frenkel’s rigid

shear model and the observed yield strength.

It is known that a sufficiently large sample whose length scale is larger than a

micrometer have preexising dislocations that can accomodate plastic deformation. In

contrast, for samples at small scale where surface-to-volume ratio is very large, most

dislocations are close or connected to free surfaces that serve as dislocation sinks, the

density of mobile dislocation is much smaller than that in the bulk [82]. Starvation

of dislocation can be achieved by either thermal annealing or mechanical annealing

as shown in Fig. 2.6. Thus, an effective way to raise the strength of materials close to


(a) (b) (c)

Figure 2.6: (a) and (b) Two consecutive in situ TEM compression tests on a FIBmicrofacbricated 160-nm-top-diameter Ni pillar with 〈111〉 orientation [89]. (a) Dark-field TEM image of the pillar before the tests; note the high initial dislocation density.(b) Dark-field TEM image of the sam pillar after the first test; the pillar is now freeof dislocations. (c) MD simulation of nanoindentation process [95]. Snapshot ofdislocation nucleation at the first plastic yield point on Au(111), Lower figure showsa two-layer-thick cross section of a (111) plane containing the partial dislocation loopon the right in the upper figure.

the ideal strength would be to reduce the length scale of the material, which reduces

the dislocation density. This has been demonstrated by series of experiments on

focused-ion-beam-carved nanopillars [83, 84]. It is shown that the flow stresses of

single crystal micrometer-sized pillars increase as the sample size reduces [83]. For

a gold nanopillars, as the pillar diameter approached hundreds of nanometers, the

measured flow strength is 800 MPa [84, 85] which is close to the ideal strength of 850

MPa - 1.4 GPa 6 predicted by first-principles calculations [86, 87]. The comparison

between micropillar experiments and theoretical ideal strength of various materials

can be found in the literature [88].

Instead of shrinking length scale of the tested sample, the reduction of high stress

contact zone under nanoscale indentor [90, 91] also allows one to study near-ideal

strength behavior. Since elasticity is governed by the same family of equations as

the electromagnetism, a spherical nanoindentor tip works like a lens, projecting the

6The ideal strength prediction significantly depend on the loading conditions assumed in thecalculation.


applied force to a focus, or the maximum shear stress zone inside sample, away from

the surface. This enables one to measure the bulk properties near the high stress

focus spot, which otherwise could be dominated by the surface. MD snapshot of

nanoindentation in Fig. 2.6 (c) shows that the plastic deformation indeed begins at

the region away from the surface. The variety of nanoindentation experiments shows

that the inferred shear strengths have at least the same order of magnitude with the

ideal strength calculations [88].

How do we interpret very high strength measured from these experiments? Cer-

tainly, it cannot be explained by the motion of preexisting dislocations, which would

cause plastic deformation at a stress several orders of magnitude smaller than ideal

strength. This suggests that dislocation nucleation plays a more important role in

determining the mechanical behaviors as materials shrink in size.

In accordance with the effort to make useful devices and materials at smaller scale,

dislocation nucleation research has gained a significant attention not only for the relia-

bility of microelectronic devices [92], but also as a responsible mechanism for incipient

plasticity in nanomaterials [88, 93, 94, 95] and nanoindentation [96, 97, 98]. To explain

experimental data, several dislocation nucleation mechanisms originated from differ-

ent sources have been proposed. For instance, there are two different explanations for

the deformation mechanism of micropillar [99]: dislocation motion and multiplication

around the single arm source (or truncated Frank-Read source) locating inside the

micropillar versus dislocation nucleation from the surface of micropillar. While ho-

mogeneous dislocation nucleation has been suggested as a responsible mechanism for

the incipient plasticity in nanoindentation, activation volume and nucleation barrier

from the nanoindentation experiments at different temperature seem to correspond

the values of heterogeneous dislocation nucleation [97].

To pinpoint the underlying mechanism, it is very important to accurately com-

pute the fundamental properties such as dislocation nucleation, multiplication, mo-

tion, and etc. One of the most important fundamental quantities of interest is the

dislocation nucleation rate I as a function of stress σ and temperature T . Continuum

and atomistic models have been used to predict dislocation nucleation rate and they

both have limitations. The applicability of continuum theory becomes questionable


because the critical nucleus can be as small as a few lattice spacings. In addition,

the continuum models are often based on linear ealsticity theory, while dislocation

nucleation typically occur at high strain conditions in which the stress-strain rela-

tion becomes non-linear. These difficulties do not exist in molecular dynamics (MD)

simulations, which can describe atomistic details of dislocation nucleation. Still, the

timescale of MD simulations is about ten orders of magnitude smaller than the ex-

perimental timescale. In the present dissertation, we study the dislocation nucleation

from perfect crystal free of preexising defects: homogeneous nucleation from bulk

and heterogeneous nucleation from the surface of nanowire. We will show that the

dislocation nucleation rate can be accurately predicted by classical nucleation theory,

the validity of which has been tested using the Ising model study.

Chapter 3

Computational Methods

Computer simulations bridge between microscopic length and time scales and the

macroscopic properites measured in the laboratory. For example, given an interatomic

interaction model, predictions of bulk properties can be made such as pair-correlation

function, equation of state, diffusion coefficient and etc. Because interatomic poten-

tial model relies on a theoretical description on the molecular bonding, we can test

the model by comparing the various properties obtained from simulations with ex-

perimental results. Using the tested model, we can carry out a virtual experiment on

the computer that are difficult or impossible in the laboratory.

Simulations can also act as an important tool to test a theory that describe and

predict kinetic reactions in materials. Theoretical derivation often relies on several

assumptions and its prediction often requires several input parameters that is hard

to obtain by experiments. The validity of a theory can be tested by conducting simu-

lations using a a toy model that satisfies the assumptions on which a theory is based.

Ultimately, we aim to have a predictive power on the rate and microscopic details in

a physical reactions such as phase transitions and deformations. For reactions hap-

pening at a rate much slower than the timescale of brute force molecular dynamics

(MD) or Monte Carlo (MC) simulation, advanced sampling methods are employed.

This chapter is devoted to the introduction to the interatomic potential, MD, MC,

and advanced samplings that are used in the this dissertation.

25

CHAPTER 3. COMPUTATIONAL METHODS 26

3.1 Ising Model

The Ising model was originally invented as a model of ferromagnetism by physicist

William Lenz [100], and the problem has been given as a doctoral research [21] for

his student Ernst Ising after whom it named. The Ising model is described by the

following Hamiltonian

H = −J∑

〈i,j〉

sisj − h∑

i

si (3.1)

where J > 0 is the coupling constant and h is the external magnetic field. The

spin variable si at site i can be either +1 (up) or −1 (down), and the sum∑

〈i,j〉 is

over nearest neighbors of the spin lattice. While variety of spin lattices have been

studied for two-dimensional (2D) and three-dimensional (3D) Ising models, we will

consider only square lattice and cubic lattice for 2D and 3D Ising model for simplicity.

The one-dimensional Ising model was solved by Ising [21] and turned out to have no

spontaneous magnetization. The 2D square lattice Ising model is much harder and

the complete analytic description was given much later by Lars Onsager [101]. Fig. 3.1

presents the magnetization of the 2D square lattice Ising model, which clearly shows

the boundary at the critical temperature Tc ∼ 2.269J/kB between ferromagnetic

phase with spontaneous magnetization and paramagnetic phase. Below the critical

temperature, majority of spins aligns in either up or down direction in the absence of

external field. Above the critical temperature, such spontaneous ordering disappears

due to thermal fluctuation. This capture the essential physics of ferromagnetism. For

the 3D cubic lattice Ising model, the magnetization can be obtained numerically and

the results are very similar to the 2D Ising model, except that the critical temperature

Tc ∼ 4.490J/kB is higher due to the increase of number of neighbor spins.

How can the Ising model, a model developed for the ferromagnetism, have an

analogy to the nucleation phenomena? For a system subjected to a positive external

field h > 0 at T < Tc, the symmetry between two ordered states is broken: a state with

positive magnetization has lower energy than a state with negative magnetization.

Suppose that we study the relaxation of the magnetization starting from an initial

state magnetized opposite to the applied field h, when the dynamics of the Ising


(a)

(b) (c)

Figure 3.1: (a) Magnetization M = 1N

∑

si of the 2D square lattice Ising model asfunction of temperature in the absence of external field, i.e. h = 0. At T < Tc =2/ ln(1 +

√2)J/kB ∼ 2.269J/kB, majority of spins in the system are spontaneously

aligned in either up or down. However, T > Tc, spontaneous ordering disappearsbecause of thermal fluctuation. (b) schematic of ferromagnetic state below Tc. (c)schematic of paramagnetic state above Tc.


model is described by a single-spin-flip Monte Carlo simulation (See Section 3.4).

Here, we can find an analogy to the nucleation phenomena, as shown in Fig. 3.2 (a).

The interface between down spin domain and up spin domain is analogous to the

liquid-solid interface in the nucleation of solid nuclei in liquid. For a planar interface,

the interface formation costs 2J of energy, as the interface energy σLS costs in the

liquid-solid interface formation 1. Because of the external field h, an up spin has self

energy −h while down spin has h, which is analogous to the solid atom having lower

chemical energy than the liquid atom at T < Tm. The energy reduction, −2h, of spin

flip can be compared with the chemical potential difference −∆µ. Because of this

analogy, the transition from negative magnetization state to positive magnetization

state occurs via nucleation of an island of up spin, as shown in the Fig. 3.2 (b). The

Ising model captures the essential physics of nucleation and has been used as a model

system for studying nucleation for several decades. We will revisit this problem later

in the Chapter 4 and discuss the test of nucleation theory in detail.

(a)

(b)

Figure 3.2: (a) Analogy between the Ising model demagnetization and nucleationphenomena. (b) The snapshot of Monte Carlo simulation of demagnetization in thepresence of an external field h > 0. The transition from the down spin dominatedmeta-stable state to the up spin dominated stable state occurs via nucleation of smallisland of up spins.

1Planar interface energy is 2J only at T = 0. At higher temperature, interface energy reducesdue to thermal fluctuation


3.2 Interatomic Potential

In order to model dislocation nucleation, we need a model to describe the interac-

tions between atoms in the crystal. The interatomic potential V is a function of the

coordinates of all atoms, i.e.

V (~r) = V (~r1, ~r2, · · · , ~rn) (3.2)

where ~rs refer to the coordinates of n atoms in the system of interest. The force ~fi

acting on atom i can be obtained by

~fi ≡ −∂V

∂~ri. (3.3)

In principle, the first-principles (ab initio) quantum calculation methods 2 can be

used to obtain the interatomic potential and the force field. In addition to the force

field, a vast amount of informations can be obtained, such as density of electronic

states and other electronic properties. However, due to large comptutation cost of

the method, it is inhibitively expensive to handle a simulation cell having more than

a few hundreds of atoms. Because the critical droplet size of nucleation can easily

be as large as a few hundred atoms and surrounding medium have more than a few

thousands atoms, the application of ab initio method is not adequate to describe

nucleation phenomena.

Compromising a modest level of the accuracy for the large improvement of effi-

ciency, various empirical methods have been developed for a long decades that can

describe various types of bonding in materials [103]. The interaction among atoms is

described by a set of simple equations that give a reasonable description of the depen-

dence on the coordinates of all atoms. The equations is tuned by a set of parameters

that are linked to essential physics of the interatomic interactions. Representative

models include Lennard-Jones (LJ) potential for inert gases, Stillinger-Weber (SW)

or Tersoff potential for semiconductors, embedded-atom-potential (EAM) for metals,

2Although various approximations may be used, these are based on theoretical considerations,not on empirical fitting.


modified Eeambedded-atom-method (MEAM) for both metals and semiconductors,

and etc. Each formalism is written in a way that captures a specific atomic bonding

mechanisms: LJ potential captures the Van der Waals bonding, SW or Tersoff poten-

tial is adequate for covalent bonding, EAM formalism describes the metallic bonding,

and MEAM formalism describes both the metallic bonding and covalent bonding.

The computational cost of these empirical formalism scales with N , the number of

atoms, linearly 3, which enables simulations of a few tens of thousands with a single

CPU. The explicit form for each interatomic model will be described in this section,

with intuitive explanation on the idea behind each formalism. Because we studied

the deformation of copper using EAM potential, more detail will be provided on the

EAM potential and its benchmark data.

Mostly, the interatomic potentials are fitted only to the elastic properties and

defect formation energies. Still, they are used to study the dynamics of phase tran-

sitions and deformation at elevated temperature. For readers interested in thermal

properties related to phase transition, we present thermal properties of various in-

teratomic potentials, such as melting point, latent heat, and diffusion coefficient in

Appendix D. An efficient free energy calculation methods to compute free energy of

solid and liquid of single component system are used to compute the melting point

and latent heat accurately.

3.2.1 List of Empirical Potentials

In 1924, John Lennard-Jones [104] first proposed a pair-wise form of potential,

V LJij (r) = 4ǫ

[

(σ

r

)1

2−(σ

r

)6]

(3.4)

where ǫ is the depth of the potential well, σ is the finite distance at which the inter-

particle potential is zero, r is the distance between the particles. The equilibrium

distance rm is 21/6σ and the potential function has the vale −ǫ. These parameters

can be fitted to reproduce experimental data.

3the ab initio methods scales with N3 or higher [102].


The r−6 term describes attraction at long range force between two induced dipoles,

i.e. Van der Waals force and the r−12 term amounts to the short range repulsion due

to Pauli exclusion principle. Although the functional form of the attractive term has

a clear physical justification, the repulsive term has no theoretical justification. r−12

is used because it is more convenient due to the relative computational efficiency of

computing r−12 as the square of r−6. This potential is particularly accurate for noble

gas atoms and neutral atoms and molecules. Because the potential depend only on

the separation distance between two atoms, it can describe a limited set of molecular

bonding. For example, it is well known that all pair potential models produce equal

values for C12 and C44, which are two different elastic constants for cubic crystals.

This is not the case for most semiconductors and metals, which manifests the necessity

of including higher order interaction.

The Stillinger-Weber potential [105] is one of the first models that are developed

to model covalent bondings in semiconductors. It is based on a two-body term and a

three-body term,

V SW(~r) = 1

2

∑

ij

φ(rij) +∑

ijk

g(rij)g(rik)

(

cos θjik +1

3

)2

(3.5)

where θjik is the angle formed by the ij bond and the ik bond, and g(r) is a decaying

function with a cutoff between the first and second neighbor shell. The first pair-wise

interaction term determines the atomic bond length, and the second term makes the

configuration with diamondlike tetrahedral structure favored. The three body term

minimized at cos θjik = −13which corresponds to the angle between two arms of sp3

orbital.

This gives a realistic description of crystalline silicon, germanium, and other semi-

conductor materials having diamond-cubic structures. However, the built-in tetrahe-

dral bias creates “transferability” problems. It does not describe the right energies

for non-diamond-cubic structures that is found at different temperature and pressure.

Reconstructions on the surface can not be captured, neither. To build more realistic

models, one should construct a model that takes into account the concept of local


environment. Nonetheless, the SW potential is capable of modeling with a high pre-

cision the structural and dynamical properties of bulk diamond structure and organic

systems having many C − H chains. Also its cheap computational cost makes the

SW potential as one of the potential that has vastly used in research since it has been

developed.

J. Tersoff [106] proposed a new approach that effectively couples two body and

higher multi atom correlations into the model. The central idea is that in real systems,

the strength of each bond depends on the local environment, i.e. an atom with many

neighbors forms weaker bonds than an atom with few neighbors. It is written in

following form,

V Tersoff(~r) = 1

2

∑

i 6=j

fC(rij)[fR(rij) + bijfA(rij)] (3.6)

where fA and fR are the attractive and repulsive pair potential, respectively, and fC

is a smooth cutoff function. The main feature of this potential lies in the bij term,

the coefficient of attractive term. bij captures the local atomic arrangement and

accentuate or diminish the attractive force relative to the repulsive force according

to the environment. The detailed formation for each function can be found in the

literature [106] .

Because the Tersoff potential considers local environments and has more functions

to fit, the scheme works in a broader spectrum of situations than the SW potential.

It has mainly been developed and used for silicon, germanium, and carbon. The

cohesive energy and the structure of diverse geometries, the elastic constants and

phonon frequencies, defect energies and migration barriers are fitted for Si, Ge, and

C atoms [106, 107]. By taking the local environment into account, it has the ability

to describe the small energy difference between fourfold sp3 bond (diamond) and

threefold sp2 bond. It has also been used to model amorphous silicon. However,

this potential does not capture a metallic low coordinated liquid of semiconductor

elements. Compared to SW potential, it is found that the qualitative feature of static

properties of liquid Silicon is in better agreement with experiments and ab initio

method [108].


While electrons in dielectric and semiconductor solids form well-localized covalent

bonds, in metals they are more diffuse and shared by many atoms. To account for

many body effect in the metal, a non-linear embedding function is included in the

embedded atom method (EAM) potential by Daw and Baskes in 1984 [109]. Its

functional form is

V EAM(~r) =∑

i<j

φ(|~ri − ~rj |) +∑

i

F (ρi) (3.7)

ρi =∑

j 6=i

f(rij) (3.8)

where rij = |~ri − ~rj |. The first term is the usual pairwise interaction that accounting

for the core electrons. ρi is the contribution to the electron charge density from

atom j at the location of atom i and function f(r) specifies the contribution of an

atom to the electron charge density as function of distance r. F (ρi) is an embedding

fucntion defining the eneergy required to embed atom i into an environment with

electron density ρi. Commonly used embedding function is F (ρ) = −A√

(ρ) as in

the Finnis-Sinclair (FS) potential [110].

Because the embedding function is non-linear, the EAM-like potentials include

many-body effects that cannot be expressed by a superposition of pair-wise interac-

tions. As a result, EAM potentials can be made more realistic than pair potentials.

For example, EAM potentials give rise to non-zero values of the Cauchy pressure

(C12 −C44) and can be fitted to accurately reproduce the elastic constants of metals.

EAM potentials have been developed for various metals in different fitting proce-

dures. To study dislocation motion in copper, we used the EAM potential developed

by Mishin et al [111]. In the next subsection, we will show benchmarks of copper

EAM model that are relevant to the study of dislocation nucleation.

All aforementioned potential models are developed to describe a single bonding

mechanism: LJ potential for Van Der Waals, SW and Tersoff potential for covalent

bonding, and EAM potential for metallic bonding. In 1989, Baskes et al [112] devel-

oped a modified version of EAM potential that can describe all the bonding mech-

anism in a single mathematical form. The idea is simple but elegant. To describe


the s, p, d, f orbitals and their angular dependence, four different electron density

functions are used when obtaining the electron density ρi. More details about this

potential can be found in the Appendix E where we will present the improvement

of pure gold and pure silicon MEAM potentials. In the Appendix F, we present the

Au-Si cross potential based on the MEAM formalism that, for the first time, is fitted

to the experimental binary phase diagram. Detailed description on the efficient free

energy calculation methods for solid and liquid alloy is also presented that are used

to construct the binary phase diagrm when developing the model.

3.2.2 The Benchmarks of EAM Copper Potential

For reasonable description of dislocation dynamics and mechanical properties of ma-

terials, following properties must be well fitted to experiments or ab initio data:

elastic constants, stacking fault energies, and thermal expansion coefficient. Mishin

et al [111] presented two parameterizations of copper potentials, EAM1 and EAM2.

In the present dissertation, we have used EAM1 which better predicts elastic con-

stants and stacking fault energies. Here, we present some benchmarks of the EAM1

copper potential that is used to model the dislocation nucleation in copper. We refer

details of EAM parametrization and references for experiments and ab initio data to

the original Mishin’s paper [111].

Table 3.1: Lattice properties of Cu predicted by the EAM potential [111].

a(A) E0 (eV) B (MPa) C11 (MPa) C12 (MPa) C44 (MPa)Experiment 3.615 -3.54 138.3 170.0 122.5 75.8

EAM 3.615 -3.54 138.3 170.0 122.5 75.8

Table 3.1 shows the lattice properties predicted from EAM potential. Because

the EAM potential is fitted to the lattice properties, it shows exact match with ex-

periments. In other words, this shows that the parametrization used in the EAM is

capable of describing the bulk elastic properties of copper accurately. One shortcom-

ing of the parametrization is that 0K values of the EAM potential are fitted to the

experimental elastic constants at 300K, which is a common problem in elastic con-

stants in most interatomic potentials. Thus, elastic constants and ideal shear strength


are underestimated by the EAM potential when perfoming molecular dynamics sim-

ulation at finite temperature.

(a) (b)

Figure 3.3: (a) The stacking pattern of the 111 planes in FCC crystals. (b) PerfectBurgers vectors b1, b2, b3 and partial Burgers vectors bp1, bp2, bp3 on the 111 plane.Figures are taken from the literature [113].

To describe the dislocation nucleation, the empirical potential must be capable

of model the slip system of copper correctly. The atomic arrangement on the 111planes in a perfect FCC crystal is illustrated in Fig. 3.3 (a). Three distinct atomic

planes are stacked upon one another in a repeating sequence (ABCABC· · · ) to for theFCC structure. Let a be the lattice constant of the FCC copper, 3.615A. Dislocation

slips on the 111 plane and the perfect Burgers vector is along crystallographically

equivalent directions of a/2[110] such as b1, b2, b3 in Fig. 3.3 (b). The perfect dislo-

cations in FCC crystals with low stacking fault energy can dissociate into two partial

dislocations separated by a stacking fault to minimize the energy. Two Shockley

partials are along a/6[211] and a/6[211] or their crystallographically equivalent direc-

tions. Thus, dislocation nucleation occurs in two step processes. Nucleation of leading

partial along a/6[211] that forms stacking fault and subsequent nucleation of trailing

partial along a/6[211] that terminates the stacking fault. Because the nucleation of

the partial Burgers vector bp=a/6[211] is a limiting step in the copper, it is important

to have a correct misfit function along this direction. It is called as the generalized

stacking fault energy (GSFE) and the GSFE of the copper EAM potential is shown

in the Fig. 3.4. It reaches maximum near the half of bp and arrives at the minimum


point at bp. The height of the maximum point is defined as unstable stacking fault

energy, and the height of the minimum point is stacking fault energy. EAM shows

very good match with experiments and ab initio data, as presented in Table 3.2.

Table 3.2: The intrinsic stacking fault energy γSF and the unstable stacking faultenergy γUSF of Cu from experiments, EAM potential [111], and ab initio calcula-tion [114]. ab initio data depend on the exchange-correlation functionals used in thestudy.

γSF (mJ/m2) γUSF (mJ/m2)Experiment 45

EAM 44.4 158ab initio 37-41 158-186

Figure 3.4: Generalized stacking fault energy on (111) plane along a/6[211] directioncalculated with EAM potential [111].

It is also found that thermal expansion and thermal softening effect are signifi-

cant in determining the temperature dependence of the dislocation nucleation rate

(See Chapter 5). As temperature rises, thermal expansion pushes neighboring atoms

further apart and makes crystallographic planes easier to shear and significantly re-

duce the free energy barrier of dislocation nucleation. Thus, it is very important to


have a correct thermal expansion coefficient to calculate the dislocation nucleation

rate as a function of temperature. Fig. 3.5 shows that the linear thermal expansion

from experiments is well reproduced by the EAM potential except the vicinity of

the zero temperature. The mismatch at low temperature range is not the artifact of

EAM potential but the limitation of classical atomistic simulation. In reality, excited

are only fraction of normal mode whose vibration energy quanta is smaller than the

thermal fluctuation, i.e. hf < kBT where h is the Planck constant and f is phonon

frequency. Such quantum effect is not taken into account in the classical molecular

dynamics simulation employed in this dissertation. Because the slope of linear ex-

pansion matches well with experiments, our simulation result still correctly capture

the relative change of free energy barrier associated with thermal expansion.

Figure 3.5: Linear thermal expansion of Cu calculated in the quasi-harmonic approx-imation (QHA) and by the Monte Carlo method using the EAM potential [111]. Themelting point of Cu (Tm) is indicated.


3.3 Molecular Dynamics Simulation

Molecular dynamics (MD) is an approach to simulate the “true” dynamics of atoms

while preserving Boltzmann’s statistics [115]. The MD method is simple in concept:

(1) obtain the acceleration ~ai of each particle from the force field ~fi. (2) integrate the

Newton’s third law ~ri = ~ai to get the trajectory of all particles as a function of time.

Because the quantum effects can be ignored at moderate high temperature, we can

obtain the trajectory from the classical equation of motion.

While the outlined idea is very simple, MD is a very powerful technique. Because

molecular systems generally consist of a vast number of particles, it is in general im-

possible to analytically trace the trajectories of particles in a system and calculate

the properties of such complex systems 4. With efficient numerical integration tech-

niques, the positions and velocities of all particles are traced on the fly. The macro-

scopic properties such as pressure and temperature can be obtained from statistical

mechanics. We can also study the microscopic reactions such as defect formation,

diffusion, crystallization, and etc.

With molecular dynamics simulation, we can simulate the dynamics of materials

subjected to different external condition, more specifically, different thermodynamic

ensembles. The simplest ensemble is the microcanonical or NVE ensemble. The NVE

ensemble is a closed system of particles that only interact each other, conserving the

number of particles (N), the total volume (V) and the total energy (E). Periodic

boundary condition and a fixed simulation cell is used to keep the volume constant.

The total energy is conserved by employing simplectic integrators.

The NVE ensemble is useful in proving the theorems in statistical mechanics nu-

merically. For example, a solid simulation cell where the positions of all atoms are

randomly disturbed will approach a thermal equilibrium where Boltzmann distribu-

tion is achieved in the velocities of particles. However, this ensemble is not adequate

in describing most laboratory experiments where temperature, pressure, or both are

kept constant. MD simulation then can be altered to allow constant temperature

simulations, instead of energy. This ensemble is called canonical or NVT ensemble.

4It is known that, for any system having more than two bodies, chaotic trajectories comes outthat can not be expressed by analytic solutions [116].


Two categories of temperature control algorithms (thermostats) exist: determinis-

tic and stochastic thermostats. Both schemes are used for all of our constant temper-

ature simulations. Nose-Hoover thermostat, a representative deterministic thermo-

stat, mimics the heat bath by adding an artificial variable associated with an artificial

mass [117]. Because it is found that the correct canonical ensemble is not sampled

for harmonic potentials such as solids at low temperature, we employed Nose-Hoover

chain thermostat [118] in our MD simulation of crystalline copper. Anderson thermo-

stat [119] and dissipative particle dynamics [120] are stochastic thermostats that add

noisy forces into the system to control the temperature. While they does not pre-

serve the natural dynamics of atoms, it works better than deterministic thermostats

in achieving thermal equilibrium, and also be useful in accelerated sampling where

stochastic trajectories are required. We use Anderson thermostat in equilibrating

copper nanowire before studying formation of dislocation loop.

The other ensemble of interest is the isothermal-isobaric NPT ensemble where

each component of stress is controlled. The pressure (or stress) is defined by the

virial stress formula which defines the instantaneous stress value for stress control in

NPT ensemble. The long time average is taken as a continuum stress that can be

compared with experimental measurements. Most popular scheme is the Parrinello-

Rahman barostat [121] where the simulation box is adjusted according to a modified

Lagrangian in order to keep the virial stress fluctuating about a constant value. Unfor-

tunately, the barostat does not work efficient at high stress regime where stress-strain

relation is non-linear. We use a iterative method to achieve a pure shear stress state

in our work, in-depth description of which is provided in the Chapter 5.

More details and comprehensive summaries regarding molecular dynamics meth-

ods can be found in the literature [122].

3.4 Monte Carlo Simulation

Monte Carlo simulation is another method to sample the configuration space accord-

ing to the Boltzmann distribution at a given temperature. Unlike the MD, Monte

Carlo (MC) simulation generates artificial trajectories spanning the configurational


space and complying with Boltzmann’s distribution. Although MC does not capture

the “true” dynamics of atoms, it has certain advantages over MD simulation. MC sim-

ulation is especially powerful when studying multicomponent systems where timescale

of configurational space sampling is much larger than timescale of MD simulation.

Typically, it takes a few pico seconds to reach thermal equilibrium in momentum

space, but it can take much longer to explore configurational space because the time

scale of diffusion is very slow. MC simulation can expedite the equilibration process

by allowing artificial moves such as interchanging two atoms and collective move of a

small groups of atoms. Many physical properties can be obtained from the ensemble

average over the phase space via MC simulation.

It is important to design a stochastic process that can sample the ensemble of

configurations according to Boltzmann’s distribution. Most MC methods simulate a

Markov process, meaning that the probability of moving to a specific state in the next

step is only a function of the current state and is independent of history. Consider an

atomic configuration R making transition to another atomic configuration R′ with

the transition probability π(R,R′). π(R,R′) satisfies the normalization condition,

∫

dR′π(R,R′) = 1. (3.9)

and for the equilibrium distribution ρ(R), it should satisfy

ρ(R) =

∫

dR′ρ(R′)π(R′,R). (3.10)

There are many transition probability matrix π(R,R′) that can reproduce the same

equilibrium distribution. Here, we will consider a more strict condition,

ρ(R)π(R,R′) = ρ(R′)π(R′,R) (3.11)

which is called the detailed balance condition. Recalling that ρ(R) ∝ exp(−V (R)/kBT )

at equilibrium, and the detailed balance condition leads to the following restriction


on the transition probabilities

π(R,R′)

π(R′,R)=

ρ(R′)

ρ(R)= exp[−β(V (R′)− V (R))]. (3.12)

where β = 1/kBT .

In this dissertation, we use the Metropolis algorithm [123] corresponding to the

following choice of π,

π(R,R′) = α(R,R′)Pacc(R,R′), forR′ 6= R (3.13)

Pacc(R,R′) ≡ min1, exp[−β(V (R′)− V (R))] (3.14)

where α(R,R′) is a symmetric density matrix. Trial movies displace atoms in a

random direction, R′ = R+ δR. If the potential energy of the trial state R′ is lower

then energy of the current state R, the move is accepted. However, if the potential

energy of the trial state is higher, it is accepted with probability Pacc. Otherwise,

the move is rejected and the system remains in the current state R. It is proved

that this algorithms samples a specific configuration R according to the Boltzmann

distribution [123], also satisfying detailed balance condition. In the MC simulation

of the Ising model, we consider the instantaneous spin configuration in the lattice, as

the analogy of the atomic configuration R. Accordingly, the trial move is a spin flip

at a randomly chosen position in the lattice.

3.5 Advanced Sampling Methods

In this section, brief descriptions on two advanced sampling techniques, forward flux

sampling and umbrella sampling, will be provided. While both methods are developed

to overcome the timescale of various dynamic processes such as protein folding and

first order phase transition, we will explain how these two techniques can be applied

to study the nucleation phenomena.

Forward flux sampling (FFS) technique [124] directly computes the absolue value


of nucleation rate without a priori knowledge of any nucleation theory. FFS is ap-

plicable to non-Markovian and non-equilibrium process. Thus, the nucleation rate

computed from FFS can be used as a benchmark for testing the validity of nucleation

theories. Umbrella sampling (US) methods can be used to obtain the free energy as

a function of the droplet size. Once the free energy curve is obtained, the nucleation

rate can be calculated from a nucleation theory that requires the free energy curve

as an input, which will be demonstrated in the last part of this section.

3.5.1 Forward Flux Sampling

(a) (b)

Figure 3.6: Schematic of forward flux sampling. (a) To compute the nucleation rateof small droplet with size n0, we count the number q that droplets larger than n0

forms for time duration t. Then, nucleation rate I0 becomes q/t. Here, it is notdesirable to count small fluctuation in n around n0 as separate events. The counterreset only after n comes back to the original basin n < nA and becomes ready tocount another event. nA can be considered as an error margin in the digital signalprocessing. (b) At each interface i, we have an ensemble of configurations havinglargest droplet size ni. N independent MC simulations are performed, starting froma randomly chosen configuration from the ensemble. Then, number M of reachingnext interface i+1 before returning back to nA is counted as successful forward flux.Then, the probability of reaching next interface P (ni+1|ni) becomes N/M .

To compute the transition rate from the initial state A to the final state B, FFS

uses a series of interfaces in the phase space defined through an order parameter

λ. State A is defined as the phase space region in which λ < λA, while state B

corresponds to λ > λn. The interfaces between A and B are defined as hyperplanes


in the phase space where λ = λi, i = 0, 1, 2, · · · , n − 1, λA < λ0 < · · · < λn. In

principle, the choice of the order parameter λ should not affect the calculated rate

constant, which means λ need not be the true reaction coordinate [124].

In the FFS method, the nucleation rate I from A to B is expressed as a multipli-

cation of two terms

I = I0 P (λn|λ0) (3.15)

where I0 is the average flux across the interface λ = λ0 (i.e. leaving state A), and

P (λn|λ0) is the probability that a trajectory leaving state A will reach state B before

returning to state A again. Because it is impossible to reach interface λ = λi+1

without reaching interface λ = λi first, the probability P (λn|λ0) can be decomposed

into a series multiplication,

P (λn|λ0) =

n−1∏

i=0

P (λi+1|λi) (3.16)

where P (λi+1|λi) is the probability that a trajectory reaching λi, having come from

A, will reach λi+1 before returning to A again. The schematic of FFS is illustrated

in the Fig. 3.6.

In this work, we set λ to be the size of the largest droplet n in the simulation cell.

The rate I0 is obtained by running a brute force Monte Carlo simulation, during which

we count how frequently droplets with size larger than λ0 are formed. An ensemble

of configurations at interface λ = λ0 (for trajectories coming from A) is stored from

this MC simulation. We set λ0 to be several times bigger than the average largest

droplet size from the MC simulation at the given condition, to collect configurations

that are uncorrelated to each other.

The next step is to run MC simulations with initial configurations taken from

the ensemble at interface λ = λ0. A fraction of the trajectories reaches interface

λ = λ1 before returning to state A. From these simulations the probability P (λ1|λ0)

is computed and an ensemble of configurations at interface λ = λ1 is created. The

process is repeated to compute the probability P (λi+1|λi) for each i = 1, · · · , n − 1.

For each (T, h) condition, the interfaces are chosen manually such that the spacing


|λi+1−λi| between interfaces increases linearly with λi, with the first spacing |λ1−λ0|large enough to have P (λ1|λ0) ≤ 10−1. The nucleation rate constant turns out to

be the same within statistical error when computed with different sets of interfaces

having larger spacings than above. The last interface λn is chosen to be 3 to 4 times

bigger than the critical nucleus size to ensure that the stable state is reached.

As an example, Fig. 3.7 plots the probability P (λi|λ0) ≡ P (λ1|λ0)P (λ2|λ1) · · ·P (λi|λi−1)

for the 2D Ising model at kBT = 1.5 and h = 0.05. In this test case, we find

I0 = 1.45×10−8MCSS−1 with λ0 = 24 from a brute force Monte Carlo simulation with

107MCSS. 15, 000 configurations are then collected at each interface, which allows

the nucleation rate to be determined within 5%. The probability of reaching interface

λ = λn from interface λ = λ0 is P (λn|λ0) = 1.92 × 10−11 with λn = 1200. Following

Eq. (3.15), the nucleation rate under this condition is IFFS = 2.78× 10−19MCSS−1.

0 100 200 300 400 500 600

10 10

10 5

10 0

λi

P( λ

i | λ0 )

0

0.2

0.4

0.6

0.8

1

P B (495) = 0.5

P B ( λ

i )

Figure 3.7: (color online) The probability P (λi|λ0) (solid line) of reaching interfaceλi from λ0 and average committor probability PB(λi) (circles) over interface λi at(kBT, h) = (1.5, 0.05) for the 2D Ising model. The 50% committor point is markedby *.

It is important to note that FFS does not require the transitions between different

interfaces to be Markovian. Neither does it require the transitions to satisfy detailed

balance, unlike other sampling methods [130, 142, 144]. Therefore, it can be used to

test the fundamental assumption of the Becker-Doring theory, which states that the

nucleation process can be coarse-grained into a one-dimensional Markov chain. FFS

would fail if there is no separation of time scale, i.e. if the time spent on a reaction


path is comparable to (instead of much shorter than) the dwell time in state A or

state B. In order to test the applicability of FFS to Ising models, we benchmark

it against brute-force MC simulations of the 3D Ising model at kBT = 0.59 Tc and

h = 0.589. The nucleation rate computed by FFS, IFFS = 4.10 × 10−10 MCSS−1,

is in good agreement with the brute-force MC result [138, 137], IMC = 5.81 × 10−10

MCSS−1, confirming the applicability of FFS.

3.5.2 Umbrella Sampling

(a) (b)

Figure 3.8: (a) For a given free energy landscape, we can sample a limited regionwithin the kBT range. (b) We can sample other regions with higher free energy withumbrella sampling that limits the Monte Carlo move within the range of bias function.

The probability of forming a droplet of size n is estimated by P (n) = Nn/N , where

Nn is the number of droplet of size n in a system containing N particles. Knowledge

of the ratio Nn/N allows us to define the Gibbs free energy ∆G(n) of forming a

droplet of size n:Nn

N= exp

[

−∆G(n)

kBT

]

. (3.17)

Using a Monte Carlo simulation, we can obtain the average value of Nn, but it is

very difficult to sample large droplets whose formation free energy is larger than the


thermal fluctuation kBT . For instance, when ∆G ∼ 10kBT corresponding to the

Boltzmann factor exp(−∆G/kBT ) ∼ 4.5 × 10−5, we would sample the droplet once

in every 22, 026 MC move, on average. It is very hard to get an accurate statistics

for the population of droplets that show up very rare in the simulation. Umbrella

sampling uses a bias function to control the size of the largest cluster in the system,

and obtain statistics in a very limited range of size. Schematic description on the

umbrella sampling is presented in the Fig. 3.8.

Figure 3.9: (From the top left cornet, in clock wide direction). Processing the rawhistogram taken from umbrella sampling simulations. (1) We first obtain the biasedrelative distribution P

′′

(n) at each window from series of umbrella sampling. (2)Unbiased relative distribution P

′

(n) can be obtained by multiplying the inverse of

weighing function, exp[1

2k(n−ni)

2

kBT], to the data of each window i. (3) From the over-

lapping histogram methods [125], we can merge the distributions into a single curve.After normalization, we obtain the absolute probability P (n). (4) The free energycurve ∆G(n) can be obtained from −kBT ln(∆G(n)).


Note that the ensemble average of a quantity A can be written as

〈A〉 =∫

dRA(R) exp(−βV (R))∫

dR exp(−βV (R)). (3.18)

It is known that the ensemble average 〈A〉 can be rewritten as

〈A〉 = 〈A/W (R)〉W〈1/W (R)〉W

(3.19)

where W is an arbitrary weighting function that is written as W (R) = exp[−βw(R)].

The subscript 〈. . .〉W indicates an ensemble average according to the biased distribu-

tion function exp[−β(V (R)+w(R))]. In this dissertation, we chose the bias function

to be harmonic function of the size of the largest cluster

w[n(R)] =1

2k[n(R)− n0]

2. (3.20)

The constant k determines the range of cluster sizes sampled in one simulation and

the parameter n0 determines the center of the sampling window. Umbrella sampling

refers to the Monte Carlo simulation using the biased distribution function to control

the size of droplet size. For each umbrella sampling, we will obtain the histogram

of Nn around n = n0, weighted by W (n) with small statistical error. The actual

distribution can be obtained by correcting the raw histogram data by 1/W (n) and the

Gibbs free energies will be determined up to a constant ∆Gi(n)/kBT +bi. Subscript i

indicates the index of the window and bi is a constant corresponding to the window i.

To determine the constant bi, we perform series of umbrella sampling whose windows

overlap with each other. This will allows us to determine the absolute value of droplet

formation energy for the wide range of size n. Fig. 3.9 shows the schematics of data

processing. Extensive discussion on the umbrella sampling methods is presented in

the literature [125].


3.5.3 Computing Rate from Becker-Doring Theory

The classical nucleation rate prediction has three input parameter; the free enerby

barrier Gc, the Zeldovich factor Γ, and the attachment rate to the critical cluster

f+c . From the droplet formation free energy as a function of size computed from

the umbrella sampling, the free energy barrier Gc and the Zeldovich factor Γ can be

obtained from the height and the curvature near the maximum, respectively.

(a) (b)

0 100 200 300 400 500 6000

10

20

30

40

50

60

70

n

F (

n)

nc = 496

Fc = 61.3

0 0.05 0.1 0.15 0.20

5

10

15

20

t (MCSS)

⟨ ∆ n

2 (t)

⟩

Figure 3.10: (color online) (a) Droplet free energy F (n) obtained by US at kBT = 1.5and h = 0.05 in the 2D Ising model. (b) Fluctuation of droplet size 〈∆n2(t)〉 as afunction of time.

As an example, Fig. 3.10 (a) shows the droplet free energy F (n) computed from

US at kBT = 1.5 and h = 0.05 for the 2D Ising model 5. The order parameter is

the size of the largest droplet, n. A parabolic bias function 0.1kBT (n− n)2 is used,

where n is the center of each sampling window, following Auer and Frenkel [129]. The

maximum of this curve indicates that the critical droplet size is nc = 496 and the free

energy barrier is Fc = 61.3. The Zeldovich factor can be calculated from the second

derivative of this curve at nc, which gives Γ = 0.0033.

To compute the f+c , it is necessary to observe the average size change of critical

droplets with size nc. We collect many configurations from the US simulation, when

the bias potential is centered at the critical droplet size. Using each configuration

5Ising model does not have pressure as a macroscopic variable. Thus, Helmholtz free energy F (n)is used to describe the droplet formation free energy at constant h and constant T .


as an initial condition, we run Monte Carlo simulations and obtain the effective

attachment rate from the following equation,

f+c =

〈∆n2(t)〉2 t

, (3.21)

where 〈∆n2(t)〉 is the mean square fluctuation of the droplet size. ∆n(t) ≡ n(t)−n(t =

0), n(t) is the droplet size at time t, and 〈〉 represents ensemble average from these

Monte Carlo simulations. The Monte Carlo simulations are stopped when |∆n(t)|exceeds a certain value. The result for the means square fluctuation 〈∆n2(t)〉 at

kBT = 1.5 and h = 0.05 is plotted in Fig. 3.10 (b), which shows a linear function of

time. From the slope of this curve, we obtain f+c = 39.1MCSS−1. A similar approach

was used by Brendel et al. [126] to compute the interface diffusion coefficient.

Combining the values of f+c , Γ, Fc and plug them into Eq. (2.7), we find that

the Becker-Doring theory would predict the nucleation rate to be IBD = 2.37 ×10−19MCSS−1, if the correct free energy function F (n) is used. This is very close

to the FFS result IFFS = 2.78 × 10−19MCSS−1 given in the previous section. Com-

parisons over a wider range of conditions are presented in the next chapter, with

extensive discussion on the validity of the classical nucleation theory.

Chapter 4

Numerical Tests of Nucleation

Theories for the Ising Models

4.1 Introduction

As discussed in Section 2.2, while CNT successfully captures many qualitative features

of nucleation events, the prediction of the nucleation rate based on CNT cannot

be compared quantitatively with experiments [13], given the gross approximations

made in the theory. During the past 50 years, many modifications and extensions

of CNT have been developed. For example, Lothe and Pound [55] considered the

contributions from extra degrees of freedom of a cluster (in addition to its size) to

its Gibbs free energy of formation. Langer [40, 56] developed a field theory to extend

the Becker-Doring steady-state solution to include the effect of other microscopic

degrees of freedom of a cluster. Zeng and Oxtoby [57] improved the temperature

dependence of the nucleation rate predicted by CNT by expressing the droplet free

energy as a functional of the radial density profile ρ(r). To date, many nucleation

theories have been developed, but it is very difficult to verify them experimentally,

due to the difficulties in measuring nucleation rates accurately. While for a theory it

is more convenient to study homogeneous nucleation in a single-component system,

such conditions are difficult to achieve in experiments [13]. Instead, experimental

measurements are usually influenced by surface structures and impurities that are

50

CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 51

difficult to control.

Computer simulations have the opportunity to probe nucleation processes in great

detail and to quantitatively check the individual components of the nucleation the-

ories. The increase of computational power and the development of advanced sam-

pling algorithms allow the calculation of nucleation rates for model systems over a

wide range of conditions [127, 128, 129, 130]. A prototypical nucleation problem is

the decay of the magnetization in the 2D or 3D Ising model, which has been stud-

ied by computer simulations for several decades [39, 47, 126, 131, 132, 133, 134, 135].

Both agreement [132, 136, 137] and disagreement [39, 47, 131, 138] between numerical

results and CNT predictions have been reported.

When the CNT predictions of nucleation rate do not agree with numerical re-

sults, several potential problems of CNT were usually discussed. For example, a

suspect is the application of surface tension of macroscopic, flat, interfaces to a

small droplet [47]. The validity of coarse-graining the many-spin system into a one-

dimensional Markov chain was also questioned [39, 47]. Nucleation theories usu-

ally express the rate in the Arrhenius form, with a free energy barrier and a pre-

exponential factor. Usually both terms are not computed in the same study. Hence,

we often cannot conclude which one causes the discrepancy between CNT and numer-

ical simulations, and how the theory should be improved. Only rarely has numerical

results lead to clear conclusions on the validity of the fundamental assumptions made

in CNT [135].

In this chapter, we present numerical results that systematically test the different

parts of the Becker-Doring theory, as applied to 2D and 3D Ising models. We compute

the nucleation rate by the forward flux sampling (FFS) method [124], which allows

the rate to be calculated over a much wider range of external field and temperature

conditions than that possible by brute force Monte Carlo simulations. To test the

individual components of CNT (Becker-Doring theory), the free energy F (n) of the

droplet as a function of droplet size n is computed using the umbrella sampling

method [129]. The kinetic prefactor of the critical cluster, f+c , which is part of

the Becker-Doring theory, is computed independently from Monte Carlo simulations

starting from the ensemble of critical clusters. The nucleation rate predicted by the


Becker-Doring theory, using the so computed F (n) and f+c as inputs, is compared

with the nucleation rate directly computed from the FFS method.

We find that, provided with the correct droplet free energy F (n), the Becker-

Doring theory predicts the nucleation rate very accurately. This confirms that the

coarse-graining of the Ising model as a one-dimensional Markov chain, as invoked in

CNT, is a very good approximation, which was also noted earlier [126, 135]. Discrep-

ancies between the droplet free energy F (n) predicted by CNT and numerical results

have been reported earlier [39, 126]. Here we show that if a logarithmic correction

term and a constant correction term are added, the theoretical prediction of droplet

free energy agrees very well with the numerical result. The logarithmic correction

term was first derived from Langer’s field theory, but was customarily put as a cor-

rection to the kinetic prefactor. Our analysis shows that this correction term should

be placed in the free energy function F (n) in order to correctly predict the size of the

critical nucleus. In 2D both the logarithmic correction term and the constant term

can be determined from existing analytic expressions and hence contain no fitting pa-

rameters. On the other hand, in 3D both the coefficient of the logarithmic correction

term and the constant term need to be treated as fitting parameters in this work.

Our analysis resolved some of the previously reported discrepancies between nu-

merical simulations and CNT. For the 2D Ising model, the logarithmic correction

term to the droplet free energy was often neglected [47, 126]. Because the logarith-

mic correction term is positive and substantial in 2D, the omission of this term would

cause CNT to grossly overestimate the nucleation rate. For the 3D Ising model, the

logarithmic correction term is much smaller relative to the other terms. However, the

temperature dependence of the surface free energy was sometimes ignored [39, 138].

While the surface free energy can be approximated as a constant at very low temper-

atures [136], it decreases appreciably with temperature above a quarter of the critical

temperature. Overestimating the surface free energy would lead to an overestimate

of the nucleation barrier and an underestimate of the nucleation rate.

The chapter is organized as follows. Section 4.2 summarizes a number of nucleation

theories and their applications to the 2D and 3D Ising model. Section 4.3 presents

the numerical methods we employ to test these theories. The numerical results are


compared with the nucleation theories in Section 4.4.

4.2 Nucleation Theories Applied to the Ising Model

As introduced in the previous chapter, the Ising model is described by the following

Hamiltonian

H = −J∑

〈i,j〉

sisj − h∑

i

si (4.1)

where J > 0 is the coupling constant and h is the external magnetic field. The spin

variable si at site i can be either +1 (up) or −1 (down), and the sum∑

〈i,j〉 is over

nearest neighbors of the spin lattice. For convenience, we set J = 1 in the following

discussions. In our simulations, we study the relaxation of the magnetization starting

from an initial state magnetized opposite to the applied field h. To be specific, we let

h > 0 and the initial state has si = −1 for most of the spins. The dynamics follows

the Metropolis single-spin-flip Monte Carlo (MC) algorithm with random choice of

trial spin. The simulation time step is measured in units of MC step per site (MCSS).

The 2D model consists of a 100× 100 square lattice and the 3D model consists of a

32 × 32 × 32 simple cubic lattice, with periodic boundary conditions (PBC) applied

to all directions. To avoid artifacts from finite simulation cell size, we consider (T, h)

conditions such that the size of the critical droplet is much smaller than simulation

cell size 1 We obtain nucleation rate per site and define free energy F such that

exp(−F/kBT ) is proportional to the cluster population per site to present results

that are invariant when the simulation cell size changes.

4.2.1 Becker-Doring Theory

To compute the nucleation rate using Eq. (2.1) and Eq. (2.7), the surface free energy

σ and bulk chemical potential difference ∆µ must be known for the Ising model. The

chemical potential difference is simply ∆µ = 2h, which is a good approximation not

only at low temperature but also near the critical temperature [139]. On the other

1We have first order estimate on the critical droplet size before running simulation.


hand, the surface free energy σ is more difficult to obtain. This is because the Ising

model is anisotropic at the microscopic scale and the free energy of a surface depends

on its orientation. Therefore, the input to the Becker-Doring theory should be an

effective surface free energy σeff , which is an average over all possible orientations

given the equilibrium shape of the droplet. σeff is a function of temperature T not

only because the surface free energy of a given orientation depends on temperature,

but also because the equilibrium shape of the droplet changes with temperature [140].

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5σ

eff

σ(10)

kBT

σ eff

Figure 4.1: (color online) Effective surface free energy σeff as a function of temperaturefor the 2D Ising model from analytic expression [47]. The free energy of the surfaceparallel to the sides of the squares, σ(10), is also plotted for comparison.

We follow the definition of Shneidman [47], which gives the analytic expression of

σeff(T ) for the 2D Ising model, as shown in Fig. 4.1. σeff(T ) is defined in such a way

that the interfacial free energy of a nucleus can be written as

F σ(n) = 2√πnσeff(T ) (4.2)

regardless of whether its equilbrium shape determined by Wulff construction is cir-

cular or not 2. The free energy of a droplet can be written as,

F 2D(n) = 2√πnσeff(T )− 2 hn (4.3)

22√πn is the circumference of a circle with area n. However, a real droplet is not compact but

consists of a mixture of up and down spins. The percentage of down-spins in the droplet as a functionof T is also absorbed in σeff(T ).


where n is the total number of up-spins in the droplet. From the maximum of F 2D(n),

we obtain the critical droplet size of the 2D Ising model,

n2Dc =

π σ2eff(T )

4 h2(4.4)

as well as the free energy barrier

F 2Dc =

π σ2eff(T )

2h(4.5)

We also obtain the Zeldovich factor defined in Eq. (2.8)

Γ2D =

√

2

kBT

h3/2

π σeff(T )(4.6)

Assuming the critical droplet has a circular shape, the attachment rate can be written

as

f+c

2D= 2β0(T )

√

π n2Dc (4.7)

where β0(T ) is the average spin-flip frequency at the boundary of the droplet. As an

approximation,

β0 ≈ exp(−σeff(T )/kBT ) (4.8)

In the temperature and field conditions considered in this study, the attachment rate

predicted by Eqs.(4.7) and (4.8) is within a factor of 2 of the value computed by

Monte Carlo simulations as shown in the section B.1. Combining all, we obtain the

nucleation rate predicted by the Becker-Doring theory

I2DBD(h, T ) = β0(T )

√

2 h

kBTexp

[

−π σ2eff(T )

2 h kBT

]

(4.9)

Given the analytic expressions for σeff(T ) in 2D, the predictions of the Becker-

Doring theory can be computed explicitly. For example, at kBT = 1.5 and h = 0.05,

we have n2Dc = 463, F 2D

c = 46.3, Γ2D = 0.0034, f+c

2D= 34.0, and I2DBD = 4.5 ×

10−15MCSS−1. The numerical results (in Section 3.5.3) under the same condition are

nc = 496, Fc = 61.3, Γ = 0.0033, f+c = 39.2, and I = 2.37 × 10−19MCSS−1. As


discussed further below, the four orders of magnitude discrepancy in the nucleation

rate mainly comes from the underestimate of F 2Dc by Eq. (4.3).

The logarithmic term in Eq. (2.9) and a constant term are needed to remove this

discrepancy. It is found that the free energy of droplet of the 2D Ising model can be

written as

F (n) = σS(n)−∆µn + τkBT lnn+ d(T ). (4.10)

To connect with the previous chapter, the logarithmic correction term accounts for

shape fluctuation effects in the droplet as suggested in the Langer theory. The con-

stant term take the self-consistency and surface energy size effects into account.

For the 3D Ising model in a simple cubic lattice, there is no analytic expression for

surface free energy for arbitrary surface orientations. A parametric expression exists

only for the (100) surface [141]. Therefore, the equilibrium shape and the equivalent

surface free energy of the 3D droplet is not known. Similar to Eq. (4.3), the free

energy of a 3D droplet can be written as,

F 3D(n) = σeff(T )αn2/3 − 2 hn (4.11)

where α = (36π)1/3 is a geometric factor expressing the surface area of a sphere with

unit volume. Contrary to the case of 2D Ising model, the analytic expression of σeff(T )

is not known in the 3D Ising model, and it will be used as a fitting parameter in our

study.

Following the same procedures as above, we obtain the critical nucleus, free energy

barrier and Zeldovich factor for the 3D Ising model,

n3Dc =

α3σ3eff(T )

27 h3(4.12)

F 3Dc =

α3σ3eff(T )

27 h2(4.13)

Γ3D =

√

9

πkBT

h2

√

α3σ3eff(T )

(4.14)

f+c

3D= β0(T )αnc

2/3 (4.15)


Finally, the nucleation rate predicted by the Becker-Doring theory is

I3DBD(h, T ) = β0(T )

√

α3σeff(T )

9π kBTexp

[

− α3σ3eff(T )

27 h2 kBT

]

(4.16)

Given that σeff(T ) is yet unknown and has to be treated as a fitting parameter, it is

more difficult to test Eq. (4.16) quantitatively.

4.2.2 Langer’s Field Theory

Langer’s field theory predicts a logarithmic correction term to the droplet free energy,

as in Eq. (2.9). In 2D Ising model, τ = 54, and this correction term not only increases

the free energy barrier, but also increases the size of the critical droplet. The critical

droplet size predicted by the field theory is,

n2D/FTc =

(√πσeff +

√

πσ2eff + 8τkBTh

4 h

)2

(4.17)

This equation is to be compared with Eq. (4.4) predicted by the Becker-Doring theory.

We will see (in Fig 4.4) that Eq. (4.17) agrees much better with numerical results

than Eq. (4.4), indicating that the field theory correction should be put in the free

energy function instead of the kinetic prefactor.

The τkBT lnn correction term also modifies the critical nucleus size in the 3D Ising

model. The analytic expression for n3Dc given by the field theory can be obtained

by solving a third order polynomial equation. The expression is omitted here to

save space. In the 3D Ising model, there have been predictions that τ depends on

temperature: τ = −19above the roughening temperature TR [46] and τ = −2

3below

TR [44]. But our numerical results do not support these predictions.

4.3 Computational Methods

We have explained that the classical nucleation theory has two parts of assumptions

and each part can be tested separately. In part I, CNT assumes that the system can be


Figure 4.2: Schematic of numerical test on the CNT nucleation rate.

coarse grained into a one-dimensional Markov chain model characterized by the size

n of the largest droplet. The steady state solution of the Markov chain predicts that

the nucleation rate to be the Eq. 2.7. In part II, CNT assumes that the droplet free

energy function can be written as the F (n) = σS(n)− n∆µ, a summation of surface

energy and volume term. When discrepancy is observed between CNT prediction and

computer simulations, it is important to know whether part I or part II is responsible

so that the theory can be modified appropriately.

Here, we present a computational method that can test two parts separately. The

schematic of our method is illustrated in the Fig. 4.2. In order to test part I of CNT,

which predicts the nucleation rate by assuming the system can be coarse grained to a

1D Markov chain, we must have an independent way to compute the nucleation rate

without relying on this assumption. It is also important to sample a wide range of

(T, h) conditions and collect sufficient statistics for every condition. This precludes the

use of brute-force Monte Carlo simulations, which become very inefficient when the

nucleation rate is low. Here, we employed the forward flux sampling method. FFS

samples rare events and computes the transition rates in nonequilibrium systems

which do not need to obey detailed balance [124]. The transition rate I has also

been proven to be independent of the choice of the order parameter, as long as it

distinguishes the initial and final states of the transition. If the nucleation rate I


obtained from the forward flux sampling matches with the CNT prediction, we can

prove the validity of the part I of CNT.

To compare the results from FFS with part I of CNT, we need to compute all

the terms in the CNT nucleation prediction, as explained in the previous chapter.

The droplet formation free energy can be obtained as a function of the droplet size

n, using umbrella sampling. The curvature near the top of the free energy curve will

determine the Zeldovich factor Γ. The attachment rate fc can be obtained from the

brute force MC simulation of an ensemble of critical size droplets.

To examine part II of CNT, we can compare the droplet free energy obtained from

umbrella sampling to the CNT expression. We have shown, from previous studies,

that CNT tend to predict the critical droplet size correctly, but the droplet formation

energy is incorrect. Because the 2D Ising model has an analytic expression for the

effective surface energy σeff (T ), direct comparison is available. Also, for very small

clusters, we can compute the exact partition function as a function of droplet size,

by identifying every possible droplet shapes.

.

4.4 Results

4.4.1 Nucleation Rate

We have computed the nucleation rates using two different methods over a wide range

of conditions: h = 0.01-0.13, T = 0.5-0.8 Tc for 2D and h = 0.30-0.60, T = 0.4-0.7 Tc

for 3D, where Tc is the critical temperature at zero field (kBTc = 2.269 in 2D and

4.512 in 3D). In the first method, the nucleation rate is directly computed by FFS. In

the second method, the nucleation rate is computed from the Becker-Doring Eq. (2.7),

but using the free energy curve obtained from US, as described in Section III.B. The

pre-exponential factor, f+c Γ, is found to have a weak dependence on T and h (see

Appendix B.1), and varies by about a factor of 2 in the entire range of T and h

considered in this study. The calculations are performed on a 3 GHz Linux cluster.

Each FFS calculation for a given (T, h) condition takes about 50 CPU-hours for the


(a) (b)

0 0.05 0.1 0.1510

−50

10−40

10−30

10−20

10−10

100

h

I (M

CS

S−1 )

kBT = 1.9 1.5

1.2

0 0.05 0.1 0.150

0.5

1

1.5

2

h

IFF

S / IB

D

(c) (d)

0.3 0.4 0.5 0.6

10−30

10−20

10−10

h

I (

MC

SS −

1 )

kBT = 2.80 2.71

2.65 2.502.35

2.20

0.3 0.4 0.5 0.60

0.5

1

1.5

2

h

IFF

S / IB

D

Figure 4.3: The nucleation rate I computed by FFS (open symbols) and Becker-Doring theory with US free energies (filled symbols) in the (a) 2D and (c) 3D Isingmodels. The ratio between nucleation rates obtained by FFS and Becker-Doringtheory at different temperatures in the (b) 2D and (d) 3D Ising models. The symbolsin (b) and (d) match those defined in (a) and (c), respectively.

2D Ising model and 200 CPU-hours for the 3D Ising model. Each US calculation

takes a similar amount of time as an FFS calculation.

As shown in Fig. 4.3, the nucleation rate over these conditions spans more than 20

orders of magnitude. Yet, most of the rates predicted by the two methods are within

50% of each other. This is a strong confirmation of Part I of the Becker-Doring theory,

i.e. Eq. (2.7). It confirms that for the purpose of computing nucleation rate, it is valid

to coarse grain the Ising model to a one-dimensional Markov chain, with the size of the

largest droplet being the reaction coordinate. Detailed balance between neighboring


states along the Markov chain, as is assumed by the Becker-Doring theory, has been

shown by a recent study [135] and our confirmation of I = f+c Γ exp

(

− Fc

kBT

)

provides

another evidence for it. This means that the Becker-Doring theory can predict the

nucleation rate of the 2D and 3D Ising models accurately, provided that the correct

free energy function F (n) is used. This is consistent with an earlier report by Brendel

et al. [126].

4.4.2 Critical Droplet Size and Shape

There are two common definitions of the critical droplets. In the first definition,

a droplet is of critical size if its probability to grow and cover the entire system is

50%. In other words, a critical droplet has a committor probability of 50%. In the

second definition, a droplet is of critical size if it corresponds to the maximum of the

free energy curve F (n). It is of interest to verify whether these two definitions are

equivalent.

(a) (b)

10−2

10−1

102

103

h

n c

kBT = 1.9

1.51.0

10−0.5

10−0.4

10−0.3

10−0.2

102

103

h

n c

kBT = 2.20

2.35

2.50

2.65

2.71

Figure 4.4: (a) For the 2D Ising model, the critical droplet size n obtained fromFFS (filled symbols) and umbrella sampling (open symbols). nc predicted by Becker-Doring theory (dotted line) and by field theoretic equation (solid line) are plotted forcomparison. (b) For the 3D Ising model, the critical droplet size n obtained fromFFS (filled symbols) and umbrella sampling (open symbols).

After each FFS simulation under a given (T, h) condition, an ensemble of 15,000


(a) (b)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

PB

P(P

B)

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

PB

P(P

B)

3D

Figure 4.5: (a) Histogram of committor probability in an ensemble of spin configura-tions with n = 496 for the 2D Ising model at kBT = 1.5 and h = 0.05. Representativedroplets are also shown, with black and white squares corresponding to +1 and −1spins, respectively. (b) Histogram of committor probability in an ensemble of spinconfigurations with n = 524 for the 3D Ising model at kBT = 2.20 and h = 0.40.

spin configurations are saved at each interface λi. The average values of PB(λi) for a

given interface can be estimated using the following recursive relation,

PB(λi) = PB(λi+1)P (λi+1|λi) (4.18)

for i = n− 1, n− 2, ...1 with the boundary condition PB(λn) = 1 [143]. By fitting the

data of PB to a smooth curve with spline interpolation, we can extract the critical

value nc for which PB = 0.5, as shown in Fig. 3.7. Some of the critical nucleus size

obtained this way are shown in Fig. 4.4 as filled symbols.

The droplet sizes that correspond to the maximum of the free energy curve ob-

tained by US are listed in Fig. 4.4 as open symbols. For both 2D and 3D Ising models,

critical size from two different methods agree with each other within 2%. This con-

firms that the two definitions for the critical nucleus are equivalent, provided that the

correct free energy curves F (n) are used. It also proves that the size of the largest

droplet is a good reaction coordinate.

Fig. 4.5(a) plots the histogram of the committor probability for an ensemble of


spin configurations with n = 496 for the 2D Ising model at kBT = 1.5 and h = 0.05.

The average committor probability of this ensemble is 49.4%. About 94% of the

spin configurations in this ensemble have committor probabilities within the range of

49± 5%. This further confirms that the size of the largest cluster, n, is a very good

reaction coordinate of the nucleation process. Fig. 4.5(b) plots the histogram of the

committor probability within an ensemble of spin configurations with n = 524 for the

3D Ising model at kBT = 2.20 and h = 0.40. The average committor probability of

this ensemble is 50%. About 80% of the spin configurations in this ensemble have

committor probabilities within the range of 50 ± 5%. The spread of the committor

probability distribution is wider than the 2D case, and is consistent with an earlier

report [39].

4.4.3 Droplet Free Energy of 2D Ising Model

The previous sections show that the Becker-Doring theory performs well as long as

the correct droplet free energy F (n) is provided. We now compare the theoretical

predictions of F (n) with numerical results by US. We will focus on 2D Ising model

in this section and will discuss F (n) in the 3D Ising model in the next section.

Fig. 4.6 plots the F (n) curves for kBT = 1.5 and h = 0.05. Numerical results

from US and predictions from the Becker-Doring theory, Eq. (4.3), and Langer’s field

theory, Eq. (4.10), are plotted together. It is clear that the logarithmic correction

term τkBT lnn from the field theory is substantial. The field theory prediction,

which contains this correction term, agrees very well with numerical US results after

a constant term is added. The free energy used in CNT, Eq. (4.3), which lacks this

correction term, is significantly lower, about orders of magnitude larger than kBT .

Obviously, if this free energy curve is used, the Becker-Doring theory will overestimate

the nucleation rate by several orders of magnitude. Our result also shows that, the

field theory predictions, though derived under the assumption of infinitesimal h, are

still valid at finite h in the range of field considered in this study.

Our results shows that the macroscopic surface free energy (at zero h) can be

safely applied to a droplet (at finite h) [137], provided that the constant correction


(a) (b)

0 100 200 300 400 500 6000

10

20

30

40

50

60

70

Eq.(6)

Eq.(8)

τ kBT ln n

d

n

F(n

)US data

0 5 10 15 20 250

5

10

15

20

25

30

Eq.(6)

Eq.(8)

τ kBT ln n

d

n

F(n

)

Eq.(D12)US data

Figure 4.6: (a) Droplet free energy curve F (n) of the 2D Ising model at kBT = 1.5and h = 0.05 obtained by US (circles) is compared with Eq. (4.10) (solid line) andEq. (4.3) (dashed line). Logarithmic correction term 5

4kBT lnn (dot-dashed line) and

the constant term d (dotted line) are also drawn for comparison. (b) Magnified view of(a) near n = 0, together with the results from analytic expressions (squares) availablefor n ≤ 17 (see Appendix B.3).

term is added (see Appendix B.3). Brendel et al. [126] reported that the effective

surface free energy exceeds that of the macroscopic surface free energy by 20%. But

this was caused by the neglect of the logarithmic correction term in that study.

Our results contradict the previous report [47] that τ is close to zero at low

temperatures (T = 0.59 Tc and 0.71 Tc) and only goes to 54near T = 0.84 Tc. In

the previous study [47], only small clusters (n < 60) are sampled without using

the umbrella sampling technique. We suspect this approach is susceptible to the

error caused by the lack of statistics at low temperatures (especially for clusters with

n > 30). Because the field theoretic correction term τkBT lnn becomes smaller at

low T , it could be masked by the statistical error. To support our finding, the free

energy curves for cluster sizes up to n = 1950 at a wide temperature range (from

0.53 Tc to 0.84 Tc) are presented in the Appendix B.4. τ = 54is necessary in the entire

temperature range to accurately describe the droplet free energy.

In the literature, the field theory correction is usually expressed as an extra pre-

exponential factor inserted into the Becker-Doring formula of the nucleation rate.


Both a pre-exponential factor and a change to the free energy curve can modify the

nucleation rate. So it may appear impossible (or irrelevant) to decide which approach

is more “correct”. However, a closer inspection shows that it is indeed possible to

tell whether the correction should be interpreted as a free energy change, or a kinetic

pre-factor. This is because self-consistency requires that the maximum of the free

energy curve F (n) should match the droplet size nc whose committor probability is

50%, as discussed in the Section 4.4.2.

Fig. 4.4 shows the critical droplet sizes nF/BDc (dotted lines), which correspond to

the maximum of the free energy curves F (n) predicted by the Becker-Doring theory,

Eq. (4.3). They are significantly smaller than the critical droplet sizes ncommc (filled

symbols) that corresponds to a 50% committor probability. With the field theory

correction term in the free energy, the critical droplet sizes nF/FTc (solid lines) agree

much better with ncommc . This result clearly shows that the field theory correction

should be placed in the free energy function F (n), instead of being a kinetic prefactor.

It is of interest to compare the various free energy expressions discussed so far

with the analytic (exact) expressions [145] for F (n) that are available for 0 ≤ n ≤ 17.

It is somewhat surprising that the field theory prediction of F (n) (after corrected by

a constant term, see Appendix B.3) agrees very well with both the numerical data

from US and the analytic expressions, for such small values of n. This is another

confirmation for the field theory prediction of the free energy curve, Eq. (4.10).

Shneidman et al. [47] also observed the effect of the logarithmic correction term,

but expressed it in terms of “size-dependent prefactor”, and suspected that it is caused

by coagulation of droplets. Our results show that this is not a coagulation (many-

droplet) effect, because the logarithmic correction term is derived by considering the

shape fluctuation of a single droplet.

In summary, the free energy expression from CNT must be modified by two terms,

i.e. a logarithmic correction term τkBT lnn from field theory and a constant term to

match the free energy of very small droplets. In 2D, both terms can be determined

completely from existing theories and contain no fitting parameters.


4.4.4 Droplet Free Energy of 3D Ising model

In the following, we will examine the functional form of the droplet free energy F (n)

in the 3D Ising model. Because there is no analytic solution to the effective surface

free energy in 3D, σeff(T ) must be treated as a fitting parameter in our analysis, which

creates more uncertainty in our conclusions. For example, we cannot unambiguously

determine the coefficient τ in the logarithmic correction term (in Eq. (4.10)) from the

numerical results. Another difficulty in determining τ is that in 3D the logarithmic

correction term is much smaller compared with the first two terms in Eq. (4.10).

(a) (b)

0 100 200 300 400 500 600 700

0

100

200

300

400

500

600

700

n

F(n

)

Eq.(25)

τ kBT ln n

US dataEq.(15)

0 5 10 15 20 25−10

0

10

20

30

40

50

60

70

n

F(n

)

Eq.(25)

τ kBT ln n

B exp(−Cn/B)

Eq.(D12)US dataEq.(15)

Figure 4.7: (a) Droplet free energy F (n) of the 3D Ising model at kBT = 2.40 andh = 0 obtained by US (circles) is compared with Eq. (4.22) (solid line) and Eq. (4.11)(dots). Logarithmic term τkBT lnn is also plotted (dot-dashed line). The differencein predictions by classical expression Eq. (4.11) and field theory Eq. (4.22) are verysmall compared to F (n) itself and cannot be observed at this scale. (b) Magnifiedview of (a) near n = 0, together with the analytic solution of small droplets (squares,see Appendix B.3) and the exponential correction term (dashed line).

To reduce the complexity from finite h, we computed droplet free energy at zero

field for a range of temperature kBT = 2.0, ..., 2.8 by US. Fig. 4.7 plots the results at

kBT = 2.40 and h = 0. We have examined a number of functional forms to see which

one best describes the numerical data of the droplet free energy.

First, we fit the data to the original Becker-Doring form, Eq. (4.11), plus a constant


correction term, i.e.,

F (n) = σeff(T )αn2/3 − 2 hn+ d(T ) (4.19)

where σeff(T ) is a free fitting parameter at each temperature. We find that Eq. (4.19)

cannot describe the droplet free energy well in the entire range of n. Since we expect

it to be more accurate in the continuum limit of large n, we fit the US data to

Eq. (4.19) only in the range of n > 50. The resulting term d(T ) is in the range

of −1.1 (at kBT = 2.0) to −2.4 (at kBT = 2.8). The error in the fit is defined as

R ≡[

1

700

750∑

i=50

(F (i)− Ffit(i))2

]1/2

, where F (i) is the numerical data from US, and

Ffit(i) is the value given by Eq. (4.19). The resulting R is in the range of 0.01-0.13

and increases with increasing temperature. Significant discrepancy between the US

data and the fit is observed in the range of n < 50, which will be further discussed

below.

The next function to be considered for the fit includes the logarithmic correction

term,

F (n) = σeff(T )αn2/3 + τ(T ) kB T lnn− 2 hn+ d(T ) (4.20)

in which τ is a free parameter for each temperature T . The error of the fit is now re-

duced to about R ≈ 0.01 for all temperatures and is now independent of temperature.

This means that the logarithmic term improves the description of the temperature

dependence of the free energy of large droplets (n > 50). But the discrepancy in

the range of n < 50 still remains. This is different from the 2D Ising model, where

Eq. (4.10) describes the droplet free energy very well even without any fitting param-

eters.

Perini et al [46] used the following functional form to fit their free energy data,

F (n) = σeff(T )αn2/3 +K(T )n1/3 + τ kB T lnn− 2 hn+ d(T ) (4.21)

where τ = −19is constrained to be a constant 3. The parameter K corresponds to

3This is against the expected temperature dependence of τ due to suppression of shape fluctuation


the extra energy of “ledges” that appear on 3D droplets. We find that the quality of

the fit using Eq. (4.21) is similar to that using Eq. (4.20) 4. However, the resulting

K(T ) is an increasing function of temperature. This is counter-intuitive because the

continuum droplet approximation is expected to be better at higher temperatures

where equilibrium droplet shape becomes more spherical. Hence we would expect

K(T ) to decrease with increasing temperature. Therefore, we believe Eq. (4.20) is a

more appropriate functional form than Eq. (4.21).

Hence, we do not include the “ledge” energy term, and will treat τ as a function

of temperature during the fitting. We also find that the fit in the range of n < 50

can be significantly improved by adding an exponential term. The data at all T and

in the entire range of 0 < n < 750 considered in this study turns out to be well fitted

by the following function.

F (n) = σeff(T )αn2/3 + τ(T ) kB T lnn− 2 hn+ A+B(T ) exp

[

− C n

B(T )

]

(4.22)

where A and C are constants independent of T , and B(T ), σeff(T ), and τ(T ) are

functions of T 5. The fitted parameters are: A = 0.06 and C = 0.59. It turns out

that B(T ) can be well described by a linear function: B(T ) = 9.12 kBT − 16.08. The

contribution of the exponential term is plotted in Fig. 4.7(b). In the range of T and h

considered in this study, the size of the critical nucleus is larger than 100. Hence the

nucleation rate predicted by CNT under these conditions is only affected by F (n) in

the range of n > 100. When Eq. (4.22) is used, the numerical values of the logarithmic

term is in the range of −5 to 0, for 2.0 ≤ kBT ≤ 2.8 and 100 < n < 750. It is the

major correction term to the classical expression of the droplet free energy, Eq. (4.11),

for n > 100. In comparison, the constant term is A = 0.06 and the magnitude of the

exponential term is less than 10−7 for n > 100.

below the roughening temperature, see Section II.A.4Perini et al [46] originally introduced the “ledge” term in order to improve the quality of the fit

in the range of n < 19. However, we found that fitting to the data in the range of n < 19 will leadto large discrepancies in the range of n > 100. Given that the droplet theory is supposed to workbetter in the continuum limit of large n, we believe the function should be fitted to data at large n.

5Eq. (4.22) also fits the data at non-zero h. However, the resulting σ3Deff from the fit slightly

increases with h. For example, σ3Deff increase by about 3% as h changes from 0 to 0.5.


(a) (b)

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

σeff

σ(100)

TR

kBT

σ eff

2 2.2 2.4 2.6 2.8

−0.25

−0.2

−0.15

−0.1

−0.05

0

TR

kBT

τ

Figure 4.8: (a) Surface free energies of the 3D Ising model as functions of temperature.Circles are fitted values of σeff from Eq. (4.22), dashed line is the expected behaviorof σeff over a wider range of temperature, and solid line is the free energy of the (100)surface [141]. Numerically fitted values of σeff from Heermann et al. [137] are plottedas +. (b) τ values that give the best fit to the free energy data from US. τ canbe roughly described by a linear function of T shown as a straight line. No abruptchange is observed near the roughening temperature TR.

Fig. 4.8(a) shows the fitted values of σeff in the temperature range of 0.4-0.65Tc.

σeff decreases with T , as expected. In the limit of large T , the difference between the

free energies of (100) and (110) surfaces diminishes, the droplet becomes spherical,

and σeff converges to the free energy of (100) surfaces. In the limit of T → 0, we

expect σeff to converge to (6/π)1/3 times the surface tension of the (100) surface. This

is because as T → 0, the shape of the droplet becomes cubical [136], and (6/π)1/3

is the surface area ratio between a sphere and a cube, both having unit volume.

The expected shape of σeff(T ) over this temperature range is plotted as a dashed

line, which is similar to the case of 2D Ising model shown in Fig. 4.1. In summary,

we expect σeff to decrease from 2.481 to 0 as temperature increases. For example,

at kBT = 2.71, σeff = 1.6. This may explain the discrepancy reported by Pan et

al. [39], in which σeff = 2 is assumed at kBT = 2.71. In Vehkamaki et al. [138],

the nucleation rate predicted by CNT was reported to have a weaker temperature

dependence than the numerical results. This is probably caused by the use of the

same surface free energy (at T = 0.59 Tc) in the entire temperature range (0.54 Tc


to 0.70 Tc). The decrease of surface energy with temperature leads to a significant

reduction of nucleation free energy barrier with temperature. This corresponds to an

anomalously large “effective entropy” of nucleation as presented in the next section,

which would be difficult to explain if the variation of surface energy were ignored.

Fig. 4.8(b) shows the fitted values of τ as a function of temperature. Over the

range of temperature considered here, τ can be approximated by a linear function of

T , τ = −0.26 kBT + 0.44. The fact that τ < 0 in 3D is consistent with theoretical

predictions. But τ is found to decrease with temperature, and no discontinuity at

the roughening temperature is observed. This is consistent with the observation that

no significant change of droplet shape occurs near the roughening temperature (see

Appendix B.2). This is contrary to the theoretical predictions of τ = −23at T < TR

and τ = −19at T > TR. The change of τ with T may be the consequence of a

gradual change of anisotropy effects as temperature changes [48]. More investigation

is needed to resolve the controversy of τ in the 3D Ising model. The difference between

2D (where τ = 5/4 remains a constant) and 3D Ising models on the behavior of τ

remains intriguing.

4.4.5 Effective Entropy of Nucleation

We study the temperature dependence of droplet free energy (or effective entropy

of nucleation) at a given h, for both 2D and 3D Ising models. In many studies,

the temperature dependence of the droplet free energy is neglected and the energy

barrier estimate at zero temperature is used to predict nucleation rate. In the current

study, we find very large temperature dependence that can change the nucleation rate

prediction by many orders of magnitude. Fig. 4.9(a) plots the droplet free energy as a

function of droplet size n for the 2D Ising model at h = 0.1 and different temperatures.

The maxima of these curves, i.e. the free energy barrier Fc, are plotted in Fig. 4.9(b).

The data can be fitted to a straight line, whose slope gives an effective entropy of

S = 43.5 kB, corresponding to the exp(S/kB) ∼ 1019, 19 orders of contribution to the

nucleation rate. An entropy of this magnitude seems anomalously large and will be

difficult to attribute to the shape fluctuation of the critical droplet. We believe that


(a) (b)

0 50 100 150 200 250 3000

10

20

30

40

50

60

70

n

F c

kBT=0.9

1.01.1

1.21.3

1.41.5

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.630

35

40

45

50

55

60

65

70

S = 43.5 kB

S = 53.4 kB

kB T

F c

US dataEq.(10)

(c) (d)

0 100 200 300 4000

50

100

150

n

F c

kBT=2.20

2.35

2.50

2.652.71

2.1 2.2 2.3 2.4 2.5 2.6 2.780

90

100

110

120

130

140

150

160

170

180

S = 143 kB

S = 152 kB

kB T

F c

US dataEq.(17)

Figure 4.9: (a) Droplet free energy as a function of droplet size n at h = 0.1 anddifferent kBT for the 2D Ising model. The critical droplet free energy is marked bycircles. (b) Critical droplet free energy (circles) from (a) as a function of kBT forthe 2D Ising model. The solid line is a linear fit of the data, and the dashed line isthe prediction of Eq. (4.5). (c) Droplet free energy as a function of droplet size n ath = 0.45 and different kBT for the 3D Ising model. (d) Critical droplet free energyfrom (c) as a function of kBT for the 3D Ising model. The solid line is a linear fit ofthe data, and the dashed line is the prediction of Eq. (4.13).

this entropy is a consequence of the temperature dependence of the effective surface

free energy σeff(T ). In CNT, the free energy barrier is linked to σeff(T ) through

Eq. (4.5). As a comparison, Fig. 4.9(b) also plots the prediction of Eq. (4.5) as

a dashed line, which gives an effective entropy of 53.4 kB. This confirms that the

anomalously large entropy is a result of the temperature dependent surface tension.

The large difference between the solid line and dashed line indicates the importance

of the logarithmic correction term in 2D.

Fig. 4.9(c) plots the droplet free energy as a function of droplet size n for the 3D

Ising model at h = 0.45 and different temperatures. The maxima of these curves,


i.e. the free energy barrier Fc, are plotted in Fig. 4.9(d). The data can be fitted to

a straight line, whose slope gives an effective entropy of S = 143 kB, corresponding

to the exp(S/kB) ∼ 1062, 62 orders of contribution to the nucleation rate. As a

comparison, Fig. 4.9(b) also plots the prediction of CNT, Eq. (4.13), as a dashed line,

which gives an effective entropy of 152 kB. Again, the anomalously large entropy is a

result of the temperature dependent surface tension.

In the next chapter, we will study the dislocation nucleation which is analogous

to the 2D Ising model. The entropic effect on the dislocation nucleation in the copper

will be extensively studied.

4.5 Summary and Discussion

In this chapter, we have used two independent methods to calculate the nucleation

rate of Ising model in 2D and 3D, in order to check independently the different as-

sumptions of the nucleation theories. The Markov chain assumption with the largest

droplet size as the reaction coordinate is found to be accurate enough to predict nu-

cleation rate spanning more than 20 orders of magnitude, provided that the correct

droplet free energy function is used. The logarithmic correction term is found to be

essential to droplet free energy in 2D. Our numerical results verified the field theory

prediction that τ = 5/4 in 2D. However, for the 3D Ising model, our numerical re-

sults are not consistent with existing theories on the coefficient τ of the logarithmic

correction term, suggesting that some important physics may still be missing in the

existing theories, such as the anistropy effect on the droplet free energy. An exponen-

tial function seems to be necessary to describe the free energy of small 3D droplets,

but it is not needed for the 2D droplets.

How do the findings in the Ising model study compares with the discussion in

the chapter 2? At a first glance, the change of critical size from logarithmic correc-

tion term seems contradict to the finding that CNT predicts the critical size with

small error. This can be explained if we consider that relative magnitude of thermal

fluctuation and the surface energy is very different between the Ising model and the


gas-liquid transition. In the Ising model, we set the nearest-neighbor interaction co-

efficient J to be 1 which means that surface energy σ is about 2 or less, while the

temperature fluctuation kBT is on the order of 1. However, the typical broken bond

costs an order of 1 eV, but the thermal fluctuation at 300K is about 0.026 eV, about

40 times smaller than the surface energy. Critical size predicted from field theory will

be significantly different from the classical nucleation theory if kBT∆µ is on the same

order of magnitude with σ2, as shown in the Eq. 4.17. In the Ising model, it is the case

and we see significant correction in the 2D Ising model critical size. On the contrary,

in the gas-liquid transition where surface energy is much larger than the thermal fluc-

tuation, the change of critical size from logarithmic term will be negligible 6. While

logarithmic term τkBT lnn does not change the droplet size much in the gas-liquid

transition, it can affect the free energy barrier and nucleation rate significantly. The

nucleation rate will be affected by a factor of exp(τkBT lnnc/kBT ) ∼ nτc . Unless the

absolute value of τ is much less than 1, this modifies the nucleation rate significantly.

From nucleation experiments results and second nucleation theorem, it is also

identified that the CNT overestimates the free energy barrier, which also seems to

contradict the results presented in this chapter where we find that the free energy

obtained from the umbrella sampling is much larger than the CNT prediction. This

originates from the difference in the free energy definition. In the Ising model simu-

lation, total number of lattice points is fixed and the number of up-spin and down-

spin changes as the nucleation proceed. Thus, the free energy is defined such that

exp(∆G(n)/kBT ) = Nn/Nlattice where Nn is the number of island that has n up-spins.

Nlattice is the total number of lattice in the simulation cell. In the 2D Ising model,

the positive constant correction term is needed to make G(1) to present the correct

population of single spin island. In the 3D Ising model where an fast decaying expo-

nential correction term was found in the small n range, the free energy curve at large

n can also be approximated by the CNT expression plus a negative constant term.

However, in the gas-liquid transition, the number of gas molecule is conserved and

the free energy is defined such that exp(∆G(n)/kBT ) = Nn/N1. For self-consistency,

6∆µ is a few to a few tens of kBT in typical gas-liquid transition experiments, as shown in theFig. 2.3. Thus, σ2 ≫ kBT∆µ because σ ≫ kBT .


G(n) must vanish at n = 1. A negative constant term correction must be subtracted

from the free energy expression ∆G(n) = σS − n∆µ to make G(1) = 0. In short, the

origin of the correction term is same: it is required to give a correct population in the

small n range.

A promising direction for future research is to numerically compute the surface

free energy of different orientations in 3D and to build the effective surface free energy

σeff from the Wulff construction. This would eliminate σeff as a fitting parameter and

would enable a more stringent test of CNT for the 3D Ising model. In addition,

3D simulations at smaller h values will allow a more direct comparison with existing

nucleation theories, most of which assume an infinitesimal h.

Chapter 5

Predicting the Dislocation

Nucleation Rate as a Function of

Temperature and Stress

5.1 Introduction

Dislocation nucleation is essential to the understanding of ductility and plastic defor-

mation of crystalline materials with sub-micrometer dimension [88, 93, 94] or under

nano-indentation [96, 97, 98], and to the synthesis of high quality thin films for micro-

electronic, optical and magnetic applications [92, 147]. The fundamental quantity of

interest is the dislocation nucleation rate I as a function of stress σ and temperature

T . Continuum [148, 149, 150] and atomistic models [151, 152, 153] have been used to

predict dislocation nucleation rate and they both have limitations. The applicability

of continuum models may be questionable because the size of the critical dislocation

nucleus can be as small as a few lattice spacing. In addition, the continuum models

are often based on linear elasticity theory, while dislocation nucleation typically occur

at high strain conditions in which the stress-strain relation becomes non-linear. These

difficulties do not arise in molecular dynamics (MD) simulations, which can reveal

important mechanistic details of dislocation nucleation. Unfortunately, the time step

of MD simulations is on the order of a femto-second, so that the time scale of MD

75

CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 76

simulations is typically on the order of a nano-second, given existing computational

resources. Therefore, the study of dislocation nucleation via direct MD simulation

has been limited to extremely high strain rate (∼ 108 s−1) conditions [151, 152]. This

is about 10 orders of magnitudes higher than the strain rate in most experimental

work and engineering applications [97, 98]. Predicting the dislocation nucleation rate

I(σ, T ) under the experimentally relevant conditions is still a major challenge.

An alternative approach is to combine reaction rate theories [36, 154, 155] with

atomistic models. Atomistic simulations can be used to compute the activation bar-

rier, which is used as an input for the reaction rate theory to predict the dislocation

nucleation rate. There are several reaction rate theories, such as the transition state

theory [154, 156] and the Becker-Doring theory [36], which lead to similar expressions

for the nucleation rate,

I(σ, T ) = Ns ν0 exp

[

−Gc(σ, T )

kBT

]

(5.1)

where Ns is the number of equivalent nucleation sites, ν0 is a frequency prefactor, Gc

is the activation Gibbs free energy for dislocation nucleation, and kB is Boltzmann’s

constant. The difference between the theories lie in the expression of the frequency

prefactor ν0. In practice, ν0 is often approximated by the Debye frequency νD of

the crystal, which is typically on the order of 1013 s−1. One could also express the

dislocation nucleation rate as a function of strain γ and temperature T . Then,

I(γ, T ) = Ns ν0 exp

[

−Fc(γ, T )

kBT

]

(5.2)

where Fc is the activation Helmholtz free energy for dislocation nucleation.

The transition state theory (TST) [154, 156] has often been combined with the

nudged elastic band (NEB) method [157] to predict the rate of rare events in solids [153,

158]. However, there exist several limitations for this approach. First, TST is known

to overestimate the rate because it does not account for the fact that a single re-

action trajectory may cross the saddle region multiple times. This deficiency can

be corrected by introducing a recrossing factor which can be computed by running


many MD simulations started from the saddle region [159]. A more serious problem

is that the NEB method, and the closely related string method [160], only deter-

mines the activation barrier at zero temperature, i.e. Gc(σ, T = 0) or Fc(γ, T = 0)

In principle, the activation barrier at finite temperature can be obtained from the

finite-temperature string method [161], which has not yet been applied to dislocation

nucleation. Both Gc and Fc are expected to decrease with T , as characterized by the

activation entropy Sc. The effect of the activation entropy is to introduce an overall

multiplicative factor of exp(Sc/kB) to the nucleation rate (See Eq. (5.13)), which can

be very large if Sc exceeds 10 kB. Until recently, the magnitude of Sc has not been

determined reliably and, within the harmonic approximation of TST, Sc is estimated

to be small (i.e. ∼ 3 kB). For example, the activation entropy of kink migration on a

30 partial dislocation in Si has recently been estimated to be less than 3 kB [162].

In this chapter, we provide a thorough discussion on the thermodynamics of dis-

location nucleation and computational methods for predicting dislocation nucleation

rate. The thermodynamic properties of activation, such as activation free energy,

activation entropy, and activation volume have been extensively discussed in the con-

text of dislocation overcoming obstacles, using continuum theory within the constant

stress ensemble [163]. Recently, there has been interest in computing these quantities

using atomistic simulations, in which it is more convenient to use the constant strain

ensemble. One of the main objectives of this study is to discuss the difference be-

tween the thermodynamic properties of activation defined in the constant stress and

the constant strain ensembles in the context of dislocation nucleation. First, we prove

that the activation Gibbs free energy Gc(σ, T ) equals the activation Helmholtz free

energy Fc(γ, T ) for dislocation nucleation, when the volume of the crystal is much

larger than the activation volume of dislocation nucleation. This leads to the intuitive

conclusion that the dislocation nucleation rate is independent of whether the crystal

is subjected to a constant stress or a constant strain loading condition. While the

equality of Fc(γ, T ) and Gc(σ, T ) quickly leads to the difference of the two activation

entropies, we provide an alternative derivation of this fact, which makes the physical

origin of this difference more transparent. Our goal is to clarify why the activation

entropy depends on the choice of ensemble while entropy itself does not. Second, we


describe the computational methods in sufficient detail so that they can be repeated

by interested readers and be adopted in their own research. We predict the nucleation

rate from the classical nucleation theory using the free energy obtained from umbrella

sampling. We have used two order parameters to characterize dislocation nucleation

and find that our previous results are independent of the choice of order parameters,

as required. Third, we compare our numerical results with several previous estimates

of the activation entropy, such as those based on the “thermodynamic compensation

law”, which states that the activation entropy is proportional to the activation en-

thalpy. We discuss the conditions at which this empirical law appears to hold (or fail)

for dislocation nucleation.

This chapter is organized as follows. Section 5.2 is devoted to the thermodynamics

of dislocation nucleation. Simulation setup and computational methods are presented

in Section 5.3. Section 5.4 presents the numerical data on activation free energy

and the frequency prefactor over a wide range of stress (strain) and temperature

conditions. Section 5.5 compares these results with previous estimates of activation

entropy and discusses the consequence of the activation entropy on experimentally

measurable quantities, such as yield stress.

5.2 Thermodynamics of Nucleation

5.2.1 Activation Free Energies

Consider a crystal of volume V subjected to stress σ at temperature T . To be specific,

we can consider σ as one of the stress components, e.g. σxy, while all other stress

components are zero. Let G(n, σ, T ) be the Gibbs free energy of the crystal when it

contains a dislocation loop that encloses n atoms. If n is very small, the dislocation

loop is more likely to shrink than to expand. On the other hand, if n is very large,

the dislocation loop is more likely to expand than to shrink. There exists a critical

loop size, nc, at which the likelihood for the loop to expand equals the likelihood to

shrink. It is also the loop size that maximizes the function G(n, σ, T ) for fixed σ and


T . The activation Gibbs free energy is defined as,

Gc(σ, T ) ≡ G(nc, σ, T )−G(0, σ, T ) (5.3)

where G(0, σ, T ) is the Gibbs free energy of the perfect crystal (containing no dislo-

cations) at stress σ and temperature T . Given Gc(σ, T ), the dislocation nucleation

rate can be predicted using Eq. (5.1).

While experimental data are usually expressed in terms of σ and T , it is often

more convenient to control strain than stress in atomistic simulations. Let γ be the

strain component that corresponds to the non-zero stress component, e.g. γxy. While

there is only one non-zero stress component, this usually corresponds to multiple non-

zero strain components. Nonetheless, the other strain components do not appear in

our discussion because their corresponding work term is zero. Thermodynamics [164]

allows us to discuss the nucleation process within the constant γ constant T ensemble,

by introducing the Helmholtz free energy F (n, γ, T ) through the Legendre transform 1.

γ(n, σ, T ) ≡ − 1

V

∂ G(n, σ, T )

∂ σ

∣

∣

∣

∣

n,T

(5.4)

F (n, γ, T ) ≡ G(n, σ, T ) + σ γ V. (5.5)

A convenient property of the Legendre transform is that it is reversible, i.e.,

σ(n, γ, T ) =1

V

∂ F (n, γ, T )

∂ γ

∣

∣

∣

∣

n,T

(5.6)

G(n, σ, T ) = F (n, γ, T )− σ γ V. (5.7)

Again, let nc be the dislocation loop size that maximizes F (n, γ, T ) for given γ and T .

In Appendix C.1, it is proven that the same nc maximizes G(n, σ, T ) and F (n, γ, T ),

so that the critical dislocation loop size does not depend on the choice of (constant

1To be more precise, σ is the Cauchy stress, and γ V is the conjugate variable to σ. We choose Vto be the volume of a reference state, i.e. the state at σ = 0. Then γ is the logarithmic (or Hencky)strain relative to the reference state [165]. Here, we are interested in the regime of 0 ≤ γ ≤ 20%,and the difference between the Hencky strain and the simple engineering strain is negligible. Hence,in the numerical test case, we are going to let γ be the engineering strain.


stress or constant strain) ensemble. The activation Helmholtz free energy is defined

as,

Fc(γ, T ) ≡ F (nc, γ, T )− F (0, γ, T ). (5.8)

Given Fc(γ, T ), the dislocation nucleation rate can be predicted using Eq. (5.2).

5.2.2 Activation Entropies

The activation Gibbs free energy Gc(σ, T ) decreases with increasing temperature T

at fixed σ, and also decreases with increasing σ at fixed T . The activation entropy,

defined as

Sc(σ, T ) ≡ − ∂Gc(σ, T )

∂T

∣

∣

∣

∣

σ

(5.9)

measures the reduction rate of Gc(σ, T ) with increasing T . Similarly, the activation

volume, defined as

Ωc(σ, T ) ≡ − ∂Gc(σ, T )

∂σ

∣

∣

∣

∣

T

(5.10)

measures the reduction rate of Gc(σ, T ) with increasing σ.

The activation enthalpy Hc is defined as,

Hc(σ, T ) = Gc(σ, T ) + T Sc(σ, T ). (5.11)

The activation entropy Sc is usually insensitive to temperature, especially in the range

of zero to room temperature, which will be confirmed by our numerical results. This

means that Hc is also insensitive to temperature and that the Gibbs free energy can

be approximated by,

Gc(σ, T ) = Hc(σ)− TSc(σ). (5.12)

Consequently, the dislocation nucleation rate in Eq. (5.1) can be rewritten as,

I(σ, T ) = Ns ν0 exp

(

Sc(σ)

kB

)

exp

(

−Hc(σ)

kBT

)

. (5.13)

Therefore, when the dislocation nucleation rate per site, I/N , at a constant stress

σ are shown in the Arrhenius plot, e.g. Fig. 5.1, the data are expected to follow


a straight line. The negative slope of the line can be identified as the activation

enthalpy Hc(σ) over kB, and the intersection of the line with the vertical axis is

ν0 exp(Sc(σ)/kB). Hence the activation entropy Sc contributes an overall multiplica-

tive factor, exp(Sc(σ)/kB), to the nucleation rate. If Sc = 3 kB, this factor is about

20 and may be considered insignificant. However, if Sc > 10 kB, this factor exceeds

104 and cannot be ignored.

0 2 4 6

x 10−3

10−20

100

1020

1040

1/T (K−1)

I (pe

r at

om)

Hc/k

B

ν0 exp(S

c/k

B)

Figure 5.1: Homogeneous dislocation nucleation rate per lattice site in Cu under pureshear stress σ = 2.0 GPa on the (111) plane along the [112] direction as a functionof T−1, predicted by Becker-Doring theory using free energy barrier computed fromumbrella sampling (See Section 5.3). The solid line is a fit to the predicted data (incircles). The slope of the line isHc/kB, while the intersection point of the extrapolatedline with the vertical axis is ν0 exp(Sc/kB). Dashed line presents the nucleation ratepredicted by ν0 exp(−Hc/kBT ), in which the activation entropy is completely ignored,leading to an underestimate of the nucleation rate by ∼ 20 orders of magnitude.

If we choose the constant strain γ ensemble, then the focus is on the activation

Helmholtz free energy Fc(γ, T ), which decreases with increasing temperature T at

fixed γ. An alternative definition of the activation entropy can be given as

Sc(γ, T ) ≡ − ∂Fc(γ, T )

∂T

∣

∣

∣

∣

γ

(5.14)

which measures the reduction rate of Fc(γ, T ) with increasing T . We can then define


the activation energy Ec(γ, T ) as,

Ec(γ, T ) = Fc(γ, T ) + T Sc(γ, T ). (5.15)

Again, since the activation entropy Sc is usually insensitive to temperature, we can

use the following approximation for the activation Helmholtz free energy,

Fc(γ, T ) = Ec(γ)− TSc(γ). (5.16)

Consequently, the dislocation nucleation rate in Eq. (5.2) can be rewritten as,

I(γ, T ) = Nν0 exp

(

Sc(γ)

kB

)

exp

(

−Ec(γ)

kBT

)

. (5.17)

Therefore, Ec(γ) and Sc(γ) can also be identified from the slope and y-intersection in

Arrhenius plot of dislocation nucleation rate at a constant strain γ.

Apparently, the above discussions in the constant σ ensemble and those in the

constant γ ensemble closely resemble each other. It may seem quite natural to expect

the two definitions of the activation entropies, Sc(σ) and Sc(γ), to be one and the

same, as long as σ and γ lie on the stress-strain curve of the crystal at temperature

T . After all, the entropy of a crystal is a thermodynamic state variable, which is

independent of whether the constant stress or constant strain ensemble is used to

describe it, and we may expect the activation entropy to enjoy the same property

too. Surprisingly, Sc(σ) and Sc(γ) are not equivalent to each other. For dislocation

nucleation in a crystal, we can show that Sc(σ) is almost always larger than Sc(γ), and

the difference between the two can be very large, e.g. 30 kB for σ < 2 GPa. The large

difference between the two activation entropies has not been noticed before. We will

present both theoretical proofs and numerical data on this difference in subsequent

sections.


5.2.3 Difference between the Two Activation Entropies

While the Gibbs free energy G(n, σ, T ) and the Helmholtz free energy F (n, γ, T ) are

Legendre transforms of each other, the activation Gibbs free energy Gc(σ, T ) and the

activation Helmholtz free energy Fc(γ, T ) are not Legendre transforms of each other.

In fact, it is proven in Appendix C.2 that Gc(σ, T ) and Fc(γ, T ) equal to each other,

in the limit of V ≫ Ωc, as long as σ and γ lie on the stress-strain curve of the perfect

crystal at temperature T , i.e. σ = (1/V ) ∂F (0, γ, T )/∂γ. This has the important

consequence that the dislocation nucleation rate predicted by Eq. (5.1) and that by

Eq. (5.2) equal to each other. This result is intuitive because the dislocation nucle-

ation rate should not depend on whether the crystal is subjected to a constant stress,

or to a constant strain that corresponds to the same stress. The thermodynamic

properties of a crystal of macroscopic size can be equivalently specified either by its

stress and temperature or by its strain and temperature, and we expect the same to

hold for kinetic properties (e.g. dislocation nucleation rate) of the crystal..

It then follows that the activation entropies, Sc(σ, T ) defined in Eq. (5.9) and

Sc(γ, T ) defined in Eq. (5.14), cannot equal to each other. In the following, we will

let σ and γ follow the stress-strain curve, σ(γ, T ), of the perfect crystal at temperature

T . Using the equality Gc(σ, T ) = Fc(γ, T ), we have,

Sc(σ, T ) ≡ − ∂Gc(σ, T )

∂T

∣

∣

∣

∣

σ

= − ∂Fc(γ, T )

∂T

∣

∣

∣

∣

σ

= − ∂Fc(γ, T )

∂T

∣

∣

∣

∣

γ

− ∂Fc(γ, T )

∂γ

∣

∣

∣

∣

T

∂γ

∂T

∣

∣

∣

∣

σ

(5.18)

= Sc(γ, T )−∂Fc(γ, T )

∂γ

∣

∣

∣

∣

T

∂γ

∂T

∣

∣

∣

∣

σ

. (5.19)

Similarly, starting from the definition of Sc(γ, T ), we can show that,

Sc(γ, T ) = Sc(σ, T )−∂Gc(σ, T )

∂σ

∣

∣

∣

∣

T

∂σ

∂T

∣

∣

∣

∣

γ

. (5.20)


Eqs. (5.19) and (5.20) are consistent with each other because of the Maxwell relation,

∂σ

∂T

∣

∣

∣

∣

γ

∂T

∂γ

∣

∣

∣

∣

σ

∂γ

∂σ

∣

∣

∣

∣

T

= −1 (5.21)

and the chain rule of differentiation,

∂Gc(σ, T )

∂σ

∣

∣

∣

∣

T

=∂Fc(γ, T )

∂γ

∣

∣

∣

∣

T

∂γ

∂σ

∣

∣

∣

∣

T

. (5.22)

Therefore, the difference between the two activation entropies is

∆Sc ≡ Sc(σ, T )− Sc(γ, T ) =∂Gc(σ, T )

∂σ

∣

∣

∣

∣

T

∂σ

∂T

∣

∣

∣

∣

γ

= − ∂F (γ, T )

∂γ

∣

∣

∣

∣

T

∂γ

∂T

∣

∣

∣

∣

σ

.(5.23)

Recall the definition of activation volume Ωc in Eq. (5.10), we have

∆Sc = −Ωc∂σ

∂T

∣

∣

∣

∣

γ

. (5.24)

Notice that Ωc is always positive and that, because of thermal softening, ∂σ∂T

∣

∣

γis

usually negative 2. Therefore ∆Sc is positive for dislocation nucleation in a crystal

under most conditions.

While the difference between the two activation entropies have not been widely

discussed, it has been pointed out by Whalley in the context of chemical reac-

tions [167, 168]. However, ∆Sc has been estimated to be rather small in chemical

reactions. The main reason is that the activation volume for most chemical reactions

is bounded. As a rough estimate, let us assume that Ωc < 100A3. Under low stress

conditions, we expect ∂σ∂T

∣

∣

γto be linear with stress, i.e.,

∂σ

∂T

∣

∣

∣

∣

γ

≈ 1

µ

∂µ

∂T

∣

∣

∣

∣

γ

· σ (5.25)

2Here we are excluding the effect of thermal expansion, in case σ refers to a normal stresscomponent, by always defining the strain relative to the zero stress state at temperature T . Eventhough most crystals exhibit thermal softening, exceptions do exist. For example C66 of α-quartzdecreases with temperature [166].


where µ is the shear modulus. Although for chemical reactions it is more appropriate

to use the bulk modulus instead of µ, this approximation is acceptable for a rough

estimate. Let us assume that µ reduces by 10% as T increases from 0 to 300 K [169],

then − 1µ

∂µ∂T

∣

∣

∣

γis approximately 3.3 × 10−4 K−1. If we further assume that σ <

100 MPa, then ∆Sc < 0.25 kB, which is negligible.

The situation is quite different for dislocation nucleation, where the activation

volume Ωc diverges as σ goes to zero. The activation volume is proportional to the

size nc of the critical dislocation loop (See Appendix C.6). Based on a simple line

tension model, it is estimated 3 to be Ωc ∝ σ−2 in the limit of σ → 0. the following

relation,

Ωc ∝ σ−2 (5.26)

Combining this with Eq. (5.25), we have,

∆Sc ∝ −1

µ

∂µ

∂T

∣

∣

∣

∣

γ

· σ−1 (5.27)

which diverges as σ goes to zero. In the relevant stress range, e.g. from 0 to 2 GPa,

∆Sc is found to be very large, easily exceeding 10 kB, for both homogeneous and het-

erogeneous dislocation nucleation, as shown in subsequent sections. The divergence of

activation volume and activation entropy in the zero stress limit is a unique property

of dislocation nucleation, which distinguishes itself from other thermally activated

processes such as dislocation overcoming an obstacle [170, 171].

While the expression for the difference between the two activation entropies,

Eq. (5.24), follows mathematically from the equality of Gc and Fc and the chain rule

of differentiation, one may still wonder whether there exists an alternative (perhaps

more physical) explanation. After all, the entropy of the crystal is a property of the

thermodynamic state and is independent of the choice of ensembles. The activation

entropy can be expressed as the difference of the entropies between the “activated”

3We can use the line tension model to give a rough estimate of the formation energy of a circulardislocation loop of radius r, i.e. Eloop = 2πrτ − bπr2σ, where τ is the dislocation line energy perunit length and b is the magnitude of Burgers vector. Maximizing Eloop with respect to r gives the

energy barrier Ec =πτ2

bσat the critical radius rc =

τbσ. The activation volume is Ωc = bπr2c = πτ2

bσ2 ,which diverges in the limit of σ → 0.


state and the “initial” (i.e meta-stable) state. It may seem puzzling that the activa-

tion entropy does not share some of the fundamental properties of entropy itself. This

question is discussed in detail in Appendix C.3. The answer is that, in the definitions

of Sc(σ) and Sc(γ), we are not taking the entropy difference between the same two

states. If we choose the same “initial” state, then two different “activated” states are

chosen depending on whether the stress or strain is kept constant during dislocation

nucleation. This is because the act of forming a critical dislocation loop introduces

plastic strain into the crystal. Following this analysis, we are lead to exactly the same

expression for ∆Sc ≡ Sc(σ)− Sc(γ) as Eq. (5.24).

A similar expression has been obtained for the difference between the point de-

fect formation entropies under constant pressure (Sp) and under constant volume

(Sv) [172]. This difference is proportional to the relaxation volume of the defect and

is negative for a vacancy and positive for an interstitial. More discussion is given in

Appendix C.3.

5.2.4 Previous Estimates of Activation Entropy

There exist several theoretical approaches that could be used to estimate the activa-

tion entropy of dislocation nucleation. We note that none of these approaches address

the fact that there are actually two different activation entropies, and, as such, they

can be equally applied to Sc(σ) and to Sc(γ) and lead to similar estimates. In this

sense, all of these approaches will lead to inconsistencies when applied to dislocation

nucleation.

An approach that is widely used in the solid state is the harmonic approximation of

the transition state theory (TST) [154], in which the activation entropy is attributed

to the vibrational degrees of freedom. In TST, the frequency prefactor is ν0 = kBT/h

where h is Planck’s constant. At T = 300 K, ν0 = 6.25 × 1012 s−1. Expanding

the energy landscape around the “initial” state (i.e. perfect crystal, or the meta-

stable state) and the “activated” state (i.e. crystal containing the critical dislocation


nucleus) up to second order, we get

ν0 exp(Sc/kB) =

∏Ni=1 ν

mi

∏N−1i=1 νa

i

(5.28)

where νmi and νa

i are the positive normal frequencies in the meta-stable state and the

“activated” state, respectively [154, 156, 173] and N is the number of normal modes

in the meta-stable state. Note that the “activated” state contains one fewer normal

frequency than the meta-stable state. A further (rather crude) approximation is often

invoked, in which it is assumed that the normal frequencies in the “activated” state

are not significantly changed from those in the meta-stable state and approximate the

entire expression in Eq. (5.28) by the Debye frequency νD of the perfect crystal. The

Debye frequency [174] is typically the highest vibrational frequency in a crystal and

is on the order of 1013 s−1. Recall that ν0 itself is also on the order of 1013 s−1 at room

temperature. This leads to the conclusion that exp(Sc/kB) would not deviate from

1 by more than one order of magnitude. This is perhaps one of the reasons for the

entropic effects to be largely ignored so far for dislocation nucleation processes. In

subsequent sections, we will show that the activation entropies are large for dislocation

nucleation and they originate from anharmonic effects. This is consistent with the

above estimate that the vibrational entropy, captured by the harmonic approximation,

makes a negligible contribution to the activation entropy for dislocation nucleation

in metals.

Alternatively, one can estimate the activation entropy by postulating that the

activation Gibbs free energy scales with the shear modulus µ of the crystal. Because

µ decreases with temperature due to the thermal softening effect, so does Gc(σ, T ),

leading to an activation entropy [175]. This approximation can be expressed more

explicitly as,

Gc(σ, T ) = Hc(σ)µ(T )

µ(0)(5.29)

where µ(T ) and µ(0) are the shear moduli of the crystal at temperature T and zero

temperature, respectively. Assuming that µ(T ) is a linear function of T , we arrive at


the following estimate for Sc(σ),

Sc(σ) = −Hc(σ)1

µ(0)

∂µ

∂T. (5.30)

A similar expression has been derived for dislocations overcoming obstacles [176, 177].

Note that 1µ(0)

∂µ∂T

is a material constant that measures the severity of the thermal

softening effect. For convenience, we can define a characteristic temperature T ∗ such

that,1

T ∗= − 1

µ(0)

∂µ

∂T(5.31)

Then we arrive at the following estimate of Sc(σ),

Sc(σ) =Hc(σ)

T ∗(5.32)

Again, if we assume µ reduces by about 10% as T increases from 0 to 300 K, then

T ∗ ≈ 3000 K. This means Sc(σ) ≈ 7.5 kB when Hc(σ) = 2 eV.

While the above analysis seems quite reasonable, the same argument can be ap-

plied to the activation Helmholtz free energy, leading to the following approximations,

Fc(γ, T ) = Ec(γ)µ(T )

µ(0)(5.33)

Sc(γ) =Ec(γ)

T ∗(5.34)

with the same T ∗ as defined in Eq. (5.31). Again, assuming T ∗ ≈ 3000 K, we have

Sc(γ) ≈ 7.5 kB when Ec(γ) = 2 eV. Given the large difference between Sc(σ) and

Sc(γ), the above two estimates cannot be both correct. In Section 5.4 we will see that

this estimate is closer to Sc(γ) than Sc(σ) for homogeneous dislocation nucleation in

Cu.

Recently, Zhu et al. [153] introduced the following approximation to the activation

Gibbs free energy for dislocation nucleation from the surface of a Cu nanorod,

Gc(σ, T ) = Hc(σ)

(

1− T

Tm

)

(5.35)


where Tm is the surface melting temperature of the nanorod and is chosen to be

700 K. This approximation was based on the so-called “thermodynamic compensation

law” [178] , or the Meyer-Neldel rule [179], which is an empirical observation that in

many thermally activated processes, Sc is proportional to Hc. It is interesting that

such an approximation leads to an expression for the activation entropy Sc(σ) that is

identical to Eq. (5.32) provided that T ∗ = Tm. This amounts to assuming that the

shear modulus decreases at a constant rate with temperature and vanishes at melting

temperature, as in Born’s theory of melting [180].

Again, while this approximation seems reasonable, the same argument can be

applied to the activation Helmholtz free energy,

Fc(γ, T ) = Ec(γ)

(

1− T

Tm

)

(5.36)

which has been used by Brochard et al. [181] This would lead to an expression for

the activation entropy Sc(γ) that is identical to Eq. (5.34) provided that T ∗ = Tm.

Clearly, these two approximations, i.e. Eqs. (5.35-5.36), cannot both be correct, at

least for the same Tm.

5.3 Computational Methods

5.3.1 Simulation Cell

We study both the homogeneous nucleation of dislocation in bulk Cu and the het-

erogeneous nucleation in a Cu nanorod. Although dislocations often nucleate het-

erogeneously at surfaces or internal interfaces, homogeneous nucleation is believed to

occur in nano-indentation [96] and in a model of brittle-ductile transition [182]. It

also provides an upper bound to the shear strength of the crystal. For simplicity, we

benchmark the computational method against brute force MD simulation and spend

most of the discussions on homogeneous nucleation. Heterogeneous nucleation will

be discussed following the homogeneous nucleation analysis.

Our model system is a Cu single crystal described by the embedded atom method


0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

3T = 0 K

100 K200 K

300 K400 K

500 K600 K

γxy

σ xy (

GP

a)

(a) (b)

[001]

σ

bp

[100]

[010]

0 0.05 0.10

1

2

3

4 T = 0 K

100 K

200 K300 K

400 K500 K

600 K

εzz

σ zz (

GP

a)

(c) (d)

Figure 5.2: Schematics of simulation cells designed for studying (a) homogeneousand (c) heterogeneous nucleation. In (a), the spheres represent atoms enclosed bythe critical nucleus of a Shockley partial dislocation loop. In (c), atoms on thesurface are colored by gray and atoms enclosed by the dislocation loop are coloredby magenta. Shear stress-strain curves of the Cu perfect crystal (before dislocationnucleation) at different temperatures for (b) homogeneous and (d) heterogeneousnucleation simulation cells.

(EAM) potential [111]. As shown in Fig. 5.2 (a), the simulation cell to study homo-

geneous dislocation is subjected to a pure shear stress along [112]. The dislocation

to be nucleated lies on the (111) plane and has the Burgers vector of a Shockley par-

tial [183], ~bp = [112]/6. The cell has dimension of 8 repeat distances along the [112]

direction, 6 repeat distances along the [111] and 3 repeat distances along the [110],

and consists of 14, 976 atoms. Periodic boundary conditions (PBC) are applied to

all three directions. To reduce artifacts from periodic image interactions, the applied

stress is always large enough so that the diameter of the critical dislocation loop is

smaller than half the width of the simulation cell.


Fig. 5.2 (b) shows the shear stress strain relationship of the perfect crystal at

different temperatures (before dislocation nucleation) that clearly shows the thermal

softening effect. The shear strain γ is the xy component of the engineering strain. The

following procedure is used to obtain the pure shear stress-strain curve because the

conventional Parrinello-Raman stress control algorithm [121] does not work properly

here due to the non-linear stress-strain relationship at large strain. At each temper-

ature T and shear strain γxy, a series of 2 ps MD simulations under the canonical,

constant temperature-constant volume (NVT) ensemble are performed. After each

simulation, all strain components except γxy are adjusted according to the average

Virial stress until σxy is the only nonzero stress component. The shear strain is then

increased by 0.01 and the process repeats until the crystal collapses spontaneously.

For heterogeneous dislocation nucleation, we study a Cu nanorod that has the

dimension of 15[100] × 15[010] × 20[001] with PBC along [001], which is shown in

Fig. 5.2 (c). When subjected to axial compression along [001], a dislocation with the

Burgers vector ~b = [112]/6 is expected to nucleate from the corner of the nanorod.

The compressive stress-strain curve is shown in Fig. 5.2 (d). An important step in

obtaining the stress-strain curve is to achieve thermal equilibrium before taking the

average of stress σzz and computing the nucleation rate at a given strain ǫzz. Due to

the free side surfaces, a nano-rod undergoes low frequency but long-lived oscillations

in x, y direction (i.e. “breathing” mode) at the initial stage of MD simulation.

This leads to very large oscillation in σzz, at a frequency that is several orders of

magnitudes smaller than the Debye frequency. We suppress this “breath” mode by

running simulation using a stochastic thermostat [119], which is more effective than

the Nose-Hoover thermostat [117] for equilibrating systems with a wide range of

eigenfrequencies.


5.3.2 Nucleation Rate Calculation

Similar to the previous chapter, we predict the nucleation rate based on the BD

theory, which expresses the nucleation rate as

IBD(γ, T ) = Ns f+c Γ exp

[

−Fc(γ, T )

kBT

]

(5.37)

where f+c is the molecular attachment rate, and Γ is the Zeldovich factor. Fc is

computed with the shape and volume of the simulation cell fixed. We assume that

the activation Helmholtz free energy Fc obtained from the finite simulation cell is

very close to the value of Fc in the infinite volume limit, which equals the activation

Gibbs free energy Gc4 (See Appendix C.2). The BD theory and TST only differs in

the frequency prefactor. Whereas TST neglects multiple recrossing over the saddle

point [154, 159] by a single transition trajectory, the recrossing is accounted for in

the BD theory through the Zeldovich factor.

0 20 40 600

0.1

0.2

0.3

0.4

0.5

0.6

n

F(n

) (e

V)

nc = 34

Fc = 0.53 eV

0 10 20 30 400

10

20

30

40

50

t (fs)

<∆

n2 (t)>

2fc+

(a) (b)

Figure 5.3: (a) The Helmholtz free energy of the dislocation loop as a function of itssize n during homogeneous nucleation at T = 300 K, σxy = 2.16 GPa (γxy = 0.135)obtained from umbrella sampling. (b) Size fluctuation of critical nuclei from MDsimulations.

4The activation Gibbs free energy at zero temperature, i.e. Gc(σ, T = 0), can be obtained fromthe stress-controlled NEB method [184]. Due to the finite size of the simulation cell, Gc(σ, T = 0)is expected to be slightly different from Fc(γ, T = 0) obtained under the constant strain condition,but such difference is ignored in this study.


The Helmholtz free energy barrier Fc is computed by umbrella sampling [125]. The

umbrella sampling is performed in Monte Carlo simulations using a bias potential that

is a function of the order parameter n, which is chosen to be number of atoms inside

the dislocation loop. The bias potential kBT (n− n)2 is super-imposed on the EAM

potential, where T = 40 K and n is the center of the sampling window. We chose T

empirically so that the width of the sampling window on the n-axis would be about

10. The umbrella sampling provides F (n) at a given γ and T . The maximum value

of the free energy curve F (n) is the activation free energy Fc and the maximizer is

the critical dislocation loop size nc. The Zeldovich factor Γ can be computed from

the definition Γ ≡(

η2πkBT

)1/2

, where η = −∂2F (n)/∂n2|n=nc.

(a) (b)

Figure 5.4: Atomistic configurations of dislocation loops at (a) 0 K and (b) 300 K.

We have used two different order parameters to recognize the dislocation nucle-

ation. First, we use a method suggested by Ngan et al. [185] to identify the formation

of dislocation loop and compute the reaction coordinate n. We labelled an atom as

“slipped” if its distance from any of its original nearest neighbors has changed by

more than a critical distance dc. We chose dc = 0.33, 0.38, and 0.43 A for T ≤ 400

K, T = 500 K, and T = 600 K, respectively, because thermal fluctuation increases

with temperature. The ”slipped” atoms are grouped into clusters; two atoms belong

to the same cluster if their distance was less than a cutoff distance rc (3.4A). The

reaction coordinate is defined as the number of atoms in the largest cluster divided

by two.

A possible problem of this reaction coordinate is that it does not take the slip

direction into account even though we are specifically interested in slip along the [112]


direction, which is parallel to the Burgers vector of the Shockley partial. Thus, we

use another order parameter to focus exclusively on the slip along the [112] direction.

As a self-consistency check, the activation free energy should be independent of this

modification of the order parameter. To compute the new order parameter during

umbrella sampling, we focus on atoms on one (111) plane, each of which has 12

neighbor atoms; three in the plane above, six in the same plane, and three in the

plane below. When the relative displacement along the [112] direction between an

atom and the center of mass position of its three neighbor atoms in the plane below

exceeds a critical distance dc = 0.35A, we label the atom as “slipped”. Here, we used

a smaller cutoff radius of rc = 3.09A to group “slipped” atoms into a cluster. The

size of the largest cluster of the slipped atoms is the reaction coordinate n.

As expected, we find that predictions of free energy barrier and nucleation rate

are independent of these two choices of the reaction coordinate. The data obtained

from these two methods match within statistical errors. In the following analysis, we

will use the data from the second order parameter, because its definition appears to

be more physical. Fig. 5.4 (a) and Fig. 5.4 (b) shows the critical cluster sampled by

the simulation of homogeneous dislocation nucleation at T = 0 K (from the string

method) and T = 300 K (from umbrella sampling), respectively. Although the dislo-

cation loop at 0 K appears symmetric, the configuration at 300 K is distorted due to

thermal fluctuation.

The attachment rate f+c is computed by direct MD simulations. From umbrella

sampling, we collected an ensemble of 500 atomic configurations for which n = nc,

and ran MD simulations using each configuration as an initial condition. The initial

velocities are randomized according to the Maxwell-Boltzmann distribution. The

mean square change of the loop size, 〈∆n2(t)〉, as shown in Fig. 5.3 (b), is fitted to a

straight line, 2f+c t, in order to extract f+

c (See section 3.5.3).


5.4 Results

5.4.1 Benchmark with MD Simulations

Before applying the BD theory to predict the nucleation rate at a wide range of applied

load and temperature, we would like to establish the applicability of the theory to

the dislocation nucleation. We benchmark the prediction of BD theory against direct

MD simulations at a relatively high stress σ = 2.16 GPa (γ = 0.135) at T = 300 K for

homogeneous nucleation. To obtain average nucleation time at the given condition, we

performed 192 independent MD simulations using the NVT ensemble with random

initial velocities. Each simulation ran for 4 ns. If dislocation nucleation occurred

during this period, the nucleation time was recorded. This information is used to

construct the function Ps(t), which is the fraction of MD simulation cells in which

dislocation nucleation has not occurred at time t, as shown in Fig. 5.5. Ps(t) can be

well fitted to the form of exp(−IMDt), from which the nucleation rate IMD is predicted

to be IMD = 2.5× 108 s−1.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

t (ns)

Ps(t

)

Figure 5.5: The fraction of 192 MD simulations in which dislocation nucleation hasnot occurred at time t, Ps(t), at T = 300 K and σxy = 2.16 GPa (γxy = 0.135).Dotted curve presents the fitted curve exp(−IMDt) with IMD = 2.5× 108s−1.

From umbrella sampling at the specified condition, we obtain the free energy

function F (n). Fig. 5.3 (a) shows the maximum of F (n), which gives the activation

free energy Fc = 0.53± 0.01 eV and the critical nucleus size nc = 34. The Zeldovich

factor, Γ = 0.051, is obtained from Γ ≡(

η2πkBT

)1/2

, where η = −∂2F (n)/∂n2|n=nc.


Using the configurations collected from umbrella sampling with n = nc as initial

conditions, MD simulations give the attachment rate f+c = 5.0 × 1014 s−1, as shown

in Fig. 5.3 (b). Because the entire crystal is subjected to uniform stress, the number

of nucleation sites is the total number of atoms, Ns = 14, 976.

Combining these data, the classical nucleation theory predicts the homogeneous

dislocation nucleation rate to be IBD = 4.8 × 108 s−1, which is within a factor of

two of the MD prediction. The difference between two is comparable to our error

bar. This agreement is noteworthy because no adjustable parameters such as the fre-

quency prefactor is involved in this comparison. It shows that the classical nucleation

theory and our numerical approach are suitable for the calculation of the dislocation

nucleation rate.

5.4.2 Homogeneous Dislocation Nucleation in Bulk Cu

Having established the applicability of nucleation theory, we now examine the ho-

mogeneous dislocation nucleation rate under a wide range of temperature and strain

(stress) conditions relevant for experiments and beyond the limited timescale of brute

force MD simulations. We find that the prefactor ν0 = Γf+c is a slowly changing func-

tion of stress and temperature. It varies by less than a factor of two for all the

conditions tested here. The average value of ν0 is about 2.5 × 1013 s−1, which is

comparable to the Debye frequency ∼ 1013 s−1.

The nucleation rate varies dominantly by the change of the activation free energy

Fc(γ, T ), which is presented as a function of γ at different T in Fig. 5.6 (a). The

zero temperature data (i.e. activation energies) are obtained from minimum energy

path (MEP) searches using a modified version of the string method, similar to that

used in the literature [153, 186]. The downward shift of Fc curves with increasing T

is the signature of the activation entropy Sc(γ). Fig. 5.6 (c) plots Fc as a function of

T at γ = 0.092. For T < 400 K, the data closely follow a straight line, whose average

slope gives Sc = 9 kB in the range of [0, 300] K. This activation entropy contributes

a significant multiplicative factor, exp(Sc/kB) ≈ 104, to the absolute nucleation rate,

and cannot be ignored.


0.08 0.1 0.12 0.140

0.5

1

1.5

2

2.5

γxy

Fc (

eV)

0K100K200K300K

0 1 2 30

0.5

1

1.5

2

2.5

σxy

(GPa)

Gc (

eV)

0K100K200K300K400K500K

(a) (b)

0 100 200 300 400 5000

0.5

1

1.5

2

2.5

T (K)

Fc (

eV)

S = 9 kB

0 100 200 300 4000

0.5

1

1.5

2

2.5

T (K)

Gc (

eV)

S = 48 kB

(c) (d)

Figure 5.6: Activation Helmholtz free energy for homogeneous dislocation nucleationin Cu. (a) Fc as a function of shear strain γ at different T . (b) Gc as a functionof shear strain σ at different T . Squares represent umbrella sampling data and dotsrepresent zero temperature MEP search results using simulation cells equilibrated atdifferent temperatures. (c) Fc as a function of T at γ = 0.092. (d) Gc as a functionof T at σ = 2.0 GPa. Circles represent umbrella sampling data and dashed linesrepresent a polynomial fit.

In Section 5.2.4, we mentioned that the activation entropy would be negligible

if only the vibrational entropy were taken into account. It is likely that the origin

of the large activation entropy is an anharmonic effect such as thermal expansion.

To examine the effect of thermal expansion, we performed a zero temperature MEP

search at γ = 0.092, but with other strain components fixed at the equilibrated

values at T = 300 K. This approach is similar to the quasi-harmonic approximation

(QHA) [187, 188] often used in free energy calculations in solids, except that, unlike

QHA, the vibrational entropy is completely excluded here. The resulting activation


energy, Ec = 2.04 eV , is indistinguishable from the activation free energy Fc =

2.05 ± 0.01 eV at T = 300 K computed from umbrella sampling. For T < 400 K,

we observe that the activation energy Ec and Fc matches well at each γ and T

condition (See Table C.1 in Appendix C.5).

Because atoms do not vibrate in the MEP search, this result shows that the

dominant mechanism for the large Sc(γ) is indeed thermal expansion, whereas the

contribution from vibrational entropy is negligible. As temperature increases, ther-

mal expansion pushes neighboring atoms further apart and weakens their mutual

interaction. This expansion makes crystallographic planes easier to shear and signif-

icantly reduces the free energy barrier for dislocation nucleation. Here, we confirm

that Sc(γ) arises almost entirely from the anharmonic effect for dislocation nucle-

ation. At T = 400 K and T = 500 K, we observe significant differences between Fc

computed from umbrella sampling and Ec computed from a zero temperature MEP

search in expanded cell. These differences must also be attributed to anharmonic ef-

fects. The activation energy Ec from the expanded cell and the activation free energy

Fc(γ, T ) at T = 400 K and T = 500 K are not plotted in Fig 5.6 (a), because they

overlap with data points at lower temperatures. All data can be found in Table C.1

in Appendix C.5.

Combining the activation Helmholtz free energy Fc(γ, T ) and the stress-strain

relations, we obtain the activation Gibbs free energy Gc(σ, T ) shown in Fig. 5.6 (b).

We immediately notice that the curves at different temperatures are more widely

apart in Gc(σ, T ) than those in Fc(γ, T ), indicating a much larger activation entropy

in the constant stress ensemble. For example, Fig. 5.6 (d) plots Gc as a function

of T at σ = 2.0 GPa, from which we can obtain an averaged activation entropy

of Sc(σ) = 48kB in the temperature range of [0, 300] K. This activation entropy

contributes a multiplicative factor of exp[Sc(σ)/kB] ≈ 1020 to the absolute nucleation

rate, as shown in Fig. 5.1.

The dramatic increases in the activation entropy when stress, instead of strain,

is kept constant is consistent with the theoretical prediction in Section 5.2. This is

caused by changing stress-strain relationship with temperature. For example, when

the shear stress is kept at σxy = 2.0 GPa, the corresponding shear strain at T = 0 K


is γxy = 0.092. But at T = 300 K, the same stress is able to cause a larger strain,

γxy = 0.113. Hence at constant stress the activation free energy decreases much faster

with temperature than that at constant strain.

5.4.3 Heterogeneous Dislocation Nucleation in Cu Nano-

Rod

0.02 0.03 0.04 0.05 0.060

0.5

1

1.5

2

εzz

Fc (

eV)

0K200K300K400K500K

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

σzz

(GPa)

Gc (

eV)

0K200K300K400K500K

(a) (b)

Figure 5.7: Activation free energy for heterogeneous dislocation nucleation from thesurface of a Cu nanorod. (a) Fc as a function of compressive strain ǫzz at differentT . (b) Gc as a function of compressive stress σzz at different T . Squares representumbrella sampling data and dots represent zero temperature MEP search results usingsimulation cells equilibrated at different temperatures.

We studied dislocation nucleation from the corner of a [001]-oriented copper

nanorod with 100 side surfaces under axial compression. While the size of dis-

location loop n is the only order parameter used in the umbrella sampling, the um-

brella sampling simulation automatically locates the dislocation nucleus at the corner

of nanorod, as shown in Fig. 5.2 (c). This is because the nucleation barrier is much

smaller for the nucleation from the corner than the surface, as found by Zhu et al [153].

We also find that the prefactor ν0 varies slowly similar to the case of homogeneous nu-

cleation, changing less than one order of magnitude at all γ and T conditions tested.

Interestingly, the average value of ν0 is about 0.5 × 1013 s−1, several times smaller

than the average prefactor homogeneous nucleation. The measured growth rate f+c

of a critical nucleus turns out to be significantly smaller (by about a factor of 3) due


to the shorter dislocation line length relative to that of the complete dislocation loop

in homogeneous nucleation.

Fig. 5.7 plots the activation free energy barrier as a function of axial compressive

strain ǫzz and compressive stress σzz. Both Fig. 5.7 (a) and (b) show the reduction

of the activation free energy with temperature, and the reduction in (b) is more

pronounced due to thermal softening. For example, at the compressive elastic strain

of ǫ = 0.03, the compressive stress is σ = 1.56 GPa at T = 0 K. The activation entropy

Sc(ǫ) at this elastic strain equals 9 kB, whereas the activation entropy Sc(σ) at this

stress equals 17 kB. Unfortunately, we could not perform the minimum energy path

search at zero temperature using an expanded cell to mimic the thermal expansion

effect due to the free surface of the nanorod.

The activation entropy difference is smaller than the homogeneous nucleation be-

cause both thermal softening and activation volume are smaller. In order for the

homogeneous nucleation to occur at room temperature, the perfect crystal must be

sheared significantly, close to the ideal shear strength. In such a high non-linear elas-

tic regime, the thermal softening effect becomes very large, as depicted in Fig. 5.2.

However, the heterogeneous nucleation is much easier to occur so that it can happen

when the nanorod is subjected to a moderate loading in which the stress-strain rela-

tion is still relatively linear. Therefore, the thermal softening effect is not as large as

the case of homogeneous nucleation. Secondly, because the applied compression stress

is not parallel to the slip direction (Schmid factor 0.471), the activation free energy is

less sensitive to the applied stress, leading to a smaller activation volume compared

with the case of homogeneous nucleation (See Appendix C.6 for more discussions on

the activation volume).

5.5 Discussion

5.5.1 Testing the “Thermodynamic Compensation Law”

With the numerical results of Gc(σ, T ), we can test the approximations Sc(σ) =

Hc(σ)/T∗ and Sc(γ) = Ec(γ)/T

∗. Specifically, we are interested in whether Sc(σ)


is proportional to Hc(σ), and whether Sc(γ) is proportional to Ec(γ), and if so,

how the coefficient T ∗ compares with the (bulk or surface) melting point of Cu.

While Eqs. (5.32) and (5.34) assume that the activation entropies do not depend on

temperature, our data show that they do vary with temperature for T ≥ 400 K.

Hence, we test the average activation entropy Sc in the range of zero to 300 K.

For homogeneous nucleation, we find that Sc(γ) can be roughly approximated by

Ec(γ)/T∗ with T ∗ ≈ 2700 K as shown in Fig. 5.8 (a), while Sc(σ) is not proportional

toHc(σ) as shown in Fig. 5.8 (c). On the other hand, for heterogeneous nucleation, we

find that Sc(ǫ) can be approximately fitted to Ec(γ)/T∗ with T ∗ = 2450 K as shown

in Fig. 5.8 (b), while Sc(σ) can be approximately fitted to Hc(σ)/T∗ with T ∗ = 930 K

as shown in Fig. 5.8 (b). Both values of the fitted T ∗ are different from the surface

melting temperature [153] of Tm = 700 K. The value of T ∗ = 2450 K also greatly

exceeds the (bulk) melting point of Cu (1358 K [189]). Hence the empirical fitting

parameter T ∗ is most likely not connected to the melting phenomenon. Fig. 5.8 shows

a consistent trend that the activation entropy increases as the activation enthalpy (or

the activation energy) increases, but the “compensation law” appears to hold for

Sc(σ) in heterogeneous nucleation and for Sc(γ) in homogeneous nucleation, but it

does not hold for Sc(σ) in homogeneous dislocation nucleation.

Because the activation entropies in dislocation nucleation mainly come from anhar-

monic effects such as thermal softening and thermal expansion, the exhibition of the

“compensation law” in Fig. 5.8(a) and (d) cannot be attributed to the usual explana-

tion [179, 190] that the activation energy is provided by multiple (small) excitations.

The breakdown of the “compensation law” for Sc(σ) in homogeneous dislocation nu-

cleation is probably caused by the elastic non-linearity at the high stress needed for

homogeneous nucleation.

We note that the empirically fitted value of T ∗ = Ec(γ)/Sc(γ) is close to the

estimated value of 3000 K, which is based on a 10% reduction of the shear modulus as

temperature increases from zero to 300 K (See Section 5.2.4). Therefore, Eq. (5.33)

can be considered as a reasonable approximation to the activation Helmholtz free


0 1 2 30

5

10

15

Ec (eV)

Sγ c/k

B

T* = 2700 K

0 1 2 30

5

10

15

Ec (eV)

Sε c/k

B T* = 2450 K

(a) homogeneous (b) heterogeneous

0 1 2 3 4 50

20

40

60

80

100

Hc (eV)

Sσ c/k

B

T* = 640 K

0 1 2 30

10

20

30

40

Hc (eV)

Sσ c/k

B T* = 930 K

(c) homogeneous (d) heterogeneous

Figure 5.8: The relation between Ec and Sc in the temperature range of zero to 300 Kfor (a) homogeneous and (b) heterogeneous nucleation. The relation between Hc andSc for (c) homogeneous and (d) heterogeneous nucleation. The solid lines representsimulation data and the dashed lines are empirical fits of the form Sc = Ec/T

∗ orSc = Hc/T

∗.

energy as a function of strain, i.e.

Fc(γ, T ) ≈ Ec(γ)µ(T )

µ(0)(5.38)

whereas Eq. (5.29) is not a good approximation forGc(σ, T ). In other words, Eq. (5.34)

can be considered as a reasonable approximation to the activation entropy Sc(γ), i.e.

Sc(γ) ≈ −Ec(γ)

µ(0)

∂µ

∂T(5.39)

whereas Eq. (5.32) is not a good approximation for Sc(σ). Appendix C.4 contains


more discussions on the approximation of Sc(σ).

5.5.2 Entropic Effect on Nucleation Rate and Yield Strength

σ (GPa)

T (

K)

10 610 1

10 −410 −9

10 −14

10610110−410−910−14

1.6 1.8 2 2.2 2.4 2.60

100

200

300

400

500

σ (GPa)

T (

K)

10 610 110 −4

10 −910 −14

10610110−410−910−14

1 1.2 1.4 1.6 1.8 2 2.20

100

200

300

400

500

(a) (b)

Figure 5.9: Contour lines of (a) homogeneous and (b) heterogeneous dislocation nu-cleation rate per site I as a function of T and σ. The predictions with and withoutaccounting for the activation entropy Sc(σ) are plotted in thick and thin lines, re-spectively. The nucleation rate of I ∼ 106 s−1 per site is accessible in typical MDtimescales whereas the nucleation rate of I ∼ 10−4 − 10−9 is accessible in typicalexperimental timescales, depending on the number of nucleation sites.

In this section, we discuss how do the activation entropies affect experimental mea-

surements. The simplest case to consider is to subject a perfect crystal to a constant

stress (i.e. creep) loading condition and measure the rate of dislocation as a function

of stress and temperature. (In practice, these kind of experiments are very difficult

to carry out [97], especially to observe homogeneous nucleation.) The data can be

plotted in the form of contour lines, similar to those shown in Fig. 5.9, which are our

theoretical predictions. To make these predictions, we use the activation Gibbs free

energy obtained from umbrella sampling. Because the frequency prefactor ν0 = f+c Γ

varies slowly with σ and T , we use average value 2.5× 1013 s−1 for the homogeneous

nucleation and 0.5 × 1013 s−1 for the heterogeneous nucleation. To show the phys-

ical effect of the large activation entropies, the thin lines plot the rate predictions

if the effect of Sc(σ) were completely neglected. Significant deviations between the

two sets of contour lines are observed. For homogeneous dislocation nucleation, at


T = 400 K and σxy = 2.0 GPa (where a thick and a thin contour line cross), we see

about 20 orders of magnitude difference between the two contours. The difference

between thick and thin curves becomes larger at smaller stress because activation en-

tropy becomes larger at smaller stress. For heterogeneous nucleation, at T = 300 K

and σzz = 1.5 GPa, the neglect of activation entropy would cause an underestimate

of the nucleation rate by 10 orders of magnitude. The smaller activation volume in

heterogeneous dislocation is manifested by the larger gaps between the contour lines

at different nucleation rates.

0 100 200 300 400 5001

1.5

2

2.5

3

T (K)

She

ar N

ucle

atio

n S

tres

s (G

Pa)

0 100 200 300 400 5000

1

2

3

4

T (K)

Com

pres

sive

Nuc

leat

ion

Str

ess

(GP

a)

(a) (b)

Figure 5.10: (a) Nucleation stress of our bulk sample (containing 14,976 atoms) underconstant shear strain loading rate γ = 10−3 and (b) nucleation stress of the nanorodunder constant compressive strain loading rate ǫ = 10−3. The strain rate 10−3 isexperimentally accessible loading rate. The solid lines are the prediction based onthe activation free energy computed by umbrella sampling. The dashed lines are thenucleation stress prediction when the activation entropy is neglected. The dotted linein (b) is the prediction based on the approximation by Zhu et al. [153].

Experimentally, it is often convenient to impose a constant strain rate to the

crystal and measure the stress-strain curve and the yield strength under the given

strain rate. If the crystal contains no pre-existing defects, then the yield strength is

the stress at which the first dislocation nucleates. The following implicit equation

for the yield strength σY has been derived by considering a nano-rod is loaded at a

constant strain rate ǫ.Gc(σY , T )

kB T= ln

kB T Ns ν0E ǫΩc(σY , T )

(5.40)


This equation is derived [153, 185] based on the assumption that the nano-rod remains

linear elastic with Young’s modulus E prior to yielding and that ν0 is insensitive to

σ and T . One may apply this equation to homogeneous nucleation case if we replace

E by the shear modulus µ and the uniaxial strain rate ǫ by γ. However, since we

observe that the crystal becomes non-linear elastic prior to dislocation nucleation (see

Fig. 5.2), we predict the yield strength numerically without assuming linear elasticity.

The stress-strain relations shown in Fig. 5.2(b) and (d) are used to extract the stress

rate given the imposed strain rate and current strain. We replace E in Eq. (5.40) by

∂σ/∂ǫ|σ=σY. The athermal nucleation stress causing dislocation nucleation at T = 0 K

is σxy = 2.8 GPa for homogeneous nucleation and σzz = 4.7 GPa for heterogeneous

nucleation, which can also be obtained from in Fig. 5.2(b) and (d). At 300 K and a

strain rate of 10−3 s−1, however, the yield strength (i.e. nucleation stress) becomes

σnucxy = 2.0 GPa for homogeneous nucleation, about 71% of the athermal nucleation

stress, and σnuczz = 1.7 GPa for heterogeneous nucleation, about 36% of the athermal

stress.

Fig. 5.10 plots our predictions of the yield strength as a function of temperature

at a strain rate of 10−3 s−1. As temperature rises, the nucleation stress decreases.

This decrease is faster in the heterogeneous nucleation, Fig. 5.10(b) than in the ho-

mogeneous nucleation, Fig. 5.10(a). This observation can be explained by the larger

activation volumes in the homogeneous nucleation than those in the heterogeneous

nucleation. We note that the predicted nucleation stress depends on both the number

of atoms in the sample and the applied strain rate. Increasing the number of atoms

has the same effect as decreasing the strain rate.

For comparison, Fig. 5.10(b) also plots the prediction by Zhu et al. [153], which

is based on the assumption of Sc(σ) = Hc(σ)/Tm with Tm = 700 K. (The yield

strength is only plotted up to T = 300 K in the original paper [153].) The two

predictions (solid and dotted line) are close to each other for T < 200 K, but their

difference becomes large for T ≥ 300 K. While the dotted line suggests that the yield

strength vanishes at T = 500 K, our prediction (solid line) shows that the nanorod

still retains 71% of its room temperature strength at 500 K. We believe this difference

is caused by the overestimate of the activation entropy when assuming Tm = 700 K


in Sc(σ) = Hc(σ)/Tm.

5.6 Summary

In this chapter, we have shown that the dislocation nucleation rate is independent of

whether a constant stress or a constant strain is applied, because Fc(γ, T ) = Gc(σ, T )

when σ and γ lie on the stress-strain curve at temperature T . This naturally results in

different activation entropies depending on whether constant stress or constant strain

ensemble is used. The difference between the two activation entropies equals the acti-

vation volume times a term that characterizes the thermal softening effect. We have

shown that the Becker-Doring theory combined with the activation free energy deter-

mined by umbrella sampling can accurately predict the rate of dislocation nucleation.

In both homogeneous and heterogeneous dislocation nucleation, a large activation

entropy at constant elastic strain is observed, and is attributed to the weakening of

atomic bonds due to thermal expansion. The activation entropy at constant stress is

even larger due to the thermal softening. Both effects are anharmonic in nature, and

emphasize the need to go beyond harmonic approximation in the application of rate

theories in solids. The “compensation law” turns out not to hold for homogeneous

dislocation nucleation, probably because of the non-linear effects at high stress condi-

tions. The “compensation law” appears to work better for heterogeneous nucleation,

probably related to the linearity of the stress-strain relation. We have predicted that

the yield stress decreases faster with temperature for the heterogeneous nucleation

than for the homogeneous nucleation.

Chapter 6

Summary and Outlook

6.1 Conclusion

This dissertation presents several contributions in proving the validity of the classical

nucleation theory and applying it to dislocation nucleation. Our work deepens the

understanding on the classical nucleation theory by proving the individual component

of the theory numerically. This gives answers to which part of the theory is valid and

what corrections are required to improve the theory. We provide an in-depth discus-

sion on the entropic effect on the dislocation nucleation and predict the nucleation

rate accurately from the validated part of CNT combined with advanced sampling

techniques.

First, we have found that the Markovian chain assumption with the largest droplet

size as the reaction coordinate is accurate enough to predict the nucleation rate span-

ning a few tens orders of magnitude, provided that the correct droplet free energy

function is used. Discrepancies observed from the existing numerical studies on the

Ising model and the classical nucleation theory are found to be originated from the

incorrect droplet free energy expression. A logarithmic correction term whose coeffi-

cient depends on the symmetry of the given system is required to take the microscopic

fluctuation of droplet into account. A constant correction term is required to adjust

the free energy curve at small droplet size range. Our results also shed a light on the

gas-liquid transition experiment results that critical supersaturation matches with

107

CHAPTER 6. SUMMARY AND OUTLOOK 108

the CNT prediction with a small error, but the temperature dependence does not.

We have investigated the dislocation nucleation both analytically and numerically.

We have shown that the dislocation nucleation rate does not depend on whether a

constant stress or a constant strain is applied. This naturally results in different ac-

tivation entropies depending on whether constant stress or constant strain ensemble

is used. Similar to the Ising model study, we have shown that the Becker-Doring

theory combined with the activation free energy determined by umbrella sampling

can accurately predict the rate of dislocation nucleation. In both homogeneous and

heterogeneous dislocation nucleation, a large activation entropy at constant elastic

strain is observed, and is attributed to the weakening of atomic bonds due to thermal

expansion. The activation entropy at constant stress is even larger due to the thermal

softening. Both effects are anharmonic in nature, and emphasize the need to go be-

yond harmonic approximation in the application of rate theories in solids. Our results

also implies that the compensation law works better for the reactions happening at

a relatively moderate stress where the linearity of stress-strain relation holds. We

believe that our methods and the general conclusions from the dislocation nucleation

study are applicable to a wide range of nucleation processes in solids that are driven

by shear stress, including cross slip, twinning and martensitic phase transformation.

6.2 Future Works

We have tested the validity of the classical nucleation theory and applied it to the dis-

location nucleation phenomena that is closely related to the mechanical deformation

of materials. A natural extension of the present dissertation is to apply the findings

and general methods used in this study to other nucleation phenomena. Especially,

we are interested in investigating the crystal nucleation in the context of nanoma-

terials synthesis. Out of many nanomaterials synthesis processes, our next research

subject will be the gold-catalyzed growth of silicon nanowire via vapor-liquid-solid

mechanism.

Many semiconductor materials can be grown in the form of nanowires (NWs)

by the VLS process, which has enabled a wide range of novel applications, such


as nanoscale electronic, optical, and chemical-sensing devices. However, many fun-

damental questions regarding the growth mechanisms still need to be answered to

achieve better control of the orientation, yield, and quality of the NWs. Much of

the studies on the VLS growth mechanism have been focused on Si NW grown from

Au catalysts, because of the relative simplicity of the Au-Si phase diagrams and the

anticipated compatibility of Si NW with existing semiconductor industry.

Atomistic simulations are expected to provide useful insights to the NW growth

and nucleation mechanisms. Because thousands of atoms are necessary for a rea-

sonable description of NW growth process, ab initio simulations are prohibitively

expensive, and classical MD or MC simulations based on empirical interatomic po-

tentials are necessary at present. The bottleneck of the atomistic investigation was

the absence of Au-Si interatomic potential. During the doctoral research, we have

developed a Au-Si interatomic potential that, for the first time, was fitted to the

experimental binary phase diagram. This means that the NW growth process in the

atomistic simulations using this model will experience thermodynamic driving forces

that are similar to those in the experiments.

Three appendices are added to describe the Au-Si potential development process.

In Appendix D, we present an efficient free energy method to compute the free energies

of pure solid and pure liquid for variety of empirical potentials. Using the free energy

calculation method, we improved the melting point and latent heat of pure gold

and pure silicon potential, which is described in Appendix E. Finally, Appendix F

presents the newly developed Au-Si interatomic potential with an efficient free energy

calculation method for solid and liquid alloies that has been used to construct the

binary phase diagram.

Overall, we have presented a wide range of research problems that can be ad-

dressed by atomistic simulations and provided numerical methods to tackle funda-

mental challenges in the simulations. This dissertation discusses about how to test an

analytic theory from numerical simulations, how to overcome the limited timescale

of molecular dynamics using advanced sampling algorithms, and how to compute the

free energies of solid and liquid alloies for testing and improving thermal properties

of interatomic model. We plan to expand and apply the numerical methods used


in this dissertation to atomistic simulation studies of various materials growth and

deformation processes.

Appendix A

Derivation of Nucleation Rate from

CNT and Nucleation Theorems

In the first section of this appendix, the steady state nucleation rate is derived using

two basic assumptions of CNT. The first assumption is that the nucleation process

can be considered as a 1D Markov chain model in which the growth and the decay

of the clusters takes place by attachment and detachment of a single particle. The

second assumption is that droplet free-energy function ∆G(n) can be written as

−n∆µ + S(n)σ and the population of droplets can be obtained from the Boltzmann

distribution.

The second section will be devoted to the proof of two nucleation theorems. We

will derive two theorems on (1) supersaturation and (2) temperature dependences of

nucleation rate that will hold regardless of the functional form of ∆G(n), using the

1D Markov chain model only. This will provide an opportunity to check the validity

of the droplet free energy function ∆G(n).

111

APPENDIX A. DERIVATIONS 112

A.1 Nucleation Rate Prediction from Classical Nu-

cleation Theory

Consider the nucleation process that proceeds with the following series of bimolecular

reactions [33].

Λn−1 + Λ1

f+n−1−−−−−−f−

n

Λn (A.1)

Λn + Λ1

f+n−−−−−−

f−

n+1

Λn+1 (A.2)

Λn and Λ1 denote a droplet consisting of n particles and a single particle, respectively.

f+n is the rate of single-particle attachment to a cluster of size n and f−

n is the rate of

loss. It is implicitly assumed that reactions of clusters with dimers, trimers, etc., are

too infrequent to be comparable with single particle attachment. Also, the history

dependence is ignored in the model, i.e. the attachment f+n and the detachment rate

f−n depend only on the cluster size n.

The kinetic model can be solved if we assume appropriate boundary conditions:

clusters consisting of n∗ atoms (n∗ being much greater than the critical size nc) are

removed from the system and replaced by an equivalent number of single particles to

ensure a constant supersaturation. n∗ must be set to a sufficiently large number such

that the probability of shrinking back to n < nc is effectively zero. This assumption

is inserted to keep the monomer concentration constant when obtaining the steady

state solution. However, in reality, the nucleation rate would decreases due to loss of

monomer in time.

Droplets consisting of n particles, i.e. Λn’s, are formed by the growth of Λn−1’s

and the decay of Λn+1, but disappear by the decay into Λn+1’s and Λn−1’s. Define

Zn(t) to be the concentration of Λn as a function of time. Introducing the net flux

from size n− 1 to size n,

Jn(t) = f+n−1Zn−1(t)− f−

n Zn(t), (A.3)


the change of the concentration Zn(t) of Λn with time can be written as

dZn(t)

dt= Jn(t)− Jn+1(t). (A.4)

In the steady state, the concentration does not change with time, i.e. dZn(t)/dt =

0. This requires

Jn(t) = Jn+1(t) = I (A.5)

where I is the steady state rate or the rate of forming cluster of any size that does

not depend on the size n. Hence, I can be considered to be the rate of forming the

critical droplet with size nc. In other words, we can write following equation in steady

state.

I = f+1 Z1 − f−

2 Z2,

I = f+2 Z2 − f−

3 Z3,

· · ·I = f+

n Zn − f−n+1Zn+1,

· · ·I = f−

n∗−1Zn∗−1 (A.6)

where Zn∗ = 0 as stated in the assumed boundary condition.

To obtain the final expression for the steady state rate I, we first multiply each

equation with a ratio of the rate constants. The first equation is multiplied by 1/f+1 ,

the second by f−2 /f

+1 f

+2 , the nth by f−

2 f−3 · · · f−

n /f+1 f

+2 · · · f+

n , and etc. When all

equations are summed up, the terms on the right hand side will be canceled out

except Z1 while the left hand side will end up to be I multiplied by a lengthy constant∑n∗−1

n=1

(

1f+n

f−

2 f−

3 ···f−

n

f+1 f+

2 ···f+n−1

)

. Rearranging the equation, we obtain the general expression

for the steady state nucleation rate

I = Z1

[

n∗−1∑

n=1

(

1

f+n

f−2 f

−3 · · ·f−

n

f+1 f

+2 · · · f+

n−1

)

]−1

. (A.7)


The expression is generally applicable for any case of nucleation process where the bi-

molecular reaction holds reasonably. The appropriate expression for the rate constant

can be obtained for each nucleation.

The rate constants f+n and f−

n are kinetic properties which cannot be obtained

from the equilibrium statistical mechanics. However, the ratio between them can be

obtained from the detailed balance condition at thermal equilibrium. Denote Xn as

the equilibrium concentration of droplets with size n. At thermal equilibrium, i.e.

I = 0, we have

f+n−1Xn−1 = f−

n Xn. (A.8)

It can be rewritten in the form of

Xn

Xn−1=

f+n−1

f−n

. (A.9)

Multiplying the ratios Xi/Xi−1 from i = 2 to n results in

Xn

X1

=n∏

i=2

(

f+i−1

f−i

)

=

(

f−2 f

−3 · · · f−

n

f+1 f

+2 · · · f+

n−1

)−1

. (A.10)

At the same time, the ratio is determined by the Boltzmann distribution

Xn

X1

= exp

(

−∆G(n)

kBT

)

(A.11)

where ∆G(n) is the formation free energy of droplet containing n particle, −n(µ1 −µ2) + S(n)σ, the Eq. (2.1).

The multiple product expression in the right hand side of Eq. (A.7) can be sim-

plified as

Υ ≡n∗−1∑

n=1

(

1

f+n

f−2 f

−3 · · · f−

n

f+1 f

+2 · · · f+

n−1

)

=n∗−1∑

n=1

(

1

f+n

exp

(

∆G(n)

kBT

))

(A.12)


and I = Z1/Υ. Suppose n∗ is significantly larger than 10, we can replace the expres-

sion into a integral, which gives

n∗−1∑

n=1

(

1

f+n

exp

(

∆G(n)

kBT

))

≈∫ n∗

1

1

f+n

exp

(

∆G(n)

kBT

)

dn. (A.13)

Note that ∆G(n) displays a maximum at n = nc and it can be expanded in a Taylor

series near the maximum

∆G(n) ≈ Gc −1

2η(n− nc)

2 (A.14)

where Gc is the maximum value of ∆G(n), or nucleation barrier and η is the negative

of the second derivative − ∂2G∂n2

∣

∣

∣

n=nc

evaluated at n = nc. Because the integral shows

very sharp peak near the maximum, we can change the range of integration from -∞to ∞. Besides, note that the f+

n is approximately proportional to the surface area

of the droplet which scales as n1−1/d, we can replace f+n by constant value f+

c ≡ f+nc.

Then, the outcome of Gaussian integral becomes

Υ =1

f+c

(

2πkBT

η

)1/2

exp

(

Gc

kBT

)

. (A.15)

where(

η2πkBT

)1/2

is known as the Zeldovich factor Γ. Finally, we obtain the steady

state nucleation rate

I = f+c ΓZ1 exp

(

− Gc

kBT

)

. (A.16)

Because the steady state distribution function Zn deviates perceptibly from the equi-

librium one, Xn, only in the vicinity of the critical size nc, we can approximate

X1 ≈ Z1. Then, the steady state rate of nucleation can be rewritten in the form

I = f+c ΓXnc

(A.17)


where Xncis the equilibrium concentration of critical nuclei,

Xnc= X1 exp

(

− Gc

kBT

)

. (A.18)

In many cases, X1 can be identified as the number of nucleation site N , in which case

we can recover the rate expression Eq. 2.7.

Prior to closing the section, we underscore that the steady state concentration of

critical nuclei Zncis different from the equilibrium concentration Xnc

. To compute

Znc, we modify the trick that is used for computing the nucleation rate I in Eq. (A.7).

Instead of summing up the series of equations up to n = n∗, we sum those only up

to n = nc. Which results in

[

nc−1∑

n=1

(

1

f+n

f−2 f

−3 · · · f−

n

f+1 f

+2 · · · f+

n−1

)

]

I = Z1 −(

f−2 f

−3 · · · f−

nc

f+1 f

+2 · · · f+

nc−1

)

Znc(A.19)

Identify that

[

nc−1∑

n=1

(

1

f+n

f−2 f

−3 · · · f−

n

f+1 f

+2 · · · f+

n−1

)

]

≈ 1

2Υ =

1

2f+ncΓexp

(

Gc

kBT

)

(A.20)

and(

f−2 f

−3 · · · f−

nc

f+1 f

+2 · · · f+

nc−1

)

= exp

(

Gc

kBT

)

. (A.21)

Substituting these expressions into the Eq. (A.19), we obtain

1

2Z1 = Z1 − Znc

exp

(

Gc

kBT

)

. (A.22)

Rearranging the equation, we obtain Znc≈ 1

2Xnc

(as compared to X1 ≈ Z1). From

Znc, Zn at n 6= nc can be obtained by using the series of equations, Eqs. (A.6), with

the detailed balance condition, Eq. (A.9).


A.2 Nucleation Theorems

In this section, we will discuss how the nucleation rate depends on temperature and

supersaturation [192, 193, 194, 195]. The results obtained in this section can be

compared with experiments on the gas-liquid transition such as Fig. 2.3.

The nucleation rate I can be written as Z1/Υ where Υ is given in Eq. (A.12).

Consider the derivative of ln I = lnZ1− lnΥ with respect to lnS (where S is defined

in Eq. 2.11) at constant T ,

∂ ln I

∂ lnS

∣

∣

∣

∣

T

=∂∆µ

∂ lnS

∣

∣

∣

∣

T

∂ ln I

∂∆µ

∣

∣

∣

∣

T

(A.23)

= kBT∂ ln I

∂∆µ

∣

∣

∣

∣

T

(A.24)

= kBT

(

∂ lnZ1

∂∆µ

∣

∣

∣

∣

T

− ∂ lnΥ

∂∆µ

∣

∣

∣

∣

T

)

(A.25)

where Eq. (2.11) is used. Imagine a droplet free energy function of ∆G(n,∆µ) =

g(n, T ) − n∆µ where g(n, T ) is an arbitrary function of n and T that includes sur-

face energy, entropic effect, and any other possible correction. n∆µ is the volume

contribution due to phase transformation. For any g(n), we have ∂∆G/∂∆µ|T =

−n. For gas-to-liquid condensation, f+n = P (2πmkBT )

−1/2S(n) where m is the

mass of the particle and S(n) is the area of nucleus having n particles. Identify-

ing P = Peq exp(∆µ/kBT ) from Eq. (2.11), we obtain the attachment rate f+n =

Peq(2πmkBT )−1/2S(n) exp(∆µ/kBT ) and thus ∂f+

n /∂∆µ|T = f+n /kBT . Using the

right hand side (RHS) of Eq. (A.12), we can rewrite

∂ lnΥ

∂∆µ

∣

∣

∣

∣

T

=1

Υ

∂Υ

∂∆µ

∣

∣

∣

∣

T

(A.26)

=1

Υ

n∗−1∑

n=1

[

1

f+n

−n

kBTexp

(

∆G(n)

kBT

)

− 1

f+n

1

kBTexp

(

∆G(n)

kBT

)]

(A.27)

Using the approximation used in the Eqs. (A.13), (A.14), and (A.15), The summation


in the RHS of Eq. (A.27) becomes

n∗−1∑

n=1

[

1

f+n

−n− 1

kBTexp

(

∆G(n)

kBT

)]

≈ 1

Γf+c

nc + 1

kBTexp

(

Gc

kBT

)

(A.28)

Combining with Eq. (A.15), we obtain

∂ lnΥ

∂∆µ

∣

∣

∣

∣

T

= −nc + 1

kBT(A.29)

Ignoring negligible ∆µ dependence of Z1, we obtain the final expression

∂ ln I

∂ lnS

∣

∣

∣

∣

T

= nc + 1 , (A.30)

which is known as the first nucleation theorem. It implies that, when the lnS depen-

dence of ln I from experiments matches with the theoretical prediction, the nucleation

theory predicts nc correctly.

Now, turn attention to the derivative of ln I = lnZ1 − lnΥ with respect to T at

constant S,∂ ln I

∂T

∣

∣

∣

∣

S

=∂ ln I

∂T

∣

∣

∣

∣

∆µ

+∂ ln I

∂∆µ

∣

∣

∣

∣

T

∂∆µ

∂T

∣

∣

∣

∣

S

. (A.31)

Notice that the second term in the RHS can be obtained from previous results, and

we have∂ ln I

∂∆µ

∣

∣

∣

∣

T

=nc + 1

kBTand

∂∆µ

∂T

∣

∣

∣

∣

S

= kB lnS (A.32)

Thus,∂ ln I

∂∆µ

∣

∣

∣

∣

T

∂∆µ

∂T

∣

∣

∣

∣

S

=(nc + 1)kBT lnS

kBT 2(A.33)

We need to compute derivative of Υ with respect to T to obtain the first term.

The T derivative of f+n at constant ∆µ becomes

∂f+n

∂T

∣

∣

∣

∣

∆µ

= −f+n

(

2∆µ/kB + T

T 2

)

(A.34)


and the T derivative of exp(∆G/kBT ) becomes

∂

∂T

∣

∣

∣

∣

∆µ

exp (∆G/kBT ) =

[

− ∆G

kBT 2+

1

kBT

∂g(n, T )

∂T

∣

∣

∣

∣

∆µ

]

exp (∆G/kBT ) (A.35)

Using the same Gaussian approximation around n = nc and combining above two

derivatives, we obtain

∂ ln I

∂T

∣

∣

∣

∣

∆µ

= − ∂ lnΥ

∂T

∣

∣

∣

∣

∆µ

(A.36)

= −−Gc + T∂g(nc, T )/∂T |∆µ + 2∆µ+ kBT

kBT 2(A.37)

=g(nc, T )− T∂g(nc, T )/∂T |∆µ − (nc + 2)∆µ− kBT

kBT 2(A.38)

This leads to the partial derivative at constant S,

∂ ln I

∂T

∣

∣

∣

∣

S

=g(nc, T )− T∂g(nc, T )/∂T |∆µ −∆µ− kBT

kBT 2(A.39)

≈ g(nc, T )− T∂g(nc, T )/∂T |∆µ

kBT 2. (A.40)

Finally, defining ∆U(n, T ) = g(n, T ) − T∂g(n, T )/∂T |∆µ, we obtain the second nu-

cleation theorem,

∂ ln I

∂T

∣

∣

∣

∣

S

=∆U(nc, T )

kBT 2. (A.41)

The ∂g(n, T )/∂T |∆µ term is much smaller than the g(n, T ) contribution in most

circumstances. g(n, T ) changes very slowly unless the surface energy per atom is

comparable to the thermal fluctuation kBT . Typically, a broken atomic bonding

costs ∼ 1 eV energy, and kBT = 0.026 eV which is much less than 1 eV 1. Thus,

we can treat ∆U as the formation energy of a droplet with size nc in the absence of

supersaturation.

1In experiments, we hardly met this condition because melting points and boiling points ofmaterials are typically low enough that kBT ≪ 1 eV. However, in a model system such as the Isingmodel, we can artificially prepare a condition where kBT is comparable to the scale of surface energy(See Chapter 4).


More detailed derivation and discussion can be found in the literature [192, 193,

194, 195]. The applicability of nucleation theorems have been shown in the Ising

model [138].

Appendix B

Additional Data on the Nucleation

Theory Test Using the Ising Model

B.1 Attachment Rate

In this appendix, we examine the dependence of the pre-exponential factor, f+c Γ, in

the nucleation rate predicted by the Becker-Doring theory, Eq. (2.7), on T and h.

In the (T, h) conditions considered in this study, both the attachment rate f+c and

the Zeldovich factor Γ vary by several orders of magnitude. However, their variations

largely cancel each other and the product f+c Γ only varies within a factor of 2, as

shown in Fig. B.1(a) and (c).

Eq. (2.8) defines the Zeldovich factor Γ in terms of the second derivative of the

droplet free energy function F (n) which is discussed in more detail in the following

appendices. Here we focus on the attachment rate f+c and evaluate the quality of the

approximations in Eqs.(4.7), (4.8) and (4.15). Fig. B.1(b) and (d) plots the ratio of the

attachment rate f+c computed from Monte Carlo and that predicted by the classical

theories, in 2D and 3D respectively. The predictions from classical theories are within

a factor of 2 of the numerical results for the entire (T, h) conditions considered in this

work. The discrepancy between theoretical and numerical results observed here can

be partly attributed to the approximation that the droplet is circular in 2D, as in

Eq. (4.7) or spherical in 3D, as in Eq. (4.15). Due to the discreteness of the Ising

121

APPENDIX B. MORE DATA ON THE NUCLEATION IN THE ISING MODEL122

model, this is obviously not the case, as shown in Appendix B.2.

(a) (b)

0 0.05 0.1 0.150.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

h

f c+ Γ

(M

CS

S−

1 )

kBT = 1.0

1.51.9

0 0.05 0.1 0.151

1.2

1.4

1.6

1.8

h

f c+

MC

/ f c+

appr

ox

kBT = 1.0

1.5

1.9

(c) (d)

0.4 0.5 0.6 0.70.5

0.6

0.7

0.8

0.9

1

1.1

h

f c+ Γ

(M

CS

S−

1 )

kBT = 2.20

2.35

2.50

2.65

2.71

0.4 0.5 0.6 0.70.7

0.8

0.9

1

1.1

1.2

1.3

1.4

h

f c+

MC

/ f c+

appr

ox

kBT = 2.20

2.35

2.50

2.65

2.71

Figure B.1: (color online) (a) The pre-exponential factor f+c Γ in 2D computed from

Monte Carlo and US. (b) The ratio between the attachment rate f+c in 2D computed

by Monte Carlo and that predicted by Eq.(4.7). (a) The pre-exponential factor f+c Γ

in 3D computed from Monte Carlo and US. (b) The ratio between the attachmentrate f+

c in 3D computed by Monte Carlo and that predicted by Eq.(4.15).

B.2 Droplet Shape

The purpose of this appendix is to examine the shape change of the droplets as

temperature changes. As temperature increases, we reduce the magnitude of the field

h, so that the size of the critical nucleus stays roughly the same. The droplets plotted


here are close to the critical size and are randomly chosen from FFS simulations.

Fig. B.2(a) shows three droplets in 2D at kBT = 1.0, 1.5 and 1.9. At kBT = 1.0,

the droplet has long facets on the boundary and a solid interior. At kBT = 1.5, the

droplet shape becomes more circular than rectangular. At kBT = 1.9, significant

fluctuation can be observed on the droplet surface. The inside of the droplet also

becomes more porous containing a number of −1 spins.

Fig. B.2(b) shows three droplets in 3D at kBT = 2.2, 2.5 and 2.71. At kBT = 2.0

(below the roughening temperature), small facets can be found on the droplet surface.

At kBT = 2.5 (near the roughening temperature), the droplet shape does not seem

to be substantially different from that at kBT = 2.0. At kBT = 2.71 (above the

roughening temperature), the surface shape becomes more irregular. The droplet

shape seems to change gradually with increasing temperature, without any sharp

transition (resembling a phase transition) at the roughening temperature TR. This

may be caused by the small size the critical droplet in this study, which prevents a

true roughening transition of its surface morphology due to its small area.

B.3 The Constant Term in Droplet Free Energy

In this appendix, we discuss how to obtain the constant correction term in the droplet

free energy function, Eq. (4.10), for the 2D Ising model by considering the exact free

energy expressions of small clusters. Shneidman et al. [47] used a similar approach

to improve the predictions of droplet distributions. A related problem was discussed

by Wilemski [146]. We will also list the free energy expressions of small 3D clusters.

Even though they cannot be used to determine the constant correction term, they

are useful for comparison purposes, as in Fig. 4.7(b).

Because the free energy expression, Eq. (2.9), is based on a continuum droplet

model, we expect it to be inaccurate for very small droplets, where the discreteness

of the lattice becomes appreciable. On the other hand, the continuum approximation

should work better for large clusters, i.e. in the continuum limit. Therefore, we

expect that Eq. (2.9) can be used to accurately predict the free energy difference

between two large droplets, F (m) − F (n), if both m ≫ 1 and n ≫ 1. This justifies


kBT = 1.0 h = 0.07

n = 525

kBT = 1.5 h = 0.05

n = 499

kBT = 1.9 h = 0.03

n = 409

(a)

kBT = 2.2 h = 0.45

n = 362

kBT = 2.5 h = 0.40

n = 396

kBT = 2.71 h = 0.35

n = 419

(b)

Figure B.2: Droplets in (a) 2D and (b) 3D Ising models randomly chosen from FFSsimulations at different (T, h) conditions. n is the size of the droplet.

the addition of a constant term in Eq. (4.10). The value of the constant term can be

determined by matching Eq. (4.10) with the exact values of F (n) for small n.

Fortunately, for small enough n, the exact expression of the droplet free energy

can be written down by enumerating all possible shapes of the droplet with size n

and summing up their contributions to the partition function. For simplicity, we will

consider the case of h = 0. For example, a droplet of n = 1 is simply an isolated spin

+1 surrounded by spins −1. The partition function of this droplet in the 2D Ising

model is,

Ω2D1 = e−8βJ (B.1)


where β ≡ 1/(kBT ). Similarly, the partition function of droplets of size 2, 3 and 4

are,

Ω2D2 = 2 e−12βJ (B.2)

Ω2D3 = 6 e−16βJ (B.3)

Ω2D4 = e−16βJ + 18 e−20βJ (B.4)

The number in front of the exponential term corresponds to the multiplicity of clusters

of a given shape. Analytic expressions for the partition functions of 2D droplets have

been obtained up to n = 17 with computer assistance [145].

We have obtained similar expressions for the droplet partition functions in the 3D

Ising model for n from 1 to 7.

Ω3D1 = e−12βJ (B.5)

Ω3D2 = 3 e−20βJ (B.6)

Ω3D3 = 15 e−28βJ (B.7)

Ω3D4 = 3 e−32βJ + 83 e−36βJ (B.8)

Ω3D5 = 48 e−40βJ + 486 e−44βJ (B.9)

Ω3D6 = 18 e−44βJ + 496 e−48βJ + 2967 e−52βJ (B.10)

Ω3D7 = 8 e−48βJ + 378 e−52βJ + 4368 e−56βJ + 18748 e−60βJ (B.11)

Given the droplet partition functions, the droplet free energy F (n) defined in this

paper can be obtained from the following equation,

e−βF (n) =Ωn

1 +∑∞

i=1Ωi(B.12)

Numerically, the summation in the denominator converges very quickly after summing

over 2 to 3 terms. As an approximation, we may write F (n) ≈ −kBT ln Ωn. But this

approximation is not invoked in Section 5.4.

The droplet free energy computed from Eq. (B.12) is used to determine the con-

stant term d in Eq. (4.10), by requiring that F (n) from the two equations matches


at a given n = n0. In this work, we have always used n0 = 1. Setting n0 to larger

values (as long as the analytic expression exists) does not change the numerical re-

sults appreciably. For example, consider the 2D Ising model at kBT = 1.5, h = 0 and

J = 1. The free energy of a droplet of n = 1 is F (1) ≈ 8, whereas Eq. (4.3) predicts

that F (1) = 2√πσeff ≈ 4.3. This means that a constant correction term d ≈ 3.7 is

needed.

B.4 Free Energy Curves F (n) for the Ising Model


(a) kBT=1.0 and h=0.05 (b) kBT=1.0 and h=0.06

0 200 400 600 800 10000

20

40

60

80

100

120

n

F (

n)

0 200 400 600 8000

20

40

60

80

100

n

F (

n)

(c) kBT=1.0 and h=0.07 (d) kBT=1.0 and h=0.08

0 100 200 300 400 500 6000

20

40

60

80

n

F (

n)

0 100 200 300 4000

20

40

60

80

n

F (

n)

(e) kBT=1.0 and h=0.09 (f) kBT=1.0 and h=0.10

0 100 200 3000

20

40

60

80

n

F (

n)

0 50 100 150 200 250 3000

10

20

30

40

50

60

n

F (

n)

Figure B.3: The free energy curve F (n) of 2D Ising system at kBT = 1.0 and (a)h = 0.05, (b) h = 0.06, (c) h = 0.07, (d) h = 0.08, (e) h = 0.09, (f) h = 0.10obtained by US (circles) is compared with Eq. (6) (solid line) and Eq. (8) (dashedline). Logarithmic correction term 5

4kBT lnn (dot-dashed line) and the constant term

d (dotted line) are also drawn for comparison.



0 200 400 600 8000

20

40

60

80

n

F (

n)

0 100 200 300 400 5000

20

40

60

80

n

F (

n)


0 100 200 300 4000

10

20

30

40

50

60

n

F (

n)

0 50 100 150 200 250 3000

10

20

30

40

50

n

F (

n)


0 50 100 150 200 2500

10

20

30

40

50

n

F (

n)

0 50 100 150 2000

10

20

30

40

n

F (

n)

(g) kBT=1.5 and h=0.10 (h) kBT=1.5 and h=0.11

0 50 100 150 2000

10

20

30

40

n

F (

n)

0 50 100 1500

10

20

30

40

n

F (

n)


(i) kBT=1.5 and h=0.12 (j) kBT=1.5 and h=0.13

0 50 1000

10

20

30

40

n

F (

n)

0 50 1000

10

20

30

40

n

F (

n)

Figure B.4: The free energy curve F (n) of 2D Ising system at kBT = 1.5 and (a)h = 0.04, (b) h = 0.05, (c) h = 0.06, (d) h = 0.07, (e) h = 0.08, (f) h = 0.09,(g) h = 0.10, (h) h = 0.11, (i) h = 0.12, (j) h = 0.13 obtained by US (circles) iscompared with Eq. (6) (solid line) and Eq. (8) (dashed line). Logarithmic correctionterm 5

4kBT lnn (dot-dashed line) and the constant term d (dotted line) are also drawn

for comparison.



0 500 1000 15000

20

40

60

80

n

F (

n)

0 500 1000 15000

20

40

60

80

n

F (

n)


0 200 400 600 800 10000

10

20

30

40

50

60

n

F (

n)

0 200 400 600 8000

10

20

30

40

50

n

F (

n)


0 200 400 6000

10

20

30

40

50

n

F (

n)

0 100 200 300 400 5000

10

20

30

40

n

F (

n)

Figure B.5: The free energy curve F (n) of 2D Ising system at kBT = 1.9 and (a)h = 0.015, (b) h = 0.017, (c) h = 0.022, (d) h = 0.025, (e) h = 0.03, (f) h = 0.035obtained by US (circles) is compared with Eq. (6) (solid line) and Eq. (8) (dashedline). Logarithmic correction term 5

4kBT lnn (dot-dashed line) and the constant term

d (dotted line) are also drawn for comparison.

Appendix C

Additional Proof and Data on the

Dislocation Nucleation

C.1 Equality of Critical Sizes nσc and nγ

c

Suppose that the Gibbs free energy G(n, σ, T ) is maximized at n = nσc , then

∂G(n, σ, T )

∂n

∣

∣

∣

∣

σ,n=nσc

= 0. (C.1)

T is held constant throughout this section. Through Legendre transform, Eq. (5.5),

we have the following property for the Helmholtz free energy F (n, γ, T )

∂F (n, γ, T )

∂n

∣

∣

∣

∣

γ,n=nσc

=∂

∂n

∣

∣

∣

∣

γ,n=nσc

[G(n, σ, T ) + σγV ]

=∂G(n, σ, T )

∂n

∣

∣

∣

∣

σ,n=nσc

+∂G(n, σ, T )

∂σ

∣

∣

∣

∣

n=nσc

∂σ

∂n+

∂σ

∂nγV

= 0− (V γ)∂σ

∂n+ (V γ)

∂σ

∂n= 0. (C.2)

By definition, F (n, γ, T ) reaches maximum at n = nγc at constant γ and T ,

∂Fc(n, γ, T )

∂n

∣

∣

∣

∣

γ,n=nγc

= 0 (C.3)

131

APPENDIX C. MORE DISCUSSION ON THE DISLOCATION NUCLEATION132

Therefore, we establish that nγc = nσ

c , i.e. the maximizer nσc of G(n, σ, T ) is also the

maximizer nγc of F (n, γ, T ).

C.2 Equality of Activation Gibbs and Helmholtz

Free Energies

The activation Gibbs free energy is the free energy difference between state 0: a

perfect crystal, and state 1: a crystal containing a critical dislocation loop under a

same shear stress σ. Because of the plastic shear deformation caused by dislocation

loop, state 1 has a higher strain (γ) than the state 0 (γ0). It has been shown that the

maximizer nσc of G(n, σ, T ) equals to the maximizer of nγ

c of F (n, γ, T ) when σ equals

σ(nc, γ, T ), as defined in Eq. (5.6). Note that we keep σ to be the stress at nc,γ, and

T . Then at the same σ, but for n = 0, the strain becomes γ0. Hence, the activation

Gibbs free energy barrier can be written as

Gc = G(nc, σ, T )−G(0, σ, T )

= F (nc, γ, T )− σγV − F (0, γ0, T ) + σγ0V (C.4)

Notice that F (nc, γ, T ) and F (0, γ0, T ) do not correspond to the same strain state,

so that their difference is not the activation Helmholtz free energy. To construct the

activation Helmholtz free energy, we subtract and add the F (0, γ, T ) term in the right

hand side,

Gc = F (nc, γ, T )− F (0, γ, T ) + F (0, γ, T )− F (0, γ0, T )− V σ(γ − γ0)

≈ Fc +∂F

∂γ

∣

∣

∣

∣

γ0,V

(γ − γ0) +1

2

∂2F

∂γ2

∣

∣

∣

∣

γ0,V

(γ − γ0)2 − σ(γV − γ0V )

= Fc +1

2

∂2F

∂γ2

∣

∣

∣

∣

γ0,V

(γ − γ0)2

= Fc +1

2V∂σ

∂γ

∣

∣

∣

∣

γ0,V

(γ − γ0)2 (C.5)


Notice that γV = −∂G(nc, σ, T )/∂σ and γ0V = −∂G(0, σ, T )/∂σ. Then (γV −γ0V ) is

equivalent to − ∂∂σ(G(nc, σ, T )−G(0, σ, T )) = −∂Gc

∂σ≡ Ωc, i.e. the activation volume.

By plugging (γ − γ0) = Ωc/V into the equation, we have

Gc = Fc +1

2

1

V

∂σ

∂γ

∣

∣

∣

∣

γ0,V

(Ω∗)2 +O(V −2)

= Fc +O(V −1) (C.6)

In the thermodynamics limit (V → ∞), we have Gc = Fc. Hence, the nucleation rate

does not depend on whether the crystal is subjected to constant stress or constant

strain loading. The equality allows us to compute the activation Gibbs free energy

Gc(T, σ) by combining the activation Helmholtz free energy Fc(T, γ) and the stress-

strain relations of the perfect crystal shown in Fig. 5.2 (b) and (d).

C.3 Physical Interpretation of Activation Entropy

Difference ∆Sc

It is well-known that the entropy is a thermodynamic stat variable that is independent

of the ensemble of choice, i.e., S(n, γ, T ) ≡ ∂F (n, γ, T )/∂T |n,γ and S(n, σ, T ) ≡∂G(n, σ, T )/∂T |n,σ equal to each other as long as σ = V −1∂F/∂γ|n,T . At the same

time, the activation entropy is just the entropy difference between the activated state

and the meta-stable state, i.e., Sc(γ, T ) = S(nc, γ, T ) − S(0, γ, T ) and Sc(σ, T ) =

S(nc, σ, T ) − S(0, σ, T ). If the entropies in two ensembles can equal each other, it

may seem puzzling how the activation entropies can be different.

The resolution of this apparent paradox is that under the constant applied stress,

the nucleation of a dislocation loop causes a strain increase. Let γ be the strain at

the state defined by n = nc, σ, and T , and γ0 be the strain at the state defined

by n = 0, σ, and T , then γ > γ0. Hence, we have S(nc, σ, T ) = S(nc, γ, T ) and


S(0, σ, T ) = S(0, γ0, T ), but S(0, γ, T ) 6= S(0, γ0, T ).

Sc(σ, T ) = S(nc, σ, T )− S(0, σ, T )

= S(nc, γ, T )− S(0, γ0, T )

= S(nc, γ, T )− S(0, γ, T ) + S(0, γ, T )− S(0, γ0, T )

= Sc(T, γ) + S(0, γ, T )− S(0, γ0, T ) (C.7)

This shows that the activation entropy difference ∆Sc ≡ Sc(σ) − Sc(γ) equals to

S(0, γ, T )−S(0, γ0, T ), which is entropy difference of the perfect crystal at two slightly

different strains.

In the limit of V → ∞, because we expect (γ − γ0) → 0, we might reach a false

conclusion that ∆Sc = (S(0, γ, T ) − S(0, γ0, T )) → 0. Instead, the correct behavior

in the thermodynamic limit can be obtained by expanding ∆Sc in a Taylor series.

S(γ)− S(γ0) =∂S

∂γ(γ − γ0) + · · ·

= − ∂σ

∂T

∣

∣

∣

∣

γ,V

V (γ − γ0) + · · · (C.8)

where the Maxwell relationship ∂S/∂γ|T = −V ∂σ/∂T |γ,V is used. The term (γV −γ0V ) equals the activation volume Ωc, and can be interpreted as plastic strain γpl due

to formation of dislocation loop times the volume of the crystal, i.e.

(γV − γ0V ) = Ωc = γplV = bAc (C.9)

where b is the magnitude of the Burgers vector and Ac is the area of the critical

dislocation loop. Using the relation (γ − γ0) = Ωc/V , we have

∆Sc = S(γ)− S(γ0) = − ∂σ

∂T

∣

∣

∣

∣

γ,V

Ωc +O(V −1) (C.10)

which is exactly the same as Eq. (5.24).

A similar expression has been obtained for the difference between point defect for-

mation entropies under constant pressure (Sp) and under constant volume (Sv) [172],


with

Sp − Sv = βBVrel (C.11)

where β ≡ V −1 ∂V∂T

∣

∣

p=0is the thermal expansion factor at zero hydrostatic pressure

p, B is the isothermal bulk modulus, and Vrel is the relaxation volume of the defect.

In Cu, the value of Sp − Sv is estimated to be −1.7 kB for a vacancy and 13.7 kB for

an interstitial [172]. Comparing Eq. (C.11) with Eq. (C.10), we note that relaxation

volume Vrel for point defects corresponds to the activation volume Ω for dislocation

nucleation, and that βB corresponds to the − ∂σ∂T

term. The similarity between these

two equations stems from the fact that they both express the entropy difference be-

tween two states, and the choice of the two states depends on whether the stress

or the strain is kept constant when the defect is introduced. On the other hand,

there are also some differences between the physics expressed by these two equations.

First, thermal expansion plays a prominent role in Eq. (C.11) because it focuses on

hydrostatic stress and strain effects. In comparison, thermal expansion does not play

a role in Eq. (C.10) because it focuses on shear stress and strain effects. Second, the

formation entropy of a point defect is the entropy difference between two metastable

states and governs equilibrium properties, e.g. density of vacancies at thermal equi-

librium. In comparison, the activation entropy is the entropy difference between a

saddle (i.e. unstable) state and a metastable state and governs kinetics, such as dis-

location nucleation rate. In addition, the saddle state (i.e. the size of the critical

nucleus) depends on stress and temperature, while such complexity does not arise in

the formation entropy of point defects.

C.4 Approximation of Sc(σ)

In this appendix, we introduce a series of simplifying approximations to estimate the

magnitude of Sc(σ) in the low temperature, low stress limit. In the temperature range

of zero to 300 K, the activation entropy is found to be insensitive to temperature.


Starting from Eqs. (5.24) and (5.39), we have

Sc(σ) = Sc(γ) + ∆Sc ≈ −Ec(γ)

µ(0)

∂µ

∂T− Ωc

∂σ

∂T

∣

∣

∣

∣

γ

(C.12)

If we assume the crystal is linear elastic, i.e. σ = µ γ, then,

Sc(σ) ≈ −Hc(σ) + Ωc(σ) · σµ(0)

∂µ

∂T(C.13)

Similar expressions can be obtained for normal (compressive) loading by replacing γ

with ǫ and replacing µ by the Young’s modulus.

To gain more intuition, we note that in the limit of σ → 0, the line tension model

estimates that Hc(σ) ∝ σ−1. In addition, in the limit of T → 0, Ωc(σ) ≈ −∂Hc/∂σ.

Under these conditions, Ωc(σ) · σ ≈ Hc(σ), so that,

Sc(σ) ≈ −2Hc(σ)

µ(0)

∂µ

∂T(C.14)

Comparing Eq. (C.14) with Eq. (5.39), we have,

Sc(σ)

Hc(σ)≈ 2

Sc(γ)

Ec(γ)(C.15)

This trend is qualitatively observed in heterogeneous nucleation, when comparing

Fig. 5.8(b) and (d), and is less clear in homogeneous nucleation, when comparing

Fig. 5.8 (a) and (c). This is probably because the stress-strain relationship is more

nonlinear in the case of homogeneous nucleation.

C.5 Activation Free Energy Data

In Table C.1 and C.2, we provide the activation free energy data at all temperature

and strain conditions in this study so that interested readers can use them as a

benchmark. We define strain with respect to the cell at 0 K, free of external loading.

For homogeneous nucleation, the strain γxy is defined as ∆x/h0y where ∆x is the


displacement of the repeat vector initially in the y-driection along the x axis at

each pure shear stress condition, and h0y is the height of the cell along the y-axis

at zero temperature without external loading. Shear stress σxy is determined from

the x-y component of the average Virial stress. For heterogeneous nucleation, we

take only the elastic strain into account. The elastic strain ǫzz at T is defined as

[Lz(σ, T )− Lz(σ = 0, T )]/L0z where Lz(σ, T ) is the length of the repeat vector along

the z-axis, Lz(σ = 0, T ) is the equilibrium length at temperature T under zero stress.

L0z = 20a0 = 72.3A is the reference length (before relaxation), where a0 is the lattice

constant of copper. The compressional stress σzz is defined by σzz = 〈F 〉/d2 where

〈F 〉 is the axial force computed from the z-z component of the average Virial stress,

and d = 15a0 = 54.225A is the reference side length of the nanorod.


T γxy 0.092 0.095 0.100 0.105 0.110 0.115 0.120 0.125 0.130 0.135 0.140 0.145

0 Kσxy 2.00 2.04 2.11 2.17 2.24 2.30 2.36 2.42 2.48 2.53 2.58 2.63Ec 2.2851 2.0866 1.7946 1.5426 1.3234 1.1315 0.9627 0.8133 0.6815 0.5648 0.4615 0.3703

100 Kσxy 1.94 1.98 2.05 2.11 2.17 2.22 2.28 2.33 2.38

Ec 2.196 2.004 1.718 1.477 1.268 1.085 0.925 0.784 0.660

200 K

σxy 1.87 1.91 1.97 2.03 2.08 2.13 2.18 2.22

Ec 2.116 1.924 1.650 1.419 1.216 1.043 0.890 0.757Fc 2.115 1.911 1.650 1.428 1.214 1.027 0.863 0.755Γ 0.062 0.064 0.067 0.070 0.072 0.073 0.073 0.073f+c 3.2 3.6 2.9 3.9 3.5 3.4 2.7 2.5

300 K

σxy 1.80 1.83 1.89 1.94 1.98 2.02 2.06 2.10

Ec 2.042 1.853 1.586 1.362 1.171 1.004 0.859 0.730Fc 2.054 1.876 1.585 1.370 1.173 1.006 0.860 0.728Γ 0.048 0.049 0.051 0.054 0.056 0.056 0.055 0.054f+c 3.6 4.5 5.1 3.6 4.4 4.2 5.1 4.4

400 K

σxy 1.71 1.74 1.79 1.84 1.88 1.91 1.95

Ec 1.968 1.793 1.529 1.312 1.124 0.962 0.824Fc 2.008 1.828 1.579 1.347 1.155 0.995 0.853Γ 0.039 0.040 0.043 0.044 0.045 0.045 0.045f+c 5.0 5.9 6.4 7.4 5.3 6.9 6.9

500 K

σxy 1.62 1.64 1.69 1.73 1.76 1.79

Ec 1.897 1.727 1.476 1.261 1.081 0.925Fc 1.939 1.782 1.548 1.341 1.167 1.010Γ 0.034 0.035 0.037 0.038 0.038 0.038f+c 7.7 6.6 8.3 9.9 7.0 7.3

Table C.1: Data for homogeneous nucleation: σxy in GPa, Ec, Ec and Fc in eV, f+c in

1014 s−1. γxy and Γ are dimensionless. The error in Ec is about 0.003 eV, due to thesmall errors in equilibrating the simulation cell to achieve the pure shear stress state.The error in Fc is about 0.5 kBT , i.e. approximately 0.01 eV, due to the statisticalerror in umbrella sampling. The error in Zeldovich factor Γ is within ±0.01. Theattachment rate f+

c has relative error of ±50%.


T = 0 Kǫzz 0.0303 0.0353 0.0403 0.0453 0.0503 0.0553 0.0603 0.0653 0.0703 0.0753 0.0803σxy 1.56 1.81 2.04 2.28 2.50 2.73 2.95 3.16 3.37 3.58 3.78Ec 1.5110 1.0383 0.7550 0.5650 0.4296 0.3277 0.2495 0.1878 0.1388 0.0993 0.0671

200 K

ǫzz 0.0312 0.0342 0.0372 0.0402 0.0432σxy 1.56 1.69 1.82 1.95 2.07Fc 1.278 1.017 0.825 0.690 0.571Γ 0.030 0.038 0.047 0.054 0.062f+c 1.1 1.0 0.91 0.84 0.85

300 K

ǫzz 0.0307 0.0337 0.0367 0.0397 0.0427σxy 1.48 1.60 1.72 1.83 1.94Fc 1.254 1.004 0.824 0.695 0.579Γ 0.021 0.028 0.033 0.045 0.049f+c 1.5 1.4 1.3 1.1 1.1

400 K

ǫzz 0.0301 0.0331 0.0361 0.0391 0.0421σxy 1.39 1.50 1.62 1.73 1.83Fc 1.258 1.003 0.839 0.700 0.578Γ 0.019 0.028 0.029 0.037 0.038f+c 1.8 1.8 1.7 1.5 1.4

500 K

ǫzz 0.0296 0.0326 0.0356 0.0386σxy 1.29 1.40 1.50 1.60Fc 1.226 1.008 0.833 0.697Γ 0.018 0.021 0.026 0.032f+c 2.6 2.0 2.0 2.0

Table C.2: Data for heterogeneous nucleation: σzz in GPa, Ec, and Fc in eV, f+c

in 1014 s−1. γxy and Γ are dimensionless. The error in Fc is about 0.5 kBT , i.e.approximately 0.01 eV, due to the statistical error in umbrella sampling. The errorin Zeldovich factor Γ is within ±0.01. The attachment rate f+

c has relative error of±50%. Notice that, due to the existence of thermal strain, the elastic strain valuesare slightly different at different temperatures.

C.6 Activation Volume and Critical Loop Size

The activation volume Ωc is defined as the derivative of activation free energy with

stress, i.e., Ωc(T, σ) = −∂Gc/∂σ|T , and measures the sensitivity of nucleation rate to

the stress. Physically, it is interpreted as plastic strain associated with the dislocation

loop times the volume of the crystal, i.e. Ωc = bAc where Ac is the area of critical

dislocation loop (See Appendix C.3). In this appendix, we use our numerical data to

test the validity of the latter interpretation (or hypothesis).

Because the activation volume measures the sensitivity of Gc(σ, T ) to applied

stress, see Eq. (5.10), it must be proportional to the Schmid factor, S, in uniaxial


0 20 40 60 80 100 1200

200

400

600

800

1000

nc

Ωc (

A3 )

200K300K400K500K

0 20 40 60 80 100 1200

200

400

600

800

nc

Ωc (

A3 )

200K300K400K500K

(a) (b)

Figure C.1: The relation between critical dislocation size nc and the activation vol-ume Ωc ≡ −∂Gc

∂σ.(a) homogeneous nucleation (b) heterogeneous nucleation. Circles

represent the activation volume obtained from the derivative of Gc with respect toσ. Squares represent the activation volume data multiplied by 1/S where S is theSchmid factor. Dashed lines are linear fits to the data.

loading. Hence, the hypothesis we wish to test is,

Ωc = nc bAa S (C.16)

where b is the magnitude of Burgers vector, Aa is the average area each atom occupy

on the 111 slip plane. Given that the lattice constant of Cu is a0 = 3.615A, we

have b = a0/√6 = 1.48 A and Aa =

√6a20/4 = 5.66 A2, so that bAa = 8.35 A3. For

the pure shear loading in our homogeneous nucleation case, the Schmid factor S = 1.

For the uniaxial loading in our heterogeneous nucleation case, S = 0.471.

Fig. C.1 plots nc versus Ωc for both homogeneous and heterogeneous dislocation

nucleation. In both cases, Ωc appears to be roughly linear with nc, as expected from

Eq. (C.16). For homogeneous nucleations under pure shear, Fig. C.1(a), the slope

of the curves is roughly Ωc/nc ≈ 10 A3, close to the expected value of 8.35 A3. For

heterogeneous nucleation under compression, Fig. C.1(a), the slopes of the curves

after correction for the Schmid factor is roughly Ωc/(nc S) ≈ 8 A3, which is similar

to the case of homogeneous nucleation. Therefore, our data confirms that the idea

that the activation volume is proportional to the size of the critical dislocation loop.

The fact that Ωc/(nc S) is somewhat smaller than bAa supports the notion that the


Burgers vector of a critical dislocation nucleus is smaller than that of a fully formed

dislocation [148, 150].

Appendix D

Comparison of Thermal Properties

Predicted by Interatomic Potential

Models

D.1 Introduction

Empirical or semi-empirical potential models play an important role in computational

materials science because many interesting processes involve the collective dynamics

of thousands of atoms, which is still too expensive for ab initio models. At the same

time, due to their (semi-) empirical nature, the potential models need to be thor-

oughly benchmarked before they can be trusted to make reliable, new, predictions.

The structural and mechanical properties of a single phase (liquid or solid) have been

extensively studied by computer simulations based on empirical potentials with con-

siderable success. There is a growing interest in applying these models to study more

complex processes, such as the catalytic growth of silicon nanowire from a eutectic

liquid droplet, which involve the transformation between different phases. For these

applications, it is very important for the potential models to provide a reasonable

description of the melting point and other thermal properties. But the empirical

potential models have not been extensively tested for these properties, mostly due to

the difficulty in accurately determining the melting point.

142

APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS143

Generally speaking, there are two ways to compute the melting point of a crystal

from atomistic simulation. In the “co-existence method”, the liquid and solid co-exist

with an interface in the simulation cell. The melting temperature is determined by

finding the temperature at which both the liquid and solid phases are stable. While

this method is easy to set up, fluctuations in the instantaneous temperature and

the slow kinetics of solid-liquid interface motion introduce statistical and systematic

errors in the estimation of the melting point [196, 197]. In the “free-energy” method,

the Gibbs free energies of the solid and liquid phases are computed as functions of

temperature, and the melting point is determined by their intersection point. The

free energy method has been applied to determine the melting point of Stillinger-

Weber(SW) model of silicon as early as 1987 [198]. Since then, several advanced

free energy methods have been developed which make free energy and melting point

calculations more efficient [199, 200], and many of them have been applied in melting

point calculations [201]. While the free energy method is more difficult to set up, we

find that it is more efficient than the interface method if we need to determine the

melting point within a very small error bar, e.g. ±1K. The difficulty in setting up

the various free energy calculations necessary for the determination of melting points

is removed by the development of an automatic computer script [202].

In this work, we show that accurate melting points can be obtained from the

state-of-the-art free-energy methods. For the first time, we present a systematic com-

parison of the melting points, latent heat, entropy and thermal expansion coefficients

of nine representative elements described by four different potential models, including

Stillinger-Weber (SW) [105, 203], embedded-atom-method (EAM) [204, 205], Finnis-

Sinclair (FS)[110] and modified-embedded-atom-method (MEAM)[206]. The com-

parison in this work identifies areas that require caution in the application of these

potential models and also suggests directions for their future improvements. Before

we begin, we shall briefly describe the differences and relationship among these po-

tential models [113]. All four models are many-body potentials, i.e. they contain

terms that cannot be written as a sum over pairs of atoms. In the SW potential, a

sum of three-body terms is introduced specifically to stabilize the tetrahedral bond

angle in the diamond-cubic crystal structure. Hence the SW potential is designed for


semiconducting crystals and is called a three-body potential. The other three models

are called many-body potentials because the potential energy cannot be written as a

sum of two-body and three-body terms. The EAM model is designed to capture the

many-body effect in metals, in which the electrons are more diffuse and shared by

more atoms than the electrons in semiconductors. For each atom, the EAM potential

contains an embedding function that describes the energy to embed this atom into

the electron background generated by its neighbors. The EAM model is widely used

to describe face-centered-cubic (FCC) metals. The FS model can be regarded as a

special type of EAM model, with its specific choice of the embedding function, and

is a commonly used model for body-centered-cubic (BCC) metals. The goal of the

MEAM model was to combine the angular dependence of covalent bonds and the

many-body effect for metallic bonds within a unified scheme, in order to provide a

basis for modelling systems (e.g. Si-Au) where both type of bonding may exist. As

a generalization to the EAM model, the embedding energy in the MEAM model de-

pends not only on the total electron density contributed by neighboring atoms, but

also their angular distribution.

Appendix D is organized as follows. In Section D.2, we present the comparison

between the predictions from different potential models with experiments. In Sec-

tion D.3, we describe the important details in our free energy calculations for the

accurate determination of melting points. We organize the appendix D in this way

because we think the results themselves, compared to the computational methods

that enabled such calculations, should be of interest to a wider audience. A brief

summary and outlook for the future research is given in Section D.4.

D.2 Comparison between Model Predictions and

Experiments

Table D.1 summarizes all the numerical results in this work. The melting point Tm,

latent heat of fusion L, and entropy of solid and liquid at melting point, SS and

SL and thermal expansion coefficient α are computed for nine pure elements. The


elements are organized into in three groups: semiconductors (Si, Ge), face-centered-

cubic (FCC) metals (Au, Cu, Ag, Pb), and body-centered-cubic (BCC) metals (Mo,

Ta, W). MEAM is the only model that has been fitted to elements in all three groups.

SW, EAM, and FS models are fitted to semiconductors, FCC metals, and BCC metals,

respectively. In the following we compare the predictions from different potential

models with experiments in these three groups separately.

D.2.1 Semiconductors: Si and Ge

The melting point of Si predicted by the MEAM model is 16% (277 K) lower than

the experimental value, whereas the prediction from the SW model is less than 1%

away from the experimental value. But the SW model for Si is fitted to the melting

point [105]. On the other hand, the SW model for Ge is not fitted to the melting

point and it grossly overestimates the melting point (by more than 100%) [203]. In

comparison, the MEAM prediction of Ge melting point is very accurate (less than

1%). The MEAM model also correctly predicts that Si has a higher melting point

than Ge. The melting point of SW-Si model is consistent with the earlier report

of 1691 ± 20 K, also using the free-energy method [198]. The melting point of the

MEAM-Si model is somewhat lower than the earlier report of 1475± 25 K, using the

co-existence method [196]. This is due to the difference in the potential models used

in both studies.1

A byproduct from the free energy calculation of the melting point is the slope

of the Gibbs free energy – temperature curve at the melting temperature, for both

solid and liquid phases. From these we can extract the entropy of the solid and

liquid phases, SS and SL at the melting point, and the latent heat of fusion from

L = Tm(SL − SS), all of which can be compared with experiments. It is interesting

to note that SS, SL and L are underestimated by MEAM-Si, SW-Si and MEAM-Ge

models, even though SW-Si and MEAM-Ge predicts melting points accurately. A

similar trend was also reported in the environment-dependent interatomic potential

1An earlier version of MEAM [112] without angular cutoff is used in Cook et al. [196] and a laterversion of MEAM [206] is used in this work. We also computed the melting point of the later versionof MEAM [206] using the co-existence method and the melting point is around 1410 K.


Table D.1: Thermal properties of various elements as predicted by several empiricalpotentials and compared with experiments [183, 207, 208]. The properties includethe melting point Tm (in K), latent heat of fusion L (in J/g), solid and liquid entropyat melting point, SS and SL (in J/mol K), and thermal expansion coefficient α (in10−6K−1) at 300 K. The MEAM∗-Au and MEAM∗-Cu entries correspond to a modi-fication of the original MEAM model by changing cmin from 2.0 to 0.8. The MEAM†

entries of BCC metals are computed by the new MEAM model that includes secondnearest neighbor interactions [209, 210].

Model Tm L SS SL α

Si MEAM 1411.3± 0.4 1309 48.74 74.79 13.6Si SW 1694.7± 0.5 1111 58.02 76.45 3.9Si Exp 1687 1650 61.765 91.562 2.6Ge MEAM 1216.2± 0.6 427 58.34 83.84 16.2Ge SW 2898.0± 1.7 847 84.07 105.30 5.8Ge Exp 1211 465 66.77 97.34 5.8

Au MEAM 1120.0± 0.6 92 77.47 93.72 2.0Au MEAM∗ 995.3± 1.3 52 84.18 94.47 16.5Au EAM 984.3± 2.3 41 85.73 94.03 13.5Au Exp 1337.3 64.9 −− −− 14.2Cu MEAM 1350.0± 1.0 368 62.70 80.19 4.5Cu MEAM∗ 1182.9± 2.2 205 69.69 80.68 16.0Cu EAM 1239.6± 2.3 164 71.78 80.17 17.3Cu Exp 1357.8 205 74.30 83.97 16.5Ag MEAM 987.1± 0.9 158 66.21 83.49 5.1Ag Exp 1234.9 103 −− −− 18.9Pb MEAM 674.7± 1.0 57 76.17 93.59 3.3Pb Exp 600.6 23.2 84.34 92.31 28.9

Mo MEAM < 1000 −− −− −− 8.6Mo MEAM† 2778.0± 10.1 153 91.98 97.28 5.3Mo FS 3062.6± 7.6 284 91.85 100.75 2.9Mo Exp 2896 290 98.10 110.52 4.8Ta MEAM < 1000 −− −− −− 9.1Ta MEAM† 2884.3± 7.9 115 102.56 109.75 5.7Ta FS 3935.7± 6.7 190 104.8 113.54 6.3Ta Exp 3290 174 111.26 122.48 6.3W MEAM < 1000 −− −− −− 6.1W MEAM† 4389.0± 9.1 161 106.24 112.98 4.2W FS 4125.6± 8.0 184 103.81 112.03 3.9W Exp 3695 192 108.90 118.52 4.5

(EDIP) of Si [201]. This implies the difficulties in describing the solid phase and

liquid phase by a single empirical model due to their fundamentally different bonding

mechanisms: The former is a low coordination semiconductor and the latter is an


intermediate coordination metal.

A point of concern is that the MEAM potential predicts a thermal expansion

coefficient (at room temperature) that is 3 to 5 times larger than experimental values.

It is possible that by adding a short range potential between Si atoms, both the

melting point and thermal expansion coefficients of the MEAM-Si model may be

improved [211]. This possibility will be explored in a future publication.

D.2.2 FCC Metals: Au, Cu, Ag and Pb

The performance of the MEAM model in FCC metals is generally satisfactory. When

comparison with the EAM model is available (Au and Cu), the MEAM model predicts

a melting point that is closer to the experimental data. However, the MEAM model

predicts a thermal expansion coefficient that is about 4 to 10 times smaller than

experimental data, exactly the opposite to the case of semiconductors.

Fortunately, by changing the angular screening factor of the MEAM potential

from the default value of cmin = 2.0 to cmin = 0.8, the thermal expansion coefficient

is greatly improved, as shown in the MEAM∗ entries in Table D.1. This modification

also improves the accuracy of latent heat and entropies of solid and liquid. The

generalized stacking fault, an important property for dislocation modelling, is also

significantly improved when cmin is changed to 0.8 [212].

Hence we suggest that the MEAM model for FCC metals can be generally im-

proved by reducing its angular screening parameter cmin. The corresponding decrease

of melting point may be compensated by adding a short range potential. This hy-

pothesis will be tested in the Appendix E.

D.2.3 BCC Metals: Mo, Ta and W

MEAM and Finnis-Sinclair (FS) models are examined for three typical BCC metals

(Mo, Ta, W). Because BCC metals generally have much higher melting points than

semiconductors and FCC metals, the simulations here experience larger statistical

fluctuations, leading to larger error bars in the predicted melting points. The FS

model overestimates melting points of Mo, Ta, and W by about 10 ∼ 20%. The


latent heat, entropy, and thermal expansion coefficient are all in good agreement

with experimental values. Hence the FS model describe the thermal properties of

BCC metal very well.

Unfortunately, the original MEAM model seems to fail dramatically in the predic-

tion of thermal properties of BCC metals. For all three elements, the MEAM model

predicts that the liquid-phase Gibbs free energy stays lower than the solid-phase

Gibbs free energy even at temperatures down to 1000 K, whereas the experimental

temperature is around 3000 K. Due to the glass transition, we are not able to obtain

the true liquid free energy at temperatures lower than 1000 K. Therefore, we are not

able to determine the melting point of the MEAM model for these BCC metals.

Fortunately, the new MEAM model [209, 210] that includes the second nearest

neighbor interactions (2NN-MEAM) seems to be much more robust than the original

MEAM model. The melting points predicted by 2NN-MEAM fall within 20% of

experimental values. The thermal expansion coefficient also becomes much closer to

experimental values. Hence 2NN-MEAM is a better model for BCC metals than the

original MEAM model. It is interesting to notice that for the 2NN-MEAM model, the

angular cut-off parameter cmin is also much smaller than that in the original MEAM

model. Therefore, reducing cmin seems to improve the behavior of MEAM models for

both FCC and BCC metals.

D.3 Free Energy Method for Melting Point Cal-

culation

Because the melting point is defined as the temperature at which Gibbs free energies

of the solid and liquid phases equal to each other, the melting point can be deterem-

ined if we know the Gibbs free energies of the two phases as functions of temperature

accurately in the neighborhood of the melting point. Since the first calculation of

melting point of Si by the free energy method two decade ago [198], several advanced

methods have been developed, such as the adiabatic switching and reversible scal-

ing [199, 200], which has made free energy calculations much more efficient. Using


these state-of-the-art methods, we find that the melting points can be obtained to a

much higher accuracy (e.g. ±1 K) than that achievable by the co-existence method.

To achieve such a high accuracy, it is important to choose carefully the beginning and

end states of the switch, as well as the switching paths, in order to reduce statistical

and systematic error in every step of computation. Because many independent free

energies need to be computed to determine the melting point, a large error in any of

these steps can undermine the overall accuracy.

Our approach to compute the melting point Tm of a pure element can be described

by the following steps.

1. Pick a temperature T1 lower than the estimated value of Tm. Find the equilib-

rium volume V1 of the crystalline solid at T1 by an MD simulation under the

NPT ensemble.

2. Determine the Helmholtz free energy Fs of the solid phase at V1 and T1. This

is done by adiabatic switching from the solid phase described by the actual

potential model to the harmonic approximation of the same potential function.

Since V1 is the equilibrium volume, i.e. pressure P = 0, Fs(T1, V1) equals to the

Gibbs free energy Gs(T1) at zero pressure.

3. Obtain the Gibbs free energy, Gs(T ), of solid phase as a function of temperature

using the reversible scaling method in the domain of T1 < T < T2, where T2 is

expected to be higher than Tm.

4. Find the equilibrium volume V2 of the liquid phase at T2 by an MD simulation

under the NPT ensemble.

5. Determine the Helmholtz free energy FL of the liquid phase at V2 and T2. This

is done by adiabatic switching from the liquid to a purely repulstive potential

and then to the ideal gas limit. Again FL(T2, V2) equals to the Gibbs free energy

GL(T2) at zero pressure.

6. Obtain the Gibbs free energy of the liquid phase as a function of temperature

using the reversible scaling method, GL(T ), in the domain of T1 < T < T2.


7. Plot GS(T ) and GL(T ) together and determine the melting temperature at

which the two curves cross.

All simulations are performed by Molecular Dynamics under periodic boundary con-

ditions in three directions and with a time step of ∆t = 0.1 fs. Every switching

simulation has the duration of 100 ps unless otherwise mentioned. Si and Ge are

modelled using a supercell with 512 atoms that is 5× 5× 5 times of a diamond-cubic

unit cell. Au, Cu, Ag, and Pb are modelled using a supercell with 500 atoms that is

4 × 4 × 4 times of an FCC unit cell. Mo, Ta and W are modelled using a supercell

with 432 atoms that is 6× 6× 6 times of a BCC unit cell.

The most challenging part of this work is probably to correctly assemble the results

from many different kinds of calculations. Fortunately, this has been automated in

the MD++ program in the form of an input script file [202]. In the following, we

describe the important details for the different steps of our calculations.

D.3.1 Solid Free Energy

The Helmholtz free energy F of a system of N atoms that can be described by a

Hamiltonian H(ri,pi) is defined by through the partition function Z,

e−βF = Z =1

N !h3N

∫ N∏

i=1

dridpi e−βH(ri,pi) (D.1)

where h is Planck’s constant, β = 1/(kBT ), T is temperature and kB is Boltzmann’s

constant. Free energy is difficult to calculate because it cannot be expressed as an

ensemble average, such as total energy, which can then be computed by MD or MC

simulations as a time average. On the other hand, the free energy difference between

two systems can be expressed in terms of an average. Hence, free energy can be

computed from the difference between the free energy of the system of interest and

that of a reference system whose free energy is known a priori. The computation

is most efficient when the reference system is very similar to the original system of

interest [113].

A widely used reference system is the Einstein crystal, in which every atom is


represented by an indepdent harmonic oscillator vibrating around its perfect lattice

positions [213]. However, a reference system that is even closer to the original system

is the harmonic approximation of the interatomic potential itself. In this work, we use

the Quasi-Harmonic-Approximation (QHA) as the reference system, whose potential

function is the Taylor expansion of the original potential function up to second order

around the equilibrium lattice positions at the given temperature T1 (i.e. allowing

thermal expansion). The free energy of the reference system is obtained by first

computing the Hessian matrix, which is the second derivatives of the potential energy

function with respect to atomic coordinates, and diagonalizing it. Let Λi be the

eigenvalues of the Hessian matrix. The eigen-frequencies of the normal modes of the

crystal are ωi =√

Λi/m, where m is the atomic mass. The Helmholtz free energy of

the reference system is

FQHA(N, V1, T1) = E0(V1)− kBT1

∑

i

lnkBT1

h wi(V1)(D.2)

where h ≡ h/(2π).

The Helmholtz free energy difference between the QHA reference system and the

real potential at T1 and V1 is computed by the adiabatic switching (AS) method [199].

SupposeH1 is the Hamiltonian of the system of interest andH0 is the reference system.

We define a new Hamiltonian parameterized by λ,

H(λ) = (1− λ)H0 + λH1 (D.3)

such that H(λ = 0) = H0 and H(λ = 1) = H1. During the AS simulation, λ

gradually changes from 0 to 1, and the Hamiltonian gradually changes from the

reference system to the system of interest. The work done during the switching, ∆W ,

provides an estimator to the free energy difference, i.e.,

F1 − F0 = ∆W ≡∫ tsw

0

∂H(λ)

∂λ

dλ(t)

d tdt (D.4)

where tsw is the total time of the switching simulation. Strictly speaking, the equality


(F1 − F0 = ∆W ) is valid only in the limit of infinitely slow switching, i.e. tsw → ∞.

For any switching performed at a finite rate, ∆W contains both statistical and sys-

tematic error. The systematic error is caused by dissipation in an irreversible process,

which makes the averaged work over many independent switching trajectories, 〈∆W 〉,greater than F1−F0 [214]. To reduce statistical error generated from finite switching

time, we employed the switching function

λ(t) = s5(70s4 − 315s3 + 540s2 − 420s+ 126) (D.5)

where s = t/tsw. This switching function makes the increase rate of λ very low both

at the beginning and at the end of the switching trajectory where the fluctuation∂H(λ)∂λ

= H1−H0 tend to be largest [113]. The switching function is also very smooth,

which was found to be important for error reduction [199].

The following details are important for the calculations of Helmholtz free energy

of the solid phase at a given temperature.

1. There are always three zero eigen-frequencies corresponding to the three rigid-

translational modes. This means that the sum over i in Eq. (D.2) should include

3(N − 1) terms. To be self-consistent, similar considerations are needed in the

calculation of the free energy of the ideal gas reference system (see next section).

2. The Nose-Hoover chain method [118] is needed in the MD simulation to ensure

ergodicity since the Hamiltonian is very close to that of a harmonic system.

3. Reverse switching simulations are required to estimate and to cancel the dissi-

pation [113]. Prior to each switching simulation, it is important to equilibrate

the system for a long enough time.

Since the solid is under zero pressure at T1 and V1, the Helmholtz free energy

Fs(T1, V1) is also the Gibbs free energy GS(T1) at pressure P = 0. In the following,

we will omit P in the Gibbs free energy expression, since the latter is always evaluated

at zero pressure in this work. The reversible scaling method is used to compute the

Gibbs free energy as a function of temperature [200]. The key idea is to multiply

the potential energy function U by a parameter λ(t), which changes smoothly with


time t during the switching simulation. The simulation is performed at a constant

temperature T0. But the work done to the system during the switching simulation

can be used to extract the free energy of the original system (with potential function

U) at a range of temperatures T = T0/λ(t). The following details are important for

a successful calculation.

1. The NPT ensemble is required to ensure zero pressure during the switch simu-

lation.

2. The Nose-Hoover chain method is required to ensure ergodicity.

3. Reverse switching should be performed to estimate and cancel the dissipation.

4. The range of λ(t) should be limited to avoid large dissipation. This means

that if the initial guess T1 is too far away from the predicted melting point

Tm, we need to repeat the previous step (compute solid free energy with quasi-

Harmonic-Approximation) at a different temperature T1 that is closer to Tm, in

order to reduce the error bar.

D.3.2 Liquid Free Energy

The ideal gas is used as the reference system to compute the Helmholtz free energy

of the liquid phase at temperature T2 and volume V2. The Helmholtz free energy of

N ideal gas particles is

Fi.g.(N, V2, T2) = −kBT2N ln(V2/Λ3)− lnN ! (D.6)

where V2 is equilbrium volume of liquid at T2 and Λ ≡ h/√2πmkBT is the thermal de

Brogile wave length. It is important to point out that we need to replace N by N −1

when using the above equation to compute the free energy of the reference system,

due to the fixed center of mass in atomistic simulations (see previous section).

To minimize dissipation which causes a systematic error in the switching simula-

tion, we should always avoid crossing any phase transition line during the adiabatic

switching. It is generally expected that a direct switching path from a liquid phase


to an ideal gas will cross the liquid-gas transition line. To avoid this, we first switch

the liquid to an intermediate reference system and then switch to the ideal gas limit.

The intermediate reference system is a collection of N particles interacting through

a purely repulsive pair potential of the following Gaussian form

φ(ri, rj) = λ ǫ exp

(−|ri − rj|2σ

)

. (D.7)

ǫ and σ are adjusted to minimize dissipation occurring when switching to and from

the real potential model.

We find that the Gaussian potential is a better reference system than the inverse-

12 potential (i.e. 1/r12) used in the literature [215]. Even though the free energy

of the inverse-12 potential is available in analytic form as a Virial expansion, the

expansion may not converge within 10 terms at the density of the silicon liquid.

While the free energy of the Gaussian potential liquid is not known analytically, it

can be easily computed by adiabatic switching to the ideal gas limit. Because the

Gaussian potential is very simple, the computational cost required in this step is

negligible compared to other steps where the real potential model (e.g. SW, MEAM)

is required. This enable us to break the switching path into many smaller steps,

reducing the systematic error caused by the large difference between the beginning

and the end states. The lack of singularity of the Gaussian potential (in contrast to

the inverse-12 potential) also improves the numerical convergence. Because we never

observe a large dissipation (i.e. the total work in the forward and reverse switching)

in our simulations, this can be taken as an empirical proof that we did not cross any

phase-transition line along the switching path.

The following details are important for the calculations of Helmholtz free energy

of the liquid phase at a given temperature.

1. Switching from the Gaussian potential to the ideal gas limit must be performed

in several steps for accuracy if a linear switching function is used. We multiply

the potential energy U by a parameter λ. λ = 1 is the original fluid with the

Gaussian potential and λ = 0 is the idea gas limit. In our simulation, we switch

from λ = 1 to λ = 10−6 in 6 steps, each time reducing λ by a factor of 10.


Further reducing of λ produces a negligible amount of work, confirming that

the ideal gas limit has been reached.

2. To minimize dissipation and statistical error, we should adjust the parameters

of the Gaussian potential to match the characteristic distance and energy scale

of the real potential. For example, σ and ǫ can be adjusted to mimic the pair

potential part of the SW potential. ǫ of 50 eV and σ of 0.7A are used with

cutoff length rc = 3.771A for the case.

3. Reverse switching must be performed to estimate and correct for dissipation.

Since the liquid is under zero pressure at temperature T2 and volume V2, the

Helmholtz free energy FL(T2, V2) is also the Gibbs free energy GL(T2). The Gibbs

free energy of the liquid phase as a function of temperature is then obtained using the

same reversible scaling method as in the previous section. The Nose-Hoover chain

method is no longer needed for the simulation in the liquid phase because the system

is far away from being harmonic and ergodicity is satisfied.

D.3.3 Melting Point and Error Estimate

After obtaining the Gibbs free energies of both the solid and liquid phases, GS(T )

and GL(T ), in the temperature range of T1 < T < T2, we fit both data into smooth

spline functions and determine the point at which the two functions cross each other.

The temperature at which the two functions cross is the melting point Tm.

The error bar on Tm is computed from the errors in the free energy estimates

in the switching simulations. Each switching simulation Si (e.g. switching between

two Hamiltonians or switching along the temperature axis) is repeated n (∼ 10)

times, which results in n independent values of the irreversible work ∆W . Given

these works in both forward and reverse directions, the free energy difference between

the two systems is estimated using an extension of the Bennett’s acceptance ratio

method [216, 217]. This estimator was shown to be unbiased (i.e. with zero systematic

error) and to have the smallest statistical error. The free energy difference ∆Fi is


obtained by solving the following equations self-consistently,

e−β∆Fi =〈(1 + eβ∆Wi+C)−1〉F〈(1 + eβ∆Wi−C)−1〉R

eC (D.8)

C = −β∆Fi + lnnF/nR (D.9)

where nF and nR are the number of independent forward and reverse switching sim-

ulations [216, 217]. This expression is exact even for very rapid switching trajectories

where linear response theory is no longer valid. The standard deviation σi of the

estimated ∆Fi can be obtained from the following equation

β2 σ2i =

Var[(1 + eβWi+C)−1]FnF 〈(1 + eβWi+C)−1〉2F

+Var[(1 + eβWi−C)−1]RnR〈(1 + eβWi−C)−1〉2R

(D.10)

where Var[x] is the variance of the random variable x.

In this work, the melting point is estimated fromm = 5 different types of switching

simulations. Assuming the error made in each switching simulation is independent of

each other, the error bar for the Gibbs’s free energy difference between the solid and

the liquid phases is estimated by,

σ(∆G) =

(

m∑

i=1

(σi)2

)1/2

(D.11)

The error bar in the melting point prediction is

σ(Tm) =σ(∆G)

SL − SS

(D.12)

where SL and SS are the entropy of the liquid and solid phases at melting point,

respectively. The entropies can be obtained from the slope of the Gibbs free energy

– temperature curve.

An example is given in Fig. D.1, which plots the Gibbs free energy as a function

of temperature for the liquid and solid phases of Si, as described by the SW potential.

The dominant source of error comes from the switching simulation between the SW

liquid to the Gaussian fluid (See Section D.5 for more details). This error contributes


1500 1600 1700 1800−5.1

−5

−4.9

−4.8

−4.7

−4.6

Temperature (K)

Fre

e E

nerg

y (e

V/a

tom

)

solidliquidsolidliquid

Figure D.1: Gibb’s free energy per atom for both the solid phase (solid line) and liquidphase (dashed line). The symbols represent data points in Broughton and Li [198]with squares for the solid phase and circles for the liquid phase.

to an uncertainly of 0.884 × 10−4 eV/atom in the liquid Gibbs free energy, which

corresponds to melting temperature uncertainty of 0.46 K. Semiconductors and FCC

metals studied here show similar error bars. The error bars of BCC metals are

considerably higher, most likely due to their high melting temperature, which leads

to larger statistical fluctuation.

Repeating each adiabatic switching simulation for n = 20 times usually brings

the error bar of the melting point to within ±1 K. The accuracy of this level can be

readily achieved in a day using a computer cluster with 30 CPUs. Due to limited

computational resource, enough computation is performed to reach ± 1 K accuracy

only for for MEAM-Si, SW-Si and MEAM-Au models. For other models, each adi-

abatic switching simulation is performed for only 5 or 6 times, leading to a larger

error bar in the predicted free energy. The finite size of the simulation cell and small

uncertainty in the determination of the equilibrium volume at finite temperature can

introduce additional error to the melting point, which is not accounted for in our

error estimate.


D.4 Summary

We have computed the melting points, latent heat, entropy and thermal expansion

coefficients for nine pure elements described by four different interatomic potential

models. The state-of-the-art free energy methods are used to determine the melt-

ing points accurately and efficiently, allowing a systematic comparison between the

potential models. The beginning and end states and the switching paths are chosen

carefully in the adiabatic switching simulations to reduce the error in the free energy

calculation. The comparison reveals several systematic trends among elements with

the same crystal structure. The MEAM model performs reasonably well in semicon-

ductors compared with the SW model, and predicts more accurate thermal properties

than the EAM model, especially the angular screening factor is adjusted. The original

MEAM model fails to predict reasonable thermodynamic properties for BCC metals.

In comparison, the FS model and the 2NN-MEAM model are more reliable for BCC

metals.

D.5 Error Estimates in Free Energy Calculations

Here, we present some intermediate free energy results of our melting point calcula-

tions. The purpose is two-fold. First, it will enable interested readers to compare

their results with ours, should they wish to adopt our computational method. Sec-

ond, it demonstrates which step is the major source of error in the final estimate of

the melting point. This allows further improvement of the accuracy and efficiency

of melting point calculations in the future. The average and standard deviation of

the reversible work accumulated in each of the 5 adiabatic switching steps (counting

forward and backward switching together) are listed in Table D.2.

Step 1 corresponds to adiabatic switching from a solid phase described by an

empirical potential to the quasi-harmonic approximation of itself. Step 2 corresponds

to switching the solid phase along the temperature axis from T1 to T1/λ1. T1 = 1600

and λ1 = 0.8 are used for SW Si and T1 = 1200 and λ1 = 0.75 are used for MEAM

Si. Step 3 corresponds to adiabatic switching from a liquid phase described by an


empirical potential to the purely repulsive liquid described by the Gaussian potential.

Step 4 corresponds to adiabatic switching from the Gaussian potential to the ideal

gas limit. Step 5 corresponds to switching the liquid phase along the temperature

axis from T2 to T2/λ2. T2 = 1800 and λ2 = 1.3 are used for SW Si and T2 = 1560 and

λ2 = 1.3 are used for MEAM Si. Table D.2 shows that the intermediate results are

similar for the SW-Si and the MEAM-Si models, both in terms of the average free

energy differences and in terms of the error bars. The major source of error comes

from the switching between the liquid phase and the purely repulsive liquid described

by the Gaussian potential (Step 3).

Table D.2: The estimated free energy difference ∆Fi and its standard deviation inthe 5 different adiabatic switching steps for the melting point calculations of SW-Siand MEAM-Si models.

SW-Si MEAM-Sii ∆Fi (eV/atom) σi (10

−4eV/atom) ∆Fi (eV/atom) σi (10−4eV/atom)

1 -0.01297 0.01 -0.00624 0.072 0.81616 0.05 1.10544 0.033 -3.68733 0.88 -3.94027 1.014 0.50080 0.45 0.41005 0.475 -1.13876 0.19 -1.21368 0.26

Appendix E

Improved Modified

Embedded-Atom Method

Potentials for Gold and Silicon

E.1 Introduction

The directed growth of semiconductor nanowires catalyzed by gold nano-particles

via the vapor-liquid-solid (VLS) mechanism has attracted a lot of interest worldwide

as a promising way to build nanoscale electronic, optical and chemical-sensing de-

vices [218, 219]. Atomistic simulations of nanowire nucleation and growth by the

VLS mechanism have been hindered by the lack of a computationally efficient model

that reliably describes the interaction between semiconductor (such as silicon) and

metal (such as gold) atoms. The tip of the growing nanowire typically contains thou-

sands of atoms, making it prohibitively expensive for first-principles models. Most of

the empirical potential models use different functional forms to describe metallic and

covalent bonds. Therefore, these models are applicable to either metals or semicon-

ductors, but not both. For example, the embedded-atom method (EAM) [109] model

captures the many-body effect in metals by a function that describes the energy re-

quired to embed an atom in the background electron density created by its neighbors.

In comparison, the Stillinger-Weber (SW) [105] and Tersoff [106] potentials for silicon

160

APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 161

use terms that include the angle between two bonds to capture the directional nature

of covalent bonds.

The modified embedded-atom method (MEAM) model was proposed to describe

both metallic and covalent bonds within the same functional form [206]. It extends

the EAM to include directional bonding by accounting for the spatial distribution of

background electron density of neighboring atoms. The MEAM potential has been

developed for a large number of metals as well as covalent semiconductors. MEAM

potentials for many binary alloys have also been developed [220, 221, 222]. The

MEAM model has many attractive properties for modelling nanowire growth. For

example, it successfully describes the change of the coordination of Si atoms from

4-fold (covalent bonding) to 6-fold (metallic bonding) at melting [196], as well as

the surface energy [223, 224] and surface segregation [225, 226] in metals. Therefore,

a promising approach to enable atomistic modelling of VLS growth is to develop a

MEAM potential that can be fitted to the binary gold-silicon phase diagram. Given

that MEAM potentials for pure gold and pure silicon have already been developed, in

principle, we only need to develop the cross-potential between gold and silicon atoms.

However, we found that the existing MEAM models of pure gold and pure silicon

have several problems that can adversely affect the binary phase diagram and the

modelling of VLS growth. In particular, the MEAM potentials underestimate the

melting points of both gold and silicon by about 200 K. There are also significant

errors in the prediction of the latent heat, which provides the driving force for the

liquid-solid phase transition. This is not surprising because the original MEAM po-

tentials were not fitted to the melting point and the latent heat. The purpose of

this study is to improve the melting point and the latent heat predictions of existing

MEAM models of gold and silicon by fine-tuning their parameters without chang-

ing the overall functional form. In addition, we monitor the predictions of several

other transport and mechanical properties that may influence various stages of the

nanowire nucleation/growth/termination process, including the diffusion coefficient,

thermal expansion coefficient, generalized stacking fault energy, and volume change

on melting. We make an effort to improve the predictions of these properties in the

development of the new MEAM potential. However, when a compromise has to be


made, we choose to fit the melting point and the latent heat more accurately at the

expense of other properties. In Appendix F, we will present an MEAM gold-silicon

cross-potential based on the improved MEAM potentials developed here.

Appendix E is organized as follows. In Section E.2, we present the limitations of

existing MEAM potentials for gold and silicon in their predictions of various thermal

and mechanical properties. In Section E.3, we describe our methods to improve the

MEAM potentials and present the results. A brief summary is given in Section E.4.

E.2 Problem Statement

The main goal of this study is to adjust the MEAM potentials for gold and sil-

icon to accurately reproduce the melting point and the latent heat. During this

process, we also monitor the predictions of the diffusion coefficient, thermal expan-

sion coefficient, generalized stacking fault energy, and volume change on melting.

Because these properties may also influence various stages of the nanowire nucle-

ation/growth/termination process, we will try to improve them, but will emphasize

the melting point and the latent heat when a compromise is necessary. In this section,

we benchmark the original MEAM potentials for gold [206] and silicon [206], as well

as the more recent second nearest-neighbor (2nn) MEAM potentials for gold [227]

and silicon [228]. First, we give a brief summary of the MEAM model.

The MEAM model describes the potential energy of a collection of atoms located

at ri, i = 1, · · · , N , by the following equation,

V (ri) =N∑

i=1

F (ρi) +

N−1∑

i=1

N∑

j=i+1

Sij φij(|ri − rj|) (E.1)

where F is the embedding function, ρi is the background electron density at ri, Sij

is a multi-body screening factor and φij is the pair potential between atoms i and j.

The pair potential function φij(r) is usually not given explicitly. Instead, it is defined

as the function that, when combined with the embedding function, reproduces the

universal equation of state (EOS) [229] for the potential energy of the reference crystal

structure.


Table E.1: Model predictions and experimental data [207, 230, 231] on thermal andmechanical properties of gold. The computation models include original MEAM [206],EAM [204], 2nn-MEAM [227] and two modifications made in this study (2nn-MEAM∗

and 2nn-MEAM†), and first-principles calculation with DFT/LDA [232]. The prop-erties include the melting point Tm (in K), latent heat of fusion L (in kJ mol−1),solid and liquid entropy at the melting point, SS and SL (in J mol−1K−1), diffusionconstant of the liquid D at Tm (in 10−9m2s−1), thermal expansion coefficient α ofthe solid (in 10−6K−1) at 300 K, ideal shear strength τc (in GPa) of the solid at zerotemperature, and volume change on melting ∆Vm/Vsolid (in %). Statistical errors areon the order of the last digit of the presented values.

Au Tm L SS SL D α τc ∆Vm/Vsolid

MEAM [206] 1120 18.2 77.5 93.7 0.6 2.0 10.7 6.6EAM [204] 984 8.2 85.7 94.0 1.5 13.5 1.68 1.6

2nn-MEAM [227] 1405 18.0 87.4 100.1 2.3 14.6 4.92 7.82nn-MEAM∗ 1173 11.2 87.7 97.2 2.2 20.8 1.81 5.12nn-MEAM† 1337 14.2 87.7 98.3 2.3 17.8 2.71 6.5

Exp’t 1337.3 12.6 88.4 97.8 – 14.2 – 5.1DFT/LDA – – – – – – 1.73

While the above functional form is similar to that of EAM [109], MEAM has two main

extensions. First, the calculation of the background electron density ρi in MEAM

accounts for the spatial arrangements of the neighboring atoms, in addition to their

distance to atom i. Second, the range of the pair potential is cut-off by a multi-body

screening function Sij that depends on the locations of atoms k that are neighbors of

both atoms i and j. The details of the MEAM formalism are well described in the

literature [206]. The multi-body screening function Sij is summarized in E.5 because

we need to adjust it in this work.

E.2.1 Limitations of the MEAM Gold Potential

The original MEAM gold potential [206] and 2nn-MEAM gold potential [227] are

fitted to two different sets of experimental elastic constants with about 10% of dis-

crepancy. We choose the more recent 2nn-MEAM gold potential as the starting point


of our model. Table E.1 shows that the 2nn-MEAM gold potential [227] overesti-

mates the melting point Tm by ∼70 K (5%) and overestimates the latent heat L by

∼6 kJ/mol (43%). These are the main problems that we will address in this study.

The melting point of pure gold is an important feature of the binary gold-silicon phase

diagram. At the same time, the latent heat provides the thermodynamic driving force

for melting and crystallization when the temperature deviates from the melting point.

Because L = Tm · (SL − SS), it is useful to monitor the entropies of the liquid phase

(SS) and the solid phase (SL) at the melting point. For the original MEAM potential

model, the overestimate of L is largely due to the underestimate of SS (by 12%),

which suggests that the gold crystal model is too “rigid”. In other words, the an-

harmonic effect is not large enough to increase the entropy of the solid phase at high

temperature. This is much more improved in the 2nn-MEAM model which uses a

smoother many body screening function. While the melting point of the more recent

2nn-MEAM potential for gold is much closer to the experimental value, the error in

the latent heat is still similar to that of the original MEAM potential. We would

like to have a better fit to both the melting point and the latent heat by adjusting

2nn-MEAM gold potential.

Table E.1 also lists the predicted values of several other properties that either

are important for certain aspects of nanowire growth, or are included to illustrate

the problem of MEAM potentials. For example, diffusion in the liquid phase may

be the rate limiting step in nanowire growth under certain experimental conditions.

The mismatch in the thermal expansion coefficient α of gold and silicon can generate

internal stress during rapid heating or cooling of the nanowire. The gross underes-

timate of α in solid gold confirms the above hypothesis that the aharmonic effect is

not large enough in the original MEAM potential. The ideal shear strength τc can

influence the probability of twinning in the gold crystal when the catalyst droplet so-

lidifies during rapid cooling. The volume change on melting ∆Vm/Vsolid can generate

internal stresses upon melting.

Artifacts of both original and 2nn MEAM potential is most clearly shown by the

(relaxed) generalized stacking fault (GSF) energy on the (111) face in the [112] shear

direction, as shown in Fig. E.1(a) and (b). We followed the procedures in [233] to


compute the GSF curves. The unphysical local minima and cusps on the GSF curve

indicate problems for these MEAM potentials, because the GSF curve is expected to

be a smooth curve, as predicted by ab initio calculations using the density functional

theory (DFT) shown in Fig. E.1(b). The maximum slope of the generalized stacking

fault curve between zero and the first local minimum is defined as the ideal shear

strength τc. 2nn-MEAM prediction of τc is about a factor of three too high compared

with ab initio results. The overestimate of L and τc suggests a fundamental problem

in the 2nn-MEAM potential for gold, which we will address in Section E.3.

(a) (b)

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Sheared Distance, d/|b| where b = [1 1 2]/2

Γ (e

V/A

ngst

rom

2 )

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05


Γ (e

V/A

ngst

rom

2 )

Figure E.1: (Color Online) Generalized stacking fault energy of different potentialmodels for gold: (a) 2nn-MEAM [227] (dashed line), 2nn-MEAM∗ (dotted line), and2nn-MEAM† (solid line); (b) EAM [204] (dashed line), MEAM [206] (dotted line) andDFT/LDA (solid line).

E.2.2 Limitations of MEAM Silicon Potentials

Table E.2 shows that the original MEAM silicon potential [206] underestimates the

melting point Tm by∼270 K (16%) and underestimates the latent heat L by∼13 kJ/mol

(27%). These are the main problems that we will address in this study. There is er-

ror cancellation in the prediction of latent heat L, because both solid entropy SS

and liquid entropy SL are underestimated. Contrary to the case of gold, the more

recent 2nn-MEAM potential for silicon predicts a much more accurate latent heat

but grossly overestimates the melting point (by ∼1000 K). We would like to have a

better fit to both the melting point and the latent heat.


Table E.2: Model predictions and experimental data [207, 230, 196] of thermal andmechanical properties of silicon. The ideal shear strength [233] obtained from firstprinciple calculation is also presented for comparison. The computational modelsinclude the original MEAM [206] and MEAM second nearest-neighbor (2nn) [228].Superscripts ∗ and † represent modifications to these models introduced in this work.SW [105] and Tersoff [106] models are also included for comparison. The variablesand their units are identical to those in Table E.1.

Si Tm L SS SL D α τc ∆Vm/Vsolid

MEAM [206] 1411 36.8 48.7 74.8 4.8 13.6 13.8 −5.3MEAM∗ 1377 36.7 48.0 74.7 6.3 13.6 13.8 −5.2MEAM† 1687 43.1 53.7 79.2 7.9 13.2 14.0 −2.7SW 1695 31.2 58.0 76.5 6.0 3.9 8.9 −6.1

Tersoff 2606 42.3 68.9 85.1 2.5 6.5 18.5 −1.62nn-MEAM [228] 2837 49.3 72.9 90.3 6.25 5.2 13.6 −7.5

Exp’t 1685 50.2 61.8 91.6 ∼ 10 2.6 −− −5.1DFT/LDA(GGA) – – – – – – 14.0(13.7) –

Table E.2 also lists the predicted values of several other properties that are in-

cluded in the benchmark of gold potentials in Table E.1. Unlike the case of gold,

the underestimate of the solid entropy by the MEAM potential [206] is accompa-

nied by a gross overestimate (instead of an underestimate) of the thermal expansion

coefficient α. Unfortunately, this is a problem that we are not able to fix in this

study (See Section E.3). Fig. E.2(a) plots the (relaxed) generalized stacking fault

energy of silicon on the shuffle-set (111) plane in the [110] shearing direction. While

the original MEAM potential predicts some unphysical oscillations, the local minima

and cusps in the gold GSF curve is no longer present. The unphysical oscillations

become even more severe in the 2nn-MEAM potential [228], indicating an underlying

problem. The maximum slope of the GSF curve is the ideal shear strength τc, the

MEAM prediction of which agrees well with ab initio results. Table E.2 also lists

the benchmark data for two other commonly used potentials: Stillinger-Weber (SW)

and Tersoff. None of the existing potentials can describe both the melting point and

the latent heat accurately. This indicates that it is very difficult to describe both


the solid phase and the liquid phase of silicon using an empirical potential, because

silicon transforms from a semiconductor to a metal when it melts. In Section E.3, we

describe our approach to improve the MEAM potential for silicon.

(a) (b)

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14


Γ (e

V/A

ngst

rom

2 )

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14


Γ (e

V/A

ngst

rom

2 )(c) (d)

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14


Γ (e

V/A

ngst

rom

2 )

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14


Γ (e

V/A

ngst

rom

2 )

Figure E.2: (Color Online) Generalized stacking fault energy of different potentialmodels for silicon. (a) MEAM [206](dashed curve), MEAM∗ (dotted curve), MEAM†

(solid curve). (b) 2nn-MEAM [228](dashed curve). (c) Tersoff [106](solid curve),SW [105] (dashed curve) (d) DFT/LDA [233] (solid curve) ,DFT/GGA [233] (dashedcurve)

E.3 Methods and Results

The main goal of this study is to improve the melting point and the latent heat

predictions of existing MEAM potentials of gold and silicon, by adjusting some of their

parameters without changing the overall functional form. The parameters we will

adjust should not affect most of the previously fitted properties, such as equilibrium


lattice constant, cohesive energy, and elastic constants. In this work, we choose

to adjust (1) the multi-body screening function and (2) the pair potential function

through the equation of state (EOS). Because the dependence of thermal properties

on these parameters is not obvious, the adjustment of the potential proceeds by

trial-and-error, which requires rapid calculation of the melting point and the latent

heat for each trial potential. This is done using the free-energy calculation method

described in Appendix D. Many physical properties are monitored during the process.

Diffusion coefficient D, thermal expansion coefficient α, volume change on melting

∆Vm/Vsolid and radial distribution function g(r) are computed by using NPT MD

simulations [196]. First, by adjusting the parameter Cmin in the multi-body screening

function, we obtain the potentials for gold and silicon that are marked as MEAM∗ in

Table E.1 and E.2. Second, by modifying the EOS, we obtain the potentials marked as

MEAM† that have much better melting point and latent heat predictions. Some other

transport and mechanical properties are also improved by the MEAM† models. In

the Appendix F, we will present a gold-silicon cross potentials based on the MEAM†

potentials presented here.

E.3.1 Multi-body Screening Function

In this section, we will discuss the physical basis for the need to change the Cmin

parameter in the multi-body screening function from its original value of 2.0 to a

lower value of 0.8 for Au and 1.85 for Si. The multi-body screening function in

MEAM allows the interaction between two atoms i and k to be screened by a third

atom j. The behavior of the multi-body screening function is controlled by two

parameters: Cmin and Cmax (for more details see E.5). The default value of Cmax is

2.8, which ensures that first nearest-neighbor interactions are completely unscreened

even for reasonably large thermal vibration. In the original MEAM model, Cmin is set

to 2.0. As a result, in most crystal structures the second nearest-neighbor interactions

are always screened by another atom. Hence the interactions are effectively cut-off

to within the first nearest-neighbors. In the 2nn-MEAM model, Cmin = 1.53 for Au

and Cmin = 1.41 for Si are used to improve properties such as the thermal expansion


coefficient and vacancy formation energy. However, as shown in the Fig. E.1(a) and

the Fig. E.2(b), it still has problem in predicting the generalized stacking fault energy

curve.

Lowering Cmin to 0.8 effectively unscreen the second nearest-neighbor interactions,

and has been found to improve several predictions of the MEAM model for FCC Ni,

including the thermal expansion coefficient α [234]. The beneficial effects of this ad-

justment have also been reported for MEAM models of other FCC metals and BCC

metals as shown in Appendix app:RyuTm. For example, the original MEAM model

for BCC metals predicts an incorrect ordering of the surface energies of low index

surfaces. It also predicts another structure to be more stable than BCC [235], as well

as anomalously low melting points. By lowering Cmin, which effectively extends the

interaction to second nearest-neighbors, the predictions of surface energy, thermal ex-

pansion coefficient [235, 236] as well as melting pointsof BCC metals are significantly

improved.

Therefore, our first step is to set Cmin = 0.8 in gold 2nn-MEAM potential and

Cmin = 1.85 in silicon MEAM potential. The resulting potentials are indicated by su-

perscript ∗ in Table E.1 and E.2 (other parameters are not changed). The latent heat

L and the volume change on melting ∆Vm/Vsolid for the 2nn-MEAM∗ gold potential

are significantly improved, indicating that the model now describes the physics of

gold more accurately. The artificial metastable states and cusps in the GSF curve

are completely removed (dotted line in Fig. E.1(a)). The maximum slope of the GSF

curve, i.e. the ideal shear strength τc, becomes very close to the ab initio result. We

find that these artificial features persist if Cmin > 0.8, whereas elastic constants and

vacancy formation energy will start to change if Cmin < 0.8. Hence Cmin = 0.8 seems

to be an optimal choice for MEAM gold. However, setting Cmin = 0.8 significantly

decreases the melting point (now about 160 K lower than experimental value) and

increase the thermal expansion coefficient (now about 40% bigger than experimental

value). This problem will be addressed in the next section by adjusting the pair

potential function.

On the other hand, changing Cmin for Si has a negligible effect on melting point Tm,

latent heat L, solid entropy SS, liquid entropy SL and thermal expansion coefficient


α. This is because the multi-body screening function has a very small effect in

the open structure of a diamond-cubic crystal, as explained in E.5. However, we

find that the liquid volume is selectively increased as Cmin is lowered, reducing the

magnitude of ∆Vm/Vsolid. Because ∆Vm/Vsolid is already close to experiment for

original MEAM Si, we lower Cmin just enough to remove the artificial oscillations in

the GSF curve (Fig. E.2(a)). The optimal value we converge to is Cmin = 1.85 for

silicon, corresponding to MEAM∗ in Table E.2. Hence, we will set Cmin = 0.8 for gold

and Cmin = 1.85 for silicon in the following discussions.

E.3.2 Pair Potential and Equation of State

In this section, we adjust the pair potential function of the MEAM potential to fit the

melting point and the latent heat of gold and silicon. As mentioned in Section E.2,

the pair potential in the MEAM models is not given explicitly, but is specified in

such a way that, when combined with the embedding function, it reproduces the

equation of state for the reference crystal structure. Hence modifying the pair poten-

tial amounts to changing the equation of state (EOS) function, which in the original

MEAM potential is expressed as,

Eu(r) = −Ec (1 + a∗) exp(−a∗) (E.2)

with a∗ =

(

9ΩB

Ec

)1/2 (r

re− 1

)

(E.3)

where r is the nearest-neighbor distance of the reference structure [229]. Ec is the co-

hesive energy, re is the equilibrium nearest-neighbor distance, Ω is the atomic volume,

and B is the bulk modulus of the reference structure.

Since the MEAM∗ potentials underestimate the melting points, we need to increase

the Gibbs free energy of the liquid phase more than that of the solid phase. An

effective way to modify the free energy difference between the two phases is to change

the EOS function at distances r where the radial distribution function g(r) of the

two phases differ the most. At the same time, the modified EOS function must

not change its value and first and second derivatives at r = re because they have


been fitted to physical properties. Specifically, Eu(re) = −Ec, dEu/dr|re = 0 and

d2Eu/dr2|re = 18ΩB/r2e . In the following, we will show that the same kind of

modifications of the EOS function can be applied to gold and silicon, in order to fit

their melting point and latent heat.

2nn-MEAM† gold potential

(a) (b) (c)

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

r(Angstrom)

G(r

)

1 2 3 4 5

−4

−3

−2

−1

0B

indi

ng E

nerg

y (e

V)

Separation (Angstrom)1100 1200 1300 1400 1500 1600

−5.2

−5

−4.8

−4.6

−4.4

Temperature (K)

Fre

e E

nerg

y (e

V/a

tom

)Figure E.3: (Color Online) (a) Pair-correlation functions of the solid (solid line) andliquid (dotted line) phases of gold described by the 2nn-MEAM [227] potential at itsmelting point. (b) The equation of state function in the 2nn-MEAM (dotted line)potential and the new 2nn-MEAM† potential (solid line). (c) The Gibbs free energy ofthe 2nn-MEAM (thick lines) and 2nn-MEAM† (thin lines) potentials for gold. Solidlines for the solid phase and dashed lines for the liquid phase.

Fig. E.3(a) plots the radial distribution function g(r) for the solid and liquid

phases of gold using the 2nn-MEAM potential at the melting point. The g(r) for the

liquid phase is greater than that in the solid phase at distances between 3.5A and

4.5A. Hence the free energy of the liquid phase can be raised more than that of the

solid phase by increasing the pair potential energy in this range. A naive approach to

modify the EOS function is to add higher order polynomials in the pre-exponential

term. Unfortunately, this approach cannot introduce sufficient changes in the melting

point without creating unphysical oscillations in the EOS function. By trial and error,

we find that the form of the EOS function leads to the desired changes in the melting

point.

Eu(r) = −Ec

(

1 + a∗ + d a∗3 + γ a∗4e−λa∗2/r)

exp(−a∗) (E.4)


d a∗3 is small correction term added in 2nn-MEAM Au potential [227] with d = 0.05

A−3. The higher order term a∗4 is multiplied by an exponential e−λa∗2/r to remove

unphysical oscillations in the range of r > re. For each pair of trial values (γ, λ),

we re-compute the free energies of the liquid and solid phases using the method

described in Appendix D. It turns out to be very difficult to find a parameter set

that simultaneously fit the experimental melting point and latent heat. Hence a

compromise must be reached. The “optimal” values we converge to are γ = −0.182

A and λ = 4.0. The resulting EOS and free energies are plotted in Fig. E.3(b) and

(c), respectively. The thermal and mechanical properties of the resulting potential,

2nn-MEAM†, are listed in Table E.1. The melting point is very close to experimental

value, and the latent heat is significantly improved over the original 2nn-MEAM.

Unfortunately, the maximum slope of the GSF curve, i.e. the ideal shear strength τc,

is now about 60% bigger than ab initio results. The generalized stacking fault (GSF)

energy for the 2nn-MEAM† model is plotted in Fig. E.1(a). The 2nn-MEAM∗ model

has better GSF curves and volume change on melting than the 2nn-MEAM† model.

But the 2nn-MEAM† model has more accurate melting point and latent heat. We

believe the latter is a better candidate on top of which we can construct a gold-silicon

cross potential for the purpose of VLS nanowire growth modelling.

MEAM† silicon potential

Fig. E.4(a) plots the radial distribution function g(r) for the solid and liquid phases

of silicon using the original MEAM potential at the melting point. The g(r) of liquid

is higher than that of solid at distances between 3 A and 3.5 A. We found that the

following form of the EOS function leads to desirable changes of the melting point.

Eu(r) = −Ec

(

1 + a∗ + γ a∗4e−λa∗2/r)

exp(−a∗) (E.5)

where the “optimal” values we found are γ = −0.36 A and λ = 16.0. The resulting

EOS and free energies are plotted in Fig. E.4(b) and (c), respectively. The thermal

and mechanical properties of the resulting potential, MEAM†, are listed in Table E.2.

The melting point and the latent heat of the MEAM† Si potential are now both


(a) (b) (c)

0 2 4 6 8 100

1

2

3

4

5

6

r(Angstrom)

G(r

)

1 2 3 4 5

−4

−3

−2

−1

0

Bin

ding

Ene

rgy

(eV

)

Separation (Angstrom)1200 1400 1600 1800 2000

−5.4

−5.2

−5

−4.8

−4.6

Temperature (K)

Fre

e E

nerg

y (e

V/a

tom

)

Figure E.4: (Color Online) (a) Pair-correlation functions of the solid (solid line) andliquid (dotted line) phases of silicon described by the MEAM [206] potential at itsmelting point. (b) The equation of state function in the original MEAM (dotted line)potential and the new MEAM† potential (solid line). (c) The Gibbs free energy ofthe MEAM (thick lines) and MEAM† (thin lines) potentials for silicon. Solid linesfor the solid phase and dashed lines for the liquid phase.

close to the experimental values. The GSF curve for the MEAM† model, as shown in

Fig. E.2(c), becomes closer to ab initio results with negligible change in ideal strength

τc. The diffusion coefficient of liquid is also improved.

However, the MEAM† silicon potential still has problems on properties other than

the melting point and the latent heat. Both solid and liquid entropies are underesti-

mated; their cancellation of error leads to a better agreement of the latent heat with

experiments. The thermal expansion coefficient of solid remains too high, similar to

the original MEAM potential. The volume change on melting becomes a factor of two

smaller than experiments. In addition, the first and second highest peaks of the liquid

g(r) occurs at 2.7A and 4.2A, respectively, deviating from the experimental values of

2.50A and 3.78A [237]. This problem existed in the original MEAM model and our

modifications do not change the location of these peaks significantly. Nonetheless,

most of the problems listed above (except the entropies) do not affect the thermo-

dynamic behavior of the potential model. With a more accurate fit to the melting

point and the latent heat, the MEAM† silicon potential is a better candidate than

the original MEAM silicon potential, on top of which we can construct a gold-silicon

cross potential that can be fitted to the binary phase diagram. We have also applied


the same approach to adjust the 2nn-MEAM potential for silicon. Unfortunately, we

have not succeeded in reproducing both the melting point and the latent heat to the

same level of agreement with experiments as in the MEAM† potential.

E.4 Summary

We have adjusted the MEAM potentials for gold and silicon, two elements with

fundamentally different bonding mechanisms, to fit the melting point and the latent

heat more accurately, by changing the multi-body screening function and equation

of state function. For both gold and silicon, the melting point and the latent heat

values are now close to experimental values. The thermal expansion coefficient and

GSF curve for gold are significantly improved, mostly by changing Cmin to 0.8 in

the multi-body screening function. The thermal expansion coefficient for silicon is

insensitive to the adjustments considered in this study and remains too high compared

with experiments. In Section E.6 we provide more benchmark data of the modified

potentials. Table E.3 lists the elastic constants, defect energy and surface energies,

which are not very different from the original MEAM potentials. The bond angle

distribution in liquid silicon also agrees well with ab initio results. The resulting

MEAM† models for gold and silicon provide a good basis for constructing a cross-

potential model which can be fitted to the binary gold-silicon phase diagram. Since

we have not changed most of the MEAM parameters, it is possible that by fitting

these parameters again one may obtain a better agreement with experimental values.

E.5 Multi-body Screening Function

Here, we give a brief summary of the multi-body screening function in the MEAM

potential and describe why lowering Cmin has a beneficial effect on many structural

properties for Au. Consider a pair of atoms i and k. The interaction between the two

is screened by a factor Sik. If Sik = 1 then the interaction is not screened; if Sik = 0

then the interaction is completely screened. Sik depends on the distribution of atoms


j that are common neighbors of i and k,

Sik =∏

j 6=i,k

Sijk (E.6)

The functional form of Sijk is constructed based on the idea that, the closer atom j

is to the segment (i.e. bond) connecting atoms i and k, the more effective it screens

their interaction. On the plane that simultaneously contain atoms i, j and k, define

a local coordinate system x-y such that the origin lies at the mid-point between i and

k, and the x-axis points from i to k. Let rik be the distance between atoms i and k

and introduce scaled coordinates X = x/rik and Y = y/rik. Now imagine a series of

ellipses that pass through atoms i and k, as shown in Fig. E.5, and parameterized by

the following equation,

X2 +Y 2

C=

1

4(E.7)

The parameter C controls the extension of the ellipse along the y-axis. If the ellipse

pass through atom j, then

C =2(Xij +Xjk)− (Xij −Xjk)

2 − 1

1− (Xij −Xjk)2(E.8)

where Xij ≡ (rij/rik)2 and Xjk ≡ (rjk/rik)

2. Fig. E.5 plots the ellipses when C = 0.5,

C = 2.0 and C = 2.8. In MEAM, one specifies the screening factor in terms of

two parameters, Cmax and Cmin. When atoms j lies outside the ellipse defined by

C = Cmax, Sijk = 1; when atom j lies inside the ellipse defined by C = Cmin, Sijk = 0.

When atom j lies in between these two ellipses, Sijk is between 0 and 1 and is a

smooth function of the coordinates of atom j. This can be achieved by introducing a

cut-off function

fc(x) =

1 x ≥ 1

(1− (1− x)4)2 0 < x < 1

0 x ≤ 0

(E.9)

and let

Sijk = fc

(

C − Cmin

Cmax − Cmin

)

(E.10)


−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

X

Y

j

ki

C=0.8

C=2.0

C=2.8

Figure E.5: (Color Online) Ellipses defined in Eq. (E.7) for different values ofC (0.8,2.0,2.8). The line segments represent nearest-neighbor bonds i-j and j-k inBCC (square), FCC (circle), and Diamond-Cubic (asterisk) crystal structures, scaledby the second nearest-neighbor distance rik. In these three crystal structures, theatom j lies on the ellipses (not shown) corresponding to C = 0.5, 1.0 and 2.0, respec-tively.

The consequences of the choices of Cmax and Cmin can be see by analyzing the

geometry of perfect crystals. For example, consider an FCC lattice. If atoms i and

k are second nearest-neighbors, and atom j is their common first nearest-neighbor,

then C = 1.0. Hence the choice of Cmin = 2.0 in the original MEAM model means

that the interaction between second nearest-neighbors in FCC crystals is completely

screened. Unfortunately, cutting off the interaction between second nearest-neighbors

leads to cusps in the generalized stacking fault energy as well as a very small thermal

expansion coefficient. Reducing Cmin to 0.8 extends the interaction range beyond

second nearest-neighbors and removes these artifacts for FCC gold.

Now consider the diamond-cubic structure. If atoms i and k are second nearest-

neighbors, and atom j is there common first nearest-neighbor, then C = 0.5. Hence

second nearest-neighbor interactions in crystalline silicon is cut-off even for very large

thermal vibration at high temperatures. This is why lowering Cmin from 2.0 to 0.8


has hardly any effect for the solid phase of silicon. However, the properties of the

liquid phase is affected by this change because the interaction range between atoms

is effectively enlarged. In this study, we change Cmin of Si to 1.85.

E.6 Further Benchmarks of the MEAM† Poten-

tials

Even though we have modified the multi-body screening function and the equation

of state, the elastic moduli, vacancy formation energy, and surface energies are not

changed significantly from the original MEAM potentials, as shown in Table E.3. The

changes of elastic and defect properties are very small due to the following reasons.

First, while lowering Cmin extends the interaction to second nearest-neighbors, these

interactions are much weaker than the first nearest-neighbor interactions. Second,

the modification of the equation of state is appreciable only at distances well beyond

the equilibrium nearest-neighbor distance. The effects on the elastic moduli, vacancy

formation energy and surface energies are negligible because they are dominated by

nearest-neighbor interactions.

Because we choose to change only a few parameters in the existing MEAM models,

our models share some of the same limitations of the existing models. One limitation is

the prediction of Si surface structure. First, the unrelaxed surface energy is about 30%

lower than ab initio results. While the MEAM potential correctly predicts the 2× 1

reconstruction of the 100 surface [228], it does not predict the (7×7) reconstruction

of the 111 surface as observed in experiments [238] and ab initio simulations [239].

The MEAM† potential predicts similar energies for the 3 × 3, 5 × 5, 7 × 7, 9 × 9

reconstructed 111 surface as the original MEAM potential. A previous report [240]

claimed that the original MEAM silicon potential predicts the (7× 7) reconstruction

as the ground state of the 111 surface. This is incorrect because the reconstructed

surface energy (1524 erg/cm2) is much higher than the unreconstructed surface energy

(1254 erg/cm2) [206].

Contrary to the case of Si, the MEAM Au potentials reproduce very well the


energies predicted by DFT for the 110 and 111 surface. A large gap is still

observed for the 100 surface.

Table E.3: The elastic constants, C11, C12 and C44 (in GPa), vacancy formation energyEv (in eV) and surface energies E100, E110, and E111 (in ergs/cm2) from experimentalmeasurements [183, 206, 241, 242], first principle calculations [241, 243, 244, 245](except silicon elastic constants computed here using DFT/LDA) and various MEAMmodels considered in this study. The unrelaxed energies are given in parenthesis. TheDFT results are from several different pseudopotentials. Surface reconstruction suchas dimer structure is not considered in calculation of relaxed surface energy of silicon.

Au C11 C12 C44 Ev E100 E110 E111

2nn-MEAM 201.5 169.7 45.4 0.91(0.96) 1083(1138) 1045(1179) 903(928)2nn-MEAM∗ 201.5 169.7 45.4 0.91(0.96) 1083(1138) 1045(1178) 903(928)2nn-MEAM† 202.1 169.5 45.4 0.91(0.96) 1084(1138) 1048(1179) 903(928)

Exp’t 201.6 169.7 45.4 0.94 1540(poly) −− −−DFT 217 171 47 0.55 1968 1098 917

Si C11 C12 C44 Ev E100 E110 E111

MEAM 164 65 76 3.28(4.04) 1742(1850) 1413(1536) 1195(1254)MEAM∗ 164 65 76 3.28(4.04) 1742(1850) 1413(1536) 1195(1254)MEAM† 164 65 76 3.56(4.06) 1744(1851) 1414(1536) 1196(1254)Exp’t 165.8 63.5 79.6 −− 1135(poly) −− −−DFT 158 65 77 2.7− 3.9 2683(2716) −− 1741(1954)

We also compared the bond angle distribution g3(θ, rm) of the silicon liquid de-

scribed by MEAM†, ab initio [246] and SW models [246] in Fig. E.6. The average

coordination numbers from ab initio and SW model are 6.5 and 4.9 at 1800 K, re-

spectively [246]. rm = 3.15A is taken to be the first local minima of the radial

distribution g(r) in our calculation. The simulation is performed at the melting point

of the MEAM† model, which is 1687 K. The MEAM† model predicts two peaks in

the g3(θ, rm) plot, consistent with ab initio simulations. In comparison, this feature

is absent in the SW model of Si, as shown in Fig. E.6. The average number of neigh-

bor atoms within a cut-off radius of rm (i.e. the coordination number) is 5.7 for the


MEAM† model. In comparison, the coordination number for the original MEAM po-

tential is 6.6 at its melting temperature of 1411 K. Hence the MEAM† model provides

a reasonable description of the solid and liquid properties of silicon.

0 50 100 1500

0.2

0.4

0.6

0.8

1

Θ(degree)

Figure E.6: (Color Online) Bond angle distribution functions of liquid phase of silicondescribed by the MEAM† (solid line) ,DFT/LDA (dashed line) [246] and SW (dottedline) [246].

Appendix F

A Gold-Silicon Potential Fitted to

the Binary Phase Diagram

F.1 Introduction

As a continuation of Appendix E, we build the gold-silicon cross potential based on

the MEAM formalism. In order to be useful for NW growth studies, the potential

needs to correctly capture the thermodynamic driving forces of crystallization. Hence

we fit the potential to the experimental binary phase diagram, which is not done for

most of the existing potential models in the literature. There have been previous

studies on the development of EAM potentials consistent with the thermodynamics

of the solid phases [247, 248]. The methods empolyed in these studies are not directly

applicable to the fitting of the solid-liquid phase boundaries, which is the main focus

of this work.

There has been an earlier attempt to develop an MEAM gold-silicon potential [249,

250]. Unfortunately, we were unable to reproduce the published data. Hence we

re-develop the MEAM potential here. Another attempt to construct a gold-silicon

potential is to use EAM and Tersoff models to describe the interaction among gold

atoms and silicon atoms, respectively, and to mix the two functional forms in an

intuitive way to model gold-silicon interactions [251]. Because the phase diagram for

these two potential models have not been calculated, it is difficult to assess whether

180

APPENDIX F. GOLD-SILICON BINARY POTENTIAL 181

they are suitable to model the VLS growth of NWs. In preparation for this work, we

have benchmarked the melting point and latent heat of the original MEAM potentials

for pure gold and silicon, and adjusted the potentials to accurately reproduce the

experimental values. Based on these improved models, the remaining task amounts

to constructing the cross-potential between gold and silicon. The fitting of the cross-

potential to experimental phase diagrams is enabled by efficient free-energy methods

to rapidly calculate the phase diagram for a given candidate potential model.

Appendix F is organized as follows. In Section F.2, we present the functional

form of our MEAM model for gold-silicon and the general procedure to determine its

parameters. In Section F.3, we present our free-energy methods to compute the binary

phase diagram from atomistic simulations. A brief summary is given in Section F.4.

Section F.5 contains further benchmark of the Au-Si cross potential by comparing its

predictions with ab initio data.

F.2 MEAM Model for Gold and Silicon

F.2.1 Functional Form

For the Au-Si cross-potential, we choose the B1 structure as the reference structure.

This is a hypothetical alloy structure because in solid state the solubility of Au in Si

(and vice versa) is very low. The EOS function used for the B1 crystal structure is

Eu(r) = −Ec

(

1 + a∗ +γ

r· a∗3

)

exp(−a∗) (F.1)

with a∗ =

(

9ΩB

Ec

)1/2 (r

re− 1

)

(F.2)

where r is the nearest-neighbor distance, Ec is the cohesive energy, re is the equilib-

rium nearest-neighbor distance, Ω is the atomic volume, and B is the bulk modulus

of the reference structure. γ is an adjustable parameter to provide additional flexi-

bility [249].


F.2.2 Determining the Parameters

We determine the parameters of the Au-Si cross-potential in three steps. First, we

perform ab initio calculations of the hypothetical B1 alloy structure to determine the

cohesive energy Ec, equilibrium nearest-neighbor distance re and bulk modulus B.

Several MEAM parameters are determined by fitting to these values, after adjusting

for the known differences between ab initio and experimental data. Second, the

substitutional impurity energies of Si in FCC Au and Au in DC Si are computed ab

initio and the data are used to adjust the electron density scaling factors ρAu0 and ρSi0 .

Third, other potential parameters are adjusted from their default values so that the

predicted binary phase diagram reproduce the experimental diagram as accurately as

possible. The parameters adjusted in the third step include γ in the EOS function

of the alloy structure, and angular cut-off parameters Cmin(i, j, k) in the multi-body

screening function Sij . The first two steps are described in this section. The method

to compute the binary phase diagram is described in Section F.3.

Ab initio calculations are performed based on the density functional theory (DFT)

using VASP [232]. We employ the ultrasoft pseudopotentials [252] within the local

density approximation, with plane-wave expansion up to a cut-off energy of 400 eV.

A FCC (DC) unit cell consisting of 4 (8) atoms is used for Au (Si). A B1 unit cell

consisting of 4 Au atoms and 4 Si atoms is used for the solid alloy with B1 structure.

For all cases, 15× 15 × 15 k-points are used with the Monkhorst-Pack scheme. The

total energy was converged within 10−4 eV. The results for the DC crystal of Si, FCC

crystal of Au, and the hypothetical B1 structure of Au-Si are given in Table F.1.

Experimental data exist for crystals of pure Si and pure Au. The differences

between experimental and ab initio data are listed in the column labelled “offset”.

This difference must be accounted for because existing MEAM models have been

fitted to experiments instead of ab initio data. The correction to the ab initio data

for the hypothetical B1 structure is obtained by averaging the differences between

experimental and ab initio data for pure Si and Au. The data after this correction

are marked with ∗ in Table F.1. These are the data that the MEAM model is fitted

to, or should be compared against. We note that these adjustment of the VASP data

is not unique and could lead to errors. The parameters re, Ec and α ≡√

9ΩB/Ec


Table F.1: Equilibrium lattice constant a, bulk modulus B, cohesive energy Ec, andcubic elastic constants C11 and C44 for DC structure of Si, FCC structure of Au, andB1 structure of Au-Si. For pure Si and Au, the differences between the experimentaland ab initio values are listed in the column labelled “offset”. Their average is theexpected “offset” value for the hypothetical B1 structure. The values marked with ∗are the ab initio values plus the correction terms given in the “offset” column. Thelast column is what the MEAM model is fitted to or predicts.

Material Exp’t DFT/LDA offset MEAM

a (A) Si (DC) 5.431 5.390 -0.041 5.431Au (FCC) 4.070 4.068 -0.005 4.073Au-Si (B1) 5.184∗ 5.161 -0.023 5.400

B (GPa) Si (DC) 98 96 2 98Au (FCC) 180 186 -6 180Au-Si (B1) 127∗ 129 -2 127

Ec (eV) Si (DC) 4.63 5.976 -1.346 4.63Au (FCC) 3.93 4.387 -0.457 3.93Au-Si (B1) 4.155∗ 5.057 -0.902 4.155

C11 (GPa) Si (DC) 164 162 2 164Au (FCC) 202 217 -15 202Au-Si (B1) 303∗ 310 -6 320

C44 (GPa) Si (DC) 76 105 -29 76Au (FCC) 45 47 -2 45Au-Si (B1) -35∗ -19 -15 14

in the EOS of the B1 reference structure are easily obtained with this approach. We

note that we intentionally fit the lattice constant of the B1 structure to a larger value

of a = 5.400A eV than the adjusted ab initio data (5.184A) (See Section F.5), because

it gives rise to better agreement with experiments on the binary phase diagram.

The electron density scaling factors, ρAu0 and ρSi0 does not affect the energy of

pure crystals but influence the interaction between Au and Si atoms. Because only

the ratio of the electron density scaling factors,ρAu0

ρSi0, is important, we have a single

parameter to fit the two dilute solution energies. The ratio is adjusted to produce

reasonable substitutional impurity energies, i.e. the energy E1 to replace a Au atom

in the FCC crystal by a Si atom, and the energy E2 to replace a Si atom in the DC

crystal by a Au atom. The MEAM and ab initio results for the impurity energies


are listed in Table F.2. It shows that our choice ofρAu0

ρSi0is the result of a compromise

between E1 and E2, because we cannot fit both of them accurately. As a result, only

E1 is fitted while E2 is overestimated. This diagreement of the MEAM result can be

due to either the adjustment of VASP data introduced in Table F.1 or inaccuracies

in MEAM formalism. Nonetheless, the MEAM predicts very low solubility of Si in

Au (< 1.3%) and Au in Si (< 10−6) in the solid phase, consistent with experimental

measurements of < 2% and < 2× 10−4. [253]

Table F.2: MEAM and ab initio (DFT/LDA) predictions of impurity energies. E1 isthe energy needed to substitute an atom in an FCC Au crystal by a Si atom. E2 isthe energy needed to substitute an atom in a DC Si crystal with by a Au atom.

MEAM DFT/LDAE1(eV) 0.636 0.634E2(eV) 3.968 1.553

The last step is to fine-tune the potential by adjusting the parameter γ in the

EOS function (for the B1 structure), and the cut-off parameters Cmin(i, j, k) in the

multi-body screening function to fit the experimental binary phase diagram as close as

possible. Because elastic constants of B2 structure, one of benchmarks for our poten-

tial model (See Section F.5), are highly sensitive to the Cmin(1, 1, 2) and Cmin(2, 2, 1),

we use only γ, Cmin(1, 2, 1), and Cmin(1, 2, 2) when adjusting free energy of liquid

alloy. Without any correction to γ, the free energy of mixing of liquid is too high

and so is the eutectic temperature. As the binding energy of Au-Si is smaller than

average of Au-Au and Si-Si binding energies, we increase γ to make the Au-Si cross

potential more repulsive, in order to reduce free energy of mixing for liquid for entire

composition range. Cmin(1, 2, 1) and Cmin(1, 2, 2) are adjusted to change the multi-

body screening effects selectively. Decreasing Cmin(1, 2, 1) (the screening factor of

Au-Si by Au) lowers the Au-rich part of liquid free energy because it reduces the

screening effects by Si atoms on Au-Au interactions. In the same way, we can in-

crease Cmin(1, 2, 2) to raise the Si-rich part of liquid free energy. We repeated these

procedures until we obtain the binary phase diagram and the free energy of mixing

close to experiments. The resulting parameters for the MEAM Au-Si cross-potential


are summarized in Table F.3.

The binary phase diagram of the resulting MEAM potential is shown in Fig. F.1,

together with the experimental phase diagram. The MEAM potential successfully

captures the eutectic behavior. The eutectic temperature (Te = 629 K) matches well

with experimental value (634 K). The eutectic composition (xe = 0.234) also agrees

well with experimental value (0.195). The boundary of the Au-rich solid phase is

not shown in the experimental phase diagram [253], but it is known that maximum

solubility of Si in Au is less than 2%, which is consistent with our value of 1.3%. We

are not able to remove the slight offset of the liquidus curve for the Au-rich branch.

This seems to be a limitation of the functional form of the MEAM potential used in

this work. Additional benchmark data of the potential is presented in Section F.5.

The method for computing the binary phase diagram for a given interatomic potential

model is presented in the following section.

Table F.3: Parameters for the Au-Si MEAM cross-potential using B1 as the referencestructure. Cmin(i, j, k) are cut-off parameters in the multi-body screening function.They describe the screening effect on the interaction between atoms of type i andj by their common neighbor of type k, where i, j, k = 1 (Au) or 2 (Si). The sameCmax(i, j, k) is used for every combination of i, j, k

Ec re αρSi0ρAu0

Cmax

4.155 2.700 5.819 1.48 2.8

Cmin(1, 1, 2) Cmin(1, 2, 1) Cmin(1, 2, 2) Cmin(2, 2, 1) γ1.9 0.95 1.85 1.0 0.26

F.3 Construction of Binary Phase Diagram

To compute the Au-Si binary phase diagram, we need the Gibbs free energy (per

atom) as a function of temperature T and composition x = xSi for three phases: (1)

FCC Au crystal with Si impurities GFCC, (2) DC Si crystal with Au impurities GDC,

and (3) liquid Au-Si alloy Gliq. At a given temperature, the range of stability in the


0 0.2 0.4 0.6 0.8 1

200

400

600

800

1000

1200

1400

1600

xSi

Tem

pera

ture

(K)

Au−Si Phase Diagram

MEAMExp

Au(s)

L

Au(s)+Si(s)

L+Si(s)

L+Au(s)

Figure F.1: Binary phase diagram of Au-Si. MEAM prediction is plotted in thickline and experimental phase diagram is plotted in thin line. L corresponds to theliquid phase. Au(s) and Si(s) correspond to the Au-rich and Si-rich solid phases,respectively.

composition axis for each phase and their mixtures is determined by the common-

tangent construction.

The free energy at a given temperature is obtained by the adiabatic switching

method [199], which computes the free energy difference between the system and a

reference whose free energy is known analytically. The change of free energy as a

function of temperature is then computed using the reversible scaling method [200].

We have used these methods to compute the free energy of single component systems

(in both solid and liquid phases) and determined their melting points in Appendix D.

In the following, we will focus on the extra complexities caused by the binary systems,

such as the configurational entropy.

F.3.1 Free Energy of Solid with Impurities

The solid free energies of pure Au (FCC) and pure Si (DC) can be computed using the

method described earlier. These correspond to GFCC(x = 0, T ) and GDC(x = 1, T ),

respectively. In calculation of GFCC and GDC as a function of x, we notice that the


solubility in the solid phase for both Si in Au and Au in Si is very low. This means

we only need to know GFCC(x, T ) in the vicinity of x = 0. Similarly, we only need to

know GDC(x, T ) in the vicinity of x = 1. In the following, we describe our approach

to obtain GFCC(x, T ). GDC(x, T ) can be obtained in a similar way.

For an FCC Au crystal containing a very low concentration of Si impurities (x ≪1), it is reasonable to assume that the impurities are not interacting with each other.

In this limit, the free energy per atom of the crystal can be approximated by [254]

GFCC(x, T ) ≈ GFCC(x = 0, T ) + x∆gimp(T )− T smix(x) (F.3)

where ∆gimp is the free energy of a single impurity, in which the configurational

entropy is ignored, i.e. only the vibrational entropy is included. smix = −kB [x ln x+

(1− x) ln(1− x)] is the configurational entropy of mixing.

We compute ∆gimp in the Eq. (F.3) using a simulation cell containing N−1 = 499

Au atoms and 1 Si atom under periodic boundary conditions (PBC). We label this

simulation cell as Cell 1 and let G1 be its free energy. G1 at a given temperature

(T0 = 254 K) 1 is computed by adiabatic switching [199] the system to its harmonic

approximation, whose free energy is known analytically. G1 as a function of temper-

ature is then computed by the reversible scaling method [200]. Similarly, we compute

the free energy G0 as a function of temperature, for a simulation cell containing 500

Au atoms. The free energy of the impurity is simply,

∆gimp(T ) = G1(T )−G0(T ) + kBT lnN. (F.4)

The kBT lnN term is added to cancel the configurational entropy contribution in

G1(T ) 2. Fig. F.5 (a) plots GFCC(x, T ) and GDC(x, T ) as a function of x at T = 700 K,

obtained using the method described above.

In several studies regarding binary phase diagram calculations, [247, 248] it is com-

mon to approximate ∆gimp(T ) = ∆himp − T∆Svib by the enthalpy ∆himp, neglecting

1To improve accuracy, we also performed an independent calculation at T0 = 629 K.2The configurational entropy of mixing in Cell 1 is S1

mix = kB ln N !(N−1)!1! = kB lnN .


the vibrational entropy contribution in ∆gimp(T ). However, we find that the contri-

bution from −T∆Svib is significant for the Au-Si system, affecting the phase diagram

significantly. Fig. F.2 plots ∆gimp(T ) and ∆himp(T ) as a function of temperature. On

one hand, the vibrational entropy change for Si impurity inside Au crystal is −4.9kB,

which increases the free energy by 26 kJ/mol at eutectic temperature. Si impurity

solubility will be significantly overestimated if we ignore the vibrational entropy. On

the other hand, for Si impurity inside Au crystal, the ∆Svib is 10kB, corresponding

to free energy decrease of 53 kJ/mol at eutectic temperature. The sign of ∆Svib is

opposite at each side, because Si-Si bonding is stiffer than Au-Au bonding. (Debye

temperature of Si is 645 K while that of Au is 165 K [174]). This result emphasizes

the importance of including vibrational entropy change in computing free energy of

solid and phase diagram.

F.3.2 Free Energy of Liquid Alloy

We first compute the Helmholtz free energy difference between the liquid alloy and

the ideal gas at a given temperature using the adiabatic switching method 3. To

improve computational efficiency, a fluid with a purely repulsive (Gaussian) potential

is used as an intermediate reference system during the switching. To obtain the free

energy of the liquid alloy at this temperature, we add this free energy difference to the

free energy of the two-component ideal gas under a fixed center-of-mass constraint,

which is [255]

Fi.g.(N1, N2) = −N1kBT lnV

Λ31

−N2kBT lnV

Λ32

+ kBT lnV

Λ3e

+ kBT ln(N1!N2!) (F.5)

where V is the volume of the simulation cell, Λi = h/√2πmikBT is the de Broglie wave

length, with i = 1 for Au and i = 2 for Si. Λe = h/√2πmekBT and me = (N1m

21 +

N2m22)/(N1m1 + N2m2) is the effective mass for the constrained degree of freedom.

The last term in Eq. (F.5) reflects the configurational entropy of mixing. Once the

free energy at a certain temperature is determined, its temperature dependence is

3Because the liquid alloy is at zero pressure, its Gibbs free energy coincides with its Helmholtzfree energy, although this is not the case for the reference ideal gas system.


(a)

250 300 350 400 450 500 550 600 650−45

−40

−35

−30

−25

−20

−15

Temperature (K)

(kJ/

mol

)

∆gimp

∆himp

(b)

250 300 350 400 450 500 550 600 650210

220

230

240

250

260

270

Temperature (K)

(kJ/

mol

)

∆gimp

∆himp

Figure F.2: Gibbs free energy ∆gimp(T ) and enthalpy ∆himp(T ) (a) a Si impuritywithin Au crystal. (b) a Au impurity within Si crystal.

obtained by reversible scaling.

Using the method above, we compute the free energy of the liquid alloy at 11

different compositions, x = 0, 0.1, 0.2, · · · , 1, and interpolate the values along the x

axis by spline fitting. The numerical error introduced in the spline fitting may have

caused the undulation of the liquidus curve in the Si-rich region of the phase diagram

in Fig. F.1. Fig. F.3(a) plots the resulting function Gliq(x, T ) at T = 1250 K. The

difference between Gliq(x, T ) and the straight line connecting the free energies of pure

Au and pure Si liquids is the free energy of mixing, which is shown in Fig. F.3(b).


The prediction of free energy from the MEAM model is in reasonable agreement with

the CALPHAD result [256].

(a)

0 0.2 0.4 0.6 0.8 1−475

−470

−465

−460

−455

−450

xSi

Fre

e en

ergy

(kJ

/mol

)

(b)

0 0.2 0.4 0.6 0.8 1−20

−15

−10

−5

0

xSi

Fre

e en

ergy

(kJ

/mol

)

MEAMCALPHAD

Figure F.3: (a) Liquid free energy Gliq(x, T ) at T = 1250 K. Circles are simulationresults, which are fitted to a spline (solid line). A straight line connecting the liquidfree energy of pure Au and pure Si is drawn for comparison. (b) The free energy ofmixing Gmix(x, T ) for the liquid phase at T = 1250 K. Predictions from the MEAMpotential is plotted in thick line, which is the difference between Gliq(x, T ) and thestraight line shown in (a). Free energy obtained from CALPHAD method [256] areplotted in thin line.

We also compare our model directly to enthalpy of mixing and excess free en-

ergy from experiments [257], which are obtained by calorimetric and Knudsen cell

method respectively as shown in Fig. F.4. While qualitative agreements can be ob-

served among MEAM, CALPHAD and experimental data, notable discrepancies can


also be observed, even though the binary phase diagrams predicted by MEAM and

CALPHAD both agree very well with experimental phase diagram. This is because

the MEAM potentials for pure Au and pure Si have ∼ 15% error in the latent heat

(See Appendix E). Therefore, the free energy difference between solid and liquid

phases in the limit of x = 0 and x = 1 are incorrectly predicted by MEAM at low

temperatures. To reproduce the experimental phase diagram, the shape of the liquid

free energy curve as a function of x predicted by MEAM must be different from that

predicted by CALPHAD at low temperatures. To remove this discrepancy, one will

have to re-fit the MEAM potential for pure Au and pure Si to obtain the latent heat

exactly.

F.3.3 Construction of Binary Phase Diagram

Given the Gibbs free energies of the three phases, GFCC(x, T ), GDC(x, T ) andGliq(x, T ),

the binary phase diagram is constructed by drawing common tangent lines between

the three curves at each temperature. An example is given in Fig. F.5 for T = 700 K.

First, a common tangent line is drawn between GFCC(x, T ) and Gliq(x, T ). The

tangent contacts the two free energy curves at x1 = 0.011 and x2 = 0.225, respectively.

This means that the Au-rich FCC (solid) phase is stable in the composition range of

x ∈ [0, x1]. The mixture of FCC solid and liquid phase is stable in the composition

range of x ∈ (x1, x2). Second, a common tangent line is drawn between Gliq(x, T )

and GDC(x, T ). The tangent contacts the two free energy curves at x3 = 0.251

and x4 ≈ 1, respectively. This means that the Si-rich DC (solid) phase is stable

in the composition range of x ∈ [x4, 1]. The mixture of DC solid and liquid phase

is stable in the composition range of x ∈ (x3, x4). The liquid phase is stable in

the composition range of x ∈ [x2, x3]. Repeating this procedure for all temperatures

allows us to construct the binary phase diagram shown in Fig. F.5(b). At the eutectic

temperature Te = 629 K, all three free energy curve shares the same tangent line

(x2 = x3)4. The tangent line contacts the liquid free energy curve at the eutectic

composition xe = 0.234. The eutectic temperature and composition of the MEAM

4At T < Te, the liquid free energy curve is entirely above the common tangent line of the twosolid free energy curves. In this case, the liquid phase is unstable at any composition.


(a)

0 0.2 0.4 0.6 0.8 1−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

xSi

∆Hliq

(kJ

/mol

)

MEAMExpCALPHAD

(b)

0 0.2 0.4 0.6 0.8 1−15

−10

−5

0

xSi

∆GliqX

S (

kJ/m

ol)

MEAMExpCALPHAD

Figure F.4: (a) Enthalpy of mixing ∆Hliq(x, T ) at T = 1373 K from experiments(circles), MEAM (thick line), and CALPHAD (thin line). (b) Excess free energy ofmixing ∆GXS

liq (x, T ) at T = 1685 K from experiments (circles), MEAM (thick line),and CALPHAD (thin line).

model is in good agreement with experimental data (634 K and 0.195).

F.4 Summary

We develop an MEAM gold-silicon potential that is fitted to the experimental bi-

nary phase diagram. The potential parameters are first fitted to ab initio data of a

hypothetical B1 alloy structure. The parameters are then adjusted to fit the substi-

tutional impurity energies in the solid phase and the binary phase diagram. The final


(a)

0 0.2 0.4 0.6 0.8 1−20

−15

−10

−5

0

5

xSi

Fre

e en

ergy

(kJ

/mol

)

GDC

Gliq

GFCC

(b)

0 0.2 0.4 0.6 0.8 1

200

400

600

800

1000

1200

1400

1600

Tem

pera

ture

(K)

xSi

Figure F.5: Common tangent method to construct binary phase diagram from freeenergy curves. (a) Gibbs free energy of the three phases, GFCC(x, T ), GDC(x, T ) andGliq(x, T ), as a function of composition x at T = 700 K. All of them are referenced tofree energies of pure Au liquid and pure Si liquid. Common tangent lines are drawnbetween GFCC(x, T ) and Gliq(x, T ) (from x1 = 0.011 to x2 = 0.225), and betweenGliq(x, T ) and GDC(x, T ) (from x3 = 0.251 to x4 ≈ 1). (b) Binary phase diagramof the MEAM Au-Si potential. The phase boundaries at T = 700 K are determinedfrom the data in (a).

potential successfully captures the eutectic behavior of gold-silicon binary system.

The eutectic temperature and composition agrees well with experimental values. The

potential is further benchmarked in other hypothetical structures such as B2 and L12

in Section F.5. The lattice constant, bulk modulus and cohesive energy are all within

15% of ab initio results. We expect the potential developed here can be used for


atomistic simulations of gold-catalyzed nucleation and growth of silicon nanowires.

Because this potential is mostly fitted to bulk properties, it can be further improved

by fitting to surface and defect properties, which are also expected to influence the

nanowire growth process. The method developed here for computing the binary phase

diagram can also be used for other binary systems that exhibit a eutectic behavior

and low solid solubility, such as gold-germanium and gold-aluminum.

F.5 Further Benchmarks

We test the transferability of the Au-Si MEAM potential by comparing it against ab

initio predictions on the energetics and elastic constants of several other hypothetical

solid structures. Table F.4 presents the equilibrium lattice constants, cohesive energy,

and elastic constants of B2 and L12 structures. The results for the B1 structure are

also included. The values of a, B and Ec for the B1 structure are used in the fitting

but the values of C11 and C44 are not. We intentionally fit the equilibrium lattice

constant of the B1 structure to a higher value of a = 5.400A than the adjusted ab

initio data (5.184A), to get better agreement with experiments on the binary phase

diagram. A reasonable binary phase diagram can also be obtained by lowering the

formation energy of B1 structure instead, and using large γ in Eq. (2). However,

we do not take this approach because it changes the elastic constants of B2 and L12

structures significantly. For all three phases, the MEAM predictions of a, B, Ec, and

C11 are within 15% of ab initio data. Notice that the ab initio model predicts C44 < 0

for all three crystal structures, indicating that they are mechanically unstable. In

comparison, the MEAM model predicts a small but positive C44, meaning that they

may be metastable in the MEAM model.


Table F.4: Comparison between MEAM and ab initio predictions on the energyand elastic properties of B1, B2 and L12 structures of Au-Si. a is the equilibriumlattice constant, B is bulk modulus, Ec is the cohesive energy, and C11 and C44 arecubic elastic constants. MEAM data should be compared with ab initio (DFT/LDA)results that have been adjusted for known differences from experimental values forpure elements.

Structure Properties DFT/LDA DFT/LDA (adjusted) MEAM

B1 a (A) 5.161 5.184 5.400B (GPa) 129 127 127Ec (eV) 5.057 4.155 4.155

C11 (GPa) 310 303 320C44 (GPa) -20 -35 15

B2 a (A) 3.202 3.226 3.370B (GPa) 130 128 114Ec (eV) 4.867 3.966 3.844

C11 (GPa) 101 101 88C44 (GPa) -3 -18 0.7

L12 a (A) 4.041 4.055 4.192B (GPa) 156 152 140Ec (eV) 4.624 3.945 4.022

C11 (GPa) 170 159 142C44 (GPa) 11 -1 19

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