The Validity of Classical Nucleation Theory and Its Application to Dislocation Nucleation

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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

This work is licensed under a Creative Commons Attribution- 3.0 United States License. http://creativecommons.org/licenses/by/3.0/us/

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

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

Wei Cai, Primary Adviser

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

Douglas Osheroff, Co-Adviser

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

Paul McIntyre

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

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 in University 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 correctly 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 mechanical strength of crystalline materials. We show that dislocation nucleation rates can be accurately predicted over a wide range of conditions using CNT with the activation free energy determined by umbrella sampling. Our data reveal very large activation entropies, which contribute a multiplicative factor of many orders of magnitude to the nucleation rate. The activation entropy at constant strain is caused by thermal expansion, with negligible contribution from the vibrational entropy. The activation 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 anharmonic 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 consistently 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 members 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 experiments 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 foundation 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 encouragement. 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 spectrum of research projects in computational physics and computational materials science. 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 computational 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 dislocation nucleation rate from the classical nucleation theory using atomistic simulations. Readers who prefer a more distilled presentation are refered to following journal articles: 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 continuum 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 potential 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 models 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 and Wei 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 ScopeoftheDissertation...... 4

2 Background and Motivation 7 2.1 BasicThermodynamicsofNucleation ...... 7 2.2 Nucleation Rate Predictions From Nucleation Theories ...... 11 2.3 NucleationExperiments ...... 14 2.4 Dislocation Nucleation and Materials Strength at Small Scale . . . . 20

3 Computational Methods 25 3.1 IsingModel ...... 26 3.2 InteratomicPotential ...... 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 AdvancedSamplingMethods ...... 41

xi 3.5.1 Forward Flux Sampling ...... 42 3.5.2 Umbrella Sampling ...... 45 3.5.3 Computing Ratefrom Becker-D¨oring 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-D¨oringTheory ...... 53 4.2.2 Langer’s Field Theory ...... 57 4.3 ComputationalMethods ...... 57 4.4 Results...... 59 4.4.1 NucleationRate...... 59 4.4.2 Critical Droplet Size and Shape ...... 61 4.4.3 DropletFreeEnergyof2DIsingModel ...... 63 4.4.4 DropletFreeEnergyof3DIsingmodel ...... 66 4.4.5 EﬀectiveEntropyofNucleation ...... 70 4.5 SummaryandDiscussion...... 72

5 Predicting the Dislocation Nucleation Rate 75 5.1 Introduction...... 75 5.2 ThermodynamicsofNucleation ...... 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 ComputationalMethods ...... 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 FutureWorks ...... 108

A Derivations 111 A.1 Nucleation Rate Prediction from Classical Nucleation Theory . . . . . 112 A.2 NucleationTheorems ...... 117

B More Data on the Nucleation in the Ising Model 121 B.1 AttachmentRate ...... 121 B.2 DropletShape...... 122 B.3 TheConstantTerminDropletFreeEnergy ...... 123 B.4 Free Energy Curves F (n)fortheIsingModel...... 126

C More Discussion on the Dislocation Nucleation 131 σ γ C.1 Equality of Critical Sizes nc and nc ...... 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 ActivationFreeEnergyData...... 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: SiandGe...... 145 D.2.2 FCCMetals:Au,Cu,AgandPb ...... 147 D.2.3 BCCMetals:Mo,TaandW...... 147

xiii D.3 Free Energy Method for Melting Point Calculation ...... 148 D.3.1 SolidFreeEnergy...... 150 D.3.2 LiquidFreeEnergy ...... 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 ProblemStatement ...... 162 E.2.1 Limitations of the MEAM Gold Potential ...... 163 E.2.2 Limitations of MEAM Silicon Potentials ...... 165 E.3 MethodsandResults ...... 167 E.3.1 Multi-body Screening Function ...... 168 E.3.2 PairPotentialandEquationofState ...... 170 E.4 Summary ...... 174 E.5 Multi-bodyScreeningFunction ...... 174 E.6 Further Benchmarks of the MEAM† Potentials...... 177

F Gold-Silicon Binary Potential 180 F.1 Introduction...... 180 F.2 MEAMModelforGoldandSilicon ...... 181 F.2.1 FunctionalForm ...... 181 F.2.2 DeterminingtheParameters ...... 182 F.3 ConstructionofBinaryPhaseDiagram ...... 185 F.3.1 Free Energy of Solid with Impurities ...... 186 F.3.2 Free Energy of Liquid Alloy ...... 188 F.3.3 ConstructionofBinaryPhaseDiagram ...... 191 F.4 Summary ...... 192 F.5 FurtherBenchmarks ...... 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 initio calculation [114]. ab initio data depend on the exchange-correlation functionalsusedinthestudy...... 36

C.1 Data for homogeneous nucleation: σxy in GPa, Ec, E˜c and Fc in eV, + 14 −1 ˜ fc in 10 s . 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 sampling. The error in Zeldovich factor Γ is within 0.01. The attachment ± rate f + has relative error of 50%...... 138 c ± + C.2 Data for heterogeneous nucleation: σzz in GPa, Ec, and Fc in eV, fc 14 −1 in 10 s . 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 + has relative error of 50%. Notice that, due c ± to the existence of thermal strain, the elastic strain values are slightly diﬀerentatdiﬀerenttemperatures...... 139

xv D.1 Thermal properties of various elements as predicted by several empirical 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- tionsofSW-SiandMEAM-Simodels...... 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 modifications made in this study (2nn-MEAM∗ and 2nn-MEAM†), and first- principles calculation with DFT/LDA [232]. The properties include the −1 melting point Tm (in K), latent heat of fusion L (in kJ mol ), solid and −1 −1 liquid entropy at the melting point, SS and SL (in J mol K ), diffu- −9 2 −1 sion constant of the liquid D at Tm (in 10 m s ), 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 lastdigitofthepresentedvalues...... 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 comparison. The computational models include 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 variables 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- 2 ergy Ev (in eV) and surface energies E100, E110, and E111 (in ergs/cm ) from experimental measurements [183, 206, 241, 242], ﬁrst principle calculations [241, 243, 244, 245] (except silicon elastic constants computed 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 reconstruction such as dimer structure is not considered in calculation of relaxed surfaceenergyofsilicon...... 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 crystalwithbyaAuatom...... 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 interaction 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 adjusted 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 contact angle θ formingawall...... 10 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 classical Becker-Döring (BD) theory. Logarithmic scales areusedforbothaxes...... 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 rippleextendingfromsidetoside...... 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 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 free of dislocations. (c) MD simulation of nanoindentation process [95]. Snapshot of dislocation 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 1 3.1 (a) Magnetization M = N si of the 2D square lattice Ising model as function of temperature in the absence of external ﬁeld, i.e. h = 0. At T < T = 2/ ln(1 + √2)J/k 2.269J/k , majority of spins in c B ∼ B 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 demagnetization 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] direction calculated with EAM potential [111]...... 36 3.5 Linear thermal expansion of Cu calculated in the quasi-harmonic approximation (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 (n n ) becomes N/M. ... 42 i+1| i 3.7 (color online) The probability P (λ λ ) (solid line) of reaching interface i| 0 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% committorpointismarkedby*...... 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 withintherangeofbiasfunction...... 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- 1 2 tained by multiplying the inverse of weighing function, exp[ 2 k(n−ni) ], 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 k T ln(∆G(n)). .. 46 − B

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) asafunctionoftime...... 48

4.1 (color online) Eﬀective surface free energy σeﬀ as a function of temperature 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 plottedforcomparison...... 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- Döring 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-Döring theory at different temperatures in the (b) 2D and (d) 3D Ising models. The symbols in (b) and (d) match those definedin(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 predicted by Becker-Döring 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) andumbrellasampling(opensymbols)...... 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 term 5 4 kBT ln n (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(seeAppendixB.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 ln n is also plotted (dot-dashed line). The difference in predictions by classical expression 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(dashedline)...... 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-D¨oring 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 ν exp( H /k T ), in which the activation entropy is 0 − c B completely ignored, leading to an underestimate of the nucleation rate by 20ordersofmagnitude...... 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 (γ =0.135). Dotted curve presents the ﬁtted curve exp( IMDt) with xy − 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 umbrella 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 linesrepresentapolynomialﬁt...... 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 equilibratedatdiﬀerenttemperatures...... 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) under constant shear strain loading rateγ ˙ = 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 nucleation stress prediction when the activation entropy is neglected. The dotted line in (b) is the prediction based on the approximation by Zhu etal.[153]...... 104

+ B.1 (color online) (a) The pre-exponential factor fc Γ in 2D computed from + Monte Carlo and US. (b) The ratio between the attachment rate fc in 2D computed by Monte Carlo and that predicted by Eq.(4.7). (a) + The pre-exponential factor fc Γ in 3D computed from Monte Carlo and + US. (b) The ratio between the attachment rate fc in 3D computed by MonteCarloandthatpredictedbyEq.(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 5 line) and Eq. (8) (dashed line). Logarithmic correction term 4 kBT ln n (dot-dashed line) and the constant term d (dotted line) are also drawn forcomparison...... 127

xxvi 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) is compared with Eq. (6) (solid line) and Eq. (8) (dashed 5 line). Logarithmic correction term 4 kBT ln n (dot-dashed line) and the constant term d (dotted line) are also drawn for comparison. . . . . 129

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 =0.035 obtained by US (circles) is compared with Eq. (6) (solid 5 line) and Eq. (8) (dashed line). Logarithmic correction term 4 kBT ln n (dot-dashed line) and the constant term d (dotted line) are also drawn forcomparison...... 130

C.1 The relation between critical dislocation size nc and the activation volume Ω ∂Gc .(a) homogeneous nucleation (b) heterogeneous nu- c ≡ − ∂σ 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 linesarelinearﬁtstothedata...... 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 fortheliquidphase...... 157

E.1 (Color Online) Generalized stacking fault energy of diﬀerent potential 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] (solidcurve) ,DFT/GGA[233](dashedcurve) ...... 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 (asterisk) 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 silicon described by the MEAM† (solid line) ,DFT/LDA (dashed line) [246] andSW(dottedline)[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 corresponds 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 simulation 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 XS free energy of mixing ∆Gliq (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 G (x, T ) (from x = 0.251 to x 1). (b) Binary phase diagram of DC 3 4 ≈ the MEAM Au-Si potential. The phase boundaries at T = 700 K are determinedfromthedatain(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 science, 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 ∆ and T . Here, ∆ 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 ∆ and T , which makes the prediction of nucleation rate very difficult. While ∆ 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 available, 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 ∆ 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 CHAPTER 1. INTRODUCTION 3

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 1.5m 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 CHAPTER 1. INTRODUCTION 4

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 computer 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 experiments, discrepancies between the CNT prediction and experimental results will be highlighted. In the last part of the chapter, we briefly introduce the concepts of dislocation and explain why dislocation nucleation plays an important role in determining the mechanical behavior of materials at small scale. CHAPTER 1. INTRODUCTION 5

Chapter 3 summarizes the computational methods used in this work. We introduce the Ising model whose demagnetization process has a close analogy to the nucleation dynamics. We review interatomic potentials that were developed to describe 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 nucleation 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 constant 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 CHAPTER 1. INTRODUCTION 6

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 1 (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 2 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( )+ S(n)σ. (2.1) − 1 − 2

The chemical potential diﬀerence ∆ = is the thermodynamics driving force 1 − 2 that induces the phase transition and becomes positive at conditions where the new

phase is thermodynamically favored, i.e. 1 >2. For example, the chemical potential

s of solid is lower than the chemical potential l 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 ∆ 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 (1 1/d)ασ d n = − (2.2) c ∆ By plugging nc into the Eq. (2.1), we ﬁnd the maximum value of ∆G, or the nucleation

barrier Gc to be ∆ [(1 1/d)ασ]d G = n = − (2.3) c c d 1 (d 1)∆d−1 − − 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 CHAPTER 2. BACKGROUND AND MOTIVATION 9

volume surface total G ∆ G c n c Free Energy

Figure 2.1: ∆G versus n showing volume and surface contributions resulting in a peak 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( G /k T ). Under small undercooling (or small supersaturation) where the driv- − c B ing force ∆ is small, the nucleation barrier Gc becomes large, because Gc is inversely proportional to ∆d−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( G /k T ) raises very quickly for deeper undercooling (or supersaturation), due − c B to the increase of the driving force ∆. 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 CHAPTER 2. BACKGROUND AND MOTIVATION 10

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 contact angle 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 σ and σ , i.e. σ > σ σ , the new phase can wet cw nw nc | cw − nw| 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 1 size of nucleus size . 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, hetero Gc is written as hetero homo Vc Gc = Gc (2.4) V0 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 CHAPTER 2. BACKGROUND AND MOTIVATION 11

homo of the complete sphere with radius rc. Gc 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 ∆ 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,

G I N f + exp c (2.5) ≈ c −k T B where N is the equivalent nucleation site and Gc is the formation free energy of + the critical droplet. fc is the attachment rate of molecules to the critical droplet. N exp( G /k T ) is the “equilibrium” population of the critical droplet. The Volmer- − c B 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 D¨oring [36], Zeldovich [37], and Frenkel [38], and remain important to date for our understanding of the nucleation process. They assumed that clusters consisting of CHAPTER 2. BACKGROUND AND MOTIVATION 12

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:

+ fn−1 Λn−1 +Λ1 ⇀ Λn ↽ − −−−−−−fn + fn Λn +Λ1 ↽ ⇀ Λn+1 (2.6) −−−− f−−−n+1

+ − Here, fn is the rate of single-particle attachment to a cluster of size n and fn 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 Döring [36] obtained a steady-state solution for the nucleation rate. Since then, the term “Becker-Döring theory” and the “classical nucleation theory” are used interchangeably in literatures. While Volmer and Weber considered the “equilibrium” population of critical droplet, Becker and Döring 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 G I = N f + Γ exp c (2.7) c −k T B where Γ is known as the Zeldovich factor [37, 38] defined by

η 1/2 ∂2G(n) Γ , η = 2 (2.8) ≡ 2πkBT − ∂ n n=nc 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. CHAPTER 2. BACKGROUND AND MOTIVATION 13

and its nucleation rate is lower. Hence the Zeldovich factor captures the multiple recrossing 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-Döring 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)= ∆ n + S σ + τk T ln n (2.9) − B

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 logarithmic correction term is τ = 5 in 2D [45] and τ = 1 in 3D [46]. However, it was later 4 − 9 CHAPTER 2. BACKGROUND AND MOTIVATION 14

predicted that the shape fluctuation of a 3D droplet should be suppressed below the roughening temperature [44], which leads to τ = 2 . Our numerical results confirm − 3 5 the τ = 4 prediction 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 5 at low temperatures and only rise to 4 at 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 assumptions: (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 potential difference ∆. 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 temperature and small chemical potential difference ∆ 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 15◦F, while the freezing point of water is 32◦F. When small ice particles were introduced into the supercooled CHAPTER 2. BACKGROUND AND MOTIVATION 15

water, however, crystallization followed immediately, with the temperature of the ice- water mixture rising to 32◦F 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

dP L eq = (2.10) dT T ∆V 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 barrier ◦ that is signiﬁcantly larger than the thermal ﬂuctuation kB T at 15 F. The immediate crystallization from the ice particle can be understood by the fast growth of a nucleus larger than the critical size. CHAPTER 2. BACKGROUND AND MOTIVATION 16

the volume change of the phase transition. The chemical potential diﬀerence between liquid and gas can be written as

∆lg = 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 . The condensation of gas occurs via the nucleation of S small liquid droplets when > 0. S The traditional method of studying gas-liquid nucleation uses a cloud chamber. At a given temperature, the supersaturation is adjusted until droplet formation is S observed. Because the nucleation rate changes so rapidly with supersaturation, but a critical value where I passes through a magnitude of order 1 (per second per cubic Sc micron) can be approximately determined instead of the absolute rate I. The critical supersaturation can then be compared with the prediction of classical nucleation Sc theory. It is found that the classical nucleation theory predicts values of that are Sc typically accurate within 10% for most substances [59]. It should be underscored that a variation of 10% in can leads to changes in I by many orders of magnitude, S because I is exponentially sensitive to the nucleation barrier which is proportional to ∆−2, i.e. (ln )−2. S 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 . The consensus of many works on the Sc nucleation rate is that the variation of nucleation with supersaturation predicted by CNT ln I A(T ) B(T )/(ln )2 (2.12) ≈ − S 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]. CHAPTER 2. BACKGROUND AND MOTIVATION 17

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 classical Becker-D¨oring (BD) theory. Logarithmic scales are used for both axes.

To test the validity of the classical nucleation theory, we can compare the experimental results with two nucleation theorems whose applicability is not restricted to the CNT assumption that droplet free energy has only surface and volume contributions 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 = n (2.13) ∂ ln c T S and (2) the temperature dependence of the nucleation rate is determined by the 4 formation energy ∆U of a droplet with size nc in the absence of supersaturation, i.e. ∂ ln I ∆U = . (2.14) ∂T k T 2 S B 4 U For the deﬁnition of ∆ , refer to Appendix A.2. CHAPTER 2. BACKGROUND AND MOTIVATION 18

What can be inferred from the experimental tests on the classical nucleation theory? Correct dependence, but wrong temperature dependence suggests that the S 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 cluster. 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)/k T ) − B 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 ln n 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 CHAPTER 2. BACKGROUND AND MOTIVATION 19

prediction where ln n 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 fc , 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) A droplet is compact with a well-deﬁned surface. CHAPTER 2. BACKGROUND AND MOTIVATION 20

2.4 Dislocation Nucleation and Materials Strength at Small Scale

Figure 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.

Nucleation has mostly been studied as an initial step of first order phase transitions. 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 CHAPTER 2. BACKGROUND AND MOTIVATION 21

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 CHAPTER 2. BACKGROUND AND MOTIVATION 22

(a) (b) (c)

Figure 2.6: (a) and (b) Two consecutive in situ TEM compression tests on a FIB microfacbricated 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 free of dislocations. (c) MD simulation of nanoindentation process [95]. Snapshot of dislocation 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. 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 the calculation. CHAPTER 2. BACKGROUND AND MOTIVATION 23

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 different 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 homogeneous 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 compute the fundamental properties such as dislocation nucleation, multiplication, motion, 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 CHAPTER 2. BACKGROUND AND MOTIVATION 24

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 relation 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 experimental 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 potential model relies on a theoretical description on the molecular bonding, we can test the model by comparing the various properties obtained from simulations with experimental 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 simulations 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 happening 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 s s h s (3.1) − i j − i i i,j where J > 0 is the coupling constant and h is the external magnetic field. The spin variable s at site i can be either +1 (up) or 1 (down), and the sum is i − i,j 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 T 2.269J/k between ferromagnetic c ∼ B 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 T 4.490J/k is higher due to the increase of number of neighbor spins. c ∼ B 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 CHAPTER 3. COMPUTATIONAL METHODS 27

(a)

(b) (c)

1 Figure 3.1: (a) Magnetization M = N 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/k 2.269J/k , majority of spins in the system are spontaneously B ∼ B aligned in either up or down. However, T > Tc, spontaneous ordering disappears because of thermal ﬂuctuation. (b) schematic of ferromagnetic state below Tc. (c) schematic of paramagnetic state above Tc. CHAPTER 3. COMPUTATIONAL METHODS 28

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 < T . The energy reduction, 2h, of spin m − flip can be compared with the chemical potential difference ∆. 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 nucleation phenomena. (b) The snapshot of Monte Carlo simulation of demagnetization 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.

1Planar interface energy is 2J only at T = 0. At higher temperature, interface energy reduces due to thermal ﬂuctuation CHAPTER 3. COMPUTATIONAL METHODS 29

3.2 Interatomic Potential

In order to model dislocation nucleation, we need a model to describe the interactions between atoms in the crystal. The interatomic potential V is a function of the coordinates of all atoms, i.e.

V ( r )= V (r , r , , r ) (3.2) { } 1 2 n

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

∂V fi . (3.3) ≡ − ∂ri

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 efficiency, 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 dependence 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. CHAPTER 3. COMPUTATIONAL METHODS 30

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 potential 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 transitions and deformation at elevated temperature. For readers interested in thermal properties related to phase transition, we present thermal properties of various interatomic 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,

σ 1 σ 6 V LJ(r)=4ǫ 2 (3.4) ij r − r 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 r is 21/6σ and the potential function has the vale ǫ. These parameters m − can be ﬁtted to reproduce experimental data.

3the ab initio methods scales with N 3 or higher [102]. CHAPTER 3. COMPUTATIONAL METHODS 31

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,

1 1 2 V SW( r )= φ(r )+ g(r )g(r ) cos θ + (3.5) { } 2 ij ij ik jik 3 ij ijk

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 θ = 1 which corresponds to the angle between two arms of sp3 jik − 3 orbital. This gives a realistic description of crystalline silicon, germanium, and other semiconductor materials having diamond-cubic structures. However, the built-in tetrahedral 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 CHAPTER 3. COMPUTATIONAL METHODS 32

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,

1 V Tersoﬀ( r )= f (r )[f (r )+ b f (r )] (3.6) { } 2 C ij R ij ij A ij i=j

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]. CHAPTER 3. COMPUTATIONAL METHODS 33

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 )= φ( r r )+ F (ρ ) (3.7) { } | i − j| i i

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 ρ . Commonly used embedding function is F (ρ) = A (ρ) as in i − 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 interactions. 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 (C C ) and can be fitted to accurately reproduce the elastic constants of metals. 12 − 44 EAM potentials have been developed for various metals in different fitting procedures. 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] developed a modified version of EAM potential that can describe all the bonding mechanism in a single mathematical form. The idea is simple but elegant. To describe CHAPTER 3. COMPUTATIONAL METHODS 34

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 materials, 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 constants 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 experiments. 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 constants in most interatomic potentials. Thus, elastic constants and ideal shear strength CHAPTER 3. COMPUTATIONAL METHODS 35

are underestimated by the EAM potential when perfoming molecular dynamics simulation at ﬁnite temperature.

(a) (b)

Figure 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].

To describe the dislocation nucleation, the empirical potential must be capable of model the slip system of copper correctly. The atomic arrangement on the 111 { } planes 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 the FCC 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 dislocations 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 directions. 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 CHAPTER 3. COMPUTATIONAL METHODS 36

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 fault energy γUSF of Cu from experiments, EAM potential [111], and ab initio calculation [114]. ab initio data depend on the exchange-correlation functionals used in the study.

2 2 γSF (mJ/m ) γUSF (mJ/m ) Experiment 45 EAM 44.4 158 ab initio 37-41 158-186

Figure 3.4: Generalized stacking fault energy on (111) plane along a/6[211] direction calculated with EAM potential [111].

It is also found that thermal expansion and thermal softening effect are significant 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 reduce the free energy barrier of dislocation nucleation. Thus, it is very important to CHAPTER 3. COMPUTATIONAL METHODS 37

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

Figure 3.5: Linear thermal expansion of Cu calculated in the quasi-harmonic approximation (QHA) and by the Monte Carlo method using the EAM potential [111]. The melting point of Cu (Tm) is indicated. CHAPTER 3. COMPUTATIONAL METHODS 38

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 impossible to analytically trace the trajectories of particles in a system and calculate the properties of such complex systems 4. With efficient numerical integration techniques, the positions and velocities of all particles are traced on the fly. The macroscopic 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 numerically. For example, a solid simulation cell where the positions of all atoms are randomly disturbed will approach a thermal equilibrium where Boltzmann distribution 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 out that can not be expressed by analytic solutions [116]. CHAPTER 3. COMPUTATIONAL METHODS 39

Two categories of temperature control algorithms (thermostats) exist: deterministic and stochastic thermostats. Both schemes are used for all of our constant temperature simulations. Nosè-Hoover thermostat, a representative deterministic thermostat, 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 Nosè-Hoover chain thermostat [118] in our MD simulation of crystalline copper. Anderson thermostat [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 methods can be found in the literature [122].

3.4 Monte Carlo Simulation

Monte Carlo simulation is another method to sample the configuration space according to the Boltzmann distribution at a given temperature. Unlike the MD, Monte Carlo (MC) simulation generates artificial trajectories spanning the configurational CHAPTER 3. COMPUTATIONAL METHODS 40

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 simulation 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)/k T ) ∝ − B at equilibrium, and the detailed balance condition leads to the following restriction CHAPTER 3. COMPUTATIONAL METHODS 41

on the transition probabilities

π(R, R′) ρ(R′) = = exp[ β(V (R′) V (R))]. (3.12) π(R′, R) ρ(R) − − 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′)P (R, R′), forR′ = R (3.13) acc P (R, R′) min 1, exp[ β(V (R′) V (R))] (3.14) acc ≡ { − − } 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 CHAPTER 3. COMPUTATIONAL METHODS 42

of nucleation rate without a priori knowledge of any nucleation theory. FFS is applicable 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 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 configurations 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 returning back to nA is counted as successful forward flux. Then, the probability of reaching next interface P (n n ) becomes N/M. i+1| i 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 CHAPTER 3. COMPUTATIONAL METHODS 43

in the phase space where λ = λ , i = 0, 1, 2, , n 1, λ < λ < < λ . In i − A 0 n 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 multiplication of two terms I = I P (λ λ ) (3.15) 0 n| 0

where I0 is the average ﬂux across the interface λ = λ0 (i.e. leaving state A), and P (λ λ ) is the probability that a trajectory leaving state A will reach state B before n| 0 returning to state A again. Because it is impossible to reach interface λ = λi+1 without reaching interface λ = λ ﬁrst, the probability P (λ λ ) can be decomposed i n| 0 into a series multiplication,

n−1 P (λ λ )= P (λ λ ) (3.16) n| 0 i+1| i i=0 where P (λ λ ) is the probability that a trajectory reaching λ , having come from i+1| i i 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 = λ before returning to state A. From these simulations the probability P (λ λ ) 1 1| 0 is computed and an ensemble of configurations at interface λ = λ1 is created. The process is repeated to compute the probability P (λ λ ) for each i = 1, , n 1. i+1| i − For each (T, h) condition, the interfaces are chosen manually such that the spacing CHAPTER 3. COMPUTATIONAL METHODS 44

λ λ between interfaces increases linearly with λ , with the ﬁrst spacing λ λ | i+1 − i| i | 1 − 0| large enough to have P (λ λ ) 10−1. The nucleation rate constant turns out to 1| 0 ≤ 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 (λ λ ) P (λ λ )P (λ λ ) P (λ λ ) i| 0 ≡ 1| 0 2| 1 i| i−1 for the 2D Ising model at kBT = 1.5 and h = 0.05. In this test case, we ﬁnd I =1.45 10−8 MCSS−1 with λ = 24 from a brute force Monte Carlo simulation with 0 × 0 107 MCSS. 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 = λ from interface λ = λ is P (λ λ )=1.92 10−11 with λ = 1200. Following n 0 n| 0 × n Eq. (3.15), the nucleation rate under this condition is IFFS =2.78 10−19 MCSS−1. × 100 1

0.8

) 5 ) 0 10 0.6 i λ λ | ( i P (495) = 0.5 B

λ B P

P( 0.4

1010 0.2 0 0 100 200 300 400 500 600 λ i

Figure 3.7: (color online) The probability P (λ λ ) (solid line) of reaching interface i| 0 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 *.

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-D¨oring 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 CHAPTER 3. COMPUTATIONAL METHODS 45

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 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.

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: N ∆G(n) n = exp . (3.17) N − k T B 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 CHAPTER 3. COMPUTATIONAL METHODS 46

thermal ﬂuctuation k T . For instance, when ∆G 10k T corresponding to the B ∼ B Boltzmann factor exp( ∆G/k T ) 4.5 10−5, we would sample the droplet once − B ∼ × 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 raw histogram taken from umbrella sampling simulations. (1) We ﬁrst obtain the biased ′′ relative 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 1 2 weighing function, exp[ 2 k(n−ni) ], to the data of each window i. (3) From the over- kB T 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 energy curve ∆G(n) can be obtained from k T ln(∆G(n)). − B CHAPTER 3. COMPUTATIONAL METHODS 47

Note that the ensemble average of a quantity A can be written as

dRA(R) exp( βV (R)) A = − . (3.18) dR exp( βV (R)) − It is known that the ensemble average A can be rewritten as A/W (R) A = W (3.19) 1/W (R) W where W is an arbitrary weighting function that is written as W (R) = exp[ βw(R)]. − The subscript ... indicates an ensemble average according to the biased distribu- W 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

1 w[n(R)] = k[n(R) n ]2. (3.20) 2 − 0

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 by1/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]. CHAPTER 3. COMPUTATIONAL METHODS 48

3.5.3 Computing Rate from Becker-D¨oring 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 + fc . 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) 70 20 F = 61.3 c 60 n = 496 c 15 50 〉 40 (t) 2 10 n F (n) ∆

30 〈 20 5 10

0 0 0 100 200 300 400 500 600 0 0.05 0.1 0.15 0.2 n t (MCSS)

Figure 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.

As an example, Fig. 3.10 (a) shows the droplet free energy F (n) computed from 5 US at kBT = 1.5 and h = 0.05 for the 2D Ising model . The order parameter is the size of the largest droplet, n. A parabolic bias function 0.1k T (n n)2 is used, B − 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 fc , 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 . CHAPTER 3. COMPUTATIONAL METHODS 49

as an initial condition, we run Monte Carlo simulations and obtain the eﬀective attachment rate from the following equation,

∆n2(t) f + = , (3.21) c 2 t 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 + −1 time. From the slope of this curve, we obtain fc = 39.1 MCSS . A similar approach was used by Brendel et al. [126] to compute the interface diffusion coefficient. + Combining the values of fc , Γ, Fc and plug them into Eq. (2.7), we find that the Becker-Döring theory would predict the nucleation rate to be IBD = 2.37 × 10−19 MCSS−1, if the correct free energy function F (n) is used. This is very close to the FFS result IFFS = 2.78 10−19 MCSS−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-Döring 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 theories. The increase of computational power and the development of advanced sampling 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 studied 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 results, 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 usually 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 numerical 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-Döring 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-Döring 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, fc , which is part of the Becker-Döring theory, is computed independently from Monte Carlo simulations starting from the ensemble of critical clusters. The nucleation rate predicted by the CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 52

+ Becker-Döring theory, using the so computed F (n) and fc 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- Döring 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 correction 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 parameters. 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 numerical simulations and CNT. For the 2D Ising model, the logarithmic correction term to the droplet free energy was often neglected [47, 126]. Because the logarithmic 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 temperatures [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 CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 53

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 s s h s (4.1) − i j − i i i,j where J > 0 is the coupling constant and h is the external magnetic field. The spin variable s at site i can be either +1 (up) or 1 (down), and the sum is over i − i,j 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 s = 1 for most of the spins. The dynamics follows i − 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/k T ) is proportional to the cluster population per site to present results − B that are invariant when the simulation cell size changes.

4.2.1 Becker-D¨oring Theory

To compute the nucleation rate using Eq. (2.1) and Eq. (2.7), the surface free energy σ and bulk chemical potential diﬀerence ∆ must be known for the Ising model. The chemical potential diﬀerence is simply ∆ =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. CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 54

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-Döring 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].

2.5 σ eff 2 σ (10) 1.5 eff σ 1

0.5

0 0 0.5 1 1.5 2 k T B

Figure 4.1: (color online) Eﬀective surface free energy σeﬀ as a function of temperature 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.

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

σ F (n)=2√πn σeﬀ (T ) (4.2) regardless of whether its equilbrium shape determined by Wulﬀ construction is circular or not 2. The free energy of a droplet can be written as,

F 2D(n)=2√πn σ (T ) 2 h n (4.3) eﬀ − 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 function of T is also absorbed in σeﬀ (T ). CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 55

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,

π σ2 (T ) n2D = eﬀ (4.4) c 4 h2

as well as the free energy barrier

π σ2 (T ) F 2D = eﬀ (4.5) c 2h

We also obtain the Zeldovich factor deﬁned in Eq. (2.8)

2 h3/2 Γ2D = (4.6) k T π σ (T ) B eff Assuming the critical droplet has a circular shape, the attachment rate can be written as +2D 2D fc =2β0(T ) π nc (4.7) where β0(T ) is the average spin-flip frequency at the boundary of the droplet. As an approximation, β exp( σ (T )/k T ) (4.8) 0 ≈ − eff B 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-Döring theory

2 h π σ2 (T ) I2D (h, T )= β (T ) exp eﬀ (4.9) BD 0 k T − 2 hk T B B

Given the analytic expressions for σeﬀ (T ) in 2D, the predictions of the Becker-

D¨oring theory can be computed explicitly. For example, at kBT =1.5 and h =0.05, we have n2D = 463, F 2D = 46.3, Γ2D = 0.0034, f +2D = 34.0, and I2D = 4.5 c c c BD × 10−15 MCSS−1. The numerical results (in Section 3.5.3) under the same condition are n = 496, F = 61.3, Γ = 0.0033, f + = 39.2, and I = 2.37 10−19 MCSS−1. As c c c × CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 56

discussed further below, the four orders of magnitude discrepancy in the nucleation 2D rate mainly comes from the underestimate of Fc 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) ∆n + τk T ln n + d(T ). (4.10) − B 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 constant 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)= σ (T ) α n2/3 2 h n (4.11) eﬀ −

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,

α3σ3 (T ) n3D = eff (4.12) c 27 h3 α3σ3 (T ) F 3D = eff (4.13) c 27 h2 9 h2 Γ3D = (4.14) πk T 3 3 B α σeff (T ) +3D 2/3 fc = β0(T )αnc (4.15) CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 57

Finally, the nucleation rate predicted by the Becker-D¨oring theory is

α3σ (T ) α3σ3 (T ) I3D (h, T )= β (T ) eﬀ exp eﬀ (4.16) BD 0 9πk T −27 h2 k T B B

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, 5 as in Eq. (2.9). In 2D Ising model, τ = 4 , 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,

2 2 2D/F T √πσeﬀ + πσeﬀ +8τkBT h nc = (4.17) 4 h

This equation is to be compared with Eq. (4.4) predicted by the Becker-D¨oring 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 ln n correction term also modiﬁes the critical nucleus size in the 3D Ising 3D model. The analytic expression for nc 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 = 1 above the roughening temperature T [46] and τ = 2 below − 9 R − 3 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 CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 58

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∆, 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 CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 59

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-Döring Eq. (2.7), but using the free energy curve obtained from US, as described in Section III.B. The + pre-exponential factor, fc 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 CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 60

(a) (b) 0 10 2

k T = 1.9 −10 B 1.5 10 1.5

) −20 −1

10 BD

/ I 1 −30 FFS

10 1.2 I I (MCSS

−40 0.5 10

−50 10 0 0 0.05 0.1 0.15 0 0.05 0.1 0.15 h h (c) (d) 2 −10 k T = 2.80 10 B 2.71 2.65 2.50 1.5 ) 2.35 −1 2.20 −20 BD

10 / I 1 FFS I I (MCSS 0.5 −30 10

0 0.3 0.4 0.5 0.6 0.3 0.4 0.5 0.6 h h

Figure 4.3: The nucleation rate I computed by FFS (open symbols) and Becker- Döring 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-Döring 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.

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-Döring 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 CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 61

states along the Markov chain, as is assumed by the Becker-Döring theory, has been shown by a recent study [135] and our confirmation of I = f +Γ exp Fc provides c − kBT another evidence for it. This means that the Becker-Döring 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)

3 10

3 2.65 10 1.0 2.35 1.5 k T = 2.20 2.71 B 2.50 c c n n k T = 1.9 B

2 10

2 10 −2 −1 −0.5 −0.4 −0.3 −0.2 10 10 10 10 10 10 h h

Figure 4.4: (a) For the 2D Ising model, the critical droplet size n obtained from FFS (filled symbols) and umbrella sampling (open symbols). nc predicted by Becker- Döring 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).

After each FFS simulation under a given (T, h) condition, an ensemble of 15,000 CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 62

(a) (b)

0.25 0.25 3D

0.2 0.2 ) ) B 0.15 B 0.15 P(P P(P 0.1 0.1

0.05 0.05

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 P P B B

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. Representative droplets are also shown, with black and white squares corresponding to +1 and 1 spins, respectively. (b) Histogram of committor probability in an ensemble of spin− conﬁgurations 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,

P (λ )= P (λ )P (λ λ ) (4.18) B i B i+1 i+1| i for i = n 1, n 2, ...1 with the boundary condition P (λ ) = 1 [143]. By fitting the − − B n 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 obtained 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 confirms 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 CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 63

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-D¨oring 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-D¨oring theory, Eq. (4.3), and Langer’s ﬁeld theory, Eq. (4.10), are plotted together. It is clear that the logarithmic correction

term τkBT ln n 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-Döring 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 CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 64

(a) (b) 70 30 US data Eq.(D12) Eq.(6) US data Eq.(6) 60 25

50 Eq.(8) 20 Eq.(8) 40 15 F(n) F(n) 30 10 20 τ τ k T ln n k T ln n B 5 B 10 d d 0 0 0 100 200 300 400 500 600 0 5 10 15 20 25 n n

Figure 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 5 Eq. (4.3) (dashed line). Logarithmic correction term 4 kBT ln n (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). ≤ 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 5 temperatures (T = 0.59 Tc and 0.71 Tc) and only goes to 4 near 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 ln n 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 5 0.53 Tc to 0.84 Tc) are presented in the Appendix B.4. τ = 4 is 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-Döring formula of the nucleation rate. CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 65

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. F/BD Fig. 4.4 shows the critical droplet sizes nc (dotted lines), which correspond to the maximum of the free energy curves F (n) predicted by the Becker-Döring theory, comm Eq. (4.3). They are significantly smaller than the critical droplet sizes nc (filled symbols) that corresponds to a 50% committor probability. With the field theory F/FT correction term in the free energy, the critical droplet sizes nc (solid lines) agree comm much better with nc . 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 ln n 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. CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 66

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) 700 70 US data Eq.(D12) Eq.(25) Eq.(15) Eq.(25) US data 600 60 Eq.(15)

500 50

40 400

30 F(n) 300 F(n) 20 200 10 100 B exp(−Cn/B) τ k T ln n 0 B τ k T ln n 0 B −10 0 100 200 300 400 500 600 700 0 5 10 15 20 25 n n

Figure 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 ln n is also plotted (dot-dashed line). The difference in predictions by classical expression 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).

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-Döring form, Eq. (4.11), plus a constant CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 67

correction term, i.e.,

F (n)= σ (T ) α n2/3 2 h n + d(T ) (4.19) eff − 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 − − 1/2 1 750 R (F (i) F (i))2 , where F (i) is the numerical data from US, and ≡ 700 − fit i=50 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)= σ (T ) α n2/3 + τ(T ) k T ln n 2 h n + d(T ) (4.20) eff B − 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 parameters. Perini et al [46] used the following functional form to fit their free energy data,

F (n)= σ (T ) α n2/3 + K(T ) n1/3 + τk T ln n 2 h n + d(T ) (4.21) eﬀ B − where τ = 1 is constrained to be a constant 3. The parameter K corresponds to − 9 3This is against the expected temperature dependence of τ due to suppression of shape ﬂuctuation CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 68

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

C n F (n)= σ (T ) α n2/3 + τ(T ) k T ln n 2 h n + A + B(T ) exp (4.22) eﬀ B − −B(T )

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 k T 16.08. The B − 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 k T 2.8 and 100 100. 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 lead to large discrepancies in the range of n > 100. Given that the droplet theory is supposed to work better in the continuum limit of large n, we believe the function should be fitted to data at large n. 5 3D Eq. (4.22) also fits the data at non-zero h. However, the resulting σeff from the fit slightly 3D increases with h. For example, σeff increase by about 3% as h changes from 0 to 0.5. CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 69

(a) (b) 3 0 σ eff T 2.5 R −0.05

T 2 σ −0.1 R (100) eff 1.5 τ −0.15 σ

1 −0.2

0.5 −0.25

0 0 1 2 3 4 5 2 2.2 2.4 2.6 2.8 k T k T B B

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 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.

Fig. 4.8(a) shows the ﬁtted values of σeﬀ in the temperature range of 0.4-0.65 Tc.

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 σ converges to the free energy of (100) surfaces. In the limit of T 0, we eff → 1/3 expect σeff to converge to (6/π) 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 CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 70

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 k T +0.44. The fact that τ < 0 in 3D is consistent with theoretical − B 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 = 2 at T < T − 3 R and τ = 1 at T > T . The change of τ with T may be the consequence of a − 9 R 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 k , corresponding to the exp(S/k ) 1019, 19 orders of contribution to the B B ∼ 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 CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 71

(a) (b)

70 70 k T=0.9 US data B 65 Eq.(10) 60 1.0 1.1 60 50 1.2 1.3 55 40 1.4 c 1.5 c F F 50 S = 43.5 k 30 B 45 20 40 S = 53.4 k B

10 35

0 30 0 50 100 150 200 250 300 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 k T n B (c) (d)

180 k T=2.20 B US data 170 Eq.(17) 150 160 2.35 150 2.50 140 100

c 2.65 c F F 2.71 130 120 S = 143 k 50 110 B 100 S = 152 k B 90

0 80 0 100 200 300 400 2.1 2.2 2.3 2.4 2.5 2.6 2.7 k T n B

Figure 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). 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, CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 72

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/k ) 1062, 62 orders of contribution to the nucleation rate. As a B ∼ 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 assumptions 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 nucleation 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 results 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 exponential 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 correction 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 CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 73

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 about0.026 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 ∆ 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 fluctuation, the change of critical size from logarithmic term will be negligible 6. While logarithmic term τkBT ln n 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(τk T ln n /k T ) nτ . Unless the B c B ∼ c 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 simulation, 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 exponential 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 ∆ is a few to a few tens of kB T in typical gas-liquid transition experiments, as shown in the 2 Fig. 2.3. Thus, σ kB T ∆ because σ kB T . ≫ ≫ CHAPTER 4. NUMERICAL TESTS OF NUCLEATION THEORIES 74

G(n) must vanish at n = 1. A negative constant term correction must be subtracted from the free energy expression ∆G(n)= σS n∆ 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 deformation 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 microelectronic, 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 barrier, 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-D¨oring theory [36], which lead to similar expressions for the nucleation rate,

G (σ, T ) I(σ, T )= N ν exp c (5.1) s 0 − k T B

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,

F (γ, T ) I(γ, T )= N ν exp c (5.2) s 0 − k T B 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 reaction trajectory may cross the saddle region multiple times. This deﬁciency can be corrected by introducing a recrossing factor which can be computed by running CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 77

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 k ). For example, the activation entropy of kink migration on a ∼ B ◦ 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 dislocation 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 context 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 between 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 CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 78

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 enthalpy. 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 CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 79

T . The activation Gibbs free energy is deﬁned as,

G (σ, T ) G(n , σ, T ) G(0, σ, T ) (5.3) c ≡ c − where G(0, σ, T ) is the Gibbs free energy of the perfect crystal (containing no dislocations) 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.

1 ∂G(n, σ, T ) γ(n, σ, T ) (5.4) ≡ −V ∂ σ n,T 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.,

1 ∂ F (n, γ, T ) σ(n, γ, T ) = (5.6) V ∂γ n,T 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 V to 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. CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 80

stress or constant strain) ensemble. The activation Helmholtz free energy is deﬁned as, F (γ, T ) F (n ,γ,T ) F (0,γ,T ). (5.8) c ≡ c −

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 ∂G (σ, T ) S (σ, T ) c (5.9) c ≡ − ∂T σ measures the reduction rate of G (σ, T ) with increasing T . Similarly, the activation c volume, defined as ∂G (σ, T ) Ω (σ, T ) c (5.10) c ≡ − ∂σ T measures the reduction rate of G (σ, T ) with increasing σ. c 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, G (σ, T )= H (σ) TS (σ). (5.12) c c − c Consequently, the dislocation nucleation rate in Eq. (5.1) can be rewritten as,

S (σ) H (σ) I(σ, T )= N ν exp c exp c . (5.13) s 0 k − k T B B 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 CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 81

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.

40 10 ν exp(S /k ) 0 c B

20 10

H /k c B 0 10 I (per atom)

−20 10

0 2 4 6 −1 −3 1/T (K ) x 10

Figure 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-D¨oring 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. ∼

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

∂F (γ, T ) S (γ, T ) c (5.14) c ≡ − ∂T γ which measures the reduction rate of Fc(γ, T ) with increasing T . We can then deﬁne CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 82

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,

F (γ, T )= E (γ) TS (γ). (5.16) c c − c

Consequently, the dislocation nucleation rate in Eq. (5.2) can be rewritten as,

S (γ) E (γ) I(γ, T )= Nν exp c exp c . (5.17) 0 k − k T B B

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. CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 83

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 Ω , as long as σ and γ lie on the stress-strain curve of the perfect ≫ c 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 nucleation 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,

∂G (σ, T ) ∂F (γ, T ) S (σ, T ) c = c c ≡ − ∂T − ∂T σ σ ∂F (γ, T ) ∂F (γ, T ) ∂γ = c c (5.18) − ∂T − ∂γ ∂T γ T σ ∂F c(γ, T ) ∂γ = Sc(γ, T ) . (5.19) − ∂γ ∂T T σ Similarly, starting from the deﬁnition of Sc(γ, T ), we can show that,

∂G (σ, T ) ∂σ S (γ, T ) = S (σ, T ) c . (5.20) c c − ∂σ ∂T T γ CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 84

Eqs. (5.19) and (5.20) are consistent with each other because of the Maxwell relation,

∂σ ∂T ∂γ = 1 (5.21) ∂T ∂γ ∂σ − γ σ T and the chain rule of diﬀerentiation,

∂G (σ, T ) ∂F (γ, T ) ∂γ c = c . (5.22) ∂σ ∂γ ∂σ T T T Therefore, the diﬀerence between the two activation entropies is

∂G (σ, T ) ∂σ ∂F (γ, T ) ∂γ ∆S S (σ, T ) S (γ, T )= c = (5.23). c ≡ c − c ∂σ ∂T − ∂γ ∂T T γ T σ Recall the deﬁnition of activation volume Ωc in Eq. (5.10), we have

∂σ ∆S = Ω . (5.24) c − c ∂T γ ∂σ Notice that Ωc is always positive and that, because of thermal softening, ∂T γ is 2 usually negative . 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 reactions [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 3 is bounded. As a rough estimate, let us assume that Ωc < 100A˚ . Under low stress ∂σ conditions, we expect ∂T γ to be linear with stress, i.e., ∂σ 1 ∂ σ (5.25) ∂T ≈ ∂T γ γ 2 Here we are excluding the effect of thermal expansion, in case σ refers to a normal stress component, by always defining the strain relative to the zero stress state at temperature T . Even though most crystals exhibit thermal softening, exceptions do exist. For example C66 of α-quartz decreases with temperature [166]. CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 85

where is the shear modulus. Although for chemical reactions it is more appropriate to use the bulk modulus instead of , this approximation is acceptable for a rough estimate. Let us assume that reduces by 10% as T increases from 0 to 300 K [169], 1 ∂µ −4 −1 then µ ∂T is approximately 3.3 10 K . If we further assume that σ < − γ × 100 MPa, then ∆S < 0.25 k , which is negligible. c B 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 Ω σ−2 in the limit of σ 0. the following c ∝ → relation, Ω σ−2 (5.26) c ∝ Combining this with Eq. (5.25), we have,

1 ∂ ∆S σ−1 (5.27) c ∝ − ∂T γ 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 heterogeneous 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 circular dislocation loop of radius r, i.e. E = 2πrτ bπr2σ, where τ is the dislocation line energy per loop − unit length and b is the magnitude of Burgers vector. Maximizing Eloop with respect to r gives the 2 2 πτ τ 2 πτ energy barrier Ec = bσ at the critical radius rc = bσ . The activation volume is Ωc = bπrc = bσ2 , which diverges in the limit of σ 0. → CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 86

state and the “initial” (i.e meta-stable) state. It may seem puzzling that the activation 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 ∆S S (σ) S (γ) as Eq. (5.24). c ≡ c − c 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 activation 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, ν = 6.25 1012 s−1. Expanding 0 × 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 CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 87

nucleus) up to second order, we get

N νm ν exp(S /k )= i=1 i (5.28) 0 c B N−1 a i=1 νi

m a where νi and νi are the positive normal frequencies in the meta-stable state and the “activated” state, respectively [154, 156, 173] and is the number of normal modes N 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 13 −1 13 −1 is on the order of 10 s . Recall that ν0 itself is also on the order of 10 s 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 of the crystal. Because

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, (T ) G (σ, T )= H (σ) (5.29) c c (0) where (T ) and (0) are the shear moduli of the crystal at temperature T and zero temperature, respectively. Assuming that (T ) is a linear function of T , we arrive at CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 88

the following estimate for Sc(σ),

1 ∂ S (σ)= H (σ) . (5.30) c − c (0) ∂T

A similar expression has been derived for dislocations overcoming obstacles [176, 177]. 1 ∂µ Note that µ(0) ∂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 1 ∂ = (5.31) T ∗ −(0) ∂T

Then we arrive at the following estimate of Sc(σ),

H (σ) S (σ)= c (5.32) c T ∗

Again, if we assume reduces by about 10% as T increases from 0 to 300 K, then T ∗ 3000 K. This means S (σ) 7.5 k when H (σ) = 2 eV. ≈ c ≈ B c While the above analysis seems quite reasonable, the same argument can be applied to the activation Helmholtz free energy, leading to the following approximations,

(T ) F (γ, T ) = E (γ) (5.33) c c (0) E (γ) S (γ) = c (5.34) c T ∗ with the same T ∗ as deﬁned in Eq. (5.31). Again, assuming T ∗ 3000 K, we have ≈ S (γ) 7.5 k when E (γ) = 2 eV. Given the large diﬀerence between S (σ) and c ≈ B c c 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,

T G (σ, T )= H (σ) 1 (5.35) c c − T m CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 89

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,

T F (γ, T )= E (γ) 1 (5.36) c c − T m 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 heterogeneous 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 CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 90

3 T = 0 K 100 K 2.5 200 K 300 K 2 400 K 500 K 1.5 600 K (GPa) xy σ 1

0.5

0 0 0.05 0.1 0.15 0.2 γ xy (a) (b) σ

4 T = 0 K [001] bp 100 K 3 200 K 300 K

(GPa) 400 K

zz 2

σ 500 K [100] 600 K 1 [010] 0 0 0.05 0.1 ε zz (c) (d)

Figure 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.

(EAM) potential [111]. As shown in Fig. 5.2 (a), the simulation cell to study homogeneous 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. CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 91

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 Nos´e-Hoover thermostat [117] for equilibrating systems with a wide range of eigenfrequencies. CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 92

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

F (γ, T ) IBD(γ, T )= N f + Γ exp c (5.37) s c − k T B

+ where fc 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 4 Gibbs free energy Gc (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.

50 0.6 F = 0.53 eV c 40 0.5 n = 34 c

0.4 30 (t)> 2

0.3 n ∆ + F(n) (eV) < 20 2f c 0.2 10 0.1

0 0 0 20 40 60 0 10 20 30 40 n t (fs) (a) (b)

Figure 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.

4 The activation Gibbs free energy at zero temperature, i.e. Gc(σ, T = 0), can be obtained from the 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. CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 93

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 k Tˆ(n n)2 is super-imposed on the EAM B − potential, where Tˆ = 40 K and n is the center of the sampling window. We chose Tˆ 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 1/2 η 2 2 the deﬁnition Γ , where η = ∂ F (n)/∂n n=n . ≡ 2πkBT − | c

(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 nucleation. 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 d . We chose d = 0.33, 0.38, and 0.43 A˚ for T 400 c c ≤ 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] CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 94

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 dislocation loop at 0 K appears symmetric, the configuration at 300 K is distorted due to thermal fluctuation. + The attachment rate fc 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, 2fc t, in order to extract fc (See section 3.5.3). CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 95

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. × 1

0.8

0.6 (t) s P 0.4

0.2

0 0 1 2 3 4 5 t (ns)

Figure 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. − × 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 ± 1/2 η 2 2 factor, Γ = 0.051, is obtained from Γ , where η = ∂ F (n)/∂n n=n . ≡ 2πkB T − | c CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 96

Using the conﬁgurations collected from umbrella sampling with n = nc as initial conditions, MD simulations give the attachment rate f + = 5.0 1014 s−1, as shown c × 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 frequency 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 homogeneous 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 =Γfc is a slowly changing function of stress and temperature. It varies by less than a factor of two for all the conditions tested here. The average value of ν is about 2.5 1013 s−1, which is 0 × 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(S /k ) 104, to the absolute nucleation rate, c B ≈ and cannot be ignored. CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 97

2.5 2.5 0K 0K 100K 100K 2 200K 2 200K 300K 300K 1.5 1.5 400K 500K (eV) (eV) c c F 1 G 1

0.5 0.5

0 0 0.08 0.1 0.12 0.14 0 1 2 3 γ σ (GPa) xy xy (a) (b)

2.5 2.5

2 2 S = 9 k B 1.5 1.5 (eV) (eV) c c F 1 G 1 S = 48 k B 0.5 0.5

0 0 0 100 200 300 400 500 0 100 200 300 400 T (K) T (K) (c) (d)

Figure 5.6: Activation Helmholtz free energy for homogeneous dislocation nucleation 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 umbrella 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.

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 CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 98

energy, E˜c = 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 E˜c 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, thermal expansion pushes neighboring atoms further apart and weakens their mutual interaction. This expansion makes crystallographic planes easier to shear and significantly reduces the free energy barrier for dislocation nucleation. Here, we confirm that Sc(γ) arises almost entirely from the anharmonic effect for dislocation nucleation. At T = 400 K and T = 500 K, we observe significant differences between Fc computed from umbrella sampling and E˜c computed from a zero temperature MEP search in expanded cell. These differences must also be attributed to anharmonic effects. The activation energy E˜c 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[S (σ)/k ] 1020 to the absolute nucleation c B ≈ 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 =0K CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 99

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

2 2 0K 0K 200K 200K 1.5 300K 1.5 300K 400K 400K 500K 500K

1 (eV) 1 (eV) c c F G

0.5 0.5

0 0 0.02 0.03 0.04 0.05 0.06 0 0.5 1 1.5 2 2.5 ε σ (GPa) zz zz (a) (b)

Figure 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.

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 umbrella 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 nucleation, changing less than one order of magnitude at all γ and T conditions tested. Interestingly, the average value of ν is about 0.5 1013 s−1, several times smaller 0 × + than the average prefactor homogeneous nucleation. The measured growth rate fc of a critical nucleus turns out to be signiﬁcantly smaller (by about a factor of 3) due CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 100

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 because 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 elastic 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 relation 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) CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 101

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 E (γ)/T ∗ with T ∗ 2700 K as shown in Fig. 5.8 (a), while S (σ) is not proportional c ≈ c to Hc(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 anharmonic 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 explanation [179, 190] that the activation energy is provided by multiple (small) excitations.

The breakdown of the “compensation law” for Sc(σ) in homogeneous dislocation nucleation 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 CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 102

15 15

10 10

B B T* = 2450 K /k /k γ c ε c

S T* = 2700 K S 5 5

0 0 0 1 2 3 0 1 2 3 E (eV) E (eV) c c (a) homogeneous (b) heterogeneous

100 40

* 80 T = 640 K 30

60 B B T* = 930 K /k /k

σ c σ c 20 S 40 S

10 20

0 0 0 1 2 3 4 5 0 1 2 3 H (eV) H (eV) c c (c) homogeneous (d) heterogeneous

Figure 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 . energy as a function of strain, i.e.

(T ) F (γ, T ) E (γ) (5.38) c ≈ c (0)

whereas Eq. (5.29) is not a good approximation for Gc(σ, T ). In other words, Eq. (5.34)

can be considered as a reasonable approximation to the activation entropy Sc(γ), i.e.

E (γ) ∂ S (γ) c (5.39) c ≈ − (0) ∂T

whereas Eq. (5.32) is not a good approximation for Sc(σ). Appendix C.4 contains CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 103

more discussions on the approximation of Sc(σ).

5.5.2 Entropic Eﬀect on Nucleation Rate and Yield Strength

−14 −9 −4 1 6 −14 −9 −4 1 6 500 10 10 10 10 10 500 10 10 10 10 10 10 6 10 10 10 6 1 1 10 10 −4 −4 10 10 400 −9 400 −9 10 10 −14 −14 300 300

T (K) 200 T (K) 200

100 100

0 0 1.6 1.8 2 2.2 2.4 2.6 1 1.2 1.4 1.6 1.8 2 2.2 σ (GPa) σ (GPa) (a) (b)

Figure 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.

In this section, we discuss how do the activation entropies affect experimental measurements. 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 = fc 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 CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 104

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 entropy 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.

4 2.5

3 2 2

1.5 1 Shear Nucleation Stress (GPa) 1 0

0 100 200 300 400 500 Compressive Nucleation Stress (GPa) 0 100 200 300 400 500 T (K) T (K) (a) (b)

Figure 5.10: (a) Nucleation stress of our bulk sample (containing 14,976 atoms) under constant shear strain loading rateγ ˙ = 10−3 and (b) nucleation stress of the nanorod under constant compressive strain loading rateǫ ˙ = 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 nucleation 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].

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 rateǫ ˙. G (σ , T ) k T N ν c Y = ln B s 0 (5.40) kB T E ǫ˙ Ωc(σY , T ) CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 105

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 and the uniaxial strain rateǫ ˙ byγ ˙ . 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 . The athermal nucleation stress causing dislocation nucleation at T =0K |σ=σY 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 nuc σxy = 2.0 GPa for homogeneous nucleation, about 71% of the athermal nucleation nuc stress, and σzz =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 homogeneous 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 CHAPTER 5. PREDICTING THE DISLOCATION NUCLEATION RATE 106

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 activation volume times a term that characterizes the thermal softening effect. We have shown that the Becker-Döring 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 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 conditions. 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 discussion 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 spanning 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 coefficient 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 activation entropies depending on whether constant stress or constant strain ensemble is used. Similar to the Ising model study, we have shown that the Becker-Döring 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 beyond 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 dislocation 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 nanomaterials 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 CHAPTER 6. SUMMARY AND OUTLOOK 109

as nanoscale electronic, optical, and chemical-sensing devices. However, many fundamental 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 reasonable description of NW growth process, ab initio simulations are prohibitively expensive, and classical MD or MC simulations based on empirical interatomic potentials 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 addressed by atomistic simulations and provided numerical methods to tackle fundamental 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 CHAPTER 6. SUMMARY AND OUTLOOK 110

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∆ + 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].

+ fn−1 Λn−1 +Λ1 ⇀ Λn (A.1) ↽ − −−−−−−fn + fn Λn +Λ1 ↽ ⇀ Λn+1 (A.2) −−−− f−−−n+1

Λn and Λ1 denote a droplet consisting of n particles and a single particle, respectively. + − fn is the rate of single-particle attachment to a cluster of size n and fn 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 fn and the detachment rate − fn 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, −

J (t)= f + Z (t) f −Z (t), (A.3) n n−1 n−1 − n n APPENDIX A. DERIVATIONS 113

the change of the concentration Zn(t) of Λn with time can be written as

dZ (t) n = J (t) J (t). (A.4) dt n − n+1

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 +Z f −Z , 1 1 − 2 2 I = f +Z f −Z , 2 2 − 3 3 I = f +Z f − Z , n n − n+1 n+1 − ∗ I = fn∗−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/f1 , the second by f −/f +f +, the nth by f −f − f −/f +f + f +, and etc. When all 2 1 2 2 3 n 1 2 n 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 1 f2 f3 fn n=1 + + + + . Rearranging the equation, we obtain the general expression fn f1 f2 fn−1 for the steady state nucleation rate

∗ −1 n −1 1 f −f − f − I = Z 2 3 n . (A.7) 1 f + f +f + f + n=1 n 1 2 n−1 APPENDIX A. DERIVATIONS 114

The expression is generally applicable for any case of nucleation process where the bimolecular reaction holds reasonably. The appropriate expression for the rate constant can be obtained for each nucleation. + − The rate constants fn and fn 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 + − fn−1Xn−1 = fn Xn. (A.8)

It can be rewritten in the form of

+ Xn fn−1 = − . (A.9) Xn−1 fn

Multiplying the ratios Xi/Xi−1 from i =2 to n results in

X n f + f −f − f − −1 n = i−1 = 2 3 n . (A.10) X f − f +f + f + 1 i=2 i 1 2 n−1 At the same time, the ratio is determined by the Boltzmann distribution

X ∆G(n) n = exp (A.11) X − k T 1 B where ∆G(n) is the formation free energy of droplet containing n particle, n( − 1 − 2)+ 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

∗ ∗ n −1 1 f −f − f − n −1 1 ∆G(n) Υ 2 3 n = exp (A.12) ≡ f + f +f + f + f + k T n=1 n 1 2 n−1 n=1 n B APPENDIX A. DERIVATIONS 115

∗ and I = Z1/Υ. Suppose n is signiﬁcantly larger than 10, we can replace the expression into a integral, which gives

∗ ∗ n −1 1 ∆G(n) n 1 ∆G(n) exp exp dn. (A.13) f + k T ≈ f + k T n=1 n B 1 n B

Note that ∆G(n) displays a maximum at n = nc and it can be expanded in a Taylor series near the maximum

1 ∆G(n) G η(n n )2 (A.14) ≈ c − 2 − c

where Gc is the maximum value of ∆G(n), or nucleation barrier and η is the negative ∂2G of the second derivative ∂n2 evaluated at n = nc. Because the integral shows − n=nc very sharp peak near the maximum, we can change the range of integration from - ∞ to . Besides, note that the f + is approximately proportional to the surface area ∞ n of the droplet which scales as n1−1/d, we can replace f + by constant value f + f + . n c ≡ nc Then, the outcome of Gaussian integral becomes

1 2πk T 1/2 G Υ= B exp c . (A.15) f + η k T c B

1/2 where η is known as the Zeldovich factor Γ. Finally, we obtain the steady 2πkB T state nucleation rate G I = f +ΓZ exp c . (A.16) c 1 −k T B 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 X Z . Then, the steady state rate of nucleation can be rewritten in the form 1 ≈ 1

+ I = fc ΓXnc (A.17) APPENDIX A. DERIVATIONS 116

where Xnc is the equilibrium concentration of critical nuclei,

G X = X exp c . (A.18) nc 1 −k T B

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 Znc is 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 1 f −f − f − f −f − f − 2 3 n I = Z 2 3 nc Z (A.19) f + f +f + f + 1 − f +f + f + nc n=1 n 1 2 n−1 1 2 nc−1 Identify that

nc−1 1 f −f − f − 1 1 G 2 3 n Υ= exp c (A.20) f + f +f + f + ≈ 2 2f + Γ k T n=1 n 1 2 n−1 nc B and f −f − f − G 2 3 nc = exp c . (A.21) f +f + f + k T 1 2 nc−1 B Substituting these expressions into the Eq. (A.19), we obtain

1 G Z = Z Z exp c . (A.22) 2 1 1 − nc k T B Rearranging the equation, we obtain Z 1 X (as compared to X Z ). From nc ≈ 2 nc 1 ≈ 1 Z , Z at n = n can be obtained by using the series of equations, Eqs. (A.6), with nc n c the detailed balance condition, Eq. (A.9). APPENDIX A. DERIVATIONS 117

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 = ln Z ln Υ with respect to ln (where is deﬁned 1 − S S in Eq. 2.11) at constant T ,

∂ ln I ∂∆ ∂ ln I = (A.23) ∂ ln ∂ ln ∂∆ S T S T T ∂ln I = k T (A.24) B ∂∆ T ∂ lnZ ∂ ln Υ = k T 1 (A.25) B ∂∆ − ∂∆ T T where Eq. (2.11) is used. Imagine a droplet free energy function of ∆G(n, ∆) = g(n, T ) n∆ 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∆ is the volume contribution due to phase transformation. For any g(n), we have ∂∆G/∂∆ = |T n. For gas-to-liquid condensation, f + = P (2πmk T )−1/2S(n) where m is the − n B mass of the particle and S(n) is the area of nucleus having n particles. Identify- + ing P = Peq exp(∆/kBT ) from Eq. (2.11), we obtain the attachment rate fn = P (2πmk T )−1/2S(n) exp(∆/k T ) and thus ∂f +/∂∆ = f +/k T . Using the eq B B n |T n B right hand side (RHS) of Eq. (A.12), we can rewrite

∂ ln Υ 1 ∂Υ = (A.26) ∂∆ Υ ∂∆ T T n∗−1 1 1 n ∆G(n) 1 1 ∆G(n) = − exp exp (A.27) Υ f + k T k T − f + k T k T n=1 n B B n B B Using the approximation used in the Eqs. (A.13), (A.14), and (A.15), The summation APPENDIX A. DERIVATIONS 118

in the RHS of Eq. (A.27) becomes

∗ n −1 1 n 1 ∆G(n) 1 n +1 G − − exp c exp c (A.28) f + k T k T ≈ Γf + k T k T n=1 n B B c B B Combining with Eq. (A.15), we obtain

∂ ln Υ n +1 = c (A.29) ∂∆ − k T T B Ignoring negligible ∆ dependence of Z 1, we obtain the ﬁnal expression

∂ ln I = n +1 , (A.30) ∂ ln c T S which is known as the ﬁrst nucleation theorem. It implies that, when the ln depen- S 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 = ln Z ln Υ with respect to T at 1 − constant S, ∂ ln I ∂ ln I ∂ ln I ∂∆ = + . (A.31) ∂T ∂T ∂∆ ∂T S ∆µ T S Notice that the second term in the RHS can be obtained from previous results, and we have ∂ ln I n +1 ∂∆ = c and = k ln (A.32) ∂∆ k T ∂T B S T B S Thus, ∂ ln I ∂∆ (n + 1)k T ln = c B S (A.33) ∂∆ ∂T k T 2 T S B We need to compute derivative of Υ with respect to T to obtain the ﬁrst term. + The T derivative of fn at constant ∆ becomes

∂f + 2∆/k + T n = f + B (A.34) ∂T − n T 2 ∆µ APPENDIX A. DERIVATIONS 119

and the T derivative of exp(∆G/kBT ) becomes

∂ ∆G 1 ∂g(n, T ) exp (∆G/k T )= + exp (∆G/k T ) (A.35) ∂T B −k T 2 k T ∂T B ∆µ B B ∆µ Using the same Gaussian approximation around n = nc and combining above two derivatives, we obtain

∂ ln I ∂ ln Υ = (A.36) ∂T − ∂T ∆µ ∆µ Gc +T∂g(nc, T )/∂T ∆µ + 2∆ + kBT = − 2| (A.37) − kBT g(nc, T ) T∂g(nc, T )/∂T ∆µ (nc + 2)∆ kBT = − | 2 − − (A.38) kBT

This leads to the partial derivative at constant , S ∂ ln I g(n , T ) T∂g(n , T )/∂T ∆ k T = c − c |∆µ − − B (A.39) ∂T k T 2 S B g(nc, T ) T∂g(nc, T )/∂T ∆µ − 2 | . (A.40) ≈ kBT

Finally, deﬁning ∆U(n, T ) = g(n, T ) T∂g(n, T )/∂T , we obtain the second nu- − |∆µ cleation theorem, ∂ ln I ∆U(n , T ) = c . (A.41) ∂T k T 2 S B The ∂g(n, T )/∂T term is much smaller than the g(n, T ) contribution in most |∆µ 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 k T = 0.026 eV which is much less than 1 eV 1. Thus, ∼ B 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 of materials are typically low enough that kB T 1 eV. However, in a model system such as the Ising ≪ model, we can artiﬁcially prepare a condition where kBT is comparable to the scale of surface energy (See Chapter 4). APPENDIX A. DERIVATIONS 120

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, fc Γ, in the nucleation rate predicted by the Becker-D¨oring theory, Eq. (2.7), on T and h. + In the (T, h) conditions considered in this study, both the attachment rate fc and the Zeldovich factor Γ vary by several orders of magnitude. However, their variations + largely cancel each other and the product fc 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 fc 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 fc 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)

0.22 1.8 1.5 0.2 1.9 1.5 1.6 ) 0.18 −1

0.16 approx + c 1.4 0.14 f / 1.9 (MCSS MC Γ

k T = 1.0 + c

B f + c f 0.12 1.2 k T = 1.0 0.1 B

0.08 1 0 0.05 0.1 0.15 0 0.05 0.1 0.15 h h (c) (d)

1.4 2.71 2.71 1.1 1.3

) 1 1.2 −1 2.65

0.9 approx 1.1 2.65 + c

0.8 2.50 f / 1 (MCSS MC Γ + c

2.50 f + c f 0.7 2.35 0.9 2.35 0.6 0.8 k T = 2.20 k T = 2.20 B B 0.5 0.7 0.4 0.5 0.6 0.7 0.4 0.5 0.6 0.7 h h

+ Figure B.1: (color online) (a) The pre-exponential factor fc Γ in 2D computed from + Monte Carlo and US. (b) The ratio between the attachment rate fc in 2D computed + by Monte Carlo and that predicted by Eq.(4.7). (a) The pre-exponential factor fc Γ in 3D computed from Monte Carlo and US. (b) The ratio between the attachment + rate fc 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 APPENDIX B. MORE DATA ON THE NUCLEATION IN THE ISING MODEL123

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 − ≫ ≫ APPENDIX B. MORE DATA ON THE NUCLEATION IN THE ISING MODEL124

kBT = 1.0 h = 0.07 kBT = 1.5 h = 0.05 kBT = 1.9 h = 0.03 n = 525 n = 499 n = 409

(a)

kBT = 2.2 h = 0.45 kBT = 2.5 h = 0.40 kBT = 2.71 h = 0.35 n = 362 n = 396 n = 419

(b)

Figure 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. 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, 2D −8βJ Ω1 = e (B.1) APPENDIX B. MORE DATA ON THE NUCLEATION IN THE ISING MODEL125

where β 1/(k T ). Similarly, the partition function of droplets of size 2, 3 and 4 ≡ B are,

2D −12βJ Ω2 = 2 e (B.2) 2D −16βJ Ω3 = 6 e (B.3) 2D −16βJ −20βJ Ω4 = e +18e (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.

3D −12βJ Ω1 = e (B.5) 3D −20βJ Ω2 = 3 e (B.6) 3D −28βJ Ω3 = 15e (B.7) 3D −32βJ −36βJ Ω4 = 3 e +83e (B.8) 3D −40βJ −44βJ Ω5 = 48e + 486e (B.9) 3D −44βJ −48βJ −52βJ Ω6 = 18e + 496e + 2967e (B.10) 3D −48βJ −52βJ −56βJ −60βJ Ω7 = 8 e + 378e + 4368e + 18748e (B.11)

Given the droplet partition functions, the droplet free energy F (n) deﬁned in this paper can be obtained from the following equation,

−βF (n) Ωn e = ∞ (B.12) 1+ i=1 Ωi

Numerically, the summation in the denominator converges very quickly after summing over 2 to 3 terms. As an approximation, we may write F (n) k T ln Ω . But this ≈ − B n approximation is not invoked in Section 5.4. The droplet free energy computed from Eq. (B.12) is used to determine the constant term d in Eq. (4.10), by requiring that F (n) from the two equations matches APPENDIX B. MORE DATA ON THE NUCLEATION IN THE ISING MODEL126

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 results 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√πσ 4.3. This means that a constant correction term d 3.7 is eﬀ ≈ ≈ needed.

B.4 Free Energy Curves F (n) for the Ising Model APPENDIX B. MORE DATA ON THE NUCLEATION IN THE ISING MODEL127

(a) kBT=1.0 and h=0.05 (b) kBT=1.0 and h=0.06

120 100

100 80 80 60 60 F (n) F (n) 40 40

20 20

0 0 0 200 400 600 800 1000 0 200 400 600 800 n n

80 80

60 60

40 40 F (n) F (n)

20 20

0 0 0 100 200 300 400 500 600 0 100 200 300 400 n n

(e) kBT=1.0 and h=0.09 (f) kBT=1.0 and h=0.10

80 60

50 60 40

40 30 F (n) F (n)

20 20 10

0 0 0 100 200 300 0 50 100 150 200 250 300 n 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 = 0.10 obtained by US (circles) is compared with Eq. (6) (solid line) and Eq. (8) (dashed 5 line). Logarithmic correction term 4 kBT ln n (dot-dashed line) and the constant term d (dotted line) are also drawn for comparison. APPENDIX B. MORE DATA ON THE NUCLEATION IN THE ISING MODEL128

(a) kBT=1.5 and h=0.04 (b) kBT=1.5 and h=0.05

80 80

60 60

40 40 F (n) F (n)

20 20

0 0 0 200 400 600 800 0 100 200 300 400 500 n n

60 50

50 40 40 30 30 F (n) F (n) 20 20

10 10

0 0 0 100 200 300 400 0 50 100 150 200 250 300 n n

(e) kBT=1.5 and h=0.08 (f) kBT=1.5 and h=0.09

50 40

40 30

30 20 F (n) F (n) 20

10 10

0 0 0 50 100 150 200 250 0 50 100 150 200 n n

(g) kBT=1.5 and h=0.10 (h) kBT=1.5 and h=0.11

40 40

30 30

20 20 F (n) F (n)

10 10

0 0 0 50 100 150 200 0 50 100 150 n n APPENDIX B. MORE DATA ON THE NUCLEATION IN THE ISING MODEL129

(i) kBT=1.5 and h=0.12 (j) kBT=1.5 and h=0.13

40 40

30 30

20 20 F (n) F (n)

10 10

0 0 0 50 100 0 50 100 n 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) is compared with Eq. (6) (solid line) and Eq. (8) (dashed line). Logarithmic correction 5 term 4 kBT ln n (dot-dashed line) and the constant term d (dotted line) are also drawn for comparison. APPENDIX B. MORE DATA ON THE NUCLEATION IN THE ISING MODEL130

(a) kBT=1.9 and h=0.015 (b) kBT=1.9 and h=0.017

80 80

60 60

40 40 F (n) F (n)

20 20

0 0 0 500 1000 1500 0 500 1000 1500 n n

60 50

50 40 40 30 30 F (n) F (n) 20 20 10 10

0 0 0 200 400 600 800 1000 0 200 400 600 800 n n

(e) kBT=1.9 and h=0.03 (f) kBT=1.9 and h=0.035

50 40

40 30

30 20 F (n) F (n) 20

10 10

0 0 0 200 400 600 0 100 200 300 400 500 n 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 = 0.035 obtained by US (circles) is compared with Eq. (6) (solid line) and Eq. (8) (dashed 5 line). Logarithmic correction term 4 kBT ln n (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

σ γ C.1 Equality of Critical Sizes nc and nc

σ Suppose that the Gibbs free energy G(n, σ, T ) is maximized at n = nc , then

∂G(n, σ, T ) =0. (C.1) ∂n σ σ,n=nc 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 ) ∂ = [G(n, σ, T )+ σγV ] ∂n σ ∂n σ γ,n=nc γ,n=nc ∂G(n, σ, T ) ∂G(n, σ, T ) ∂σ ∂σ = + + γV ∂n σ ∂σ σ ∂n ∂n σ,n=nc n=nc ∂σ ∂σ = 0 (Vγ) +(Vγ) =0. (C.2) − ∂n ∂n

γ By deﬁnition, F (n, γ, T ) reaches maximum at n = nc at constant γ and T ,

∂F (n, γ, T ) c = 0 (C.3) ∂n γ γ,n=nc 131 APPENDIX C. MORE DISCUSSION ON THE DISLOCATION NUCLEATION132

γ σ σ Therefore, we establish that nc = nc , i.e. the maximizer nc of G(n, σ, T ) is also the γ maximizer nc 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 σ γ maximizer nc of G(n, σ, T ) equals to the maximizer of nc 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

G = G(n , σ, T ) G(0, σ, T ) c c − = F (n ,γ,T ) σγV F (0,γ , T )+ σγ V (C.4) c − − 0 0

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,

G = F (n ,γ,T ) F (0,γ,T )+ F (0,γ,T ) F (0,γ , T ) V σ(γ γ ) c c − − 0 − − 0 ∂F 1 ∂2F F + (γ γ )+ (γ γ )2 σ(γV γ V ) ≈ c ∂γ − 0 2 ∂γ2 − 0 − − 0 γ0,V γ0,V 2 1 ∂ F 2 = Fc + (γ γ0) 2 ∂γ2 − γ0,V 1 ∂σ 2 = Fc + V (γ γ0) (C.5) 2 ∂γ − γ0,V APPENDIX C. MORE DISCUSSION ON THE DISLOCATION NUCLEATION133

Notice that γV = ∂G(n , σ, T )/∂σ and γ V = ∂G(0, σ, T )/∂σ. Then (γV γ V ) is − c 0 − − 0 equivalent to ∂ (G(n , σ, T ) G(0, σ, T )) = ∂Gc Ω , i.e. the activation volume. − ∂σ c − − ∂σ ≡ c By plugging (γ γ ) = Ω /V into the equation, we have − 0 c 1 1 ∂σ G = F + (Ω∗)2 + O(V −2) c c 2 V ∂γ γ0,V −1 = Fc + O(V ) (C.6)

In the thermodynamics limit (V ), we have G = F . Hence, the nucleation rate →∞ c c 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 and S(n, σ, T ) ≡ |n,γ ≡ ∂G(n, σ, T )/∂T equal to each other as long as σ = V −1∂F/∂γ . At the same |n,σ |n,T time, the activation entropy is just the entropy diﬀerence between the activated state and the meta-stable state, i.e., S (γ, T ) = S(n ,γ,T ) S(0,γ,T ) and S (σ, T ) = c c − c S(n , σ, T ) S(0, σ, T ). If the entropies in two ensembles can equal each other, it c − 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 APPENDIX C. MORE DISCUSSION ON THE DISLOCATION NUCLEATION134

S(0, σ, T )= S(0,γ , T ), but S(0,γ,T ) = S(0,γ , T ). 0 0

S (σ, T ) = S(n , σ, T ) S(0, σ, T ) c c − = S(n ,γ,T ) S(0,γ , T ) c − 0 = S(n ,γ,T ) S(0,γ,T )+ S(0,γ,T ) S(0,γ , T ) c − − 0 = S (T,γ)+ S(0,γ,T ) S(0,γ , T ) (C.7) c − 0

This shows that the activation entropy difference ∆S S (σ) S (γ) equals to c ≡ c − c S(0,γ,T ) S(0,γ , T ), which is entropy difference of the perfect crystal at two slightly − 0 different strains. In the limit of V , because we expect (γ γ ) 0, we might reach a false → ∞ − 0 → conclusion that ∆S = (S(0,γ,T ) S(0,γ , T )) 0. Instead, the correct behavior c − 0 → in the thermodynamic limit can be obtained by expanding ∆Sc in a Taylor series.

∂S S(γ) S(γ ) = (γ γ )+ − 0 ∂γ − 0 ∂σ = V (γ γ )+ (C.8) − ∂T − 0 γ,V where the Maxwell relationship ∂S/∂γ = V ∂σ/∂T is used. The term (γV |T − |γ,V − pl γ0V ) equals the activation volume Ωc, and can be interpreted as plastic strain γ due to formation of dislocation loop times the volume of the crystal, i.e.

(γV γ V ) = Ω = γplV = bA (C.9) − 0 c c

where b is the magnitude of the Burgers vector and Ac is the area of the critical dislocation loop. Using the relation (γ γ ) = Ω /V , we have − 0 c ∂σ ∆S = S(γ) S(γ ) = Ω + O(V −1) (C.10) c − 0 − ∂T c γ,V 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], APPENDIX C. MORE DISCUSSION ON THE DISLOCATION NUCLEATION135

with S S = βBV (C.11) p − v rel where β V −1 ∂V is the thermal expansion factor at zero hydrostatic pressure ≡ ∂T p=0 p, B is the isothermal bulk modulus, and Vrel is the relaxation volume of the defect. In Cu, the value of S S is estimated to be 1.7 k for a vacancy and 13.7 k for p − v − B B 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 term. The similarity between these − ∂T two equations stems from the fact that they both express the entropy difference between 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 equilibrium. 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 dislocation 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. APPENDIX C. MORE DISCUSSION ON THE DISLOCATION NUCLEATION136

Starting from Eqs. (5.24) and (5.39), we have

E (γ) ∂ ∂σ S (σ) = S (γ) + ∆S c Ω (C.12) c c c ≈ − (0) ∂T − c ∂T γ If we assume the crystal is linear elastic, i.e. σ = γ, then,

H (σ) + Ω (σ) σ ∂ S (σ) c c (C.13) c ≈ − (0) ∂T

Similar expressions can be obtained for normal (compressive) loading by replacing γ with ǫ and replacing by the Young’s modulus. To gain more intuition, we note that in the limit of σ 0, the line tension model → estimates that H (σ) σ−1. In addition, in the limit of T 0, Ω (σ) ∂H /∂σ. c ∝ → c ≈ − c Under these conditions, Ω (σ) σ H (σ), so that, c ≈ c 2H (σ) ∂ S (σ) c (C.14) c ≈ − (0) ∂T

Comparing Eq. (C.14) with Eq. (5.39), we have,

S (σ) S (γ) c 2 c (C.15) Hc(σ) ≈ Ec(γ)

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. 0 For homogeneous nucleation, the strain γxy is deﬁned as ∆x/hy where ∆x is the APPENDIX C. MORE DISCUSSION ON THE DISLOCATION NUCLEATION137

displacement of the repeat vector initially in the y-driection along the x axis at 0 each pure shear stress condition, and hy 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 [L (σ, T ) L (σ =0, T )]/L0 where L (σ, T ) is the length of the repeat vector along z − z z z the z-axis, Lz(σ =0, T ) is the equilibrium length at temperature T under zero stress. 0 ˚ Lz = 20a0 = 72.3A is the reference length (before relaxation), where a0 is the lattice constant of copper. The compressional stress σ is deﬁned by σ = F /d2 where zz zz 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. APPENDIX C. MORE DISCUSSION ON THE DISLOCATION NUCLEATION138

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 σ 2.00 2.04 2.11 2.17 2.24 2.30 2.36 2.42 2.48 2.53 2.58 2.63 0 K xy Ec 2.2851 2.0866 1.7946 1.5426 1.3234 1.1315 0.9627 0.8133 0.6815 0.5648 0.4615 0.3703 σ 1.94 1.98 2.05 2.11 2.17 2.22 2.28 2.33 2.38 100 K xy E˜c 2.196 2.004 1.718 1.477 1.268 1.085 0.925 0.784 0.660 σxy 1.87 1.91 1.97 2.03 2.08 2.13 2.18 2.22 E˜c 2.116 1.924 1.650 1.419 1.216 1.043 0.890 0.757 200 K Fc 2.115 1.911 1.650 1.428 1.214 1.027 0.863 0.755 Γ 0.062 0.064 0.067 0.070 0.072 0.073 0.073 0.073 + fc 3.2 3.6 2.9 3.9 3.5 3.4 2.7 2.5 σxy 1.80 1.83 1.89 1.94 1.98 2.02 2.06 2.10 E˜c 2.042 1.853 1.586 1.362 1.171 1.004 0.859 0.730 300 K Fc 2.054 1.876 1.585 1.370 1.173 1.006 0.860 0.728 Γ 0.048 0.049 0.051 0.054 0.056 0.056 0.055 0.054 + fc 3.6 4.5 5.1 3.6 4.4 4.2 5.1 4.4 σxy 1.71 1.74 1.79 1.84 1.88 1.91 1.95 E˜c 1.968 1.793 1.529 1.312 1.124 0.962 0.824 400 K Fc 2.008 1.828 1.579 1.347 1.155 0.995 0.853 Γ 0.039 0.040 0.043 0.044 0.045 0.045 0.045 + fc 5.0 5.9 6.4 7.4 5.3 6.9 6.9 σxy 1.62 1.64 1.69 1.73 1.76 1.79 E˜c 1.897 1.727 1.476 1.261 1.081 0.925 500 K Fc 1.939 1.782 1.548 1.341 1.167 1.010 Γ 0.034 0.035 0.037 0.038 0.038 0.038 + fc 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, fc in 14 −1 10 s . γxy and Γ are dimensionless. The error in E˜c 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 sampling. The error in Zeldovich factor Γ is within 0.01. The ± attachment rate f + has relative error of 50%. c ± APPENDIX C. MORE DISCUSSION ON THE DISLOCATION NUCLEATION139

ǫ 0.0303 0.0353 0.0403 0.0453 0.0503 0.0553 0.0603 0.0653 0.0703 0.0753 0.0803 T =0K zz σxy 1.56 1.81 2.04 2.28 2.50 2.73 2.95 3.16 3.37 3.58 3.78 Ec 1.5110 1.0383 0.7550 0.5650 0.4296 0.3277 0.2495 0.1878 0.1388 0.0993 0.0671 ǫzz 0.0312 0.0342 0.0372 0.0402 0.0432 σxy 1.56 1.69 1.82 1.95 2.07 200 K Fc 1.278 1.017 0.825 0.690 0.571 Γ 0.030 0.038 0.047 0.054 0.062 + fc 1.1 1.0 0.91 0.84 0.85 ǫzz 0.0307 0.0337 0.0367 0.0397 0.0427 σxy 1.48 1.60 1.72 1.83 1.94 300 K Fc 1.254 1.004 0.824 0.695 0.579 Γ 0.021 0.028 0.033 0.045 0.049 + fc 1.5 1.4 1.3 1.1 1.1 ǫzz 0.0301 0.0331 0.0361 0.0391 0.0421 σxy 1.39 1.50 1.62 1.73 1.83 400 K Fc 1.258 1.003 0.839 0.700 0.578 Γ 0.019 0.028 0.029 0.037 0.038 + fc 1.8 1.8 1.7 1.5 1.4 ǫzz 0.0296 0.0326 0.0356 0.0386 σxy 1.29 1.40 1.50 1.60 500 K Fc 1.226 1.008 0.833 0.697 Γ 0.018 0.021 0.026 0.032 + fc 2.6 2.0 2.0 2.0

+ Table C.2: Data for heterogeneous nucleation: σzz in GPa, Ec, and Fc in eV, fc 14 −1 in 10 s . 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 fc has relative error of 50%. Notice that, due to the± existence of thermal strain, the elastic strain values are± 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., Ω (T, σ)= ∂G /∂σ , and measures the sensitivity of nucleation rate to c − c |T 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 APPENDIX C. MORE DISCUSSION ON THE DISLOCATION NUCLEATION140

1000 200K 200K 300K 800 300K 400K 400K 800 500K 500K 600 ) ) 3 600 3 (A (A c c 400 Ω 400 Ω

200 200

0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 n n c c (a) (b)

Figure C.1: The relation between critical dislocation size nc and the activation volume Ω ∂Gc .(a) homogeneous nucleation (b) heterogeneous nucleation. Circles c ≡ − ∂σ 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.

loading. Hence, the hypothesis we wish to test is,

Ωc = nc b Aa 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 a = 3.615A˚, we { } 0 ˚ 2 ˚2 ˚3 have b = a0/√6=1.48 A and Aa = √6a0/4=5.66 A , so that b Aa = 8.35 A . 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 Ω /n 10 A˚3, close to the expected value of 8.35 A˚3. For c c ≈ heterogeneous nucleation under compression, Fig. C.1(a), the slopes of the curves after correction for the Schmid factor is roughly Ω /(n S) 8 A˚3, which is similar c c ≈ 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 b Aa supports the notion that the APPENDIX C. MORE DISCUSSION ON THE DISLOCATION NUCLEATION141

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 comparison 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 comparison 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 potential 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 APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS144

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 depends 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 APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS145

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 = T (S S ), all of which can be compared with experiments. It is interesting m L − S 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 later version of MEAM [206] is used in this work. We also computed the melting point of the later version of MEAM [206] using the co-existence method and the melting point is around 1410 K. APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS146

Table D.1: Thermal properties of various elements as predicted by several empirical 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 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 second nearest neighbor interactions [209, 210].

Model Tm L SS SL α Si MEAM 1411.3 0.4 1309 48.74 74.79 13.6 Si SW 1694.7 ± 0.5 1111 58.02 76.45 3.9 Si Exp 1687± 1650 61.765 91.562 2.6 Ge MEAM 1216.2 0.6 427 58.34 83.84 16.2 Ge SW 2898.0 ± 1.7 847 84.07 105.30 5.8 Ge Exp 1211± 465 66.77 97.34 5.8 Au MEAM 1120.0 0.6 92 77.47 93.72 2.0 Au MEAM∗ 995.3 ±1.3 52 84.18 94.47 16.5 Au EAM 984.3 ± 2.3 41 85.73 94.03 13.5 Au Exp 1337±.3 64.9 14.2 −− −− Cu MEAM 1350.0 1.0 368 62.70 80.19 4.5 Cu MEAM∗ 1182.9 ± 2.2 205 69.69 80.68 16.0 Cu EAM 1239.6 ± 2.3 164 71.78 80.17 17.3 Cu Exp 1357±.8 205 74.30 83.97 16.5 Ag MEAM 987.1 0.9 158 66.21 83.49 5.1 Ag Exp 1234±.9 103 18.9 −− −− Pb MEAM 674.7 1.0 57 76.17 93.59 3.3 Pb Exp 600±.6 23.2 84.34 92.31 28.9 Mo MEAM < 1000 8.6 Mo MEAM† 2778.0 10.1 −−153 91−−.98 97−−.28 5.3 Mo FS 3062.6± 7.6 284 91.85 100.75 2.9 Mo Exp 2896± 290 98.10 110.52 4.8 Ta MEAM < 1000 9.1 −− −− −− Ta MEAM† 2884.3 7.9 115 102.56 109.75 5.7 Ta FS 3935.7 ± 6.7 190 104.8 113.54 6.3 Ta Exp 3290± 174 111.26 122.48 6.3 W MEAM < 1000 6.1 W MEAM† 4389.0 9.1 −−161 106−−.24 112−−.98 4.2 W FS 4125.6 ± 8.0 184 103.81 112.03 3.9 W 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 APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS147

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 hypothesis 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 ∼ APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS148

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 prediction 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 scaling [199, 200], which has made free energy calculations much more efficient. Using APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS149

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 NP T 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

expected to be higher than Tm.

4. Find the equilibrium volume V2 of the liquid phase at T2 by an MD simulation under the NP T 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

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 conditions 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, { }

1 N e−βF = Z = dr dp e−βH({ri,pi}) (D.1) N!h3N i i i=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 APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS151

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 k T F (N,V , T )= E (V ) k T ln B 1 (D.2) QHA 1 1 0 1 − B 1 hw¯ (V ) i i 1 whereh ¯ 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].

Suppose H1 is the Hamiltonian of the system of interest and H0 is the reference system. We deﬁne a new Hamiltonian parameterized by λ,

H(λ)=(1 λ) H + λH (D.3) − 0 1 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.,

tsw ∂H(λ) dλ(t) F F = ∆W dt (D.4) 1 − 0 ≡ ∂λ d t 0

where tsw is the total time of the switching simulation. Strictly speaking, the equality APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS152

(F F = ∆W ) is valid only in the limit of infinitely slow switching, i.e. t . 1 − 0 sw →∞ For any switching performed at a finite rate, ∆W contains both statistical and systematic 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 F F [214]. To reduce statistical error generated from finite switching 1 − 0 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) = H H tend to be largest [113]. The switching function is also very smooth, ∂λ 1 − 0 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 dissipation [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 APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS153

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 simulation.

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

F (N,V , T )= k T N ln(V /Λ3) ln N! (D.6) i.g. 2 2 − B 2{ 2 − } where V is equilbrium volume of liquid at T and Λ h/√2πmk T is the thermal de 2 2 ≡ B 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 simulation, 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 APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS154

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

ri rj φ(ri, rj)= λ ǫ exp −| − | . (D.7) 2σ 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. APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS155

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

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 APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS156

obtained by solving the following equations self-consistently,

(1+eβ∆Wi+C )−1 e−β∆Fi = F eC (D.8) (1+eβ∆Wi−C )−1 R C = β∆F + ln n /n (D.9) − i F R where nF and nR are the number of independent forward and reverse switching simulations [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

βWi+C −1 βWi−C −1 2 2 Var[(1 + e ) ]F Var[(1 + e ) ]R β σî = + (D.10) n (1+eβWi+C )−1 2 n (1+eβWi−C )−1 2 F F R R where Var[x] is the variance of the random variable x. In this work, the melting point is estimated from m = 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,

m 1/2 2 σ(∆G)= (ˆσi) (D.11) i=1 The error bar in the melting point prediction is

σ(∆G) σ(T )= (D.12) m S S L − S

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 APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS157

−4.6 solid −4.7 liquid solid −4.8 liquid

−4.9

−5 Free Energy (eV/atom)

−5.1 1500 1600 1700 1800 Temperature (K)

Figure 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.

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 adiabatic 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. APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS158

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 melting 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 semiconductors 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 calculations. 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 APPENDIX D. THERMAL PROPERTIES FROM INTERATOMIC POTENTIALS159

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 in the 5 different adiabatic switching steps for the melting point calculations of SW-Si and MEAM-Si models. SW-Si MEAM-Si −4 −4 i ∆Fi (eV/atom) σî (10 eV/atom) ∆Fi (eV/atom) σî (10 eV/atom) 1 -0.01297 0.01 -0.00624 0.07 2 0.81616 0.05 1.10544 0.03 3 -3.68733 0.88 -3.94027 1.01 4 0.50080 0.45 0.41005 0.47 5 -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 devices [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 thousands 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 semiconductors, 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 required 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 potentials 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 changing 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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 162

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 silicon to accurately reproduce the melting point and the latent heat. During this process, we also monitor the predictions of the diffusion coefficient, thermal expansion coefficient, generalized stacking fault energy, and volume change on melting. Because these properties may also influence various stages of the nanowire nucleation/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 r , i =1, , N, by the following equation, i

N N−1 N V ( r )= F (ρ )+ S φ ( r r ) (E.1) { i} i ij ij | i − j| i=1 i=1 j=i+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. APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 163

Table 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 modifications made in this study (2nn-MEAM∗ and 2nn-MEAM†), and first-principles calculation with DFT/LDA [232]. The prop- −1 erties include the melting point Tm (in K), latent heat of fusion L (in kJ mol ), −1 −1 solid and liquid entropy at the melting point, SS and SL (in J mol K ), diffusion −9 2 −1 constant of the liquid D at Tm (in 10 m s ), thermal expansion coefficient α of −6 −1 the solid (in 10 K ) 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.

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 6.6 EAM [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 7.8 2nn-MEAM∗ 1173 11.2 87.7 97.2 2.2 20.8 1.81 5.1 2nn-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 – 5.1 DFT/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 discrepancy. We choose the more recent 2nn-MEAM gold potential as the starting point APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 164

of our model. Table E.1 shows that the 2nn-MEAM gold potential [227] overestimates the melting point T by 70 K (5%) and overestimates the latent heat L by m ∼ 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 = T (S S ), it is useful to monitor the entropies of the liquid phase m L − S (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 anharmonic 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 underestimate 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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 165

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) 0.05 0.05 0.045

0.04 0.04

0.035 ) 2 ) 2 0.03 0.03 0.025

0.02 0.02 (eV/Angstrom Γ

0.015 (eV/Angstrom Γ

0.01 0.01

0.005

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Sheared Distance, d/|b| where b = [1 1 2]/2 Sheared Distance, d/|b| where b = [1 0 1]/2

Figure E.1: (Color Online) Generalized stacking fault energy of diﬀerent potential 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).

E.2.2 Limitations of MEAM Silicon Potentials

Table E.2 shows that the original MEAM silicon potential [206] underestimates the melting point T by 270 K (16%) and underestimates the latent heat L by 13 kJ/mol m ∼ ∼ (27%). These are the main problems that we will address in this study. There is error 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. APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 166

Table 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 comparison. The computational models include 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 variables and 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 5.3 MEAM∗ 1377 36.7 48.0 74.7 6.3 13.6 13.8 −5.2 MEAM† 1687 43.1 53.7 79.2 7.9 13.2 14.0 −2.7 − SW 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 −1.6 2nn-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 5.1 DFT/LDA(GGA) – – – – ∼– – 14.0(13.7)−− −–

Table E.2 also lists the predicted values of several other properties that are included 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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 167

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) 0.14 0.14

0.12 0.12

0.1 ) 2 ) 0.1 2

0.08 0.08

0.06 0.06 (eV/Angstrom Γ 0.04 (eV/Angstrom Γ 0.04

0.02 0.02

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Sheared Distance, d/|b| where b = [1 0 1]/2 Sheared Distance, d/|b| where b = [1 0 1]/2 (c) (d)

0.14 0.14

0.12 0.12 ) )

2 0.1 2 0.1

0.08 0.08

0.06 0.06 (eV/Angstrom (eV/Angstrom

Γ 0.04 Γ 0.04

0.02 0.02

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Sheared Distance, d/|b| where b = [1 0 1]/2 Sheared Distance, d/|b| where b = [1 0 1]/2

Figure 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)

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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 168

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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 169

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 adjustment 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 expansion 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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 170

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 potential amounts to changing the equation of state (EOS) function, which in the original MEAM potential is expressed as,

Eu(r)= E (1 + a∗) exp( a∗) (E.2) − c − 9 ΩB 1/2 r with a∗ = 1 (E.3) E r − c e 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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 171

been fitted to physical properties. Specifically, Eu(r ) = E , dEu/dr = 0 and e − c |re d2Eu/dr2 = 18Ω B/r2. In the following, we will show that the same kind of |re e 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)

3.5 0 −4.4

3 −1 −4.6 2.5

2 −2 −4.8 G(r) 1.5 −3 1 −5 Free Energy (eV/atom) Binding Energy (eV) −4 0.5

0 −5.2 0 2 4 6 8 10 1 2 3 4 5 1100 1200 1300 1400 1500 1600 r(Angstrom) Separation (Angstrom) Temperature (K)

Figure 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.

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. ∗2 Eu(r)= E 1+ a∗ + d a∗3 + γ a∗4e−λa /r exp( a∗) (E.4) − c − APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 172

d a∗3 is small correction term added in 2nn-MEAM Au potential [227] with d =0.05 −3 ∗2 A˚ . The higher order term a∗4 is multiplied by an exponential e−λa /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.

∗2 Eu(r)= E 1+ a∗ + γ a∗4e−λa /r exp( a∗) (E.5) − c − 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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 173

(a) (b) (c)

6 0 −4.6

5 −1 −4.8 4 −2 3 −5 G(r) −3 2 −5.2 Free Energy (eV/atom) 1 Binding Energy (eV) −4

0 −5.4 0 2 4 6 8 10 1 2 3 4 5 1200 1400 1600 1800 2000 r(Angstrom) Separation (Angstrom) Temperature (K)

Figure 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. 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 underestimated; 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 thermodynamic 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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 174

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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 175

j that are common neighbors of i and k,

Sik = Sijk (E.6) j=i,k

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, Y 2 1 X2 + = (E.7) C 4 The parameter C controls the extension of the ellipse along the y-axis. If the ellipse pass through atom j, then

2(X + X ) (X X )2 1 C = ij jk − ij − jk − (E.8) 1 (X X )2 − ij − jk where X (r /r )2 and X (r /r )2. Fig. E.5 plots the ellipses when C =0.5, ij ≡ ij ik jk ≡ jk ik 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 1 x 1 ≥ 4 2 fc(x)=  (1 (1 x) ) 0

j 0.5

i k Y 0 C=0.8 C=2.0 −0.5 C=2.8

−1 −1 −0.5 0 0.5 1 X

Figure 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 (asterisk) 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.

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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 177

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 APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 178

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 energy 2 Ev (in eV) and surface energies E100, E110, and E111 (in ergs/cm ) from experimental measurements [183, 206, 241, 242], ﬁrst principle calculations [241, 243, 244, 245] (except silicon elastic constants computed 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 reconstruction such as 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 described 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, respectively [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 neighbor atoms within a cut-oﬀ radius of rm (i.e. the coordination number) is 5.7 for the APPENDIX E. MEAM POTENTIALS FOR PURE AU AND PURE SI 179

MEAM† model. In comparison, the coordination number for the original MEAM potential 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.

0.8

0.6

0.4

0.2

0 0 50 100 150 Θ(degree)

Figure E.6: (Color Online) Bond angle distribution functions of liquid phase of silicon described by the MEAM† (solid line) ,DFT/LDA (dashed line) [246] and SW (dotted line) [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

γ Eu(r)= E 1+ a∗ + a∗3 exp( a∗) (F.1) − c r − 9 ΩB 1/2 r with a∗ = 1 (F.2) E r − c e where r is the nearest-neighbor distance, Ec is the cohesive energy, re is the equilibrium 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]. APPENDIX F. GOLD-SILICON BINARY POTENTIAL 182

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 Au Si initio and the data are used to adjust the electron density scaling factors ρ0 and ρ0 . 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 r , E and α 9ΩB/E e c ≡ c APPENDIX F. GOLD-SILICON BINARY POTENTIAL 183

Table 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. Material Exp’t DFT/LDA offset MEAM a (A˚) Si (DC) 5.431 5.390 -0.041 5.431 Au (FCC) 4.070 4.068 -0.005 4.073 Au-Si (B1) 5.184∗ 5.161 -0.023 5.400 B (GPa) Si (DC) 98 96 2 98 Au (FCC) 180 186 -6 180 Au-Si (B1) 127∗ 129 -2 127 Ec (eV) Si (DC) 4.63 5.976 -1.346 4.63 Au (FCC) 3.93 4.387 -0.457 3.93 Au-Si (B1) 4.155∗ 5.057 -0.902 4.155 C11 (GPa) Si (DC) 164 162 2 164 Au (FCC) 202 217 -15 202 Au-Si (B1) 303∗ 310 -6 320 C44 (GPa) Si (DC) 76 105 -29 76 Au (FCC) 45 47 -2 45 Au-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. Au Si The electron density scaling factors, ρ0 and ρ0 does not affect the energy of pure crystals but influence the interaction between Au and Si atoms. Because only Au ρ0 the ratio of the electron density scaling factors, Si , is important, we have a single ρ0 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 APPENDIX F. GOLD-SILICON BINARY POTENTIAL 184

Au ρ0 are listed in Table F.2. It shows that our choice of Si is the result of a compromise ρ0 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 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. MEAM DFT/LDA E1(eV) 0.636 0.634 E2(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 potential 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 APPENDIX F. GOLD-SILICON BINARY POTENTIAL 185

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 reference structure. 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 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 Si ρ0 Ec re α Au Cmax ρ0 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) γ 1.9 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 APPENDIX F. GOLD-SILICON BINARY POTENTIAL 186

Au−Si Phase Diagram

1600 MEAM Exp 1400 1200 L 1000 800 L+Si(s) L+Au(s) 600

Temperature(K) Au(s) 400 Au(s)+Si(s) 200

0 0.2 0.4 0.6 0.8 1 x Si

Figure 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 corresponds to the liquid 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 APPENDIX F. GOLD-SILICON BINARY POTENTIAL 187

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]

G (x, T ) G (x =0, T )+ x ∆g (T ) Ts (x) (F.3) FCC ≈ FCC imp − mix 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. s = k [x ln x + mix − B (1 x) ln(1 x)] is the conﬁgurational entropy of mixing. − − We compute ∆g in the Eq. (F.3) using a simulation cell containing N 1 = 499 imp − 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 1 (T0 = 254 K) is computed by adiabatic switching [199] the system to its harmonic approximation, whose free energy is known analytically. G1 as a function of temperature 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,

∆g (T )= G1(T ) G0(T )+ k T ln N. (F.4) imp − B

The kBT ln N term is added to cancel the conﬁgurational entropy contribution in 1 2 G (T ) . 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 common to approximate ∆g (T ) = ∆h T ∆S by the enthalpy ∆h , neglecting imp imp − vib imp 1 To improve accuracy, we also performed an independent calculation at T0 = 629 K. 2 1 N! The conﬁgurational entropy of mixing in Cell 1 is Smix = kB ln (N−1)!1! = kB ln N. APPENDIX F. GOLD-SILICON BINARY POTENTIAL 188

the vibrational entropy contribution in ∆gimp(T ). However, we find that the contribution from T ∆S is significant for the Au-Si system, affecting the phase diagram − vib 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 4.9k , − B 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]

V V V Fi.g.(N1, N2)= N1kBT ln 3 N2kBT ln 3 + kBT ln 3 + kBT ln(N1! N2!) (F.5) − Λ1 − Λ2 Λe

where V is the volume of the simulation cell, Λi = h/√2πmikBT is the de Broglie wave 2 length, with i = 1 for Au and i = 2 for Si. Λe = h/√2πmekBT and me = (N1m1 + 2 N2m2)/(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 Helmholtz free energy, although this is not the case for the reference ideal gas system. APPENDIX F. GOLD-SILICON BINARY POTENTIAL 189

(a) −15 ∆g imp ∆h −20 imp

−25

−30 (kJ/mol)

−35

−40

−45 250 300 350 400 450 500 550 600 650 Temperature (K) (b) 270

260

250

240 (kJ/mol)

230

220 ∆g imp ∆h imp 210 250 300 350 400 450 500 550 600 650 Temperature (K)

Figure 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. 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). APPENDIX F. GOLD-SILICON BINARY POTENTIAL 190

The prediction of free energy from the MEAM model is in reasonable agreement with the CALPHAD result [256].

(a)

−450

−455

−460

−465

Free energy (kJ/mol) −470

−475 0 0.2 0.4 0.6 0.8 1 x Si (b)

−5

−10

−15 Free energy (kJ/mol) MEAM CALPHAD −20 0 0.2 0.4 0.6 0.8 1 x Si

Figure F.3: (a) Liquid free energy Gliq(x, T ) at T = 1250 K. Circles are simulation 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.

We also compare our model directly to enthalpy of mixing and excess free energy from experiments [257], which are obtained by calorimetric and Knudsen cell method respectively as shown in Fig. F.4. While qualitative agreements can be observed among MEAM, CALPHAD and experimental data, notable discrepancies can APPENDIX F. GOLD-SILICON BINARY POTENTIAL 191

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 ) and Gliq(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, x ]. The mixture of FCC solid and liquid phase is stable in the composition ∈ 1 range of x (x , x ). Second, a common tangent line is drawn between G (x, T ) ∈ 1 2 liq and GDC(x, T ). The tangent contacts the two free energy curves at x3 = 0.251 and x 1, respectively. This means that the Si-rich DC (solid) phase is stable 4 ≈ in the composition range of x [x , 1]. The mixture of DC solid and liquid phase ∈ 4 is stable in the composition range of x (x , x ). The liquid phase is stable in ∈ 3 4 the composition range of x [x , x ]. Repeating this procedure for all temperatures ∈ 2 3 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 4 (x2 = x3) . The tangent line contacts the liquid free energy curve at the eutectic

composition xe = 0.234. The eutectic temperature and composition of the MEAM

4 At T

(a) 2

−1

−2

−3 (kJ/mol) liq

H −4 ∆ −5

−6 MEAM −7 Exp CALPHAD −8 0 0.2 0.4 0.6 0.8 1 x Si (b) 0

−5 (kJ/mol) XS liq G ∆ −10

MEAM Exp CALPHAD −15 0 0.2 0.4 0.6 0.8 1 x Si

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 of XS mixing ∆Gliq (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 binary 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 substitutional impurity energies in the solid phase and the binary phase diagram. The final APPENDIX F. GOLD-SILICON BINARY POTENTIAL 193

(a)

G −5 liq

−10 G G FCC DC

Free energy (kJ/mol) −15

−20 0 0.2 0.4 0.6 0.8 1 x Si (b)

1600 1400 1200 1000 800 600 Temperature(K) 400 200

0 0.2 0.4 0.6 0.8 1 x Si

Figure 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). 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 APPENDIX F. GOLD-SILICON BINARY POTENTIAL 194

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. APPENDIX F. GOLD-SILICON BINARY POTENTIAL 195

Table 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 energy, and C11 and C44 are cubic elastic constants. MEAM data should be compared with ab initio (DFT/LDA) results that have been adjusted for known diﬀerences from experimental values for pure elements. Structure Properties DFT/LDA DFT/LDA (adjusted) MEAM B1 a (A˚) 5.161 5.184 5.400 B (GPa) 129 127 127 Ec (eV) 5.057 4.155 4.155 C11 (GPa) 310 303 320 C44 (GPa) -20 -35 15 B2 a (A˚) 3.202 3.226 3.370 B (GPa) 130 128 114 Ec (eV) 4.867 3.966 3.844 C11 (GPa) 101 101 88 C44 (GPa) -3 -18 0.7 L12 a (A˚) 4.041 4.055 4.192 B (GPa) 156 152 140 Ec (eV) 4.624 3.945 4.022 C11 (GPa) 170 159 142 C44 (GPa) 11 -1 19 Bibliography

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