Liquid-Solid Systems out of Equilibrium: Phase-Field Crystal Studies of Solidification, Melting, and Plasticity Joel M. Berry Department of Physics McGill University, Montreal October 2011 A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Ph.D. c Joel M. Berry 2011. i Abstract Dynamic processes in nonequilibrium liquid-solid systems are studied over mesoscopic time scales and atomistic length scales using phase-field crystal (PFC) models. Vari- ous freezing and melting transitions are examined in two and three dimensions, and microscopic phenomena responsible for solid-phase plasticity are investigated. A pri- mary focus is on the issue of describing atomistic dynamics over time scales that are generally inaccessible to conventional approaches. Glass forming dynamics in supercooled liquids near a glass transition are studied numerically, and the central features of the transition, including a number of be- haviors previously undemonstrated within PFC / classical density functional theory simulations, are successfully reproduced. A connection between the liquid dynamic correlation length and transition fragility is identified, and a physically motivated time scaling applied to the simulation data is shown to generate qualitative agreement with basic glass transition phenomenology across 12 orders of magnitude in time. The competing processes of amorphous precursor nucleation and crystallization in diffusion-dominated spinodal and non-spinodal simple liquids are also examined. Melting and premelting transitions in defected body-centered cubic solids are studied numerically, and a localized melting theory based on defect elastic energies is for- mulated. Basic features of the dynamic phase separation patterns that develop in growing heteroepitaxially strained alloy films are also outlined based on numerical simulations of a binary PFC model. Finally, dislocation dynamics are examined in strained periodic systems. The central features of dislocation glide, climb, and an- nihilation are shown to naturally emerge within PFC models, and the dynamics of individual dislocations are found to reduce to a simple generalized equation of motion. iii R´esum´e Des proc´ed´esdynamiques dans des syst`emesliquide-solide non-´equilibr´essont ´etudi´es au cours d’´echelles de temps m´esoscopiques et d’´echelles de longueur atomistiques en utilisant des mod`eles “phase-field crystal” (PFC). Diverses transitions de cong´elation et de fusion sont examin´ees en deux et trois dimensions, et les ph´enom`enesmicro- scopiques responsables de la plasticit´edes phases solides sont ´etudi´ees. Un accent est mis sur la probl´ematique des dynamiques atomistiques au cours d’´echelles de temps qui sont g´en´eralement inaccessibles aux approches conventionnelles. Les dynamiques de formation vitreuse dans les liquides metastables surfondus pr`esd’une transition vitreuse sont ´etudi´esnum´eriquement, et les caract´eristiquescen- trales de la transition, y compris un certain nombre de comportements qui n’ont pas ´et´ed´emontr´eespr´ec´edemment par les mod`elesPFC / simulations de la th´eorie clas- sique densit´efonctionnelle, sont reproduites avec succ`es. Un lien entre la longueur de corr´elation dynamique liquide et la fragilit´eest identifi´e,et il est d´emontr´epar une normalisation de temps physiquement motiv´e,appliqu´eeaux donn´eesde simulation, qu’il y a une correspondance qualitative avec des bases ph´enom`enesde transition vitreuse sur 12 ordres de grandeurs de temps. Les proc´ed´es concurrentiels de la nucl´eation de pr´ecurseursamorphes et de la cristallisation domin´eespar la diffusion dans les liquides simples spinodaux et non- spinodaux sont aussi examin´es.Les transitions de fusion et pr´e-fusion dans des solides cubiques centr´esayant des d´efauts sont ´etudi´esnum´eriquement, et une th´eorie de fusion localis´ee bas´ee sur les ´energies ´elastiques des d´efauts est formul´ee. Des car- act´eristiques de base des motifs de la s´eparation de phase dynamique qui se d´eveloppent pendant la croissance des films tendus heteroepitaxiellement sont ´egalement etudi´esen utilisant des simulations num´eriques d’un mod`elebinaire PFC. Enfin, les dynamiques des dislocations sont examin´eesdans les syst`emes p´eriodiques tendus, ainsi il est d´emontr´eque les caract´eristiquesfondamentales des proc´ed´esde glisse, d’escalade, et d’annihilation ´emergent naturellement des mod`eles PFC. v Acknowledgements This dissertation was completed with direct and indirect contributions from a number of people and organizations. It will be a difficult task to adequately thank those who have helped me through this process, but I hope that the following lines serve as a beginning. I would first like to express my deepest gratitude to my thesis advisor, Prof. Mar- tin Grant. Martin has given me time, guidance, encouragement, support, and access to his expertise without lapse over the past years, despite so many other demands on his time as Chair of Physics and then as Dean of Science. His patience, humor, in- sight, humility, and willingness to let me work with freedom and without unnecessary judgement have been greatly appreciated. I would also like to express my deepest gratitude to Prof. Ken Elder. Ken gave me the opportunity to pursue my interest in physics as a final year engineering student at a nearby university, and has acted as my unofficial thesis co-advisor while I have studied at McGill. He has contributed heavily to the direction, development, and solution of much of the work presented in this dissertation. He has also set an excellent professional and personal example to which I can only hope to aspire. To the Grant group postdocs - Dan Vernon, Sami Majaniemi, and Makiko Nono- mura: Thank you for the advice, encouragement, and many helpful discussions. I am continually in awe of your ability and knowledge. I would also like to thank the Grant group students that I have had the pleasure of knowing along the way - Cristiano, David, Smiso, Jens, and Dominique. I have had the fortune as well of learning from and collaborating with Prof. Nik Provatas while at McMaster University, and more recently, of discussing research with Prof. J¨org Rottler at UBC. Of the many excellent and generous professors at McGill, I would like to thank the following for sharing their time with me in class lectures or one- on-one discussions: Prof. Paul Tupper, Prof. Maria Kilfoil, Prof. Jorge Vi˜nals,Prof. Michael Hilke, Prof. Guy Moore, Prof. Dik Harris, and Prof. Zaven Altounian. Juan Gallego deserves a special acknowledgement for expertly solving the large number of supercomputing issues that I brought his way and for humoring my regular requests for more nodes. Thank you as well to the others in the Physics Department who vi have patiently helped me over the past years: Paula Domingues, Diane Koziol, Louise Decelles, Elizabeth Shearon, Sonia Vieira, Lauren Kay, and Paul Mercure. To my family: Mom and dad - thank you for letting me grow in my own way and for being so unbelievably supportive. Katie, Rob, Kevin, and Leah - thank you for being such positive sisters and brothers to your strange middle sibling. Tiffany, thank you for making my time in the city of Montreal not just bearable, but genuinely happy. Your belief in me has helped more than you know. Thanks also to some physics-enabled friends: Till, Steve, Carolyn, Alex, Allison, and James. In addition to the McGill University Physics Department, this research was sup- ported by a Richard H. Tomlinson Fellowship, a Carl Reinhardt Fellowship, a Schulich Graduate Fellowship, and computing resources made available by CLUMEQ/Compute Canada. Contents Abstract . i R´esum´e . iii Acknowledgements . v Table of contents . vii List of figures . xi List of symbols . xv 1 Introduction 1 1.1 Modeling Materials Classically . 3 1.1.1 Atomistic and discrete approaches . 3 1.1.2 Continuum approaches . 5 1.1.3 Multiscale modeling . 7 1.2 Background, Goals, and Main Findings . 9 1.2.1 Liquid-solid transitions . 9 1.2.2 Crystal plasticity . 16 1.3 Dissertation Outline . 17 1.4 Publications . 18 2 An Introduction to Phase-Field Crystal Modeling 19 2.1 Relevant Length and Time Scales . 19 2.2 The PFC Functional . 20 2.2.1 Relation to conventional phase-field models . 22 2.2.2 Derivation from classical density functional theory . 22 2.2.3 Static model properties . 26 vii viii Contents 2.2.4 Binary PFC functionals . 32 2.3 Dynamics . 36 2.3.1 Nonconserved relaxation: Model A . 37 2.3.2 An accelerated Model A . 38 2.3.3 Conserved diffusion: Model B . 38 2.3.4 Conserved nonlinear diffusion: Overdamped DDFT . 39 2.3.5 Inertial dynamics . 39 2.3.6 Higher-order models: Hydrodynamics of isothermal solids . 41 2.3.7 Binary systems: Coupled dynamics . 42 2.4 Overview and Research Directions . 42 3 Grain Boundary Melting and Premelting 45 3.1 Phenomenology and Literature . 45 3.2 Model Equations and Notation . 47 3.3 Simulation Results . 48 3.4 Local Melting Equations . 53 3.4.1 Isolated dislocations . 53 3.4.2 Low angle boundaries . 56 3.4.3 High angle boundaries . 59 3.5 Coexistence and the Canonical Ensemble . 60 3.6 Grand Canonical Ensemble . 63 3.7 Conclusions . 66 4 Dislocation Dynamics 67 4.1 Phenomenology and Literature . 67 4.2 Model Equations and Notation . 68 4.3 Simulation Method . 69 4.3.1 Discretization, initial and boundary conditions . 69 4.3.2 Strain application . 70 4.3.3 Symmetries and time scales . 71 4.3.4 Simulation output: A preliminary example . 72 4.4 Results and Analysis . 74 4.4.1 Equilibrium dislocation geometry . 74 Contents ix 4.4.2 Glide: Constant applied shear rate dynamics . 75 4.4.3 Climb: Constant applied strain rate dynamics . 85 4.4.4 Annihilation . 90 4.5 Conclusions .