Field Theoretic Method in Phase Transformations

This book describes a novel and popular method for the theoretical and computational study of phase transformations and materials processing in condensed and soft matter. The field theoretic method for the study of phase transformations in material systems, also known as the phase-field method, allows one to analyze different stages of transformations within a unified framework. It has received significant attention in the materials science community due to many recent successes in solving or illuminating important problems. In a single volume, this book addresses the fundamentals of the method starting from the basics of the field theoretic method along with its most important theoretical and computational results and some of the most advanced recent results and applications. Now in a revised and expanded second edition, the text is updated throughout and includes material on the classical theory of phase transformations. This book serves as both a primer in the area of phase transformations for those new to the field and as a guide for the more seasoned researcher. It is also of interest to historians of physics. Preface What Is This Book About? How Is This Edition Different from the First One? Who Is This Book For? Historical Note Nomenclature References Contents About the Author Part I: Classical Theories of Phase Equilibria and Transformations Chapter 1: Thermodynamic States and Their Stabilities 1.1 Laws of Thermodynamics 1.2 Thermodynamic Stability of Equilibrium States 1.3 Dynamic Stability of States 1.4 Stability of Heterogeneous States 1.5 Analysis of Dynamic Stability in Terms of Normal Modes Exercises References Chapter 2: Thermodynamic Equilibrium of Phases 2.1 Definition of a Phase and Phase Transition 2.2 Conditions of Phase Equilibrium 2.3 Ehrenfest Classification of Phase Transitions 2.4 Phase Coexistence and Gibbsian Description of an Interface Exercises Reference Chapter 3: Examples of Phase Transitions 3.1 Crystallization (Freezing)-Melting 3.2 Martensitic Transition 3.3 Magnetic Transitions 3.4 Ferroelectric Transition Exercises References Chapter 4: Isothermal Kinetics of Phase Nucleation and Growth 4.1 JMAK Theory of Nucleation and Growth 4.1.1 Theory of Thermally Activated Nucleation and Growth 4.2 Classical Nucleation Theories 4.2.1 Gibbsian Theory of Capillarity 4.2.2 Frenkel ́s Distribution of Heterophase Fluctuations in Unsaturated Vapor 4.2.3 Becker-Döring Kinetic Theory of Nucleation in Supersaturated Vapor 4.2.4 Zeldovich Theory of Nucleation* 4.3 Critique of Classical Nucleation Theories Exercises References Chapter 5: Coarsening of Second-Phase Precipitates 5.1 Formulation of the Problem 5.1.1 Rate Equation 5.1.2 Condition of Local Equilibrium 5.1.3 Mass Conservation Condition 5.1.4 Particle Size Distribution Function 5.1.5 Initial, Boundary, and Stability Conditions 5.2 Resolution of the Problem 5.2.1 Unchanging Supersaturation 5.2.2 Asymptotic Analysis of Bulk Properties 5.2.3 Asymptotic Analysis of the Distribution Function 5.3 Domain of Applicability and Shortcomings of the Theory 5.3.1 Quasi-Steady Diffusion Field 5.3.2 Local Equilibrium 5.3.3 Absence of Nucleation 5.3.4 Mean-Field Approximation 5.3.5 Disregard of Coalescence 5.3.6 Spherical Shape 5.3.7 Asymptotic Regime 5.3.8 Small Particles Exercises References Chapter 6: Spinodal Decomposition in Binary Systems Exercises Chapter 7: Thermal Effects in Kinetics of Phase Transformations 7.1 Formulation of the Stefan Problem 7.1.1 Stefan Boundary Condition 7.1.2 Thermodynamic Equilibrium Boundary Condition 7.1.3 Phase Equilibrium 7.1.4 Initial Condition 7.1.5 Far Field Condition 7.2 Plane-Front Stefan Problem 7.2.1 Diffusion Regime of Growth 7.2.2 Adiabatic Crystallization 7.2.3 Critical Supercooling Crystallization 7.2.4 General Observations 7.3 Ivantsov ́s Theory* 7.4 Morphological Instability* 7.5 Dendritic Growth 7.5.1 Analysis of Classical Theories of Crystallization 7.6 Numerical Simulations of Dendritic Growth 7.6.1 Cellular Automata Method 7.6.2 Simulation Results 7.6.3 Conclusions 7.7 Rate of Growth of New Phase Exercises References Part II: The Method Chapter 8: Landau Theory of Phase Transitions 8.1 Phase Transition as a Symmetry Change: The Order Parameter 8.2 Phase Transition as a Bifurcation: The Free Energy 8.3 Classification of the Transitions 8.4 The Tangential Potential 8.5 Other Potentials 8.6 Phase Diagrams and Measurable Quantities 8.6.1 First-Order Transitions 8.6.2 Second-Order Transitions 8.7 Effect of External Field on Phase Transition Exercises References Chapter 9: Heterogeneous Equilibrium Systems 9.1 The Free Energy 9.1.1 Gradients of Order Parameter 9.1.2 Gradients of Conjugate Fields* 9.1.3 Field-Theoretic Analogy 9.2 Equilibrium States 9.3 One-Dimensional Equilibrium States 9.3.1 General Properties 9.3.2 Classification 9.3.3 Thermomechanical Analogy 9.3.4 Type-e Solutions: General Properties 9.3.5 Type-e1 Solution: Bifurcation From the Transition State 9.3.6 Type-e3 Solution: Approach to Thermodynamic Limit 9.3.7 Type-e4 Solution: Plane Interface 9.3.8 Interfacial Properties: Gibbs Adsorption Equation* 9.3.9 Type-n4 Solution: Critical Plate-Instanton 9.4 Free Energy Landscape* 9.5 Multidimensional Equilibrium States 9.5.1 Ripples: The Fourier Method* 9.5.2 Sharp-Interface (Drumhead) Approximation 9.5.3 The Critical Droplet: 3d Spherically Symmetric Instanton 9.6 Thermodynamic Stability of States: Local Versus Global* 9.6.1 Type-e4 State: Plane Interface 9.6.2 General Type-e and Type-n States 9.6.3 3d Spherically Symmetric Instanton Exercises References Chapter 10: Dynamics of Homogeneous Systems 10.1 Evolution Equation: The Linear Ansatz 10.2 Dynamics of Small Disturbances in the Linear-Ansatz Equation 10.2.1 Spinodal Instability and Critical Slowing Down 10.2.2 More Complicated Types of OPs 10.3 Evolution of Finite Disturbances by the Linear-Ansatz Equation 10.4 Beyond the Linear Ansatz 10.5 Relaxation with Memory 10.6 Other Forces Exercises References Chapter 11: Evolution of Heterogeneous Systems 11.1 Time-Dependent Ginzburg-Landau Evolution Equation 11.2 Dynamic Stability of Equilibrium States 11.2.1 Homogeneous Equilibrium States 11.2.2 Heterogeneous Equilibrium States 11.3 Motion of Plane Interface 11.3.1 Thermomechanical Analogy 11.3.2 Polynomial Solution 11.3.3 Selection Principle 11.3.4 Morphological Stability 11.4 Emergent Equation of Interfacial Dynamics 11.4.1 Nonequilibrium Interface Energy 11.5 Evolution of a Spherical Droplet 11.6 Domain Growth Dynamics 11.7 *Thermomechanical Analogy Revisited Exercises References Chapter 12: Thermodynamic Fluctuations 12.1 Free Energy of Equilibrium System with Fluctuations 12.2 Correlation Functions 12.3 Levanyuk-Ginzburg Criterion 12.4 Dynamics of Fluctuating Systems: The Langevin Force 12.4.1 Homogeneous Langevin Force 12.4.2 Inhomogeneous Langevin Force 12.5 Evolution of the Structure Factor 12.6 *Drumhead Approximation of the Evolution Equation 12.6.1 Evolution of the Interfacial Structure Factor 12.6.2 Nucleation in the Drumhead Approximation 12.7 Homogeneous Nucleation in Ginzburg-Landau System 12.8 *Memory Effects: Non-Markovian Systems Exercises References Chapter 13: Multi-Physics Coupling: Thermal Effects of Phase Transformations 13.1 Equilibrium States of a Closed (Adiabatic) System 13.1.1 Type-E1 States 13.1.2 *Type-E2 States 13.2 Generalized Heat Equation 13.3 Emergence of a New Phase 13.4 Non-isothermal Motion of a Curved Interface 13.4.1 Generalized Stefan Heat-Balance Equation 13.4.2 Surface Creation and Dissipation Effect 13.4.3 Generalized Emergent Equation of Interfacial Dynamics 13.4.4 Gibbs-Duhem Force 13.4.5 Interphase Boundary Motion: Heat Trapping Effect 13.4.6 APB Motion: Thermal Drag Effect 13.5 Variability of the System: Length and Energy Scales 13.6 Pattern Formation 13.6.1 One-Dimensional Transformation 13.6.2 Two-Dimensional Transformation Exercises References Chapter 14: Validation of the Method 14.1 Physical Consistency 14.2 Parameters of FTM 14.3 Boundaries of Applicability Exercises Part III: Applications Chapter 15: Conservative Order Parameter: Theory of Spinodal Decomposition in Binary Systems 15.1 Equilibrium in Inhomogeneous Systems 15.2 Dynamics of Decomposition 15.3 Evolution of Small Disturbances 15.4 Pattern Formation 15.5 Role of fluctuations Exercises References Chapter 16: Complex Order Parameter: Ginzburg-Landau ́s Theory of Superconductivity 16.1 Order Parameter and Free Energy 16.2 Equilibrium Equations 16.3 Surface Tension of the Superconducting/Normal Phase Interface Exercise References Chapter 17: Multicomponent Order Parameter: Crystallographic Phase Transitions 17.1 Invariance to Symmetry Group 17.2 Inhomogeneous Variations 17.3 Equilibrium States Exercises References Chapter 18: ``Mechanical ́ ́ Order Parameter Reference Chapter 19: Continuum Models of Grain Growth 19.1 Basic Facts About Growing Grains 19.2 Multiphase-Field Models 19.3 Orientational Order Parameter Field Models 19.4 Phase-Field Crystal References Appendix A: Coarse-Graining Procedure Appendix B: Calculus of Variations and Functional Derivative Appendix C: Orthogonal Curvilinear Coordinates Appendix D: Lagrangian Field Theory Appendix E: Eigenfunctions and Eigenvalues of the Schrödinger Equation and Sturm ́s Comparison Theorem Appendix F: Fourier and Legendre Transforms Fourier Transform Legendre Transform Appendix G: Stochastic Processes The Master and Fokker-Plank Equations Decomposition of Unstable States Diffusion in Bistable Potential Autocorrelation Function The Langevin Approach Appendix H: Two-Phase Equilibrium in a Closed Binary System Appendix I: On the Theory of Adsorption of Sound in Liquids Epilogue: Challenges and Future Prospects References Appendix A Appendix B Appendix E Appendix F Appendix G Index