PREFACE

1.1 What Is This Book About?

1.2 Who Is This Book For?

1.3 Historical Note

1.4 Nomenclature

References

PART I: Classical Theories of Phase Transformations

CHAPTER 1: Thermodynamic Equilibrium of Phases

1.1 Definition of a Phase and Phase Transition

1.3 Theory of Capillarity

Problems

References

CHAPTER 2: Ehrenfest Classification of Phase Transitions

Problems

CHAPTER 3: Isothermal Kinetics of Phase Transformations

3.1 JMAK Theory of Nucleation and Growth

3.2 Classical Nucleation Theories

3.2.1 Frenkel's Distribution

3.2.2 Becker-Döring Theory

3.2.3 Zeldovich Theory

Problems

References

CHAPTER 4: Stefan Problem

Problems

References

CHAPTER 5: Stability of States

5.3.1 Thermodynamic Stability

5.3.1 Dynamic Stability

5.3.3 Morphological Stability

Problems

References

CHAPTER 6: Dendritic Growth

Problems

References

CHAPTER 7: Coarseningof Second Phase Precipitates

Problems

References

CHAPTER 8: Magnetic Transitions

Problems

References

PART II: The Method

CHAPTER 1: Landau Theory of Phase Transitions

1.1 The Order Parameter: Phase Transition as a Symmetry Change

1.2 The Free Energy: Phase Transition as a Bifurcation

1.3 The Tangential Potential

1.4 Phase Diagrams and Measurable Quantities

1.4.1 First-Order Transitions

1.4.2 Second-Order Transitions

1.5 Effect of External Field on Phase Transition

Problems

References

CHAPTER 2: Heterogeneous Equilibrium Systems

2.1 The Free Energy

2.1.1 Gradient Energy Contributions

2.1.2 Gradients of Temperature and Pressure

2.1.3 Gradients of Conjugate Fields

2.2 Equilibrium States

2.3 One-Dimensional Solutions of Equilibrium Equation

2.3.1 Thermo-Mechanical Analogy

2.3.2 Classification of States

2.3.3 Type-e1 States: Bifurcation off the Transition State

2.3.4 Type-e3 States: Approach to Thermodynamic Limit

2.3.5 Type-e4 State: Plane Interface

2.3.7 Type-n4 State: Critical Plate-Instanton

2.4 Free Energy Landscape

2.5 Multidimensional Equilibrium States

2.5.1 Multidimensional Close-to-Homogeneous Equilibrium States

2.5.2 Quasi One-Dimensional Equilibrium States: Sharp Interface (Drumhead) Approximation

2.6 Thermodynamic Stability of States: Local versus Global

2.6.1 Type-e4 State: Plane Interface

2.6.2 General Type-e and Type-n States

2.6.3 3d Spherically Symmetric Instanton

Problems

References

CHAPTER 3: Dynamics of Homogeneous Systems

3.1 Evolution Equation: The Linear Ansatz

3.2 Solutions of the Linear-Ansatz Dynamic Equation

3.2.1 Evolution of Small Disturbances

3.2.2 More complicated types of OPs

3.2.3 Critical Slowing Down

3.2.4 Non-Linear Evolution

3.3 Beyond the Linear Ansatz

3.4 Relaxation with Memory

3.5 Other Forces

Problems

References

CHAPTER 4: Evolution of Heterogeneous Systems

4.1 Time-Dependent Ginzburg-Landau Evolution Equation

4.2.1 Homogeneous Equilibrium States

4.2.2 Heterogeneous Equilibrium States

4.3 Motion of a Plane Interface

4.3.1 Thermo-Mechanical Analogy

4.3.2 Polynomial Solution

4.3.3 Morphological Stability

4.4.1 Non-Equilibrium Interface Energy

4.4.2 Evolution of a Spherical Droplet

4.5 Dynamics of Domain Growth

Problems

References

CHAPTER 5: Thermodynamic Fluctuations

5.2 Levanyuk-Ginsburg Criterion

5.3 Dynamics of Fluctuating Systems: Langevin Force

5.4 Evolution of the Structure Factor

5.5 Drumhead Approximation of the Evolution Equation

5.5.1 Evolution of the Interfacial Structure Factor

5.5.2 Nucleation in the Drumhead Approximation

Problems

References

CHAPTER 6: Concluding Remarks

6.1 Parameters of the Method

6.2 Boundaries of Applicability of the Method

Problems

References

PART III: Applications

1.1 Conservative Order Parameter: Theory of Spinodal Decomposition

1.1.1 Thermodynamic Equilibrium in Binary Systems

1.1.2 Equilibrium in Inhomogeneous Systems

1.1.3 Dynamics of Decomposition in Binary Systems

1.1.4 Evolution of Small Disturbances

1.1.5 Role of fluctuations

1.2.1 Order Parameter and Free Energy

1.2.2 Equilibrium Equations

1.2.3 Surface Tension of the Superconducting/Normal Phase Interface

1.3 Multicomponent Order Parameter: Crystallographic Phase Transitions

1.3.1 Invariance to Symmetry Group

1.3.2 Inhomogeneous Variations

1.4 Memory Effects: Non-Markovian Systems

1.5 "Mechanical" Order Parameter

Problems

References

CHAPTER 2: Multi-Physics Coupling: Thermal Effects of Phase Transformations

2.1 Equilibrium States of a Closed (Adiabatic) System

2.1.2 Type-E2 States

2.2 Generalized Heat Equation

2.3 Emergence of a New Phase

2.4 Motion of Interfaces: Drumhead (Sharp Interface) Approximation

2.4.1 Generalized Stefan Heat-Balance Equation

2.4.2 Generalized Kinetic Equation

2.5 Length and Energy Scales

2.6 Pattern Formation

2.6.1 One-Dimensional Transformation

2.6.2 Two-Dimensional Transformation

2.7 Thermo-Mechanical Analogy

Problems

References

CHAPTER 3: Extensions of the Method

3.1 Cellular Automata Method: "Poor Man's Phase Field"

3.2 Phase-Field Models of Grain Growth

3.2.1 Multiphase Field Models

3.2.1 Orientational Order-Parameter Field Models

3.3 Phase-Field Models of Dislocations and Voids

3.4 Phase-Field Crystal

Problems

References

EPILOGUE

APPENDIX A: Coarse-Graining Procedure

APPENDIX B: Calculus of Variations and Functional Derivative

APPENDIX C: Orthogonal Curvilinear Coordinates

APPENDIX D: Classical Mechanics and Lagrangian Field Theory

APPENDIX E: Eigenfunctions and Eigenvalues of The Schrödinger Equation and Sturm's Comparison Theorem

APPENDIX F: Fourier and Legendre Transforms

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: The Stefan Problem

APPENDIX J: "On the Theory of Adsorption of Sound in Liquids"

By L. I. Mandelshtam and M. A. Leontovich

APPENDIX K: Thermodynamically Consistent Heat Equation

SUBJECT INDEX