Field Theoretic Method in Phase Transformations (Lecture Notes in Physics, 1016)

✍ Scribed by Alexander Umantsev

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 504
• Edition: 2
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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