Field Theoretic Method in Phase Transformations

ISBN-13: 9783031296048 / Angielski

Alexander Umantsev

Opis

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.

Spis treści

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

1.2  Gibbs Phase Rule

1.3  Theory of Capillarity

Problems

References

CHAPTER 2: Ehrenfest Classification of Phase Transitions

Problems

References

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: Coarsening of 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

2.3.6 Interfacial Properties: Gibbs Adsorption Equation

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

2.5.3    Critical Droplet—3d Spherically-Symmetric Instanton

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       Dynamic Stability of Equilibrium States

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       Motion of Curved Interfaces: Sharp Interface (Drumhead) Approximation

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

5.1       Free Energy of Equilibrium System with 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

CHAPTER 1:  More Complicated Systems

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       Complex Order Parameter: Ginzburg-Landau’s Theory of Superconductivity

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

1.3.3    Equilibrium States

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

2.1.1    Type-E1 States

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

2.4.3    Gibbs-Duhem Force 2.4.4    Inter-Phase Boundary Motion: Heat Trapping 2.4.5    APB Motion: Thermal Drag

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

Challenges and Future Prospects

 

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

O autorze

Alex Umantsev is a Professor of Materials Physics in the Department of Chemistry, Physics, and Materials Science at Fayetteville State University in North Carolina.  He earned his doctorate in 1986 in Moscow (Russia) and worked as a research associate at Northwestern University in the early 1990s.  After that he began his teaching career. His research interests are in the areas of materials theory and multiscale modeling of phase transformations in traditional small-molecule metallic or ceramic systems to crystallization of macromolecules of polymers and proteins. He has always been interested in the processing-structure-properties relations of materials ranging from their production to the analysis of their failure.