Field Theoretic Method in Phase Transformations

Field Theoretic Method in Phase Transformations

by Alexander Umantsev


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Product Details

ISBN-13: 9781461414865
Publisher: Springer New York
Publication date: 04/20/2012
Series: Lecture Notes in Physics , #840
Edition description: 2012
Pages: 344
Product dimensions: 6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1: Introduction.- What Is This Book About?.- Who Is This Book For?.- Historical Note.- Nomenclature.- Acknowledgements.- 2: Landau Theory of Phase Transitions.- 2.1 A Phase and Phase Transition.- 2.2 Phase Transition as Symmetry Change: the Order Parameter.- 2.3 Phase Transition as a Catastrophe: the Free Energy.- 2.4 Ehrenfest Classification.- 2.5 The Tangential Potential.- 2.6 Phase Diagrams and Measurable Quantities.- 2.6.1 First-Order Transitions.- 2.6.2 Second-Order Transitions.- 2.7 Effect of External Field on Phase Transition.- 3: Heterogeneous Equilibrium Systems.- 3.1 Theory of Capillarity.- 3.2 The Free Energy.- 3.3 Equilibrium States.- 3.4 One-Dimensional Solutions of Equilibrium Equation.- 3.4.1 Classification.- 3.4.2 Type-e1 States: Bifurcation off the Transition State.- 3.4.3 Type-e3 States: Approach to Thermodynamic Limit.- 3.4.4 Type-e4 State: Plane Interface.- 3.4.5 Interfacial Properties: Gibbs Adsorption Equation.- 3.4.6 Type-n4 State: Critical Plate—Instanton.- 3.5 Free Energy Landscape.- 3.6 Multidimensional Equilibrium States.- 3.6.1 Quasi One-Dimensional States: Drumhead (Sharp Interface) Approximation.- 3.6.2 Critical Droplet—3d Spherically-Symmetric Instanton.- 3.6.3 Small Deviations From Homogeneous Equilibrium States: Fourier Method.- 3.7 Thermodynamic Stability of States: Local versus Global.- 3.7.1 Type-e4 State: Plane Interface.- 3.7.2 General Type-e and Type-n States.- 3.7.3 3d Spherically-Symmetric Instanton.- 3.8 Gradients of Conjugate Fields.- 4: Dynamics of Homogeneous Systems.- 4.1 Evolution Equation: The Linear Ansatz.- 4.2 Solutions of the Linear-Ansatz Dynamic Equation.- 4.2.1 Evolution of Small Disturbances.- 4.2.2 More complicated types of OPs.- 4.2.3 Critical Slowing Down.- 4.2.4 Non-Linear Evolution.- 4.3 Beyond the Linear Ansatz.- 4.4 Relaxation with Memory.- 4.5 Other Forces.- 5: Evolution of Heterogeneous Systems.- 5.1 Time-Dependent Ginzburg-Landau Evolution Equation.- 5.2 Motion of Plane Interfaces.- 5.3 Dynamic Stability of Equilibrium States .- 5.3.1 Homogeneous Equilibrium States.- 5.3.2 Heterogeneous Equilibrium States.- 5.3.3 Morphological Stability of Moving Plane Interface.- 5.4 Motion of Curved Interfaces: Drumhead (Sharp Interface) Approximation.- 5.4.1 Non-Equilibrium Interface Energy.- 5.4.2 Evolution of a Spherical Droplet.- 5.5 Dynamics of Domain Growth.- 6: Thermo-Mechanical Analogy.- 7: Thermodynamic Fluctuations.- 7.1 Classical Nucleation Theory.- 7.2 Free Energy of Equilibrium System with Fluctuations.- 7.3 Levanyuk-Ginsburg Criterion.- 7.4 Dynamics of Fluctuating Systems: Langevin Force.- Evolution of the Structure Factor.- Drumhead Approximation of the Evolution Equation.- Evolution of the Interfacial Structure Factor.- Nucleation in the Drumhead Approximation.- 8: More Complicated Systems.- 8.1 Conservative Order Parameter: Theory of Spinodal Decomposition.- 8.1.1 Thermodynamic Equilibrium in Binary Systems.- 8.1.2 Equilibrium in Inhomogeneous Systems.- 8.1.3 Dynamics of Decomposition in Binary Systems.- 8.1.4 Evolution of Small Disturbances.- 8.1.5 Role of fluctuations.- 8.2 Complex Order Parameter: Ginzburg-Landau’s Theory of Superconductivity.- Order Parameter and Free Energy.- Equilibrium Equations.- 8.2.3 Surface Tension of the Superconducting/Normal Phase Interface.- Multicomponent Order Parameter: Crystallographic Phase Transitions.- Invariance to Symmetry Group.- Inhomogeneous Variations.- 8.3.3 Equilibrium States.- 8.4 Memory Effects: Non-Markovian Systems.- 8.5 “Mechanical” Order Parameter.- 9: Thermal Effects: Coupling to “Hydrodynamic” Variables.- 9.1 Equilibrium States of a Closed (Adiabatic) System.- 9.1.1 Type-E1 States.- Type-E2 States.- Generalized Heat Equation.- Emergence of a New Phase.- Motion of Interfaces: Drumhead Approximation.- 9.4.1 Generalized Stefan Heat-Balance Equation.- 9.4.2 Generalized Kinetic Equation.- 9.4.3 Gibbs-Duhem Force.- 9.4.4 Inter-Phase Boundary Motion: Heat Trapping.- 9.4.5 APB Motion: Thermal Drag.- Length and Energy Scales.- Pattern Formation.- 1-Dimensional Transformation.- 2-Dimensional Transformation.- 10: Transformations in Real Materials.- 10.1 Parameters of FTM.- 10.2 Boundaries of Applicability of FTM.- 11: Extensions of the Method.- 11.1 Cellular Automata Method: “Poor Man’s Phase Field”.- 11.2 Continuum Models of Grain Growth.- Multiphase Field Models.- Orientational Order Parameter Field Models.- Phase-Field Crystal.- 11.3 Epilog: Successes Stories.- 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.- 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 K: “On the Theory of Adsorption of Sound in Liquids” By L. I. Mandelshtam and M. A. Leontovich.- Index.

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