# Transformations of Materials

- Dimitri D Vvedensky

- September 2019

### Description

Phase transformations are among the most intriguing and technologically useful phenomena in materials, particularly with regard to controlling microstructure. After a review of thermodynamics, this book contains chapters on Brownian motion and the diffusion equation, diffusion in solids based on transition-state theory, spinodal decomposition, nucleation and growth, instabilities in solidification, and diffusionless transformations. Each chapter includes exercises and the solutions are available in a separate manual.

Based on the notes from a graduate course taught in the Centre for Doctoral Training on Theory and Simulation of Materials, which was attended by students with undergraduate degrees in physics, mathematics, chemistry, materials science and engineering, this book is written to accommodate diverse backgrounds.

### About Editors

Dimitri D Vvedensky is a professor of physics in the Department of Physics at Imperial College London. He obtained his BS in mathematics at the University of Maryland, and his SM and PhD in materials science at the Massachusetts Institute of Technology (MIT). He has been on the faculty at Imperial since 1985.

### Table of Contents

Preface

1 Overview of Thermodynamics

1.1 Basic Concepts and Terminology

1.1.1 Systems and Boundaries

1.1.2 Equilibrium and State Variables

1.1.3 Processes

1.2 The Laws of Thermodynamics

1.2.1 The Zeroth Law of Thermodynamics

1.2.2 The First Law of Thermodynamics

1.2.3 The Second Law of Thermodynamics

1.2.4 The Third Law of Thermodynamics

1.3 Fundamental Equations

1.3.1 The Gibbs Function

1.3.2 The Helmholtz Function

1.4 Thermal, Mechanical, and Chemical Equilibria

1.5 Phase Equilibria

1.5.1 The Clausius–Clapeyron Equation

1.5.2 The Gibbs Phase Rule

1.6 Summary

Further Reading

Exercises

2 Brownian Motion, Random Walks, and the Diffusion Equation

2.1 Random Walks and Brownian Motion

2.1.1 Brownian Motion

2.1.2 The Random Walk and the Diffusion Equation

2.2 Fick's Laws and the Diffusion Equation

2.2.1 Fick's First Law

2.2.2 The Continuity Equation

2.2.3 Fick's Second Law and the Diffusion Equation

2.3 Fundamental Solution of the Diffusion Equation

2.3.1 Differential Equation for Fourier Components

2.3.2 The Fundamental Solution

2.3.3 Solution of the Initial-Value Problem

2.4 Examples

2.5 Summary

Further Reading

Exercises

3 Atomic Diffusion in Solids

3.1 Defects in Solids

3.1.1 Point Defects

3.1.2 Line Defects

3.1.3 Plane Defects

3.1.4 Volume Defects

3.2 Thermodynamics of Point Defects

3.3 Diffusion Mechanisms

3.4 Transition-State Theory

3.4.1 Assumptions of Classical Transition-State Theory

3.4.2 Equilibrium Statistical Mechanics

3.4.3 The Dividing Surface

3.4.4 The Rate Constant

3.4.5 The Harmonic Approximation

3.5 Analysis of Diffusion Experiments

3.5.1 Diffusion Processes

3.5.2 Arrhenius Diagrams

3.6 Summary

Further Reading

Exercises

4 Spinodal Decomposition

4.1 The Bragg–Williams Model

4.2 The Phase Diagram

4.2.1 Thermodynamic Stability

4.2.2 Stable, Metastable, and Unstable Phases

4.2.3 Kinetics of Unmixing

4.3 The Cahn–Hilliard Equation

4.3.1 Spatially-Varying Concentrations

4.3.2 The Fundamental Equations

4.3.3 Functional Derivative of the Free Energy

4.3.4 Evolution Equation for the Concentration

4.4 Experiments on Spinodal Decomposition

4.5 Summary

Further Reading

Exercises

5 Nucleation and Growth

5.1 Classical Nucleation Theory

5.1.1 Metastability

5.1.2 Homogeneous formation of nuclei

5.2 Nucleation Rate of Solid-State Transformations

5.3 Homogeneous versus Heterogeneous Nucleation

5.3.1 Heterogeneous Nucleation on a Surface

5.3.2 Elastic Effects in Solid State Nucleation

5.4 Overall Transformation Rate

5.4.1 Kolmogorov–Johnson–Mehl–Avrami Theory

5.4.2 Derivation of the KJMA Equation

5.4.3 Determination of Nucleation Mechanisms

5.4.4 Graphene

5.5 Summary

Further Reading

Exercises

6 Instabilities of Solidification Fronts

6.1 Solidification of a Pure Liquid

6.1.1 The Heat Equation

6.1.2 Velocity of the Liquid-Solid Interface

6.1.3 The Gibbs–Thomson Equation

6.1.4 Equations for the Dimensionless Temperature

6.2 Motion of a Spherical Solidification Front

6.2.1 Solution for Shape-Preserving Growth

6.3 Linear Stability of Spherical Front

6.3.1 Solution of the Heat Equation

6.3.2 The Gibbs–Thomson Relation

6.3.3 The Conservation Equation

6.3.4 The Dispersion Relation

6.3.5 Numerical Simulation of Two-Dimensional Instabilities

6.4 Constitutional Supercooling

6.5 Summary

Further Reading

Exercises

7 Diffusionless Transformations

7.1 Martensitic Transformations

7.1.1 Crystallographic Considerations

7.1.2 Free Energy Changes

7.2 Shape-Memory Alloys and Pseudoelasticity

7.3 Theory of Pseudoelasticity

7.3.1 Ginzburg-Landau Free Energy Functional

7.3.2 Nucleation of Critical 'True Twin' Droplets

7.4 Summary

Further Reading

Exercises

Bibliography

Appendices

A Contour Integrals for the Fundamental Solution

B Curvature of Plane Curves

C Integrals for Droplet Nucleation### Bibliographic

Paperback ISBN: 9780750330107

Ebook ISBN: 9781643276199

DOI: 10.1088/2053-2571/ab191e

Publisher: Morgan & Claypool Publishers