Phase Transitions in Materials

Description

The new edition of this popular textbook provides a fundamental approach to phase transformations and thermodynamics of materials. Explanations are emphasised at the level of atoms and electrons, and it comprehensively covers the classical topics from classical metallurgy to nanoscience and magnetic phase transitions. The book has three parts, covering the fundamentals of phase transformations, the origins of the Gibbs free energy, and the major phase transformations in materials science. A fourth part on advanced topics is available online. Much of the content from the first edition has been expanded, notably precipitation transformations in solids, heterogeneous nucleation, and energy, entropy and pressure. Three new chapters have been added to cover interactions within microstructures, surfaces, and solidification. Containing over 170 end-of-chapter problems, it is a valuable companion for graduate students and researchers in materials science, engineering, and applied physics. Phase Transitions in Materials, 2nd Edition Contents Preface Notation Part I: Basic Thermodynamics and Kinetics of Phase Transformations 1 Introduction 1.1 What Is a Phase Transition? 1.2 Atoms and Materials 1.3 Pure Elements 1.3.1 Melting: A Discontinuous Phase Transition 1.3.2 Structural Symmetry and Continuous Phase Transitions 1.4 Alloys: Unmixing and Ordering 1.5 What Is a Phase Transformation? 1.5.1 Diffusional and Diffusionless 1.5.2 Continuous and Discontinuous Transformations 1.6 Brief Review of Thermodynamics and Kinetics 1.6.1 Partition Function 1.6.2 Free Energy 1.6.3 Kinetic Master Equation Problems 2 Temperature–Composition Phase Diagrams 2.1 Intuition and Expectations about Alloy Thermodynamics 2.1.1 Overview of the Approach 2.1.2 Free Energies of Alloy Phases 2.1.3 Thermodynamics of Solutions: Qualitative 2.1.4 Hume-Rothery rules 2.2 Free Energy Curves, Solute Conservation, and the Lever Rule 2.2.1 Free Energy versus Composition 2.2.2 Conservation of Solute and the Lever Rule 2.3 Common Tangent Construction 2.4 Continuous Solid Solubility Phase Diagram 2.5 Eutectic and Peritectic Phase Diagrams 2.5.1 Eutectic Phase Diagram 2.5.2 Peritectic Phase Diagram 2.5.3 More Complex Phase Diagrams 2.6 Ternary Phase Diagrams 2.7 Free Energy of a Solid Solution 2.7.1 Parameters for Atom Configurations and Bonds 2.7.2 Partition Function and Free Energy 2.8 Unmixing Phase Diagram 2.8.1 Instability against Unmixing 2.8.2 Features of the Unmixing Phase Diagram 2.9 Order–Disorder Phase Diagram 2.9.1 Long-Range Order Parameterization 2.9.2 Temperature Dependence of Long-Range Order 2.10 Alloy Phase Diagrams Problems 3 Diffusion 3.1 Processes of Atom Movements in Crystals 3.1.1 Mechanisms 3.1.2 Temperature Dependence of Diffusion 3.1.3 Ionic Crystals 3.2 The Diffusion Equation 3.2.1 Atom Jumps, Fluxes, and Fick’s First Law 3.2.2 The Diffusion Equation 3.3 Gaussian and Error Functions in One Dimension 3.3.1 Gaussian Solution 3.3.2 Green’s Function Solution 3.4 Fourier Series Solutions to the Diffusion Equation 3.4.1 Separation of Variables in Cartesian Coordinates 3.4.2 Boundary Conditions and Initial Conditions 3.4.3 Orthogonality and Initial Conditions 3.5 Bessel Functions and Other Special Function Solutions 3.5.1 General Approach for Special Function Solutions 3.5.2 Bessel Function Solutions to the Diffusion Equation 3.5.3 Boundary Conditions, Initial Conditions, Orthogonality Problems 4 Nucleation 4.1 Nucleation Phenomena and Terminology 4.2 Critical Nucleus 4.2.1 Fluctuations 4.2.2 Surface Energy 4.2.3 Critical Radius 4.2.4 Temperature Dependence 4.3 Heterogeneous Nucleation 4.3.1 General Features of Heterogeneous Nucleation 4.3.2 Activation Barrier for Heterogeneous Nucleation 4.3.3 Surface Energies in Heterogeneous Nucleation 4.4 Free Energy Curves and Nucleation 4.4.1 Shifts of Free Energy Curves 4.4.2 Effects of Elastic Energy on Nucleation 4.5 The Nucleation Rate 4.5.1 Clusters of Atoms 4.5.2 Atom Additions and Subtractions 4.5.3 Steady-State Nucleation Rate 4.5.4 Time for a Random Walk across the Critical Size 4.5.5 Temperature Dependence of Nucleation Rate 4.6 Time-Dependent Nucleation 4.6.1 Numerical Example of Time-Dependent Nucleation 4.6.2 Analytical Results for Time-Dependent Nucleation 4.7 Nucleation in Multicomponent Systems Problems 5 Effects of Diffusion and Nucleation on Phase Transformations 5.1 Nonequilibrium Processing of Materials 5.1.1 Diffusion Lengths 5.1.2 Cooling Techniques 5.2 Alloy Solidification with Suppressed Diffusion in the Solid 5.2.1 Alloy Solidification with Solute Partitioning 5.2.2 Alloy Solidification: Suppressed Diffusion in the Solid Phase 5.2.3 Scheil Analysis of Mixing in Liquid, Suppressed Diffusion in Solid 5.3 Alloy Solidification with Suppressed Diffusion in Both Solid and Liquid 5.3.1 Concentration Gradients in the Liquid 5.3.2 Practical Issues in Alloy Solidification, Dendrite Instability 5.4 Time, Temperature, and Transformation 5.5 Glasses and Liquids 5.5.1 Glass Formation 5.5.2 The Glass Transition 5.6 Kinetics near Equilibrium Problems Part II: The Atomic Origins of Thermodynamics and Kinetics 6 Energy 6.1 Atomic Schrödinger Equations and Formalism 6.2 Molecular Orbital Theory of Diatomic Molecules 6.2.1 Molecular Schrödinger Equation 6.2.2 Molecular Wavefunctions of Bonding and Antibonding States 6.2.3 Origin of the Bond Energy 6.2.4 Ionicity and Electronegativity 6.3 Electronic Bands and the Tight-Binding Model 6.3.1 Translational Symmetry and Phase Factors 6.3.2 Tight Binding on a Linear Chain 6.4 Free and Nearly-Free Electrons 6.4.1 A Gas of Free Electrons in a Big Box 6.4.2 Electronic Heat Capacity 6.5 Some Electronic Structures of Materials 6.5.1 From Molecules to Metals 6.5.2 Fermi Surfaces 6.6 Elastic Constants and the Interatomic Potential 6.6.1 Metallic Bonding 6.6.2 General Interatomic Potential 6.7 Linear Elasticity 6.7.1 Strains 6.7.2 Stresses 6.7.3 Elastic Response 6.7.4 Elastic Energy 6.8 Misfitting Particle 6.8.1 Eshelby Cycle and Elastic Energy 6.8.2 Strain Fields for Misfitting Sphere in Infinite Matrix 6.8.3 Elastic Springs in Series 6.8.4 Energy of Misfitting Sphere in Infinite Matrix 6.8.5 Energies for Other Shapes of Misfitting Particles in an Infinite Matrix Problems 7 Entropy 7.1 Counting and Entropy 7.1.1 Static and Dynamical Sources of Entropy 7.1.2 Counting Atom Configurations in a Random Solid Solution 7.1.3 Phonon Statistics 7.2 Short-Range Order and the Pair Approximation 7.3 Materials Structures and Properties Described by Clusters 7.3.1 Correlation Functions, ξ 7.3.2 Interaction Functions, f 7.3.3 Thermodynamics with Cluster Approximations 7.4 Concept of Vibrational Entropy 7.4.1 Changes in Springs 7.4.2 Changes in Masses 7.5 Phonon Thermodynamics 7.5.1 Oscillators and Normal Modes 7.5.2 Partition Function 7.5.3 Vibrational Entropy of a Harmonic Crystal 7.6 Bond Proportion Model 7.6.1 Local Approximations for Bonds and Springs 7.6.2 Bond Proportion Model and Unmixing on the Ising Lattice 7.6.3 Bond Proportion Model and Ordering on the Ising Lattice 7.7 Bond-Stiffness-versus-Bond-Length Model 7.7.1 Phonon Frequencies and Bond Lengths 7.7.2 Extending the Bond Proportion Model 7.7.3 Other Effects on Bond Stiffness Problems 8 Pressure 8.1 Materials under Pressure at Low Temperatures 8.1.1 Gases (for Comparison) 8.1.2 Solids (for Comparison) 8.2 Thermal Pressure, a Step beyond the Harmonic Model 8.3 Free Energies and Phase Boundaries under Pressure 8.3.1 Clausius–Clapeyron Equation 8.3.2 Characteristic Pressure of a Solid 8.4 Chemical Bonding and Antibonding under Pressure 8.5 Pressure-Driven Phase Transitions 8.5.1 Effects of Pressure on Free Energy 8.5.2 Two-Level System under Pressure 8.5.3 Simultaneous High P and T 8.6 Activation Volume Problems 9 Interactions in Microstructures and Constrained Equilibrium 9.1 Solid-State Amorphization 9.2 Self-Trapping 9.2.1 Examples of Self-Trapping 9.2.2 Polarons 9.3 Thermodynamics of Complex Materials 9.3.1 Multiferroic and Complex Materials 9.3.2 Thermodynamic Relationships 9.4 Partitioning of Energy in Polycrystals and Single Crystals 9.4.1 Magnetic and Elastic Energy 9.4.2 Thermal Stresses in Anisotropic Microstructures 9.5 Coherency Strains in Chemical Unmixing 9.5.1 Chemical Effects on Elastic Fields 9.5.2 Free Energies with Coherency Stresses 9.6 Coupling between Unmixing Processes 9.6.1 Crystal Lattice and Interstitial Lattice 9.6.2 Energies of Interactions between Substitutional and Interstitial Solutes 9.6.3 Free Energies with Interactions between Substitutional and Interstitial Solutes 9.6.4 Stability near Equilibrium 9.6.5 Critical Temperature in a Coupled System 9.6.6 Examples of Unmixing with Interactions between Substitutional and Interstitial Solutes 9.6.7 Metal Hydride Alloys 9.7 Factoring the Partition Function 9.7.1 Kinetic Energy Is Separable 9.7.2 Potential Energy and Separability 9.7.3 Natural or Convenient Coordinates Problems 10 Atom Movements with the Vacancy Mechanism 10.1 Random Walk and Correlations 10.1.1 Correlation Factor (General Features) 10.1.2 Correlation Factor for Atoms with Uncorrelated Vacancy Jumps 10.2 Correlation Factors for Atoms and Vacancies in Alloys 10.2.1 Correlation Factor for a Vacancy in a Heterogeneous Alloy 10.2.2 Correlation Factors for Ordered Alloys 10.3 Phenomena in Alloy Diffusion 10.3.1 Marker Velocity 10.3.2 Interdiffusion Coefficient 10.3.3 Variable D̃(c) 10.4 Diffusion in a Potential Gradient 10.5 Diffusion in a Temperature Gradient 10.5.1 Jump Frequency and Temperature 10.5.2 Vacancies, Solutes, and Interstitials in Temperature Gradients 10.6 Nonthermodynamic Equilibrium in Driven Systems 10.7 Vineyard’s Theory of Diffusion 10.7.1 Degrees of Freedom for Atoms in a Crystal 10.7.2 Ensemble-Averaged Jump Rate 10.7.3 Transition States 10.7.4 Transition Rate 10.7.5 Harmonic Vibrations 10.7.6 Diffusion Coefficient D(T) Problems Part III: Types of PhaseTransformations 11 Thermodynamics and Phase Transitions at Surfaces 11.1 Surface Structure 11.1.1 Surface Reconstruction 11.1.2 Terraces, Ledges, Kinks, and Roughness 11.2 Thermodynamic Roughening Transition 11.2.1 High-Temperature Approach to Tc 11.2.2 Low-Temperature Approach to Tc 11.2.3 Entropy and Dimensionality for Line Defects 11.3 Surface Structure and Kinetics 11.3.1 Screw Dislocation Mechanism 11.3.2 Layer or Island Growth 11.4 Energies of Grain Boundaries and Interfaces 11.4.1 Grain Boundary Structure 11.4.2 Grain Boundary Energy 11.4.3 Chemical Energy of a Precipitate Interface 11.5 Anisotropic Surface Energy 11.5.1 Surface Structures 11.5.2 Wulff Construction 11.5.3 Wulff Construction with Grain Boundaries 11.6 Reactions at Surfaces 11.6.1 Linear and Parabolic Oxidation 11.6.2 Phases in Concentration Gradients 11.6.3 Phase Growth in Thin Films 11.7 Gas Adsorption Problems 12 Melting 12.1 Structure and Thermodynamics of Melting 12.1.1 Liquids and Solids 12.1.2 Premelting 12.1.3 Superheating 12.1.4 Free Energy and Latent Heat 12.2 Chemical Trends of Melting 12.3 Free Energy of a Solid 12.3.1 Energy, Thermal Expansion, Phonon Softening 12.3.2 Separable Contributions to the Entropy 12.3.3 Contributions to the Entropy of fcc Aluminum 12.4 Entropy of a Liquid 12.5 Thermodynamic Condition for the Melting Temperature 12.5.1 Entropy Difference of Solid and Liquid 12.5.2 Lindemann Rule for Tm 12.5.3 Correlation of Tm with Bulk Modulus 12.6 Glass Transition 12.6.1 The Kauzmann Paradox 12.6.2 Potential Energy Landscape (PEL) 12.6.3 Fragile and Strong 12.6.4 Entropy and Heat Capacity 12.7 Two Dimensions 12.7.1 Two-Dimensional Models 12.7.2 Kosterlitz–Thouless Melting Transition Problems 13 Solidification 13.1 Solidification Microstructures 13.1.1 Casting Microstructure 13.1.2 Heat Flow and Interface Motion 13.1.3 Dendrite Instability 13.2 Alloy Solidification with Suppressed Diffusion in the Liquid 13.3 Constitutional Supercooling 13.3.1 Concept of Constitutional Supercooling 13.3.2 Critical Temperature Gradient 13.4 Cellular and Dendritic Microstructures 13.4.1 Formation of Cells (and Dendrites) 13.4.2 Growth of a Columnar Dendrite 13.5 Dendrite Growth with Solute Segregation 13.5.1 Scheil Equation in a Model of Dendrite Growth 13.5.2 Forward Velocity of Solidification Profile 13.6 Surface Energy 13.6.1 Effects of Surface Energy on Undercooling and Growth Velocity 13.6.2 Heat Transport 13.6.3 Growth Velocity with Surface Energy 13.6.4 Combined Undercoolings 13.7 Developments in Solidification Science 13.7.1 Stability Analysis of Solid–Liquid Interface 13.7.2 Summary of Morphological Transitions with G, v Problems 14 Phase Transformations with Interfaces: 1. Microstructure 14.1 Guinier–Preston Zones and Precipitation Sequences 14.1.1 Thin Plates 14.1.2 Quenched-in Vacancies 14.1.3 Coherency and Strains 14.1.4 Precipitation Sequence 14.2 Precipitation at Grain Boundaries and Defects 14.2.1 Collector Plate Mechanism 14.2.2 Precipitate-Free Zone 14.2.3 Other Sites for Heterogeneous Nucleation 14.3 The Eutectoid Transformation and Pearlite 14.3.1 The Iron–Carbon Phase Diagram 14.3.2 Pearlite: a Eutectoid Transformation 14.4 Heat Treatments of Steel 14.4.1 Austenitizing 14.4.2 Decomposition of Austenite 14.4.3 TTT Diagrams of Steels 14.4.4 Martensite 14.4.5 Tempering, Carbides, and Retained Austenite 14.5 The Kolmogorov–Johnson–Mehl–Avrami Growth Equation 14.5.1 Growth Rate before Particle Contact 14.5.2 Growth Rate after Particle Contact 14.5.3 Nucleation Processes 14.6 Coarsening 14.6.1 Two Particles 14.6.2 Self-Similarity of the Microstructure during Coarsening Problems 15 Phase Transformations with Interfaces: 2. Energetics and Kinetics 15.1 Interface Thermodynamics and Kinetics 15.1.1 Interfaces in Phase Transformations 15.1.2 Diffusion across the Interface 15.1.3 Net Flux across a Moving Interface 15.1.4 Chemical Composition of the New Phase 15.2 Atomistic Model of Interface Motion 15.3 Local Nonequilibrium at Fast Interfaces 15.3.1 Assumption of Local Equilibrium 15.3.2 Description of Solute Drag 15.3.3 Analysis of Local Nonequilibrium and Solute Drag 15.4 Elastic Energy and Shape of Growing Plate-Like Precipitates 15.5 Elastic Energy and Solute Atoms 15.5.1 Context 15.5.2 Components of the Elastic Model 15.5.3 Elastic Energy of a Solid Solution 15.5.4 The Bitter–Crum Theorem 15.5.5 Coherency Stresses 15.5.6 Free Energy of Coherent Precipitation Problems 16 Spinodal Decomposition 16.1 Concentration Fluctuations and the Free Energy of Solution 16.2 A Square Gradient Term in the Free Energy 16.2.1 Justification for the Square of the Composition Gradient 16.2.2 Derivation for the Square of the Composition Gradient 16.2.3 Effects of the Square of the Composition Gradient 16.2.4 Preview of the Cahn Approach to Spinodal Decomposition 16.3 Constrained Minimization of the Free Energy 16.3.1 Calculus of Variations 16.3.2 Constraint of Constant Composition 16.3.3 Minimizing a Constrained Free Energy Functional 16.4 The Diffusion Equation 16.4.1 Diffusion Driven by a Gradient in Chemical Potential 16.4.2 Fourier Transform Solution 16.5 Effects of Elastic Energy on Spinodal Decomposition 16.5.1 Thermodynamic Effects of Elastic Energy 16.5.2 Kinetic Effects of Elastic Energy Problems 17 Phase Field Theory 17.1 Spatial Distribution of Phases and Interfaces 17.1.1 Phases as Distinct Volumes with Abrupt Interfaces 17.1.2 Phases as Field Quantities with Diffuse Interface 17.2 Order Parameters as Field Quantities 17.2.1 Kinetics of Conserved and Nonconserved Order Parameters 17.2.2 Ginzburg–Landau Equation 17.2.3 An Unmixing Phase Diagram with Landau Theory 17.2.4 Extensions to Multiple Fields 17.3 Domain Boundary Structure 17.3.1 General Features of Domain Boundaries 17.3.2 Finding the Optimal Boundary Profile 17.3.3 Boundary Profile with Ginzburg–Landau Potential 17.4 Domain Boundary Kinetics 17.4.1 Nonconservative Dynamics 17.4.2 Stability of a Flat Interface 17.4.3 Curvature and Growth Kinetics 17.4.4 Stability of Curved Boundaries with Saddle Points Problems 18 Method of Concentration Waves and Chemical Ordering 18.1 Structure in Real Space and Reciprocal Space 18.1.1 First Brillouin Zone 18.1.2 Reciprocal Lattice 18.1.3 Ordered Structures Described by ConcentrationWaves 18.2 Symmetry and the Star 18.3 The Free Energy in k-Space with Concentration Waves 18.3.1 Using ConcentrationWaves to Construct a Free Energy Functional 18.3.2 Physical Interpretation of b(k̄ ͐) 18.3.3 Special Points 18.4 Symmetry Invariance of Free Energy and Landau–Lifshitz Rule for Second-Order Phase Transitions 18.4.1 ConcentrationWaves and Lattice Translations 18.4.2 The Landau–Lifshitz Criterion 18.4.3 Applications of the Landau–Lifshitz Criterion 18.5 Thermodynamics of Ordering in the Mean Field Approximation with Long-Range Interactions 18.5.1 Thermodynamic Formulation 18.5.2 Small-Amplitude ConcentrationWaves 18.5.3 Critical Temperature 18.5.4 Formulation in k-Space Problems 19 Diffusionless Transformations 19.1 Dislocations, Mechanisms, and Twinning 19.1.1 Structure of a Dislocation 19.1.2 Dislocation Glide 19.1.3 Elastic Energy of a Dislocation (Self-Energy) 19.1.4 Stacking Faults in fcc Crystals 19.1.5 Twinning 19.2 Martensite 19.2.1 Features of Martensite 19.2.2 Transformation Mechanisms 19.2.3 Crystallographic Theory of Martensite 19.3 Landau Theory of Displacive Phase Transitions 19.3.1 Displacive Instability 19.3.2 Free Energy 19.3.3 Critical Temperature and Equilibrium Displacement 19.3.4 First-Order Landau Theory 19.4 Crystal Instabilities and Phonons 19.4.1 Criteria for Crystal Stability 19.4.2 Soft Phonons in bcc Structures 19.4.3 Phonons and Entropy Problems 20 Thermodynamics of Nanomaterials 20.1 Energies of Atoms at Grain Boundaries in Nanocrystals 20.1.1 Dimensional Scaling and Its Validity 20.1.2 Comparison of Grain Boundary Structure and Amorphous Structure 20.2 Gibbs–Thomson Effect 20.2.1 Chemical Potential 20.2.2 Melting of Nanoparticles 20.2.3 Phase Boundaries 20.3 Atomic Structures of Nanocrystals 20.3.1 Thermal Instabilities of Nanoparticles 20.3.2 Chemical Configurations within Alloy Nanocrystals 20.4 Electron Energies in Nanomaterials 20.4.1 Energies of Free Electrons in Nanostructures 20.4.2 Tight-Binding Model in Two Dimensions 20.4.3 Density of States in Two Dimensions 20.5 Entropy of Nanomaterials 20.5.1 Configurational Entropy from Microstructural Degrees of Freedom 20.5.2 Vibrational Entropy of Nanostructures 20.6 Magnetic Nanoparticles Problems 21 Magnetic and Electronic Phase Transitions 21.1 Overview of Magnetic and Electronic Phase Transitions 21.1.1 Some Magnetic Structures and Phase Transitions 21.1.2 Electronic Transitions 21.1.3 Complex Materials 21.2 Exchange Interactions 21.2.1 Exchange in the Heisenberg Model 21.2.2 Quantum Mechanical Exchange 21.2.3 Exchange Hole 21.2.4 Exchange and Shapes of Wavefunctions 21.2.5 Hund’s Rule and Stoner Criterion for Ferromagnetism 21.2.6 Slater–Pauling Curve 21.3 Thermodynamics of Ferromagnetism 21.3.1 Local Moment Ising Problem 21.3.2 Critical Temperature (Curie Temperature) 21.3.3 High-Temperature Magnetic Susceptibility 21.3.4 Low-Temperature Behavior (Classical) 21.4 Spin Waves 21.5 Thermodynamics of Antiferromagnetism 21.5.1 Néel Transition 21.5.2 Antiferromagnetic Susceptibility 21.6 Dzyaloshinskii–Moriya Interactions and Skyrmions 21.6.1 D-M Vector Interaction 21.6.2 Topological Protection 21.6.3 Magnetic Phase Diagram 21.7 Thermodynamics of Ionic Crystals 21.8 Ferroelectric Transition 21.9 Domains 21.9.1 Minimizing the Energy in External Fields 21.9.2 Bloch Walls Problems Further Reading References Index