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Contents

Preface page iiiList of tables xiiiNotation xiv

Part I Basic Thermodynamics and Kinetics ofPhase Transformations 1

1 Introduction 31.1 What Is a Phase Transition? 31.2 Atoms and Materials 41.3 Pure Elements 61.4 Alloys – Unmixing and Ordering 91.5 Transitions and Transformations 121.6 Brief Review of Thermodynamics and Kinetics 15Problems 19

2 Essentials of T-c Phase Diagrams 212.1 Overview of the Approach 212.2 Intuition and Expectations about Alloy Thermodynamics 232.3 Free Energy Curves, Solute Conservation, and the Lever Rule 272.4 Common Tangent Construction 302.5 Continuous Solid Solubility Phase Diagram 322.6 Solid Solutions 332.7 Unmixing Phase Diagrams 382.8 Eutectic and Peritectic Phase Diagrams 412.9 Ternary Phase Diagrams 452.10 Long-Range Order in the Point Approximation 472.11 Alloy Phase Diagrams 51Problems 53

3 Diffusion 563.1 The Diffusion Equation 573.2 Gaussian and Error Function Solutions to the 1D Diffusion Equation 613.3 Fourier Series Solutions to the Diffusion Equation 66

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8 Contents

3.4 Bessel Functions and other Special Function Solutions to theDiffusion Equation 71

3.5 Kinetic Master Equation and Equilibrium 743.6 Linear Kinetic Response 76Problems 77

4 Nucleation 804.1 Terminology and Issues 814.2 Critical Nucleus 824.3 Heterogeneous Nucleation 854.4 Free Energy Curves, Elastic Energy 884.5 The Nucleation Rate 914.6 Time-Dependent Nucleation 98Problems 101

5 Effects of Diffusion and Nucleation on Phase Transformati ons 1035.1 Non-Equilibrium Processing of Materials 1045.2 Alloy Solidification with Solute Partitioning 1075.3 Alloy Solidification – Suppressed Diffusion in the Solid Phase 1085.4 Alloy Solidification – Suppressed Diffusion in the Solid and Liquid 1135.5 Practical Issues for Alloy Solidification and Evaporation 1155.6 Heat Flow and Kinetics 1175.7 Nucleation Kinetics 1195.8 Glass Formation 1205.9 Solid-State Amorphization and Suppressed Diffusion in a Solid

Phase 1225.10 Reactions at Surfaces 1245.11 The Glass Transition 129Problems 131

Part II The Atomic Origins of Thermodynamics andKinetics 135

6 Energy 1376.1 Molecular Orbital Theory of Diatomic Molecules 1376.2 Electronic Energy Bands in Solids 1446.3 Elastic Constants and the Interatomic Potential 1566.4 Linear Elasticity 1606.5 Misfitting Particle 1646.6 Surface Energy 169Problems 172

9 Contents

7 Entropy 1757.1 Static and Dynamic Sources of Entropy 1767.2 Short-Range Order and the Pair Approximation 1777.3 Local Atomic Structures Described by Clusters 1817.4 Thermodynamics with Cluster Approximations 1837.5 Concept of Vibrational Entropy 1857.6 Phonon Statistics 1887.7 Lattice Dynamics and Vibrational Entropy 1907.8 Bond Proportion Model 1937.9 Bond-Stiffness-versus-Bond-Length Model 201Problems 204

8 Pressure 2088.1 Materials under Pressure at Low Temperatures 2088.2 Thermal Pressure, a Step Beyond the Harmonic Model 2138.3 Free Energies and Phase Boundaries under Pressure 2158.4 Chemical Bonding and Antibonding under Pressure 2178.5 Two-Level System under Pressure 2208.6 Activation Volume 223Problems 224

9 Atom Movements with the Vacancy Mechanism 2269.1 Random Walk and Correlations 2269.2 Phenomena in Alloy Diffusion 2359.3 Diffusion in a Potential Gradient 2449.4 Non-Thermodynamic Equilibrium in Driven Systems 2489.5 Vineyard’s Theory of Diffusion 252Problems 258

Part III Types of Phase Transformations 261

10 Melting 26310.1 Free Energy and Latent Heat 26310.2 Chemical Trends of Melting 26410.3 Free Energy of a Solid 26510.4 Entropy of a Liquid 27310.5 Thermodynamic Condition for Tm 27610.6 Glass Transition 27810.7 Two Dimensions 281Problems 283

11 Transformations Involving Precipitates and Interfaces 28511.1 Guinier–Preston Zones 285

10 Contents

11.2 Surface Structure and Thermodynamics 28711.3 Surface Structure and Kinetics 29311.4 Chemical Energy of a Precipitate Interface 29611.5 Elastic Energy and Shape of Growing Precipitates 29811.6 Precipitation at Grain Boundaries and Defects 30111.7 The Eutectoid Reaction and Ferrous Metallurgy 30411.8 The Kolmogorov-Johnson-Mehl-Avrami Growth Equation 31011.9 Coarsening 313Problems 316

12 Spinodal Decomposition 31812.1 Concentration Fluctuations and the Free Energy of Solution 31812.2 Adding a Square Gradient Term to the Free Energy F(c) 32012.3 Constrained Minimization of the Free Energy 32512.4 The Diffusion Equation 33012.5 Effects of Elastic Strain Energy 332Problems 335

13 Phase Field Theory 33613.1 Spatial Distribution of Phases and Interfaces 33613.2 Solidification 33913.3 Ginzburg–Landau Equation and Order Parameters 34113.4 Interfaces Between Domains 344Problems 353

14 Method of Concentration Waves and Chemical Ordering 35414.1 Structure in Real Space and Reciprocal Space 35414.2 Symmetry and the Star 36114.3 The Free Energy in k-Space with Concentration Waves 36414.4 Symmetry Invariance of Free Energy and Landau–Lifshitz Rule

for Second-Order Phase Transitions 36814.5 Thermodynamics of Ordering in the Mean Field Approximation

with Long-Range Interactions 372Problems 377

15 Diffusionless Transformations 37915.1 Dislocations and Mechanisms 38015.2 Twinning 38515.3 Martensite 38715.4 The Crystallographic Theory of Martensite 39315.5 Landau Theory of Displacive Phase Transitions 39615.6 First-Order Landau Theory 40115.7 Phonons and Structural Collapse 40315.8 Soft Phonons in BCC Structures 404

11 Contents

Problems 408

16 Thermodynamics of Nanomaterials 41016.1 Grain Boundary Structure 41116.2 Grain Boundary Energy 41216.3 Gibbs–Thomson Effect 41516.4 Energies of Free Electrons Confined to Nanostructures 41716.5 Configurational Entropy of Nanomaterials 41916.6 Vibrational Entropy 42216.7 Gas Adsorption 42416.8 Characteristics of Magnetic Nanoparticles 42716.9 Elastic Energy of Anisotropic Microstructures 429Problems 431

17 Magnetic and Electronic Phase Transitions 43317.1 Overview of Magnetic and Electronic Phase Transitions 43417.2 Exchange Interactions 43917.3 Correlated Electrons 44317.4 Thermodynamics of Ferromagnetism 44617.5 Spin Waves 45017.6 Thermodynamics of Antiferromagnetism 45217.7 Ferroelectric Transition 45617.8 Domains 458Problems 460

18 Phase Transitions in Quantum Materials 46218.1 Bose-Einstein Condensation 46218.2 Superfluidity 46518.3 Condensate Wavefunction 46818.4 Superconductivity 47118.5 Quantum Critical Behavior 480Problems 483

Part IV Advanced Topics 485

19 Low Temperature Analysis of Phase Boundaries 48719.1 Ground State Analysis for T = 0 48819.2 Richards, Allen, Cahn Ground State Maps 49019.3 Low, but Finite Temperatures 49019.4 Analysis of Equiatomic bcc Alloys 49619.5 High Temperature Expansion of the Partition Function 497Problems 499

12 Contents

20 Cooperative Behavior Near a Critical Temperature 50120.1 Critical Exponents 50220.2 Critical Slowing Down 50220.3 The Rushbrooke Inequality 50420.4 Scaling Theory 50520.5 Scaling and Decimation 50720.6 Partition Function for One-Dimensional Chain 50820.7 Partition Function for Two-Dimensional Lattice 512Problems 516

21 Elastic Energy of Solid Precipitates 51721.1 Transformation Strains and Elastic Energy 51821.2 Real Space Approach 52021.3 k-Space Approach 522Problems 525

22 Statistical Kinetics of Ordering Transformations 52722.1 Ordering Transformations with Vacancies 52822.2 B2 Ordering with Vacancies in the Point Approximation 53022.3 Vacancy Ordering 53522.4 Kinetic Paths 536Problems 542

23 Diffusion, Dissipation, and Inelastic Scattering 54423.1 Atomic Processes and Diffusion 54423.2 Dissipation and Fluctuations 54823.3 Inelastic Scattering 55123.4 Phonons and Quantum Mechanics 554Problems 558

24 Vibrational Thermodynamics of Materials at High Tempera tures 55924.1 Lattice Dynamics 56024.2 Harmonic Thermodynamics 56524.3 Quasiharmonic Thermodynamics 56624.4 Thermal Effects Beyond Quasiharmonic Theory 56924.5 Anharmonicity and Phonon-Phonon Interactions 57124.6 Electron-Phonon Interactions and Temperature 575Problems 577

Further Reading 579References 583Index 595

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Preface

Content

This book explains the thermodynamics and kinetics of most of the importantphase transitions in materials science. It is a textbook, so the emphasis is on ex-planations of phenomena rather than a scholarly assessment of their origins. Thegoal is explanations that are concise, clear, and reasonably complete. The level anddetail are appropriate for upper division undergraduate students and graduatestudents in materials science and materials physics. The book should also be use-ful for researchers who are not specialists in these fields. The book is organized forapproximately linear coverage in an graduate-level course. The four parts of thebook serve different purposes, however, and should be approached differently.

Part I presents topics that all graduate students in materials science must know.1

After a general overview of phase transitions, the statistical mechanics of atomarrangements on a lattice is developed. The approach uses a minimum amountof information about interatomic interactions, avoiding detailed issues at the levelof electrons. Statistical mechanics on an Ising lattice is used to understand alloyphase stability for basic behaviors of chemical unmixing and ordering transitions.This approach illustrates key concepts of equilibrium T-c phase diagrams, and isextended to explain some kinetic processes. Essentials of diffusion, nucleation, andtheir effects on kinetics are covered in Part I.

Part II addresses the origins of materials thermodynamics and kinetics at thelevel of atoms and electrons. Electronic and elastic energy are covered, and thedifferent types of entropy, especially configurational and vibrational, are presentedin the context of phase transitions. Effects of pressure, combined with temperature,are explained with a few concepts of chemical bonding. The kinetics of atommovements are developed for diffusion in solids, and from the statistical kineticsof the atom-vacancy interchange.

Part III is the largest. It describes many of the important phase transformationsin materials, with the concepts used to understand them. Topics include melting,phase transformations by nucleation and growth, spinodal decomposition, freez-ing and phase fields, continuous ordering, martensitic transformations, phenom-ena in nanomaterials, phase transitions involving electrons or spins, and quantumphase transitions. These different phase transitions in materials are covered at dif-ferent breadths and depths based on their richness or importance, although thisreflects my own bias. Many topics from metallurgy and ceramic engineering arecovered, although the connection between processing and properties is less em-phasized, allowing for a more concise presentation than in traditional texts. PartIII includes a number of topics from condensed matter physics that were selectedin part because they give new insights into materials phenomena.

1 The author asks graduate students to explain some of the key concepts at a blackboard during theirPh.D. candidacy examinations.

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Part IV presents topics that are more modern, but have proved their importance.Low and high temperature treatments of the partition function, the renormalizationgroup, scaling theory, a k-space formulation of elastic energy, nonequilibrium statesin crystalline alloys, fluctuations and dissipation, and some complexities of hightemperature thermodynamics are presented. The topics in Part IV are explained ata fundamental level, but unlike Parts I through III, for conciseness in Part IV thereare some omissions of methods and steps.

The book draws a distinction between phase transformations and phase tran-sitions. Phase transitions are thermodynamic phenomena based on free energyalone, whereas phase transformations include kinetic processes that alter the lifecycle of the phase change. Phase transitions originate from discontinuities in freeenergy functions, so much of the text focuses on formulating free energies fordifferent systems. The book formulates statistical mechanics models for differentphase transitions, sometimes using an Ising lattice, which is well suited for suchanalysis and finds reuse for different phase transitions. Other topics that recur inthe text are Landau theory in various forms, the topic of domains, the square gra-dient energy, the effect of curvature on nucleation, and dynamics with the kineticmaster equation. Sometimes the thermodynamics of phase transitions is developedwith the partition function, although the classical equation G = E−TS+PV is usedwidely, and it is assumed that the reader has some familiarity with the terms inthis expression. For the kinetics of phase transformations, there is some traditionalpresentation of diffusion and nucleation, but the kinetic master equation is alsoused throughout the text.

Many topics in phase transitions and related phenomena are not covered in thistext. These include: other mechanisms of atom movements (and their effects onkinetics), polymer flow and dynamics, including reptation, phase transitions influid systems including phenomena near the critical temperature, massive trans-formations. Also beyond the scope of the book are computational methods that areincreasingly important for studies of phase transformations in materials, including:Monte Carlo methods, molecular dynamics methods (classical and quantum), anddensity functional theory with extensions to phenomena at finite temperatures.

The field of phase transitions is huge, and continues to grow. This text is asnapshot of phase transitions in materials in the year 2013, composed from theangle of the author. Impressively, this field continues to offer a rich source of newideas and results for both fundamental and applied research, and parts of it willlook different in a decade or so. I expect, however, that many core topics will remainthe same – the free energy of materials will remain the central concept, surroundedby issues of kinetics.

Teaching

I use this text for a graduate-level course taken by Ph.D. students in both mate-rials science and in applied physics at the California Institute of Technology. The10-week course, which includes approximately 30 hours of classroom lectures, isoffered in the third academic quarter as part of a one-year sequence. The first two

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quarters in this sequence cover thermodynamics and statistical mechanics, so thestudents are familiar with the use of the partition function to obtain thermodynamicquantities, and have seen basic concepts from quantum statistical mechanics suchas the Fermi–Dirac distribution. Familiarity with some concepts from solid-statephysics and chemistry is certainly helpful, as is prior exposure to diffusion andtransport, but the text develops many of the important concepts as needed.

In the one-quarter graduate-level course at Caltech, I cover all topics in PartsI and II, moving in sequence through these chapters. Time limitations force aselection of topics from Parts III and IV, but I typically cover more of Part III thanPart IV. For example, this year I covered Chapters 10, 11, 12, parts of 13, 15, 16, 19,and selections from 20, 22, 24. It may be unrealistic to cover all the content in thebook in a 15-week semester with 45 hours of lectures. An instructor can certainlyexercise discretion in selecting topics for the second half of her course.

Most of the problems at the end of each chapter have been used for weeklystudent assignments, and this experience has helped to improve their wordingand content. The majority of these problems make use of concepts explained inthe text, fill in the explanations of concepts, or extend analyses. Others developnew concepts not described in the chapter, but these problems usually includelonger explanations and hints that may be worth reading even without workingthe problem. None of the problems are intended to be particularly difficult, andsome can be answered quickly once the main idea emerges. I usually assign five orsix problems every week during the term. An expanding online solutions manualis available to course instructors whose identity can be verified. Please ask me forfurther information.

Acknowledgments

I thank J.J. Hoyt for collaborating with me on a book chapter about phase equi-libria and phase transformations that prompted me to get started on this book.Jeff has since published a fine book on phase transformations in materials that isavailable at low cost from McMaster Innovation Press.

The development of the topic of vibrational entropy would not have been pos-sible without the contributions of my junior collaborators at Caltech, especiallyL. Anthony, L.J. Nagel, H.N. Frase, A.F. Yue, M.E. Manley, P.D. Bogdanoff, J.Y.Y.Lin, T.L. Swan–Wood, A.B. Papandrew, O. Delaire, M.S. Lucas, M.G. Kresch, M.L.Winterrose, J. Purewal, C.W. Li, T. Lan, L. Mauger and S.J. Tracy. Today several ofthem are taking this field into new directions.

Important ideas have come from stimulating conversations over the years withA. van de Walle, V. Ozolins, G. Ceder, M. Asta, L.-Q. Chen, D.D. Johnson, D. deFontaine, A.G. Khachaturyan, A. Zunger, P. Rez, K. Samwer and W.L. Johnson.This work was supported by the NSF under award DMR-0520547.

Brent FultzPasadena, California

3 September 2013

PART I

BASICTHERMODYNAMICS AND

KINETICS OF PHASETRANSFORMATIONS

The field of phase transitions is rich, vast, and continues to grow. Thistext covers parts of the field relevant to materials physics, but manyconcepts and tools of phase transitions in materials are used elsewherein the larger field of phase transitions. Likewise, new methods from thelarger field are beginning to be applied to studies of materials.

Part I of the book covers essential topics of free energy, phase diagrams,diffusion, nucleation, and a few classic phase transformations that havebeen part of the historical backbone of materials science. In essence,the topics in Part I are the thermodynamics of how atoms prefer to bearranged when brought together at various temperatures, and how theprocesses of atom movements control the rates and even the structuresthat are formed during phase transformations. The topics in Part I arelargely traditional ones, but formulating the development in terms ofstatistical mechanics and in terms of the kinetic master equation allowsmore rigor for some topics, and makes it easier to incorporate a higherlevel of detail from Part II into descriptions of phase transitions in PartsIII and IV.

1 Introduction

1.1 What Is a Phase Transition?

A phase transition is an abrupt change in a system that occurs over a small range ina control variable. For thermodynamic phase transitions, typical control variablesare the “intensive variables” of temperature, pressure, or magnetic field. Thermo-dynamic phase transitions in materials and condensed matter, the subject of thisbook, occur when there is a singularity in the free energy function of the material,or in one of the derivatives of the free energy function.1 Accompanying a phasetransition are changes in some physical properties and structure of the material,and changes in properties or structure are the usual way that a phase transition isdiscovered. There is a very broad range of systems that can exhibit phase transi-tions, extending from atomic nuclei to traffic flow or politics. For many systems it isa challenge to find reliable models of the free energy, however, so thermodynamicanalyses are not available.

Our focus is on thermodynamic phase transitions in assemblages of many atoms.How and why do these groups of atoms undergo changes in their structures withtemperature and pressure? In more detail, we often find it useful to considerseparately:

• nuclei, which have charges that define the chemical elements,• nuclear spins and their orientations,• electrons that occupy states around the nuclei, and• electron spins, which may have preferred orientations with respect to other spins.

Sometimes a phase transition involves only one of these entities. For example, atlow temperatures (microKelvin), the weak energy of interaction between nuclearspins can lead to nuclear spin alignments. An ordered state of aligned nuclearspins may have the lowest energy, and may be favored thermodynamically atlow temperatures. Temperature disrupts these delicate alignments, however, andthermodynamics favors a disordered nuclear magnetic structure at modest tem-peratures. The free energy, F, changes with temperature when the nuclear spinsare aligned, but the functional form of this curve of Ford(T) is not the same asFdis(T) for the disordered state at higher temperature. At the critical temperatureof the ordering transition there is a switch from one curve to another, or perhapsthe second derivative d2F/dT2 has a kink. Order-disorder phase transitions are

1 A brief review of free energy is given in Sect. 1.6.2.

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4 Introduction

enlightening, and have spawned several creative methods to understand how anorder parameter, energy, and entropy depend on temperature.

Sometimes phase transitions involve multiple physical entities. Electrons of op-posite spin can be coupled together by a wave of nuclear vibration (a phonon).These Cooper pairs can condense into a superconducting state at low tempera-tures. Perhaps electron charge or spin fluctuations couple the electrons in hightemperature superconductors, although the mechanism is not fully understoodtoday. Much of the fascination with phase transitions such as superconductivity iswith the insight they give into the interactions between the electrons and phonons,or the electron charges and spins. While these are indeed important subjects forstudy, they are to some extent diversions from the main topic of phase transitions.Likewise, delving deeper into the first example of nuclear spin alignments at lowtemperatures reveals that the information about the alignment of one nucleus iscarried to a nearby nucleus by the conduction electrons, and these hyperfine inter-actions between nuclei and electrons are an interesting topic in their own right.

In a study of phase transitions, it is easy to lose track of the forest if we focus on theinteresting trees within it. Throughout much of this text, the detailed interactionsbetween the entities of matter are replaced with simplifying assumptions thatfacilitate mathematical modeling. Sometimes the essence of the phase transitionis captured well with such a simple model. Other times the discrepancies proveinteresting in their own right. Perhaps surprisingly, the same mathematical modelreappears in explanations of phase transitions involving very different types ofmatter. A phase transition is an “emergent phenomenon,” meaning that it displaysfeatures that emerge from interactions between numerous individual entities, andthese large-scale features can occur in systems with very different microscopicinteractions. The study of phase transitions has become a respected field of sciencein its own right, and Chapter 20, for example, presents some concepts from thisfield that need not be grounded in materials phenomena.

1.2 Atoms and Materials

An interaction between atoms is a precondition for a phase transition in a material(and, in fact, for having a material in the first place). Atoms interact in interestingways when they are brought together. In condensed matter there are liquids ofvarying density, and numerous types of crystal structures. Magnetic moments formstructures of their own, and the electron density can show spatial modulations. Ingeneral, chemical bonds are formed when atoms are brought together. The energyof interatomic interactions is dominated by the energy of the electrons, whichare usually assumed to adapt continuously (“adiabatically”) to the positions ofthe nuclei. The nuclei, in turn, tend to position themselves to allow the lowestenergy of the material, which means that nuclei move around to let the electronsfind low-energy states. Nevertheless, once we know the electronic structure of a

2 Essentials of T-c Phase Diagrams

This Chapter 2 explains the concepts behind T-c phase diagrams, which are mapsof the phases that exist in an alloy of chemical composition c at temperature T.1

A T-c phase diagram displays the phases in thermodynamic equilibrium, andthese phases are present in the amounts f , and with chemical compositions thatminimize the total free energy of the alloy. The emphasis in this Chapter 2 is onderiving T-c phase diagrams from free energy functions F(c,T).2 The constraint ofsolute conservation is expressed easily as the “lever rule.” The minimization ofthe total free energy leads to the more subtle “common tangent construction” thatselects the equilibrium phases at T from the F(c) curves of the different phases. Forbinary alloys, the shapes of F(c) curves and their dependence on temperature areused to deduce eutectic, peritectic, and continuous solid solubility phase diagrams.Some features of ternary alloy phase diagrams are also discussed.

If atoms occupy sites on a lattice throughout the phase transformation, freeenergy functions can be calculated with a minimum set of assumptions about howdifferent atoms interact when they are brought together. Because the key features ofphase diagrams can be obtained with general types of interactions between atoms,systems with very different types of chemical bonding, e.g., both oil in water andiron in copper, can show similar phase transitions. In these unmixing cases, theindividual atoms or molecules prefer their like species as neighbors (see Fig. 1.4).The opposite case of a preference for unlike atom neighbors leads to chemicalordering at low temperatures, which requires the definition of an order parameter,L. These generalization of chemical interactions should not be lamented for theirloss of rigor, but celebrated as a way to identify phenomena common to many phasetransitions. Such “emergent” behavior can be missed if there is too much emphasison the electronics of chemical bonding and atom vibrations. Nevertheless, wemust be wise enough to know the predictive power available at different levels ofgeneralization.

2.1 Overview of the Approach

Temperature promotes disorder in a material, favoring higher-entropy phases suchas liquids, but the chemical bond energy favors ordered crystals at low tempera-

1 Pressure is assumed constant, or negligible.2 The usage of phase diagrams is deferred to Chapter 5.

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22 Essentials of T-c Phase Diagrams

tures. This information alone is sufficient to predict a phase change with tempera-ture, but adding a bit more detail about the energy and entropy rewards us withconsiderably more information about alloy phase diagrams, and this is the essenceof the present chapter.

The thermodynamic functions of the alloy, E and S, depend on the spatial ar-rangement of the atoms in the alloy.3 In what follows, a minimal atomistic modelis constructed to calculate thermodynamic functions and predict the equilibriumphases in alloys at different combinations of T and chemical composition c. Thisminimal model successfully predicts the main features of T-c phase diagrams forbinary A-B alloys (A and B are the two species of atoms). Sometimes the free pa-rameters in this approach, the pairwise atomic bond energies, can be tuned to fitphase transitions in a specific material, and sometimes the alloy thermodynamicscan be extrapolated successfully into regions where no experimental data exists.Convenience for calculation is a virtue and a priority for this chapter, but we mustbe aware of the two types of risks it entails

1. A good parameterization of atom positions may require more detail than ispossible with a simple model.

2. Even if the parameterizations of atom positions are excellent, calculating theenergy or entropy from these parameterizations may require more sophisticatedmethods.

Part II of this book addresses these two issues in more detail. Nevertheless, thesimple models developed in this chapter are often useful semiquantitatively, andare useful benchmarks for assessing the value added by more sophisticated treat-ments.

This book will not emphasize methods to calculate the energy of the phases, E, asis done today by electronic structure calculations, such as density functional theorywith plane wave pseudopotentials.4 Nevertheless, because the energies of theelectrons in a material are set by the positions of the nuclei (i.e., the configurationsof atoms), a good parameterization of atom configurations can accommodate moreadvanced analyses, too. Here we obtain E from local atomic arrangements, typicallyas a sum of energies of chemical bonds between the nearest-neighbor pairs of atoms.The entropy of an alloy also depends on atom configurations, often local ones.

For alloy thermodynamics we need expressions for E(c) and S(c), for which theHelmholtz free energy is

F(c,T) = E(c) − T[Sconfig(c) + Svib(c,T)

]. (2.1)

In a first approximation, the electronic energy E(c) is assumed to be independent oftemperature, as is the way configurations are counted to obtain the configurationalentropy Sconfig(c). If E(c) and Sconfig(c) depend only on local atom configurations

3 This restates the paradigm that the atom arrangements in a material determine its properties, in-cluding thermodynamic properties.

4 These methods are powerful, and will grow in importance to materials science. This is a large topic,however, and deserves a course of its own.

23 Intuition and Expectations about Alloy Thermodynamics

in a crystal, it is efficient to use a fixed lattice and pairwise chemical bonds. Forsolid solution phases, it is also convenient to assume statistical randomness ofatom occupancies of the sites on the lattice.5 Two important T-c phase diagrams,unmixing and ordering, can be obtained readily by minimizing such an F(c,T)on an Ising lattice. More generally, there is competition between multiple phasesthat cannot be placed naturally on the same lattice, so each phase has a differentF(c,T). Minimizing the total free energy by varying the compositions and fractionsof the phases requires the “common tangent” algorithm that allows us to completethe set of the five main types of T-c phase diagrams for binary alloys. With theseassumptions, the two general risks listed above can be stated more specifically

1. Chapter 7 goes beyond the assumed random environment of an average atom,i.e., the “point approximation.” One approach is to use probabilities of local“clusters” of atoms, working up systematically from chemical composition(point), to numbers of atom pairs (pair), then tetrahedra. Usually the energyand entropy of materials originate from local characteristics of atom configura-tions, so this approach has an excellent record of success.

2. How accurately can we account for the energy and entropy of the material? Forexample, the electronic energy, which provides the thermodynamic E, need notbe confined to neighboring pairs of atoms. The energy of delocalized electrons ina box or in a periodic crystal is introduced in Chapter 6. This Chapter 2 ignoresthe entropy caused by the thermal vibrations of atoms, Svib(c,T), the entropyfrom disorder in magnetic spins, Smag(c,T), and the entropy from disorder inelectron state occupancies, Sel(c,T). These will have to be assessed later.6

2.2 Intuition and Expectations about AlloyThermodynamics

2.2.1 Free Energies of Alloy Phases

Alloys often transform to different phases at different temperatures. We have al-ready mentioned the liquid at high temperatures and ordered structures at lowtemperature. Here, listed from high to low temperature, are some typical phasesthat are found for an alloy system (“system” meaning a range of compositions forspecified chemical elements), with considerations of the Helmholtz free energy,F(c) = E(c) − TS(c), for each phase

• The phase of maximum entropy dominates at the highest temperatures. Formost materials this is a gas of isolated atoms, but at lower temperatures most

5 Or, analogously for an order parameter, the sublattice occupancies are assumed random.6 A textbook could start to develop phase diagrams by using Svib(c,T) and ignoring Sconfig(c), but this

would be unconventional.

24 Essentials of T-c Phase Diagrams

Box 2.1 Configurational and Dynamical Sources of Entropy

We take the statistical mechanics approach to entropy, and count the configu-rations of a system with equivalent macrostates, giving the Ω for Eq. 1.10. Forcounting:

• Procedures to count atom arrangements (or spin arrangements) are indepen-dent of temperature.

• On the other hand, procedures to count states explored with thermal excita-tions, such as atom vibrations or electronic excitations, do depend on tempera-ture.

This separation works well for the thermodynamics of crystals at modest temper-atures. If atom mobility becomes extremely rapid while some vibrations becomevery slow, as when a liquid changes its viscosity with temperature, its reliabilitymay be in doubt.

The reader should know that at a temperature of 1,000 K or so, Svib is typically anorder-of-magnitude larger than Sconfig (compare Figs. 2.9a and 10.3). It is possi-ble for Svib to be similar for different phases, however, so the difference in entropycaused by vibrations is not always dominant in a phase transition. Nevertheless,Svib usually does depend on atom configurations, and this is developed for in-dependent harmonic oscillations in Section 7.8. At high temperatures, however,vibrations change their frequencies, and normal modes of vibration interact witheach other, so the statistical mechanics of harmonic vibrations needs significantmodifications (Chapter 24).

alloys form a liquid phase with continuous solubility of A and B-atoms. (Thereare cases of chemical unmixing in the liquid phase, however, for which thethermodynamics of Eq. 2.19 is again relevant.)

• At low-to-intermediate temperatures, the equilibrium phases and their chemicalcompositions depend in detail on the free energy versus composition curvesFξ(c) for each phase ξ. Usually there are chemical unmixings, and the differ-ent chemical compositions frequently prefer different crystal structures. Thechemical unmixings may not require precise stoichiometries (e.g., a precisecomposition of A2B3) because some spread of compositions may provide en-tropy to make off-stoichiometric compositions favorable (e.g., A2−δB3+δ).

• At the lowest temperatures, the equilibrium state for a general chemical compo-sition is a combination of crystalline phases, usually with precise stoichiome-tries. These often correspond to crystal structures that have unique sites forthe different atom species, and a high degree of long-range order.7

7 At low temperatures, atoms usually do not have enough mobility to form these precise structures,so some chemical disorder is often observed.

3 Diffusion

In solids, atoms move by a process of diffusion. The vacancy mechanism for diffu-sion in crystals was presented in Section 1.5.3 and illustrated with Fig. 1.7. Mentionwas made of interstitial diffusion and interstitialcy diffusion. Mass transport inglasses and liquids can also occur by atomic-level diffusion, but for gases or flu-ids of low viscosity there are larger-scale convective currents with dynamics quitedifferent from diffusion.1

The diffusion equation has the same mathematical form as the equation for heatconduction, if solute concentration is replaced by heat or by temperature. The heatequation has been known for centuries, and methods for its solution have a longhistory in classical mathematical physics. Some of these methods are standardfor diffusion in materials, such as the basic solutions of Gaussian functions anderror functions for one-dimensional problems. This Chapter 3 also presents themethod of separation of variables for three-dimensional problems with Cartesianand cylindrical coordinates. The Laplacian is separable in nine other coordinatesystems, each with their own special functions and orthogonality relationships, butthese are beyond the scope of this book. For the problems in ellipsoidal coordinates,for example, the reader may consult classic texts in mathematical physics (e.g., (19)).Today finite element methods are practical for many problems, and often provemore efficient than analytical methods.

Because diffusion depends on atomic-scale processes, changes in the local atomicstructure during diffusion can depreciate the diffusion equation because the “diffu-sion constant,” D, is not constant. This can be a serious problem when using the dif-fusion equation to describe the kinetics of a phase transformation. By deriving thediffusion equation from the kinetic master equation, however, we can later replacethe assumption of random atomic jumps with an assumption of chemically-biasedjumps to predict the kinetics of chemical ordering or mixing. This is the subject ofChapter 22. This Chapter 3 concludes by showing how the kinetic master equationcan lead to thermodynamic equilibrium.

1 Convective currents can be driven by differences in density, such as the rising of a hot liquid in agravitational field.

56

57 The Diffusion Equation

3.1 The Diffusion Equation

Writing the kinetic master equation in the form of Eq. 1.25 motivates a matrixdescription of the kinetic processes

W≈

(∆t) N∼

(t) = N∼

(t + ∆t) (3.1)

where N∼

(t) is a column vector that we lay out along the bins of Fig. 1.9a, andW≈

(∆t) is a two-dimensional matrix that gives the new contents after time ∆t. Twosuch matrix elements are shown in Fig. 1.9b. This approach has an advantage fornumerical computations. If ∆t is small, after m intervals of ∆t the new contents ofthe bins will be

[W≈

(∆t)]m

N∼

(t) = N∼

(t +m∆t) . (3.2)

The following assumptions are fundamental to the diffusion equation, and toour construction of a kinetic master equation for diffusion. They are importantto remember whenever using the diffusion equation for a problem in materialsscience.

• all atoms have the same jump probability (unaffected by the presence of otheratoms)

• if an atom has probability δ of jumping out of a bin in Fig. 1.9, it has an equalprobability δ/2 of going left or right (in three dimensions the probability isshared as δ/6 between left, right, up, down, in, out)

• an atom can jump only into an adjacent bin (but this is not an essential assump-tion for obtaining the diffusion equation as shown by Problem 2 in Chapter9)

For our matrix equation we first arrange the two column vectors in correspon-dence with the bins in Fig. 1.9a, n, and their contents N

N∼

(t) =[N1(t),N2(t)...Nn−1(t),Nn(t),Nn+1(t)...

]. (3.3)

For the structure of W≈

(∆t), first assume zero atom jumps in the time ∆t. In this caseW≈

(∆t) must be the identity matrix, I≈, with all 1’s on its diagonal, and 0’s elsewhere.

The operation of this identity matrix on the vector N∼

(t) preserves the contents ofall bins at time t + ∆t, so in this case I

≈N∼

(t) = N∼

(t + ∆t).

Next, assume each atom has only a small probability δ of leaving its bin in thetime interval ∆t. The probability of it remaining in the bin is therefore 1− δ, and itsprobability of entering an adjacent bin is δ/2. Likewise, the probability of an atomentering a bin from an adjacent bin is also δ/2. The W-matrix is close to diagonal,

4 Nucleation

As discussed in Section 1.5.2, phase transformations can occur continuously ordiscontinuously. The discontinuous case begins with the appearance of a small butdistinct volume of material having a structure and composition that differ fromthose of the parent phase.1 A discontinuous transition can be forced by symmetry,as formalized for some cases in Sect. 14.4. There is no continuous way to rearrangethe atoms of a liquid into a crystal, for example. The new crystal must appearin miniature in the liquid, a process called “nucleation.” If the nucleation event issuccessful, this crystal will grow. The process of nucleation is an early step for mostphase transformations in materials. It has many variations, but two key conceptscan be appreciated immediately.

• Because the new phase and the parent phase have different structures, theremust be an interface between them. The atom bonding across this interface isnot optimal,2 so the interfacial energy must be positive. This surface energy ismost significant when the new phase is small, because a larger fraction of itsatoms are at the interface. Surface energy plays a key role in nucleation.

• For nucleation of a new phase within a solid, a second issue arises when the newphase differs in shape or specific volume from the parent phase. The mismatchcreates an elastic field that costs energy. This is not an issue for nucleation ina liquid or gas, since the surrounding atoms can flow out of the way.

An issue for nucleation with chemical unmixing is the time required for diffusionof the different chemical species of atoms, and this time can be long if atomsmust move long distances between incipient nuclei (sometimes called “embryos”).The addition of atoms to embryos is largely a kinetic phenomenon, although thetendency of atoms to remain on the embryos is a thermodynamic one. A steady-state rate of forming “critical” nuclei that can grow is calculated. The chapterends with a discussion of the transient time after a quench from high temperaturewhen the distribution of solute relaxes towards the equilibrium distribution forsteady-state nucleation.

1 The nucleus may or may not have the structure and composition of the final phase because thetransformation may occur in stages.

2 If the structure of the interfacial atoms and bonds were favorable, the new phase would take thislocal atomic structure.

80

81 Terminology and Issues

tFig. 4.1 Binary phase diagram depicting a quench path from a temperature with pureα-phase to a temperature where some β-phase will nucleate.

4.1 Terminology and Issues

Nucleation can occur without a change in crystal structure. Consider an A-rich A-B alloy having the α-phase at high temperature, as shown in the unmixing phasediagram of Fig. 2.10. Suppose the alloy is quenched (cooled quickly) to a tem-perature such as 0.4zV/kB, where the equilibrium state would have mostly A-richα′-phase, plus some B-rich α′′-phase. For some compositions, the B-rich α′′-phasemay nucleate as small zones or “precipitates” in an A-rich matrix.3 In this case, theunderlying crystal lattice remains the same while the solute atoms coalesce. A dif-ferent case is shown with Fig. 4.1 for the unmixing of solute in a eutectic alloy. Herethe precipitation of β-phase in an alloy cooled rapidly from the α-phase requiresboth a redistribution of chemical elements and a different crystal structure. In boththese examples of nucleation, the parent phase is “supersaturated” immediatelyafter the quench, and is unstable against forming the new phase.

“Homogeneous” nucleation occurs when nuclei form randomly throughout thebulk material; i.e. without preference for location. “Heterogeneous” nucleationrefers to the formation of nuclei at specific sites. In solid→solid transformations,heterogeneous nucleation occurs on grain boundaries, dislocation lines, stackingfaults, or other defects or heterogeneities. When freezing a liquid, the wall of thecontainer is a common heterogeneous site. For the nucleation of solid phases,heterogeneous nucleation is more common than homogeneous.

The precipitate phase can be “coherent” or “incoherent” with the surroundingmatrix. Figure 4.2a illustrates an incoherent nucleus. The precipitating β-phase hasa crystal structure different from the parent α-phase, and there is little registry of

3 A “matrix” is an environment in which a new phase develops.

5 Effects of Diffusion and Nucleationon Phase Transformations

A phase diagram is a construction for thermodynamic equilibrium, a static state,and therefore contains no information about how much time is needed beforethe phases appear with their equilibrium fractions and compositions. It mightbe assumed that the phases found after practical times of minutes or hours willbe consistent with the phase diagram, since most phase diagrams were deducedfrom experimental measurements on such time scales.1 However, a number ofnonequilibrium phenomena such as those described in this chapter are well known,and were likely taken into account when a T-c phase diagram was prepared.

For rapid heating or cooling, the kinetic processes of atom rearrangements of-ten cause deviations from equilibrium, and some of these nonequilibrium effectsare described in this Chapter 5. In general, first one seeks to understand the ef-fects of the slowest processes. For faster heating or cooling, however, sometimesthe slowest processes are inactive, so the next-slowest processes become impor-tant. Approximately, this Chapter 5 follows a course from slower to faster kineticprocesses.

1 On the other hand, this does not necessarily mean that all phases on phase diagrams are in factequilibrium phases. Exceptions are found, especially at temperatures below about half the liquidustemperature.

Box 5.1 Thermodynamics and Kinetics

Two necessary conditions for a phase transformation to occur in a materialare

• a driving force from a reduction of free energy, and

• a mechanism for atoms to move towards their equilibrium positions.

The phase diagram gives information about first condition, but most phasechanges in materials occur by diffusional motions of atoms.

103

104 Effects of Diffusion and Nucleation on Phase Transformations

5.1 Non-Equilibrium Processing of Materials

5.1.1 Diffusion Lengths

The diffusional motion of atoms is thermally activated, and is more rapid at highertemperatures. An important consideration is the characteristic time τ for diffusionover a characteristic length x (see Eq. 3.48)

τ =x2

D(T), (5.1)

where D(T) is the diffusion coefficient.Minimizing the time when D(T) is large serves to minimize x, so one way to

classify either kinetic phenomena or methods of materials processing is by effec-tive cooling rate. Table 5.1 shows some kinetic phenomena that are suppressed bycooling at increasingly rapid rates. Insights into the phenomena listed in Table 5.1are obtained with a rule of thumb that diffusion coefficients near the melting tem-perature of many (metallic) materials are D(Tm) ∼ 10−8 cm2/s.2 For an interatomicdistance of typically 2× 10−8 cm, Eq. 5.1 gives a characteristic time of 4× 10−8 s. Forultrafast cooling, it seems possible to suppress all atom diffusion in solids belowthe melting temperature, suppressing crystal nucleation and growth. This allowsthe formation of amorphous metallic elements, but these are highly unstable andmay persist only at cryogenic temperatures. At 2/3 of the melting temperature, thetime scale for suppressing atom motion at the atomic scale is increased by a factorof 104 or so, owing to a decrease in D(T).

5.1.2 Quenching Techniques

Some practical methods to achieve high cooling rates are also listed in Table 5.1.The cooling rates are approximate, since these depend on the sample thickness andthermal conductivity, which vary with the particular material and the equipment.Like the diffusion of atoms, the diffusion of heat has a quadratic relationshipbetween the characteristic cooling time τ and the sample thickness, x, as in Eq. 5.1,where D(T) is now a thermal diffusivity. Approximately, the techniques listed inTable 5.1 make thicknesses of materials that are proportional to the square root ofthe inverse cooling rate. Samples from melt spinning are tens of microns thick, andsamples from laser surface melting are often less than 0.1µm, for example.

Iced brine quenching is an older technique, where a sample at elevated tempera-ture is quickly immersed in a solution of rocksalt in water, cooled with ice. The saltserves to elevate the boiling temperature and improve the thermal conductivity.The rate of cooling depends on the thickness of the sample. The quench rate alsodepends on the formation of bubbles of water vapor on the sample surface, which

2 Sometimes an estimate of D ∼ 10−5 cm2/s for the liquid proves useful, too.

105 Non-Equilibrium Processing of Materials

Table 5.1 Cooling Rates, Methods, Typical Kinetic Phenomena

infinitesimal 10−6 K/s geologic cooling equilibrium (sometimes)slow 100 K/s suppressed diffusion in solid

101 K/s casting dendrites103 K/s iced brine quench suppressed precipitation

medium 104 K/s suppressed diffusion in liquid105 K/s melt spinning extended solid solubility

fast 106 K/s piston-anvil quench metallic glasses109 K/s laser surface melting

ultrafast 1010 K/s amorphous elements1011 K/s physical vapor deposition

− shock wave melting− high-energy ball milling nanocrystallinity, glass formation− heavy ion irradiation chemical mixing, glass formation

suppress thermal contact to the water bath. Stirring the mixture can improve thecooling rate.

In melt spinning, a steady stream of liquid metal is injected onto the outer surfaceof a spinning wheel of cold copper, for example. The liquid cools quickly whenin contact with the wheel, making a solid ribbon that is thrown off the wheeland spooled. This method is suitable for high volume production. The liquid metalshould have modest wetting of the spinning wheel, and optimizing the parametersof the system can be challenging.

Piston-anvil quenching, sometimes known as “splat quenching,” uses a pair ofcopper plates that impact a liquid droplet from two sides. The alloy is typicallymelted by levitation melting in an induction coil. When the radiofrequency heatingcurrent is stopped, the liquid droplet falls under gravity past an optical sensor thattriggers the pistons. Like melt spinning, the sample is thin, perhaps 20 to 30 microns,but wetting properties are less of a concern.

Laser surface melting can be performed with either a pulsed or continuouslaser. The sample may be moved in a raster pattern under laser illumination so asignificant area can be treated. Once melted, the surface is cooled by the underlyingsolid material, and the thinner the melted region, the faster the cooling.

Physical vapor deposition may use a high temperature heater to evaporate amaterial under vacuum. The evaporated atoms move ballistically towards the coldsurface of a substrate. When these atoms are deposited on the cold substrate,their thermal energy is removed quickly, in perhaps a hundred atom vibrations(approximately 10−11 s), leading to very high cooling rates when the deposition rateis not too rapid and the substrate is isolated thermally from the hot evaporator.

PART II

THE ATOMIC ORIGINSOF THERMODYNAMICS

AND KINETICS

Free energy is a central topic of this book because a phase transitionoccurs in a material when its free energy, or a derivative of its free energy,has a singularity. Chapter 2 showed how to use the dependence of freeenergy on composition or order parameter to obtain thermodynamicphase diagrams. Chapters 3 and 4 discussed the kinetics of diffusion andnucleation, which can be calculated with an activated state rate theorythat uses a free energy of activation. Chapter 5 showed how the freeenergies of equilibrium phases and the free energies of activation giverise to competition between thermodynamic and kinetic phenomena inphase transformations such as alloy solidification, glass formation, andthin film reactions.

The Gibbs free energy isG = E − TS + PV .

Chapter 6 discusses the sources of energy of materials that are impor-tant for phase transitions. The next Chapter 7 addresses the importantsources of entropy, and Chapter 8 discusses effects of pressure. Finally,Chapter 9 explains chemical effects on diffusion in alloys, which dependon the free energy of an activated state. This coverage of E, S, P, and∆G∗ comprises Part II of the book.

6 Energy

This Chapter 6 explains the different types of energies that are important for thethermodynamics of materials phases and materials microstructures, and sometechniques for calculating them. It begins with the chemical bond between twoatoms – a fundamentally quantum mechanical phenomenon that depends on thecoherent interference of an electron wavefunction with itself, giving an electrondensity that is not a linear sum of densities from two separate atoms. In a periodicsolid or in a large box for electrons, the number of electron states depends on awavevector k, which can be used to obtain the spectrum of electron energies. Theconcepts presented here are important, but quantitative results require quantumchemical computer calculations.

At a more general, but more phenomenological level, interatomic potentials aredescribed and used to explain the elastic behavior of solids. The elastic energyof a misfitting solid particle in a matrix is discussed, and this misfit energy isgenerally important for precipitation reactions in solid materials. Surface energyis also described, along with the Wulff construction for predicting the shapes ofcrystals and precipitates.

6.1 Molecular Orbital Theory of DiatomicMolecules

6.1.1 Interacting Atoms

Start with two isolated atoms, A and B. There are states for a single electron abouteach atom of energy ǫA and ǫB, set by the Schrodinger equations

− ~2

2m∇2ψA(~r) + VA(~r)ψA(~r) − ǫAψA(~r) = 0 , (6.1)

− ~2

2m∇2ψB(~r) + VB(~r)ψB(~r) − ǫBψB(~r) = 0 , (6.2)

where ψA and ψB are single-electron wavefunctions at atoms A and B. As isolatedatoms, their nuclei are far apart. Now bring the nuclei close enough together so theirwavefunctions overlap. Our goal is to understand what the individual electronsdo in the presence of both atoms, and understand the chemical bond in the newdiatomic molecule.

We seek single-electron wavefunctions for the diatomic molecule. The potential

137

138 Energy

proves to be a real challenge because the potential for one electron depends onthe presence of the second electron. The effect of the second electron is to pusharound the first electron, but this alters the potential and wavefunction of thesecond electron. Iterative methods are the most accurate for this problem, but hereassume that the total potential is simply the sum of potentials of the isolated atoms

V(~r) = VA(~r) + VB(~r) . (6.3)

This approach does not always work, especially when there are large electrontransfers between atoms, which alter the atomic potentials. The approach worksbest when the overlap of the atom wavefunctions is small, and the potentials tend toretain their original character. We make a related assumption that a single electronis in a wavefunction ψ constructed from the original atomic wavefunctions

ψ(~r) = cAψA(~r) + cBψB(~r) . (6.4)

It is important to remember that ψ pertains to a single electron, so the coefficientscA and cB are less than 1 (the atomic wavefunctions ψA and ψB accommodate oneelectron each). Thisψ is a “molecular orbital” for one electron. We started with twoelectrons though, so we need to find two molecular orbitals. To do so, lay out themolecular Schrodinger equation twice and do two standard tricks: 1) multiply byψ∗A(~r) and ψ∗B(~r)

− ~2

2mψ∗A(~r)∇2ψ(~r) + ψ∗A(~r)V(~r)ψ(~r) − ǫψ∗A(~r)ψ(~r) = 0 , (6.5)

− ~2

2mψ∗B(~r)∇2ψ(~r) + ψ∗B(~r)V(~r)ψ(~r) − ǫψ∗B(~r)ψ(~r) = 0 , (6.6)

and 2) integrate

〈A|H|A〉cA + 〈A|H|B〉cB − ǫ(cA + 〈A|B〉cB) = 0 , (6.7)

〈B|H|A〉cA + 〈B|H|B〉cB − ǫ(〈B|A〉cA + cB) = 0 , (6.8)

where the integrals are written in Dirac notation. Equations 6.7 and 6.8 can bearranged as a matrix equation

〈A|H|A〉 − ǫ 〈A|H|B〉 − ǫ〈A|B〉

〈B|H|A〉 − ǫ〈B|A〉 〈B|H|B〉 − ǫ

cA

cB

=

0

0

. (6.9)

6.1.2 Definitions and Conventions

Before solving Eq. 6.9 for ǫ and then for cA and cB, we evaluate some terms andchange notation. The integrals 〈A|B〉 and 〈B|A〉 are not zero – the wavefunctionsare centered on different atoms, but the tails of these wavefunctions overlap. Theseare “overlap integrals,” defined as S

S ≡ 〈A|B〉 = 〈B|A〉 . (6.10)

7 Entropy

Without entropy to complement energy, thermodynamics would have the impactof one hand clapping. The epithaph on Boltzmann’s monument shown in Fig. 7.1

S = kB lnΩ , (7.1)

is an equation for entropy of breathtaking generality. Here it is modernized slightly,with kB as the Boltzmann constant. The nub of the problem is the numberΩ, whichcounts the ways of finding the internal coordinates of a system for thermodynamically-equivalent macroscopic states. Physical questions are, “What do we count for Ω,and how do we count them?”

Sources of entropy are listed in Table 7.1, and some were discussed in Chapter1. Configurational entropy in the point approximation was used extensively inChapter 2, and Section 17.4 accounts for magnetic entropy in essentially the sameway. This Chapter 7 shows how the configurational entropy of chemical disorderor magnetic disorder can be calculated more accurately with cluster expansionmethods. The other major source of entropy is vibrational entropy, and its origin isexplained in Section 7.5. Critical temperatures of ordering and unmixing, calculatedpreviously with configurational entropy alone, are then adapted to include effectsof vibrational entropy.

tFig. 7.1 The Boltzmann monument in Vienna. The constant k is related to kB by the factor2.3026 if log denotes log10. Our notation uses Ω instead of W.

175

176 Entropy

Table 7.1 Sources of Entropy in Materials

Matter Structural Configurations

nuclei sites for the nuclei (the electrons will adapt)electrons sites for electrons in mixed-valent compounds

electron spins orientational disorder (magnetic disorder)

Energy Dynamics

nuclei vibrations (phonons)electrons excitations across Fermi level (electronic heat capacity)

electron spins spin waves (magnons)

For metals there is a heat capacity and entropy from thermal excitations ofelectrons near the Fermi surface, but as discussed in Section 6.2.4 this is often asmall effect because not many electrons are available for these excitations. Theelectronic entropy of a metal increases with temperature, but at high temperaturesthe electronic excitations may interact with phonons, and phonons interact withother phonons as discussed in Chapter 24.

7.1 Static and Dynamic Sources of Entropy

It is often reasonable to separate the internal coordinates of a material into configu-rational ones and dynamical ones. As an example, when the numberΩ enumeratesthe ways to arrange atoms on the sites of a crystal, the method to calculateΩ doesnot depend on temperature. We used this idea extensively in Chapter 2. On theother hand, when the numberΩ for dynamical coordinates counts the intervals ofvolume explored as atoms vibrate, this Ω increases with temperature, as does thevibrational entropy.

Configurational entropies of atoms or spins undergo changes during chemicalordering or magnetic phase transitions, respectively.1 Configurational entropy waslargely understood by Gibbs, who presented some of the combinatoric calculationsof entropy that are used today (23). The calculation ofΩ is more difficult when thereare partial correlations over short distances, but cluster approximation methodshave proved powerful and accurate (53)-(55), and are presented in Section 7.2. Inessence, new local variables are added to the list of composition and long-range-

1 Electronic entropy can also have a configurational component in mixed-valent systems. Nuclearspins undergo ordering transitions at low temperatures, too, although at most temperatures ofinterest in materials physics the nuclear spins are fully disordered and their entropy does not changewith temperature.

177 Short-Range Order and the Pair Approximation

order parameter to describe more precisely the atom configurations on lattice sites.Again, although the configurational variables have different equilibrium values atdifferent temperatures, temperature does not alter the combinatorial method forcalculating Ωwith the configurational variables.

Dynamical entropy grows with temperature as dynamical degrees of freedom ofa solid, such as normal modes of vibration, are excited more strongly by thermal en-ergy.2 With increasing temperature more phonons are created, and the vibrationalexcursions of atomic nuclei are larger. Fundamentally, the entropy from dynamicalsources increases with temperature because with stronger excitations of dynam-ical degrees of freedom, the system explores a larger volume in the hyperspace3

spanned by position and momentum coordinates, as discussed in Section 7.5. Thisvolume, normalized by a quantum volume if necessary, is the Ω for Eq. 7.1. For aphase transition, what is important is not the total vibrational entropy so much asthe difference in vibrational entropy between the two phases.4

7.2 Short-Range Order and the PairApproximation

Section 2.10 presented a thermodynamic analysis of the order-disorder transitionin the point approximation, which assumed that all atoms on a sublattice weredistributed randomly. This assumption is best in situations when 1) the temperatureis very high, so the atoms are indeed randomly distributed on the sublattice, 2)the temperature is very low, and only a few antisite atoms are present, or 3) ahypothetical case when the coordination number of the lattice goes to infinity. Formore interesting temperatures around the critical temperature, for example, it ispossible to improve on this assumption of sublattice randomness by systematicallyallowing for short-range correlations between the positions of atoms.

For example, a deficiency of the point approximation for ordering is illustratedwith Fig. 7.2. We see that the numbers of A-atoms and B-atoms on each sublattice areequal, so the LRO parameter L = (R−W)/(N/2) = 0. Nevertheless, there is obviouslya high degree of order within each of the two domains. It might be tempting to re-define the sublattices within each domain, allowing for a large value of L, but thisgets messy when the domains are small. The standard approach to this problem is

2 Temperature also drives electronic excitations to unoccupied states, and when many states areavailable the electronic entropy is large. Spin excitations are another source of entropy, but care mustbe taken when counting them if the configurations of spin disorder are already counted.

3 This is frequently called a “phase space,” not to be confused with geometric properties of crystallo-graphic phases.

4 The “Kopp–Neumann rule” from the nineteenth century states that the heat capacity of a compoundis the sum of atomic contributions from its elements. By this rule, the vibrational entropy of a solidphase depends only on its chemical composition, and not on its structure. This rule is not helpfulfor understanding the thermodynamics of phase transitions. Furthermore, the Kopp–Neumann rulehas inconsistencies when picking an atomic heat capacity for carbon, for example.

8 Pressure

Historically there has been comparatively little work on how phase transitionsin materials depend on pressure, as opposed to temperature. For experimentalwork on materials, it is difficult to achieve pressures of thermodynamic impor-tance, whereas high temperatures are obtained easily. The situation is reversed forcomputational work. The thermodynamic variable complementary to pressure isvolume, whereas temperature is complemented by entropy. It is comparativelyeasier to calculate the free energy of materials with different volumes, as opposedto calculating all different sources of entropy.

Recently there have been rapid advances in high pressure experimental tech-niques, often driven by interest in the geophysics of the Earth. Nevertheless, newmaterials are formed under extreme conditions of pressure and temperature, andsome such as diamond can be recovered at ambient pressures. The use of pres-sure to tune the electronic structure of materials can be a useful research tool forfurthering our understanding of materials properties. Sometimes the changes ininteratomic distances caused by pressure can be induced by chemical modificationsof materials, so experiments at high pressures can point directions for materialsdiscovery.

This Chapter 8 begins with basic considerations of the thermodynamics of ma-terials under pressure, and how phase diagrams are altered by temperature andpressure together. Volume changes can also be induced by temperature, and theconcept of “thermal pressure” from non-harmonic phonons is explained. The elec-tronic energy accounts for most of the PV contribution to the free energy, and thereis a brief description of how electron energies are altered by pressure. The chapterends with a discussion about using pressure to investigate kinetic processes, andthe meaning of an activation volume.

8.1 Materials under Pressure at LowTemperatures

The behavior of solids under pressure, at least high pressures that induce substan-tial changes in volume, is more complicated than the behavior of gases. Neverthe-less, it is useful to compare gases to solids to see how the thermodynamic extensivevariable, V, depends on the thermodynamic intensive variables T and P.

208

209 Materials under Pressure at Low Temperatures

8.1.1 Gases (for comparison)

Recall the equation of state for an ideal gas comprised of non-interacting atoms

PV = NkBT . (8.1)

Non-ideal gases are often treated with two Van der Waals corrections:

• The volume for the gas is a bit less than the physical volume it occupies becausethe molecules themselves take up space. The quantity V in Eq. 8.1 is replacedby V −Nb, where b is an atomic parameter with units of volume.

• An attractive interaction between the gas molecules tends to increase the pressurea bit. This can be considered as a surface tension that pulls inwards on agroup of gas molecules. The quantity P in Eq. 8.1 is replaced by P + a(N/V)2.The quadratic dependence of 1/V2 is expected because the number of atomsaffected goes as 1/V, and the force between them may also go as 1/V. (Also, ifthe correction went simply as 1/V, it would prove uninteresting in Eq. 8.1.)

The Van der Waals equation of state (EOS) is

[P + a

N2

V2

][V −Nb

]= NkBT . (8.2)

Equation 8.2 works surprisingly well for the gas phase when the parameters a

and b are small and the gas is “gas-like.” Equation 8.2 can be converted to thisdimensionless form

P = TV − 1

− 1V2 , (8.3)

with the definitionsP ≡ P

b2

a, (8.4)

V ≡ V

Nb, (8.5)

T ≡ kBTb

a. (8.6)

Figure 8.1 shows the Van der Waals EOS of Eq. 8.2 for a fixed a and b, but withvarying temperature. At high temperatures the behavior approaches that of an idealgas, with P ∝ T/V (Eq. 8.1). More interesting behavior occurs at low temperatures.It can be shown that below the critical pressure and critical temperature

Pcrit =1

27a

b2 , (8.7)

kBTcrit =8

27a

b, (8.8)

a two-phase coexistence between a high-density and a low-density phase appears.This is point “C” in Fig. 8.1a, for which the volume is

Vcrit = 3 b N . (8.9)

210 Pressure

tFig. 8.1 (a) Isothermals of the Van der Waal’s equation of state Eq. 8.2, plotted withrescaled variables of Eqs. 8.4 - 8.6. (b) Maxwell construction for T = 0.26a/(bkB).

At lower temperatures, such as kBT = 0.26a/b shown in Fig. 8.1b, the samepressure corresponds to three different volumes V1, V2, and V3. We can ignore V2

because it is unphysical – at V2 an increase in pressure causes an expansion (andlikewise, the material shrinks if pressure is reduced). Nevertheless, the volumesV1 and V3 can be interpreted as the specific volumes of a liquid and as a gas,respectively. We find the pressure that defines V1 and V3 from the condition thatthe chemical potentials of the gas and liquid are equal in equilibrium, i.e., µ3 = µ1.Along a P(V) curve, the change in chemical potential is 1/N

∫P dV. Starting at a

chemical potential of µ1 at the point V1 in Fig. 8.1b

µ3 = µ1 +1N

∫ V3

V1

P(V) dV . (8.10)

The integral must be zero if µ3 = µ1. The areas above and below the horizontal linein Fig. 8.1b must therefore be equal, and this “Maxwell construction” defines thepressure of the horizontal line.

A dimensionless ratio can be formed from Eqs. 8.7, 8.8, 8.9

Pcrit Vcrit

NkB Tcrit=

38. (8.11)

Rescaled appropriately, the Van der Waals equations of state for all gases are thesame. This is approximately true in practice, although the dimensionless ratiois lower than 3/8, often around 0.25 to 0.3, and varies for different gases. Somecharacteristics of a generic gas are presented in Table 8.1, for comparison with thecharacteristics of a solid.

211 Materials under Pressure at Low Temperatures

Table 8.1 Pressures and Temperatures of Gasesand Solids

Gas Solid

Pressure P > 0 P > −Pcoh

Temperature T > 0 T ≥ 0

Stresses isotropic anisotropic

TypicalPressure 1 Atm = 0.1 MPa 10 GPa = 105 Atm

This instability of the Van der Waals EOS below a critical temperature can beused to model a pressure-induced liquifaction, for example, or the liquid-gas phaseboundary at constant pressure. The approach has problems with quantitative de-tails, but it gives the essential behavior, and is worthy of more study than givenhere.

8.1.2 Solids (for comparison)

The ideal gas behavior shown at the top of Fig. 8.1a, i.e., P ∝ T/V for large T,V, is never appropriate for a solid. At P = 0, for example, the solid has a finitevolume. Table 8.1 shows that on the scale of familiar pressures in gases, a solidis essentially incompressible. More familiar are small compressions of solids andelastic behavior, where typical materials deform as springs. The bulk modulus ofa solid, B,

B ≡ −VdP

dV, (8.12)

is typically a few times 100 GPa, and the elastic energy per unit volume underuniform dilation is

Eel =12

B δ2 , (8.13)

where δ is the fractional change in volume.Equation 8.12 can be handy as a definition of B if the elastic energy is needed, and

not individual strains or stresses of Section 6.4. The elastic constants originate fromsecond derivatives of the interatomic potentials, which give tensorial “springs”between atoms as explained in Sect. 6.3. These springs are loaded in differentdirections when different stresses are applied to a material, but all strains are linearwith stresses, and the macroscopic response of the material is still that of a spring.It is possible to relate the interatomic force constants to the macroscopic elastic

Index

Abrikosov vortex, 477activated state and equilibrium, 78activated state rate theory, 76activation barrier, 529activation energy, 223, 224, 257activation entropy, 257activation volume, 223activity (chemical), 431adatom, 288adiabatic approximation, 576Ag, 413Al, 413, 563Al-Li, 201Al-Pb, 87allotriomorph, 86, 172, 301alloy phases, 21alloy supercooling, 119alumina, 126aluminum, 126amorphization, 122amorphous phase, 130, 414anharmonicity, 204, 571, 574anisotropy, 162, 211anisotropy energy, 459anisotropy gap energy, 455annealing, 279anomalous diffusion, 15antibonding, 139, 217antiferromagnetism, 434, 452antiphase boundary profile, 347antiphase domain boundary, 178, 234, 344antisite atom, 487antisymmetric wavefunction, 439APB width, 344aspect ratio, 300athermal, 389atom configurations, 34atomic ordering, 47Atomium, 405attempt frequency, 257, 529Au, 413austenite, 304, 387austenitizing, 309autocatalytic, 304autocorrelation function, 545average, 157

Avrami equation, 312

b(k), 366B2, 181, 359, 376, 378, 496, 538

second order, 372B2 structure, 359B32, 359, 496, 538Ba, 267bainite, 305, 310ball milling (high energy), 106ballistic jump, 249, 254, 542band structure, 147, 152, 154, 220basis vector, 358BaTiO3, 457bcc, 357, 490, 496

2nn forces, 404BCS theory, 471Becker and Doring, 91Bessel function, 72bin, 531binary alloy, 33binomial expansion, 35, 195binomial probability, 258bin, 18Bloch T3/2 law, 452Bloch wall, 428, 458Bloch’s theorem, 146, 450blocking temperature, 429Bogers–Burgers double shear, 392Boltzmann, 175Boltzmann factor, 16, 221Boltzmann substitution, 65bond counting, 298bond energy, 141bond integral, 139, 147bond length, 201, 203bond proportion model, 193bond stiffness, 194

chemical effects, 204bond stiffness vs. bond length model, 201bonding, 139, 153, 217Born–Oppenheimer approximation, 190, 473Born–von Karman model, 190, 194, 560Bose-Einstein condensation, 462Bose-Einstein factor, 462bosons, 188, 451, 462boundary conditions, 66, 95, 149

595

596 Index

boxel, 14Bragg–Williams approximation, 47, 50, 199branch, 562Brillouin zone, 355, 562bulk modulus, 157, 211, 279, 566Burgers vector, 380

conservation of, 384

CV and CP, 566Cahn, John Werner, 326Cahn–Hilliard equation, 324calculus of variations, 326, 520calorimetry, 407, 566carbon dioxide, 9carbon nanotubes, 419casting, 340Cauchy’s first law, 521Cayley tree, 461Ce, 8, 225, 575cementite, 304characteristic width, 347charge density, 440chemical bonding, 137chemical bonding energy, 34chemical environment, 528, 529chemical factor, 247chemical potential, 17, 244, 246, 462chemical spinodal, 333chemisorption, 424chessboard, 182, 453, 488classical vs. quantum behavior, 469Clausius–Clapeyron equation, 215, 408cluster, 91, 93cluster expansion, 181, 204Co, 413coarse-graining, 507coarsening, 302, 313, 352coherence, 477, 482, 554coherent spinodal, 333coincidence boundary, 412collector plate, 301collision cascade, 106collisions, 93, 244colonies, 306combinatorial factor, 204common tangent construction, 30commutation, 554completeness, 555complex conjugate, 365complex exponential, 359complex material, 438composition fluctuation, 250, 319, 321compressibility, 211computer program

complexity and entropy, 480computer simulation, 300, 343concentration gradient, 330

concentration profile, 114concentration profile, stability of, 319concentration wave, 354

infinitesimal, 374, 376stability, 376

condensate wavefunction, 468conduction electron polarization, 456conductivity, 245configurational coordinates, 176configurational energy, 35configurational entropy, 35, 37, 177

nanostructure, 419confined electrons, 418confocal ellipsoidal coordinates, 56conservation of solute, 29conservative dynamics, 348constititutional supercooling, 119constraint, 44, 328, 530continuous, 80convection, 56convolution, 376, 524cooling rate, 104Cooper pair, 4, 471, 570cooperative transition, 447, 501coordinate systems, 56, 71coordinates, 175coordination number, 34core electron polarization, 442correlated electrons, 444correlation

entropy of liquid, 274correlation factor, 233, 257, 259

heterogeneous alloy, 232table, 231tracer, 230

correlation function, 182correlation length, 503, 506Coulomb correlation, 445Coulomb energy, 441covalency, 142, 143Cr, 575critical condition, 209critical field, 477critical free energy of formation, 84critical nucleus, 82, 101critical point, 8critical radius, 84, 126

strain effects, 90critical slowing down, 504critical temperature, 38, 39, 50, 193, 198, 375, 448,

501, 534change with ∆Svib, 193displacive transition, 399ordering, 200, 203unmixing, 195, 197, 250

critical undercooling, 98

597 Index

critical wavevector, 332cry of tin, 387crystal field theory, 139, 218crystallization, 130, 371crystallographic variant, 400CsCl, 359Cu, 267, 413Cu3Au, 359Cu-Al, 387Cu-Co, 13, 82Cu-Zn, 387CuAu, 359CuPt, 378Curie temperature, 304, 434, 447, 448, 456Curie-Weiss law, 449curvature, 294, 348, 351curvilinear coordinates, 350CuZr, 130cylindrical coordinates, 72

Debye model, 565Debye–Waller factor, 556decimation, 507deformation potential, 473degrees of freedom, 42, 252, 507

microstructure, 419dendrite, 119, 340density, 502density of states, 147, 149, 451determinant, 139deuterium, 247devitrification, 130diatomic molecule, 137diffractometry, 525diffusion, 12, 104, 108, 313, 330, 387, 552

anomalous, 15chemical potential, 314distance, 66ideal gases, 236in inhomogeneous alloy, 236liquid, 113marker velocity, 237zone, 126

diffusion coefficient, 236, 257, 545diffusion equation, 59diffusionless transformation, 387dilation, electron energy, 472dilute impurity, 53dimensionality, 101, 349, 419dimensionless integral, 452Dirac δ-function, 61, 63Dirac notation, 138discontinuous, 80dislocation, 82, 283, 291, 303, 380, 411

Burgers vector, 380core, 382core radius, 283

edge, 380fcc and hcp, 383glide, 381groups of, 411mixed, 381partial, 384plastic deformation, 380reactions, 383screw, 293, 380self energy, 382tilt boundary, 411

dispersion, 562dispersion forces, 424displacement parameter, 400displacement vector, 160displacive ferroelectric, 437displacive instability, 396dissipation, 544, 548distribution coefficient, 110divacancy, 491divergence, 350, 502DO3, 359, 538domain, 178, 458domain boundary, 348domain wall, 476driven systems, 249driving force, 77Duwez, Pol, 281dynamical coordinates, 176dynamical matrix, 563Dzyaloshinskii–Moriya interaction, 456

Edisonian testing, 6Ehrenfest’s equations, 216eigenvalues, 564eigenvectors, 564

of dynamical matrix, 563Einstein convention, 518Einstein model, 194, 195, 398elastic, 160

anisotropy, 429energy, 212, 298, 332, 429, 517, 567field, 304scattering, 556waves, 562

electric dipole moment, 456electrical conductivity, 153, 245electron energy, 157electron gas, 148electron levels, 220electron transfer, 143electron-phonon coupling parameter, 474electron-phonon interaction, 472, 575electron-phonon interactions, 569, 575electron-to-atom ratio, 26, 154electronegativity, 26, 143, 144, 204electronic DOS, 147, 153, 474, 575

598 Index

electronic energydimensionality, 418

electronic entropy, 568electronic heat capacity, 151electronic oxides, 446electronic structure, 22electrons, 3Eliashberg coupling function, 474embryo, 80emergent phenomenon, 4energy, 17, 38, 196

elastic, 163surface, 83, 170

energy gap, 475energy landscape, 252, 279energy versus matter, 463ensemble average, 228, 545enthalpy, 52entropy, 16, 17, 38, 52, 196, 388

configurational, 52, 388electronic, 52, 151magnetic, 52, 388microstructure, 410vibrational, 191, 388, 559

equation of state, 209equations of motion, 561equiaxed, 91equilibrium, 533

kinetic, 75, 77non-thermodynamic, 249thermodynamic, 74

equilibrium interatomic separation, 158equilibrium shape, 84error function, 65Eshelby cycle, 164Euler equation, 328, 345eutectic, 42, 54, 107eutectoid, 45, 305evaporation, 116exchange energy, 428, 439

spin, 508exchange hole, 441exchange interaction, 447, 514extreme conditions, 217

fcc, 357, 490Fe, 26, 413, 442

nanocrystal, 423Fe3Al, 359Fe3O4, 436Fe-C, 387Fe-Cr, 13Fe-Ni, 388, 390, 558Fermi

energy, 150level, 154, 456surface, 153, 474, 577

wavevector, 150Golden Rule, 554

Fermi-Dirac statistics, 373ferrimagnetism, 434ferrite, 304ferroelectric, 437, 456ferromagnet, 507ferromagnetism, 434, 442, 447Fick’s laws, 59finite element methods, 56first Brillouin zone, 355first phase to form, 127first-order transition, 7fluctuation, 82, 319

concentration, 331fluctuation–dissipation theorem, 549fluid, 56, 505flux, 60, 61flux of atoms, 330fluxon, 477force, 244force constants, 560forging, 309Fourier Series, 66Fourier transform, 376Fourier-Bessel expansion, 73Fowler correction factor, 179Frank–Van der Merwe growth, 295free electron model, 148, 156free energy, 27

alloy phases, 23curvature, 39k-space, 364phonon, 190spinodal unmixing, 249thermal expansion, 568

free energy vs. composition, 27freezing, 6, 7, 107, 263Friedel, 284, 407Friedel oscillation, 456frustration, 435functional, 327

G-P zone, 285gases, 209Gauss integrals, 253, 268Gauss’s theorem, 521Gaussian, 96, 553Ge, 575generating function, 259Gibbs, 91, 176Gibbs free energy, 18Gibbs phase rule, 42, 45, 338Gibbs–Thomson effect, 415Gibbs-Thomson effect, 89Ginzburg–Landau Equation, 341, 347, 476glass, 120, 129

599 Index

strong or fragile, 131, 280transition, 129

glide, 381Goodenough-Kanamori rules, 455Gruneisen parameter, 201, 203, 214, 224, 571gradient, 60, 350gradient energy, 322, 323grain boundary, 85, 171, 301, 410, 411

energy, 412width, 412

grain boundary allotriomorph, 86, 172gravity, 225Green’s function, 64, 517, 524ground state, 464ground state maps, 490growth equation, 310growth rate

interface, 117Guinier-Preston zones, 90

H+2 molecular ion, 142H2 , 465H2 molecule, 173habit plane, 388Hamiltonian, 560hard core repulsion, 158harmonic approximation, 192, 560, 571harmonic model, 188, 256harmonic oscillator, 190, 397Hartree-Fock wavefunction, 440hcp, 408He, 463He II, 467heat, 557heat capacity, 216

quasiharmonic, 566heat of formation, 28heat treating, 309Heisenberg model, 282, 439, 450Helmholtz free energy, 17, 196heterogeneous nucleation, 81, 303heteronuclear molecule, 142HfO2 , 575high vacuum evaporation, 116high-temperature expansion, 497high-temperature limit, 192homogeneous nucleation, 81homogeneous precipitate, 302Hooke’s law, 162, 518Hubbard model, 445Hume–Rothery rules, 25Hund’s rule, 442hydrogen, 15, 188, 424hydrogen atom, 173hyperbolic functions, 499hypereutectoid steel, 309hyperfine interaction, 4, 442

hyperspace, 185, 186, 252hypoeutectoid steel, 309hysteresis, 388

ice, 224iced brine quenching, 104ideal gas, 209, 246ideal solution, 38imaginary frequency, 398incoherent approximation, 556inconsistency

pair approximation, 178incubation time, 99independent variables, 537indistinguishability, 274inelastic scattering, 544, 551infinitesimal displacement, 399infinitesimal order, 370initial conditions, 66instability eigenvector, 404insulator, 153integration by parts, 327interatomic potential, 158interchange, 528interchange energy, 36interdiffusion, 126interface concentration, 126interface energy, 297, 324interface velocity, 128interference, 468intergranular fracture, 303, 339internal interface, 303internal stress, 430interstitial, 15, 387interstitialcy mechanism, 15Invar, 222inversion, 361ion beam bombardment, 106ionicity, 142, 143Ising lattice, 10, 195, 198, 282, 420, 446island growth, 295isoelectronic, 143isothermal, 389isothermal compressibility, 216isotope, 188isotopic fractionation, 206isotropy, 162iteration, 511

Josephson junction, 470jump rate, 529jump sequence, 230jumping beans, 258

K, 157, 172, 267k-space, 149, 358k-vector

as quantum number, 145Kauzmann paradox, 130

600 Index

KCl, 173kinetic arrest, 542kinetic energy, 142, 156, 254, 575kinetic master equation, 18, 74, 531kinetic path, 536, 541kinetic stability, 410kinetics, 5, 76, 226, 527kink, 288, 481Kirkendall voids, 237Kopp–Neumann rule, 177Kosterlitz–Thouless transition, 282Kurdjumov–Sachs, 390

L10 structure, 359L11, 378L12, 359

first order, 371laboratory frame, 237Lagrange multiplier, 328lamellar spacing, 308Landau theory, 396

first order, 401function, 397potential, 402vibrational entropy, 402

Landau-Lifshitz criterion, 370Langevin equation, 548Langmuir isotherm, 425Laplacian, 56, 350

separable, 71laser processing, 105latent heat, 7, 19, 263, 306lattice connectivity, 513lattice dynamics, 560lattice gas, 431lattice mismatch, 82Laue condition, 357layer-by-layer growth, 295ledeburite, 305ledge, 288ledges and growth, 287, 293length scale, 512Lennard-Jones potential, 158, 172lenticular precipitate, 171lever rule, 27, 107

differential form, 110levitation melting, 105Li, 275Lifshitz, Slyozov, Wagner, 315Lindemann rule, 277linear elasticity, 160linear oxidation, 125linear response, 76, 250, 547linearize near Tc, 533liquid, 6, 211, 273, 501liquid free energy, 27liquidus, 32, 55

local chemical environment, 535local spins, 447long-range interactions, 375long-range order, 48, 198, 506longitudinal branch, 562Lorentzian, 553lower bainite, 305LRO, 47, 448, 527

Mf and Ms, 389magnetic field, 44magnetic

flux, 458nanoparticles, 431order, 341phase transitions, 433susceptibility, 435, 454systems, 501torque, 454

magnetism, 222, 427, 476magnetite, 435, 436magnetization, 408, 449, 504magnetocrystalline anisotropy, 429magnetoelastic energy, 343magnon, 452many-body theory, 256marker velocity, 237martensite, 12, 304, 310, 379, 407

crystallography, 393magnetic field effect, 438

martensite transformation mechanism, 392Martin, Georges, 249master equation, 18, 74, 527, 528matrix stress, 166Maxwell construction, 210Maxwell relationship, 18, 216McMillan expression for Tc, 475mean field approximation, 373, 453mean-squared displacement, 226mechanical attrition, 414melt spinning, 105melting, 6, 7, 225, 263, 371

atom displacements, 277melting point, 32melting temperatures, table of, 266memory, 229Mendeleev number, 28metallic bond, 156metallic glass, 121, 129metallic radius, 26metastable, 288, 319, 541methane, 424Metropolis algorithm, 542Mg, 26microstructure, 6, 313, 337, 410

in steels, 310midrib, 389

601 Index

Miedema, A.R., 28misfit energy, 518misfitting particle, 164

ellipsoid, needle, plate, 168, 524Mo, 267, 575mobility, 77, 245, 349, 536models, 4, 281molecular dynamics, 258molecular orbital, 138molecular wavefunction, 140monomer, 93

attachment to cluster, 93Monte Carlo, 540morphology, 91Morse potential, 158moving boundary, 113multicomponent alloy, 537multiphonon scattering, 557

item Na, 157, 212Nabarro, Frank, 168nano-dots, 419nanomaterial, 410nanostructure, 417

low-energy modes, 422phonon broadening, 424

NaTl, 359Nb3Sn, 387Neel temperature, 454neutron, 554Newton’s law, 548Nishiyama, 391NiTi, 407, 414Nobel prize, 507nonconservative dynamics, 348nonconserved quantity, 341nonequilibrium, 536nonequilibrium cooling, 110nonlinear oscillator, 398normal coordinates, 190, 252, 573normal modes, 186, 194, 561nose, 120nucleation, 12, 40, 80, 119, 126, 285, 320

and growth, 12, 80, 120, 285edges and corners, 87elastic energy, 89grain boundary, 85heterogeneous, 81, 85homogeneous, 81in concentration gradient, 126rate, 91, 97temperature dependence, 97time-dependent, 98

nuclei, 3nucleus

coherent, incoherent, 81, 82critical, 82

numerical calculations, 515

occupancy variable, 373Ohm’s law, 245Olson–Cohen model, 392omega phase, 405Onnes, 471optical mode, 562optical phonon, 458orbitals, 218order parameter, 338, 361, 529, 537ordered domain, 178ordering, 47, 359orientation relationship, 390orthogonality

Bessel functions, 73sine functions, 69

overcounting, 34, 194overlap integral, 138oxidation, 124oxides

electronic, 446

Peclet number, 340pair approximation, 503, 527

consistency, 177pair potentials, 36pair variable, 178, 527parabolic oxidation, 125paraelectric, 437parallelopipedon, 525paramagnetism, 434parent phase, 83partial differential equation, 66partial dislocation, 384particle in a box, 148, 463partition function, 15, 190, 487

at low T, 493configurational, 34, 195harmonic oscillator, 190high temperature, 497ordering, 199spin, 508, 512

partitioning ratio, 109passivation of surface, 126Pauling, Linus, 144Pd, 413Pd-V, 201pearlite, 304peeling, 296percolation threshold, 351periodic boundary condition, 145, 561periodic box, 451periodic minimal surface, 351, 352peritectic, 42peritectoid, 45perovskite, 457Pettifor, David, 28, 143

602 Index

phase, 327, 469compositions, 107definition, 336fractions, 107

phase boundary with T, 495phase diagram

T-c, 9continuous solid solubility, 32eutectic, 41, 44peritectic, 41, 44unmixing, 38

phase factor, 144, 468phase field theory, 336phase space, 274phase stability, 21phase transition

concept, 10ordering vs. unmixing, 10transition vs. transformation, 12

Phillips–Van Vechten, 143phonon, 544, 554

damping, 424density of states, 191dispersion, 562entropy, 191, 566free energy, 190polarization, 563thermodynamics, 188, 571

phonon dispersion, 575phonon DOS, 191, 564phonon-phonon interaction, 569, 575physical vapor deposition, 105physisorption, 424pipe diffusion, 303piston-anvil quenching, 105Planck distribution, 191, 565plastic deformation, 387, 430plate precipitate, 298, 524PMN, 437point approximation, 47, 372, 453, 503, 527, 528,

530Poisson ratio, ν, 159polariton, 458polarization vector, 562polaron, 445pole mechanism, 385, 386polycrystalline material, 420polymer, 144position-sensitive atom probe, 12potential energy, 142, 575potential gradient, 244potential well, 254Potts model, 282pre-melting, 264precipitate, 81, 109, 518

ellipsoidal, 89, 168

growth, 108lens shape, 91needle, 168shape, 298, 522strain effects, 89

precipitate-free zone, 302preferred structure, 172pressure, 8, 17, 157, 208

thermal, 214primitive lattice translation vector, 355probability, 16probability density, 468processing of materials, 6product rule for derivatives, 520proeutectoid ferrite, 309prototype structures, 359pseudo-binary, 45pseudostable state, 541Pt, 413Pythagorean theorem, 350Python code, 511PZT, 437

quadratic formula, 401quantum dot structures, 296quantum level separation, 149quantum matter, 462quantum volume, 273quartic, 401quasielastic scattering, 553quasiharmonic model, 207, 214, 269, 571quasistatic, 258quench, 81

r-space, 358radiation defects, 542random solid solution, 34random walk, 96, 226, 544rare earth metals, 173reaction coordinates, 278recalescence, 264reciprocal lattice, 355reciprocal space, 354reconstruction (surface), 287recursion relation, 510reflection, 361relativistic self energy, 387relaxation, 544relaxation (surface), 287relaxation time, 547relaxor ferroelectric, 437renormalization group, 511rescaling, 509Richard’s rule, 102, 276Richards, Allen, Cahn ground states, 490Riemann zeta function, 452right and wrong neighbors, 49rigid band model, 154

603 Index

roots of Bessel function, 73Rose equation of state, 267rotation, 361roton, 468roughening transition, 291Ruderman-Kittel-Kasuya-Yosida oscillation, 456Rushbrooke inequality, 504

Sackur–Tetrode equation, 274saddle point, 252, 280, 367, 541scaling of interatomic potential, 267scaling theory, 505, 507scattering law, 551Scheil Equation, 111Schrodinger equation, 138, 469Schwarz P-surface, 351, 352second sound, 467second-order phase transition, 216, 370, 400self-force constant, 561self-similar, 315, 414, 507semiconductor, 153separation of variables, 66sextic, 401shape factor

strain effects, 90shape memory alloy, 407shear transformation, 391shell model, 576shock wave processing, 106Shockley partial dislocation, 384, 385short-range order, 178, 506short-range structure, 183shuffle, 393Si, 15, 128, 275, 288, 575singularity, 3, 185, 501, 502SiO2 , 131Slater–Pauling curve, 156Sm, 267small displacements, 560SnO2, 575social network, 461soft mode, 404solid mechanics, 160solid solution, 10, 25, 33solid-on-solid model, 292solid-state amorphization, 122solidification, 338, 340

practical, 116solidus, 32, 55solute conservation, 29, 447solute partitioning, 121special points, 363, 367sphere, misfitting, 165spherodite, 305spin, 3, 181spin exchange energy, 508spin excitations, 177

spin wave, 450, 454spin-orbit coupling, 456spinodal decomposition, 13, 38, 40, 184, 198, 249,

318, 366spinwave, 434spiral growth, 293splat quenching, 105spring, 163, 212sputtering, 105square gradient energy, 13, 298, 477square lattice, 47, 198, 513SRO, 180, 527stability, 153stabilization of austenite, 392stacking fault energy, 385star of wavevector, 361state variable, 15, 74, 536static concentration wave, 354statistical kinetics, 226statistical mechanics, 35, 188statistical variations, 551steady-state, 94steel, 304, 387Stirling approximation, 35, 189, 196, 421Stoner criterion, 443strain, 160strain field, 343Stranski–Krastanov growth, 295stress, 160stress-free strain, 518stress-free transformation strain, 298striped structure, 488structure factor rules, 357structure map, 28structure of solids, 4structure-property relation, 183strukturbericht, 359sublattice, 47, 198substrate mismatch, 295superconductor, 4, 471, 570superexchange, 455superferromagnetism, 429superfluid, 463, 465superparamagnetism, 429supersaturation, 81surface growth, 293surface energy, 13, 83, 169, 170, 295, 298, 415, 522

chemical contribution, 296surface reactions, 124surface-to-volume ratio, 323symmetry, 6, 400, 562

breaking, 541operations, 361

tanh, 448Taylor series, 560ternary alloy, 537

604 Index

ternary phase diagram, 45terrace, 288Th, 275thermal conductivity, 104thermal de Broglie wavelength, 274, 463thermal electronic excitations, 472thermal expansion, 158, 268, 430, 505, 566, 567

coefficient, 216free energy, 568

thermal pressure, 214thermodynamic identity, 17thermodynamic square, 18thermodynamics, 5, 15, 52thin film, 116

growth, 295reactions, 124, 127

thin plate, 525third law of thermodynamics, 466Ti, 405Ti-Nb, 387tie-lines, 45tight binding, 146tiling, 35time average, 228time constants, 535time-dependent nucleation, 98time-temperature-transformation diagram, 120TiO2 , 575tracer, 230transient structure, 538transition metal, 158transition metal silicides, 128transition rate, 75, 554translation, 361translational invariance, 369translational symmetry, 144transverse branch, 562truncation of cluster expansion, 183TTT diagram, 120twinning, 379, 391, 408

mechanism, 385two-level system, 220two-phase mixture, 467

umklapp process, 575undercooling, 120unit cell, 358universal behavior, 502universal potential curve, 267unmixing, 318, 366upper bainite, 305

V, 575V3Si, 387vacancy, 230, 288, 491

concentration, 53diffusion, 56jump, 528

mechanism, 14ordering, 535relaxation, 530trap, 233, 259

Van der Waals, 209, 424Van Hove function, 551Van Hove singularity, 422, 563vapor pressure, 116variational calculus, 326vector identity, 520velocity

of interface, 340of boundary, 350

velocity-velocity correlation function, 544Verwey transition, 436vibrational entropy, 257, 559

concept, 185vicinal surface, 170, 411Vineyard, George, 252virtual phonon, 473viscosity, 131void and vacancies, 316Volmer–Weber growth, 91, 295vortex tube, 477

W, 413, 575W-matrix, 75, 531Walser–Bene rule, 129water, 9wave, 359waves in crystals, 355Weiss theory, 222Wigner–Seitz radius, 277Wulff construction, 170

x-ray diffraction, 358x-y model, 282

Young’s modulus, 159

Zeldovich factor, 93, 96zero-point energy, 207, 465zone refining, 132Zr, 405ZrO2, 571, 575