Phase Transitions in Materials - Brent Fultz

Contents

page iii xiii xiv

Preface List of tables Notation

Part I Basic Thermodynamics and Kinetics of Phase Transformations

7

1

1 Introduction 1.1 What Is a Phase Transition? 1.2 Atoms and Materials 1.3 Pure Elements 1.4 Alloys – Unmixing and Ordering 1.5 Transitions and Transformations 1.6 Brief Review of Thermodynamics and Kinetics Problems

3 3 4 6 9 12 15 19

2 Essentials of T-c Phase Diagrams 2.1 Overview of the Approach 2.2 Intuition and Expectations about Alloy Thermodynamics 2.3 Free Energy Curves, Solute Conservation, and the Lever Rule 2.4 Common Tangent Construction 2.5 Continuous Solid Solubility Phase Diagram 2.6 Solid Solutions 2.7 Unmixing Phase Diagrams 2.8 Eutectic and Peritectic Phase Diagrams 2.9 Ternary Phase Diagrams 2.10 Long-Range Order in the Point Approximation 2.11 Alloy Phase Diagrams Problems

21 21 23 27 30 32 33 38 41 45 47 51 53

3 Diffusion 3.1 The Diffusion Equation 3.2 Gaussian and Error Function Solutions to the 1D Diffusion Equation 3.3 Fourier Series Solutions to the Diffusion Equation

56 57 61 66

Contents

8

3.4

Bessel Functions and other Special Function Solutions to the Diffusion Equation 3.5 Kinetic Master Equation and Equilibrium 3.6 Linear Kinetic Response Problems

71 74 76 77

4 Nucleation 4.1 Terminology and Issues 4.2 Critical Nucleus 4.3 Heterogeneous Nucleation 4.4 Free Energy Curves, Elastic Energy 4.5 The Nucleation Rate 4.6 Time-Dependent Nucleation Problems

80 81 82 85 88 91 98 101

5 Effects of Diffusion and Nucleation on Phase Transformations 5.1 Non-Equilibrium Processing of Materials 5.2 Alloy Solidification with Solute Partitioning 5.3 Alloy Solidification – Suppressed Diffusion in the Solid Phase 5.4 Alloy Solidification – Suppressed Diffusion in the Solid and Liquid 5.5 Practical Issues for Alloy Solidification and Evaporation 5.6 Heat Flow and Kinetics 5.7 Nucleation Kinetics 5.8 Glass Formation 5.9 Solid-State Amorphization and Suppressed Diffusion in a Solid Phase 5.10 Reactions at Surfaces 5.11 The Glass Transition Problems

103 104 107 108 113 115 117 119 120 122 124 129 131

Part II The Atomic Origins of Thermodynamics and Kinetics 135 6 Energy 6.1 Molecular Orbital Theory of Diatomic Molecules 6.2 Electronic Energy Bands in Solids 6.3 Elastic Constants and the Interatomic Potential 6.4 Linear Elasticity 6.5 Misfitting Particle 6.6 Surface Energy Problems

137 137 144 156 160 164 169 172

9

Contents

7 Entropy 7.1 Static and Dynamic Sources of Entropy 7.2 Short-Range Order and the Pair Approximation 7.3 Local Atomic Structures Described by Clusters 7.4 Thermodynamics with Cluster Approximations 7.5 Concept of Vibrational Entropy 7.6 Phonon Statistics 7.7 Lattice Dynamics and Vibrational Entropy 7.8 Bond Proportion Model 7.9 Bond-Stiffness-versus-Bond-Length Model Problems

175 176 177 181 183 185 188 190 193 201 204

8 Pressure 8.1 Materials under Pressure at Low Temperatures 8.2 Thermal Pressure, a Step Beyond the Harmonic Model 8.3 Free Energies and Phase Boundaries under Pressure 8.4 Chemical Bonding and Antibonding under Pressure 8.5 Two-Level System under Pressure 8.6 Activation Volume Problems

208 208 213 215 217 220 223 224

9 Atom Movements with the Vacancy Mechanism 9.1 Random Walk and Correlations 9.2 Phenomena in Alloy Diffusion 9.3 Diffusion in a Potential Gradient 9.4 Non-Thermodynamic Equilibrium in Driven Systems 9.5 Vineyard’s Theory of Diffusion Problems

226 226 235 244 248 252 258

Part III Types of Phase Transformations

261

10 Melting 10.1 Free Energy and Latent Heat 10.2 Chemical Trends of Melting 10.3 Free Energy of a Solid 10.4 Entropy of a Liquid 10.5 Thermodynamic Condition for Tm 10.6 Glass Transition 10.7 Two Dimensions Problems

263 263 264 265 273 276 278 281 283

11 Transformations Involving Precipitates and Interfaces 11.1 Guinier–Preston Zones

285 285

10

Contents 11.2 Surface Structure and Thermodynamics 11.3 Surface Structure and Kinetics 11.4 Chemical Energy of a Precipitate Interface 11.5 Elastic Energy and Shape of Growing Precipitates 11.6 Precipitation at Grain Boundaries and Defects 11.7 The Eutectoid Reaction and Ferrous Metallurgy 11.8 The Kolmogorov-Johnson-Mehl-Avrami Growth Equation 11.9 Coarsening Problems

287 293 296 298 301 304 310 313 316

12 Spinodal Decomposition 12.1 Concentration Fluctuations and the Free Energy of Solution 12.2 Adding a Square Gradient Term to the Free Energy F(c) 12.3 Constrained Minimization of the Free Energy 12.4 The Diffusion Equation 12.5 Effects of Elastic Strain Energy Problems

318 318 320 325 330 332 335

13 Phase Field Theory 13.1 Spatial Distribution of Phases and Interfaces 13.2 Solidification 13.3 Ginzburg–Landau Equation and Order Parameters 13.4 Interfaces Between Domains Problems

336 336 339 341 344 353

14 Method of Concentration Waves and Chemical Ordering 14.1 Structure in Real Space and Reciprocal Space 14.2 Symmetry and the Star 14.3 The Free Energy in k-Space with Concentration Waves 14.4 Symmetry Invariance of Free Energy and Landau–Lifshitz Rule for Second-Order Phase Transitions 14.5 Thermodynamics of Ordering in the Mean Field Approximation with Long-Range Interactions Problems

354 354 361 364

15 Diffusionless Transformations 15.1 Dislocations and Mechanisms 15.2 Twinning 15.3 Martensite 15.4 The Crystallographic Theory of Martensite 15.5 Landau Theory of Displacive Phase Transitions 15.6 First-Order Landau Theory 15.7 Phonons and Structural Collapse 15.8 Soft Phonons in BCC Structures

379 380 385 387 393 396 401 403 404

368 372 377

Contents

11

Problems

408

16 Thermodynamics of Nanomaterials 16.1 Grain Boundary Structure 16.2 Grain Boundary Energy 16.3 Gibbs–Thomson Effect 16.4 Energies of Free Electrons Confined to Nanostructures 16.5 Configurational Entropy of Nanomaterials 16.6 Vibrational Entropy 16.7 Gas Adsorption 16.8 Characteristics of Magnetic Nanoparticles 16.9 Elastic Energy of Anisotropic Microstructures Problems

410 411 412 415 417 419 422 424 427 429 431

17 Magnetic and Electronic Phase Transitions 17.1 Overview of Magnetic and Electronic Phase Transitions 17.2 Exchange Interactions 17.3 Correlated Electrons 17.4 Thermodynamics of Ferromagnetism 17.5 Spin Waves 17.6 Thermodynamics of Antiferromagnetism 17.7 Ferroelectric Transition 17.8 Domains Problems

433 434 439 443 446 450 452 456 458 460

18 Phase Transitions in Quantum Materials 18.1 Bose-Einstein Condensation 18.2 Superfluidity 18.3 Condensate Wavefunction 18.4 Superconductivity 18.5 Quantum Critical Behavior Problems

462 462 465 468 471 480 483

Part IV Advanced Topics 19 Low Temperature Analysis of Phase Boundaries 19.1 Ground State Analysis for T = 0 19.2 Richards, Allen, Cahn Ground State Maps 19.3 Low, but Finite Temperatures 19.4 Analysis of Equiatomic bcc Alloys 19.5 High Temperature Expansion of the Partition Function Problems

485 487 488 490 490 496 497 499

Contents

12

20 Cooperative Behavior Near a Critical Temperature 20.1 Critical Exponents 20.2 Critical Slowing Down 20.3 The Rushbrooke Inequality 20.4 Scaling Theory 20.5 Scaling and Decimation 20.6 Partition Function for One-Dimensional Chain 20.7 Partition Function for Two-Dimensional Lattice Problems

501 502 502 504 505 507 508 512 516

21 Elastic Energy of Solid Precipitates 21.1 Transformation Strains and Elastic Energy 21.2 Real Space Approach 21.3 k-Space Approach Problems

517 518 520 522 525

22 Statistical Kinetics of Ordering Transformations 22.1 Ordering Transformations with Vacancies 22.2 B2 Ordering with Vacancies in the Point Approximation 22.3 Vacancy Ordering 22.4 Kinetic Paths Problems

527 528 530 535 536 542

23 Diffusion, Dissipation, and Inelastic Scattering 23.1 Atomic Processes and Diffusion 23.2 Dissipation and Fluctuations 23.3 Inelastic Scattering 23.4 Phonons and Quantum Mechanics Problems

544 544 548 551 554 558

24 Vibrational Thermodynamics of Materials at High Temperatures 24.1 Lattice Dynamics 24.2 Harmonic Thermodynamics 24.3 Quasiharmonic Thermodynamics 24.4 Thermal Effects Beyond Quasiharmonic Theory 24.5 Anharmonicity and Phonon-Phonon Interactions 24.6 Electron-Phonon Interactions and Temperature Problems

559 560 565 566 569 571 575 577

Further Reading References Index

579 583 595

3

Preface Content This book explains the thermodynamics and kinetics of most of the important phase transitions in materials science. It is a textbook, so the emphasis is on explanations of phenomena rather than a scholarly assessment of their origins. The goal is explanations that are concise, clear, and reasonably complete. The level and detail are appropriate for upper division undergraduate students and graduate students in materials science and materials physics. The book should also be useful for researchers who are not specialists in these fields. The book is organized for approximately linear coverage in an graduate-level course. The four parts of the book 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 atom arrangements on a lattice is developed. The approach uses a minimum amount of information about interatomic interactions, avoiding detailed issues at the level of electrons. Statistical mechanics on an Ising lattice is used to understand alloy phase stability for basic behaviors of chemical unmixing and ordering transitions. This approach illustrates key concepts of equilibrium T-c phase diagrams, and is extended to explain some kinetic processes. Essentials of diffusion, nucleation, and their effects on kinetics are covered in Part I. Part II addresses the origins of materials thermodynamics and kinetics at the level of atoms and electrons. Electronic and elastic energy are covered, and the different types of entropy, especially configurational and vibrational, are presented in the context of phase transitions. Effects of pressure, combined with temperature, are explained with a few concepts of chemical bonding. The kinetics of atom movements are developed for diffusion in solids, and from the statistical kinetics of the atom-vacancy interchange. Part III is the largest. It describes many of the important phase transformations in materials, with the concepts used to understand them. Topics include melting, phase transformations by nucleation and growth, spinodal decomposition, freezing and phase fields, continuous ordering, martensitic transformations, phenomena in nanomaterials, phase transitions involving electrons or spins, and quantum phase transitions. These different phase transitions in materials are covered at different breadths and depths based on their richness or importance, although this reflects my own bias. Many topics from metallurgy and ceramic engineering are covered, although the connection between processing and properties is less emphasized, allowing for a more concise presentation than in traditional texts. Part III includes a number of topics from condensed matter physics that were selected in 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 their Ph.D. candidacy examinations.

4

Part IV presents topics that are more modern, but have proved their importance. Low and high temperature treatments of the partition function, the renormalization group, scaling theory, a k-space formulation of elastic energy, nonequilibrium states in crystalline alloys, fluctuations and dissipation, and some complexities of high temperature thermodynamics are presented. The topics in Part IV are explained at a fundamental level, but unlike Parts I through III, for conciseness in Part IV there are some omissions of methods and steps. The book draws a distinction between phase transformations and phase transitions. Phase transitions are thermodynamic phenomena based on free energy alone, whereas phase transformations include kinetic processes that alter the life cycle of the phase change. Phase transitions originate from discontinuities in free energy functions, so much of the text focuses on formulating free energies for different systems. The book formulates statistical mechanics models for different phase transitions, sometimes using an Ising lattice, which is well suited for such analysis and finds reuse for different phase transitions. Other topics that recur in the text are Landau theory in various forms, the topic of domains, the square gradient energy, the effect of curvature on nucleation, and dynamics with the kinetic master equation. Sometimes the thermodynamics of phase transitions is developed with the partition function, although the classical equation G = E − TS + PV is used widely, and it is assumed that the reader has some familiarity with the terms in this expression. For the kinetics of phase transformations, there is some traditional presentation of diffusion and nucleation, but the kinetic master equation is also used throughout the text. Many topics in phase transitions and related phenomena are not covered in this text. These include: other mechanisms of atom movements (and their effects on kinetics), polymer flow and dynamics, including reptation, phase transitions in fluid systems including phenomena near the critical temperature, massive transformations. Also beyond the scope of the book are computational methods that are increasingly important for studies of phase transformations in materials, including: Monte Carlo methods, molecular dynamics methods (classical and quantum), and density functional theory with extensions to phenomena at finite temperatures. The field of phase transitions is huge, and continues to grow. This text is a snapshot of phase transitions in materials in the year 2013, composed from the angle of the author. Impressively, this field continues to offer a rich source of new ideas and results for both fundamental and applied research, and parts of it will look different in a decade or so. I expect, however, that many core topics will remain the same – the free energy of materials will remain the central concept, surrounded by issues of kinetics.

Teaching I use this text for a graduate-level course taken by Ph.D. students in both materials science and in applied physics at the California Institute of Technology. The 10-week course, which includes approximately 30 hours of classroom lectures, is offered in the third academic quarter as part of a one-year sequence. The first two

5

quarters in this sequence cover thermodynamics and statistical mechanics, so the students are familiar with the use of the partition function to obtain thermodynamic quantities, and have seen basic concepts from quantum statistical mechanics such as the Fermi–Dirac distribution. Familiarity with some concepts from solid-state physics and chemistry is certainly helpful, as is prior exposure to diffusion and transport, 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 Parts I and II, moving in sequence through these chapters. Time limitations force a selection of topics from Parts III and IV, but I typically cover more of Part III than Part 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 the book in a 15-week semester with 45 hours of lectures. An instructor can certainly exercise 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 weekly student assignments, and this experience has helped to improve their wording and content. The majority of these problems make use of concepts explained in the text, fill in the explanations of concepts, or extend analyses. Others develop new concepts not described in the chapter, but these problems usually include longer explanations and hints that may be worth reading even without working the problem. None of the problems are intended to be particularly difficult, and some can be answered quickly once the main idea emerges. I usually assign five or six problems every week during the term. An expanding online solutions manual is available to course instructors whose identity can be verified. Please ask me for further information.

Acknowledgments I thank J.J. Hoyt for collaborating with me on a book chapter about phase equilibria 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 is available at low cost from McMaster Innovation Press. The development of the topic of vibrational entropy would not have been possible without the contributions of my junior collaborators at Caltech, especially L. 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 of them are taking this field into new directions. Important ideas have come from stimulating conversations over the years with A. van de Walle, V. Ozolins, G. Ceder, M. Asta, L.-Q. Chen, D.D. Johnson, D. de Fontaine, 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 Fultz Pasadena, California 3 September 2013

PART I

BASIC THERMODYNAMICS AND KINETICS OF PHASE TRANSFORMATIONS The field of phase transitions is rich, vast, and continues to grow. This text covers parts of the field relevant to materials physics, but many concepts and tools of phase transitions in materials are used elsewhere in the larger field of phase transitions. Likewise, new methods from the larger 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 have been part of the historical backbone of materials science. In essence, the topics in Part I are the thermodynamics of how atoms prefer to be arranged when brought together at various temperatures, and how the processes of atom movements control the rates and even the structures that are formed during phase transformations. The topics in Part I are largely traditional ones, but formulating the development in terms of statistical mechanics and in terms of the kinetic master equation allows more rigor for some topics, and makes it easier to incorporate a higher level of detail from Part II into descriptions of phase transitions in Parts III 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 in a control variable. For thermodynamic phase transitions, typical control variables are the “intensive variables” of temperature, pressure, or magnetic field. Thermodynamic phase transitions in materials and condensed matter, the subject of this book, 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 phase transition 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 is discovered. There is a very broad range of systems that can exhibit phase transitions, extending from atomic nuclei to traffic flow or politics. For many systems it is a challenge to find reliable models of the free energy, however, so thermodynamic analyses 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 with temperature and pressure? In more detail, we often find it useful to consider separately: • • • •

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, at low temperatures (microKelvin), the weak energy of interaction between nuclear spins can lead to nuclear spin alignments. An ordered state of aligned nuclear spins may have the lowest energy, and may be favored thermodynamically at low temperatures. Temperature disrupts these delicate alignments, however, and thermodynamics favors a disordered nuclear magnetic structure at modest temperatures. The free energy, F, changes with temperature when the nuclear spins are aligned, but the functional form of this curve of Ford (T) is not the same as Fdis (T) for the disordered state at higher temperature. At the critical temperature of the ordering transition there is a switch from one curve to another, or perhaps the second derivative d2 F/dT2 has a kink. Order-disorder phase transitions are 1

3

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

4

Introduction enlightening, and have spawned several creative methods to understand how an order parameter, energy, and entropy depend on temperature. Sometimes phase transitions involve multiple physical entities. Electrons of opposite spin can be coupled together by a wave of nuclear vibration (a phonon). These Cooper pairs can condense into a superconducting state at low temperatures. Perhaps electron charge or spin fluctuations couple the electrons in high temperature superconductors, although the mechanism is not fully understood today. Much of the fascination with phase transitions such as superconductivity is with the insight they give into the interactions between the electrons and phonons, or the electron charges and spins. While these are indeed important subjects for study, 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 low temperatures reveals that the information about the alignment of one nucleus is carried to a nearby nucleus by the conduction electrons, and these hyperfine interactions 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 the interesting trees within it. Throughout much of this text, the detailed interactions between the entities of matter are replaced with simplifying assumptions that facilitate mathematical modeling. Sometimes the essence of the phase transition is captured well with such a simple model. Other times the discrepancies prove interesting in their own right. Perhaps surprisingly, the same mathematical model reappears in explanations of phase transitions involving very different types of matter. A phase transition is an “emergent phenomenon,” meaning that it displays features that emerge from interactions between numerous individual entities, and these large-scale features can occur in systems with very different microscopic interactions. The study of phase transitions has become a respected field of science in its own right, and Chapter 20, for example, presents some concepts from this field 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 interesting ways when they are brought together. In condensed matter there are liquids of varying density, and numerous types of crystal structures. Magnetic moments form structures of their own, and the electron density can show spatial modulations. In general, chemical bonds are formed when atoms are brought together. The energy of interatomic interactions is dominated by the energy of the electrons, which are usually assumed to adapt continuously (“adiabatically”) to the positions of the nuclei. The nuclei, in turn, tend to position themselves to allow the lowest energy of the material, which means that nuclei move around to let the electrons find low-energy states. Nevertheless, once we know the electronic structure of a

Essentials of T-c Phase Diagrams

2

This Chapter 2 explains the concepts behind T-c phase diagrams, which are maps of 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, and these phases are present in the amounts f , and with chemical compositions that minimize the total free energy of the alloy. The emphasis in this Chapter 2 is on deriving T-c phase diagrams from free energy functions F(c, T).2 The constraint of solute conservation is expressed easily as the “lever rule.” The minimization of the total free energy leads to the more subtle “common tangent construction” that selects the equilibrium phases at T from the F(c) curves of the different phases. For binary alloys, the shapes of F(c) curves and their dependence on temperature are used 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, free energy functions can be calculated with a minimum set of assumptions about how different atoms interact when they are brought together. Because the key features of phase 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 and iron in copper, can show similar phase transitions. In these unmixing cases, the individual 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 chemical ordering at low temperatures, which requires the definition of an order parameter, L. These generalization of chemical interactions should not be lamented for their loss of rigor, but celebrated as a way to identify phenomena common to many phase transitions. Such “emergent” behavior can be missed if there is too much emphasis on the electronics of chemical bonding and atom vibrations. Nevertheless, we must be wise enough to know the predictive power available at different levels of generalization.

2.1 Overview of the Approach Temperature promotes disorder in a material, favoring higher-entropy phases such as liquids, but the chemical bond energy favors ordered crystals at low tempera1 2

21

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

Essentials of T-c Phase Diagrams

22

tures. This information alone is sufficient to predict a phase change with temperature, but adding a bit more detail about the energy and entropy rewards us with considerably more information about alloy phase diagrams, and this is the essence of the present chapter. The thermodynamic functions of the alloy, E and S, depend on the spatial arrangement of the atoms in the alloy.3 In what follows, a minimal atomistic model is constructed to calculate thermodynamic functions and predict the equilibrium phases in alloys at different combinations of T and chemical composition c. This minimal model successfully predicts the main features of T-c phase diagrams for binary A-B alloys (A and B are the two species of atoms). Sometimes the free parameters in this approach, the pairwise atomic bond energies, can be tuned to fit phase transitions in a specific material, and sometimes the alloy thermodynamics can be extrapolated successfully into regions where no experimental data exists. Convenience for calculation is a virtue and a priority for this chapter, but we must be aware of the two types of risks it entails 1. A good parameterization of atom positions may require more detail than is possible with a simple model. 2. Even if the parameterizations of atom positions are excellent, calculating the energy or entropy from these parameterizations may require more sophisticated methods. Part II of this book addresses these two issues in more detail. Nevertheless, the simple models developed in this chapter are often useful semiquantitatively, and are useful benchmarks for assessing the value added by more sophisticated treatments. This book will not emphasize methods to calculate the energy of the phases, E, as is done today by electronic structure calculations, such as density functional theory with plane wave pseudopotentials.4 Nevertheless, because the energies of the electrons in a material are set by the positions of the nuclei (i.e., the configurations of atoms), a good parameterization of atom configurations can accommodate more advanced analyses, too. Here we obtain E from local atomic arrangements, typically as 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 the Helmholtz free energy is h i 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 of temperature, as is the way configurations are counted to obtain the configurational entropy Sconfig (c). If E(c) and Sconfig (c) depend only on local atom configurations 3 4

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

Intuition and Expectations about Alloy Thermodynamics

23

in a crystal, it is efficient to use a fixed lattice and pairwise chemical bonds. For solid solution phases, it is also convenient to assume statistical randomness of atom 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 phases that cannot be placed naturally on the same lattice, so each phase has a different F(c, T). Minimizing the total free energy by varying the compositions and fractions of the phases requires the “common tangent” algorithm that allows us to complete the set of the five main types of T-c phase diagrams for binary alloys. With these assumptions, 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 energy and entropy of materials originate from local characteristics of atom configurations, so this approach has an excellent record of success. 2. How accurately can we account for the energy and entropy of the material? For example, the electronic energy, which provides the thermodynamic E, need not be confined to neighboring pairs of atoms. The energy of delocalized electrons in a box or in a periodic crystal is introduced in Chapter 6. This Chapter 2 ignores the entropy caused by the thermal vibrations of atoms, Svib (c, T), the entropy from disorder in magnetic spins, Smag (c, T), and the entropy from disorder in electron state occupancies, Sel (c, T). These will have to be assessed later.6

2.2 Intuition and Expectations about Alloy Thermodynamics 2.2.1 Free Energies of Alloy Phases Alloys often transform to different phases at different temperatures. We have already mentioned the liquid at high temperatures and ordered structures at low temperature. Here, listed from high to low temperature, are some typical phases that are found for an alloy system (“system” meaning a range of compositions for specified 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. For most materials this is a gas of isolated atoms, but at lower temperatures most 5 6

Or, analogously for an order parameter, the sublattice occupancies are assumed random. 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 configurations of a system with equivalent macrostates, giving the Ω for Eq. 1.10. For counting: • Procedures to count atom arrangements (or spin arrangements) are independent of temperature. • On the other hand, procedures to count states explored with thermal excitations, such as atom vibrations or electronic excitations, do depend on temperature. This separation works well for the thermodynamics of crystals at modest temperatures. If atom mobility becomes extremely rapid while some vibrations become very slow, as when a liquid changes its viscosity with temperature, its reliability may be in doubt. The reader should know that at a temperature of 1,000 K or so, Svib is typically an order-of-magnitude larger than Sconfig (compare Figs. 2.9a and 10.3). It is possible for Svib to be similar for different phases, however, so the difference in entropy caused by vibrations is not always dominant in a phase transition. Nevertheless, Svib usually does depend on atom configurations, and this is developed for independent harmonic oscillations in Section 7.8. At high temperatures, however, vibrations change their frequencies, and normal modes of vibration interact with each other, so the statistical mechanics of harmonic vibrations needs significant modifications (Chapter 24).

alloys form a liquid phase with continuous solubility of A and B-atoms. (There are cases of chemical unmixing in the liquid phase, however, for which the thermodynamics of Eq. 2.19 is again relevant.) • At low-to-intermediate temperatures, the equilibrium phases and their chemical compositions depend in detail on the free energy versus composition curves Fξ (c) for each phase ξ. Usually there are chemical unmixings, and the different chemical compositions frequently prefer different crystal structures. The chemical unmixings may not require precise stoichiometries (e.g., a precise composition of A2 B3 ) because some spread of compositions may provide entropy to make off-stoichiometric compositions favorable (e.g., A2−δ B3+δ ). • At the lowest temperatures, the equilibrium state for a general chemical composition is a combination of crystalline phases, usually with precise stoichiometries. These often correspond to crystal structures that have unique sites for the 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 diffusion in crystals was presented in Section 1.5.3 and illustrated with Fig. 1.7. Mention was made of interstitial diffusion and interstitialcy diffusion. Mass transport in glasses and liquids can also occur by atomic-level diffusion, but for gases or fluids of low viscosity there are larger-scale convective currents with dynamics quite different from diffusion.1 The diffusion equation has the same mathematical form as the equation for heat conduction, if solute concentration is replaced by heat or by temperature. The heat equation has been known for centuries, and methods for its solution have a long history in classical mathematical physics. Some of these methods are standard for diffusion in materials, such as the basic solutions of Gaussian functions and error functions for one-dimensional problems. This Chapter 3 also presents the method of separation of variables for three-dimensional problems with Cartesian and cylindrical coordinates. The Laplacian is separable in nine other coordinate systems, each with their own special functions and orthogonality relationships, but these 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 prove more efficient than analytical methods. Because diffusion depends on atomic-scale processes, changes in the local atomic structure during diffusion can depreciate the diffusion equation because the “diffusion constant,” D, is not constant. This can be a serious problem when using the diffusion equation to describe the kinetics of a phase transformation. By deriving the diffusion equation from the kinetic master equation, however, we can later replace the assumption of random atomic jumps with an assumption of chemically-biased jumps to predict the kinetics of chemical ordering or mixing. This is the subject of Chapter 22. This Chapter 3 concludes by showing how the kinetic master equation can lead to thermodynamic equilibrium.

1

56

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

The Diffusion Equation

57

3.1 The Diffusion Equation Writing the kinetic master equation in the form of Eq. 1.25 motivates a matrix description 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, and ∼

W (∆t) is a two-dimensional matrix that gives the new contents after time ∆t. Two ≈

such matrix elements are shown in Fig. 1.9b. This approach has an advantage for numerical computations. If ∆t is small, after m intervals of ∆t the new contents of the bins will be  m (3.2) W (∆t) N(t) = N(t + m∆t) . ∼

≈

∼

The following assumptions are fundamental to the diffusion equation, and to our construction of a kinetic master equation for diffusion. They are important to remember whenever using the diffusion equation for a problem in materials science. • all atoms have the same jump probability (unaffected by the presence of other atoms) • if an atom has probability δ of jumping out of a bin in Fig. 1.9, it has an equal probability δ/2 of going left or right (in three dimensions the probability is shared 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 assumption for obtaining the diffusion equation as shown by Problem 2 in Chapter 9) For our matrix equation we first arrange the two column vectors in correspondence with the bins in Fig. 1.9a, {n}, and their contents {N} h i 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 case ≈

W (∆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 of ∼

all 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 the time interval ∆t. The probability of it remaining in the bin is therefore 1 − δ, and its probability of entering an adjacent bin is δ/2. Likewise, the probability of an atom entering 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 or discontinuously. The discontinuous case begins with the appearance of a small but distinct volume of material having a structure and composition that differ from those 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 rearrange the atoms of a liquid into a crystal, for example. The new crystal must appear in miniature in the liquid, a process called “nucleation.” If the nucleation event is successful, this crystal will grow. The process of nucleation is an early step for most phase transformations in materials. It has many variations, but two key concepts can be appreciated immediately.

• Because the new phase and the parent phase have different structures, there must be an interface between them. The atom bonding across this interface is not optimal,2 so the interfacial energy must be positive. This surface energy is most significant when the new phase is small, because a larger fraction of its atoms 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 new phase differs in shape or specific volume from the parent phase. The mismatch creates an elastic field that costs energy. This is not an issue for nucleation in a 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 diffusion of the different chemical species of atoms, and this time can be long if atoms must move long distances between incipient nuclei (sometimes called “embryos”). The addition of atoms to embryos is largely a kinetic phenomenon, although the tendency of atoms to remain on the embryos is a thermodynamic one. A steadystate rate of forming “critical” nuclei that can grow is calculated. The chapter ends with a discussion of the transient time after a quench from high temperature when the distribution of solute relaxes towards the equilibrium distribution for steady-state nucleation. 1 2

80

The nucleus may or may not have the structure and composition of the final phase because the transformation may occur in stages. If the structure of the interfacial atoms and bonds were favorable, the new phase would take this local atomic structure.

Terminology and Issues

81

t

Fig. 4.1

Binary phase diagram depicting a quench path from a temperature with pure α-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 AB alloy having the α-phase at high temperature, as shown in the unmixing phase diagram of Fig. 2.10. Suppose the alloy is quenched (cooled quickly) to a temperature such as 0.4zV/kB , where the equilibrium state would have mostly A-rich α′ -phase, plus some B-rich α′′ -phase. For some compositions, the B-rich α′′ -phase may nucleate as small zones or “precipitates” in an A-rich matrix.3 In this case, the underlying crystal lattice remains the same while the solute atoms coalesce. A different case is shown with Fig. 4.1 for the unmixing of solute in a eutectic alloy. Here the precipitation of β-phase in an alloy cooled rapidly from the α-phase requires both a redistribution of chemical elements and a different crystal structure. In both these examples of nucleation, the parent phase is “supersaturated” immediately after the quench, and is unstable against forming the new phase. “Homogeneous” nucleation occurs when nuclei form randomly throughout the bulk material; i.e. without preference for location. “Heterogeneous” nucleation refers to the formation of nuclei at specific sites. In solid→solid transformations, heterogeneous nucleation occurs on grain boundaries, dislocation lines, stacking faults, or other defects or heterogeneities. When freezing a liquid, the wall of the container 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 surrounding matrix. Figure 4.2a illustrates an incoherent nucleus. The precipitating β-phase has a 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.

Effects of Diffusion and Nucleation on Phase Transformations

5

A phase diagram is a construction for thermodynamic equilibrium, a static state, and therefore contains no information about how much time is needed before the phases appear with their equilibrium fractions and compositions. It might be assumed that the phases found after practical times of minutes or hours will be consistent with the phase diagram, since most phase diagrams were deduced from experimental measurements on such time scales.1 However, a number of nonequilibrium 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 often cause deviations from equilibrium, and some of these nonequilibrium effects are described in this Chapter 5. In general, first one seeks to understand the effects of the slowest processes. For faster heating or cooling, however, sometimes the slowest processes are inactive, so the next-slowest processes become important. Approximately, this Chapter 5 follows a course from slower to faster kinetic processes.

1

Box 5.1

On the other hand, this does not necessarily mean that all phases on phase diagrams are in fact equilibrium phases. Exceptions are found, especially at temperatures below about half the liquidus temperature.

Thermodynamics and Kinetics Two necessary conditions for a phase transformation to occur in a material are • 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 phase changes in materials occur by diffusional motions of atoms. 103

Effects of Diffusion and Nucleation on Phase Transformations

104

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 higher temperatures. An important consideration is the characteristic time τ for diffusion over 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 effective cooling rate. Table 5.1 shows some kinetic phenomena that are suppressed by cooling at increasingly rapid rates. Insights into the phenomena listed in Table 5.1 are obtained with a rule of thumb that diffusion coefficients near the melting temperature of many (metallic) materials are D(Tm ) ∼ 10−8 cm2 /s.2 For an interatomic distance of typically 2 × 10−8 cm, Eq. 5.1 gives a characteristic time of 4 × 10−8 s. For ultrafast cooling, it seems possible to suppress all atom diffusion in solids below the melting temperature, suppressing crystal nucleation and growth. This allows the formation of amorphous metallic elements, but these are highly unstable and may persist only at cryogenic temperatures. At 2/3 of the melting temperature, the time scale for suppressing atom motion at the atomic scale is increased by a factor of 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 and thermal conductivity, which vary with the particular material and the equipment. Like the diffusion of atoms, the diffusion of heat has a quadratic relationship between 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 in Table 5.1 make thicknesses of materials that are proportional to the square root of the inverse cooling rate. Samples from melt spinning are tens of microns thick, and samples from laser surface melting are often less than 0.1 µm, for example. Iced brine quenching is an older technique, where a sample at elevated temperature is quickly immersed in a solution of rocksalt in water, cooled with ice. The salt serves to elevate the boiling temperature and improve the thermal conductivity. The rate of cooling depends on the thickness of the sample. The quench rate also depends 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.

Non-Equilibrium Processing of Materials

105

Table 5.1 Cooling Rates, Methods, Typical Kinetic Phenomena infinitesimal 10−6 K/s slow 100 K/s 101 K/s 103 K/s medium 104 K/s 105 K/s fast 106 K/s 109 K/s ultrafast 1010 K/s 1011 K/s − − −

geologic cooling casting iced brine quench melt spinning piston-anvil quench laser surface melting

equilibrium (sometimes) suppressed diffusion in solid dendrites suppressed precipitation suppressed diffusion in liquid extended solid solubility metallic glasses amorphous elements

physical vapor deposition shock wave high-energy ball milling heavy ion irradiation

melting nanocrystallinity, glass formation chemical mixing, glass formation

suppress thermal contact to the water bath. Stirring the mixture can improve the cooling rate. In melt spinning, a steady stream of liquid metal is injected onto the outer surface of a spinning wheel of cold copper, for example. The liquid cools quickly when in contact with the wheel, making a solid ribbon that is thrown off the wheel and spooled. This method is suitable for high volume production. The liquid metal should have modest wetting of the spinning wheel, and optimizing the parameters of the system can be challenging. Piston-anvil quenching, sometimes known as “splat quenching,” uses a pair of copper plates that impact a liquid droplet from two sides. The alloy is typically melted by levitation melting in an induction coil. When the radiofrequency heating current is stopped, the liquid droplet falls under gravity past an optical sensor that triggers 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 continuous laser. The sample may be moved in a raster pattern under laser illumination so a significant area can be treated. Once melted, the surface is cooled by the underlying solid material, and the thinner the melted region, the faster the cooling. Physical vapor deposition may use a high temperature heater to evaporate a material under vacuum. The evaporated atoms move ballistically towards the cold surface 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 rate is not too rapid and the substrate is isolated thermally from the hot evaporator.

PART II

THE ATOMIC ORIGINS OF THERMODYNAMICS AND KINETICS Free energy is a central topic of this book because a phase transition occurs 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 free energy on composition or order parameter to obtain thermodynamic phase diagrams. Chapters 3 and 4 discussed the kinetics of diffusion and nucleation, which can be calculated with an activated state rate theory that uses a free energy of activation. Chapter 5 showed how the free energies of equilibrium phases and the free energies of activation give rise to competition between thermodynamic and kinetic phenomena in phase transformations such as alloy solidification, glass formation, and thin film reactions. The Gibbs free energy is G = E − TS + PV . Chapter 6 discusses the sources of energy of materials that are important for phase transitions. The next Chapter 7 addresses the important sources of entropy, and Chapter 8 discusses effects of pressure. Finally, Chapter 9 explains chemical effects on diffusion in alloys, which depend on 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 the thermodynamics of materials phases and materials microstructures, and some techniques for calculating them. It begins with the chemical bond between two atoms – a fundamentally quantum mechanical phenomenon that depends on the coherent interference of an electron wavefunction with itself, giving an electron density that is not a linear sum of densities from two separate atoms. In a periodic solid or in a large box for electrons, the number of electron states depends on a wavevector k, which can be used to obtain the spectrum of electron energies. The concepts presented here are important, but quantitative results require quantum chemical computer calculations. At a more general, but more phenomenological level, interatomic potentials are described and used to explain the elastic behavior of solids. The elastic energy of a misfitting solid particle in a matrix is discussed, and this misfit energy is generally important for precipitation reactions in solid materials. Surface energy is also described, along with the Wulff construction for predicting the shapes of crystals and precipitates.

6.1 Molecular Orbital Theory of Diatomic Molecules 6.1.1 Interacting Atoms Start with two isolated atoms, A and B. There are states for a single electron about each atom of energy ǫA and ǫB , set by the Schrodinger ¨ equations ~2 2 ∇ ψA (~r) + VA (~r)ψA (~r) − ǫA ψA (~r) = 0 , (6.1) 2m ~2 2 ∇ ψB (~r) + VB (~r)ψB (~r) − ǫB ψB (~r) = 0 , (6.2) − 2m where ψA and ψB are single-electron wavefunctions at atoms A and B. As isolated atoms, their nuclei are far apart. Now bring the nuclei close enough together so their wavefunctions overlap. Our goal is to understand what the individual electrons do in the presence of both atoms, and understand the chemical bond in the new diatomic molecule. We seek single-electron wavefunctions for the diatomic molecule. The potential −

137

Energy

138

proves to be a real challenge because the potential for one electron depends on the presence of the second electron. The effect of the second electron is to push around the first electron, but this alters the potential and wavefunction of the second electron. Iterative methods are the most accurate for this problem, but here assume 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 electron transfers between atoms, which alter the atomic potentials. The approach works best when the overlap of the atom wavefunctions is small, and the potentials tend to retain their original character. We make a related assumption that a single electron is in a wavefunction ψ constructed from the original atomic wavefunctions ψ(~r) = cA ψA (~r) + cB ψB (~r) .

(6.4)

It is important to remember that ψ pertains to a single electron, so the coefficients cA and cB are less than 1 (the atomic wavefunctions ψA and ψB accommodate one electron each). This ψ is a “molecular orbital” for one electron. We started with two electrons though, so we need to find two molecular orbitals. To do so, lay out the molecular Schrodinger ¨ equation twice and do two standard tricks: 1) multiply by ψ∗A (~r) and ψ∗B (~r) ~2 ∗ ψ (~r)∇2 ψ(~r) + ψ∗A (~r)V(~r)ψ(~r) − ǫψ∗A (~r)ψ(~r) = 0 , 2m A ~2 ∗ ψ (~r)∇2 ψ(~r) + ψ∗B (~r)V(~r)ψ(~r) − ǫψ∗B (~r)ψ(~r) = 0 , − 2m B and 2) integrate −

hA|H|AicA + hA|H|BicB − ǫ(cA + hA|BicB ) = 0 ,

hB|H|AicA + hB|H|BicB − ǫ(hB|AicA + cB ) = 0 ,

(6.5) (6.6)

(6.7) (6.8)

where the integrals are written in Dirac notation. Equations 6.7 and 6.8 can be arranged as a matrix equation            hA|H|Ai − ǫ hA|H|Bi − ǫhA|Bi cA  0     =   .  (6.9)                  hB|H|Ai − ǫhB|Ai hB|H|Bi − ǫ cB 0

6.1.2 Definitions and Conventions Before solving Eq. 6.9 for ǫ and then for cA and cB , we evaluate some terms and change notation. The integrals hA|Bi and hB|Ai are not zero – the wavefunctions are centered on different atoms, but the tails of these wavefunctions overlap. These are “overlap integrals,” defined as S S ≡ hA|Bi = hB|Ai .

(6.10)

7

Entropy Without entropy to complement energy, thermodynamics would have the impact of one hand clapping. The epithaph on Boltzmann’s monument shown in Fig. 7.1 S = kB ln Ω ,

(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 number Ω, which counts the ways of finding the internal coordinates of a system for thermodynamicallyequivalent 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 Chapter 1. Configurational entropy in the point approximation was used extensively in Chapter 2, and Section 17.4 accounts for magnetic entropy in essentially the same way. This Chapter 7 shows how the configurational entropy of chemical disorder or magnetic disorder can be calculated more accurately with cluster expansion methods. The other major source of entropy is vibrational entropy, and its origin is explained in Section 7.5. Critical temperatures of ordering and unmixing, calculated previously with configurational entropy alone, are then adapted to include effects of vibrational entropy.

t

Fig. 7.1

175

The Boltzmann monument in Vienna. The constant k is related to kB by the factor 2.3026 if log denotes log10 . Our notation uses Ω instead of W.

Entropy

176

Table 7.1 Sources of Entropy in Materials Matter nuclei electrons electron spins Energy nuclei electrons electron spins

Structural Configurations sites for the nuclei (the electrons will adapt) sites for electrons in mixed-valent compounds orientational disorder (magnetic disorder) Dynamics vibrations (phonons) excitations across Fermi level (electronic heat capacity) spin waves (magnons)

For metals there is a heat capacity and entropy from thermal excitations of electrons near the Fermi surface, but as discussed in Section 6.2.4 this is often a small effect because not many electrons are available for these excitations. The electronic entropy of a metal increases with temperature, but at high temperatures the electronic excitations may interact with phonons, and phonons interact with other 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 configurational ones and dynamical ones. As an example, when the number Ω enumerates the ways to arrange atoms on the sites of a crystal, the method to calculate Ω does not depend on temperature. We used this idea extensively in Chapter 2. On the other hand, when the number Ω for dynamical coordinates counts the intervals of volume explored as atoms vibrate, this Ω increases with temperature, as does the vibrational entropy. Configurational entropies of atoms or spins undergo changes during chemical ordering or magnetic phase transitions, respectively.1 Configurational entropy was largely understood by Gibbs, who presented some of the combinatoric calculations of entropy that are used today (23). The calculation of Ω is more difficult when there are partial correlations over short distances, but cluster approximation methods have proved powerful and accurate (53)-(55), and are presented in Section 7.2. In essence, new local variables are added to the list of composition and long-range1

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

Short-Range Order and the Pair Approximation

177

order parameter to describe more precisely the atom configurations on lattice sites. Again, although the configurational variables have different equilibrium values at different temperatures, temperature does not alter the combinatorial method for calculating Ω with the configurational variables. Dynamical entropy grows with temperature as dynamical degrees of freedom of a solid, such as normal modes of vibration, are excited more strongly by thermal energy.2 With increasing temperature more phonons are created, and the vibrational excursions of atomic nuclei are larger. Fundamentally, the entropy from dynamical sources increases with temperature because with stronger excitations of dynamical degrees of freedom, the system explores a larger volume in the hyperspace3 spanned by position and momentum coordinates, as discussed in Section 7.5. This volume, normalized by a quantum volume if necessary, is the Ω for Eq. 7.1. For a phase transition, what is important is not the total vibrational entropy so much as the difference in vibrational entropy between the two phases.4

7.2 Short-Range Order and the Pair Approximation Section 2.10 presented a thermodynamic analysis of the order-disorder transition in the point approximation, which assumed that all atoms on a sublattice were distributed randomly. This assumption is best in situations when 1) the temperature is 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) a hypothetical case when the coordination number of the lattice goes to infinity. For more interesting temperatures around the critical temperature, for example, it is possible to improve on this assumption of sublattice randomness by systematically allowing for short-range correlations between the positions of atoms. For example, a deficiency of the point approximation for ordering is illustrated with Fig. 7.2. We see that the numbers of A-atoms and B-atoms on each sublattice are equal, so the LRO parameter L = (R−W)/(N/2) = 0. Nevertheless, there is obviously a high degree of order within each of the two domains. It might be tempting to redefine the sublattices within each domain, allowing for a large value of L, but this gets messy when the domains are small. The standard approach to this problem is 2

3 4

Temperature also drives electronic excitations to unoccupied states, and when many states are available the electronic entropy is large. Spin excitations are another source of entropy, but care must be taken when counting them if the configurations of spin disorder are already counted. This is frequently called a “phase space,” not to be confused with geometric properties of crystallographic phases. The “Kopp–Neumann rule” from the nineteenth century states that the heat capacity of a compound is the sum of atomic contributions from its elements. By this rule, the vibrational entropy of a solid phase depends only on its chemical composition, and not on its structure. This rule is not helpful for understanding the thermodynamics of phase transitions. Furthermore, the Kopp–Neumann rule has inconsistencies when picking an atomic heat capacity for carbon, for example.

8

Pressure Historically there has been comparatively little work on how phase transitions in materials depend on pressure, as opposed to temperature. For experimental work on materials, it is difficult to achieve pressures of thermodynamic importance, whereas high temperatures are obtained easily. The situation is reversed for computational work. The thermodynamic variable complementary to pressure is volume, whereas temperature is complemented by entropy. It is comparatively easier to calculate the free energy of materials with different volumes, as opposed to calculating all different sources of entropy. Recently there have been rapid advances in high pressure experimental techniques, often driven by interest in the geophysics of the Earth. Nevertheless, new materials are formed under extreme conditions of pressure and temperature, and some such as diamond can be recovered at ambient pressures. The use of pressure to tune the electronic structure of materials can be a useful research tool for furthering our understanding of materials properties. Sometimes the changes in interatomic distances caused by pressure can be induced by chemical modifications of materials, so experiments at high pressures can point directions for materials discovery. This Chapter 8 begins with basic considerations of the thermodynamics of materials under pressure, and how phase diagrams are altered by temperature and pressure together. Volume changes can also be induced by temperature, and the concept of “thermal pressure” from non-harmonic phonons is explained. The electronic energy accounts for most of the PV contribution to the free energy, and there is a brief description of how electron energies are altered by pressure. The chapter ends with a discussion about using pressure to investigate kinetic processes, and the meaning of an activation volume.

8.1 Materials under Pressure at Low Temperatures The behavior of solids under pressure, at least high pressures that induce substantial changes in volume, is more complicated than the behavior of gases. Nevertheless, it is useful to compare gases to solids to see how the thermodynamic extensive variable, 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 = NkB T .

(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 because the molecules themselves take up space. The quantity V in Eq. 8.1 is replaced by V − Nb, where b is an atomic parameter with units of volume. • An attractive interaction between the gas molecules tends to increase the pressure a bit. This can be considered as a surface tension that pulls inwards on a group of gas molecules. The quantity P in Eq. 8.1 is replaced by P + a(N/V)2. The quadratic dependence of 1/V 2 is expected because the number of atoms affected goes as 1/V, and the force between them may also go as 1/V. (Also, if the correction went simply as 1/V, it would prove uninteresting in Eq. 8.1.) The Van der Waals equation of state (EOS) is h

P+a

i N2 ih V − Nb = NkB T . 2 V

(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 this dimensionless form T 1 P= − , (8.3) V − 1 V2 with the definitions

b2 , a V , V≡ Nb b T ≡ kB T . a P≡P

(8.4) (8.5) (8.6)

Figure 8.1 shows the Van der Waals EOS of Eq. 8.2 for a fixed a and b, but with varying temperature. At high temperatures the behavior approaches that of an ideal gas, 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 1 a , 27 b2 8 a , = 27 b

Pcrit = kB Tcrit

(8.7) (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

t

Fig. 8.1

Pressure

(a) Isothermals of the Van der Waal’s equation of state Eq. 8.2, plotted with rescaled variables of Eqs. 8.4 - 8.6. (b) Maxwell construction for T = 0.26a/(bkB). At lower temperatures, such as kB T = 0.26a/b shown in Fig. 8.1b, the same pressure 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 (and likewise, the material shrinks if pressure is reduced). Nevertheless, the volumes V1 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 that the chemical potentials of the gas and liquid are equal in equilibrium, i.e., µ3 = µ1 . R 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 1 µ3 = µ1 + N

Z

V3

P(V) dV .

(8.10)

V1

The integral must be zero if µ3 = µ1 . The areas above and below the horizontal line in Fig. 8.1b must therefore be equal, and this “Maxwell construction” defines the pressure of the horizontal line. A dimensionless ratio can be formed from Eqs. 8.7, 8.8, 8.9 3 Pcrit Vcrit = . NkB Tcrit 8

(8.11)

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

211

Materials under Pressure at Low Temperatures

Table 8.1 Pressures and Temperatures of Gases and Solids Gas

Solid

Pressure

P>0

P > −Pcoh

Temperature

T>0

T≥0

Stresses

isotropic

anisotropic

Typical Pressure

1 Atm = 0.1 MPa

10 GPa = 105 Atm

This instability of the Van der Waals EOS below a critical temperature can be used to model a pressure-induced liquifaction, for example, or the liquid-gas phase boundary at constant pressure. The approach has problems with quantitative details, but it gives the essential behavior, and is worthy of more study than given here.

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 finite volume. Table 8.1 shows that on the scale of familiar pressures in gases, a solid is essentially incompressible. More familiar are small compressions of solids and elastic behavior, where typical materials deform as springs. The bulk modulus of a solid, B, dP , (8.12) B ≡ −V dV is typically a few times 100 GPa, and the elastic energy per unit volume under uniform dilation is 1 Eel = B δ2 , (8.13) 2 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 from second derivatives of the interatomic potentials, which give tensorial “springs” between atoms as explained in Sect. 6.3. These springs are loaded in different directions when different stresses are applied to a material, but all strains are linear with 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, 477 activated state and equilibrium, 78 activated state rate theory, 76 activation barrier, 529 activation energy, 223, 224, 257 activation entropy, 257 activation volume, 223 activity (chemical), 431 adatom, 288 adiabatic approximation, 576 Ag, 413 Al, 413, 563 Al-Li, 201 Al-Pb, 87 allotriomorph, 86, 172, 301 alloy phases, 21 alloy supercooling, 119 alumina, 126 aluminum, 126 amorphization, 122 amorphous phase, 130, 414 anharmonicity, 204, 571, 574 anisotropy, 162, 211 anisotropy energy, 459 anisotropy gap energy, 455 annealing, 279 anomalous diffusion, 15 antibonding, 139, 217 antiferromagnetism, 434, 452 antiphase boundary profile, 347 antiphase domain boundary, 178, 234, 344 antisite atom, 487 antisymmetric wavefunction, 439 APB width, 344 aspect ratio, 300 athermal, 389 atom configurations, 34 atomic ordering, 47 Atomium, 405 attempt frequency, 257, 529 Au, 413 austenite, 304, 387 austenitizing, 309 autocatalytic, 304 autocorrelation function, 545 average, 157

595

Avrami equation, 312 b(k), 366 B2, 181, 359, 376, 378, 496, 538 second order, 372 B2 structure, 359 B32, 359, 496, 538 Ba, 267 bainite, 305, 310 ball milling (high energy), 106 ballistic jump, 249, 254, 542 band structure, 147, 152, 154, 220 basis vector, 358 BaTiO3 , 457 bcc, 357, 490, 496 2nn forces, 404 BCS theory, 471 Becker and D¨oring, 91 Bessel function, 72 bin, 531 binary alloy, 33 binomial expansion, 35, 195 binomial probability, 258 bin, 18 Bloch T 3/2 law, 452 Bloch wall, 428, 458 Bloch’s theorem, 146, 450 blocking temperature, 429 Bogers–Burgers double shear, 392 Boltzmann, 175 Boltzmann factor, 16, 221 Boltzmann substitution, 65 bond counting, 298 bond energy, 141 bond integral, 139, 147 bond length, 201, 203 bond proportion model, 193 bond stiffness, 194 chemical effects, 204 bond stiffness vs. bond length model, 201 bonding, 139, 153, 217 Born–Oppenheimer approximation, 190, 473 Born–von K´arm´an model, 190, 194, 560 Bose-Einstein condensation, 462 Bose-Einstein factor, 462 bosons, 188, 451, 462 boundary conditions, 66, 95, 149

Index

596

boxel, 14 Bragg–Williams approximation, 47, 50, 199 branch, 562 Brillouin zone, 355, 562 bulk modulus, 157, 211, 279, 566 Burgers vector, 380 conservation of, 384 CV and CP , 566 Cahn, John Werner, 326 Cahn–Hilliard equation, 324 calculus of variations, 326, 520 calorimetry, 407, 566 carbon dioxide, 9 carbon nanotubes, 419 casting, 340 Cauchy’s first law, 521 Cayley tree, 461 Ce, 8, 225, 575 cementite, 304 characteristic width, 347 charge density, 440 chemical bonding, 137 chemical bonding energy, 34 chemical environment, 528, 529 chemical factor, 247 chemical potential, 17, 244, 246, 462 chemical spinodal, 333 chemisorption, 424 chessboard, 182, 453, 488 classical vs. quantum behavior, 469 Clausius–Clapeyron equation, 215, 408 cluster, 91, 93 cluster expansion, 181, 204 Co, 413 coarse-graining, 507 coarsening, 302, 313, 352 coherence, 477, 482, 554 coherent spinodal, 333 coincidence boundary, 412 collector plate, 301 collision cascade, 106 collisions, 93, 244 colonies, 306 combinatorial factor, 204 common tangent construction, 30 commutation, 554 completeness, 555 complex conjugate, 365 complex exponential, 359 complex material, 438 composition fluctuation, 250, 319, 321 compressibility, 211 computer program complexity and entropy, 480 computer simulation, 300, 343 concentration gradient, 330

concentration profile, 114 concentration profile, stability of, 319 concentration wave, 354 infinitesimal, 374, 376 stability, 376 condensate wavefunction, 468 conduction electron polarization, 456 conductivity, 245 configurational coordinates, 176 configurational energy, 35 configurational entropy, 35, 37, 177 nanostructure, 419 confined electrons, 418 confocal ellipsoidal coordinates, 56 conservation of solute, 29 conservative dynamics, 348 constititutional supercooling, 119 constraint, 44, 328, 530 continuous, 80 convection, 56 convolution, 376, 524 cooling rate, 104 Cooper pair, 4, 471, 570 cooperative transition, 447, 501 coordinate systems, 56, 71 coordinates, 175 coordination number, 34 core electron polarization, 442 correlated electrons, 444 correlation entropy of liquid, 274 correlation factor, 233, 257, 259 heterogeneous alloy, 232 table, 231 tracer, 230 correlation function, 182 correlation length, 503, 506 Coulomb correlation, 445 Coulomb energy, 441 covalency, 142, 143 Cr, 575 critical condition, 209 critical field, 477 critical free energy of formation, 84 critical nucleus, 82, 101 critical point, 8 critical radius, 84, 126 strain effects, 90 critical slowing down, 504 critical temperature, 38, 39, 50, 193, 198, 375, 448, 501, 534 change with ∆Svib , 193 displacive transition, 399 ordering, 200, 203 unmixing, 195, 197, 250 critical undercooling, 98

Index

597

critical wavevector, 332 cry of tin, 387 crystal field theory, 139, 218 crystallization, 130, 371 crystallographic variant, 400 CsCl, 359 Cu, 267, 413 Cu3 Au, 359 Cu-Al, 387 Cu-Co, 13, 82 Cu-Zn, 387 CuAu, 359 CuPt, 378 Curie temperature, 304, 434, 447, 448, 456 Curie-Weiss law, 449 curvature, 294, 348, 351 curvilinear coordinates, 350 CuZr, 130 cylindrical coordinates, 72 Debye model, 565 Debye–Waller factor, 556 decimation, 507 deformation potential, 473 degrees of freedom, 42, 252, 507 microstructure, 419 dendrite, 119, 340 density, 502 density of states, 147, 149, 451 determinant, 139 deuterium, 247 devitrification, 130 diatomic molecule, 137 diffractometry, 525 diffusion, 12, 104, 108, 313, 330, 387, 552 anomalous, 15 chemical potential, 314 distance, 66 ideal gases, 236 in inhomogeneous alloy, 236 liquid, 113 marker velocity, 237 zone, 126 diffusion coefficient, 236, 257, 545 diffusion equation, 59 diffusionless transformation, 387 dilation, electron energy, 472 dilute impurity, 53 dimensionality, 101, 349, 419 dimensionless integral, 452 Dirac δ-function, 61, 63 Dirac notation, 138 discontinuous, 80 dislocation, 82, 283, 291, 303, 380, 411 Burgers vector, 380 core, 382 core radius, 283

edge, 380 fcc and hcp, 383 glide, 381 groups of, 411 mixed, 381 partial, 384 plastic deformation, 380 reactions, 383 screw, 293, 380 self energy, 382 tilt boundary, 411 dispersion, 562 dispersion forces, 424 displacement parameter, 400 displacement vector, 160 displacive ferroelectric, 437 displacive instability, 396 dissipation, 544, 548 distribution coefficient, 110 divacancy, 491 divergence, 350, 502 DO3 , 359, 538 domain, 178, 458 domain boundary, 348 domain wall, 476 driven systems, 249 driving force, 77 Duwez, Pol, 281 dynamical coordinates, 176 dynamical matrix, 563 Dzyaloshinskii–Moriya interaction, 456 Edisonian testing, 6 Ehrenfest’s equations, 216 eigenvalues, 564 eigenvectors, 564 of dynamical matrix, 563 Einstein convention, 518 Einstein model, 194, 195, 398 elastic, 160 anisotropy, 429 energy, 212, 298, 332, 429, 517, 567 field, 304 scattering, 556 waves, 562 electric dipole moment, 456 electrical conductivity, 153, 245 electron energy, 157 electron gas, 148 electron levels, 220 electron transfer, 143 electron-phonon coupling parameter, 474 electron-phonon interaction, 472, 575 electron-phonon interactions, 569, 575 electron-to-atom ratio, 26, 154 electronegativity, 26, 143, 144, 204 electronic DOS, 147, 153, 474, 575

Index

598

electronic energy dimensionality, 418 electronic entropy, 568 electronic heat capacity, 151 electronic oxides, 446 electronic structure, 22 electrons, 3 Eliashberg coupling function, 474 embryo, 80 emergent phenomenon, 4 energy, 17, 38, 196 elastic, 163 surface, 83, 170 energy gap, 475 energy landscape, 252, 279 energy versus matter, 463 ensemble average, 228, 545 enthalpy, 52 entropy, 16, 17, 38, 52, 196, 388 configurational, 52, 388 electronic, 52, 151 magnetic, 52, 388 microstructure, 410 vibrational, 191, 388, 559 equation of state, 209 equations of motion, 561 equiaxed, 91 equilibrium, 533 kinetic, 75, 77 non-thermodynamic, 249 thermodynamic, 74 equilibrium interatomic separation, 158 equilibrium shape, 84 error function, 65 Eshelby cycle, 164 Euler equation, 328, 345 eutectic, 42, 54, 107 eutectoid, 45, 305 evaporation, 116 exchange energy, 428, 439 spin, 508 exchange hole, 441 exchange interaction, 447, 514 extreme conditions, 217 fcc, 357, 490 Fe, 26, 413, 442 nanocrystal, 423 Fe3 Al, 359 Fe3 O4 , 436 Fe-C, 387 Fe-Cr, 13 Fe-Ni, 388, 390, 558 Fermi energy, 150 level, 154, 456 surface, 153, 474, 577

wavevector, 150 Golden Rule, 554 Fermi-Dirac statistics, 373 ferrimagnetism, 434 ferrite, 304 ferroelectric, 437, 456 ferromagnet, 507 ferromagnetism, 434, 442, 447 Fick’s laws, 59 finite element methods, 56 first Brillouin zone, 355 first phase to form, 127 first-order transition, 7 fluctuation, 82, 319 concentration, 331 fluctuation–dissipation theorem, 549 fluid, 56, 505 flux, 60, 61 flux of atoms, 330 fluxon, 477 force, 244 force constants, 560 forging, 309 Fourier Series, 66 Fourier transform, 376 Fourier-Bessel expansion, 73 Fowler correction factor, 179 Frank–Van der Merwe growth, 295 free electron model, 148, 156 free energy, 27 alloy phases, 23 curvature, 39 k-space, 364 phonon, 190 spinodal unmixing, 249 thermal expansion, 568 free energy vs. composition, 27 freezing, 6, 7, 107, 263 Friedel, 284, 407 Friedel oscillation, 456 frustration, 435 functional, 327 G-P zone, 285 gases, 209 Gauss integrals, 253, 268 Gauss’s theorem, 521 Gaussian, 96, 553 Ge, 575 generating function, 259 Gibbs, 91, 176 Gibbs free energy, 18 Gibbs phase rule, 42, 45, 338 Gibbs–Thomson effect, 415 Gibbs-Thomson effect, 89 Ginzburg–Landau Equation, 341, 347, 476 glass, 120, 129

Index

599

strong or fragile, 131, 280 transition, 129 glide, 381 Goodenough-Kanamori rules, 455 Gruneisen ¨ parameter, 201, 203, 214, 224, 571 gradient, 60, 350 gradient energy, 322, 323 grain boundary, 85, 171, 301, 410, 411 energy, 412 width, 412 grain boundary allotriomorph, 86, 172 gravity, 225 Green’s function, 64, 517, 524 ground state, 464 ground state maps, 490 growth equation, 310 growth rate interface, 117 Guinier-Preston zones, 90 H+ molecular ion, 142 2 H2 , 465 H2 molecule, 173 habit plane, 388 Hamiltonian, 560 hard core repulsion, 158 harmonic approximation, 192, 560, 571 harmonic model, 188, 256 harmonic oscillator, 190, 397 Hartree-Fock wavefunction, 440 hcp, 408 He, 463 He II, 467 heat, 557 heat capacity, 216 quasiharmonic, 566 heat of formation, 28 heat treating, 309 Heisenberg model, 282, 439, 450 Helmholtz free energy, 17, 196 heterogeneous nucleation, 81, 303 heteronuclear molecule, 142 HfO2 , 575 high vacuum evaporation, 116 high-temperature expansion, 497 high-temperature limit, 192 homogeneous nucleation, 81 homogeneous precipitate, 302 Hooke’s law, 162, 518 Hubbard model, 445 Hume–Rothery rules, 25 Hund’s rule, 442 hydrogen, 15, 188, 424 hydrogen atom, 173 hyperbolic functions, 499 hypereutectoid steel, 309 hyperfine interaction, 4, 442

hyperspace, 185, 186, 252 hypoeutectoid steel, 309 hysteresis, 388 ice, 224 iced brine quenching, 104 ideal gas, 209, 246 ideal solution, 38 imaginary frequency, 398 incoherent approximation, 556 inconsistency pair approximation, 178 incubation time, 99 independent variables, 537 indistinguishability, 274 inelastic scattering, 544, 551 infinitesimal displacement, 399 infinitesimal order, 370 initial conditions, 66 instability eigenvector, 404 insulator, 153 integration by parts, 327 interatomic potential, 158 interchange, 528 interchange energy, 36 interdiffusion, 126 interface concentration, 126 interface energy, 297, 324 interface velocity, 128 interference, 468 intergranular fracture, 303, 339 internal interface, 303 internal stress, 430 interstitial, 15, 387 interstitialcy mechanism, 15 Invar, 222 inversion, 361 ion beam bombardment, 106 ionicity, 142, 143 Ising lattice, 10, 195, 198, 282, 420, 446 island growth, 295 isoelectronic, 143 isothermal, 389 isothermal compressibility, 216 isotope, 188 isotopic fractionation, 206 isotropy, 162 iteration, 511 Josephson junction, 470 jump rate, 529 jump sequence, 230 jumping beans, 258 K, 157, 172, 267 k-space, 149, 358 k-vector as quantum number, 145 Kauzmann paradox, 130

Index

600

KCl, 173 kinetic arrest, 542 kinetic energy, 142, 156, 254, 575 kinetic master equation, 18, 74, 531 kinetic path, 536, 541 kinetic stability, 410 kinetics, 5, 76, 226, 527 kink, 288, 481 Kirkendall voids, 237 Kopp–Neumann rule, 177 Kosterlitz–Thouless transition, 282 Kurdjumov–Sachs, 390 L10 structure, 359 L11 , 378 L12 , 359 first order, 371 laboratory frame, 237 Lagrange multiplier, 328 lamellar spacing, 308 Landau theory, 396 first order, 401 function, 397 potential, 402 vibrational entropy, 402 Landau-Lifshitz criterion, 370 Langevin equation, 548 Langmuir isotherm, 425 Laplacian, 56, 350 separable, 71 laser processing, 105 latent heat, 7, 19, 263, 306 lattice connectivity, 513 lattice dynamics, 560 lattice gas, 431 lattice mismatch, 82 Laue condition, 357 layer-by-layer growth, 295 ledeburite, 305 ledge, 288 ledges and growth, 287, 293 length scale, 512 Lennard-Jones potential, 158, 172 lenticular precipitate, 171 lever rule, 27, 107 differential form, 110 levitation melting, 105 Li, 275 Lifshitz, Slyozov, Wagner, 315 Lindemann rule, 277 linear elasticity, 160 linear oxidation, 125 linear response, 76, 250, 547 linearize near Tc , 533 liquid, 6, 211, 273, 501 liquid free energy, 27 liquidus, 32, 55

local chemical environment, 535 local spins, 447 long-range interactions, 375 long-range order, 48, 198, 506 longitudinal branch, 562 Lorentzian, 553 lower bainite, 305 LRO, 47, 448, 527 Mf and Ms , 389 magnetic field, 44 magnetic flux, 458 nanoparticles, 431 order, 341 phase transitions, 433 susceptibility, 435, 454 systems, 501 torque, 454 magnetism, 222, 427, 476 magnetite, 435, 436 magnetization, 408, 449, 504 magnetocrystalline anisotropy, 429 magnetoelastic energy, 343 magnon, 452 many-body theory, 256 marker velocity, 237 martensite, 12, 304, 310, 379, 407 crystallography, 393 magnetic field effect, 438 martensite transformation mechanism, 392 Martin, Georges, 249 master equation, 18, 74, 527, 528 matrix stress, 166 Maxwell construction, 210 Maxwell relationship, 18, 216 McMillan expression for Tc , 475 mean field approximation, 373, 453 mean-squared displacement, 226 mechanical attrition, 414 melt spinning, 105 melting, 6, 7, 225, 263, 371 atom displacements, 277 melting point, 32 melting temperatures, table of, 266 memory, 229 Mendeleev number, 28 metallic bond, 156 metallic glass, 121, 129 metallic radius, 26 metastable, 288, 319, 541 methane, 424 Metropolis algorithm, 542 Mg, 26 microstructure, 6, 313, 337, 410 in steels, 310 midrib, 389

Index

601

Miedema, A.R., 28 misfit energy, 518 misfitting particle, 164 ellipsoid, needle, plate, 168, 524 Mo, 267, 575 mobility, 77, 245, 349, 536 models, 4, 281 molecular dynamics, 258 molecular orbital, 138 molecular wavefunction, 140 monomer, 93 attachment to cluster, 93 Monte Carlo, 540 morphology, 91 Morse potential, 158 moving boundary, 113 multicomponent alloy, 537 multiphonon scattering, 557 item Na, 157, 212 Nabarro, Frank, 168 nano-dots, 419 nanomaterial, 410 nanostructure, 417 low-energy modes, 422 phonon broadening, 424 NaTl, 359 Nb3 Sn, 387 N´eel temperature, 454 neutron, 554 Newton’s law, 548 Nishiyama, 391 NiTi, 407, 414 Nobel prize, 507 nonconservative dynamics, 348 nonconserved quantity, 341 nonequilibrium, 536 nonequilibrium cooling, 110 nonlinear oscillator, 398 normal coordinates, 190, 252, 573 normal modes, 186, 194, 561 nose, 120 nucleation, 12, 40, 80, 119, 126, 285, 320 and growth, 12, 80, 120, 285 edges and corners, 87 elastic energy, 89 grain boundary, 85 heterogeneous, 81, 85 homogeneous, 81 in concentration gradient, 126 rate, 91, 97 temperature dependence, 97 time-dependent, 98 nuclei, 3 nucleus coherent, incoherent, 81, 82 critical, 82

numerical calculations, 515 occupancy variable, 373 Ohm’s law, 245 Olson–Cohen model, 392 omega phase, 405 Onnes, 471 optical mode, 562 optical phonon, 458 orbitals, 218 order parameter, 338, 361, 529, 537 ordered domain, 178 ordering, 47, 359 orientation relationship, 390 orthogonality Bessel functions, 73 sine functions, 69 overcounting, 34, 194 overlap integral, 138 oxidation, 124 oxides electronic, 446 P´eclet number, 340 pair approximation, 503, 527 consistency, 177 pair potentials, 36 pair variable, 178, 527 parabolic oxidation, 125 paraelectric, 437 parallelopipedon, 525 paramagnetism, 434 parent phase, 83 partial differential equation, 66 partial dislocation, 384 particle in a box, 148, 463 partition function, 15, 190, 487 at low T, 493 configurational, 34, 195 harmonic oscillator, 190 high temperature, 497 ordering, 199 spin, 508, 512 partitioning ratio, 109 passivation of surface, 126 Pauling, Linus, 144 Pd, 413 Pd-V, 201 pearlite, 304 peeling, 296 percolation threshold, 351 periodic boundary condition, 145, 561 periodic box, 451 periodic minimal surface, 351, 352 peritectic, 42 peritectoid, 45 perovskite, 457 Pettifor, David, 28, 143

Index

602

phase, 327, 469 compositions, 107 definition, 336 fractions, 107 phase boundary with T, 495 phase diagram T-c, 9 continuous solid solubility, 32 eutectic, 41, 44 peritectic, 41, 44 unmixing, 38 phase factor, 144, 468 phase field theory, 336 phase space, 274 phase stability, 21 phase transition concept, 10 ordering vs. unmixing, 10 transition vs. transformation, 12 Phillips–Van Vechten, 143 phonon, 544, 554 damping, 424 density of states, 191 dispersion, 562 entropy, 191, 566 free energy, 190 polarization, 563 thermodynamics, 188, 571 phonon dispersion, 575 phonon DOS, 191, 564 phonon-phonon interaction, 569, 575 physical vapor deposition, 105 physisorption, 424 pipe diffusion, 303 piston-anvil quenching, 105 Planck distribution, 191, 565 plastic deformation, 387, 430 plate precipitate, 298, 524 PMN, 437 point approximation, 47, 372, 453, 503, 527, 528, 530 Poisson ratio, ν, 159 polariton, 458 polarization vector, 562 polaron, 445 pole mechanism, 385, 386 polycrystalline material, 420 polymer, 144 position-sensitive atom probe, 12 potential energy, 142, 575 potential gradient, 244 potential well, 254 Potts model, 282 pre-melting, 264 precipitate, 81, 109, 518 ellipsoidal, 89, 168

growth, 108 lens shape, 91 needle, 168 shape, 298, 522 strain effects, 89 precipitate-free zone, 302 preferred structure, 172 pressure, 8, 17, 157, 208 thermal, 214 primitive lattice translation vector, 355 probability, 16 probability density, 468 processing of materials, 6 product rule for derivatives, 520 proeutectoid ferrite, 309 prototype structures, 359 pseudo-binary, 45 pseudostable state, 541 Pt, 413 Pythagorean theorem, 350 Python code, 511 PZT, 437 quadratic formula, 401 quantum dot structures, 296 quantum level separation, 149 quantum matter, 462 quantum volume, 273 quartic, 401 quasielastic scattering, 553 quasiharmonic model, 207, 214, 269, 571 quasistatic, 258 quench, 81 r-space, 358 radiation defects, 542 random solid solution, 34 random walk, 96, 226, 544 rare earth metals, 173 reaction coordinates, 278 recalescence, 264 reciprocal lattice, 355 reciprocal space, 354 reconstruction (surface), 287 recursion relation, 510 reflection, 361 relativistic self energy, 387 relaxation, 544 relaxation (surface), 287 relaxation time, 547 relaxor ferroelectric, 437 renormalization group, 511 rescaling, 509 Richard’s rule, 102, 276 Richards, Allen, Cahn ground states, 490 Riemann zeta function, 452 right and wrong neighbors, 49 rigid band model, 154

Index

603

roots of Bessel function, 73 Rose equation of state, 267 rotation, 361 roton, 468 roughening transition, 291 Ruderman-Kittel-Kasuya-Yosida oscillation, 456 Rushbrooke inequality, 504 Sackur–Tetrode equation, 274 saddle point, 252, 280, 367, 541 scaling of interatomic potential, 267 scaling theory, 505, 507 scattering law, 551 Scheil Equation, 111 Schrodinger ¨ equation, 138, 469 Schwarz P-surface, 351, 352 second sound, 467 second-order phase transition, 216, 370, 400 self-force constant, 561 self-similar, 315, 414, 507 semiconductor, 153 separation of variables, 66 sextic, 401 shape factor strain effects, 90 shape memory alloy, 407 shear transformation, 391 shell model, 576 shock wave processing, 106 Shockley partial dislocation, 384, 385 short-range order, 178, 506 short-range structure, 183 shuffle, 393 Si, 15, 128, 275, 288, 575 singularity, 3, 185, 501, 502 SiO2 , 131 Slater–Pauling curve, 156 Sm, 267 small displacements, 560 SnO2 , 575 social network, 461 soft mode, 404 solid mechanics, 160 solid solution, 10, 25, 33 solid-on-solid model, 292 solid-state amorphization, 122 solidification, 338, 340 practical, 116 solidus, 32, 55 solute conservation, 29, 447 solute partitioning, 121 special points, 363, 367 sphere, misfitting, 165 spherodite, 305 spin, 3, 181 spin exchange energy, 508 spin excitations, 177

spin wave, 450, 454 spin-orbit coupling, 456 spinodal decomposition, 13, 38, 40, 184, 198, 249, 318, 366 spinwave, 434 spiral growth, 293 splat quenching, 105 spring, 163, 212 sputtering, 105 square gradient energy, 13, 298, 477 square lattice, 47, 198, 513 SRO, 180, 527 stability, 153 stabilization of austenite, 392 stacking fault energy, 385 star of wavevector, 361 state variable, 15, 74, 536 static concentration wave, 354 statistical kinetics, 226 statistical mechanics, 35, 188 statistical variations, 551 steady-state, 94 steel, 304, 387 Stirling approximation, 35, 189, 196, 421 Stoner criterion, 443 strain, 160 strain field, 343 Stranski–Krastanov growth, 295 stress, 160 stress-free strain, 518 stress-free transformation strain, 298 striped structure, 488 structure factor rules, 357 structure map, 28 structure of solids, 4 structure-property relation, 183 strukturbericht, 359 sublattice, 47, 198 substrate mismatch, 295 superconductor, 4, 471, 570 superexchange, 455 superferromagnetism, 429 superfluid, 463, 465 superparamagnetism, 429 supersaturation, 81 surface growth, 293 surface energy, 13, 83, 169, 170, 295, 298, 415, 522 chemical contribution, 296 surface reactions, 124 surface-to-volume ratio, 323 symmetry, 6, 400, 562 breaking, 541 operations, 361 tanh, 448 Taylor series, 560 ternary alloy, 537

Index

604

ternary phase diagram, 45 terrace, 288 Th, 275 thermal conductivity, 104 thermal de Broglie wavelength, 274, 463 thermal electronic excitations, 472 thermal expansion, 158, 268, 430, 505, 566, 567 coefficient, 216 free energy, 568 thermal pressure, 214 thermodynamic identity, 17 thermodynamic square, 18 thermodynamics, 5, 15, 52 thin film, 116 growth, 295 reactions, 124, 127 thin plate, 525 third law of thermodynamics, 466 Ti, 405 Ti-Nb, 387 tie-lines, 45 tight binding, 146 tiling, 35 time average, 228 time constants, 535 time-dependent nucleation, 98 time-temperature-transformation diagram, 120 TiO2 , 575 tracer, 230 transient structure, 538 transition metal, 158 transition metal silicides, 128 transition rate, 75, 554 translation, 361 translational invariance, 369 translational symmetry, 144 transverse branch, 562 truncation of cluster expansion, 183 TTT diagram, 120 twinning, 379, 391, 408 mechanism, 385 two-level system, 220 two-phase mixture, 467 umklapp process, 575 undercooling, 120 unit cell, 358 universal behavior, 502 universal potential curve, 267 unmixing, 318, 366 upper bainite, 305 V, 575 V3 Si, 387 vacancy, 230, 288, 491 concentration, 53 diffusion, 56 jump, 528

mechanism, 14 ordering, 535 relaxation, 530 trap, 233, 259 Van der Waals, 209, 424 Van Hove function, 551 Van Hove singularity, 422, 563 vapor pressure, 116 variational calculus, 326 vector identity, 520 velocity of interface, 340 of boundary, 350 velocity-velocity correlation function, 544 Verwey transition, 436 vibrational entropy, 257, 559 concept, 185 vicinal surface, 170, 411 Vineyard, George, 252 virtual phonon, 473 viscosity, 131 void and vacancies, 316 Volmer–Weber growth, 91, 295 vortex tube, 477 W, 413, 575 W-matrix, 75, 531 Walser–Ben´e rule, 129 water, 9 wave, 359 waves in crystals, 355 Weiss theory, 222 Wigner–Seitz radius, 277 Wulff construction, 170 x-ray diffraction, 358 x-y model, 282 Young’s modulus, 159 Zeldovich factor, 93, 96 zero-point energy, 207, 465 zone refining, 132 Zr, 405 ZrO2 , 571, 575