Magnetism in materials originates with electron spins and their alignments. Groups of spins develop patterns and structures at low temperatures through interactions with each other. With temperature, pressure, and magnetic field, these spatial patterns of electron spins are altered, and several trends can be understood by thermodynamic considerations.

This chapter describes how magnetic structures change with temperature. The emphasis is on magnetic moments localized to individual atoms, as may arise from unpaired 3d electrons at an iron atom, for example. The strong intraatomic exchange interaction gives an atom a robust magnetic moment, but the magnetic moments at adjacent iron atoms interact through interatomic exchange interactions. Interatomic exchange interactions are often weaker, having energies comparable to thermal energies. Interatomic exchange is analogous to chemical bonding between pairs of atoms in a binary alloy that develops chemical order. The critical temperature of chemical ordering Tc corresponds to the Curie temperature for a magnetic transition TC, and short-range chemical order above the Tc finds an analog in the Curie–Weiss law for paramagnetic susceptibility above TC. For chemical ordering the atom species are discrete types, whereas magnetic moments can vary in strength and direction as vector quantities. This extra freedom allows for diverse magnetic structures, including antiferromagnetism, ferrimagnetism, frustrated structures, and spin glasses.

As discussed in Sect. 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. 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, 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 a concern for nucleation in a liquid or gas, since the surrounding atoms can flow out of the way.

Figures 1.5c,d and 1.6a,b illustrate the difference between chemical unmixing that occurs by nucleation and growth (the topic of the previous Chapter 11) and spinodal decomposition (the topic of Chapter 12). Nucleation creates a distinct surface between the new phase and the parent phase, and the two phases differ significantly in their chemical composition or structure. In addition to the surface energy, an elastic energy is often important, too.

Spinodal decomposition does not involve a surface in the usual sense because it begins with infinitesimally small changes in composition. Nevertheless, there is an energy cost for gradients in composition, specifically the square of the gradient, since a region with a large composition gradient begins to look like an interface. The “square gradient energy” is an important new concept presented in this chapter, but it is also essential to phase field theory and to the Ginzburg–Landau theory of superconductivity.

At the end of Sect. 2.7 on unmixing phase diagrams, it was pointed out that there are conceptual problems with a free energy that is concave downwards because the alloy is unstable, but the free energy pertains to equilibrium states. An unstable free energy function may prove useful for short times, however. Taking a kinetic approach, we use the thermodynamic tendencies near equilibrium to obtain a chemical potential to drive a diffusion flux that causes unmixing.

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. 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.

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 nonharmonic phonons is explained.

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 a 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. 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.

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. 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 is on deriving T–c phase diagrams from free energy functions F(c, T). 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”, which 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.

The field of phase transitions is rich and 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 now being 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.

Section 6.5 gave an introduction to the elastic energy that is generated in a solid material when an internal region transforms into a new phase of different size or shape. Both the new particle and the surrounding matrix are distorted, and the positive elastic energy tends to suppress the phase change. Elastic energy can be large, and usually influences the thermodynamics, nucleation, growth, and morphology of solid–solid phase transformations, especially at low temperatures. Section 15.4 explained how the selection of a habit plane for a martensite plate is so dominated by the elastic energy that the problem is reduced to a set of geometrical conditions based on the transformation strain. Detailed calculations of the elastic energy are difficult, however, and analytical results are not practical in most cases when the elastic constants of the new precipitate differ from those of the matrix. Even with the assumption that the elastic constants are equal for both phases, the solid mechanics of optimizing the shape of the precipitate for minimum elastic energy is an advanced topic. Crystallographic anisotropy is essential for understanding the orientation relationship between precipitate and matrix, and proper tensorial analysis is required for calculating the elastic energy.

Chapter 21 describes some of the methods for calculating the elastic energy of solid–solid phase transformations. A first approach finds a condition on the elastic field in real space that can guide the search for the minimum elastic energy.

Part III describes the important and established families of phase transitions in materials. Chapters 10–16 describe structural and chemical phase transformations of materials that occur by movements of atoms. These include heterogeneous first-order transitions such as melting and precipitation, and spinodal decomposition and ordering that may occur homogeneously as second-order phase transitions. Martensite and other displacive phase transitions are the subject of Chapter 15, and microstructural and nanostructural aspects of phase transformations are covered in Chapter 16. All these phase transitions involving atom rearrangements are historical figures in the field of materials science, and new phenomena are often explained with reference to them.

Chapter 17 describes some of the major phase transitions involving electrons and spins that occur inside materials. Electronic and magnetic phase transitions can sometimes be understood with similar approaches as phase transformations involving atom rearrangements, but some aspects of electronic or magnetic excitations are not classical. This is emphasized in Chapter 18, which ends by touching on quantum criticality. Research on quantum phenomena such as superconductivity is often reliant on controlling the structures of materials. Likewise, results from condensed matter physics offer new insights into phase transformations in materials.