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.

So far, Part III has described melting, precipitation, and unmixing phase transformations. For alloys, these all require the diffusion of atoms over moderate distances, for which continuum diffusion equations provide much of the essential behavior. Ordering transformations also require the movements of individual atoms, but during ordering the atom movements are over such short distances that a diffusion equation is not appropriate. Nevertheless, the atoms move independently (often by a vacancy mechanism), and the configurational entropy can undergo large changes with only a few jumps per atom.

This chapter describes diffusionless transformations, in which the atoms in a crystal move cooperatively, and the crystal is distorted into a new shape. The atoms may not move at exactly the same time, but the transformation is very fast, and does not require a vacancy mechanism for the motions of individual atoms. Diffusionless transformations include “twinning,” in which a crystal transforms into a different variant of the same type of crystal. Martensitic transformations are changes in crystal structure that occur by shears and dilatations, but again without long-range diffusion. Vacancy migration is not important for either twinning or martensitic transformations. Because the atoms do not move with individual independence, the change in configurational entropy is small. The entropy of martensitic transitions is primarily vibrational, sometimes with magnetic entropy as in the case of iron.

Most phase transformations in materials occur by nucleation and growth, where a small particle of the new phase nucleates in the parent phase, and then grows as atoms diffuse to its surface. Especially at low temperatures, the first new particles can be quite small, and their crystal structures may differ from the equilibrium phases that form with increased temperature or longer times. A nucleus of a distinct phase must have an interface to the surrounding parent phase. This interface between the precipitate and matrix has an atomic structure and chemical composition that is not simply a termination of bulk crystal of the precipitate and the matrix. For free surfaces and for interfaces between crystals, this chapter explains important aspects of surface energy, surface thermodynamics, and kinetic processes at surfaces. Most solid-solid phase transformations require some consideration of elastic energy, and the balance between surface energy, elastic energy, and volume free energy is altered as a precipitate grows and changes its shape.

Chapter 4 presented concepts of nucleation in a homogeneous medium or at a generic surface, but additional microstructural aspects of nucleation are presented here. Diffusional transport to the new phase often controls the rate of growth, and the rate of the phase transformation. Some examples are given, but numerous nucleation and growth transformations in different materials have received significant study. The eutectoid transformation is described, and its role in steel metallurgy is discussed in brief.

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. 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:

1. • nuclei, which have charges that define the chemical elements,

2. • nuclear spins and their orientations,

3. • electrons that occupy states around the nuclei, and

4. • electron spins, which may have preferred orientations with respect to other spins.