Phase field theory treats the phases in materials as fields inside a material, as opposed to tracking the motions of interfaces during phase transformations. The interface sharpness is determined by a balance between bulk free energies and the square gradients of the fields. Treating phases as fields has advantages for the computational materials science of microstructural evolution, and some kinetic mechanisms are described. The different equations for the evolution of a conserved order parameter (e.g., composition) and a nonconserved order parameter (e.g., spin orientation) are discussed. The structure of an interface, especially its width, is analyzed for the typical case of an antiphase domain boundary. The Ginzburg–Landau equation is presented, and the effects of curvature on interface stability are discussed. Some aspects of the dynamics of domain growth are described.

Chapter 6 covers the internal energy E, which is the first term in the free energy, F = E – TS. The internal energy originates from the quantum mechanics of chemical bonds between atoms. The bond between two atoms in a diatomic molecule is developed first to illustrate concepts of bonding, antibonding, electronegativity, covalency, and ionicity. The translational symmetry of crystals brings a new quantum number, k, for delocalized electrons. This k-vector is used to explain the concept of energy bands by extending the ideas of molecular bonding and antibonding to electron states spread over many atoms. An even simpler model of a gas of free electrons is also developed for electrons in metals. Fermi surfaces of metals are described. The strength of bonding depends on the distance between atoms. The interatomic potential of a chemical bond gives rise to elastic constants that characterize how a bulk material responds to small deformations. Chapter 6 ends with a discussion of the elastic energy generated when a particle of a new phase forms inside a parent phase, and the two phases differ in specific volume.

The physical origins of entropy are explained. Configurational entropy in the point approximation was used previously, but Chapter 7 shows how configurational entropy can be calculated more accurately with cluster expansion methods, and the pair approximation is developed in some detail. Atom vibrations are usually the largest source of entropy in materials, and the origin of vibrational entropy is explained in Section 7.4. Vibrational entropy is used in new calculations of the critical temperatures of ordering and unmixing, which were done in Chapter 2 with configurational entropy alone. For metals there is a heat capacity and entropy from thermal excitations of electrons near the Fermi surface, and this increases with temperature. At high temperatures, electron excitations can alter the vibrational modes, and there is some discussion about how the different types of entropy interact.

The quantum mechanical exchange interaction gives rise to magnetic moments and their interactions in materials, which give rise to patterns and structures in the orientations of magnetic moments at low temperatures. With increasing temperature, pressure, and magnetic field, magnetic structures are altered, and Chapter 21 describes several trends that can be understood by thermodynamics. The critical temperature of magnetic ordering, the Curie temperature TC, is calculated. Compared to chemical ordering, the strengths and alignments of magnetic moments have more degrees of freedom, allowing for diverse magnetic structures. These include ferrimagnetism, frustrated structures, and spin glasses. The vectorial character of spin interactions can give rise to localized spin structures such as skyrmions. An electromechanical phase transition can occur when the energy for a displacement of positive and negative ions in a unit cell is comparable to thermal energies. This ferroelectric transition has some similarities to the ferromagnetic transition, but is described by Landau theory. Domains in ferroelectric and ferromagnetic materials can reduce the energy in surrounding elastic and magnetic fields, andthe width of a boundary between two magnetic domains is estimated.

Most solid-to-solid phase transformations are much more interesting than just the growth of a small, homogeneous particle of the new phase. For reasons of both kinetics and thermodynamics, the new particles evolve in crystal structure, chemical composition, interface structures, defects, elastic energies, and shapes. Chapter 14 gives an overview of processes that occur during the nucleation and growth of a new phase from a parent phase. It covers essential features of precipitation in a solid, with a few traditional examples from steels, such as the pearlite transformation, and examples of precipitation sequences in aluminum alloys. Much of the content is central to physical metallurgy. The Kolmogorov-Johnson-Mehl-Avrami model of the rates of nucleation and growth transformations is presented. The late-stage coarsening process is also discussed in terms of the self-similarity of the microstructure.

Chapter 2 explains T–cphase diagrams, which are maps of equilibrium alloy phases in a space spanned by temperature T and chemical composition c. The emphasis is on deriving T–c phase diagrams by minimizing the total free energy of an alloy with two or three phases. The lever rule and common tangent constructions are developed. Some basic ideas about chemical interactions and entropy are used to justify of the free energies of alloy phases at different temperatures. For binary alloys, the shapes of free energy versus composition 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 discussed. If atoms are confined to sites on an Ising lattice, free energy functions can be calculated with a minimum set of assumptions about the energies of different atomic configurations. These generalizations of chemical interactions are useful for identifying phenomena common to unmixing and ordering transitions, but warnings about their limitations are presented.

Phase transitions are driven by pressure as well as temperature, and the use of pressure to tune the electronic structures of materials can help further our understanding of materials properties. 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 driven by temperature through thermal expansion, and the concept of “thermal pressure” from nonharmonic phonons is explained. Electronic energy is responsible for big contributions of +PV to the free energy, and this chapter describes how electron energies are altered by pressure. Cross-terms between temperature and pressure are discussed. The chapter ends with a discussion of kinetic processes under pressure, and the concept of an activation volume.

This chapter explains why atom jumps with a vacancy mechanism are not random, even if the vacancy itself moves by random walk. In an alloy with chemical interactions strong enough to cause a phase transformation, the vacancy frequently resides at energetically favorable locations, so any assumption of random walk can be seriously in error. When materials with different diffusivities are brought into contact, their interface is displaced with time because the fluxes of atoms across the interface are not equal in both directions. Even the meaning of the interface, or at least its position, requires new concepts. An applied field can bias the diffusion process towards a particular direction, and such a bias can also be created by chemical interactions between atoms. When thermal atom diffusion occurs in parallel with atom jumps forced without thermal activation, a steady state can be calculated, but it is not a state of thermodynamic equilibrium. Finally, the venerable statistical mechanics model of diffusion by Vineyard is described.