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.