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.

Interactions between different physical processes often make rich contributions to phase transformations in materials. The slow kinetics of one physical process can alter the thermodynamics of another process, confining it to a “constrained equilibrium,”sometimes a local minimum of free energy called a “metastable” state. A first example is the formation of a glass, which we approach with the simplest assumption that some state variables remain constant, while others relax towards equilibrium. Sometimes “self-trapping” occurs, when the slowing of a one variable enables the relaxation of a slower second variable coupled to it, and this relaxation impedes changes of the first variable. Couplings between interstitial and substitutional concentration variables are shown to alter the unmixing of both. Coherency stresses in two-phase materials are described. This chapter develops thermodynamic relationships between the different degrees of freedom of multiferroic materials, with a focus on the extensive variables that are closer to the atoms and electrons. The chapter concludes by addressing more deeply the meaning of “separability,” showing some of its formal thermodynamic consequences.

Chapter 15 develops further the concepts underlying precipitation phase transformations that were started in Chapter 14. Atoms move across an interface as one of the phases grows at the expense of the other. The interface, an essential feature of having two adjacent phases, has an atomic structure and chemical composition that are set by local thermodynamic equilibrium, but interface velocity constrains this equilibrium. Interactions of solute atoms with the interface can slow the interface velocity by "solute drag." When an interface moves at a high velocity, chemical equilibration by solute atoms does not occur in the short time when the interface passes by. These issues also pertain to rapid solidification, and extend the ideas of Chapter 13. Solid–solid phase transformations also require consideration of elastic energy and how it evolves during the phase transformation. The balance between surface energy, elastic energy, and chemical free energy is altered as a precipitate grows larger, so the optimal shape of the precipitate changes as it grows. The chapter ends with some discussion of the elastic energy of interstitial solid solutions and metal hydrides.