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.

Here nanomaterials are defined as materials with structural features of approximately 10 nm or smaller, i.e., tens of atoms across. Unique physical properties of nanomaterials originate from one or two of their essential features: (1) nanomaterials have high surface-to-volume ratios, and a large fraction of atoms located at, or near, surfaces; (2) nanomaterials confine electrons, phonons, excitons, or polarons to relatively small volumes, altering their energies. Chapter 20 focuses on the thermodynamic functions of nanostructures that determine whether a nanostructure can be synthesized, or if a nanostructure is adequately stable at a modest temperature. The internal energy of nanomaterials is increased by the surfaces, interfaces, or large composition gradients. A nanostructured material generally has a higher entropy than bulk material, however, and at finite temperature the entropy contribution to the free energy can help to offset the higher internal energy term in the free energy F = E – TS. Chapter 20 discusses the structure of nanomaterials, the thermodynamics of interfaces in nanostructures, electron states in nanostructures, and the entropy of nanostructures.