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.

An ordered structure can be described as a static concentration wave, which varies from site to site on a crystal. Crests denote B-atoms and troughs denote the A-atoms, for example. A solid solution has zero amplitude of the concentration wave, so the amplitude of the concentration wave, η, serves as a long-range order parameter. With concentration waves, the free energy is transformed from real space to k-space. The concentration waves accommodate the symmetry of the ordered structure, and how it differs from the high temperature solid solution. A subtle analysis by Landau and Lifshitz shows that if a second-order phase transition is possible (i.e., the ordered structure evolves from the disordered with infinitesimal amplitude at the critical temperature), the translational symmetry of the free energy sets an elegant condition for the wavevectors of the ordered structure. Chapter 18 ends with a more general formulation of the freeenergy in terms of static concentration waves, which is an important example of how Fourier transform methods can treat long-range interactions in materials thermodynamics.

Chapter 3 begins by describing mechanisms of atomic diffusion in crystals, with emphasis on how their rates depend on temperature. Characteristic diffusion lengths and times are explained. The diffusion equation is derived for the chemical composition in space and time, c(r,t). The mathematics for solving the diffusion equation in one dimension are developed by standard approaches with Gaussian functions and error functions. The method of separation of variables is presented for three-dimensional problems in Cartesian and cylindrical coordinates. Typical boundary value problems for diffusion are solved with Fourier series and Bessel functions.

Spinodal decomposition of a solid solution begins with infinitesimally small changes in composition. Nevertheless, there is an energy cost for gradients in composition, specifically the square of the gradient. This “square gradient energy” is an important new concept presented in this chapter, and it is also essential to phase field theory (Chapter 17). An unstable free energy function is a conceptual challenge, but it proves useful for short times. Taking a kinetic approach, the thermodynamic tendencies near equilibrium are used to obtain a chemical potential to drive the diffusion flux of spinodal unmixing. This chapter follows the classic approach of John Cahn by adding a term to the free energy that includes the square of the composition gradient. Lagrange multipliers are used in the diffusion equation for the chemical potential, and compositional unmixing is described by Fourier transformation. There is also an elastic energy that increases with the extent of unmixing, and gives the “coherent spinodal” on the unmixing phase diagram.

Chapter 5 uses concepts of diffusion and nucleation to understand phase transformations in ways beyond a simple usage of equilibrium phase diagrams. A number of nonequilibrium phenomena are described, which show how to understand some phase transformations that have impediments from nucleation and diffusion. In general, the slowest processes are first to cause deviations from states of equilibrium. For faster heating or cooling, however, sometimes the slowest processes are fully suppressed, and the next-slowest processes become important. Nonequilibrium processes in alloy freezing are explained, as is the glass transition. Approximately, Chapter 5 progresses from slower to faster kinetic processes. However, the last section discusses why kinetic processes based on activated state rate theory should bring materials to thermodynamic equilibrium.

Diffusionless transformations occur when atoms in a crystal move cooperatively and nearly simultaneously, distorting the crystal into a new shape. The martensite transformation is the most famous diffusionless transformation, owing to its importance in steel metallurgy. In a martensitic transformation the change in crystal structure occurs by shears and dilatations, and the atom displacements accommodate the shape of the new crystal. The atoms do not move with independent degrees of freedom, so the change in configurational entropy is negligible or small. The entropy of a martensitic transformation is primarily vibrational (sometimes with electronic entropy, or magnetic entropy for many iron alloys). This chapter begins with a review of dislocations, and how their glide motions can give crystallographic shear. Some macroscopic and microscopic features of martensite are then described, followed by a two-dimensional analog for a crystallographic theory that predicts the martensite “habit plane” (the orientation of a martensite plate in its parent crystal). Displacive phase transitions are explained more formally with Landau theories having anharmonic potentials and vibrational entropy. Phonons are discussed from the viewpoint of soft modes and instabilities of bcc structures that may be relevant to diffusionless transformations.

Phase transformations often begin by nucleation, where a small but distinct volume of material forms with a structure and composition that differ from those of the parent phase. An unfavorable surface bounds the new phase, giving rise to a barrier that must be overcome before thefluctuation in structure and composition can become a stable, growing region of new phase. Chapter 4 develops the thermodynamics of forming a nucleus, with emphasis on the characteristic size and undercooling that are required. Homogeneous and heterogeneous nucleation are explained. The temperature dependence of nucleation is explained. The time dependence of nucleation is discussed in terms of the shape of the free energy barrier that must be crossed by a growing nucleus. There is some discussion of nucleation in multicomponent alloys.

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.

This chapter introduces key concepts that are developed in this textbook. It describes the concept of microstructure and other features of materials that undergo interesting changes with temperature or pressure. These changes are motivated by the thermodynamic free energy, but require a kinetic mechanism for atoms to move. Chemical unmixing and ordering on a crystal lattice are described, and the kinetics of diffusion by vacancies is explained. The free energy is used to explain melting. A summary of essential aspects of thermodynamics and kinetics is given at the end of the chapter, including basic ideas of statistical mechanics and the kinetic master equation.

The structures and dynamics of surfaces affects the chemical reactivity and growth characteristics of materials. Chapter 11 describes atomistic structures of surfaces of crystalline materials, and describes how a crystal may grow by adding atoms to its surface. Most inorganic materials are polycrystalline aggregates, and their crystals of different orientation make contact at “grain boundaries.” Some features of atom arrangements at grain boundaries are explained, as are some aspects of the energetics and thermodynamics of grain boundaries. Grain boundaries alter both the internal energy and the entropy of materials. Surface energy varies with crystallographic orientation, and this affects the equilibrium shape of a crystal. The interaction of gas atoms with a surface, specifically the topic of gas physisorption, is presented.

Chapter 12 discusses the enthalpy and entropy of solid and liquid phases near the melting temperature Tm, and highlights rules of thumb, such as the tendency for the entropy of melting to be similar for different materials. Correlations between Tm and the amplitude of thermal displacements of atoms (“Lindemann rule”), and between Tm and the bulk modulus are presented, but these correlations are semiquantitative at best. Richard's rule for the entropy of melting is more robust. Interface behavior during melting is covered in more detail, including premelting. At a temperature well below Tm, a glass undergoes a type of melting called a “glass transition” which is discussed in more detail in this chapter.Some features of melting in two dimensions are described, which are quite different from melting in three dimensions.