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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, and the 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–c phase 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.
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 free energy 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 the fluctuation 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.