J. W. Cahn’s research while affiliated with Washington DC VA Medical Center and other places

In many solid-state phase changes, atoms in a crystal rearrange themselves to form several phases differing in composition and order, while retaining the topology of the original crystal [l]. Although the crystal is distorted, the individual sites that atoms occupy and the lattice can still be identified with equivalent positions in an undistorted crystal. An equilibrium with respect to atom rearrangement, keeping the crystal intact, is called a coherent equilibrium. Such an equilibrium has been observed and its properties can be computed [1–3]. Coherent equilibrium is usually compared with an incoherent phase equilibrium in which each of the same phases form separate crystals that do not interact elastically with the other phases, but exchange atoms. For large enough particles, coherent equilibrium is metastable with respect to incoherent equilibrium, and loss of coherency is a spontaneous process.

Geometrical constructions, such as the tangent construction on the molar free energy for determining whether a particular composition of a solution, is stable, are related to similar tangent constructions on the orientation-dependent interfacial energy for determining stable interface orientations and on the orientation dependence of the crystal growth rate which tests whether a particular orientation appears on a growing crystal. Subtle differences in the geometric constructions for the three fields arise from the choice of a metric (unit of measure). Using results from studies of extensive and convex functions we demonstrate that there is a common mathematical structure for these three disparate topics, and use this to find new uses for well-known graphical methods for all three topics. Thus the use of chemical potentials for solution thermodynamics is very similar to known vector formulations for surface thermodynamics, and the method of characteristics which tracks the interfaces of growing crystals; the Gibbs-Duhem equation is analogous to the Cahn-Hoffman equation. The Wulff construction for equilibrium crystal shapes can be modified to construct a ``phase shape'' from solution free energies that is a potentially useful method of numerical calculations of phase diagrams from known thermodynamical data.

this paper we review recent progress in the use of variational methods in which even extreme anisotropies, non-differentiability and nonlinearities are treated in a straightforward manner and which have created new insights on how microstructures evolve and the kinetics of such processes. [1, 2, 3]). Microstructural evolution theories often start with constitutive relations and empirical data which provide equations and principles for microstructural evolution; however these often need to be checked for consistency with the laws of thermodynamics that place limits on, for instance, elastic coefficients and chemical rate constants. Thermodynamic laws alone are not sufficient to derive the dynamics of microstructural evolution. Thermodynamics does provide total or integral energy, entropy, and free energies that change monotonically in time under appropriate experimental conditions. Functions that are monotonic (i.e., either never increasing or never decreasing) are called Lyapunov functions.[?] If extremal principles could be applied to the time-rate at which an appropriate total (free) energy decreases (or entropy function increases at constant total internal energy) then it would be possible to derive dynamical equations. Recent methods for doing this have focussed on the choice and theoretical construction of an inner product as the vehicle for incorporating the kinetics JOM Article 2 into the variational framework, called a gradient flow, that indicates the `direction' in which a system can move to decrease such quantities as fast as possible in a given time increment. The gradient flow can be used to determine the instantaneous `direction ' which becomes encoded in a partial differential equations (PDE) describing the kinetics of evolving microstructures; or, as w...

Recent progress in variational methods helps to provide general principles for microstructural evolution. Especially when several processes are interacting, such general principles are useful to formulate dynamical equations and to specify rules for evolution processes. Variational methods provide new insight and apply even under conditions of nonlinearity, nondifferentiability, and extreme anisotropy. Central to them is the concept of gradient flow with respect to an inner product. This article shows, through examples, that both well-known kinetic equations and new triple junctions motions fit in this context.

Geometrical constructions, such as the tangent construction on the molar free energy for determining whether a particular composition of a solution is stable, are related to similar tangent constructions on the orientation-dependent interfacial energy for determining stable interface orientations and on the orientation dependence of the crystal growth rate which tests whether a particular orientation appears on a growing crystal. Subtle differences in the geometric constructions for the three fields arise from the choice of a metric (unit of measure). Using results from studies of extensive and convex functions, we demonstrate that there is a common mathematical structure for these three disparate topics and use this to find new uses for well-known graphical methods for all three topics. Thus, the use of chemical potentials for solution thermodynamics is very similar to known vector formulations for surface thermodynamics and the method of characteristics which tracks the interfaces of growing crystals; the Gibbs-Duhem equation is analogous to the Cahn-Hoffman equation. The Wulff construction for equilibrium crystal shapes can be modified to construct a “phase shape” from solution free energies that is a potentially useful method of numerical calculations of phase diagrams from known thermodynamical data.

The evolution of two-dimensional shapes to equilibrium shapes is investigated for two kinetic mechanisms, surface diffusion and surface attachment limited kinetics. Qualitative differences are found that may be used in experiments for easy distinction among the two mechanisms, and find topological changes not expected for the corresponding isotropic problems. We take advantage of the mathematical developments for surface evolution and equilibration problems when surface energy anisotropy is “crystalline”, so extreme that crystals are fully faceted. We confirm the prediction that with this anisotropy these problems are more easily solvable than for lesser anisotropies, and the techniques developed may even be useful for approximating isotropic problems.

The nonlinear equations that couple diffusion and stress are solved by computation for one-dimensional spinodal decomposition and coarsening in a thin plate. The vicinity of two critical values of the stress parameter are explored. At small values of elastic self-stresses, coarsening changes to exponentially fast from the exponentially slow dynamics expected for one-dimensions in the absence of stress. Coarsening is by rapid thickening of a single layer of a different phase from each of the two plate surfaces, leading to bending of the plate. Even though the diffusion equation is the same for bulk and thin plate and changes type at the coherent spinodal transition, in a thin plate the changes near this transition are gradual; the difference in behavior is due to the boundary conditions. Furthermore elasto-chemical equilibria through this transition are completely continuous. If the elastic term is large compared to the wetting term, surface wetting layers of phase of lower energy are found to disappear late in the coarsening.

The Kolmogorov-Johnson-Mehl-Avrami theory is an exact statistical solution for the expected fraction transformed in a nucleation and growth reaction in an infinite specimen, when nucleation is random in the untransformed volume and the radial growth rate after nucleation is constant until impingement. Many of these restrictive assumptions are introduced to facilitate the use of statistics. The introduction of “phantom nuclei” and “extended volumes” are constructs that permit exact estimates of the fraction transformed. An alternative, the time cone method, is presented that does not make use of either of these constructs. The method permits obtaining exact closed form solutions for any specimen that is convex in time and space, and for nucleation rates and growth rates that are both time and position dependent. Certain types of growth anisotropies can be included. The expected fraction transformed is position and time dependent. Expressions for transformation kinetics in simple specimen geometries such as plates and growing films are given, and are shown to reduce to expected formulas in certain limits.

The evolution of two-dimensional shapes to equilibrium shapes is investigated for two kinetic mechanisms, surface diffusion and surface attachment limited kinetics. Qualitative differences are found that may be used in experiments for easy distinction among the two mechanisms, and find topological changes not expected for the corresponding isotropic problems. We take advantage of the mathematical developments for surface evolution and equilibration problems when surface energy anisotropy is "crystalline," so extreme that crystals are fully faceted. We confirm the prediction that with this anisotropy these problems are more easily solvable than for lesser anisotropies, and the techniques developed may even be useful for approximating isotropic problems.

Some recent developments in the applications of thermodynamics to problems is physical metallurgy have required adaptation that are in the inventive spirit of classical thermodynamics. Diffuse interfaces and the problems that crystal lattices pose are taken as examples.