Microstructural evolution involves complex, coupled, and often nonlinear processes. Even the description of the dynamics for isolated microstructural evolution processes can be quite complicated. Familiar derivations of such processes involve simplifying assumptions of isotropy, differentiability, regularity and linearization. Such assumptions are often unwarranted in materials systems of interest, and lead to inadequate descriptions of evolving microstructures. It would be useful to have general principles for the kinetics of microstructural evolution, that could be used to formulate dynamical equations and develop general rules about individual microstructural processes that can also be used when several such processes are interacting. In 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]).
Microstructures 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 energy, entropy, and free energies that behave like Lyapunov functions (functions that monotonically change in time). In this manner, thermodynamics seems to provide a variational principle. If extremal principles could be applied to the rate at which some total (free) energy decreases then it would be possible to derive dynamical equations. The recent methods have focussed on the choice and theoretical construction of an inner product as the vehicle for incorporating the kinetic principles into the variational process, called a gradient flow, that indicates the `direction' in which a system can move to decrease a quantity-in particular, total free energy-as much as possible in a small fixed time step, and thereby derive kinetic equation for the evolution of microstructures. We review the concepts and illustrate them with a small set of inner products, and show how the choice of inner product in a gradient flow bestows a particular set of dynamical equations.
Such a development starts with the thermodynamics and guarantees that any equation derived will be consistent with its laws. The inner product gives a kinetic measure of the time it would take for a system to evolve from one microstructure to another. The gradient flow then chooses the evolution that will decrease free energy as much as possible in a given time step. How distance between points is encoded in the inner (dot) product is already familiar from simple vector algebra. The concept has been extended to describe distances between two microstructures.
Thus gradient flows operate on fields that describe a microstructure, rather than points. For example the concentration, C, as a function of position is a field , as is the local order parameter . A flow can be thought of as a velocity field describing the time-rate of change of a field; with C the flow is Fick's law, or one of its many modifications, with a composition dependent diffusion coefficient. Another example is the motion of diffuse surfaces in such fields. Those spatial positions where a field has a uniform constant value (e.g., ) are called level sets and form structures with one less dimension. Thus, for example, a surface can be written as a level set of a composition or order parameter field. The flow of a field yields a motion of a level set and therefore can be used to track the evolution of an interface or grain boundary, even when these are anisotropic enough to be faceted and have corners and edges. In this manner, the mathematics of flows on fields can be used to describe quite disparate nonlinear phenomena-spinodal decomposition or grain boundary motion-in a common framework. Even when these variational methods only recover known kinetic laws, they guarantee conformance with thermodynamics and often lead to new insights.