Fundamentals of phase field theory

Phase field theory provides an alternative method for solving dynamical problems involving crystallization from a melt. The sharp solid liquid interface of the classical model is replaced by a diffuse interface by introducing an auxiliary variable phi, the phase field, that indicates the phase. The variable phi varies from some value, say 0 in the solid, to another value, say 1 in the liquid. It changes continuously from 0 to I over a thin region of space, the diffuse interface. Equations for the time evolution of phi as well as those for temperature T and compositional fields omega (for alloys) are derived from postulated functionals for entropy, energy and chemical species. For a sufficiently thin diffuse interface, these equations account for the Gibbs-Thomson dependence of melting point on local interface curvature and also for linear interface attachment kinetics. They can be solved to generate complicated solidification patterns, such as occur during cellular and dendritic growth. They can also be generalized to include multicomponent diffusion, fluid convection, order-disorder transformations in crystals, and relative crystallographic orientation (grain boundaries) in polycrystalline materials.