Phase Field Computations of Single-Needle Crystals, Crystal Growth, and Motion by Mean Curvature

Abstract

The phase field model for free boundaries consists of a system of parabolic differential equations in which the variables represent a phase (or “order”) parameter and temperature, respectively. The parameters in the equations are related directly to the physical observables including the interfacial width $\epsilon $, which we can regard as a free parameter in computation. The phase field equations can be used to compute a wide range of sharp interface problems including the classical Stefan model, its modification to incorporate surface-tension and/or surface kinetic terms, the Cahn–Allen motion by mean curvature, the Hele–Shaw model, etc. Also included is anisotropy in the equilibrium and dynamical forms generally considered by materials scientists. By adjusting the parameters, the computations can be varied continuously from single-needle dendritic to faceted crystals. The computational method consists of smoothing a sharp interface problem within the scaling of distinguished limits of the phase field equations that preserve the physically important parameters. The two-dimensional calculations indicate that this efficient method for treating these stiff problems results in very accurate interface determination without interface tracking. These methods are tested against exact and analytical results available in planar waves, faceted growth, and motion by mean curvature up to extinction time. The results obtained for the single-needle crystal show a constant velocity growth, as expected from laboratory experiments.