Morphology: from sharp interface to phase field models

Abstract

Over the last 50 years, there has been tremendous progress in the quantification of crystal growth morphology. In the 1950s, the dynamics of crystal growth from the melt was based on the sharp interface model (interface of zero thickness separating solid and liquid), often under the assumption of isotropy. Ivantsov had discovered analytical solutions to the Stefan problem for the special class of shapes known as quadric surfaces (ellipsoids, hyperboloids and paraboloids, including their special cases spheres, cylinders and planes). But in the 1960s, these solutions were shown to be morphologically unstable, resulting in cellular and dendritic growth forms that had long been known to exist from experimental work. Sharp interface models were used to model these growth forms, but it was necessary to include corrections of the interface temperature for capillarity and curvature (Gibbs–Thomson equation) in order to avoid instabilities at all wavelengths and to set the size scale of the resulting morphologies. Except for the case of total interface control, for which exact solutions even for facetted crystals had been provided by Frank using the method of characteristics, little could be done analytically to treat anisotropies. By the 1980s, our reliance on the sharp interface model began to change with the adaptation by Langer and others of diffuse interface models, of the Cahn–Hilliard type, to solve dynamical problems. This class of models, now known as phase field models, replaced the sharp interface model by the solution in the entire computational domain of coupled partial differential equations for thermal and compositional fields and for an auxiliary variable that keeps track of the phase. Moreover, the phase field equations incorporate automatically the Gibbs Thomson equation, anisotropy and even departures from local equilibrium (interface kinetics) asymptotically for a sufficiently thin diffuse interface. But it has been only in about the last decade that massive improvements in computing power have rendered the numerical solution of the phase field model tractable. By means of this model, complex morphologies and related phenomena over a vast range of length scales can now be studied.