A geometric model for anisotropic crystal growth

Abstract

Equilibrium crystal shapes are defined uniquely by the Wulff construction. The classical kinematic theory of crystal growth, due mainly to Frank and Chernov, provides a mathematically equivalent prescription for the limiting growth shape. To connect these two well studied states, we derive a local geometric growth model and examine the transient shape evolution of an equilibrium form containing both facets and rough regions. Our model is appropriate to the weakly driven growth of a two-dimensional single crystal with n-gonal symmetry and arbitrary closed initial shape. The model links disparate kinetic processes determined by the local interfacial structure to the isotropic growth drive, and reproduces the experimentally observed transition from a partly rounded equilibrium shape to a highly faceted crystal which we term 'global kinetic faceting'. We solve for the transient shape dynamics globally, and locally, and in the latter case present a curvature evolution equation valid for any local growth law. Both approaches show that, during kinetic faceting, rough orientations grow out of existence with decreasing curvature.