Model for faceting in a kinetically controlled crystal growth

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Abstract
A two-dimensional anisotropic nonlinear evolution equation is derived to model the formation of facets and corners in the course of kinetically controlled crystal growth. The equation is solved numerically in particular cases corresponding to the faceting of [001], [111], and [110] growing crystal surfaces, and the formation of hill-and-valley structures in the form of square, triangular, and rhombic pyramids; grooves are observed as well. The pyramidal slopes far from the vertices are found analytically, and in particular cases exact solutions of the equation are found. The pyramidal structures coarsen in time, and the rate of coarsening is studied.