Crystal growth, coarsening and the convective Cahn-Hilliard equation

The coarsening dynamics of a faceted vicinal crystalline surface growing into its melt by attachment kinetics is considered. The convective Cahn-Hilliard equation (CCH) is derived as a small amplitude expansion of such surface evolutions restricted to 1-D morphologies, with the local surface slope serving as the order parameter. A summary of the sharp interface theory for CCH, that follows from a matched asymptotic analysis, is also presented [26]. It takes the form of a nearest neighbor interaction between two non-symmetrically related phase boundaries (kink and anti-kink). The resulting coarsening dynamical system CDS for the phase boundaries exhibits novel coarsening mechanisms. In particular, binary coalescence of phase boundaries is impossible. Also, ternary coalescence occurs only through two kinks meeting an anti-kink resulting in a kink (kink-ternary); the alternative of two anti-kinks meeting a kink is impossible. This behavior stands in marked contrast to the Cahn-Hilliard equation CH where binary coalescence of phase boundaries is generic. Numerical simulations of CCH are presented to validate the predictions of the sharp interface theory CDS.