Mean-field theory for coarsening faceted surfaces

A mean-field theory is developed for the scale-invariant length distributions observed during the coarsening of generic one-dimensional faceted surfaces. This theory closely follows the Lifshitz-Slyozov-Wagner theory of Ostwald ripening in two-phase systems, but the mechanism of coarsening in faceted surfaces requires the addition of convolution terms recalling work on particle coalescence, and induces an unexpected coupling between the convolution and the rate of facet loss. As a generic framework, the theory concisely illustrates how the universal processes of facet removal and neighbor merger are moderated by the system-specific mean-field velocity describing average rates of length change. For a simple, example facet dynamics associated with the directional solidification of a binary alloy, agreement between the predicted scaling state and that observed after direct numerical simulation of 40 000 000 facets is reasonable given the limiting assumption of noncorrelation between neighbors; relaxing this assumption is a clear path forward toward improved quantitative agreement with data in the future.