Faceted interfaces in directional solidification

We consider the directional solidification, in two dimensions, of a dilute binary alloy having a large anisotropy of surface energy, (i.e., orientations with negative surface stiffness), where the sample is pulled in the highest-energy direction. Linear stability analysis reveals that the planar state is thermodynamically prohibited, leading to a search for faceted solutions. Below the critical pulling speed associated with constitutional supercooling, a small-wavelength assumption allows the reduction of interface dynamics to a single PDE. Matched asymptotic analysis then reveals a family of faceted interface profiles, while variational arguments confirm a small optimal wavelength. Questions on dynamic behavior lead to the derivation of a gradient-descent dynamics and an associated facet-velocity law. This reveals that faceted steady solutions are stable in the absence of supercooling, while coarsening replaces cell formation as the instability mechanism when supercooling is reached.