When a large anisotropy of surface energy is added to standard models of surface
evolution, faceting can result which, counterintuitively, allows the evolution to
be reduced to a simple dynamical system.
We examine this phenomenon first in the context of directional solidification of a dilute binary alloy, where evolution follows a simple geometric law associated with facet height. In the supercooled regime, this law leads to coarsening -- the gradual increase in system lengthscales as small facets shrink and vanish.
Simulations of evolving faceted surfaces are demonstrated in one and two dimenstions using a novel computational geometry tool, which reveals a rich variety of topological behavior among neighboring facets. Although observed behavior is complex, the surface quickly settles into a scale-invariant state, where statistical distributions of scaled geometric quantities exhibit constant shape even as lengthscales continuously increase.
After showing some surface evolution movies and examples of scale-invariant data,
we conclude with a brief mean-field theory which qualitatively describes the scaling
state. This simple result illustrates the more general goal of characterizing
interface behavior at long times.