Continuum theory of nanostructure decay via a microscale condition.

The morphological relaxation of faceted crystal surfaces is studied via a continuum approach. Our formulation includes (i) an evolution equation for the surface slope that describes step line tension, g1, and step repulsion energy, g3; and (ii) a condition at the facet edge (a free boundary) that accounts for discrete effects via the collapse times, t(n), of top steps. For initial cones and t(n) approximately t(n)4, we use t(g) from step simulations and predict self-similar slopes in agreement with simulations for any g = g3/g1 > 0. We show that for g >> 1, (i) the theory simplifies to an equilibrium-thermodynamics model; (ii) the slope profiles reduce to a universal curve; and (iii) the facet radius scales as g(-3/4).