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, tn, of top steps. For initial cones and tn ≈ 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.