Continuum Theory of Nanostructure Decay Via a Microscale Condition

Abstract

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[approximate]t-tilde n⁴, we use t-tilde(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.