Continuum approach to self-similarity and scaling in morphological relaxation of a crystal with a facet

The morphological relaxation of axisymmetric crystal surfaces with a single facet below the roughening transition temperature is studied analytically for diffusion-limited (DL) and attachment-detachment-limited (ADL) kinetics with inclusion of the Ehrlich-Schwoebel barrier. The slope profile F(r,t), where r is the polar distance and t is time, is described via a nonlinear, fourth-order partial differential equation (PDE) that accounts for step line-tension energy g(1) and step-step repulsive interaction energy g(3); for ADL kinetics, an effective surface diffusivity that depends on the step density is included. The PDE is derived directly from the step-flow equations and, alternatively, via a continuum surface free energy. The facet evolution is treated as a free-boundary problem where the interplay between g(1) and g(3) gives rise to a region of rapid variations of F, a boundary layer, near the expanding facet. For long times and g(3)/g(1)< O(1) singular perturbation theory is applied for self-similar shapes close to the facet. For DL kinetics and a class of axisymmetric shapes, (a) the boundary-layer width varies as (g(3)/g(1))(1/3), (b) a universal ordinary differential equation (ODE) is derived for F, and (c) a one-parameter family of solutions of the ODE are found; furthermore, for a conical initial shape, (d) distinct solutions of the ODE are identified for different g(3)/g(1) via effective boundary conditions at the facet edge, (e) the profile peak scales as (g(3)/g(1))(-1/6), and (f) the change of the facet radius from its limit as g(3)/g(1)-> 0 scales as (g(3)/g(1))(1/3). For ADL kinetics a boundary layer can still be defined, with thickness that varies as (g(3)/g(1))(3/8). Our scaling results are in excellent agreement with kinetic simulations.