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 sDLd and attachment-detachment-limited sADLd kinetics with inclusion of the Ehrlich-Schwoebel barrier. The slope profile Fsr , td, where r is the polar distance and t is time, is described via a nonlinear, fourth-order partial differential equation sPDEd that accounts for step line-tension energy g1 and step-step repulsive interaction energy g3; 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 g1 and g3 gives rise to a region of rapid variations of F, a boundary layer, near the expanding facet. For long times and g3 /g1,Os1d singular perturbation theory is applied for self-similar shapes close to the facet. For DL kinetics and a class of axisymmetric shapes, sad the boundary-layer width varies as sg3 /g1d, sbd a universal ordinary differential equation sODEd is derived for F, and scd a one-parameter family of solutions of the ODE are found; furthermore, for a conical initial shape, sdd distinct solutions of the ODE are identified for different g3 /g1 via effective boundary conditions at the facet edge, sed the profile peak scales as sg3 /g1d, and sfd the change of the facet radius from its limit as g3 /g1 →0 scales as sg3 /g1d. For ADL kinetics a boundary layer can still be defined, with thickness that varies as sg3 /g1d. Our scaling results are in excellent agreement with kinetic simulations.