Discrete and Continuum Relaxation Dynamics of Faceted Crystal Surface in Evaporation Models

Abstract

We study linkages of two scales in the relaxation of an axisymmetric crystal with a facet in evaporation-condensation kinetics. The macroscale evolution is driven by the motion of concentric circular, repulsively interacting line defects (steps) which exchange atoms with the vapor. At the microscale, the step velocity is proportional to the variation of the total step free energy, leading to large systems of differential equations for the step radii. We focus on two step flow models. In one model (called M1) the discrete mobility is simply proportional to the upper-terrace width; in another model (M2) the mobility is altered by an extra geometric factor. By invoking self-similarity at long time, we numerically demonstrate that (i) in M1, discrete slopes follow closely a continuum thermodynamics approach with “natural boundary conditions” at the facet edge; (ii) in contrast, predictions of M2 deviate from results of the above continuum approach; and (iii) this discrepancy can be eliminated via a continuum boundary condition with a geometry-induced jump for top-step collapses. At the macroscale, both step models give rise to free-boundary problems for a second-order, parabolic partial differential equation, which we study via the subgradient formalism. We discuss the interpretation of the facet height as shock and prove convergence of the solution of each discrete scheme to the (weak) entropy solution of a conservation law if steps do not interact.