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

Abstract

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.

Role of chemical potential in relaxation of faceted crystal structure

“…It has been demonstrated by simulations for a faceted axisymmetric crystal structure under diffusion-limited kinetics that the continuum slope determined in the reference case is not consistent with step motion [8]. For an analogous result in evaporation-condensation kinetics, see [15]. In fact, it has been realized that the facet is a special region where discrete effects, especially collapses of steps, may dramatically influence the large-scale surface morphology [4].…”

“…An alternate scenario avoiding Eq. (13) was proposed in [15] for an evaporation-condensation model in a radial geometry. In this case, the evolution PDE takes the form ∂h/∂t = −ν g 1 divξ everywhere [see Eq.…”

“…This perspective, especially the use of discontinuities for thermodynamic variables at the facet edge, has been inspired by a recent analysis of an evaporation-condensation dynamics model [15]. In [15], the jump is introduced only for the radial flux generating the continuum-scale chemical potential. Here, we need additional boundary conditions (since the PDE is of higher order) [16][17][18].…”

“…It is of some interest to compare our approach to previous continuum models where atomic steps are annihilated on top of facets, e.g., [3][4][5][7][8][9]15]. For example, in [5,7] only the continuity of the surface chemical potential, which yields results not consistent with step flow, is considered.…”

“…For example, in [5,7] only the continuity of the surface chemical potential, which yields results not consistent with step flow, is considered. In [3,4,9,15] the entire many-step system is simulated for reconciling continuum predictions with step motion. In [8], a continuum-type equation is formulated from an ensemble of step configurations, and the facet boundary is not treated explicitly.…”