Person:

Stone, Howard A.

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.

The development of surface grooves at grain boundaries that intersect a planar surface is analyzed for the case that the evolution occurs below the thermodynamic roughening transition by evaporation–condensation processes. The dynamics are described by a nonlinear partial differential equation that has a similarity solution, so the resulting groove profile is described by a nonlinear ordinary differential equation. An approximate analytical solution to the nonlinear problem is obtained and is in excellent agreement with the numerical solution. The depth and width of the groove varies as t^1/2, where t is time, analogous to the classical results valid above the thermodynamic roughening temperature. In addition, the approximate analytical solution provides an explicit relation between the groove width and the dihedral angle, and is in sufficiently good agreement with the numerical results as to make such numerical solutions unnecessary for this problem. The results demonstrate explicitly how the groove shape depends on the functional form of the slope-dependent surface mobility.

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 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<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 (g3∕g1)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 g3∕g1 via effective boundary conditions at the facet edge, (e) the profile peak scales as (g3∕g1)−1∕6 , and (f) the change of the facet radius from its limit as g3∕g1→0 scales as (g3∕g1)1∕3 . For ADL kinetics a boundary layer can still be defined, with thickness that varies as (g3∕g1)3∕8 . Our scaling results are in excellent agreement with kinetic simulations.