Coupled surface and grain boundary motion: a travelling wave solution

Abstract

Existence and uniqueness are proven for a travelling wave solution for a problem in which motion by mean curvature is coupled with surface diffusion. This problem pertains to a bicrystal in a “quarter-loop” geometry in which one grain grows at the expense of the other, and the internal grain boundary between the two crystals contacts the exterior surface at a “groove root” or “tri-junction” where various balance laws hold. Far in front and behind the groove root the overall height of the bicrystal is assumed to be unperturbed. Whereas in a previous paper (Acta Mater. 51 (2003) 1981) a partially linearized formulation was considered for which explicit solutions could be found, here we treat the fully nonlinear problem. Employing an angle formulation and a scaled arc-length parameterization, we reduce the problem to the solution of a third order ODE with a jump condition at the origin. Existence is proven if m, the ratio of the exterior surface energy to the surface energy of the grain boundary, is less than about ≈ 0.92 . Uniqueness of these solutions is demonstrated within the class of single-valued solutions. A numerical comparison is made with the solution of the partially linearized formulation found earlier for the sake of illustration.