Coupled surface and grain boundary motion: nonclassical traveling-wave solutions

Abstract

We study a geometric free-boundary problem for a bicrystal in which a grain boundary is attached at a groove root to the exterior surface of the bicrystal. Mathematically, this geometric problem couples motion by mean curvature of the grain boundary with surface diffusion of the exterior surface. If the groove root effects are localized, it is realistic to look for traveling-wave solutions. We show that traveling-wave solutions can be determined via solutions of Problem $P_{\Psi}$ in which $$\Psi_{sss}= \sin \Psi, \quad s \in (-\infty, \, 0) \cup (0, \, \infty), \quad \Psi(\pm \infty)=0,$$ and appropriate jump conditions are prescribed at $s=0$. We prove existence of solutions to Problem $P_{\Psi}$ for all $m,$ $0 \le m <2$, where $m$ denotes the ratio of the surface energies of the grain boundary and of the exterior surface. We show numerically that for $\,\approx\! 1.81 < m \le 2$, the corresponding solutions to the original geometric problem are not single-valued as functions of $x$, where $x$ varies along the unperturbed exterior surface of the bicrystal. We refer to these solutions as "nonclassical traveling-wave solutions."