A traveling wave solution for coupled surface and grain boundary motion

Abstract We study the coupled motion of a grain boundary in a bicrystal which is attached at a “groove root” to an exterior surface which evolves according to surface diffusion in a “quarter loop” geometry (Dunn et al., Trans. Am. Inst. Min. Engrs. 185 (1949) 708), and prove the existence of a unique traveling wave solution for various partially linearized formulations. Our results complement and complete the earlier analysis by Mullins (Acta metall. 6 (1958) 414) where the groove root and the velocity as a function of groove depth were determined. We demonstrate that the net effect of the coupling to the exterior surface is to reduce the overall velocity relative to a freely moving grain boundary by a factor which is small (≈3.5%) for typical parameter values. For extreme values of the parameters, the coupling may cause an increase in the overall grain boundary velocity. No “jerky” or “stop and go” motion is predicted by our solution, and we conjecture why this is so.