The Influence of the Exterior Surface on Grain Boundary Mobility Measurements

Abstract

Polycrystalline materials are materials made up of a typically large number of crystals (``grains''), whose boundaries (``grain boundaries'') define an interior network which effectively determines the microstructure of the specimen. As the network is not at equilibrium, it evolves in time to reduce the overall energy of the specimen. Since the grain boundaries couple to the exterior surface of the specimen, the evolution of the microstructure is influenced by the motion of the exterior surface. In thin specimens, this effect can be appreciable. Within the framework of the classical Mullins' model, grain boundaries move by mean curvature motion, $V_n=A\kappa$, and the exterior surface evolves by surface diffusion, $V_n=-B\Delta_s\kappa$, where $V_n$ and $\kappa$ denote the normal velocity and the mean curvature of the respective evolving surfaces, and $\Delta_s$ is the surface Laplacian. Accurately estimating the kinetic parameters, $A$ and $B$, constitutes an important characterization of material properties. A classical way to determine the “reduced mobility,” $A$, is to make measurements based on a specially constructed “half-loop bicrystalline geometry.” In this geometry there are two grains; one grain is embedded within the other and recedes at a roughly constant rate. This rate may be used to estimate $A$ if the effects of the exterior surface may be neglected. Here we focus on the effects of the exterior surface on grain boundary mobility measurements in the context of the half-loop bicrystalline geometry. Asymptotics, based on a small grain boundary surface energy to exterior surface energy ratio and a thin specimen aspect ratio, are used to predict the coupled grain boundary and exterior surface motion, as well as a leading order correction to mobility measurements.