Coupled grain boundary and surface diffusion in a polycrystalline thin film constrained by substrate

it has recently been shown [1] that constrained grain boundary diffusion in a stressed polycrystalline thin film causes exponential relaxation of grain boundary traction, leading to formation of crack-like diffusion wedges via mass transport between the grain boundaries and the free surface of the film. It was assumed in [1] that surface diffusion is very fast compared to grain boundary diffusion and surface tension is very large so that the film surface remains perfectly flat during the diffusion process. In this paper, we try to relax some of these assumptions by extending the analysis of [1] to coupled grain boundary and surface diffusion. We use Mullins' equation to describe surface evolution, with mass conservation and continuity of chemical potential strictly enforced at the junctions between grain boundaries and the free surface. The surface diffusion is treated as a matter source or sink for the grain boundary diffusion and, assuming the surface slope is small, we neglect the effect of surface slope on the normal stress and chemical potential along the grain boundary. No sliding or diffusion is allowed at the film/substrate interface. We develop singular integral equation techniques to numerically solve the integro-differential governing equations for the strongly coupled deformation and diffusion problem. The characteristic time for constrained grain boundary diffusion is found to scale with the cubic power of the film thickness and depends on an effective diffusivity combining surface diffusivity and grain boundary diffusivity. The results indicate that constrained grain boundary diffusion leads to the formation of crack-like grain boundary diffusion wedges.