A Mathematical Model for Diffusion-induced Grain Boundary Motion

Abstract

For a film of metallic alloy immersed in a suitable vapour, we use a system of four coupled nonlinear differential equations to model the steady diffusion-induced motion of: a grain boundary, the surfaces of the two grains, and the triple junction where they all meet. One of the equations models diffusion along the moving grain boundary; another models the force balance which determines its speed. The remaining two equations model diffusion in the surfaces of the two grains. The equations are linked by boundary condi-tions at the triple junction. The resulting system of differential equations and boundary conditions is solved here for the case of 'trailing' grain boundaries (ones where the growing crystal grain develops as a layer beneath the surface of the specimen rather than filling up the entire space between the two sur-faces) in a limit where the elastic driving force is very small. The main result is that for small values of ∆c, defined as the (experimentally controllable) jump in mole fraction of solute at the triple junction, the growth velocity of the trailing grain is approximately proportional to (∆c) 4 , but for large positive ∆c the velocity is approximately proportional to (∆c) 5 . The thickness of the trailing grain is approximately proportional to (∆c) −2 for small ∆c and to (∆c) −8/3 for large. There is a negative value of ∆c beyond which the model predicts that the velocity and thickness are independent of ∆c, but this result should be treated with caution because the solution may be unstable.