Fast grain boundary diffusion and rate-limiting surface exchange reactions in polycrystalline materials

Analytical solutions to the diffusion equations for fast grain boundary diffusion in polycrystalline solids of finite thickness bounded by two parallel surface planes are presented. The grain boundary diffusion coefficient may be higher by several orders of magnitude than the bulk (volume) diffusion coefficient. The microstructure of the polycrystalline sample has been modeled by means of an isolated grain boundary as well as an array of parallel grain boundaries. The surface concentration has been assumed to vary continuously during the diffusion process, as the surface exchange reactions of the diffusing species are considered to be fairly slow and equilibrium between the surface and the constant diffusion source (gas phase) is not attained. Two different cases for surface diffusion at the interface polycrystalline sample/diffusion source are taken into account, viz. negligible and fast surface diffusion. The pertinent analytical solutions allow the calculation of two-dimensional diffusion profiles in thin films and average concentration profiles for semi-infinite diffusion systems. In addition, the time dependence of the total amount of diffusing species exchanged between the diffusion source and the solid sample has been calculated. The theoretical results are interpreted in terms of Harrison’s classification of diffusion regimes. The parallel grain boundary model enables the calculation of diffusion profiles for both type-A and type-B kinetics. In the case of type-A kinetics expressions for the effective diffusion coefficient as well as the effective surface exchange coefficient are proposed, which depend on the corresponding bulk values, the microstructure of the specimen, and the respective surface condition.