Surface exchange reactions and fast grain boundary diffusion in polycrystalline materials: Application of a spherical grain model

Laplace transforms of the solution functions to the diffusion equations for surface exchange reactions and fast grain boundary diffusion in polycrystalline materials of finite thickness have been derived by applying a spherical grain model. Diffusion profiles have been calculated for semi-infinite diffusion systems as well as thin films by application of numerical Laplace inversion. The surface exchange reaction at the surface of the sample (e.g. oxide ceramics) in contact with the constant diffusion source (e.g. gas phase) is assumed to be fairly slow such that the diffusion source is not in equilibrium with the surface during the diffusion anneal. Two limiting cases for the surface conditions are taken into account, viz. fast surface diffusion and a uniform ratio of the surface exchange coefficient/diffusion coefficient. The calculated profiles refer to Harrison's type-A diffusion kinetics. Apart from expressions for the effective diffusion coefficient, analogous relations for the effective surface exchange coefficient are proposed. Relaxation curves for the total amount of diffusant exchanged with the diffusion source are discussed in terms of the diameter of the spherical grains (average grain size), surface exchange coefficient, bulk and grain boundary diffusion coefficient, respectively.