The enhancement of the concentration of a defect in the space-charge region near to a grain boundary, in an intrinsic ionic system, was used to specify the change in the corresponding diffusion coefficient as a function of distance, x, normal to the boundary. This zone of increased and continuously varying diffusivity facilitated enhanced transport along the boundary, in addition to the expected contribution which arose from the differing structure at the boundary core. A 2-dimensional diffusion equation was established for the concentration in the space-charge region. This was solved numerically using finite-difference methods, and was integrated normal to the boundary to give the average concentration gradient along the boundary. The gradients were analyzed as if they were experimental data by using a recent solution, to the conventional model for grain-boundary diffusion, in which the interface was treated as a thin slab of half-width, a, and enhanced diffusivity, D', embedded in a material with bulk diffusivity, D. The analysis provided effective values for the conventional diffusion parameter, β = a[(D'/D) - 1]/√(Dt), which described the effect of space charge upon enhanced diffusion along the boundary. The calculations were repeated for a number of values of the parameter, Zeφ(∞)/kT, where Ze was the effective charge of the defect and φ(∞) was the bulk electric potential far from the boundary. In finite-difference calculations, the value of the annealing time, t, in β was set equal to the product of the square of the Debye length, divided by the bulk diffusion coefficient. Given the value of φ(∞), the present results therefore permitted the estimation of the effect of space charge, upon preferential diffusion along a grain boundary, in terms of β. If the Debye length was known for the material, or could be measured, the value of β could be easily converted into an effective value for the conventional grain-boundary diffusion product.

Calculation of the Contribution to Grain Boundary Diffusion in Ionic Systems that arises from Enhanced Defect Concentrations Adjacent to the Boundary. Y.C.Chung, C.K.Kim, B.J.Wünsch: Journal of Applied Physics, 2000, 87[6], 2747-52