Calculation of the contribution to grain boundary diffusion in ionic systems that arises from enhanced defect concentrations adjacent to the boundary

The enhancement of the concentration of a defect in the space-charge region near 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 enhanced and continuously varying diffusivity serves as a vehicle for enhanced transport along the boundary in addition to the expected contribution arising from the different structure at the boundary core. A two-dimensional diffusion equation was established for concentration c(x,y,t) in the space-charged region, solved numerically using the Crank-Nicolson finite-difference method, and integrated normal to the boundary to provide the average concentration gradient (c) over bar(y,t) along the boundary-the solute distribution usually measured by experiment. The present gradients were analyzed in the same fashion as experimental data using a recent solution to the conventional model for the grain-boundary diffusion problem in which the interface is treated as a thin slab of half-width a and enhanced diffusivity D-' imbedded in a material with bulk diffusivity D. The analyses provided effective values for the conventional diffusion parameter beta = [(D'/D) - 1]left perpendiculara/Dtright perpendicular that describe the influence of space charge on enhanced diffusion along the boundary. The calculations were repeated for a number of values of the parameter Z(i)e phi(infinity)/kT, where Z(i)e is the effective charge of the defect and phi(infinity) is bulk electric potential far from the boundary. In the finite-difference calculations the value of the "annealing time," t, that appears in beta was set equal to the product of the square of the Debye length, delta(2), divided by the bulk diffusion coefficient, D. Given a value of phi(infinity), the present results thus permit an estimation of the effect of space charge on preferential diffusion along a grain boundary in a material in terms of beta. If the Debye length is known for the material or can be measured, the value of beta can readily be converted to an effective value of the conventional grain boundary diffusion product aD'/D. (C) 2000 American Institute of Physics. [S0021-8979(00)07806-3].