A detailed mathematical analysis was made of isolated and multiple grain boundary diffusion through thin films of finite thickness. The thin film was bounded by 2 permeable and parallel surface planes, and a constant flux of diffusing particles through both surfaces was assume to exist during the entire process. The flux was maintained at a low level in order to guarantee the applicability of Fick’s second law, with constant and position-independent diffusion coefficients. The boundary conditions which were considered were those found in symmetrical electrochemical cells for the determination of the chemical diffusion coefficients of mixed conductors. Fourier-Laplace transforms and Fourier analysis were used to solve Fick’s second law. Concentration profiles were calculated for various ratios of grain boundary diffusivity to volume diffusivity. Diffusion in polycrystalline thin films was considerably enhanced by an increase in the latter ratio, as well as by a decrease in the grain boundary distance. The results for multiple grain boundaries were compared with those which were obtained for the isolated grain boundary model. Satisfactory agreement between the concentration profiles in these 2 cases was found when the grain boundary distance was sufficiently large or the diffusion time fairly short.

W.Preis, W.Sitte: Journal of Applied Physics, 1996, 79[6], 2986-3002