Grain boundary diffusion through thin films. Application to permeable surfaces

Abstract

A detailed mathematical analysis of isolated as well as multiple grain boundary diffusion through thin films of finite thickness has been performed. The thin film is bounded by two permeable and parallel surface planes, and a constant flux of diffusing particles through both surfaces is maintained during the entire diffusion process. The flux is kept small in order to guarantee the applicability of Fick’s second law of diffusion with constant and position independent diffusion coefficients. The boundary conditions considered can be found in symmetric electrochemical cells applied to the determination of chemical diffusion coefficients of mixed conductors. Both Fourier–Laplace transforms and Fourier analysis techniques are used to solve Fick’s second law of diffusion. Various concentration profiles for different parameter sets (grain boundary/volume diffusion coefficient ratio and grain boundary distance) have been calculated. The diffusion process in polycrystalline thin films is considerably enhanced by an increase of the grain boundary/volume diffusion coefficient ratio as well as a decrease of the grain boundary distance. The results for multiple grain boundaries are compared with those obtained from the isolated grain boundary model. A satisfactory coincidence between the concentration profiles of these two models is achieved when the grain boundary distance is sufficiently high or the diffusion time fairly small.