Analysis of Three-dimensional Grain-boundary Diffusion

Abstract

Grain boundaries play an important role in the material transfer in polycrystalline solids. Formerly, the diffusion processes in the solids were analyzed using homogeneous models or the one-dimensional grain boundary model which had been given by Fisher. In this study, we discuss a three-dimensional grain boundary model which represents actual gr ain boundary structure more properly than the conventional models do. Fig.1 shows the schematic of the model and Equations(1)∼(5) represent the diffusion equations for normal grain boundary 1, normal grain boundary 2, parallel grain boundaries, the junctions of these grain boundaries and for the inside of grains, respectively. Equations (6)∼(12) are the boundary conditions. The concentration distributions for the thin film model were calculated using a numerical method with' finite-difference equations. One of the calculated concentration distributions is sho-wrii n Fig.2.
The diffusion species flows rapid ly via grain boundaries and penetrate into grains when the ratio, of the diffusivity in grain to that in grain boundary (D_G/D_B) is less than 10^-3. When this ratio is more than 10^-2, the diffusion profile is similar to that in homogeneous medium which have. no grain boundaries. The concentration near the substrate of the film lowered when the diffusivity in grain increased (Fig.3) or when grain boundary thickness decreased (Fig.4). The concentration diptribu. tion in grain boundary was not largely affected by the number of grains (Fig.5). By comparing the results of three-dimensional grain boundary model with the two-dimensional model, it is known that the grain boundary normal to the film surface largely contributes to increase in the amount of solute content in the film.