On the Theory of Diffusion in Inhomogeneous Media: Long Times of the Process

Analytical solutions to the problem of one-dimensional diffusion in a semiinfinite inhomogeneous medium have been obtained with an arbitrary (monotonically decreasing) dependence of the diffusion coefficient D(x) on the distance to its boundary for long annealing times, when the characteristic scale of changes in D(x) is smaller than the diffusion-zone width. It has been shown that under certain conditions the diffusion zone becomes similar in the shape of the concentration profile to the two-phase zone with a moving boundary. The main characteristics of the process, such as the concentration profiles and the position and velocity of the diffusion front, have been determined. In order to test the approach and to establish the ranges of its applicability, the approximate solutions obtained have been compared numerically with the known exact solution for the case of exponential dependence D(x). Within the approach developed, some features of diffusion near nonequilibrium grain boundaries in nanostructured materials have been discussed.