Steady-state diffusion with the smooth step dependence of the diffusion coefficient on concentration

The steady-state diffusion processes are investigated theoretically. Exact solutions of one-dimensional steady-state quasi-linear diffusion equation with smooth step dependence of the diffusion coefficient on concentration in various smooth step functions (linear step, exponential step, logistic, power saturation step) are obtained. The proposed theory is applied to describe peculiarities of diffusion-activated recrystallisation at steady-state stage at long times of diffusion annealing. The diffusant concentration monotonically decreases with an increasing depth of diffusant penetration. The smallest penetration depth of the diffusant is the characteristic of the logistic model. The analytical estimations of the diffusion coefficient with known recrystallisation front position are given. The position of recrystallisation front decreases monotonically with an increase in the model parameter, which corresponds to the smoothness of the step and the rate of change of the diffusion coefficient.