Nonlinear Diffusion Equation with Diffusion Coefficient Directly Proportional to Concentration of Impurities

Abstract

We analyzed a diffusion model based on the assumption that the sufficient condition for the mass flux at point x + L to be different from zero is a nonzero value of the impurity gradient and of impurity concentration at point x. In our model, the length of the jump of diffusing particles from one equilibrium position to another has a defined value L. By describing variation of impurity concentration with time when the frequency of the jumps depends on coordinates and L, the nonlinear diffusion equation was derived. We found that the diffusion coefficient in this nonlinear equation is directly proportional to the concentration of impurities, as it had been proposed in earliest papers. The derived nonlinear diffusion equation was solved numerically for the case of spherical symmetry.

Analytical Solution of Nonlinear Diffusion Equation Describing Interdiffusion of Gases

“…Using nonlinear diffusion theory for impurities diffusion in gases [6] we can transform linear diffusion equation into nonlinear [5,6] applying Eq. (3) for self-diffusion of first kind molecules.…”

“…Here we introduced, instead D 1 the nonlinear coefficient D 1n for nonlinear diffusion of impurities [6]. In this case we can transform Eq.…”

“…Those complicated mass transport problems for multicomponent systems can be realized [2] using the Darken equations [3] and numerical methods. We consider the binary diffusion of gases applying the mathematical methods and analytical solutions, derived for the nonlinear diffusion equation [4][5][6]:…”

“…Diffusion equation was introduced [5] using first Fick's law and requiring that diffusion of impurities must emanate with finite velocity and maximum penetration depth into the crystal. This nonlinear diffusion equation in the spherical-symmetric case [6]:…”

“…was applied for high-density impurities spread from spherical source into the gases [6] with diffusion coefficient D 0 . Here N 1 is maximum density of impurities and solvent, σ is the collision cross-section, L is the free path of diffusing particles.…”