Numerical analysis on the formation of Cottrell atmosphere using Fokker–Planck equation

Abstract

In this study, we develop a new theoretical model for the quantitative prediction of atomic diffusion in a crystalline material. Concentration of the solute atom is expressed by a probability density function whose time evolution is governed by the Fokker–Planck equation. The stochastic differential equation describes the two competitive processes; drift motion by the stress field and thermal diffusion by the Brownian motion. Non-singular stress field by a lattice defect is obtained from nonlinear elastoplasticity on Riemann–Cartan manifold. Numerical integration is conducted using the finite difference method which satisfies the numerical stability criteria. Present analysis demonstrates the formation of Cottrell atmosphere around a straight edge dislocation. The probability density reaches to an equilibrium distribution and the time evolution satisfies the Cottrell’s theoretical prediction quantitatively. The mean diffusion path is predicted from the streamline which includes the surface effects. We also demonstrate the dislocation pipe diffusion, i.e., accelerated diffusion along a dislocation line.

Keywords

• fokker–planck equation
• cottrell atmosphere
• diffusion of solute atoms
• probability density
• stress field