A mathematical analysis of diffusion in dislocations. I. Application to concentration 'tails'

Abstract

A mathematical description of dislocation diffusion is presented that avoids the grosser approximations employed by previous authors. Dislocations are represented as cylindrical 'pipes' of radius a, perpendicular to a surface y=0, within which the diffusion coefficient D' is very much greater than the coefficient D elsewhere. With the sole assumption that the concentration within the pipe is uniform radially (because D'>>D) and equal to the concentration outside at r=a, the authors calculate the total mean concentration (c) at depth y for diffusion (i) from a constant concentration source at y=0 and (ii) from a finite thin source deposited at y=0. In both cases, log (c) varies almost exactly linearly with y at distances y>or approximately=4(Dt)^1/2 and with exactly the same slope. Absolute values of (c) are much greater, by 1 to several powers of 10, than those previously calculated by Mimkes and Wuttig (1970). From the slope and the extrapolated intercept, at y=0, of this linear 'tail' region can be estimated D'a^2/D and the dislocation density d.