A mathematical analysis of diffusion in dislocations. III. Diffusion in a dislocation array with diffusion zone overlap

For pt.II see ibid., vol.15, p.3445 (1982). The authors consider diffusion from a surface y=0 into a solid containing a regular array of d dislocations per unit area, all normal to the surface and each represented as a pipe of radius a within which the diffusion coefficient D' is very much greater than that, D, for diffusion in a regular crystal. They calculate the mean concentrations (c(y)) at depths y, as determined in conventional sectioning experiments, for both constant concentration and thin-finite-source conditions at y=0. (c) is evaluated as a function of eta =y/(Dt)¹2/, alpha =a/(Dt) ¹2/, Delta =D'/D, epsilon ²= pi a²d and of epsilon / alpha , the ratio of the mean diffusion length, (Dt)¹2/, to the effective half-spacing between dislocations, ( pi d)^-12/. Full account is taken, through an appropriate Wigner-Seitz-like boundary condition, of the effects of the mutual overlap of the diffusion zones around dislocations. For epsilon / alpha <or approximately=1, values of (c) do not differ by more than a few per cent from those previously calculated by the authors for well spaced non-interacting dislocations; for epsilon / alpha <or approximately=0.4 the two solutions are numerically identical. As epsilon / alpha increases above one the effects of overlap rapidly become increasingly important. They examine the nature of the penetration plots, log(c) versus y², and the effective diffusion coefficients, D_eff, deduced from them. The case epsilon / alpha <or=1 was discussed previously by the authors. The plots show maximum curvature around epsilon / alpha approximately=5 but would be difficult experimentally to distinguish from linear when epsilon / alpha >or approximately=10. For this range D_eff is found to be closely represented by the Hart relation, D_eff=D(1+ pi a²d( Delta -1)), the condition of validity of which can now be specified more precisely than heretofore.