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

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Abstract
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 2, and the effective diffusion coefficients, Deff, deduced from them. The case epsilon / alpha or approximately=10. For this range Deff is found to be closely represented by the Hart relation, Deff=D(1+ pi a2d( Delta -1)), the condition of validity of which can now be specified more precisely than heretofore.