An extension was made to the theory of the coefficient of grain-boundary diffusion for the case of relatively large annealing times, when the width of the saturated sub-boundary zone exceeds the characteristic scale of decreasing coefficient of diffusion but no overlap of diffusion zones occurred in the bulk of nanograins. An analysis of the diffusion problem in this case led to solutions that in form were analogous to the solution of the problem of one-dimensional diffusion along grain boundaries, corresponding to the C-regime of annealing in conventional polycrystals but with time-dependent effective parameters (the grain-boundary diffusion width and the grain-boundary diffusion coefficient). It was shown that the allowance for the existence of a sub-boundary region of enhanced diffusion led to a decrease in the depth of the diffusant penetration along the boundary and to a simultaneous increase in the average sheet concentration. Estimates of these diffusion characteristics for nanocrystalline copper were given. Results of numerical calculations of the diffusion problem were presented, which made it possible to establish the field of the applicability of the approach suggested.

On the Theory of Grain-Boundary Diffusion in Nanostructured Materials Under Conditions of Saturation of the Subboundary Region by the Diffusant. A.G.Kesarev, V.V.Kondratev, I.L.Lomaev: The Physics of Metals and Metallography, 2011, 112[1], 44-52