The problem of calculating the effective diffusivity in nanocrystalline materials was addressed. Two simple 2-dimensional grain patterns were constructed and several equations were derived from the Hart–Mortlock and Maxwell–Garnett equations. Monte Carlo computer simulation was used to test the validity of the various equations. It was shown that one of the derived equations provided an excellent description of the effective diffusivities up to a volume fraction of atoms in grain boundaries of 40%.

Diffusion in Nanocrystalline Materials. I.V.Belova, G.E.Murch: Journal of Physics and Chemistry of Solids, 2003, 64[5], 873-8