The effective long-range long-time tracer diffusivity Deff for interstitial diffusion of hydrogen through heterogeneous systems was studied theoretically for model systems consisting of isolated grains of G embedded in a matrix of M. Different solubilities of hydrogen in these two materials as well as different diffusivities were allowed for. Additionally, modified diffusion barriers at the phase boundaries were included in the diffusion model. The effect of different sizes, arrangements, and forms of the grains was also considered. Deff was determined by Monte Carlo simulations on simple lattice models of the systems described above. An equilibrium distribution of hydrogen atoms among the two constituent materials was assumed. The main interest was focused on whether and how Deff may be related to mesoscopic or macroscopic quantities characterizing the heterogeneous system and its constituent materials, such as the volume fractions of the two materials, the fraction of lattice sites in the immediate vicinity of the phase boundary, the hydrogen concentrations cG and cM in the grains and in the matrix and the respective hydrogen diffusivities DG(cG) and DM(cM). In order to obtain good estimates for these relations in terms of analytic formulas, an attempt was made to model a heterogeneous system by a network of diffusion elements connected in series and in parallel, in analogy to an electric network. The properties of the basic connections, in parallel and in series, were studied on layered structures, for which analytic expressions for Deff could be derived. The network formulas for different grain-matrix systems were tested by comparing with results of Monte Carlo simulations. In general, the network formulas describe the corresponding Monte Carlo results for Deff fairly well. It was found that differences in the hydrogen solubilities in the two phases as well as modified energy barriers at the phase boundaries could have dramatic effects upon Deff.

Diffusion of Hydrogen in Heterogeneous Systems. A.Herrmann, L.Schimmele, J.Mössinger, M.Hirscher, H.Kronmüller: Applied Physics A, 2001, 72[2], 197-208