Diffusion of hydrogen in heterogeneous systems

The effective long-range long-time tracer diffusivity D-eff for interstitia l diffusion of hydrogen through het erogeneous systems was studied theoreti cally for model systems consisting of isolated grains of material G embedde d in a matrix of material M. Different solubilities of hydrogen in these tw o materials as well as different diffusivities are allowed for. Additionall y, 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. D-eff was determined by Monte Carlo (MC) s imulations on simple lattice models of the systems described above. An equi librium distribution of hydrogen atoms among the two constituent materials was assumed. Our main interest was focused on whether and how D-eff may be related to mesoscopic or macroscopic quantities characterizing the heteroge neous 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 c(G) and c(M) in the gr ains and in the matrix and the respective hydrogen diffusivities D-G(c(G)) and D-M(c(M)) In order to obtain good estimates for these relations in term s of analytic formulas, we attempted to model a heterogeneous system by a n etwork of diffusion elements connected in series and in parallel, in analog y to an electric network. The properties of the basic connections, in paral lel and in series, were studied on layered structures, for which analytic e xpressions for D-eff could be derived. The network formulas for different g rain-matrix systems were tested by comparing with results of MC simulations . In general, the net work formulas describe the corresponding MC results f or D-eff fairly well. It was found that differences in the hydrogen solubil ities in the two phases as well as modified energy barriers at the phase bo undaries may have dramatic effects on D-eff.