Diffusion of hydrogen in heterogeneous systems

The effective long-range long-time tracer diffusivity D_eff for interstitial diffusion of hydrogen through heterogeneous systems was studied theoretically for model systems consisting of isolated grains of material G embedded in a matrix of material M. Different solubilities of hydrogen in these two materials as well as different diffusivities are 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. D_eff was determined by Monte Carlo (MC) simulations on simple lattice models of the systems described above. An equilibrium 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 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 c_G and c_M in the grains 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 terms of analytic formulas, we attempted 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 D_eff could be derived. The network formulas for different grain–matrix systems were tested by comparing with results of MC simulations. In general, the network formulas describe the corresponding MC results for D_eff 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 may have dramatic effects on D_eff. Received: 19 September 2000 / Accepted: 6 November 2000 / Published online: 9 February 2001