A compact formula for the effective diffusivity of two-dimensional, anisotropic porous media with surface diffusion and interacting phases

Abstract

The effective diffusivity of two-dimensional, anisotropic porous materials with surface diffusion is studied. The continuum model of Albaalbaki and Hill (2012), which couples diffusion in the bulk and surface domains via interfacial exchange fluxes, is implemented to couple the phases. Using a cell model, a new analytical solution is developed for aligned fibres with elliptic cross-sections and arbitrary orientation with respect to the mean gradient. The anisotropic boundary-value-problem is solved using an isotropic approximation to furnish concentration distributions in the three phases. Therefore, the model is more accurate near the isotropic limit and at lower inclusion volume fractions. When surface diffusion is significant, the present anisotropic model reproduces the isotropic model of Albaalbaki and Hill (2012) for unit aspect ratio and a variety of physical parameters. For a sphere with negligible surface flux, the model agrees with Maxwell’s theory, and reproduces the model of Akbari et al. (2013) with various aspect ratios. To test the model for several parameters and other aspect ratios, direct numerical computations of the effective diffusivity, using spatially periodic unit cells, are undertaken, and a comparison with experimental data is presented. This model serves as a two-dimensional solution for the effective diffusivity of dilute anisotropic structures with surface diffusion.