Influence of rough surfaces on electrolytic conduction in porous media

There are many examples of porous media in which electrical transport is associated with a pore fluid of conductivity ${\ensuremath{\sigma}}_{f}$ and a constant excess surface conductivity ${\ensuremath{\Sigma}}_{s}$. In all such systems the relation between the effective conductivity ${\ensuremath{\sigma}}_{\mathrm{eff}}$ and ${\ensuremath{\sigma}}_{f}$ is nonlinear due to intrinsic geometrical effects. By calculating ${\ensuremath{\sigma}}_{\mathrm{eff}}$ for a sequence of geometrical models, we show that the character of this relation depends on the degree of surface roughness. In particular, we consider (1) a three-dimensional consolidated sphere pack with a smooth pore-grain interface, (2) a two-dimensional model in which a self-affine fractal interface is generated by a random-walk algorithm, and (3) a two-dimensional model with a fractal Koch-curve interface. For each of these models, the behavior of ${\ensuremath{\sigma}}_{\mathrm{eff}}$ is shown to be reasonably well described by a simple Pad\'e approximant based on four independently measurable geometrical parameters. Our analysis provides a physical explanation for the puzzling fact that often only a fraction of the total surface charge (as determined by chemical titration) contributes to electrolytic conduction.