Unified model of conductivity and membrane potential of porous media

It is shown explicitly, for the first time, how the microgeometry influences the membrane potential $E_m$ of a porous medium, in which the pore walls carry an electric double layer. A unified model of the electrical conductivity ${\sigma }_{\mathrm{eff}}$ and the membrane potential is obtained. It is shown that ${\sigma }_{\mathrm{eff}={\sigma }_{\mathrm{cation}+{\sigma }_{\mathrm{anion}}}}$, and $E_m$∝${\mathrm{\sigma }}_{\mathrm{cation}}$(${\mathrm{\sigma }}_{\mathrm{cation}}$+${\mathrm{\sigma }}_{\mathrm{anion}}$), for any arbitrary tortuosity. The effective tortuosities for the macroscopic cation and anion conductivities, ${\sigma }_{\mathrm{cation}}$ and ${\sigma }_{\mathrm{anion}}$, are distinct, as they depend on the microgeometries of the pore and the surface and on the surface charges. For a positively charged surface, ${\sigma }_{\mathrm{cation}}$ increases nonlinearly as a function of ${\sigma }_w$, the conductivity of the electrolyte which fills the pore space, whereas the ${\sigma }_{\mathrm{anion}}$ varies linearly. As ${\sigma }_w$ increases, the dominant paths for the cation current shift from surface to pore, and, therefore, are subjected to different tortuosities at different ${\sigma }_w$, while the tortuosity for the anion does not vary with ${\sigma }_w$. The use of the electrical conductivity and the membrane potential as the probes of the microstructure is explored.