Electrolytic conduction in periodic arrays of insulators with charges

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Abstract
We compute dc electrical conductivity σ for a periodic array of charged cylinders immersed in an electrolyte in order to elucidate conduction in brine saturated porous medium containing clay particles. We extend a method due to Lord Rayleigh to charged systems and include diffusion currents. The key dimensionless parameter ξ=Ω+r+/(N0a) that represents the surface effects is the surface ion density Ω+ times the ratio r+ of the average diffusion coefficient in the double layer to that outside divided by the bulk density N0 times the particle radius a. We show that σ is a nonlinear function of ξ and hence of brine conductivity σw and thus provide the first explanation, resting on first principles, of this well known experimental fact. With a given geometry, when Ω+→0, or r+→0 or when N0≫1, the parameter ξ→0 and one obtains a linear dependence of σ on σw. The constant charge system have Ω+ constant, but when the surface potential ζ is constant, Ω+=2δN0eζ/2 varies with N0, δ being the screening distance.

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