Electrolytic conduction past arrays of charged insulating spheres

The dc electrical conductivity σ for a three-dimensional (3D) periodic array of charged spheres immersed in an electolyte is computed by augmenting a method due to Lord Rayleigh to include diffusion currents and by using Fixman’s method for the double layer. The key dimensionless parameter that represents the surface effects emerges from Fixman’s boundary conditions, and is given by ξ=Ω+r+/(N0a), where Ω+ is the surface ion number density, r+ is the ratio of the average diffusion coefficient in the double layer to that outside, N0 is the bulk ion density far from the double layer, and a is the particle radius. This calculation extends our previous calculation for 2D and supports our previous assertion that the 3D calculation differs only in details. We find that σ is a nonlinear function of ξ and hence of the electrolyte conductivity σw. σ vs σw shows the experimentally observed bend at low salinity. At high salinity, σ depends linearly on σw, resembling the empirical relations that are commonly used in exploration geophysics. In addition, our theory explains the observed cation dependencies, the particle size dependence of the surface effect, and the geometry dependence of the parameters in the empirical laws. For a large range of salinity, the theory is in good agreement with the data on latex suspensions both at high and low particle concentrations. Data on ion exchange resins are best explained by assuming that the particles are conductive. The nonspherical shape of clay particles rules out quantitative application of the theory; nevertheless, qualitative agreement is obtained. In particular, it is shown that for clay suspensions the isoconductance point, σ=σw, is weakly dependent on concentration, but depends on cation type and size of particle.