Low-frequency dielectric response of a periodic array of charged spheres in an electrolyte solution

We study the low-frequency dielectric response of charged spheres arranged in a cubic lattice and immersed in an electrolyte solution. We focus on the influence of the out-of-phase current in the ionic charge neutral regime. In the thin double-layer limit, we use Fixman's boundary condition at the outer surface of the double layer to capture the direct interaction between the electric field and the flow of the ions. For periodic conditions, we combine the methods developed by Lord Rayleigh and by Korringa, Kohn and Rostoker. When the charged spheres occupy a very small volume fraction, smaller than one percent, our solution becomes consistent with the Maxwell Garnett mixing formula together with the single particle polarization response, as expected, because inter-particle interactions become less prominent in the dilute limit. By contrast, the inter-particle interaction, computable in the periodic geometry, greatly alters the dielectric response when charged spheres occupy even a slightly higher volume fraction, as low as two percent. Our results imply that the signature of the dielectric response of a system consisting of densely packed charged spheres immersed in the electrolyte can differ drastically from a dilute suspension.