Low-frequency dielectric response of a periodic array of charged spheres in an electrolyte solution: The simple cubic lattice

Abstract

We study the low-frequency dielectric response of highly 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 regime where the ionic charge is neutral. We consider the case where the charged spheres have no surface conductance and no frequency-dependent surface capacitance. Hence, the frequency dispersion of the dielectric constant is dominated by the effect of neutral currents outside the electric double layer. In the thin double-layer limit, we use Fixman's boundary condition at the outer surface of the double layer to capture interaction between the electric field and the flow of the ions. For periodic conditions, we combine the methods developed by Lord Rayleigh for understanding the electric conduction across rectangularly arranged obstacles and by Korringa, Kohn and Rostoker for the electronic band structure computation. 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 greatly alters the dielectric response even when charged spheres occupy only two percent of the volume. We found that the characteristic frequency shifts to a higher value compared to that derived from the single-particle polarization response. At the same time, the low-frequency dielectric enhancement, a signature of charged spheres immersed in an electrolyte, becomes less prominent for the periodic array of charged spheres. Our results imply that the signature of the dielectric response of a system consisting of densely packed charged spheres immersed in an electrolyte can differ drastically from a dilute suspension.