Dielectric enhancement due to electrochemical double layer: Thin double layer approximation

Abstract

The dielectric constant and conductivity of a dilute ensemble of immobile, spherical particles with fixed surface (zeta) potential Φo, immersed in an electrolytic solution, is obtained in the thin double layer approximation δ≪a, δ being the thickness of the double layer, and a the radius of the particles. Equations of motion for coions and counter-ions are solved by the method of matched asymptotics. The equations of motions, linearized in the applied electric field Eo and with coefficients that are functions of the unperturbed potential (zeroth order in Eo), are solved to second order in (δ/a). The term giving enhancement in the real part of the effective dielectric constant of the ensemble ε1e, is second order in δ/a; but the series converges if (δ/a)t2/(1–t2)≪1, where t=tanh(eΦo/kBT), e being the ionic charge, kB the Boltzmann constant, and T the absolute temperature. The static value of ε1e, to this order, is ε1e∼36fε1t2/(1–t2)2, where f is the volume fraction of particles, ε1 the real part of the dielectric constant of the solution. When Φo→∞, therefore, t→1, ε1e diverges as ε1e∼9/4fε′ exp(eΦo/kBT). The present treatment is free from the approximations of previous analytical results. When applicable, the theory agrees well with experiments over three decades in frequency, with one adjustable parameter Φo. Comparison with other theories are made.