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

The dielectric constant and conductivity of a dilute ensemble of immobile, spherical particles with fixed surface (zeta) potential Φ_o, immersed in an electrolytic solution, is obtained in the thin double layer approximation δ≪α, δ being the thickness of the double layer, and α 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 E_o and with coefficients that are functions of the unperturbed potential (zeroth order in E_o), are solved to second order in (δ/α). The term giving enhancement in the real part of the effective dielectric constant of the ensemble ε′_e, is second order in δ/α; but the series converges if (δ/α)t²/(1-t²)≪1, where t = tanh(eΦ_o/k_BT), e being the ionic charge, k _B the Boltzmann constant, and T the absolute temperature. The static value of ε′_e to this order, is ε′ _e∼36fε′t²/(1-t²)², where f is the volume fraction of particles, ε′ the real part of the dielectric constant of the solution. When Φ_o→∞, therefore, t→1, ε′_e diverges as ε′ _e∼9/4fε′ exp[eΦ_o/k_BT). 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 Φ_o. Comparison with other theories are made.