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

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Abstract
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≪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 fieldE o and with coefficients that are functions of the unperturbed potential (zeroth order in E o ), are solved to second order in (δ/a). The term giving enhancement in the real part of the effective dielectric constant of the ensemble ε1 e , is second order in δ/a; but the series converges if (δ/a)t 2/(1–t 2)≪1, where t=tanh(eΦ o /k B T), e being the ionic charge, k B the Boltzmann constant, and T the absolute temperature. The static value of ε1 e , to this order, is ε1 e ∼36fε1 t 2/(1–t 2)2, where f is the volume fraction of particles, ε1 the real part of the dielectric constant of the solution. When Φ o →∞, therefore, t→1, ε1 e diverges as ε1 e ∼9/4fε′ exp(eΦ o /k B T). 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.