Dielectric Enhancement from Non-Insulating Particles with Ideally Polarized Interfaces and Zero $\zeta$-Potential I: Exact Solution

Abstract

We solve exactly the dielectric response of a non-insulating sphere of radius $a$ suspended in symmetric, univalent electrolyte solution, with ideally-polarizable interface but without significant $\zeta$-potential. We then use this solution to derive the dielectric response of a dilute random suspension of such spheres, with volume fraction $f\ll1$, within the Maxwell-Garnett Effective Medium Approximation. Surprisingly, we discover a huge dielectric enhancement in this bare essential model of dielectric responses of solids in electrolyte solution: at low frequency $\omega\tau_D \ll (\lambda/a) / (\sigma_w / \sigma_s+1/2)$, the real part of the effective dielectric constant of the mixture is $1-(3f/2)+(9f/4)(a/\lambda)$. Here $\sigma_{w/s}$ is the conductivity of the electrolyte solution/solids, $\lambda$ is the Debye screening length in the solution, $\tau_D=\lambda^2/D$ is the standard time scale of diffusion and $D$ is the ion diffusion coefficient. As $\lambda$ is of the order nm even for dilute electrolyte solution, even for sub-mm spheres and low volume fraction $f=0.05$ the huge geometric factor $a/\lambda$ implies an over $10^4$-fold enhancement. Furthermore, we show that this enhancement produces a significant low frequency ($\omega\tau_D\ll1$) phase shift $\tan\theta = \mathrm{Re}~ \epsilon(\omega) / \mathrm{Im} ~\epsilon(\omega)$ in a simple impedance measurement of the mixture, which is usually negligible in pure electrolyte solution. The phase shift has a scale-invariant maximum $\tan\theta_{\mathrm{max}}=(9/4)f/(2\sigma_w/\sigma_s+1)$ at $\omega_{\mathrm{max}}=(2D/\lambda a)/(2\sigma_w/\sigma_s+1)$. We provide a physical picture of the enhancement from an accumulation of charges in a thin Externally Induced Double Layer (EIDL) due to the blocking boundary conditions on interfaces.