Dielectric Enhancement from Non-Insulating Particles with Ideally Polarized Interfaces and Zero $ζ$-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 $ζ$-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 $ωτ_D \ll (λ/a) / (σ_w / σ_s+1/2)$, the real part of the effective dielectric constant of the mixture is $1-(3f/2)+(9f/4)(a/λ)$. Here $σ_{w/s}$ is the conductivity of the electrolyte solution/solids, $λ$ is the Debye screening length in the solution, $τ_D=λ^2/D$ is the standard time scale of diffusion and $D$ is the ion diffusion coefficient. As $λ$ 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 $a/λ$ implies an over $10^4$-fold enhancement. Furthermore, we show that this enhancement produces a significant low frequency ($ωτ_D\ll1$) phase shift $\tanθ= \mathrm{Re}~ ε(ω) / \mathrm{Im} ~ε(ω)$ in a simple impedance measurement of the mixture, which is usually negligible in pure electrolyte solution. The phase shift has a scale-invariant maximum $\tanθ_{\mathrm{max}}=(9/4)f/(2σ_w/σ_s+1)$ at $ω_{\mathrm{max}}=(2D/λa)/(2σ_w/σ_s+1)$. 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.