Differential capacitance of the electric double layer: The interplay between ion finite size and dielectric decrement

TL;DR

This paper analytically investigates how ion finite size and dielectric decrement influence the electric double layer's capacitance, revealing saturation effects, asymmetry, and shape transitions in capacitance profiles under various conditions.

Contribution

It derives a generalized analytical framework for the differential capacitance considering both ion size and dielectric decrement effects, including saturation criteria and shape transitions.

Findings

Counter-ion saturation depends on ion size and dielectric decrement.

Capacitance profile exhibits camel-shape at low salinity and transitions to uni-modal at high salinity.

Nonlinear dielectric models predict higher saturation concentrations than linear models.

Abstract

We study the electric double layer by combining the effects of ion finite size and dielectric decrement. At high surface potential, both mechanisms can cause saturation of the counter-ion concentration near a charged surface. The modified Grahame equation and differential capacitance are derived analytically for a general expression of a permittivity epsilon(n) that depends on the local ion concentration, n, and under the assumption that the co-ions are fully depleted from the surface. The concentration at counter-ion saturation is found for any epsilon(n), and a criterion predicting which of the two mechanisms (steric vs. dielectric decrement) is the dominant one is obtained. At low salinity, the differential capacitance as function of surface potential has two peaks (so-called camel-shape). Each of these two peaks is connected to a saturation of counter-ion concentration caused either by dielectric decrement or by their finite size. Because these effects depend mainly on the counter-ion concentration at the surface proximity, for opposite surface-potential polarity either the cations or anions play the role of counter-ions, resulting in an asymmetric camel-shape. At high salinity, we obtain and analyze the crossover in the differential capacitance from a double-peak shape to a uni-modal one. Finally, several nonlinear models of the permittivity decrement are considered, and we predict that the concentration at dielectrophoretic saturation shifts to higher concentration than those obtained by the linear decrement model.