Debye-Hückel Free Energy of an Electric Double Layer with Discrete Charges Located at a Dielectric Interface.

Guilherme Volpe Bossa,Sylvio May

doi:10.3390/membranes11020129

Guilherme Volpe Bossa, Sylvio May

Open Access

https://doi.org/10.3390/membranes11020129

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Abstract

Poisson–Boltzmann theory provides an established framework to calculate properties and free energies of an electric double layer, especially for simple geometries and interfaces that carry continuous charge densities. At sufficiently small length scales, however, the discreteness of the surface charges cannot be neglected. We consider a planar dielectric interface that separates a salt-containing aqueous phase from a medium of low dielectric constant and carries discrete surface charges of fixed density. Within the linear Debye-Hückel limit of Poisson–Boltzmann theory, we calculate the surface potential inside a Wigner–Seitz cell that is produced by all surface charges outside the cell using a Fourier-Bessel series and a Hankel transformation. From the surface potential, we obtain the Debye-Hückel free energy of the electric double layer, which we compare with the corresponding expression in the continuum limit. Differences arise for sufficiently small charge densities, where we show that the dominating interaction is dipolar, arising from the dipoles formed by the surface charges and associated counterions. This interaction propagates through the medium of a low dielectric constant and alters the continuum power of two dependence of the free energy on the surface charge density to a power of 2.5 law.

Highlights

Charged interfaces in aqueous solution give rise to the formation of an electric double layer—a diffuse cloud of co- and counter-ions that screen the interfacial charges [1,2]
Some studies have focused on the pressure that acts across electrolytes due to the presence of discrete charges [28,35,38], yet—perhaps somewhat surprisingly—there has not been an attempt so far to compute the free energy of a single planar dielectric interface with discrete charges and compare this with the corresponding free energy derived for a continuous charge distribution
Charges are often located at dielectric interfaces between a salt-containing aqueous solution and a salt-free medium of low dielectric constant

Summary

Introduction

Charged interfaces in aqueous solution give rise to the formation of an electric double layer—a diffuse cloud of co- and counter-ions that screen the interfacial charges [1,2]. Some studies have focused on the pressure that acts across electrolytes due to the presence of discrete charges [28,35,38], yet—perhaps somewhat surprisingly—there has not been an attempt so far to compute the free energy of a single planar dielectric interface with discrete charges and compare this with the corresponding free energy derived for a continuous charge distribution. Filling this gap on the level of the linearized Debye-Hückel model is the goal of the present work

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