THE DIFFUSE LAYER CORRECTION TO THE DISCRETE-ION EFFECT IN ELECTRIC DOUBLE LAYER THEORY

Abstract

The screening effect of the diffuse layer on the self-atmosphere (or discrete-ion) potential of a counter-ion adsorbed from aqueous electrolyte on to a plane charged interface is investigated. A theory based on Fourier–Bessel integrals and the approximate method of Loeb and Williams for treating the potential distribution in the diffuse layer is presented. The distribution of counter-ions in the adsorption plane surrounding a specified ion adsorbed in the plane is expressed in terms of a two-dimensional grand canonical ensemble and this is approximated by a 'revised cut-off model'. The 'Grahame cut-off model' is an approximation to the adsorbed counter-ion distribution based on a canonical ensemble and we present arguments to show that this interpretation is incorrect. We also show that the radius of the revised disc is, in general, smaller than that of the Grahame disc but approaches it at large charge densities (σβ) of counter-ions in the Stern inner region. Comparison is made with an alternative approach to this discrete-ion effect by Buff and Stillinger, who also applied a Fourier–Bessel technique to the mercury – aqueous electrolyte system. Their results provide a lower estimate to the magnitude of the self-atmosphere potential, whereas our revised cut-off model gives an upper estimate. A third method, which assumes the adsorbed ions to be situated on a hexagonal lattice, is also discussed and is found to give a potential larger than the upper estimate of the revised cut-off approach.It is shown that the effect of the diffuse layer is to diminish the discrete-ion potential at the adsorbed ion. When the electrolyte concentration tends to zero the results with the Grahame model reduce to those derived in an earlier paper from the method of multiple electrostatic images. Calculations on the discrete-ion potential at an adsorbed counter-ion are given for metallic and dielectric surfaces in contact with an aqueous 1–1 electrolyte solution and the parameters of the inner region are chosen to be consistent with known data on mercury and silver iodide systems. It is shown in both cases that on dilute salt solutions [Formula: see text] the diffuse layer screening is small compared to the unscreened term except at small charge densities of counter-ions in the inner region.