Theory of Double-Layer Differential Capacitance in Electrolytes

Abstract

A theory of the double layer in uni-univalent unadsorbed electrolytes is developed and used to analyze Grahame's experimental measurements of differential capacitance for NaF in water at 0° to 85°C and KF in methanol at 25°C. Excellent agreement with experiment is obtained except in the region of strong anodic polarization; this disagreement is tentatively ascribed to specific adsorption of anions, an effect not quantitatively considered in the present work. Although the quantities calculated herein relate to the entire double layer as, of course, do Grahame's data, the Gouy—Chapman theory of the diffuse part of the double layer (without dielectric saturation) is adequate in the present situation for all concentrations considered and has been used throughout. Consequently, the degree of agreement between theory and experiment found reflects primarily upon the applicability of the present theory of the inner layer. In the absence of specific adsorption this region is taken to be a hexagonally close-packed charge-free monolayer of solvent, physically adsorbed on the mercury electrode by dipole image forces. Adsorption anisotropy can lead to some dielectric saturation in the inner layer even at the electrocapillary maximum, the point of zero electrode charge. Neglecting association in the monolayer, the inner-layer dielectric constant and its dielectric saturation properties are calculated under three situations—where dipole image contributions are neglected, where the monolayer dipoles are imaged in the mercury electrode only, and where they are additionally imaged in an equipotential plane on the other side of the layer. The last case leads to an infinite set of images and to infinite series which are summed. These treatments all lead to much smaller dielectric constants and saturation constants than are found for bulk solvent. Comparison with values of these constants obtained from fitting the theory to the experimental data using a digital computer yields reasonably close agreement. New equations for the dependence of inner-layer thickness, volume, and dielectric constant on pressure and electric field, are derived and applied. The electrostatic pressure in this region is shown to consist of a capacitor-plate compressive term and an electrostrictive term, the latter originating only from the distortional and not the orientational polarization of the inner layer. As with the dielectric properties, the compressibility of the inner region found from curve fitting is of the right order of magnitude for both water and methanol solvents. The hump which occurs in water at low temperatures and small anodic polarization is attributed to the interplay of specific adsorption and dielectric saturation. Finally, it is pointed out that the usual method of separation of the inner-layer capacitance from the total differential capacitance by assuming the former to be in series with the diffuse-layer capacitance is unjustified in regions of appreciable specific adsorption, where the inner layer is no longer charge free.