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.

A self-consistent treatment is presented of the change in work function, or surface potential, produced by the adsorption of an immobile or mobile regular array of adsorbate entities on a plane (usually conducting) adsorbing surface. The adsorbed entities may be polarizable atoms, ions, or molecules, and the molecules and ions may have orientable permanent dipole moments, not assumed fixed in direction. The depolarizing field at a given adsorbed element arising from the total polarization of all its surrounding elements is taken into account in all cases, as is the possible presence of an average charge on the adsorbing surface. A distinction is introduced between the ``natural'' field polarizing a single adsorbate entity and the similar effective field leading to some time-average permanent dipole polarization. General formulas for surface potential in terms of surface charge and surface coverage are derived for all cases and compared with earlier, less general treatments of the same cases. The results are applied to electrolyte double-layer measurements and surface-potential determinations.

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A convenient general method for calculating potentials and fields arising from planar arrays of discrete adions under a variety of imaging conditions is described and illustrated. Adions are perfectly imaged by one conducting plane (single imaging) and are also imaged by a dielectric constant discontinuity at a plane on their other side. The method employs only solutions of the single imaging problem, is readily applied without a computer, and is pertinent to the usual electrolyte compact layer adjoining either an electrode or a dielectric material, which may be air. The single image solutions used in calculating more complex imaging results may be exact values obtained from a previous rather complicated approach, or for ease in calculation, may frequently be approximate but quite accurate values calculated by a simple method described herein. Using the exact approach, one can calculate, for the full range of the dielectric reflection parameter, fields and potentials along any line perpendicular to the conducting plane. Here we are primarily concerned with potentials and fields along the line through a removed adion, and the approximate single imaging solution is especially useful. Although we apply the method to regular hexagonal arrays, in the ....' " 1 latter case it is equally apphcable to arrays described by. Grahame s partla ly smeared, cut-off model for single imaging. Some comparison with the results of this model is presented. In addition to calculating and illustrating the variation of field and potential within the compact layer and the adjoining dielectric medium, we have examined in detail the difference between the micropotential and the macropotential for many different imaging situations. The present study includes the previously treated single conductive plane imaging and also the (infinite) conductive-conductive imaging situations as special cases. It is found that special care is needed to describe the latter situation by the present model. Finally, the effect of possible conductive imaging by the electrolyte diffuse layer is considered qualitatively.The system we shall consider in this paper is the electrolyte compact double layer (1). We shall assume it consists of a monolayer of ions (effective charge zve and average surface charge density qz) bounded on one side by a plane interface which we shall call the electrode-surface plane (ESP), generally (but not always) associated with an adsorbing conductor, and on the other side by an imaginary plane marking the points of closest approach of the charge centroids of ions in the electrolyte, or diffuse layer. The plane of closest approach is known as the outer Helmholtz plane (OHP), and the plane passing through the charge centroids of the adions in the monolayer is the inner Helmholtz plane (IHP). We shall define to be the distance between the OHP and the IHP and ~ to be the IHP-ESP separation; d ---/9 + "r is therefore the total thickness of the compact layer. REGION

For an infinite hexagonal array of ions adsorbed on a conducting plane of infinite extent, the thermal fluctuation from strict lattice ordering in the neighborhood of a given adion is considered. The ions are imaged in the uniform, conducting adsorbent and are assumed to move freely in the plane; they thus arrange themselves in a perfect hexagonal array a t absolute zero temperature. By use of the accurate planar potential seen by one adion moving in the field arising from a n infinite number of fixed hexagonally arrayed surrounding ions, the root-meansquare (r.m.s.) amplitude of planar vibration of the ion relative to its neighbors is approximately determined for several values of nearest neighbor distances between ions, rl. On the basis of these results, we find, for example, that for a distance, P, between the center of charge of a n ion and the imaging plane of 3 A, an ionic valence z, of unity, and a n effective dielectric constant of e, a hexagonal array with rl = 15 A is stable up t o a temperature, To, of approximately 1760/e OK while one with rl = 21 A is stable up to about 760/e OK. Results apply t o adsorption from either a gas or liquid phase and, as well, to a n array of real dipoles adsorbed on a nonconducting surface. The appropriate values for e are of the order of 2 and 6, respectively, for adsorption from gas or from aqueous electrolytes. For various rl's, explicit expressions for T o are obtained which depend on e, 8, and z, . -

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Articles you may be interested inEffects of image charges, interfacial charge discreteness, and surface roughness on the zeta potential of spherical electric double layers A selfconsistent study of ion adsorption and discrete charge effects in the electrical double layerThe Ershler infinite imaging model for adsorption of ions from electrolytes is discussed ill detail. An incorrect derivation of the basic equations for the micropotential is critically examined and a correct derivation provided. ~he potential arising from discret~ adions and their images is determin~d exactly based upon a method which heretofore has only been applIed to crystal field problems. Finally the accurate results obtained using a digital computer are compared with the approximate results of othe~ treatments.