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A general theory of the interaction of two charged identical colloidal particles of arbitrary shape is developed. An expression for the Helmholtz free energy of the electric double layers is obtained by the methods of statistical mechanics. The condition that there is equilibrium between the ions adsorbed on the surfaces of the colloidal particles and those dissolved in the dispersion medium is accounted for by requiring that the free energy of the whole system be a minimum with respect to variation of the ionic density on the surfaces.The theory presented here is a further development of the work of Verwey and Overbeek. The conclusions of this paper are that in dilute sols, to which the present investigations are restricted, the results of these authors require extension in two directions. First, there is a correction to the mutual energy of two particles, due to the Coulomb interaction of the ions in the bulk of the solution. Secondly, no special assumption concerning the relation between the surface potential (or charge) and interparticle separation need be introduced. The equations set up to determine the free energy of interaction at the same time yield the ‘adsorption isotherm’ for the ion type which is common to the solution and the surface of the particle.

An extension is proposed to the theory of Verwey and Overbeek on the force between two colloidal particles in a dilute sol. The effect of the Coulomb interaction of the ions in the dispersion medium is considered. It is shown that this contributes an additional repulsion term to the mutual free energy of the two particles. This correction term behaves as an inverse power and is the predominant effect at large separations, although it is a second-order effect at small separations. For parallel plates, this power is the fourth, and for two spheres, the sixth. A tentative calculation shows that, at least for thin plates and low electrolyte concentrations, our new repulsion will outweigh the attractive London-van der Waals force considered by Verwey and Overbeek, and more recently by Casimir and Polder, when the plates are far apart.

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