Double-layer interaction of two charged colloidal spherical particles of a concentrated dispersion in a medium of low dielectric constant. Part 2.—A cell model

Abstract

The double-layer force between two neighbouring, charged, identical, spherical colloidal particles O₁ and O₂ of a concentrated dispersion in a hydrocarbon medium is investigated. Only one ionic species, constituting the counter-ion, is present in the hydrocarbon at a very low concentration. It is assumed that the surface charge on each particle is fixed and uniformly distributed over the surface. We seek the mean force between O¹ and O₂ averaged over all positions of counter-ions and other particles. Introducing a cell model, the two particles are enclosed by a symmetrical surface S₀ such that the net charge contained by S₀ is zero. The position of the surface S₀ is determined by assuming that the normal derivative of the mean electrostatic potential is zero on it. Because the concentration of counter-ions is sufficiently small to give an almost uniform charge density in the hydrocarbon medium enclosed by S₀, the volume contained in S₀ equals twice the mean volume per particle in the dispersion. In order to isolate electrically the region contained within S₀ as much as possible from the surroundings, an additional condition on S₀ is that the potential variation on it be small. The potential satisfies the Poisson–Boltzmann equation in the hydrocarbon medium and the Laplace equation inside the particles, the solutions to these equations being expressed in terms of multipole expansions about the two particle centres. Up to quadrupole terms, the double layer force between the two particles is found to be attractive at separations corresponding to the mean separation between neighbouring particles in the dispersion, although this force becomes repulsive at smaller separations. The force calculations indicate that the electric double layer will not produce stability, in contrast to its role in the Deryaguin–Landau–Verwey–Overbeek theory for dilute aqueous suspensions.