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The theory of the interaction of large, high-charge bearing, identical particles immersed in an electrolytic medium has been placed on a firm footing by Verwey and Overbeek.1 If an attempt is made to generalize this theory to nonidentical particles, its conclusions become physically unacceptable and a revision of some of its basic assumptions is necessitated. In the discussion which follows, an acquaintance with the Verwey-Overbeek theory is assumed. 1. Application of the Verwey-Overbeek Theory to Two Nonidentical Particles.Consider two infinite parallel plates (the first at x = 0, the second at x = 1) in an ionic medium. The electric potential between them is found from the solution of the Poisson-Boltzmann (P.B.) equation, as developed by Debye and Hickel. Verwey and Overbeek2 impose the boundary conditions that the surface potentials, ,6(0) and t(l), are independent of the interaction, i.e., the separation distance 1. A family of curves results from the above, giving {t(x) as a function of 1; these are shown in Figure 1 for two identical particles, 4t(O) = 4t(l). It is seen that the minimum of +,(x) increases with increasing interaction and that the curve approaches a horizontal line in the limit as 1 approaches zero. An examination of the equation shows that the derivative with respect to 1, and hence the charge densities of the two plates (from Gauss's law), approach zero for large interaction. But, if we now fix 4'(0) $ 4/(1) and cause these two nonidentical plates to approach, A,(x) must clearly acquire a larger and larger slope which will ultimately become infinite. The charge densities therefore also become unbounded, a conclusion which is physically unacceptable. The above physical argument (represented in Fig. 2) can be confirmed by an examination of the first and second integrations of the P.B. equation. Further, if we solve the linear approximation of the P.B. equation, we find that the force between these plates becomes infinite and attractive atl = 0. These results point up the need for a revision of the theory. In particular, they indicate the need for a revision of the condition that the surface potentials remain independent of the particle interaction. This boundary condition is justified in the Verwey-Overbeek theory by the following argument. The charge on the surface has its origin in the adsorption of potential-determining ions from the solution. There must exist, therefore, a drop in the chemical potential AA between these two phases. The adsorption is opposed by the electric potential difference, At(0). At equilibrium these two must balance, i.e.,

AN APPLICATION OF THE POISSON-BOLTZMANN EQUATION TO NONIDENTICAL, CHARGED PARTICLES