The discrete-ion effect and mutual antagonism in a hydrophobic sol containing A 1-1 + 2-2 electrolyte mixture

The so-called discreteness-of-charge effect (or discrete-ion effect) for adsorbed counter-ions is incorporated into the stability theory of lyophobic colloids of Derjaguin and Landau and Verwey and Overbeek, for a coagulating 1-1 + 2-2 electrolyte mixture in an aqueous medium. The interaction of two parallel negatively charged colloidal plates in a large volume of the dispersion medium is considered and the approximation of linear superposition of the double layer potential distributions is employed. The Stern-Grahame-Devanathan model is used for the inner region and the Gouy-Chapman theory is applied to the diffuse layer. It is assumed that both the univalent and divalent cations (counter-ions) of the electrolyte mixture are adsorbed on the plate walls and, for simplicity, that the two ion types are situated at the same distance from the wall on the inner Helmholtz plane. Corrections due to the self-atmosphere potentials of the adsorbed ions (discrete-ion effect) and to the entropy effect arising from ion size are introduced into the adsorption isotherms of the two types of counter-ions. It is shown that the theory predicts the phenomenon of mutual antagonism when the self-atmosphere potential of the adsorbed divalent cation is sufficiently large. Among the conditions that favor antagonism are strong adsorption of the divalent cations and low integral capacity of the inner region. Plots of the coagulation curves in the (n, p) plane, where n and p are the concentrations of 1-1 and 2-2 electrolytes, respectively, show the usual pattern of antagonism, as obtained experimentally.