Schulze-Hardy rule revisited

Abstract

The classical Schulze-Hardy rule suggests that the critical coagulation concentration (CCC) decreases as the inverse sixth power of the counterion valence. While this dependence can be derived from the theory of Derjaguin, Landau, Verwey, and Overbeek (DLVO), this derivation relies on unrealistic assumptions. In particular, one cannot assume that the electrolytes are symmetric, since one normally works with the better soluble asymmetric electrolytes. For such electrolytes, however, it is essential to distinguish between multivalent counterions and coions. For multivalent counterions, one must consider their strong tendency towards adsorption to the oppositely charged substrates, which leads to low charge densities. In this situation, the CCC increases with the surface charge density, inducing the strong decrease of the CCC with valence. For multivalent coions, the substrates are typically highly charged. In this case, the CCC decreases with increasing ionic valence and is in fact inversely proportional to the valence. This dependence is referred to as the inverse Schulze-Hardy rule.