Charge-regulation modelling of the Schulze–Hardy rule and related coagulation effects

The DLVO theory of colloid stability provided an explanation of the long-standing Schulze–Hardy rule, i.e. that the critical coagulation concentration (c.c.c.) is proportional to some inverse power of the counter-ion valency (z). For particular limiting cases the inverse power exponent was calculated to be either 2 or 6. By using the charge-regulation approach to double-layer interaction for real surfaces with identifiable surface sites, dissociation equilibria, new insights into the link between physiochemical parameters and the Schulze–Hardy exponent are obtained. The model also produces an important result, viz. that the total double-layer potential for the isolated particle at the c.c.c. solution condition, while different for each of a series of real materials, is independent of counter-ion valency (z).