The Critical Coagulation Concentration of Counterions: Spherical Particles in Asymmetric Electrolyte Solutions

The ratio of the critical coagulation concentration (CCC) of counterions is evaluated for spherical particles and asymmetric electrolytes. A perturbation method is adopted to solve the Poisson-Boltzmann equation governing the electrical potential distribution of the system under consideration. On the basis of the result obtained, an approximate expression for the CCC is derived. Another approach based on the Derjaguin approximation is also used to estimate the CCC. We show that the CCC ratio of counterions is a complicated function of the valences of the ion species in the liquid phase and the sizes of particles. Depending upon the thickness of the Debye length, the CCC ratio of counterions for various combinations of electrolytes can be estimated. The classic Schulze-Hardy rule for planar particles in a symmetric electrolyte solution can be recovered as a limiting case of the present model. If the surface potential is low, the effect of curvature on the CCC ratio of counterions is negligible.