Ion size effect on counterion condensation around a spherical colloidal particle in a salt-free medium containing only counterions

A theory of counterion condensation around a spherical colloidal particle in a salt-free medium containing only counterions developed by Imai and Oosawa as reported by Busseiron Kenkyu 52:42, 1952 ibid. 59:99, 1953) and later by Ohshima (J Colloid Interface Sci 247:18, 2002) is extended to incorporate the ion size effect on the basis of the modified Poisson-Boltzmann equation using the Carnahan-Starling activity coefficient of counterions. The potential distribution depends on the particle surface charge, the particle volume fraction, and the average counterion volume fraction. It is shown that for a dilute particle suspension, the counterion condensation occurs, leading to a decrease in the particle surface potential when the following two conditions are both satisfied: (i) the scaled particle surface charge is greater than the logarithm of the inverse of the particle volume fraction and (ii) the average counterion volume fraction is much smaller than the particle volume fraction.