Effective Screened Coulomb Charge of Spherical Colloidal Particles

Abstract

The application of the well-known DLVO pair potential and its variations in the literature to the evaluation of the charge on macroions routinely results in an “effective” charge considerably smaller than the “bare” surface charge. Various theories have been proposed for “charge renormalization” on the basis of either surface effects (viz., counterion “condensation”) or equivalence of linearized and nonlinearized electrostatic potentials at the surface of a computation cell in the numerical analysis of the Poisson−Boltzmann equation. In the present discourse, a model for the effective charge is proposed that is based on the equivalence of the screened Coulomb macroion−counterion interaction to the thermal energy kBT. The distance at which these interactions are equal, Rtherm, defines a distance partition for the counterions into two populations: those tightly associated with the macroion and contributing to the charge reduction and those “free” in solution and contributing to the properties of the surrounding medium. The characteristics of this model are in good agreement with other theoretical approaches based on more elaborate solutions to the Poisson−Boltzmann equation and computer simulations. More important, this relatively simple model is in agreement with the experimentally determined trends relating the effective charge to the titration charge. Limitations of the proposed model, as well as the screened Coulomb potential, are also discussed.