| The objective of this project is to study by computer simulation the effective charge of spherical colloidal particle and the structure of the ion cloud using Brownian dynamics. The simulations are reported for monovalent counterions about the parent macroion for the different values of the surface charge, Zp and the radius, ap. The counterion distribution functions, gpc(r), were used to determine a “thermal radius”, rtherm, defined by the condition, gpc(rtherm) = exp(1), when the interaction energy of the counterion with the parent macroion is equal to the thermal energy, kBT. The thermodynamically bound counterions are defined as those counterions that are within and up to the radial distance from the macroion center at which the reduced mean electrostatic potential energy equals the thermal energy. The effective charge is assumed to consist of the intrinsic macroion charge reduced by the integrated thermodynamically bound counterion charge. An “effective charge”, Zeff, was thus obtained by including with the bare charge, Zp, and the equilibrium distribution of counterions that lie within the distance r therm. These data, represented as Zeff/Zp versus Zp/ap, may be characterized into three categories and are compared with four literature models. The results are shown to follow the trend in the experimental data summarized by Roberts, O'Dea, and Osteryoung (Roberts et al., 1998) indicated: (1) a steep initial slope for Z p/ap < 15 nm−1 (category I); (2) a “transition” region 15 nm−1 < Zp /ap < 40 nm−1 with a variable slope (category II); and (3) a shallow terminal slope for Z p/ap > 40 nm−1 (category III).; The dynamics of the counterions in each category was inferred from the ratio Δr/|Δr|, where Δr is the difference in the radial displacement of the counterion at the initial and final positions, and |Δr| is the magnitude of the vector difference between these locations. It was thus shown that the counterions in categories I and II are highly mobile whereas they are somewhat restricted to the vicinity of the macroion surface in category III macroions. These results are compared with theories of “charge renormalization” in the literature. Implications of the current model are discussed. |
