Structure of the ion cloud and the effective charge of spherical colloidalparticles: Brownian dynamics simulations

Brownian dynamics (BD) simulations are reported for monovalent counterions about the parent macroion for different values of the surface charge Z(p) a nd the radius a(p). The counterion distribution functions g(pc)(r) thus obt ained were used to determine a "thermal radius" r(therm) defined by the con dition g(pc)(r(therm)) = exp(1), viz., when the interaction energy of the c ounterion with the parent macroion is equal to the thermal energy k(B)T. An "effective charge" Z(eff) was thus obtained by including with the bare cha rge Z(p) the equilibrium distribution of counterions that lie within the di stance r(therm). These data, represented as Z(eff)/Z(p) versus Z(p)/a(p), a re shown to follow the trend in the experimental data summarized by Roberts , O'Dea, and Osteryoung (Anal. Chem., 1998, 70, 3667). On the basis of the shape of this plot, three macroion classifications were indicated: (1) a st eep initial, slope for Z(p)/a(p) < 30 nm(-1) (category I); (2) a "transitio n" region 30 nm(-1) < Z(p)/a(p) < 90 nm(-1) with a variable slope (category II); and (3) a shallow terminal slope for Z(p)/a(p) > 90 nm(-1) (category III). The dynamics of the counterions in each category was inferred from th e ratio Delta r/\Delta r\, where Delta r is the difference in the radial di splacement of the counterion at the initial and final positions and \Delta r\ is the magnitude of the vector difference between these locations. It wa s thus shown that the counterions in categories I and II are highly mobile whereas they are somewhat restricted to the vicinity of the macroion surfac e in category III macroions. These results are compared with theories of "c harge renormalization" in the literature, and implications of the current m odel are discussed.