Overcharging

Here we mainly consider situations where charge inversion occurs, i.e., when the polyelectrolyte is sufficiently adsorbed so that the net charge of the complex changes sign. We shall begin with the salt-free case. In Fig. 14 the net charge of the complex \Q*\, which is equal to \Q - q \, is represented as a function of particle charge Q. Our results are in reasonably good agreement with the NS model, where the net inverted charge is predicted to be proportional to Q1/2 (and subsequently to the op/am ratio, which is also presented in Fig. 14). According to our MC simulations, when op/am < 6, two-tailed configurations are achieved with two polymer arms extended in opposite directions from the particle, overcharging is not observed. Data points situated above the theoretical curve overestimate the charge inversion. The excess of monomers at the macroion surface is due to latency in the first-order phase transition and tail formation. By further increasing the size of the chain above a critical value, a certain amount of the

FIG. 14 Net charge \Q* = \Q - N"s q \ variation and overcharging intensity as a function of particle charge and Cp/om ratio when C = 0M. When the polyelectrolyte size is large enough, Monte Carlo data are found asymptotically to be in good agreement with the Nguyen-Shklovskii model. Points situated below the theoretical curve underestimate overcharging. Polyelectrolyte chains that are not large enough to produce a tail in solution overestimate overcharging.

FIG. 14 Net charge \Q* = \Q - N"s q \ variation and overcharging intensity as a function of particle charge and Cp/om ratio when C = 0M. When the polyelectrolyte size is large enough, Monte Carlo data are found asymptotically to be in good agreement with the Nguyen-Shklovskii model. Points situated below the theoretical curve underestimate overcharging. Polyelectrolyte chains that are not large enough to produce a tail in solution overestimate overcharging.

adsorbed monomers is spontaneously ejected (removed) from the particle surface to form a tail radially outward from the center of the particle. A decrease of | Q* | is then observed. The effect of the presence of a monovalent salt is now discussed by considering Fig. 15, which reports the |Q* | values as a function of Q for Ci values equal to 0.001, 0.01, and 0.03 M, respectively. Charge inversion increases with charge screening, clearly demonstrating that monomers condense more on the particle in a salty solution. Once again, our MC results are in good agreement with the analytical theory, which predicts a linear dependance of |Q* | with Q. It is worth noting that when op/om > 6 and N = 100, the data do not fit the theoretical model because of the limited size of the polyelectrolyte, which is not large enough to form a tail in solution.

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