Ion Condensation in the Electric Double Layer and the Corresponding Poisson-Boltzmann Effective Surface Charge

The effective surface charge obtained by fitting the Poisson-Boltzmann approximation to measured data is related to the actual surface charge. Evaluation of the formally exact expression, which depends upon the wall-ion direct correlation function, gives a precise measure of the amount of ion condensation in the planar double layer. Results are presented for 0.001, 0.01, and 0.1 M monovalent and divalent restricted primitive model electrolytes using the singlet hypernetted chain closure with the first bridge diagram. In general, the apparent surface charge is less than the actual surface charge, due to the influence of ion-ion correlations. Thus, the Poisson-Boltzmang approximation underestimates the actual surface charge and overestimates the amount of counterion binding. At high surface charge densities and apparent surface charge saturates, and there is a maximum surface charge density that a given electrolyte will appear to support. At surface charge densities beyond this and high concentrations (e.g. 0.1 M divalent, 75 %1* per unit surface charge), charge reversal occurs, due to overscreening by the counterions in the first layer adjacent to the surface.