A more accurate fluctuation potential problem is considered in the modified Poisson-Boltzmann theory. Numerical solutions of the resulting new equation are compared with those of the previous theory and 1 : 1, 1 : 2 , 2 : 2 Monte Carlo calculations. The new modified Poisson-Boltzmann equation is found to be much more successful in predicting the Monte Carlo results, especially for the higher-valency electrolytes.

A modified Poisson-Boltzmann equation in diffuse double-layer theory which includes estimates for the fluctuation and exclusion volume terms is solved numerically for a 1 : 1 restricted-model electrolyte. Two estimates of the exclusion-volume term are considered at three electrolyte concentrations with varying surface charge when image terms are neglected. Details are given of the numerical calculation and double-layer properties such as the mean electrostatic potential, the ion-wall distribution functions and the differential capacitance are presented. Comparisons are made with the Gouy-Chapman-Stern theory and with some preliminary hypernetted chain (HNC) calculations.

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Monte Carlo simulations of linear polyelectrolyte solutions containing mixed valency simple ions in the cylindrical cell model are reported. The equilibrium distributions of the simple ions and the osmotic pressure of the solution are calculated at various concentrations of the monomer units of the polyelectrolyte. Specifically, the following systems are studied—monovalent counterions with added 2:2 salt, divalent counterions with added 1:1 salt, and systems containing mono- and divalent counterions only, and mono- and trivalent counterions only. The simulation results are compared with the corresponding predictions from the Poisson–Boltzmann and modified Poisson–Boltzmann theories applied to the cell model. It is seen that upto moderate concentrations of the polyion, the modified Poisson–Boltzmann theory provides a very good description of the systems with deviations occurring at higher concentrations. The theory also reproduces the charge reversal observed in the simulations when strongly correlated counterions overscreen the charge of the polyion. On the other hand, the classical Poisson–Boltzmann results begin to show discrepencies from the Monte Carlo results at relatively lower concentrations. Comparisons of the simulated osmotic pressures with available experimental results confirm the validity of the cell model in a spectrum of practical situations of interest.

The behavior of the differential capacitance of a planar electric double layer containing a restricted primitive model electrolyte in the neighborhood of zero surface charge is investigated by theory and simulation. Previous work has demonstrated that at zero surface charge the differential capacitance has a minimum for aqueous electrolytes at room temperature but can have a maximum for molten salts and ionic liquids. The transition envelope separating the two situations is found for a modified Poisson-Boltzmann theory and a Poisson-Boltzmann equation corrected for the exclusion volume term. Good agreement is found between simulation and the modified Poisson-Boltzmann theory in the neighborhood of the envelope at the reduced temperature of 0.8, while the exclusion volume corrected Poisson-Boltzmann theory shows correct qualitative trends.