Exact Analytic Result of Contact Value for the Density in a Modified Poisson-Boltzmann Theory of an Electrical Double Layer

For a simple modified Poisson-Boltzmann (SMPB) theory, taking into account the finite ionic size, we have derived the exact analytic expression for the contact values of the difference profile of the counterion and co-ion, as well as of the sum (density) and product profiles, near a charged planar electrode that is immersed in a binary symmetric electrolyte. In the zero ionic size or dilute limit, these contact values reduce to the contact values of the Poisson-Boltzmann (PB) theory. The analytic results of the SMPB theory, for the difference, sum, and product profiles were compared with the results of the Monte-Carlo (MC) simulations [Bhuiyan, L. B.; Outhwaite, C. W.; Henderson, D. J. Electroanal. Chem. 2007, 607, 54; Bhuiyan, L. B.; Henderson, D. J. Chem. Phys. 2008, 128, 117101], as well as of the PB theory. In general, the analytic expression of the SMPB theory gives better agreement with the MC data than the PB theory does. For the difference profile, as the electrode charge increases, the result of the PB theory departs from the MC data, but the SMPB theory still reproduces the MC data quite well, which indicates the importance of including steric effects in modeling diffuse layer properties. As for the product profile, (i) it drops to zero as the electrode charge approaches infinity, (ii) the speed of the drop increases with the ionic size, and these behaviors are in contrast with the predictions of the PB theory, where the product is identically 1.