Poisson-Boltzmann equation integrated form

The TPE-HNC/MS theory reduces to an integral form of the nonlinear Poisson-Boltzmann equation in the limit of point ions [8,44]. Hence, in that limit agreement between the two methods is exact. For a 0.1 M, 1 1 electrolyte separating plates with surface potentials of 70 mV, Lozada-Cassou and Diaz-Herrera [8] show excellent agreement between the TPE-HNC/MS theory and the Poisson-Boltzmann equation. The agreement becomes very poor, however, at a higher concentration of 1 M. In addition, like the Monte Carlo and AHNC results, the TPE-HNC/MS theory predicts attractive interactions at sufficiently high potentials and/or salt concentrations, and such effects are missed entirely by the Poisson-Boltzmann equation. 

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