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In a continuum model of the solvation of charged molecules in an aqueoussolvent, the classical Poisson-Boltzmann (PB) theory is generalized to includethe solute point charges and the dielectric boundary that separates thehigh-dielectric solvent from the low-dielectric solutes. With such a setting,we construct an effective electrostatic free-energy functional of ionicconcentrations, where the solute point charges are regularized by a reactionfield. We prove that such a functional admits a unique minimizer in a class ofadmissible ionic concentrations and that the corresponding electrostaticpotential is the unique solution to the boundary-value problem of thedielectric-boundary PB equation. The negative first variation of this minimumfree energy with respect to variations of the dielectric boundary defines thenormal component of the dielectric boundary force. Together with thesolute-solvent interfacial tension and van der Waals interaction forces, suchboundary force drives an underlying charged molecular system to a stableequilibrium, as described by a variational implicit-solvent model. We developan $L^2$-theory for the continuity and differentiability of solutions toelliptic interface problems with respect to boundary variations, and derive anexplicit formula of the dielectric boundary force. With a continuumdescription, our result of the dielectric boundary force confirms amolecular-level prediction that the electrostatic force points from thehigh-dielectric and polarizable aqueous solvent to the charged molecules. Ourmethod of analysis is general as it does not rely on any variationalprinciples.

The Calculus of Boundary Variations and the Dielectric Boundary Force in the Poisson-Boltzmann Theory for Molecular Solvation