Dielectric Boundary Force in Molecular Solvation with the Poisson-Boltzmann Free Energy: A Shape Derivative Approach

In an implicit-solvent description of molecular solvation, the electrostatic free energy is given through the electrostatic potential. This potential solves a boundary-value problem of the Poisson-Boltzmann equation in which the dielectric coefficient changes across the solute-solvent interface-the dielectric boundary. The dielectric boundary force acting on such a boundary is the negative first variation of the electrostatic free energy with respect to the location change of the boundary. In this work, the concept of shape derivative is used to define such variations and formulas of the dielectric boundary force are derived. It is shown that such a force is always in the direction toward the charged solute molecules.