  We study a phase-field variational model for the solvaiton of charged
molecules with an implicit solvent. The solvation free-energy functional of all
phase fields consists of the surface energy, solute excluded volume and
solute-solvent van der Waals dispersion energy, and electrostatic free energy.
The surface energy is defined by the van der Waals--Cahn--Hilliard functional
with squared gradient and a double-well potential. The electrostatic part of
free energy is defined through the electrostatic potential governed by the
Poisson--Boltzmann equation in which the dielectric coefficient is defined
through the underlying phase field. We prove the continuity of the
electrostatics---its potential, free energy, and dielectric boundary
force---with respect to the perturbation of dielectric boundary. We also prove
the $\Gamma$-convergence of the phase-field free-energy functionals to their
sharp-interface limit, and the equivalence of the convergence of total free
energies to that of all individual parts of free energy. We finally prove the
convergence of phase-field forces to their sharp-interface limit. Such forces
are defined as the negative first variations of the free-energy functional; and
arise from stress tensors. In particular, we obtain the force convergence for
the van der Waals--Cahn--Hilliard functionals with minimal assumptions.
