Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation
We study a phase-field variational model for the solvation 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 the dielectric boundary. We also prove the Γ -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.
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