Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation

TL;DR

This paper establishes the mathematical convergence of phase-field models for molecular solvation to sharp-interface models, including free energy and boundary forces, providing rigorous foundations for implicit solvent simulations.

Contribution

It proves the $ ext{Gamma}$-convergence of phase-field free energies and the convergence of electrostatic and boundary forces to their sharp-interface limits, under minimal assumptions.

Findings

Proved continuity of electrostatic potential and free energy with respect to dielectric boundary perturbations.

Established $ ext{Gamma}$-convergence of phase-field free energies to sharp-interface models.

Demonstrated convergence of phase-field forces to sharp-interface forces.

Abstract

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.