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We construct a new phase-field model for the solvation of charged moleculeswith a variational implicit solvent. Our phase-field free-energy functionalincludes the surface energy, solute-solvent van der Waals dispersion energy,and electrostatic interaction energy that is described by the Coulomb-fieldapproximation, all coupled together self-consistently through a phase field. Byintroducing a new phase-field term in the description of the solute-solvent vander Waals and electrostatic interactions, we can keep the phase-field valuescloser to those describing the solute and solvent regions, respectively, makingit more accurate in the free-energy estimate. We first prove that ourphase-field functionals $\Gamma$-converge to the corresponding sharp-interfacelimit. We then develop and implement an efficient and stable numerical methodto solve the resulting gradient-flow equation to obtain equilibriumconformations and their associated free energies of the underlying chargedmolecular system. Our numerical method combines a linear splitting scheme,spectral discretization, and exponential time differencing Runge-Kuttaapproximations. Applications to the solvation of single ions and a two-platesystem demonstrate that our new phase-field implementation improves theprevious ones by achieving the localization of the system forces near thesolute-solvent interface and maintaining more robustly the desirable hyperbolictangent profile for even larger interfacial width. This work provides a schemeto resolve the possible unphysical feature of negative values in thephase-field function found in the previous phase-field modeling (cf. H. Sun, etal. J. Chem. Phys., 2015) of charged molecules with the Poisson--Boltzmannequation for the electrostatic interaction.

A New Phase-Field Approach to Variational Implicit Solvation of Charged Molecules with the Coulomb-Field Approximation