Communications in Mathematical Sciences

Volume 16 (2018)

Number 5

A new phase-field approach to variational implicit solvation of charged molecules with the Coulomb-field approximation

Pages: 1203 – 1223



Yanxiang Zhao (Department of Mathematics, George Washington University, Washington, D.C., U.S.A.)

Yanping Ma (Department of Mathematics, Loyola Marymount University, Los Angeles, California, U.S.A.)

Hui Sun (Department of Mathematics and Statistics, California State University, Long Beach, Calif., U.S.A.)

Bo Li (Department of Mathematics and Quantitative Biology Graduate Program, University of California at San Diego)

Qiang Du (Department of Applied Physics and Applied Mathematics, Columbia University, New York, N.Y., U.S.A.)


We construct a new phase-field model for the solvation of charged molecules with a variational implicit solvent. Our phase-field free-energy functional includes the surface energy, solute-solvent van der Waals dispersion energy, and electrostatic interaction energy that is described by the Coulomb-field approximation, all coupled together self-consistently through a phase field. By introducing a new phase-field term in the description of the solute-solvent van der Waals and electrostatic interactions, we can keep the phase-field values closer to those describing the solute and solvent regions, respectively, making it more accurate in the free-energy estimate. We first prove that our phase-field functionals $\Gamma$-converge to the corresponding sharp-interface limit. We then develop and implement an efficient and stable numerical method to solve the resulting gradient-flow equation to obtain equilibrium conformations and their associated free energies of the underlying charged molecular system. Our numerical method combines a linear splitting scheme, spectral discretization, and exponential time differencing Runge-Kutta approximations. Applications to the solvation of single ions and a two-plate system demonstrate that our new phase-field implementation improves the previous ones by achieving the localization of the system forces near the solute-solvent interface and maintaining more robustly the desirable hyperbolic tangent profile for even larger interfacial width. This work provides a scheme to resolve the possible unphysical feature of negative values in the phase-field function found in the previous phase-field modeling (cf. H. Sun, et al. J. Chem. Phys., 2015) of charged molecules with the Poisson–Boltzmann equation for the electrostatic interaction.


phase field model, exponential time differencing, implicit solvation, $\Gamma$-convergence

2010 Mathematics Subject Classification

65K10, 65M70, 65Z05

Received 11 October 2017

Received revised 4 April 2018

Accepted 4 April 2018

Published 19 December 2018