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A new phase-field approach to variational implicit solvation of charged molecules with the Coulomb-field approximation. (English) Zbl 1407.65234

Summary: 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., “A self-consistent phase-field approach to implicit solvation of charged molecules with Poisson-Boltzmann electrostatics”, J. Chem. Phys. 143, 243110 (2015)]) of charged molecules with the Poisson-Boltzmann equation for the electrostatic interaction.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
65Z05 Applications to the sciences
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
35Q35 PDEs in connection with fluid mechanics
78A30 Electro- and magnetostatics
76T20 Suspensions
35Q20 Boltzmann equations