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

Abstract

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 Γ-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. *

Energy Stable Semi-implicit Schemes for Allen–Cahn–Ohta–Kawasaki Model in Binary System

“…Moreover, we will investigate some higher order schemes and perform energy stability analysis if possible. Some other numerical methods, such as exponential time differencing based schemes, coupled with Fourier spectral discretization in space will also be under our consideration in the future as it is an efficient and stable numerical method which has been successfully applied to other gradient flow type dynamics [13,23,3].…”

“…On the other hand, a more traditional choice f (φ) = φ with f ′ (φ) = 1 will induce a local surface tension force (the first two terms on the right hand side of (1.11)) against a global repulsive force (the last term on the right hand side of (1.11)). To balance such two forces, the φ might lose the desired tanh profile and results in either unphysical negative values in the proximity of the interface (see [3]) or values not equal 0 or 1 away from the interface (see Figure 4.1). Secondly, since the new f (φ) results in a much better tanh profile, consequently the new model will describe the interfacial structures more accurately and lead to a better estimate of the free energy.…”

“…However, the existing works mainly focus on the Cahn-Hilliard dynamics, namely, the H −1 gradient flow dynamics of the OK energy (1.1) with f (φ) = φ. Our work is advantageous of studying the force balance in the process of phase separation and emphasizing the force localization near the interface, and could have potential applications in other complex dynamics involving interfacial structures such as cell motility [12] and implicit solvation [3].…”

The phase field method for geometric moving interfaces and their numerical approximations

“…For example, [400] presented phase field models for the implicit solvation of charged molecules with a coupling to Poisson-Boltzmann electrostatics. Various refined phase field models,numerical methods on both spatial discretization and time integration have also been studied [300,126,461].…”

Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

“…However, the existing works mainly focus on the Cahn-Hilliard dynamics, namely, the H −1 gradient flow dynamics of the OK/ON energy with f (ϕ) = ϕ. Our work is advantageous of studying the force balance in the process of phase separation and emphasizing the force localization near the interface, and could have potential applications in other complex dynamics involving interfacial structures such as cell motility [5] and implicit solvation [41]. Besides, some other nonlocal extensions of the Allen-Cahn and Cahn-Hilliard equations and the related numerical methods have been studied [15,16,17,22].…”