Motion of a Cylindrical Dielectric Boundary

Abstract

The interplay between geometry and electrostatics contributes significantly to hydrophobic interactions of biomolecules in an aqueous solution. With an implicit solvent, such a system can be described macroscopically by the dielectric boundary that separates the high-dielectric solvent from low-dielectric solutes. This work concerns the motion of a model cylindrical dielectric boundary as the steepest descent of a free-energy functional that consists of both the surface and electrostatic energies. The effective dielectric boundary force is defined and an explicit formula of the force is obtained. It is found that such a force always points from the solvent region to solute region. In the case that the interior of a cylinder is of a lower dielectric, the motion of the dielectric boundary is initially driven dominantly by the surface force but is then driven inward quickly to the cylindrical axis by both the surface and electrostatic forces. In the case that the interior of a cylinder is of a higher dielectric, the competition between the geometrical and electrostatic contributions leads to the existence of equilibrium boundaries that are circular cylinders. Linear stability analysis is presented to show that such an equilibrium is only stable for a perturbation with a wavenumber larger than a critical value. Numerical simulations are reported for both of the cases, confirming the analysis on the role of each component of the driving force. Implications of the mathematical findings to the understanding of charged molecular systems are discussed.

Diffused Solute-Solvent Interface with Poisson--Boltzmann Electrostatics: Free-Energy Variation and Sharp-Interface Limit

“…Let n denote the unit normal to the interface Γ pointing from the solvent region Ω w to solute region Ω p . We have [5,9–11,28,44]…”

LS-VISM: A software package for analysis of biomolecular solvation

“…To numerically solve (III.5), we develop a CCIM. [46,[65][66][67] This method is based on the coupling interface method (CIM) [65] and a compact scheme for irregular geometry. [68] We cover the whole computational box by a uniform finite-difference grid.…”

Stability of a Cylindrical Solute-Solvent Interface: Effect of Geometry, Electrostatics, and Hydrodynamics

“…Consequently, by these and the expression of Laplacian in cylindrical coordinates, we can rewrite the boundary-value problem of Poisson’s equation (1.4) for the electrostatic potential ϕ = ϕ ( r, z, t ) as the elliptic interface problem (2.6) [14, 28]. Note that the third and fourth equations in (2.6) are the continuities of the electrostatic potential ϕ and the normal component of electric displacement − ε Γ( t ) ∂ n ϕ , respectively, across the dielectric boundary r = R ( z, t ).…”

“…These are positive constants and satisfy in general ${\epsilon }_{\mathrm{m}}^{\prime }<{\epsilon }_{\mathrm{w}}^{\prime }$. The electrostatic potential determines the normal component of effective dielectric boundary force [7, 8, 14, 29] $$f_{\text{ele}}=\frac{1}{2}\left(\frac{1}{{\epsilon }_{\mathrm{normalw}}}-\frac{1}{{\epsilon }_{\mathrm{normalm}}}\right){\mid {\epsilon }_{\mathrm{\Gamma }}\nabla \varphi ·\mathrm{boldn}\mid }^2+\frac{1}{2}false({\epsilon }_{\mathrm{m}}-{\epsilon }_{\mathrm{w}}false){\mid false(I-\mathrm{boldn}\otimes \mathrm{boldn}false)\nabla \varphi \mid }^2,$$…”

“…In our previous work [14], we studied the linear stability of a cylindrical dielectric boundary in the relaxation of surface energy and electrostatic energy. We found the dispersion relation ω ( k ) = − γ 0 k 2 for k ≫ 1.…”