Solving fluctuation-enhanced Poisson–Boltzmann equations

Abstract

Electrostatic correlations and fluctuations in ionic systems can be described within an extended Poisson-Boltzmann theory using a Gaussian variational form. The resulting equations are challenging to solve because they require the solution of a non-linear partial differential equation for the pair correlation function. This has limited existing studies to simple approximations or to one-dimensional geometries. In this paper we show that the numerical solution of the equations is greatly simplified by the use of selective inversion of a finite difference operator which occurs in the theory. This selective inversion preserves the sparse structure of the problem and leads to substantial savings in computer effort. In one and two dimensions further simplifications are made by using a mixture of selective inversion and Fourier techniques.

Beyond Poisson–Boltzmann: fluctuations and fluid structure in a self-consistent theory

“…Numerical methods have recently been developed to solve them [6,7,16,18]. In particular, in comparison with the MC simulations, [6] and [7] identified the validity regime of the equations for liquids confined to slit and cylindrical nanopores, respectively.…”

Understanding Depletion Induced Like-Charge Attraction from Self-Consistent Field Model

“…The divergent part of the self-Green's function is eliminated by numerically solving G 0 (ω;x,x) by the same scheme, then G− G 0 is convergent with the refinement of the mesh size. The self energy is then determined by the inverse Fourier transform [61].…”

Modified Poisson--Nernst--Planck Model with Accurate Coulomb Correlation in Variable Media

“…The Green's function G P λ (r, r ′ ) can be viewed as the inverse of the modified Helmholtz operator, which is still hard to solve by numerical methods. Since we are only interested in the diagonal components, i.e., the self Green's function G P λ (r, r), the selected inversion algorithm [51] can be employed for the purpose [29]. Although the self energy is well-defined, however it ignores the excluded-volume effect and would lead to the instability for strong correlated systems due to the ionic collapse.…”

“…Another promising routine is by introducing a self-energy correction to the potential of mean force using the generalized Debye-Hückel equation, which can be derived under the point-charge approximation (PCA) by the Debye closure of the BBGKY hierarchy [24], random phase approximation in density functional theory [25] or the Gaussian variational field theory [26][27][28] for equilibrium systems. The PDEs under the PCA are unstable [29] for strong-coupling systems because cations and anions may collapse when the interaction is strong. To go beyond PCA, Wang [30] used the variational field theory to derive a generalized Debye-Hückel equation for finite-sized ions with Gaussian charge distribution, which first revealed the importance of the ionic self energy for mobile ions with finite size and obtained the Born solvation energy in variable dielectric media by performing expansion using ionic radius as small parameter, and provides the theoretical framework for accounting for these many-body effects.…”