Electrostatics

Abstract

The electrostatic potential of a charge distribution is a solution of Poisson’s equation which, if mobile charges are taken into account, like for an electrolyte or semiconductor, turns into the Poisson–Boltzmann equation. In this chapter we discretize the Poisson and the linearized Poisson–Boltzmann equation by finite volume methods which are applicable even in case of discontinuous dielectric constant. We solve the discretized equations iteratively with the method of successive overrelaxation. The solvation energy of a charged sphere in a dielectric medium is calculated to compare the accuracy of several methods. This can be studied also in a computer experiment. Since the Green’s function is analytically available for the Poisson and Poisson–Boltzmann equations, alternatively the method of boundary elements can be applied, which can reduce the computer time, for instance for solvation models. A computer experiment simulates a point charge within a spherical cavity and calculates the solvation energy with the boundary element method.

Problems

Problem 18.1 Linearized Poisson–Boltzmann Equation

This computer experiment simulates a homogeneously charged sphere in a dielectric medium (Fig. 18.19). The electrostatic potential is calculated from the linearized Poisson Boltzmann equation (18.65) on a cubic grid of up to $100^{3}$ points. The potential $\varPhi (x)$ is shown along a line through the center together with a log-log plot of the maximum change per iteration

$$\begin{aligned} |\varPhi ^{(n+1)}(\mathbf {r})-\varPhi ^{(n)}(\mathbf {r})| \end{aligned}$$

(18.114)

as a measure of convergence.

Fig. 18.19

Charged sphere in a dielectric medium

Explore the dependence of convergence on

• the initial values which can be chosen either $\varPhi (\mathbf {r})=0$ or from the analytical solution

$$\begin{aligned} \varPhi (\mathbf {r})=\left\{ \begin{array}{c} \frac{Q}{8\pi \epsilon \epsilon _{0}a}\frac{2+\epsilon (1+\kappa a)}{1+\kappa a}-\frac{Q}{8\pi \epsilon _{0}a^{3}}r^{2}\quad \text {for }r<a\\ \frac{Qe^{-\kappa (r-a)}}{4\pi \epsilon _{0}\epsilon (\kappa a+1)r}\quad \text { for }r>a. \end{array}\right. \end{aligned}$$

(18.115)

• the relaxation parameter $\omega$ for different combinations of $\epsilon$ and $\kappa$

• the resolution of the grid

Problem 18.2 Boundary Element Method

In this computer experiment the solvation energy of a point charge within a spherical cavity (Fig. 18.20) is calculated with the boundary element method (18.93).

Fig. 18.20

Point charge inside a spherical cavity

The calculated solvation energy is compared to the analytical value from (18.104)

$$\begin{aligned} E_{solv}=\frac{Q^{2}}{8\pi \epsilon _{0}R}\sum _{n=1}^{\infty }\frac{s^{2n}}{R^{2n}}\frac{(\epsilon _{1}-\epsilon _{2})(n+1)}{n\epsilon _{1}+(n+1)\epsilon _{2}} \end{aligned}$$

(18.116)

where R is the cavity radius and s is the distance of the charge from the center of the cavity.

Explore the dependence of accuracy and convergence on

• the damping parameter $\omega$

• the number of surface elements ($6\times 6\cdots 42\times 42)$ which can be chosen either as $d\phi d\theta$ or $d\phi d\,\cos \theta$ (equal areas)

• the position of the charge