Exact solutions of the Nonlinear PB Eq.

Solution of the Poisson-Boltzmann (PB) Equation, which is necessary when there are mobile ions surrounding fixed charge distributions, is difficult because of the nonlinearities in the equation associated with the mobile ion Boltzmann factors. The PB Eq. can only be solved analytically for highly symmetric systems, e.g., charged planar plates in electrolytic solutions [36]. For example, consider a single plate with surface charge density $\sigma$ surrounded by a 1:1 electrolyte with bulk density $\bar n_+ = \bar n_- = \bar n$.For concreteness, let the plate be located at x=0 (and infinite in extent along y and z directions), and let the electrolyte be monovalent (so q is the magnitude of the electron charge). Finally, let the dielectric constant $\epsilon$ be uniform throughout space. It is useful to define a dimensionless electrical potential $\theta = \beta q \phi$ and a dimensionless distance $u=\kappa x$,where $\kappa^{-1}$ is a length known as the Debye screening length (see below) prescribed by $\kappa= \sqrt{8\pi \beta \bar n q^2 /\epsilon}$. In terms of these variables the PB equation reads:

$$\frac {d^2 \theta(u)}{du^2} = {\sinh} (\theta (u))$$ (32)

[The value of the plate charge density $\sigma$ enters only as a boundary condition which is used to fix the integration constant associated with the solution of Eq. (32).]

The solution to this differential equation, subject to the boundary condition that $\theta \rightarrow 0$ as $\vert u\vert \rightarrow \infty$ is:

$$\theta(u) = 2 \ln \left [ \frac {1+\tanh(\theta_0 /4)e^{-\vert u\vert}} {1-\tanh(\theta_0 /4)e^{-\vert u\vert}} \right ]$$ (33)

The integration constant $\theta_0$ gives the value of the electric potential at the surface of the plate. It determines the slope of the $\phi$ field, i.e. the electric field, at the plate, which is in turn easily related to the surface charge on the plate via Gauss's law. (Typical results are illustrated in Fig. 2.) Note that the spatial extent of the ionic atmosphere is on the order of $\kappa^{-1}$, the Debye screening length, hence the name. It is also worth noting that the solution to Eq. (32) becomes particularly simple in the limit of weak electric fields, such that $\theta \ll 1$ everywhere. Then the the r.h.s. of Eq. (32) can be replaced by $\theta (u)$,and the physically relevant solution of the differential equation reads $\theta(u) = \theta_0 e^{-\vert u\vert}$. This is the familiar exponential screening familiar from 1-d Debye-Hückel theory [43], which directly reflects the Debye screening length.

While it is appealing to seek analytical forms for the potential field associated with the PB equation, numerical techniques must be utilized for all but the simplest systems. Real-space lattice methods are extremely useful for this purpose, as will be discussed below.