Solution of the PB Eq. in the weak-field limit

Before proceeding to discuss general numerical techniques for solving the PB Eq. in 3-d, it is worth noting a special limit of this equation alluded to above, namely when the electric fields are weak enough that $\beta q\phi \ll 1$throughout the relevant spatial region. The PB Equation can consequently be linearized by expanding the Boltzmann factors to lowest nonvanishing (namely linear) order in $\phi$. For simplicity let us specialize to the common case that $\epsilon$ is constant and the bulk density of $\pm$ simple ions is the same, namely $\bar n$.Then the linearized PB Equation reads:

$$\nabla^2 \phi (\vec r) = \frac {-4\pi}{\epsilon} \rho_s (\vec r) + \kappa^2 \phi (\vec r)$$ (34)

where $\kappa$ is the inverse Debye screening length introduced above. This equation can be inverted to give an explicit solution for $\phi$ in terms of the Fourier transform of the static free charge density $\hat \rho_s (\vec k) = \int d\vec r \rho_s (\vec r) e^{i \vec k \cdot \vec r}$, specifically

$$\phi (\vec r) = \frac {1}{2\pi^2\epsilon} \int d\vec k \frac... ...at \rho_s (\vec k)}{k^2 + \kappa^2} e^{-i \vec k \cdot \vec r}$$ (35)

An important special case of this formula is when the static charge density is a point source corresponding to charge Q and location $\vec r_0$, i.e. $\rho_s (\vec r) = Q \delta (\vec r - \vec r_0)$. Then

$$\phi(\vec r) = \frac {Q}{\epsilon} \frac {e^{-\kappa \vert\vec r - \vec r_0\vert}} {\vert\vec r - \vec r_0\vert}$$ (36)

This exponentially damped Coulomb potential with screening length $\kappa^{-1}$ (also known as the Yukawa potential, due to its relevance in nuclear physics) is a characteristic feature of ionic screening. Clearly, the larger the ionic strength of the solution, the larger the screening. Moreover, the solution to Eq. 34 for a collection of point charge sources is just a sum of Yukawa potentials, one associated with each source. Thus the Yukawa potential corresponding to a given Debye screening length $\kappa^{-1}$ is the modification of the ``bare" Coulomb potential needed to account for ionic screening in the weak electric field limit. Note that the exponential screening renders the Yukawa potential short ranged, so that explicit summation of energetic contributions associated with this potential is easier than for the bare Coulomb analog.