# Poisson-Boltzmann equation

The Poisson-Boltzmann equation describes the ion distribution in an electrolyte solution outside a charged interface. It relates the mean-field potential to the concentration of electrolyte.

## Derivation

The Poisson equation reads

<math> \epsilon_0 \epsilon_r \nabla^2 \phi = \rho_{(free ions)} </math>

where the charge distribution is

<math> \rho_{(free ions)} = e \sum_i z_i c_i </math>

where <math>c_i</math> is the concentration. Assuming that the energy of each ion is due to only the electrostatic potential, the Boltzmann distribution dictates that

<math> c_i = c_{i0} Exp(\frac{-z_i e \phi}{k T}) </math>

where <math>z_i</math> is the ion valency, and <math>c_{i0}</math> is the concentration where <math>\phi = 0</math>, usually taken to be the bulk concentration. Combining these three equations yields the Poisson-Boltzmann equation

<math> \epsilon_0 \epsilon_r \nabla^2 \phi = - e \sum_i z_i c_{i0} Exp(\frac{-z_i e \phi}{k T}) </math>.