# Difference between revisions of "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.

## Short derivation

$\epsilon_0 \epsilon_r \nabla^2 \phi = \rho_{(free ions)}$

where the charge distribution is

$\rho_{(free ions)} = e \sum_i z_i c_i$.

Assuming that the energy of each ion is due to only the electrostatic potential, the Boltzmann distribution dictates that

$c_i = c_i0 Exp(\frac{-z_i e \phi, k T})$