# Poisson-Boltzmann equation

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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

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>.

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>