# 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

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

where the charge distribution is

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

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

$c_i = c_{i0} Exp(\frac{-z_i e \phi}{k T})$

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

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

## References

Evans, D.F. <underline>The Colloidal Domain: where physics, chemistry, and biology meet</underline>. Pg. 131-132. New York:Wiley-VCH, 1999.