Poisson–Boltzmann equation

The Poisson–Boltzmann equation describes the distribution of the electric potential in solution in the presence of one or more charged surfaces. This distribution is important to determine how the electrostatic interactions will affect the molecules in solution.

It is expressed as a differential equation of the electric potential ${\displaystyle \psi }$, which depends on the solvent permitivity ${\displaystyle \varepsilon }$, the solution temperature ${\displaystyle T}$, and the mean concentration of each ion species ${\displaystyle c_{i}^{0}}$:

${\displaystyle \nabla ^{2}\psi =-{\frac {1}{\varepsilon }}\sum _{i}c_{i}^{0}q_{i}\exp \left({\frac {-q_{i}\psi (x,y,z)}{k_{B}T}}\right)}$

The Poisson–Boltzmann equation is derived via mean-field assumptions. From the Poisson–Boltzmann equation many other equations have been derived with a number of different assumptions.