Poisson–Boltzmann equation

The Poisson–Boltzmann equation describes the distribution of the electric potential in solution in the direction normal to a charged surface. 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.