# Difference between revisions of "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. | 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== |

The Poisson equation reads | The Poisson equation reads | ||

− | <math>\epsilon_0 \epsilon_r \nabla^2 \phi = \rho_{(free ions)}</math> | + | |

+ | <math> | ||

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

+ | </math> | ||

+ | |||

where the charge distribution is | where the charge distribution is | ||

− | <math>\rho_{free ions) = e \sum_i z_i c_i</math> | + | |

+ | <math> | ||

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

+ | </math> | ||

+ | |||

+ | where <math>c_i</math> is the concentration. 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> | ||

+ | |||

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

+ | |||

+ | <math> | ||

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

+ | </math>. | ||

+ | |||

+ | ==References== | ||

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

## Latest revision as of 21:33, 20 November 2009

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

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>

where <math>c_i</math> is the concentration. 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>

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

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

## References

Evans, D.F. __The Colloidal Domain: where physics, chemistry, and biology meet__. Pg. 131-132. New York:Wiley-VCH, 1999.