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 | ||
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</math> | </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 | + | 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> | <math> | ||
\epsilon_0 \epsilon_r \nabla^2 \phi = - e \sum_i z_i c_{i0} Exp(\frac{-z_i e \phi}{k T}) | \epsilon_0 \epsilon_r \nabla^2 \phi = - e \sum_i z_i c_{i0} Exp(\frac{-z_i e \phi}{k T}) | ||
| − | </math> | + | </math>. |
Revision as of 21:22, 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>.
