Difference between revisions of "Poisson-Boltzmann equation"
Chakraborty (Talk | contribs) |
Chakraborty (Talk | contribs) |
||
| (16 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
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> | <math> | ||
| − | \epsilon_0 \epsilon_r \ | + | \epsilon_0 \epsilon_r \nabla^2 \phi = \rho_{(free ions)} |
</math> | </math> | ||
| + | |||
where the charge distribution is | where the charge distribution is | ||
| + | |||
<math> | <math> | ||
| − | \rho_{free ions) = e \sum_i z_i c_i | + | \rho_{(free ions)} = e \sum_i z_i c_i |
</math> | </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.
