Difference between revisions of "Poisson-Boltzmann equation"

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\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>.
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==References==
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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.