Nernst-Planck Equations and the Electroneutrality and Donnan Equilibrium Assumptions

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Abstract
The Nernst-Planck flux equations and Poisson's equation are used to describe the transport of ions across a membrane carrying a fixed charge. The resulting problem is studied using a perturbation theory in order to derive the so-called “electroneutrality” condition as a certain limiting case. It is found that the electroneutrality condition is a consequence of Poisson's equation when a certain dimensionless parameter is small. It is also shown that a Donnan equilibrium at the membrane boundaries is a consequence of the Nernst-Planck equations when this dimensionless parameter is small, and a separate assumption of such an equilibrium is redundant. The small parameter can be interpreted as the ratio of a Debye length and the membrane thickness.

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