Cell model and Poisson-Boltzmann theory: A brief introduction

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The cell-model and its treatment on the Poisson-Boltzmann level are two important concepts in the theoretical description of charged macromolecules. In this brief contribution to the Proceedings of the NATO-ASI on ``Electrostatic Effects in Soft Matter and Biophysics'', which took place in Les Houches from Oct. 1-13, 2000, we provide an introduction to both ideas and summarize a few important results which can be obtained from them. We outline the sequence of approximations which ultimately lead to the cell-model. Then we present two exact results, namely, an expression for the osmotic pressure and a formula for the ion density at the surface of the macromolecule, known as the contact value theorem. The next section provides a derivation of the Poisson-Boltzmann equation from a variational principle and the assumption of a product state. Afterwards we apply Poisson-Boltzmann theory to the cell-model of linear polyelectrolytes. In particular, the behavior of the exact solution in the limit of zero density is compared to the concept of Manning condensation. Finally, we demonstrate how a system described by a cell model can be coupled to a salt reservoir, i.e., how the so-called Donnan equilibrium is established.