Cell Model and Poisson-boltzmann Theory: a Brief Introduction

The cell-model and its treatment on the Poisson-Boltzmann level are two important concepts in the theoretical description of charged macro-molecules. In this brief contribution to Ref. [1] we provide an introduction to both ideas and summarize a few important results which can be obtained from them. Our article is organized as follows: Section 1 outlines the sequence of approximations which ultimately lead to the cell-model. Section 2 is devoted to 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. Section 3 provides a derivation of the Poisson-Boltzmann equation from a variational principle and the assumption of a product state. Section 4 applies 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, Section 5 shows 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. Our main motivation is to compile in a concise form a few of the basic concepts which form the arena for more advanced theories, treated in other lectures of this volume. Many of the basic concepts are discussed at greater length in a review article by Katchalsky [2], which we warmly recommend.