The Donnan model derived from microstructure

The ideal Donnan potential of an ionized polyelectrolyte medium is shown to be an approximate solution to a system of Poisson-Boltzmann (PB) equations for a periodic array of charged plates in an electrolyte bath. This result, derived using homogenization and scaling methods, demonstrates that the macrocontinuum, thermodynamic Donnan, and statistical mechanical PB models describe the same phenomenon: electrostatic repulsion between fixed-charged groups (albeit at different length scales). The Donnan approximation is accurate at low ionic strength (i.e., where the Debye length is much larger than the separation between charged plates), but is less faithful at physiologic and higher ionic strength. This work also provides a framework for relating theories of electrostatic repulsive interactions formulated at microscopic and macroscopic length scales.