Increased Donnan exclusion in charged polymer networks at high salt concentrations

The swelling of univalent and multivalent charged polymeric networks in electrolytic solutions is studied using a classical thermodynamic model. Such systems were first modeled by Donnan, who derived an expression for the chemical potential of the ions by introducing an electric potential that is commonly referred to as the Donnan potential. This well-established theory leads to a simple quadratic relationship for the partitioning of ions between the network and the external solution. When the concentration of fixed charges in the swollen gel is large enough, the electrolyte in the external solution is "excluded" from the gel (commonly referred to as Donnan exclusion). In the standard Donnan theory, and in virtually all subsequent theories, the magnitude of Donnan exclusion decreases with increasing electrolyte concentration in the external solution. Our model predicts this is not necessarily true; we show that the magnitude of Donnan exclusion increases with increasing electrolyte concentration over a broad range of parameter space (average chain length between crosslinks, fraction of charged monomers in the network, the nature of the interactions between the ions, solvent molecules and polymer chains, and ion concentration in the external solution). We also present explicit bounds for the validity of Donnan's original theory. Model predictions are compared to simulations and experimental data obtained for a cationic gel immersed in electrolytic solutions of salts containing univalent and bivalent cations.