The effect of divalent vs. monovalent ions on the swelling of mucin-like polyelectrolyte gels: governing equations and equilibrium analysis.

We introduce a comprehensive model of a mucin-like polyelectrolyte gel swelling-deswelling which includes the ion-mediated crosslinking of polymer strands and the exchange of divalent and monovalent ions in the gel. The gel is modeled as a multi-phase mixture which accounts for the polymer and solvent volume fractions and velocities as well as ionic species concentrations. Motion is determined by force balances involving viscous, drag, and chemical forces. The chemical forces are derived from a free energy which includes entropic contributions as well as the chemical and electrostatic interactions among the crosslinked polymer, uncrosslinked polymer, and the ionic solvent. The unified derivation produces all the classical effects (van't Hoff osmotic pressure, Donnan equilibrium potential, Nernst-Planck motion of ions) as well as expressions for Flory interaction parameter and the standard free energy parameters that explicitly depend on the gel chemistry and crosslink structure. For this model, we show how the interplay between ionic bath concentrations, ionic binding, and transient divalent crosslinking leads to a variety of swelled and deswelled phases/phase transitions. In particular, we show how the absorption of divalent ions can lead to a massive deswelling of the gel. We conclude that the unique properties of mucin-like gels can be explained by their ionic binding affinities and transient divalent crosslinking.