Condensation of Counterions Gives Rise to Contraction Transitions in a One-Dimensional Polyelectrolyte Gel

The equilibrium volume of a polyelectrolyte gel results from a balance between the tendency to swell caused by outbound polymer/counterion diffusion along with Coulomb interactions on the one hand; and, on the other, the elastic resilience of the cross-linked polymer network. Direct Coulomb forces contribute both to non-ideality of the equilibrated Donnan osmotic pressure, but also to stretching of the network. To isolate the effect of polyelectrolyte expansion, we have analyzed a “one-dimensional” version of a gel, a linear chain of charged beads connected by Hooke’s law springs. As in the range of weak Coulomb strengths previously studied, the springs are significantly stretched by the repulsive interactions among the beads even when the Coulomb strength is strong enough to cause condensation of counterions. There is a quasi-abrupt transition from a stretched state to a partially collapsed state in a transition range between weak and strong Coulomb strengths. Fluctuations between stretched and contracted conformations occur within the transition range. As the solvent quality decreases past the transition range, a progressive collapse can result if the condensed counterions strengthen the spring constant