Abstract
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.
Highlights
The swelling and deswelling transitions of gels have long been the subject of wide-ranging experimental and theoretical research, motivated both by a large number of practical applications and by the intrinsic interest in gels as systems subject to the fundamental physical laws of phase transitions [1,2,3,4,5,6,7,8,9]
As the solvent quality decreases past the transition range, a progressive collapse can result if the condensed counterions strengthen the spring constant
At the cost of additional modeling, we found that the radius of gyration, another measure of swelling, increases as the solvent becomes poorer
Summary
The swelling and deswelling transitions of gels have long been the subject of wide-ranging experimental and theoretical research, motivated both by a large number of practical applications and by the intrinsic interest in gels as systems subject to the fundamental physical laws of phase transitions [1,2,3,4,5,6,7,8,9]. In addition, the polymer chains are stretched by direct Coulomb charge repulsion, contributing toward the equilibrated volume. The equilibrium spring lengths, or, equivalently, the equilibrium charge spacing on the chain, was always greater than the Bjerrum length (defined below) In this range, there is no counterion condensation, and Debye-Hückel electrostatics provides an accurate account of the bead-bead repulsions [17,18]. The observed behavior in poorer solvents is expected with Debye-Hückel interactions, which do not draw counterions in toward the chain very strongly, and only partially counteract the increased bead-bead repulsions in conditions of increased Coulomb strength. We consider the range of strong Coulomb interactions where the equilibrium spring length is smaller than the Bjerrum length, and counterions condense on the chain. We will introduce additional assumptions, qualitatively similar to ion-pairing, that allow the bead-spring assembly to contract toward small values of the spring length
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