Elastic Modulus and Equilibrium Swelling of Polyelectrolyte Gels Michael Rubinstein † and Ralph H. Colby* Imaging Research and Advanced Development, Eastman Kodak Company, Rochester, New York 14650-2109 Andrey V. Dobrynin † Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627 Jean-Francois Joanny Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg Cedex, France Received August 15, 1995 X ABSTRACT: We present a scaling theory for the modulus G of polyelectrolyte gels as a function of strand length between cross-links, monomer concentration c, salt concentration cs, and preparation conditions (monomer concentration c0 and salt concentration cs°). The theory assumes affine displacement of the junction points when the concentration is changed. With no added salt (cs ) cs° ) 0), we predict a new concentration dependence of the modulus G ∼ c 5/6 . In the high-salt limit, we predict the usual concentration dependence for uncharged polymers but a novel dependence on salt concentration, G ∼ c 7/12 cs 1/4 . We also predict the modulus to decrease as charge is added to the gel strands. The predicted effects of added salt and charge on modulus have recently been observed by Candau and co-workers. At low concentrations, we discuss the strong stretching of network strands and its effect on modulus and equilibrium swelling. 1. Introduction Networks made from charged polymers are quite common in both nature and industry. Polyelectrolyte gels, with a single sign of charge covalently bonded to the polymer chain, are capable of swelling to much greater extents than their uncharged counterparts because of the high osmotic pressure due to dissociated counterions. Thus they are used 1 as superabsorbent materials (e.g., diapers), as ion-exchange resins, and also as the carrier for novel drug delivery that targets specific organs. Early attempts to model the swelling of polyelectrolyte gels 2,3 encompassed much of the important physics, as it was recognized that swelling was determined by a balance between the osmotic pressure of free ions acting to swell the gel and the elasticity of the gel that restricts swelling. The osmotic part has long been understood in terms of the Donnan equilibrium. 4 We use a scaling model for the configu- ration of intrinsically flexible polyelectrolyte chains in semidilute solution 5-8 and our recent ideas relating network strand configuration and modulus 9 to construct a scaling theory for the modulus and swelling of poly- electrolyte gels. We consider polyelectrolyte gels, prepared by ran- domly cross-linking a semidilute solution of linear polyelectrolyte chains (at monomer number density c 0 ). The solvent must have a high dielectric constant (e.g., water) so that at least some of the counterions dissociate from the charged chain, leaving A monomers between effective charges. We also allow for the possibility of salt in the preparation state, with concentration c s ° (number density). The cross-linking creates a network of charged strands between cross-link junctions, with an average number N of monomers (of size b) per strand. We focus on the limit with many charges per strand (N . A). The number density of free ions in the prepara- tion state is the sum of the uncondensed counterions (c 0 /A) and any dissociated salt ions (2c s °, since we assume, for simplicity, monovalent salt). The shear modulus of the network in its preparation state is identical to the modulus of an uncharged polymer, simply kT per strand. This well-known result holds for all strongly cross- linked gels, independent of the charge on the chain and salt concentration. For weakly cross-linked gels, with a strand length larger than a critical length for en- tanglement N e , the modulus is determined by the number density of entanglement strands, 9 and eq 1 holds (roughly) with N replaced by N e . In what follows, we focus on strongly cross-linked gels, but all results are also valid in the weakly cross-linked regime (N > N e ) by merely replacing N with N e , the entanglement strand length in the preparation state. We start by reviewing our recent results for the configuration of polyelectrolyte chains in semidilute solution (Section 2). Since we assume cross-linking does not change the chain configurations, these results apply directly to the strands of a polyelectrolyte gel in its preparation state. Section 3 describes stretching a polyelectrolyte chain by pulling its ends, as this happens to the strands of a polyelectrolyte gel when it is swollen. In section 4 we consider the concentration dependence of the modulus of polyelectrolyte gels at fairly high concentrations, where the strands of the network are weakly stretched. Since the strong osmotic pressure arising from counterion entropy can cause polyelectro- lyte gels to reach very low concentrations, in sections 5 * To whom correspondence should be addressed. Permanent address: Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802. † Permanent address: Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290. X Abstract published in Advance ACS Abstracts, December 1, 1995. G 0 = c 0 N kT (1) 398 Macromolecules 1996, 29, 398-406 0024-9297/96/2229-0398$12.00/0 © 1996 American Chemical Society