Elastic Modulus and Equilibrium Swelling of Near-Critical Gels

Scaling ideas are used to predict the modulus G₀ of gels, just above the point of incipient gel formation, in the reaction bath as a function of the proximity to the gel point ∊. The concentration dependence of the modulus when the gel is diluted in a good solvent is also calculated and used to predict the maximum swelling Q, obtained from the gel swollen at equilibrium with pure solvent. The Ginzburg criterion separates the critical (∊ < ∊_G) and mean-field (∊ > ∊_G) percolation regimes. We derive a new criterion for entanglement ∊_E, which leads us to expect three regimes of behavior. Close to the gel point (for ∊ < ∊_G) critical percolation applies to an unentangled gel: G₀ ∼ ∊^2.6 and Q ∼ ∊^−1.1. For ∊_G < ∊ < ∊_E we predict a mean-field unentangled regime with G₀ ∼ ∊³ and Q ∼ ∊^−8/5. For ∊ > ∊_E entanglements raise the modulus and restrict the swelling of the mean-field gels, with G₀ ∼ ∊^14/3 and Q ∼ ∊^−13/5.