Gelation Time of Network-Forming Polymer Solutions with Reversible Cross-Link Junctions of Variable Multiplicity

Abstract

The gelation time tg necessary for a solution of functional (associating) molecules to reach its gel point after a temperature jump, or a sudden concentration change, is theoretically calculated on the basis of the kinetic equation for the stepwise cross-linking reaction as a function of the concentration, temperature, functionality f of the molecules, and multiplicity k of the cross-link junctions. It is shown that quite generally tg can be decomposed into the product of the relaxation time tR and a thermodynamic factor Q. They are functions of a single scaled concentration x≡λ(T)ϕ, where λ(T) is the association constant and ϕ is the concentration. Therefore, the superposition principle holds with λ(T) as a shift factor of the concentration. Additionally, they all depend on the rate constants of the cross-link reaction, and hence it is possible to estimate these microscopic parameters from macroscopic measurements of tg. The thermodynamic factor Q is shown to depend on the quench depth. It generates a singularity of logarithmic divergence as the temperature (concentration) approaches the equilibrium gel point, while the relaxation time tR changes continuously across it. Gelation time tg obeys a power law tg−1∼xn in the high concentration region, whose power index n is related to the multiplicity of the cross-links. The retardation effect on the gelation time due to the reversibility of the cross-linking is explicitly calculated for some specific models of cross-linking to find the rate-controlling steps in order for the minimization of the gelation time to be easier in the gel processing. For a micellar cross-linking covering a wide range of the multiplicity, as seen in hydrophobically-modified water-soluble polymers, tR is shown to obey a formula similar to the Aniansson–Wall law.