Gelation Time of Network-Forming Polymer Solutions with Ion-Binding Cross-Link Junctions

The gelation time tg of the solutions of functional polymers carrying ligands capable of forming metal-coordinated cross-link junctions is theoretically studied as a function of the concentration of the polymers and added metal ions. Quite generally, tg is given by the product of relaxation time tR and thermodynamic factor Q. An ion-coordinated cross-link is characterized by the variable number k of ligands bound to the metal ion with the maximum coordination number s. Coupled rate equations of stepwise coordination with temperature-dependent rate constants are derived for the reactivity pk (k = 1, 2, ..., s) of the ligands. From the solutions of the rate equations, tg, tR, and Q are obtained as functions of the phase point on the ternary phase plane of the polymer/ion/solvent mixture. The thermodynamic factor Q is shown to exhibit a singularity of logarithmic divergence as the ion concentration approaches its gel-point value, while the relaxation time tR changes continuously across it. The concentration ratio of the added ions relative to the ligands that makes the gelation time the shortest is found. The lines of the shortest gelation time (SGT) are drawn on the ternary phase plane together with the re-entrant sol–gel–sol transition lines. The crossing point of these two lines is a critical gel point. Rate equations are numerically solved by the Runge–Kutta method for a given upper bound s = 6 of the coordination number to study the effect of cooperativity and of chelate formation on the gelation time. For the Langmuir type binding to equivalent sites, analytical solutions for the SGT are found.