Cluster aggregation and fragmentation kinetics model for gelation

Gelation can occur in polymer, hydrogel, and colloid systems that undergo reversible aggregation-fragmentation (crosslinking accompanied by breakage). Gelation, characterized by rapid divergence of weight-average molecular weight and viscosity due to initial network formation, can be reversed if conditions change. In this paper, reversible aggregation and fragmentation in the pre-gelation time period are modeled with distribution kinetics. Moment equations are obtained from the population balance equation, and solved for eight different rate kernels. We identify the cases for which gelation is possible and obtain the critical values for the rate constants that allow gelation. The model provides a good simulation of published experimental data for aggregation and degradation of plasticized wheat gluten during thermo-mechanical treatments. We also evaluate two closure approximations based on Gamma and log-normal distributions, and conclude that log-normal closure predicts all five possible steady states, in agreement with the Vigil-Ziff criterion, and Gamma closure predicts only three. However, Gamma closure approximates the steady state either closely or exactly, whereas log-normal closure only poorly approximates the steady-state distribution.