Sol-Gel Transition of Loop-Rich Networks

Polymer gelation is a complex phase transition where a space-spanning network emerges through the collective aggregation of precursor chains and junctions. Classically, this process is modeled as percolation on a Cayley tree (Flory-Stockmayer theory) or on well-defined lattices (3D Percolation). Real networks, however, contain a distribution of cyclic topologies and spatial inhomogeneities that are not captured by these ideal analogies. This work characterizes the critical behavior below the gel point for end-linked poly(ethylene glycol) gels to explore the impact of intramolecular cyclization on the universality classes of these gels. At high concentrations, the critical exponents of these gels deviate from the predictions of Flory-Stockmayer and percolation theory. Furthermore, a family of distinct critical exponents emerges upon dilution, as loop-rich clusters suppress the gel point to higher conversions and enhance the molecular weight divergence in the critical regime, resembling a clustered-percolation process. A similar acceleration is observed in purely-topological kinetic Monte Carlo simulations that recover Flory-Stockmayer theory in the high-concentration limit, demonstrating the intrinsic impact of local cyclization on the sol-gel transition.