Crossover between mean-field and critical scaling near the gel point

Gelation of linear precursor chains via random crosslinking is accompanied by the formation of finite clusters. The number and size distributions of the clusters have been predicted by the Flory-Stockmayer mean-field theory, which neglects the formation of loops. The theory breaks down in the close vicinity of the gel point, and its predictions are replaced with a set of critical scaling exponents. We explore the crossover between the mean-field and critical scaling by conducting coarse grained simulations for a mixture of reversibly crosslinked precursors and clusters. Molecular weights, backbone stiffness, and monomer bulkiness of the precursors are tuned systematically. The simulated radii of gyration of clusters exhibits distinct behaviors, which collapse onto a master curve upon normalization by an emergent scale, thermal blob . The thermal blob measures the magnitude of the excluded volume interaction of clusters. It only depends on the invariant degree of polymerization, which quantifies the extent of inter-chain overlap. The master curve constructed using this thermal blob reveals a surprisingly gradual crossover between the mean-field and critical regimes and indicates that majority of the crosslinked systems of experimental interests fall within the critical regime.