From real polymers to random graphs: percolation thresholds in associative polymer solutions

Abstract

Sol-gel transitions are ubiquitous in soft matter and biological systems, yet their thresholds are often poorly captured by classical Flory-Stockmayer theory because spatial organization and loop formation are neglected. Here, we combine molecular dynamics simulations with random graph and random geometric graph models to determine the respective roles of topology and geometry in reversible associative polymer solutions. We show that a coordinate-free random graph recovers the mean-field Flory-Stockmayer limit, whereas a random geometric graph quantitatively reproduces the shifted percolation thresholds observed in molecular dynamics simulations when the detection radius is chosen according to the polymer conformational size. This geometric mapping remains quantitatively valid for linear chains with regularly spaced binding sites over a broad range of chain stiffness. At the microscopic level, we identify primary loops formed already in the pre-gel regime as the dominant source of the deviation from mean-field predictions. Near the gel point, the cluster-size statistics obtained from simulations and random geometric graphs are consistent with the universality class of three-dimensional percolation. These results establish random geometric graphs as a minimal predictive framework for describing topological transitions in reversible associative polymer solutions and show that gelation and network formation can be inferred directly from single-chain conformational information.