Quantifying Cyclic Topology in Polymer Networks Using 3D Nets

Polymer networks invariably possess topological inhomogeneities in the form of molecular loops which critically affect their macroscopic properties. Existing theories describe such topological defects using perturbations to acyclic tree models. However, at large enough sizes, loops must become highly overlapping to satisfy finite density and conversion criteria. Systematic comparisons of these loops across different simulation approaches has proven challenging. To address this, a new formalism to model polymer networks is developed based on the mathematical concept of 3D nets. A cycle counting algorithm is designed to characterize local as well as global cyclic topology. Comparison of the topological similarity of networks formed by different simulation algorithms using a distance-like metric reveals the fundamental cycle size of each network which depends on the topological proximity of crosslinkers across chains during bond formation. This parameter can help identify distinct categories of network simulation algorithms and can be tuned to simulate a wide array of topologically diverse networks starting from suitable 3D nets. This approach can be generalized to interconnected systems beyond polymer networks, which enables more detailed quantification of the cyclic topology and thus facilitates studying their topology-property correlations.