In the classical phantom network theory, the shear modulus of a polymer network is derived assuming the underlying network has a treelike topology made up of identical strands. However, in real networks, defects such as dangling ends, cyclic defects, and polydispersity in strand sizes exist. Moreover, studies have shown that cyclic defects, or loops, are intrinsic to polymer networks. In this study, we illustrate a general framework for calculating the rubber elasticity of phantom networks with arbitrary defects. Closed form solutions for the elastic effectiveness of strands near isolated loops and dangling ends are obtained, and it was found that under classical assumptions of phantom network theory loops with order ≥3 have zero net impact on the overall elasticity. However, when a simple approximation for strand prestrain is considered, the modified network theory agrees well with experimentally measured moduli of PEG gels.

Gels formed by coupling two different four-arm star polymers lead to polymer networks with high strength and low spatial heterogeneity. However, like all real polymer networks, these materials contain topological defects which affect their properties. In this study, kinetic graph theory and Monte Carlo simulation are used to investigate the structure and cyclic defects formed via A–B type end-linking of symmetric tetra-arm star polymer precursors. While loops constituting of odd number of junctions are forbidden by precursor chemistry, the amount and the correlation of secondary loops are found to increase with decreasing precursor concentration. This suppresses gelation, and the delay of gel point is quantitatively predicted by the topological simulations. Furthermore, comparison with network formed with asymmetric bifunctional–tetrafunctional precursors revealed that the behavior of loops consisting of 2n junctions in the symmetric system is analogous to the behavior of loops consisting of n junctions in the asymmetrical system, suggesting analogies between chemically distinct networks.

Accurate prediction of the gel point for real polymer networks is a long-standing challenge in polymer chemistry and physics that is extremely important for applications of gels and elastomers. Here, kinetic Monte Carlo simulation is applied to simultaneously describe network topology and growth kinetics. By accounting for topological defects in the polymer networks, the simulation can quantitatively predict experimental gel point measurements without any fitting parameters. Gel point suppression becomes more severe as the primary loop fraction in the networks increases. A topological homomorphism theory mapping defects onto effective junctions is developed to qualitatively explain the origins of this effect, which accurately captures the gel point suppression in the low loop limit where cooperative effects between topological defects are small.

To predict and understand the properties of polymer networks, it is necessary to quantify network defects. Of the various possible network defects, loops are perhaps the most pervasive and yet difficult to directly measure. Network disassembly spectrometry (NDS) has previously enabled counting of the simplest loopsprimary loopsbut higher-order loops, e.g., secondary loops, have remained elusive. Here, we report that the introduction of a nondegradable tracer within the NDS framework enables the simultaneous measurement of primary and secondary loops in end-linked polymer networks for the first time. With this new “NDS2.0” method, the concentration dependences of the primary and secondary loop fractions are measured; the results agree well with a purely topological theory for network formation from phantom chains. In addition, semibatch monomer addition is shown to decrease both primary and secondary loops, though the latter to a much smaller extent. Finally, using the measured primary and secondary loop fractions, we were able to predict the shear storage modulus of end-linked polymer gels via real elastic network theory (RENT).

The failure properties of a polymer network, including toughness, ultimate strain, and ultimate stress, are some of the most critical properties for network performance. The polymer networks often contain various topological defects, such as primary loops and dangling ends, which have a noticeable effect on these properties. This work focuses on understanding the effect of these defects on the fracture strength of a material by expanding the classical Lake−Thomas theory to account for such defects under the assumption that each defect is unaffected by the presence of other defects in its environment. A Flory−Stockmayer gel point criterion is combined with the improved theory to identify the incipience of fracture. The predictions demonstrate that although the presence of defects weakens the material by reducing the tearing energy, the overall network elongation depends strongly on the primary loop fraction. Specifically, a transition from a low ultimate-strain to a significantly high ultimate-strain behavior is predicted. The addition of a kinetic theory for bond scission predicts that the sharpness of this transition is a strong function of the strain rate. To experimentally test these predictions, a series of poly(ethylene glycol) (PEG) gels with previously characterized primary loop fractions were synthesized. Remarkably, the measured tearing energies agree quite well with the theoretical predictions and also suggest the onset of the low to high extensibility transition.