Extending the Real Elastic Network Theory to Account for Cooperative Effect of Cyclic Defects

The real elastic network theory, or RENT, has provided insights into how cyclic defects within a phantom network lead to significant lowering of the modulus of polymer networks. However, RENT is a linear theory derived under the ideal loop gas assumption, which is strictly true only in the limit of infinite dilution of defects, rendering RENT inapplicable for networks with nonnegligible loop fraction. To go beyond the linear regime, the cooperative effects of multiple loops are investigated. First, the multi-defect effect is studied through the Virial expansion, in which the behavior of adjacent loops is compared to isolated loops. In most cases, it was found that the Virial coefficients vanish identically except for the first order terms, indicating that the results for the linear approximation are exact. Next, a nonperturbative probability-based theory was developed to capture the nonlinear behaviors. As expected, the nonlinear theory gave predictions almost identical to that of the linear theory in the small loop fraction regime. However, as loop fraction is increased, the nonlinear theory predicts a significant negative deviation from the linear theory, qualitatively matching the behavior observed in experiments.