Plateau Moduli of Several Single-Chain Slip-Link and Slip-Spring Models

We calculate the plateau moduli of several single-chain slip-link and slip-spring models for entangled polymers. In these models, the entanglement effects are phenomenologically modeled by introducing topological constraints such as slip-links and slip-springs. The average number of segments between two neighboring slip-links or slip-springs, $N_{0}$, is an input parameter in these models. To analyze experimental data, the characteristic number of segments in entangled polymers $N_{e}$ estimated from the plateau modulus is used instead. Both $N_{0}$ and $N_{e}$ characterize the topological constraints in entangled polymers, and naively $N_{0}$ is considered to be the same as $N_{e}$. However, earlier studies showed that $N_{0}$ and $N_{e}$ (or the plateau modulus) should be considered as independent parameters. In this work, we show that due to the fluctuations at the short time scale, $N_{e}$ deviates from $N_{0}$. This means that the relation between $N_{0}$ and the plateau modulus is not simple as naively expected. The plateau modulus (or $N_{e}$) depends on the subchain-scale details of the employed model, as well as the average number of segments $N_{0}$. This is due to the fact that the subchain-scale fluctuation mechanisms depend on the model rather strongly. We theoretically calculate the plateau moduli for several single-chain slip-link and slip-spring models. Our results explicitly show that the relation between $N_{0}$ and $N_{e}$ is model-dependent. We compare theoretical results with various simulation data in the literature, and show that our theoretical expressions reasonably explain the simulation results.