Fluctuation in entanglement positions via elastic slip-links.

We consider the spatiotemporal fluctuation of slip-link positions via the implementation of elastic slip-links. The level of description is similar to our previously proposed slip-link model, wherein we use the entanglement position in space as dynamic variables, and the number of Kuhn steps between entanglements. However, since it is a mean-field, single-chain description it has some relevance to the slip-spring simulations of Likhtman, and the phantom chain model for cross-linked networks. It might also provide a connection between slip-links and tubes. Two implementations are possible, depending on whether or not the slip-links are allowed to pass through one another. If a boundary condition on the dynamics preventing such passage is imposed, then the plateau modulus is unchanged from perfectly rigid slip-links. Only the dynamics is changed. On the other hand, for phantom slip-links the distribution of the number of entanglements changes from Poisson. Furthermore, requiring normalization of the distribution function sets a constraint on how loose the virtual springs for the elastic slip-link are. These restrictions appear to be in agreement with parameter values used for the slip-spring simulations, although nonphantom slip-links were used there. The results are completely analogous to what was found by James and Guth for ideal elastic networks, whose derivation is repeated here. Our earlier rigid slip-link model is recovered as a limiting case.