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.
For type-A polymer chains having type-A dipoles parallel along the chain backbone (such as cis-polyisoprene), a theoretical analysis was conducted for the rheodielectric response to relate this response to the chain dynamics. The rheodielectric response in the shear gradient direction (y direction) under steady shear was analyzed on the basis of a Langevin equation. It turned out that the relaxation time is exactly the same for the rheodielectric relaxation function and the end-to-end vector autocorrelation function defined in the shear gradient direction and that the relaxation mode distribution also coincides for these functions at least up to second order of the shear rate (corresponding to the lowest order of nonlinearities of these functions). Consequently, the Green-Kubo theorem holds satisfactorily, and the rheodielectric intensity is proportional to the squared chain size in y direction, hR 2 y i, averaged over the time-independent conformational distribution function under steady shear. The situation is more complicated under large amplitude oscillatory strain (LAOS) because the conformational distribution function f LAOS is synchronized with LAOS to oscillate at the LAOS frequency, X. The rheodielectric response under LAOS was found to detect this oscillation of f LAOS being coupled with the oscillation of the electric field, E(t) ¼ E 0 sin xt, and thus, split into a series of components oscillating at frequencies x and x AE bX (b ¼ 1, 2, …). Consequently, the rheodielectric intensity under LAOS, evaluated from the component oscillating at x, is no longer proportional to hR 2 y i. However, the relative mode distribution and relaxation time of this component can be directly related to those of the end-to-end vector correlation averaged over a nonoscillatory part of f LAOS .
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