Thermodynamically consistent incorporation of entanglement spatial fluctuations in the slip-link model.

We evaluate the thermodynamic consistency of the anisotropic mobile slip-link model for entangled flexible polymers. The level of description is that of a single chain, whose interactions with other chains are coarse grained to discrete entanglements. The dynamics of the model consist of the motion of entanglements through space and of the chain through the entanglements, as well as the creation and destruction of entanglements, which are implemented in a mean-field way. Entanglements are modeled as discrete slip links, whose spatial positions are confined by quadratic potentials. The confinement potentials move with the macroscopic velocity field, hence the entanglements fluctuate around purely affine motion. We allow for anisotropy of these fluctuations, described by a set of shape tensors. By casting the model in the form of the general equation for the nonequilibrium reversible-irreversible coupling from nonequilibrium thermodynamics, we show that (i) since the confinement potentials contribute to the chain free energy, they must also contribute to the stress tensor, (ii) these stress contributions are of two kinds: one related to the virtual springs connecting the slip links to the centers of the confinement potentials and the other related to the shape tensors, and (iii) these two kinds of stress contributions cancel each other if the confinement potentials become anisotropic in flow, according to a lower-convected evolution of the confinement strength or, equivalently, an upper-convected evolution of the shape tensors of the entanglement spatial fluctuations. In previous publications, we have shown that this cancellation is necessary for the model to obey the stress-optical rule and the Green-Kubo relation, and simultaneously to agree with plateau modulus predictions of multichain models and simulations.