Fluctuating Junctions of Physically Cross-Linked Networks in a Single-Chain Model. – Consistency Between the Relaxation Modulus for Small Deformations and Green-Kubo Predictions

Abstract

In theoretical attempts of predicting the viscoelastic behavior of polymer networks, such as chemical gels of rubbers, physical gels of associating polymers, and entangled polymer melts, the affine assumption is often employed to describe the junction (or node) motion of the networks under flows or deformations. That is, the motion of the junctions is assumed to obey the same deformation tensor as that used for the macroscopic deformation applied to the network surface. While the affine assumption captures the essential feature of the node dynamics somehow on average, neglect of the spatial (and temporal) fluctuations of junctions around the affine motion gives rise to several fundamental problems. One example is an overestimation of the plateau modulus for a given network topology when the affine assumption is employed. Also the viscoelastic relaxation spectrum depends on the extent of junction fluctuations. One way to implement the junction fluctuation into the theoretical model is to introduce the virtual spring that connects a point of the chain, which is supposed to be a member of a network junction, and the background field that is assumed to deform affinely. However, in singlechain models of polymer networks, the inclusion of the virtual spring brings about several fundamental problems. There have been some arguments about whether to include explicitly the stress σv originating from the virtual springs into the expression of the total stress σ of the network. Likhtman neglected σv in his slip-spring model for entangled polymer melts and assumed that the total stress σ is given by the stress from the chain σc alone because otherwise the contribution from the virtual springs is double counted. On the other hand, as Ramírez et al. noticed, the neglect of σv leads to a disagreement between the dynamic modulus of the chain part Gc(t ) obtained by applying a small external deformation to the network and that obtained from the Green-Kubo (GK) formula of the linear response theory Fluctuating Junctions of Physically Cross-Linked Networks in a Single-Chain Model. – Consistency Between the Relaxation Modulus for Small Deformations and Green-Kubo Predictions