Effects of fluctuations of cross-linking points on viscoelastic properties of associating polymer networks

We study the effects of fluctuations of cross-linking points, or junctions, on the dynamic mechanical and viscoelastic properties of polymer networks formed by multisticker associating polymers on the basis of a single-chain approach. Fluctuations of junctions are implemented by introducing virtual springs. We consider two possible cases for the treatment of virtual springs: the direct contribution from virtual springs is either neglected or included in the stress. We show that, if neglected, the fluctuation of junctions decreases (or softens) the dynamic modulus over a wide range of frequencies. This result agrees qualitatively with the result of several multichain models that predicts the decrease of the static or plateau modulus. We also show that the fluctuation accelerates the associative Rouse mode at low frequencies originating from the association/dissociation process of stickers. These results are apparently reasonable, but it is expected that there are some errors arising from thermodynamical inconsistency due to the neglect of virtual-spring contribution from the total stress against the virtual work principle and the second law of thermodynamics. On the other hand, if the direct contribution from the virtual springs is included in the stress, thermodynamics is satisfied but the plateau modulus does not change, contrary to the multichain prediction. The softening occurs only at the low-frequency regime. Thus, each method has both merits and demerits, and hence the treatment of junction fluctuations in the framework of single-chain approaches requires care and further investigation.