Linear viscoelastic properties of transient networks formed by associating polymers with multiple stickers

We have developed a single-chain theory that describes dynamics of associating polymer chains carrying multiple associative groups (or stickers) in the transient network formed by themselves and studied linear viscoelastic properties of this network. It is shown that if the average number $N¯$ of stickers associated with the network junction per chain is large, the terminal relaxation time $τA$ that is proportional to $τXN¯2$ appears. The time $τX$ is the interval during which an associated sticker goes back to its equilibrium position by one or more dissociation steps. In this lower frequency regime $ω<1/τX$⁠, the moduli are well described in terms of the Rouse model with the longest relaxation time $τA$⁠. The large value of $N¯$ is realized for chains carrying many stickers whose rate of association with the network junction is much larger than the dissociation rate. This associative Rouse behavior stems from the association/dissociation processes of stickers and is different from the ordinary Rouse behavior in the higher frequency regime, which is originated from the thermal segmental motion between stickers. If $N¯$ is not large, the dynamic shear moduli are well described in terms of the Maxwell model characterized by a single relaxation time $τX$ in the moderate and lower frequency regimes. Thus, the transition occurs in the viscoelastic relaxation behavior from the Maxwell-type to the Rouse-type in $ω<1/τX$ as $N¯$ increases. All these results are obtained under the affine deformation assumption for junction points. We also studied the effect of the junction fluctuations from the affine motion on the plateau modulus by introducing the virtual spring for bound stickers. It is shown that the plateau modulus is not affected by the junction fluctuations.