Reptation Relaxation

Equations of the previous subsection describe relaxation of large-scale conformation of the macromolecule due to diffusive motion of particles through the sea of segments, which is valid, as we considered in Chapters 4 and 5, for weakly entangled systems. For highly entangled system, when 

One has no results for this case derived consequently from the basic equations (7.6) with local anisotropy. Instead, to find conformational relaxation equation, we shall use the Doi-Edwards model, which approximate the large-scale conformational changes of the macromolecule due to reptation. The mechanism of relaxation in the Doi-Edwards model was studied thoroughly (Doi and Edwards 1986 Ottinger and Beris 1999), which allows us to write down the simplest equation for the conformational relaxation for the strongly entangled systems 

We assume in the above equation, that anisotropy of environment is possible. 

If one neglect the latter, the relaxation equation takes the simpler form 

One can compare equations (7.29) and (7.30) with equations (7.25) and (7.26) to see that the only difference between this and previous case is the difference in relaxation times, which for the strongly entangled systems, according to formula (4.37), are 

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Persistent Motion by Partial Desorption and Reptating Relaxation. 168... 

These are exactly the known results (Doi and Edwards 1986, p. 196). The time behaviour of the equilibrium correlation function is described by a formula which is identical to formula for a chain in viscous liquid (equation (4.34)), while the Rouse relaxation times are replaced by the reptation relaxation times. In fact, the chain in the Doi-Edwards theory is considered as a flexible rod, so that the distribution of relaxation times naturally can differ from that given by equation (4.36) the relaxation times can be close to the only disentanglement relaxation time r[ep. 

A particular choice of the coefficients ae = 0.3 and a = 0.06 determines the value T = 417 r for the relaxation time of the first mode, which is close to the reptation relaxation time 370 r. The calculated relaxation times of the third mode 73 = 315 r is a few times as much as the corresponding reptation relaxation time 41.1 r, which indicates that the dependence of the relaxation times on the mode label is apparently different from the law (4.36). It is clearly seen in Fig. 7, where the dependence of the relaxation times of the first six modes of a macromolecule on the coefficient of internal anisotropy is shown. The relaxation times of different modes are getting closer to each other with increase of the coefficient of internal anisotropy. The values of the largest relaxation time of the first mode for different molecular weights are shown in Fig. 8. The results demonstrate a drastic decrease in values of the largest relaxation times for strongly entangled systems induced by introduction of local anisotropy. 

Though the reptation relaxation times are defined by equation (4.37), the weights pa of the contributions of separate relaxation processes remain unknown, and in fact, the replacement is forbidden, so that we prefer, as an initial approximation, to consider evaluation of dynamic modulus without any modification. 

In panels (a)-(c), comparison of the blend moduli calculated from the model (curves Equations 3.69 through 3.73) with the moduli data (symbols) for various high-M polyisoprene/ poly(p-tert butyl styrene) (PI/PtBS) blends as indicated. The sample code nmnbers of the blends indicate 10 M of the components. The model considers the cooperative Rouse equilibration and successive constraint release (CR)/reptation relaxation of the component chains, and the model parameters summarized in Table 3.1 were determined experimentally. (Redrawn, with permission, from Watanabe, H., Q. Chen, Y. Kawasaki, Y. Matsumiya, T. Inoue, and O. Urakawa. 2011. Entanglement dynamics in miscible polyisoprene/poly(p-fert-butylstyrene) blends. Macromolecules 44 1570-1584). 

The molecular aspects of interdigitation of Unear entangled polymers (M > Me) during welding of polymer interfaces are summarized for a symmetric interface in Table 2. The properties are expressed as a time-dependent function of reptation relaxation time (Tr M ). Thus, for some general property H ... 

According to Doi and Edwards [9, p. 196] the time behaviour of the equilibrium correlation function is described by a formula which is similar to formula for the Rouse chain but the Rouse relaxation times replaced by the reptation relaxation times... 

The characteristic reptation relaxation time ra in our previous notations can be written as... 

In order to characterize the reduction in chain mobility in an entangled melt quantitatively, one can use the characteristic reptation relaxation time, Tj.gp, introduced by deGeiines [155] and Doi and Edwards [156]. The is given for an entangled chain as ... 

The reptation model envisages the polymer chain to be enclosed within an initial tube out of which it gradually escapes by wriggling in a snake-like manner ( reptating reptare to creep). The reptation relaxation time Tr corresponds to the time when about 70% of the chain has escaped from the initial tube. During this time, the interfacial thickness increases with time t, in proportion to f, in contrast with the relationship for Fickian diffusion. The relaxation time Tr is proportional to the cube of molecular weight. 

Chain relaxation and diffusion mechanisms relaxation time (r) and molecular weight (M) relationships. Me is molecular weight between chain entanglements Te is the Rouse relaxation time between chain entanglements tro is the Rouse relaxation time of the whole chain Tr is the Reptation relaxation time... 

The sharpness of the predicted peaks is due, to a small degree, to the use of a single relaxation time for each component of the bidisperse melt. This deficiency can easily be fixed by including the full reptation relaxation spectrum for each component. That is, for P(f) we can generalize Eq. 6.25 for the double reptation model to include two components ... 

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