Reptation model relaxation times

In connection with a discussion of the Eyring theory, we remarked that Newtonian viscosity is proportional to the relaxation time [Eqs. (2.29) and (2.31)]. What is needed, therefore, is an examination of the nature of the proportionality between the two. At least the molecular weight dependence of that proportionality must be examined to reach a conclusion as to the prediction of the reptation model of the molecular weight dependence of viscosity. 

There are three basic time scales in the reptation model [49]. The first time Te Ml, describes the Rouse relaxation time between entanglements of molecular weight Me and is a local characteristic of the wriggling motion. The second time Tro M, describes the propagation of wriggle motions along the contour of the chain and is related to the Rouse relaxation time of the whole chain. The important... 

The scaling results above all pertain to local segmental relaxation, with the exception of the viscosity data in Figure 24.5. Higher temperature and lower times involve the chain dynamics, described, for example, by Rouse and reptation models [22,89]. These chain modes, as discussed above, have different T- and P-dependences than local segmental relaxation. 

The sum over weighted relaxation times is heavily dominated by the longest time (the reptation time) r gp=L /7T Dp. Because of this the frequency-dependent dissipative modulus, G"(cd) is expected to show a sharp maximum The higher modes do modify the prediction from that of a single-mode Maxwell model, but only to the extent of reducing the form of G"(a>) to the right of the maximum from ccr to In fact, experiments on monodisperse linear polymers... 

There were different generalisations of the reptation-tube model, aimed to soften the borders of the tube and to take into account the underlying stochastic dynamics. It seems that the correct expansion of the Doi-Edwards model, including the underlying stochastic motion and specific movement of the chain along its contour - the reptation mobility as a particular mode of motion, is presented by equations (3.37), (3.39) and (3.41). In any case, the introduction of local anisotropy of mobility of a particle of chain, as described by these equations, allows one to get the same effects on the relaxation times and mobility of macromolecule, which are determined by the Doi-Edwards model. 

Derived from linear approximation of the equations (3.37), the equilibrium correlation function (4.29), defines two conformation relaxation times r+ and r for every mode. The largest relaxation times have appeared to be unrealistically large for strongly entangled systems, which is connected with absence of effect of local anisotropy of mobility. To improve the situation, one can use the complete set of equations (3.37) with local anisotropy of mobility. It is convenient, first, to obtain asymptotic (for the systems of long macromolecules) estimates of relaxation times, using the reptation-tube model. 

Each point is calculated as the asymptotic value of the rate of relaxation for large times (see examples of dependences in Fig. 6) for a macromolecule of length M = 25Me (x = 0.04, B = 429, ij) = 8.27) with the value of the coefficient of external local anisotropy ae = 0.3. The dashed lines reproduce the values of the relaxation times of the macromolecule due to the reptation-tube model. The labels of the modes are shown at the lines. Adapted from Pokrovskii (2006). 

In other cases, several discrete relaxation times or distributions of relaxation times can be found [39]. This is typically the case if the stress relaxation is dominated by reptation processes [42 ]. The stress relaxation model can explain why surfactant solutions with wormlike micelles never show a yield stress Even the smallest applied stress can relax either by reptation or by breakage of micelles. For higher shear rates those solutions typically show shear thinning behaviour and this can be understood by the disentanglement and the orientation of the rod-like micelles in the shear field. 

The Rouse model is the earliest and simplest molecular model that predicts a nontrivial distribution of polymer relaxation times. As described below, real polymeric liquids do in fact show many relaxation modes. However, in most polymer liquids, the relaxation modes observed do not correspond very well to the mode distribution predicted by the Rouse theory. For polymer solutions that are dilute, there are hydrodynamic interactions that affect the viscoelastic properties of the solution and that are unaccounted for in the Rouse theory. These are discussed below in Section 3.6.1.2. In most concentrated solutions or melts, entanglements between long polymer molecules greatly slow polymer relaxation, and, again, this is not accounted for in the Rouse theory. Reptation theories for entangled... 

The storage and loss moduli, G and G", are obtained from the relaxation spectrum in the usual way—that is, using G = Gi[co rl/(l + co zl)] G — G,[mT /(l -P The longest relaxation mode of the relaxation modulus in Eq. (3-67) is the dominant one it accounts for 96% of the zero-shear viscosity. Thus, the reptation model predicts that for a nearly monodisperse melt, the relaxation spectrum is dominated by a single relaxation time, T = Ta. This is in reasonable accord with experimental data at low and moderate frequencies (see the dashed line in Fig. 3-29). As the frequency increases, however, there... 

The reptation ideas discussed above will now be combined with the relaxation ideas discussed in Chapter 8 to describe the stress relaxation modiihis G t) for an entangled polymer melt. On length scales smaller than the tube diameter a, topological interactions are unimportant and the dynamics are similar to those in unentangled polymer melts and are described by the Rouse model. The entanglement strand of monomers relaxes by Rouse motion with relaxation time Tg [Eq. (9.10)] ... 

In the simple reptation model, there is a delay in relaxation (the rubbery plateau) between te and the reptation time of the chain trep [Eq. (9.11)]. By restricting the chain s Rouse motions to the tube, the time the chain takes to diffuse a distance of order of its size is longer than its Rouse time by a factor of 6 N/N. This slowing arises because the chain must move along the confining tube. The reptation time of the chain trep — 0.2 s is measured experimentally as the reciprocal of the frequency at which G = G" in Fig. 9.3 at low frequency (see Problem 9.8). In practice, this time is determined experimentally and tq, Te and Tr are determined from Trep-... 

The stress relaxation modulus in the reptation model is proportional to the fraction of original tube remaining at time t (see Fig. 9.1). As time goes on, sections of the original tube are abandoned when the chain end first visits them. Such a problem is called a first-passage time problem. 

The longest relaxation time in this model is the reptation time required for the chain to escape from its tube... 

The simple reptation model does not properly account for all the relaxation modes of a chain confined in a tube. This manifests itself in all measures of terminal dynamics, as the longest relaxation time, diffusion coefficient and viscosity all have stronger molar mass dependences than the reptation model predicts. Tn Sections 9.4.5 and 9.6.2, more accurate ana-... 

On length scales larger than the tube diameter, topological interactions are important and the motion is described by the reptation model with the chain relaxation time given by the reptation time ... 

The stress relaxation modulus then decays exponentially at the reptation time [Eq. (9.22)]. The terminal relaxation time can be measured quite precisely in linear viscoelastic experiments. Hence, Eq. (9.82) provides the simplest direct means of testing the Doi fluctuation model and evaluating... 

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