Strongly entangled system relaxation times

The mechanism of small-scale (fast) relaxation of conformation of the macromolecule does not change at the transition from weakly to strongly entangled systems the times of relaxation are defined by formulae (4.31). However, one has to take into account, that ip weakly entangled systems, whereas ip 1 for strongly entangled systems, so that one has for the largest of the fast relaxation times... 

It is instructive to compare the system of equations (3.46) and (3.47) with the system (3.37). One can see that both the radius of the tube and the positions of the particles in the Doi-Edwards model are, in fact, mean quantities from the point of view of a model of underlying stochastic motion described by equations (3.37). The intermediate length emerges at analysis of system (3.37) and can be expressed through the other parameters of the theory (see details in Chapter 5). The mean value of position of the particles can be also calculated to get a complete justification of the above model. The direct introduction of the mean quantities to describe dynamics of macromolecule led to an oversimplified, mechanistic model, which, nevertheless, allows one to make correct estimates of conformational relaxation times and coefficient of diffusion of a macromolecule in strongly entangled systems (see Sections 4.2.2 and 5.1.2). However, attempts to use this model to formulate the theory of viscoelasticity of entangled systems encounted some difficulties (for details, see Section 6.4, especially the footnote on p. 133) and were unsuccessful. 

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. 

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. 

Each point is calculated as asymptotic value of the rate of relaxation for large times (see examples of dependences in Fig. 6) for different molecular weights with corresponding values of the parameters B and if. The values of the coefficients of local anisotropy are ae = 0.3, ae = 0.06 for the circles and ae = 0.3, ae = 0.15 for the squares. The solid line depicts analytical results for linear approximation. The dashed lines with the slope 1 reproduce the well-known dependence n M3 for the relaxation time of macromolecules in strongly entangled systems. Adapted from Pokrovskii (2006). 

While the law with index 3.4 for viscosity is valid in the whole region above Mc, the dependence of terminal relaxation time is different for weakly and strongly entangled systems (Ferry 1980) and determines the second critical point M ... 

The difference in the molecular-weight dependence of the terminal relaxation time can be attributed to the change of the mechanisms (diffusive and repta-tion, correspondingly) of conformational relaxation in these systems. Further on in this section, we shall calculate dynamic modulus and discuss characteristic quantities both for weakly and strongly entangled systems. 

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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