Strongly Entangled Systems

For strongly entangled systems (M > 10Me), the requirement of self-consistency is fulfilled identically, while the quantity x is connected with the intermediate length (see Section 5.1.2, formula (5.8)), or (as we shall see in Section 6.4.4, formula (6.55)) with the length of the macromolecule between adjacent entanglement Me, that is... 

In the last case, the parameter has the meaning of the ratio of a length of macromolecule between adjacent entanglements to the length of the macromolecule (see Section 5.1.2). The parameters t, x, , and Me appear to be equivalent for the strongly entangled systems. One of these parameters is used to describe polymer dynamics in either interpretation. 

For the systems of long macromolecules (strongly entangled systems), the requirements of universality and self-consistency allow us to write practically identical asymptotic relations (5.17) and (6.53) between the parameter y, introduced in Section 3.3.1, and the ratio E/B, which allows us to write for this case... 

For the weakly entangled systems, one can expect, that the ratio E/B, that is the parameter of internal viscosity is small. It can be demonstrated in Section 4.2.3, that transition point from weakly to strongly entangled systems occurs at E B. To describe these facts, one can use any convenient approximate function for the measure of internal resistance, for example, the simple formula... 

Now the dependencies of the phenomenological parameters B and E on the concentration of polymer c can also be given. From the above relations, it follows, for example, that for the strongly entangled systems... 

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). 

The transition point can be different, if one uses different modes, but only the transition point for the first mode is considered here. For the strongly entangled systems, according to relation (5.17), ijj = 7t2/x, so that, at %l> 1,... 

One can consider the parameter B to be a function of x and, taking equations (3.17), (3.25) and empirical value 6 = 2.4 into account, finds a solution of the equation, and estimate the value of the transition point between weakly and strongly entangled systems as... 

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... 

M < M, weakly entangled systems M > M, strongly entangled systems... 

The results of estimation of coefficient of self-diffusion due to simulation for macromolecules with different lengths are shown in Fig. 12. The introduction of local anisotropy practically does not affect the coefficient of diffusion below the transition point M, the position of which depends on the coefficient of local anisotropy. For strongly entangled systems (M > M ), the value of the index —2 in the reptation law is connected only with the fact of confinement of macromolecule, and does not depend on the value of the coefficient of local anisotropy. At the particular value ae = 0.3, the simulation reproduces the results of the conventional reptation-tube model (see equation (5.21)) and corresponds to the typical empirical situation (M = 10Me). 

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 critical value of molecular weight can be identified with the transition point between weakly and strongly entangled systems, the position of which was estimated in Sections 4.2.3 and 5.1.2 as... 

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. 

The expression (6.45) for the stress tensor can be applied to both weakly and strongly entangled systems, but, let us note, that the macromolecular dynamics is different in these cases. We use the expression (6.45) to calculate the stress tensor for entangled systems in linear approximation of macromolecular dynamics. Using expressions for moments (4.17), (4.20) and (4.21) one obtains... 

To begin with, let us consider the simple case, when C can be neglected in comparison to ( B in equations (6.47), which can be done, if one considers low-frequency properties of the systems with long macromolecules - the strongly entangled systems. In this case, according to (4.32) and (4.33), we have... 

Note that the first and the second terms in (6.49) at w —> oo have the orders of magnitudes nTib 2 and nTy-1, respectively. The ratio of the quantities is very small for systems of long macromolecules, so that the contribution of the first, conformation branch to the linear viscoelasticity is negligibly small at X x. Note also that, for strongly entangled systems, at x X or Af M, as it was shown in Section 4.2.3, conformational relaxation cannot be occurred via the diffusive mechanism (considered here), but via the reptation mechanism, so that the first term in equation (6.49) ought to be replaced by other term, for example, in the form... 

One can see that the approximation of the theory, based on the linear dynamics of a macromolecule, is not adequate for strongly entangled systems. One has to introduce local anisotropy in the model of the modified Cerf-Rouse modes or use the model of reptating macromolecule (Doi and Edwards 1986) to get the necessary corrections (as we do in Chapters 4 and 5, considering relaxation and diffusion of macromolecules in entangled systems). The more consequent theory can be formulated on the base of non-linear dynamic equations (3.31), (3.34) and (3.35). 

A preliminary estimate of which, according to (5.8), can be interpreted as the ratio of the square of the tube diameter (2 )2 to the mean square end-to-end distance (R2)o, shows that x AC 1 for strongly entangled systems. For large N, this enables us to replace summation by integration and, according to the rules of Appendix G, to obtain expressions for the characteristic quantities... 

Thus in the mesoscopic approximation or, in other words, in the mean-field approximation, the dynamic shear modulus of the melt or the concentrated solution of the polymer (strongly entangled systems) is represented by a function of a small number of parameters... 

The small deviation of the derived value of the index 3 from the empirical value 3.4 (see equations (6.43) and (6.44)) gave rise to the hopes that some improvements of the model could bring the correct results, at least, for strongly entangled systems. However, it appeared that the results delivered by the model far from empirical results (6.43) and (6.44) more, than one could earlier imagine (Altukhov et al. 2004). To appreciate these results... 

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... 

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... 

However, the effect of anisotropy of the environment is expressed differently. One can see from formula (5.17), that the parameter ip is big in the case of strongly entangled system (y < y ), so that, according to equation (7.22),... 

The set of internal variables is usually determined when considering a particular system in more detail. For concentrated solutions and melts of polymers, for example, a set of relaxation equation for internal variables were determined in the previous chapter. One can see that all the internal variables for the entangled systems are tensors of the second rank, while, to describe viscoelasticity of weakly entangled systems, one needs in a set of conformational variables xfk which characterise the deviations of the form and size of macromolecular coils from the equilibrium values. To describe behaviour of strongly entangled systems, one needs both in the set of conformational variables and in the other set of orientational variables w fc which are connected with the mean orientation of the segments of the macromolecules. 

To simplify the situation, one can keep only one internal variables with the smallest number from each set, that is x k and u k. It allows one to specify equations (8.28) for this case and to write a set of constitutive equations for two internal variables - the symmetric tensors of second rank. The particular case of general equations are equations (9.24)-(9.27) - constitutive equations for strongly entangled system of linear polymer. For a weakly entangled system, one can keep a single internal variable to obtain an approximate... 

Thus, two sets of constitutive relations are formulated. The systems of equations both (9.19)-(9.22), applicable to the weakly entangled systems, and (9.19) and (9.24)-(9.26), applicable to the strongly entangled systems, include, through equations (9.23) and (9.27), the tensors of global anisotropy... 

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