Weakly Entangled Systems

In steady state, the quantity ufk can be calculated as an expansion in powers of velocity gradients. Up to the accuracy to the second order, equation (7.38) is followed by the relation 

One uses equation (7.28) for xf to be convinced that the expansion begins with the second-order terms 

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In the case of the bulk polymer, the requirement of self-consistency of the theory states that the relaxation time r can be interpreted as a characteristic of the whole system. Properties of the system will be calculated in Sections 6.3.2 and 6.4.3, which allows one to estimate relaxation time r and quantity y. It will be demonstrated that, for weakly entangled systems (2Me < M < 10Me), the quantity x has the self-consistent value... 

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... 

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 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... 

Watanabe (1999, p. 1354) has deducted that, according to experimental data for polystyrene/polystyrene blends, when the matrix is a weakly entangled system, terminal time of relaxation depends on the lengths of macromolecules as... 

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... 

This is exactly the molecular-weight dependence of conformational relaxation times of polymer in non-entangled state and for the region of diffusive mobility (see equation (4.41), weakly-entangled system). 

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... 

For the weakly entangled system, the steady-state modulus depends on the molecular weight of polymer as M 1, while for strongly entangled system, the steady-state modulus does not depend on the molecular weight of polymer, which is consistent with typical experimental data for concentrated polymer systems (Graessley 1974). The expression for the modulus is exactly the same as for the plateau value of the dynamic modulus (equations (6.52) and (6.58)) Expressions (9.42) lead to the following relation for the ratio of the normal stresses differences... 

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