Single-Mode Description of Entangled System

Notwithstanding the simplifying assumptions in the dynamics of macromolecules, the sets of constitutive relations derived in Section 9.2.1 for polymer systems, are rather cumbersome. Now, it is expedient to employ additional assumptions to obtain reasonable approximations to many-mode constitutive relations. It can be seen that the constitutive equations are valid for the small mode numbers a, in fact, the first few modes determines main contribution to viscoelasticity. The very form of dependence of the dynamical modulus in Fig. 17 in Chapter 6 suggests to try to use the first modes to describe low-frequency viscoelastic behaviour. So, one can reduce the number of modes to minimum, while two cases have to be considered separately. 

It is clear that at transition from many modes to a single mode, weight coefficients for mode contributions into the stress tensor have to be introduced. One has to require correspondence of some specified quantities to the same ones calculated within the many-mode theory. The procedure eliminates the 

we refer to constitutive relations (9.19)-(9.22) which describe the behaviour of the system with moderate concentration of polymer and/or the systems with shorter macromolecules, when the characteristic parameters of the system are satisfied to conditions 

Every mode contains two relaxation processes, described by the relaxation equations (9.20) and (9.21). One can retain one relaxation equation from each relaxation branch only, so that the two relaxation equations have to be considered 

The relaxation equations (9.46) and (9.47) describe the joint non-linear relaxation of the two variables which appear to be weakly connected with each other through the term with the small quantity ip in equation (9.47). 

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