Self-Consistency of the Mesoscopic Approach

One can notice that the dissipative terms in the dynamic equation (3.11) (taken for the case of zero velocity gradients, z/jj = 0) have the form of the resistance force (D.3) for a particle moving in a viscoelastic liquid, while the memory functions are (with approximation to the numerical factor) fading memory functions of the viscoelastic liquid. The macromolecule can be considered as moving in a viscoelastic continuum. In the case of choice of memory functions (3.15), the medium has a single relaxation time and is characterised by the dynamic modulus 

On the other hand, the properties of the system as a whole can be calculated and the macroscopic dynamic modulus can be determined. Here the question of the relation between the postulated micro-viscoelasticity and the resulting macro-viscoelasticity appears. The answer requires a properly formulated self-consistency condition. Simple speculations show that equality of the micro- and macro-viscoelasticity cannot be obtained. Nevertheless, it is natural to require the equality of relaxation times of micro- and macroviscoelasticities. It will be shown in this section that this condition can be satisfied. 

we shall consider in detail the characteristic quantities the viscosity coefficient rj and the elasticity coefficient i/, defined by expansion (6.11), and the value of the dynamic modulus on the plateau Ge. The latter can be calculated as the limiting value of the modulus at frequencies satisfying the relation 

The estimation of the main terms of expansion of dynamic modulus (6.49) determine the expressions for the terminal quantities 

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 

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