Fraenkel segment

Eq. (18.17) is simply the definition of the Doi-Edwards damping function with the independent-alignment approximation (Chapters 8 and 12). The damping functions of Doi and Edwards with and without the independent-alignment approximation both explain well the experimental results of a well-entangled nearly monodisperse polymer system (Fig. 9.2). Note that here the unit vector u° represents the orientation of a Fraenkel segment as opposed to the orientation associated with an entanglement strand or primitive step in the Doi-Edwards theory. 

The above analysis can be extended to Fraenkel chains longer than a dumbbell. As detailed in Appendix 17.B, by comparing the simulation results obtained with and without the contribution from couplings between different segments, the above conclusion has been shown to be applicable to a multiple-segment Fraenkel chain as well. The basic natme of the fast mode and that of the slow mode are further confirmed by the studies of the step strain-simulated Gs t) curves as detailed below. 

Note For the mean square end-to-end vector of a Fraenkel chain, which is a static property, there is no correlation between different segments, just as in the case of the freely jointed chain (Chapter 1). 

---