No Chain Stretch the Doi-Edwards Equation

Doi and Edwards noted that since % = 2 is expected to be much smaller than the reptation time Tj, then for flows that are fast compared to the rate of reptation 1/Tj, but slow compared to the rate of retraction 1 /t, one can assume that the chains remain completely retracted during flow i.e., there is no chain stretch. Under this assumption, Doi and Edwards, in a seminal series of papers [12-15] derived the famous constitutive equation that bears their name. The Doi-Edwards (DE) constitutive equation, introduced in Section 10.3.4, is written as  

This memory function contains the same distribution of relaxation times as the hnear viscoelastic theory of reptation. Thus, from Eq. 6.27, we have 

for reptation, this relaxation spectrum is dominated by the longest relaxation time, we can, with reasonable accuracy, simplify Eq. 11.2 to 

Here is the plateau modulus discussed in Sections 5.8,5.9, and 6.3.2. The nonlinear aspects 

The DE equation was the first detailed molecular theory for the rheology of polymer melts and since its introduction has been the basis for almost all theories for the dynamics and rheology of entangled polymers. The predictions of the DE equation have been explored in detail in many publications (for a review, see [ 1,4,17,18]). Summing up these reviews, we can say that while the DE equation inspired all that followed and does capture some aspects of the rheology of melts, even for the simplest case of monodisperse linear polymers it is not a realistic, quantitative, theory except in a few special types of deformation. 

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