Independent alignment assumption

The Doi—Edwards Theory. Much of the excitement that came during the early years of the Doi-Edwards (DE) tube model (52-56) for reptation of polymer chains revolved about the fact that the use of the independent alignment assumption resulted in a special form of the K-BKZ model. Hence, much of the machinery that was developed in the 1960s and early 1970s to test the K-BKZ model could be implemented to test the DE model. The next sections discuss the molecular basis of the DE model, the independent alignment assumption, and how well monodis-perse polymers follow the DE version of the K-BKZ model. 

The framework for examining arbitrary deformation histories for the rep-tation fluid has now been established and one can obtain a constitutive law for the stress response to arbitrary deformation histories. While the DE model can provide a more general constitutive equation than that to be developed now, the more general form requires numerical solution. The approximation known as the independent alignment assumption (lA) results in a closed form solution that gives a special case of the K-BKZ theory developed previously. 

Two-Step Stress Relaxation Experiments. As discnssed above in the context of the K-BKZ model, the two-step stress relaxation experiment is a very important tool in the assessment of constitutive relations. The same has been true in the nse of the two-step experiment to assess the DE model. Here, the tests in the context of the independent alignment assumption are first examined. The important aspect of the experiments performed by Osaki and co-workers is that (123,129-133) the work was performed on monodisperse polymers. The work of Venerus and Burghardt, which is further built on the work of Osaki, will also be discussed. Finally, a proposed form of constitutive model that goes beyond the independent alignment assumption and gives a tractable set of predictions for the double-step experiments is examined. 

In addition, because much of the data in the literature include comparisons with the DE model both with and without independent alignment, the equations for the form of the DE model that Doi proposed by suppressing the independent alignment assumption are presented. Here the results are referred to as DE-NIA. In this case for two-step stress relaxation histories recall equations 59 (DE-IA or K-BKZ)... 

Half-Step Deformation. In the discussion of the K-BKZ model, the half-step deformation was covered. Here two sets of data that go beyond what was described above are discussed. First, in the Osaki data shown in Figure 52, it is seen that the DE model with independent alignment fits the shear stress response, but not the normal stress response. Also, the theory without the independent alignment assumption does not fit the shear data. The normal stress deviation from the model prediction is interesting, as this was a special history for which the K-BKZ and... 

Coppola et al. [142] calculated the dimensionless induction time, defined as the ratio of the quiescent nucleation rate over the total nucleation rate, as a function of the strain rate in continuous shear flow. They used AG according to different rheological models the Doi-Edwards model with the independent alignment assumption, DE-IAA [143], the linear elastic dumbbell model [154], and the finitely extensible nonlinear elastic dumbbell model with Peterlin s closure approximation, FENE-P [155]. The Doi-Edwards results showed the best agreement with experimental dimensionless induction times, defined as the time at which the viscosity suddenly starts to increase rapidly, normalized by the time at which this happens in quiescent crystallization [156-158]. 

Step to Zero Deformation. Figures 50 and 51 show the response for a two-step history in which one first applies a step strain yi for some time ti and then returns the material to zero deformation, y2 = 0. The results are, again, from Os-aki (138). Figure 50 shows the results when the first step duration t = 20 s, and it is seen that both the shear and normal stress responses are poorly represented by either the DE model with independent alignment (solid lines) or the predictions without the independent alignment assumption (dashed lines). Figure 51... 

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