Convective Constraint Release CCR

The original Doi-Edwards model predicted that the shear stress in steady shear increases from zero and goes through a maximum. This type of behavior has never been observed, and this remained a basic deficiency of tube models until relatively recently when lanniruberto and Marrucci [76] introduced the concept of convective constraint release (CCR). In steady shear flow, molecules on neighboring streamlines are moving at different speeds, and this carries away entanglements at a rate comparable to the reciprocal of the shear rate. An early version of this idea that predates the tube model was presented in 1965 by Graessley [4]. 

From the point of view of tube models, the two key elements of nonlinear behavior are tube orientation and tube or chain stretch. The former nonlinearity can be probed using shear flow, but shear flows are not effective in generating significant chain stretch. As we have seen, chain stretch in shear is strongly suppressed by the mechanism of convective constraint release (CCR) up to extremely high shear rates. The CCR mechanism of relaxation is qualitatively much less important in extensional flows than in shear flows, because in the former molecules on neighboring streamlines move at the same velocity. Thus, extensional flows are of particular importance in the study of nonlinear viscoelasticity. 

This quantity fcj is the total rate of constraint release produced by the motion of chain j. This constraint release is produced by both convective constraint release (CCR), given by AT S - Aj /Aj, and reptative constraint release, given by 1/Zj. Rep tative constraint release... 

Tube models have been used to predict this material function for linear, monodisperse polymers, and a so-called standard molecular theory [159] gives the prediction shovm in Fig. 10.17. This theory takes into account reptation, chain-end fluctuations, and thermal constraint release, which contribute to linear viscoelasticity, as well as the three sources of nonlinearity, namely orientation, retraction after chain stretch and convective constraint release, which is not very important in extensional flows. At strain rates less than the reciprocal of the disengagement (or reptation) time, molecules have time to maintain their equilibrium state, and the Trouton ratio is one, i.e., % = 3 7o (zone I in Fig. 10.17). For rates larger than this, but smaller than the reciprocal of the Rouse time, the tubes reach their maximum orientation, but there is no stretch, and CCR has little effect, with the result that the stress is predicted to be constant so that the viscosity decreases with the inverse of the strain rate, as shown in zone II of Fig. 10.17. When the strain rate becomes comparable to the inverse of the Rouse time, chain stretch occurs, leading to an increase in the viscosity until maximum stretch is obtained, and the viscosity becomes constant again. Deviations from this prediction are described in Section 10.10.1, and possible reasons for them are presented in Chapter 11. 

Equations 11.23 through 11.26 are the counterparts to Eqs. 11.14 through 11.17 of the MED theory. In Eq. 11.23, the CCR term is just Ir S, similar to the CCR term K S - XlA) in the MED theory, but without the transient chain retraction rate A/ /I. (In Eq. 11.23, an absolute value must be taken of the CCR term at S to keep its value positive, while in the MED theory, this term is kept positive through the stretch equation 11.16.) The expression Eq. 11.23 for the orientational relaxation time contains not only the reptation time and the rate of convective constraint release k S, but also the stretch time t. This guarantees that even for velocity gradients greater than 1 /Tj, the rate of orientational relaxation remains bounded by 1 /Tj. This effectively switches off the CCR effect for fast flows, and so functions in much the same way as the switch function/(A) in the MED theory. Hence, no explicit switch function is present in Eq. 11.23. 

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