Convective constraint release

The earliest tube models included only the simplest nonlinearities, that is, convective constraint release was neglected (since its importance was not clearly recognized), and the retraction was assumed to occur so fast relative to the rate of flow that the chains were assumed to remain imstretched. The linear relaxation processes of constraint release and primitive path fluctuations were also ignored, so that the model contained only one linear relaxation mechanism, namely reptation, and only the nonlinearity associated with large orientation of tube segments, but no stretch. Subsequent models added the omitted relaxation phenomena, one at a time, and in what follows we present the most important constitutive models that included these effects, starting with models for monodisperse linear polymers. 

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M. H. Wagner, P. Rubio, and H. Bastian, The Molecular Stress Function Model for Polydisperse Polymer Melts with Dissipative Convective Constraint Release, J. Rheol., 45, 1387-1412 (2001). 

Problem 3.11 A theory incorporating convective constraint release and chain stretch into the Doi-Edwards model gives the constitutive equations below (Larson et al. 1998) ... 

In the above, X is the chain stretch, which is greater than unity when the flow is fast enough (i.e., y T, > 1) that the retraction process is not complete, and the chain s primitive path therefore becomes stretched. This magnifies the stress, as shown by the multiplier X in the equation for the stress tensor a, Eq. (3-78d). The tensor Q is defined as Q/5, where Q is defined by Eq. (3-70). Convective constraint release is responsible for the last terms in Eqns. (A3-29a) and (A3-29c) these cause the orientation relaxation time r to be shorter than the reptation time Zti and reduce the chain stretch X. Derive the predicted dependence of the dimensionless shear stress On/G and the first normal stress difference M/G on the dimensionless shear rate y for rd/r, = 50 and compare your results with those plotted in Fig. 3-35. 

Fig. 7.6 Illustration of convective constraint release responsible for the shearthinning region between...

The molecular dynamics theories need to make a proper combination to describe the rheological behaviors of polymer melt in various regions of shear rates (Bent et al. 2003). Above 1/t the convective constraint release dominates the rheological behaviors of polymers in shear flow, and thus explains the shear-thinning phenomenon. Beyond 1/t, the extensional deformation reaches saturation, and the shear flow becomes stable, entering the second Newtonian-fluid region, as demonstrated in Fig. 7.6. 

The molecular theory of extensional viscosity of polymer melts is again based oti the standard tube model. It considers the linear viscoelastic factors such as reptation, tube length fluctuations, and thermal constraint release, as well as the nonlinear viscoelastic factors such as segment orientations, elastic contractimi along the tube, and convective constraint release (Marrucci and lannirubertok 2004). Thus, it predicts the extensional stress-strain curve of monodispersed linear polymers, as illustrated in Fig. 7.12. At the first stage, the extensional viscosity of polymer melts exhibits the Newtonian-fluid behavior, following Trouton s ratio... 

An alternative approach is that convective constraint release can be regarded as due to a hopping motion of the tube, most simply following Rouse dynamics [32]. This has been developed analytically by Milner, McLeish and Likhtman [33], whose theory predicts a monotonically increasing shear stress with increasing shear strain rate, with no stress maximum for polymer melts. 

Graham, R.S., likhtman, A.E., McLeish, T.C.B., and Milner, S.T. (2003) Microscopic theory oflineai entangled polymer chains under rapid deformation induding chain stretch and convective constraint release. f Rhed., 47 (5), 1171-1200. 

It is of interest to note that Wagner s Eq. 10.10 predicts shear thinning, even though it does not seem to contain any feature that would represent the effect of convective constraint release. We will see later in this chapter that any form of the damping function h y) that decreases monotonically to a small value around a strain of unity will provide a rough estimate of the viscosity function. Thus, the form precise of the damping function has only a weak effect on shear thinning, and this is consistent with predictions of tube models that incorporate convective constraint release. 

It is the portion of the response curves around the maxima that are of primary interest in the characterization of nonlinear behavior, because this is where chain stretch has its most pronoimced effect. At longer times convective constraint release becomes dominant. Wagner etal. [23] used start-up of shear flow to evaluate the molecular stress function model for nonlinear behavior in which chain stretch and tube diameter are strain dependent. This theory was found to be suitable for describing an HDPE having a broad molecular weight distribution and an LDPE with random long-chain branching. 

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. 

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. 

Wagner, M. H., Rubio, P. Bastian, H. The molecular stress function model for polydisperse polynners melts with dissipative convective constraint release. /. Rheol. (2001) 45, pp. 1387-1412... 

X-X)-, this product accounts for the effect of convective constraint release on the length... 

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