Extensional Tube model predictions

Figure 10.17 Standard tube-model prediction of the extensional viscosity function. In zone I the Trouton ratio is three (tj = 3 tj, while in zone II there is orientation but no stretch, and CCR has little effect, so the stress is constant. When the strain rate reaches 1/Tp, chain stretch occurs and the extensional viscosity rises until maximum stretch is achieved,and the stress becomes proportional to the strain rate. From Marrucci and lanniruberto [159].

Tube models predict that in a linear, monodisperse polymer, the extensional viscosity should decrease with strain rate at moderate rates, pass through a minimum, and then rise as chain stretch begins to occur. At a sufficiently high rate, the chain reaches its maximum stretch, and the extensional viscosity should approach a final plateau. However, strain rates capable of generating chain stretch in linear molecules are very high and out of the range of most experimental methods. Some aspects of this behavior, however, have been observed in experimental data. 

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... 

Fig. 7.12 Illustration of extensional viscosity versus the extensional rate curve predicted by the molecular theory based on the standard tube model for the stable extensional flow of linear polymers. Starting from the low extensional rate, the viscosity first keeps in 3%, then decays, after deformation begins to increase, till to saturation (Marrucci and lannirubertok 2004) (Readapted with permission)...

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. 

Figure 4 shows the transient evolution of the extensional viscosity at varying strain rates for PHA Sample I. The plots follow the start-up to reach a steady state plateau value. At short times, the curves fall on top of one another. At high strain rates, we observe a slight decrease in the steady state extensional viscosity for both PHA samples. This could indicate that the two PHA samples are strain-rate thinning, as predicted by the Doi-Edwards tube model [9]. The Trouton Ratio (Tr), defined as the ratio of the extensional viscosity to the zero-shear viscosity of the polymer, was also investigated for the two PHA samples. For both PHA samples, the Trouton Ratio maintained the Newtonian value of Tr=3 at low strain rates. 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

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