Shear Tube model predictions

The tube model predictions for polymer melts have been extensively compared against experimental data. While there appears to be qualitative agreement between theory and experiment, the agreement is not quantitative and several discrepancies between theory and experiment exist. For example, the theoretical predictions of the zero-shear viscosity, jo are about 15 times larger 4han experimental values. Also, the molecular weight dependence of qo is typically stronger than the qo predicted by... 

This model, often referred to as the upper convective Maxwell model, is weakly non-linear in that it predicts a first normal stress, but no shear thinning effects, i.e, the shear stress increases linearly with shear rate so that the viscosity is independent of shear rate. Combining Eqs. 2, 4, 5 and 6, we see that the tube model predicts the viscosity to be. 

The original tube model predicts that the spectrum of relaxation times, defined by Eq. 6.27, is quite narrow, i.e it is dominated by the largest relaxation time Tj. In fact, 98% of the zero-shear viscosity can be attributed to the slowest relaxation mode, which is controlled by the longest relaxation time i.e., to the first term in Eq. 6.29. Note that the reptation time, is... 

Turning to the behavior of typical melts, it is found that the damping function is not nearly as sensitive to molecular structure as are the linear viscoelastic properties, e.g. the storage and loss moduli. The rubberlike liquid, as well as the tube model, predict that the ratio of the first normal stress difference to the shear stress in step shear should be equal to the strain at all strains, and this is in fact observed. The other quantity measured in simple shear experiments is the second normal stress difference, but this is difficult to measure and few data are available. Of the shear histories other than step strain than have been used to study nonlinear viscoelasticity, start-up of steady simple shear has been the most used. If the shear rate is sufficiently large, some degree of chain stretch can be generated in the early stages. 

It became clear in the early development of the tube model that it provided a means of calculating the response of entangled polymers to large deformations as well as small ones [2]. Some predictions, especially in steady shear flow, lead to strange anomaUes as we shall see, but others met with surprising success. In particular the same step-strain experiment used to determine G(t) directly in shear is straightforward to extend to large shear strains y. In many cases of such experiments on polymer melts both Hnear and branched, monodisperse and polydisperse,the experimental strain-dependent relaxation function G(t,Y) may be written... 

Clearly, the extrapolated front factor Ci — in principle — offers no upper limit and would be greater than 1 for (of course umealistic) very large Gn/Gc. In the case of usual values Gn/Gc 1, the difference between Cj and the free-fluctuation phantom limit result Cj = 0.5, if the functionality f is 4, plays only a minor role. On the other hand, the theory of restricted junction fluctuations limits Ci to values smaller (or equal to) 1, and the upper bound corresponds to the affine-deformation phantom limit where complete suppression of junction fluctuation has been assum-mj4i-44) Furthermore, Cj passes through a maximum for values of Cj in the range between 0.5 and 1, and the sum of the reduced parameters Cj and Cj, the reduced shear modulus g = G/v kT c, -t- 03, is limited to the range 0.5 g g 1 The different predictions of the tube model with deformation-dependent constraints and of the model of restricted junction fluctuations concerning the c, — C2. relation are shown in Fig. 9. 

Other theories which are not based on the reptation/tube model have also been developed. While some make an ansatz about how a polymer moves in a melt, others are more microscopic. Schweizer uses a modecoupling approach. His theory predicts the emergence of a plateau shear... 

According to the Rouse model, n is thus proportional to M, which is indeed verified for all polymers in the domain of low molar masses—that is, lower than the critical molar mass between entanglements. From the observation that the Newtonian viscosity (t)o) of a polymer melt is the product of the shear modulus G times frep—that is, the time for a chain to emerge from the tube—the reptation model predicts... 

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. 

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]. 

The predictions of simplified tube models that include CCR, such as the MLD model [27], and the models of Likhtman etal [28, 37, 38] and of lanniruberto and Marrucci [31-33], discussed in Section 11.3.4, as well as the related theory of Fang ef a/. [29], have been compared to experimental data on start-up of shear flow for entangled monodisperse polymer solutions. 

The simplest nonlinear tube model is the classical Doi-Edwards (DE) constitutive equation for linear polymers, which accounts for reptation and affine rotation of tube segments. The Doi-Edwards equation predicts thinning in both shear and extension, because it accounts for orientation of tube segments, but it is unable to predict extension thickening because it neglects the stretching of tube segments. Inclusion of tube stretch leads to the Doi-Edwards-Marrucci-... 

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. 

The original, fixed tube model for linear chains does not consider the chain stretch and thus predicts the shear and elongational viscosities to scale as nd / (e) e", ... 

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