Zero-shear-rate viscosity theory

Graessley s theory, though satisfactory for linear polymers, has not yet been shown to apply to branched polymers. Fujimoto and co-workers (65) attempted to apply it to comb-shaped polystyrenes, but obtained only poor agreement with experiment. They attributed this to the failure of the assumption that the state of entanglement is the same in branched polymers as in linear ones. It is not surprising that this theory fails, for (in common with earlier theories) it predicts that the zero shear-rate viscosity of all branched polymers will be lower than that of linear ones, contrary to experiment. 

For polymeric fiuids, early Idnetic-theory workers (40) attempted to calculate the zero-shear-rate viscosity of dilute solutions by modeling the polymer molecules as elastic dumbbells. Later the constants in the Rivlin-Ericksen (17) expansion were obtained for dumbbells (41, 42) and other more complex models and only recently have the kernel functions in the memory integral expansions been obtained (43), This rapidly expanding field has been summarized recently in a monograph (44) here, too, molecular dynamics simulation may prove fhiitful (45),... 

De Gennes (1971) postulated that polymer molecules were constrained to move along a tube formed by neighbouring molecules. In a deformed melt, the ends of the molecules could escape from the tube by a reciprocating motion (reptation), whereas the centre of the molecule was trapped in the tube. When the chain end advanced, it chose from a number of different paths in the melt. This theory predicts that the zero-shear rate viscosity depends on the cube of the molecular weight. However, in the absence of techniques to image the motion of single polymer molecules in a melt, it is hard to confirm the theory. 

Before describing the reptation theory quantitatively, we first examine part of the rich set of behaviors that any theory of polymer chain dynamics in the melt state needs to be able to describe. One of the most interesting aspects of pol5uner melt and solution behavior is that once the material is well entangled, the viscoelastic behavior is quasi-imiversal. First, the zero shear rate viscosity rjo is observed to vary with the molecular weight to a very strong power (9,57) ... 

The Rouse theory is clearly not applicable to polymer melts of a molar mass greater than (M ) for which chain entanglement plays an important role. This is obvious from a comparison of eqs (6.40)-(6.42) and experimental data (Figs 6.13 and 6.14) and from the basic assumptions made. However, for unentangled melts, i.e. melts of a molar mass less than (M ), both the zero-shear-rate viscosity and recoverable shear compliance have the same molar mass dependence as was found experimentally (Figs 6.13 and 6.14). The Rouse model does not predict any shear-rate dependence of the shear viscosity, in contradiction to experimental data. 

Masao Doi and Sam F. Edwards (1986) developed a theory on the basis of de Genne s reptation concept relating the mechanical properties of the concentrated polymer liquids and molar mass. They assumed that reptation was also the predominant mechanism for motion of entangled polymer chains in the absence of a permanent network. Using rubber elasticity theory, Doi and Edwards calculated the stress carried by individual chains in an ensemble of monodisperse entangled linear polymer chains after the application of a step strain. The subsequent relaxation of stress was then calculated under the assumption that reptation was the only mechanism for stress release. This led to an equation for the shear relaxation modulus, G t), in the terminal region. From G(t), the following expressions for the plateau modulus, the zero-shear-rate viscosity and the steady-state recoverable compliance are obtained ... 

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

A major goal in the physics of polymer melts and concentrated solutions is to relate measurable viscoelastic constants, such as the zero shear viscosity, to molecular parameters, such as the dimensions of the polymer coil and the intermolecular friction constant. The results of investigations to this end on the viscosity were reviewed in 1955 (5). This review wiU be principaUy concerned with advances made since in both empirical correlation (Section 2) and theory of melt flow (Section 3). We shall avoid data confined to shear rates so high that the zero shear viscosity cannot be reliably obtained. The shear dq endent behavior would require an extensive review in itself. 

Figure 3.66 shows the steady-shear viscosity for a polymer system at three molar masses. Note the plateau in viscosity at low shear rates (or the zero-shear viscosity). Also note how the zero-shear viscosity scales with to the power 3.4. (This is predicted by Rouse theory (Rouse, 1953).) Figure 3.67 shows the viscosity and first normal-stress difference for a high-density polyethylene at 200 C. Note the decrease in steady-shear viscosity with increasing shear rate. This is termed shear-thinning behaviour and is typical of polymer-melt flow, in which it is believed to be due to the polymer chain orientation and non-affine motion of polymer chains. Note also that the normal-stress difference increases with shear rate. This is also common for polymer melts, and is related to an increase in elasticity as the polymer chain motion becomes more restricted normal to flow at higher shearing rates. 

By Cox-Mertz rule, for polymeric solutirms at low frequency and shear rate in accordance with the simple fluid theory, the zero shear viscosity can be estimated by [2, 20]... 

Enough experimental data are now available that a clear picture of polymer behavior in extension is emerging. For Newtonian liquids, it is found, by both experiment and theory, that the extensional viscosity equals exactly three times the shear viscosity. For non-Newtonian polymer melts, though, the extensional viscosity can exceed three time the zero-shear viscosity by more than a factor of ten. The extensional viscosity, however, is a function of the stretch rate and is frequently a decreasing function of the stretch rate, especially at large values of the stretch rate (see Figure 5). 

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