Rheological models extensional viscosity

Rheological models have been described for steady shear viscosity function, normal stress difference function, complex viscosity function, dynamic modulus function and the extensional viscosity function. The variation of viscosity with temperature and pressure is also discussed. 

Polymers with LCB exhibit a completely different rheological behaviour, which has been formulated in the pom-pom viscoelastic model, based on a phased reptation process of the side and backbone chains of the system. This model has been tested against the behaviour of different LDPE types, where shear and extensional viscosity are taken into account (Figure 5.9). 

We have seen that rheometers capable of accurate measiuements of extensional flow properties are limited to use at low Hencky strain rates, usually well below 10 s . In order to reach higher strain rates, the drawdown of an extruded filament ( melt spinning ) and the converging flow into an orifice die or capillary have been used to determine an apparent extensional viscosity . Since the stress and strain are not imiform in these flows, it is necessary to model the flow in order to interpret data in terms of material functions or constants. And such a simulation must incorporate a rheological model for the melt under study, but if a reliable rheological model were available, the experiment would not be necessary. This is the basic problem with techniques in which the kinematics is neither controlled nor known with precision. It is necessary to make a rather drastically simplified flow analysis to interpret the data in terms of some approximate material function. 

It is clear that viscoelastic fluids require a constitutive equation that is capable of describing time-dependent rheological properties, normal stresses, elastic recovery, and an extensional viscosity which is independent of the shear viscosity. It is not clear at this point exactly as to how a constitutive equation for a viscoelastic fluid, when coupled with the equations of motion, leads to the prediction of behavior (i.e., velocity and stress fields) which is any different from that calculated for a Newtonian fluid. As the constitutive relations for polymeric fluids lead to nonlinear differential equations that cannot easily be solved, it is difficult to show how their use affects calculations. Furthermore, it is not clear how using a constitutive equation, which predicts normal stress differences, leads to predictions of velocity and stress fields which are significantly different from those predicted by using a Newtonian fluid model. Finally, there are numerous possibilities of constitutive relations from which to choose. The question is then When and how does one use a viscoelastic constitutive relation in design calculations especially when sophisticated numerical methods such as finite element methods are not available to the student at this point For the... 

Figure 3.32 The predictions of the Doi-Edwards integral model for the normalized uniaxial extensional (or elongational) viscosity rj and for the viscometric shear coefficients r)(y) and i(y). Also shown are the predictions of the differential model, Eq. (3-77). (From Larson, 1984b, with permission from the Journal of Rheology.)--------------------------------------------------...

The Phan-Thien/Tanner constitutive equation does not represent the state of the art in modeling melt flow at the time of this writing, but it is adequate to illustrate the response of melts of flexible polymers in complex flows and it has a mathematical structure that does not differ substantively from other equations with a firmer basis in molecular theory. Furthermore, it has been widely used in simulation studies to date. Hence, we will use it for illustrative purposes in this text, recognizing that it is likely to be replaced as the preferred constitutive equation for applications. The minimum rheological information required for simulations is thus the temperature-dependent linear viscoelastic spectrum and the temperature-dependent viscosity as a function of shear rate. Extensional data should be used, but they are often unavailable when the PTT equation is employed it is therefore common to select a reasonable value of to describe the extensional response. 

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