Extensional material function

From the extensional material function data shown in Chapters 2 and 4, we know that extensional response can differ very much from shear. For example, at low extension rate tensile viscosity typically obeys Trouton s rule, but at higher extension rates it shows very little of the thinning so common for shear viscosity (Figure 2.1.3). Sometimes even thickening is observed (Figures 4.2.5-4.2.7). Such behavior is not unexpected from structural theories for rodlike suspensions (Chapter 10) and for polymers (Chapter 11). 

The remaining four geometries in Figure 7.1.2 all represent attempts to measure extensional material functions on lower viscosity liquids. Their strengths and weaknesses will be discussed in the following sections. 

As with the shear rheometers, we develop the basic working equations to convert measured forces and displacements into stresses and strains. These in turn are used to calculate extensional material functions. 

Effects of prehistory and nonconstant i obviate interpretation of results in terms of simple extensional material function, but the method is still useful... 

We define a material function rj, commonly called the elongational or extensional viscosity, through the primary normal stress difference % — %iT, thus, for the case of F e), it is given by... 

By contrast, quite different results have been obtained with dilute polymer solutions. Here the extensional viscosity may be as much as thousand times the shear viscosity. Measurement of extensional viscosity of such mobile liquids is far more difficult than shear viscosity, or even impossible. According to Barnes et al. (General references, 1993) "The most that one can hope for is to generate flow which is dominated by extension and then to address the problem of how best to interpret the data in terms of material functions that are Theologically meaningful". An example of the difficulties that arise with the measurement of extensional viscosity is shown In Fig. 16.21 for a Round Robin test... 

De Gennes PG (1979) Scaling concepts in polymer physics. Cornell University Press, Ithaca Dealy JM (1994) Official nomenclature for material functions describing the response of a viscoelastic fluid to various shearing and extensional deformations. J Rheol 38 179-191 Debbaut B, Crochet MJ (1988) Extensional effects in complex flows. J Non-Newtonian Fluid Mech 30 169-184... 

Eiastic Materiai Functions. The material functions that are used here for an isotropic, elastic material are the shear modulus G, the extensional modulus E, the bulk modulus K, and the Poisson s ratio v. However, any two of these provides the full set of information needed to describe such a material, as they are not all independent. For simple deformations, the definitions of these moduli are... 

From Equation 10.21, the dimensions of viscosity are stress multiplied by time, and in the SI system viscosity is measured in units of pascal-seconds (Pas). For polymer melts and solutions, the fluid behavior is non-Newtonian and Equation 10.21 must be modified to allow the viscosity to become a material function of the shear rate. Similarly, material elements may be deformed by pulling on opposite sides of the cube with an equal force this constitutes an extensional deformation that may be characterized by an extensional viscosity. 

J. M. Dealy, Official nomenclature for material functions describing the response of a viscoelastic fluid to various shearing and extensional deformations, J. Rheol. 28, 181-195 (1984). 

Extensive reviews [6-10] and a monograph [11] summarize the literature covering significant aspects of extensional flows in various commercial processes, theoretical treatment for ttie hydrod)mamics of such flows and different methods of determining material functions such as uniaxial, biaxial and planar extensional viscosities. 

The material function of prime importance in extensional flow is the extensional viscosity whidi is basically a measure of the resistance of the material to flow when stress is applied to extend it. 

Material functions must however be considered with respect to the mode of deformation and whether the applied strain is constant or not in time. Two simple modes of deformation can be considered simple shear and uniaxial extension. When the applied strain (or strain rate) is constant, then one considers steady material functions, e.g. q(y,T) or ri (e,T), respectively the shear and extensional viscosity functions. When the strain (purposely) varies with time, the only material functions that can realistically be considered from an experimental point of view are the so-called dynamic functions, e.g. G ((D,y,T) and ri (a), y,T) or E (o),y,T) and qg(o),y, T) where the complex modulus G (and its associated complex viscosity T] ) specifically refers to shear deformation, whilst E and stand for tensile deformation. It is worth noting here that shear and tensile dynamic deformations can be applied to solid systems with currently available instruments, whUst in the case of molten or fluid systems, only shear dynamic deformation can practically be experimented. There are indeed experimental and instrumental contingencies that severely limit the study of polymer materials in the conditions of nonlinear viscoelasticity, relevant to processing. 

Rheological measurements are performed so as to obtain a test fluid s material functions. Under viscometric flows we have seen that the shear viscosity and the primary and secondary normal stress differences suffice to rheologically characterize the fluid. If the flow field is extensional and the material is able to attain a state of dynamic equilibrium, then one measures the extensional viscosity otherwise, we measure the extensional viscosity growth or decay functions. In this section, we will examine steady and dynamic shear plus uniaxial extensional tests, since these make up the majority of routine rheological characterization. 

If we want to find out how a fluid behaves under extension, we have to somehow grip and stretch it. Experimentally, this is much more difficult than the shear arrangement, especially if the fluid has a low viscosity. Earlier (see Section 5) we saw that it is possible to classify steady extensional flows under the categories of uniaxial, biaxial and planar flows. We will now examine uniaxial testing, since this mode is more commonly employed as a routine characterization tool. Here we encounter two approaches the first seeks to impart a uniform extensional field and back out a true material function, while the second employs a mixed flow field that is rich in its extensional component (e.g. converging flows) and use it to back out a measured property of the fluid which is somehow related to its extensional viscosity. 

Further, it is shown how the unification technique can be extended to other rheological material functions, such as normal stress difference, dynamic viscoelastic parameters, and extensional viscosity, to obtain coalesced curves which are grade and temperature invariant. 

Comparisons of results from rheometers and indexers are essential in evaluating what material function dominates the indexer s response. Such comparisons can help us to determine when an indexer may give us useful rheological data, as in the case of squeezing flow. Figure 6.4.8. This ability becomes even more important in the next chapter, where we shall see that indexers are the only choice for extensional measurements on low viscosity fluids. 

In the following sections, in addition to giving the work ing equations for determining these material functions, we discuss corrections, applicability, and limitations for each of the methods depicted in Figure 7.1.2. Further information on extensional rheometry can be found in the references at the end of this chapter. Reviews by Meissner (1985,1987), a monograph by Petrie (1979), and Dealy s book (1982) are also recommend. ... 

Table 7.9.1 summarizes the main flow geometries that have been tested as extensional ifaeomemrs. They are listed from top to bottom in the order of this chapter, but also fiom rheometer to indexer, and generally from use with more viscous to less viscous test samples. The key advantages and disadvantages of each method are noted. The types of material function that these rheometers and indexers can measure were summarized in Figure II.3. 

Chapters 5-7, which describe shear and extensional rheometry, give the most important deformation geometries and derive the working equations for each. These equations permit conversion of measured quantities like force, torque, pressure, and angular velocity to stress and strain on the sample. Such stress and strain data allow us to determine rheological material functions, which are needed to evaluate the parameters in particular constitutive equations. 

In Chapter 4 it was explained that the linear elastic behavior of molten polymers has a strong and detailed dependency on molecular structure. In this chapter, we will review what is known about how molecular structure affects linear viscoelastic properties such as the zero-shear viscosity, the steady-state compliance, and the storage and loss moduli. For linear polymers, linear properties are a rich source of information about molecular structure, rivaling more elaborate techniques such as GPC and NMR. Experiments in the linear regime can also provide information about long-chain branching but are insufficient by themselves and must be supplemented by nonlinear properties, particularly those describing the response to an extensional flow. The experimental techniques and material functions of nonlinear viscoelasticity are described in Chapter 10. 

Most experimental studies of melt behavior involve shearing flows, and we saw in Chapter 5 that linear viscoelastic behavior is a rich source of information about molecular structure. However, no matter how many material functions we determine in shear, outside the regime of linear viscoelasticity such information cannot be used to predict behavior in other types of deformation, ie., for any other flow kinematics. A class of flows that is of particular importance in commercial processing is extensional flow. In this type of flow, material elements are stretched very rapidly along streamlines. Nonlinear behavior in extensional deformations provides information about structural features of molecules that are not revealed by shear data. 

The material function usually reported in extensional flow rheometry is the tensile stress growth coefficient defined by Eq. 10.92. 

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. 

Melt behavior has been studied using uniaxial (also called simple or tensile), biaxial, and planar extensional flows [9, Ch. 6]. However, only the first two of these are in general use and will be discussed here. A uniaxial extensional rheometer is designed to generate a deformation in which either the net tensile stress Tg or the Hencky strain rate e (defined by Eq. 10.89) is maintained constant. The material functions that can, in principle, be determined are the tensile stress growth coefficient / (f, ), the tensile creep compliance, andthetensile... 

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. 

We conclude that while entry flow subjects some fluid elements to large rates of elongation, the rate of elongation is not uniform in space, so this flow field is not useful in determining a well-defined material function that describes the response of a material to extensional flow. 

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