Shear material functions

Data for a 60 mole % PHB/PET system are shown in Figure 20, We observe at all strain levels that G continues to relax to zero rather than approach a plateau as would be the case when a yield stress exists. The relaxation modulus also seems to be highly stain dependent which is in contrast to the fact that dynamic and steady shear material functions agree so well. For flexible chain polymers the strain dependence of G is associated with the rate of loss of entanglements. For LCP it is not clear as to the significance of the strain dependence of G. 

All these rheometers can be used to measure one or more of the shear material functions discussed in Chapters 1-4 ... 

We recall that N2 is a steady shear material function and only depends on shear rate. Since y is independent of r... 

Nonhomogeneous flow, only steady shear material functions Entrance corrections entail more data collection... 

Chapters S and 6 have described and evaluated the important shear rheometers and indexers. If measurements of torques, forces, and velocities or pressures and flow rates are made and interpreted properly, all these rheometers should measure the same shear material functions. To conclude these two chapters, we make a few comparisons between rheometers to emphasize this point. Table 6.5.1 highlights some of the strengths and weaknesses of each. 

In Section 3.1 we define two basic flows used in the characterization of polymeric fluids along with the appropriate material functions. These basic flows are also found in polymer processes. In Section 3.2 several constitutive equations capable of describing the viscoelastic behavior of polymer melts are presented. The emphasis in this section is on manipulating these equations for flows in which the deformation history is known. In this section we have added discussion of fiber suspensions as they are commonly processed to yield materials with increased stiffness and strength. In Section 3.3 an introduction into the methods for measuring rheological properties is presented. In Section 3.4 several useful relationships between material functions are presented. These relationships (or correlations) are important as they allow one to obtain estimates, for example, of steady shear material functions from linear viscoelastic data. Because... 

Steady Shear Material Functions for the Giesekus Model. The Giesekus model (Giesekus, 1982) is a... 

The model contains four parameters a, Ai, A2, and r]o and a = al(l - A,2/A,i). The model is capable of describing many of the observed rheological properties of polymeric fluids. Show that the steady shear material functions are... 

C.3 Fit of Giesekus Model to Rheological Data. The steady shear material functions for the Giesekus model are given in Problem 3B.5. Find the parameters in this model which give the best fit of the steady shear and dynamic oscillatory data at 170 °C given for LDPE in Appendix A.l, Tables A.l and A.2. 

These flow features are of importance in a great number of technical processes, especially for high process velocities when extremely high shear rates can be observed. For polymeric systems this can lead to a so-called non-Newtonian behaviour, i.e. the rheological material functions become dependent on the shear or elongational rate. 

Deviation from laminar shear flow [88,89],by calculating the material functions r =f( y),x12=f( Y),x11-x22=f( y),is assumed to be of a laminar type and this assumption is applied to Newtonian as well as viscoelastic fluids. Deviations from laminar flow conditions are often described as turbulent, as flow irregularities or flow instabilities. However, deviation from laminar flow conditions in cone-and-plate geometries have been observed and analysed for Newtonian and viscoelastic liquids in numerous investigations [90-95]. Theories have been derived for predicting the onset of the deviation of laminar flow between a cone and plate for Newtonian liquids [91-93] and in experiments reasonable agreements were found [95]. 

The theoretical basis for spatially resolved rheological measurements rests with the traditional theory of viscometric flows [2, 5, 6]. Such flows are kinematically equivalent to unidirectional steady simple shearing flow between two parallel plates. For a general complex liquid, three functions are necessary to describe the properties of the material fully two normal stress functions, Nj and N2 and one shear stress function, a. All three of these depend upon the shear rate. In general, the functional form of this dependency is not known a priori. However, there are many accepted models that can be used to approximate the behavior, one of which is the power-law model described above. 

As a result, we find for sols that the divergence of the above zero shear viscosity rj0 and of two other linear viscoelastic material functions, first normal stress coefficient and equilibrium compliance 7°, depends on the divergence... 

The critical gel equation is expected to predict material functions in any small-strain viscoelastic experiment. The definition of small varies from material to material. Venkataraman and Winter [71] explored the strain limit for crosslinking polydimethylsiloxanes and found an upper shear strain of about 2, beyond which the gel started to rupture. For percolating suspensions and physical gels which form a stiff skeleton structure, this strain limit would be orders of magnitude smaller. 

Polymer rheology can respond nonllnearly to shear rates, as shown in Fig. 3.4. As discussed above, a Newtonian material has a linear relationship between shear stress and shear rate, and the slope of the response Is the shear viscosity. Many polymers at very low shear rates approach a Newtonian response. As the shear rate is increased most commercial polymers have a decrease in the rate of stress increase. That is, the extension of the shear stress function tends to have a lower local slope as the shear rate is increased. This Is an example of a pseudoplastic material, also known as a shear-thinning material. Pseudoplastic materials show a decrease in shear viscosity as the shear rate increases. Dilatant materials Increase in shear viscosity as the shear rate increases. Finally, a Bingham plastic requires an initial shear stress, to, before it will flow, and then it reacts to shear rate in the same manner as a Newtonian polymer. It thus appears as an elastic material until it begins to flow and then responds like a viscous fluid. All of these viscous responses may be observed when dealing with commercial and experimental polymers. 

These liquids are known as Ostwald-de Waele fluids. Figure 5 depicts a typical course of such a flow curve. Figure 6 shows a dimensionless standardized material function of two pseudoplastic fluids often used in biotechnology. It proves that they behave similarly with respect to viscosity behavior under shear stress. 

Elastic behavior of liquids is characterized mainly by the ratio of first differences in normal stress, Ni, to the shear stress, t. This ratio, the Weissenberg number Wi = Nih, is usually represented as a function of the rate of shear y. Figure 7 depicts flow curves of some viscoelastic fluids, and Figure 8 presents a dimensionless standardized material function of these fluids. It again verifies that they behave similarly with respect to viscoelastic behavior under shear stress. 

Figure 8 Dimensionless standardized material function of the fluids in Figure 7, verifying the similar viscoelastic behavior under shear stress. (From Ref. 12.)...

The response of simple fluids to certain classes of deformation history can be analyzed. That is, a limited number of material functions can be identified which contain all the information necessary to describe the behavior of a substance in any member of that class of deformations. Examples are the viscometric or steady shear flows which require, at most, three independent functions of the shear rate (79), and linear viscoelastic behavior (80,81) which requires only a single function, in this case a relaxation function. The functions themselves must be determined experimentally for each substance. 

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... 

Equations 1.2 to 1.4 represent material functions under large deformations (e.g., continuous shear of a fluid). One may recall a simple experiment in an introductory physics course where a stress (a) is applied to a rod of length Z, in a tension mode and that results in a small deformation AL. The linear relationship between stress (ct) and strain (j/) (also relative deformation, y = AL/L) is used to define the Young s modulus of elasticity E (Pa) ... 

Since the linear viscoelasticity of a material is described with a material function G(t), any experiment which gives full information on G(t) is sufficient it is not necessary to give the stresses corresponding to various strain histories. We will restrict the discussion to incompressible isotropic materials. In this case, different types of deformation such as elongation and shear give equivalent information in the range of linear viscoelasticity. Several types of experiments measure relaxation modulus, creep compliance, complex modulus etc which are equivalent to the relaxation modulus (1). 

It is known that incompressible newtonian fluids at constant temperature can be characterized by two material constants the density p and the viscosity T. The characterization of a purely viscous nonnewtonian fluid using the power law model (or any of the so-called generalized newtonian models) is relatively straightforward. However, the experimental description of an incompressible viscoelastic nonnewtonian fluid is more complicated. Although the density can be measured, the appropriate expression for r poses considerable difficulty. Furthermore there is some uncertainty as to what other properties need to be measured. In general, for viscoelastic fluids it is known that the viscosity is not constant but depends on shear rate, that the normal stress differences are finite and depend on shear rate, and that the stress may also depend on the preshear history. To characterize a nonnewtonian fluid, it is necessary to measure the material functions (apparent viscosity, normal stress differences, etc.) in a relatively simple or standard flow. Standard flow patterns used in characterizing nonnewtonian fluids are the simple shear flow and shear-free flow. 

Shear Flow Material Functions. A simple shear flow is given by the velocity field... 

Here t, 4, and 4 2 are three important material functions of a nonnewtonian fluid in steady shear flow. Experimentally, the apparent viscosity is the best known material function. There are numerous viscometers that can be used to measure the viscosity for almost all nonnewtonian fluids. Manipulating the measuring conditions allows the viscosity to be measured over the entire shear rate range. Instruments to measure the first normal stress coefficients are commercially available and provide accurate results for polymer melts and concentrated polymer solutions. The available experimental results on polymer melts show that , is positive and that it approaches zero as y approaches zero. Studies related to the second normal stress coefficient 4 reveal that it is much smaller than 4V and, furthermore, 4 2 is negative. For 2.5 percent polyacrylamide in a 50/50 mixture of water and glycerin, -4 2/4 i is reported to be in the range of 0.0001 to 0.1 [7]. 

Conservation Equations. In the above section, the material functions of nonnewtonian fluids and their measurements were introduced. The material functions are defined under a simple shear flow or a simple shear-free flow condition. The measurements are also performed under or nearly under the same conditions. In most engineering practice the flow is far more complicated, but in general the measured material functions are assumed to hold. Moreover, the conservation principles still apply, that is, the conservation of mass, momentum, and energy principles are still valid. Assuming that the fluid is incompressible and that viscous heating is negligible, the basic conservation equations for newtonian and nonnewtonian fluids under steady flow conditions are given by... 

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