Normal Stress Differences in Shear

Steady state shear viscosity and primary normal stress coefficient for low density polyethylene melt T and from the Kaye-Bernstein, Kearsley, Zapas (K-BKZ) equation wim the double exponential damping function, eq4.4.13 (solid lines) and with the single exponential, eq4.4.12 (dotted Une). Data at different temperatures have been shifted to one master curve by ar T). Replotted from Laun (1978). 

---

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

The Eyring analysis does not explicity take chain structures into account, so its molecular picture is not obviously applicable to polymer systems. It also does not appear to predict normal stress differences in shear flow. Consequently, the mechanism of shear-rate dependence and the physical interpretation of the characteristic time t0 are unclear, as are their relationships to molecular structure and to cooperative configurational relaxation as reflected by the linear viscoelastic behavior. At the present time it is uncertain whether the agreement with experiment is simply fortuitous, or whether it signifies some kind of underlying unity in the shear rate dependence of concentrated systems of identical particles, regardless of their structure and the mechanism of interaction. 

Prediction of the second normal stress difference in shear and thermodynamic consistency obviously requires the use of a different strain measure including of the Cauchy strain tensor in the form of the K-BKZ model. With the ratio of second to first normal stress difference as a new parameter, Wagner and Demarmels [32] have shown that this is also necessary for accurate prediction of other flow situations such as equibiaxial extension, for example. 

The temporary network model predicts many qualitative features of viscoelastic stresses, including a positive first normal stress difference in shear, gradual stress relaxation after cessation of flow, and elastic recovery of strain after removal of stress. It predicts that the time-dependent extensional viscosity rj rises steeply whenever the elongation rate, s, exceeds 1/2ti, where x is the longest relaxation time. This prediction is accurate for some melts, namely ones with multiple long side branches (see Fig. 3-10). (For melts composed of unbranched molecules, the rise in rj is much less dramatic, as shown in Fig. 3-39.) However, even for branched melts, the temporary network model is unrealistic in that it predicts that rj rises to infinity, whereas the data must level eventually off. A hint of this leveling off can be seen in the data of Fig. 3-10. A more realistic version of the temporary network model... 

Some of the manifestations of viscoelasticity are delayed relaxation of stress after cessation of flow phase shift between stress and strain rate in oscillatory shear flow shear thinning (decrease of viscosity) at shear rates exceeding the reciprocal of the longest relaxation time and normal stress differences in shear flow, whose magnitudes are related to the relaxation time spectrum. A very convenient observation for experimentalists is that there is a close similarity between the shear viscosity and first normal stress difference as functions of shear rate and the corresponding parameters, complex viscosity and storage modulus, as functions of frequency in a small amplitude oscillatory shear. 

P.K. Currie, personal communication (1980), "The First Normal Stress Difference in Shear of Nematic Liquid Crystals," to be submitted to Mol. 

The first manifestation of nonlinear behavior with increasing strain or strain rate is the appearance of normal stress differences in shearing deformation. For steady-state shear flow at small shear rates, several nonlinear models ° predict the relation for the primary normal stress difference given as equation 62 of Chapter 1, which, combined with equation 54, gives... 

The entire discussion above is concerned with getting accurate shear viscosity data. It is also possible to use slit geometry to obtain information on the normal stress differences in shear. As with a capillary, extrudate swell occurs as liquid leaves a slit. Again, starting fi-om an integral model, a relation similar to eq. 6.2.27 can be derived... 

Eqs. 11.50 and 11.51 for the orientation tensor are easy to solve and give almost the same result as the original Eq. 11.45, except that the new equation predicts a zero second normal stress difference in shearing flows. This deficiency can be corrected by inclusion of an addition term similar to that in the Giesekus model, as is included in the extended pom-pom model described in Verbeeten et al. [ 100] see Section 11.6.2.4. We note that a diiferential equation for the orientation tensor for linear polymers was also given recently by lanniruberto and Marrucci [46]. 

The total thrust force F produced during shear can be related to the normal stress difference in shear. Integrating Equation 8.37 yields PJf) = xjf) -p(r), where P (f) is the integration constant and, physically, is the net pressure exerted by the sheared fluid on a surface with normal vector n. Substituting for p in Equation 8.35 and integrating the resulting equation from rtoR yields... 

Clearly, the first normal stress difference in shear is a positive quantity. 

Figure 14.8 First normal stress difference in shear for lUPAC A LDPE at 130°C. (From Ref. 4.)...

Obtain an expression for the first normal stress difference in shear at steady state according to the upper-convected Maxwell equation. Is there a solvent eontribution in this case ... 

Kaye, A., Lodge, A. S. and Vale, D. G., 1968. Determination of normal stress difference in steady shear flow. Rheol. Acta 7, 368-379. 

In packed beds of particles possessing small pores, dilute aqueous solutions of hydroly2ed polyacrylamide will sometimes exhibit dilatant behavior iastead of the usual shear thinning behavior seen ia simple shear or Couette flow. In elongational flow, such as flow through porous sandstone, flow resistance can iacrease with flow rate due to iacreases ia elongational viscosity and normal stress differences. The iacrease ia normal stress differences with shear rate is typical of isotropic polymer solutions. Normal stress differences of anisotropic polymers, such as xanthan ia water, are shear rate iadependent (25,26). 

Meissner J, Garbella W, Hosteller J (1989) Measuring normal stress differences in polymer melt shear flow. J Rheol 33 843-864... 

The high normal stress differences in comparison to the shear stress cause flow phenomena which may influence many technical processes. One example is the Weissenberg Effect (see Fig. 3.10), which arises when a shaft rotates within a viscoelastic fluid. The first normal stress difference leads to a pressure distribution which causes the fluid to climb up the stirrer shaft. This effect occurs when processing polymer color dispersions or mixing cake dough. 

FIG. 16.32 First normal stress difference vs. shear rate of 16.4wt% PBLG in m-cresol. NB the positive and negative regions are indicated by + and —. After Kiss and Porter (1978,1980). Courtesy John Wiley Sons, Inc. 

FIG. 16.34 The first and second normal stress differences in the shear flow of a 12.5% nematic solution of PBLG in m-cresol. From Magda et al. (1991). Courtesy The American Chemical Society. 

Using this derivative in the former UCM model, Johnson and Segalman proposed a model [48] that improves the predictions especially in shear, leading to normal stress differences and shear viscosity which are now shear rate dependent. Unfortunately, although it appears to be attractive in shear, the use of such a derivative can lead to some physical paradoxes that will be discussed... 

The viscous and elastic properties of orientable particles, especially of long, rod-like particles, are sensitive to particle orientation. Rods that are small enough to be Brownian are usually stiff molecules true particles or fibers are typically many microns long, and hence non-Brownian. The steady-state viscosity of a suspension of Brownian rods is very shear-rate- and concentration-dependent, much more so than non-Brownian fiber suspensions. The existence of significant normal stress differences in non-Brownian fiber suspensions is not yet well understood. 

This scaling law, Eq. (9-48), implies that all components of the stress tensor are linear in the shear rate. Consider for example, a constant-shear-rate experiment. At steady state, not only is the shear stress predicted to be proportional to the shear rate, but so also is the first normal stress difference N This prediction has been nicely confirmed in recent experiments by Takahashi et al. (1994), who studied mixtures of silicon oil and hydrocarbon-formaldehyde resin. Both these fluids are Newtonian, and have the same viscosity, around 10 Pa s. Figure 9-18 shows that both the shear stress o and the first normal stress difference N = shear rate, so that the shear viscosity rj = aly and the so-called normal viscosity rjn = N /y are constants. The first normal stress difference in this mixture must be attributed entirely to the presence of interfaces, since the individual liquids in the mixture have no measurable normal stresses. A portion of the shear stress also comes from the interfacial stress. Figure 9-19 shows that the shear and normal viscosities are both maximized at a component ratio of roughly 50 50. At this component ratio, the interfacial term accounts for roughly half the total shear stress. 

Figure 11.12 Shear viscosity and first normal stress difference versus shear rate for 17% PBLG (molecular weight 350,000) in /n-cresol. The circled triangles are negative N values. (From Kiss and Porter, reprinted with permission fromMol. Cryst. Liq. Cryst. 60 267, Copyright 1980, Gordon and Breach Publishers.)...

It should be noted that as t becomes large the lowest order term in the coefficient of the K-term is just 60, that is one half the zero-shear-rate value of the primary normal stress function. A similar result was obtained by Bird and Marsh (7) and by Carreau (14) from the slowly varying flow expansions of two continuum models. Hence the time-dependent behavior of the shear stress is related to the steady-state primary normal stress difference in the limit of vanishingly small shear rate. 

A typical dependence of the first normal stress difference on shear rate is shown in Figme 1.15 for a series of polystyrene-in-toluene solutions. Usually, the rate of decrease of [ri with shear rate is greater than that of the apparent... 

A more fundamental difference is observed in the normal stress difference in steady shear flow ... 

---