Viscosity maximum extensional

Figure 1.5 Effect of the maximum extensional viscosity to the drag reduction, FENE-P model.

Thus, one of the differences between the predictions of the pom-pom model and the DEMG model for linear molecules is the inequality in Eq. 11.46. In steady imiaxial extension, this criterion produces a saturation value of the stress and a local maximum in the extensional viscosity, while in the DEMG model, the stress has no saturation value. (If finite extensibility is included in the DEMG model, then there is a saturation in viscosity, but no viscosity maximum.) Experimental data of McLeish et al. [97] for a melt of polyisoprene H molecules confirm the predicted decrease in extensional viscosity at high extension rate see Fig. 11.25. 

The effect of the melt temperature on the vortex size development has been studied experimentally as well as theoretically. The most important results are depicted in Figme 7. It is obvious, that the vortex area primarily increases, reaches a maximum and then it decreases again with increasing temperature. This behavior can be explained by the temperature dependency of the nonmonotonic shape of the planar extensional viscosity predicted by e improved mWM model, which is depicted in Figure 8. In more detail, the planar extensional viscosity maximum moves fi om low extensional strain rates to higher ones for increasing melt temperatures. This seems to be the driving mechanism for the maximum appearance in the vortex size vs. temperature flmction. 

Another type of experiment involves a fluid filament being drawn upward against gravity from a reservoir of the fluid (101,213,214), a phenomenon often called the tubeless siphon. The maximum height of the siphon is a measure of the spinnabiUty and extensional viscosity of the fluid. Mote quantitative measures of stress, strain, and strain rate can be determined from the pressure difference and filament diameter. A more recent filament stretching device ia which the specimen is held between two disks that move apart allows measurements ia low viscosity Hquids (215). AH of these methods are limited to spinnable fluids under small total strains and strain rates. High strain rates tend to break the column or filament. 

In Fig. 15.27, the transient extensional viscosity of a low-density polyethylene, measured at 150 °C for various extensional rates of strain, is plotted against time (Munstedt and Laun, 1979). Qualitatively this figure resembles the results of the Lodge model for a Maxwell model in Fig. 15.26. For small extensional rates of strain (qe < 0.001 s ) 77+(f) is almost three times rj+ t). For qe > 0. 01 s 1 r/+ (f) increases fast, but not to infinite values, as is the case in the Lodge model. The drawn line was estimated by substitution of a spectrum of relaxation times of the polymer (calculated from the dynamic shear moduli, G and G") in Lodge s constitutive equation. The resulting viscosities are shown in Fig. 15.28 after a constant value at small extensional rates of strain the viscosity increases to a maximum value, followed by a decrease to values below the zero extension viscosity. 

In Sect. 15.4 it was shown how the shear thinning behaviour of the viscosity could be described empirically with the aid of many suggestions found in literature. It was not mentioned there that the first normal stress coefficient also shows shear thinning behaviour. In this Sect. 15.5 it became clear that also the extensional viscosity is not a constant, but depending on the strain rate upon increasing the strain rate qe the extensional viscosity depart from the Trouton behaviour and increases (called strain hardening) to a maximum value, followed by a decrease to values below the zero extensional viscosity. It has to be emphasised that results in literature may show different behaviour for the extensional behaviour, but in many cases this is due to the limited extensions used,... 

In the same way, but much more complicated, with a damping function depending on the Hencky strain, it proved to be possible to calculate the transient extensional viscosity as a function of qe. The result is illustrated in Fig. 15.30 for the same polymer. It shows that extensional viscosity remains finite and increases with increasing strain rate up to a maximum at qe = 2 s, after which it decreases again. The calculated lines coincide quite well with the experiments, but the calculated viscosities are somewhat too high. 

Another effect of the variation of the extensional viscosity is the maximum extend-ibility. For polymers like high-density polyethylene, the rapid increase of the extensional viscosity during the spinning process limits the obtainable spin-draw ratio that is the ratio between the winding velocity and the velocity in the orifice. Examples can be found in an article of Han and Lamonte (1972). 

As discussed in Section 12.3.3, unusual time- and shear-rate-dependencies have been reported for some wormy micellar solutions at dilute concentrations—for example, 1-5 mTAB/NaSal. At higher concentrations, 7-250 mM, of a similar surfactant, tetrade-cyltrimethyammonium bromide in NaSal, the extensional viscosity increases with in-creasing extension rate until a maximum is reached, and extension thinning then follows Thomme and Warr 1994). Prud homme and Warr interpret the maximum as the critical... 

When describing dilatant behavior, the maximum stretch rate, e, in the converging flow at the contraction is a better parameter, but more difficult to be calculated. Instead of the term stretch rate, other authors also used deformation rate (e.g., Chauveteau, 1981) or elongational rate (e.g.. Sorbic, 1991). The shear-thickening viscosity is also called elongational viscosity (often referred to as the Trouton viscosity Sorbie, 1991) or extensional viscosity in the literature. James and McLaren (1975) reported that for a solution of polyethylene oxide (a flexible coil, water-soluble polymer physically similar to HPAM), the onset of elastic behavior at maximum stretch rates was of the order of 100 s and shear rates of the order of 1000 s. In this instance, the stretch rate is about 10 times lower than the shear rate. However, some authors use shear rate instead of stretch rate in defining the Deborah number—for example, Delshad et al. (2008). 

If modified coefficients, Tjg and Deg, are used, as suggested in Refs. [1, 2] on the basis of the FENE dumbbell model. Fig. lb is obtained. The onset behaviour is described by the onset Deborah number, De o e,0 " with g 0 critical elongation rate of the porous media flow and T = relaxation time of the polymer solution, whilst the maximum value of the attainable increase of the extensional viscosity in normalized form only depends on the... 

However, Sreenivasan and White (89) point out that the connection between fluctuating strain rates and large extensional viscosity is circumstantial. Further polymer coils can only be partially stretched in a random field of strain rate. Sreenivasan and White (89) point out that the elastic theory proposed by de Gennes (84) is compatible with at least two experimental observations ie, the dependence of drag reduction onset on polymer concentration and maximum drag reduction asymptote. 

Vinogradov et used a more sophisticated setup compared to that of Cogswell, and showed the equivalence of extensional viscosity data obtained using a constant stretch rate instrument and a constant stress instrument for molten polystyrene. An unexpected feature of the constant stress results was that the strain rate was found to decrease initially, as expected, but it then exhibited a minimum before becoming constant. This implies a maximum in the tensile stress growth coefficient and, in that respect, this behavior was similar to that of linear low density polyethylene as reported by Schlund and Utracki< ) among others. 

The linear metallocene polyethylene (mPE) reference, polymer Clb P5, shows stress-independent steady-state viscosity throughout the stress range measured, whereas the LDPE shows strain hardening behavior typical for that range. At low stress, the response is equal to three times the LVE shear viscosity. Increasing the tensile stress leads to strain hardening up to a maximum stress, after which the response becomes extension thinning [121]. In contrast to the LDPE, the steady-state extensional viscosity ( /e) of branched mPE polymer C4 P1 appears... 

All else being equal, an increase in flow-rate for an elastic liquid, or an increase in the level of elasticity (measured as, say, the first normal-stress difference) produces the effect on the flow pattern in a contraction shown in figure 30. The complex flow pattern shown on the right of the figure is very susceptible to flow instabilities. The onset of these instabilities dictates the maximum flow-rate possible for an extrudate emerging from the end of the die having a smooth, acceptable surface. (See chapter 17 for a discussion of the role of extensional viscosity in this kind of flow.)... 

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