Viscosity shift factor

The viscosity shift factor relates the viscosity p of the solid material to the density q and the temperature T. By applying the assumption that the product of absolute temperature and density is roughly constant, the Williams, Landel and Ferry (WLF) equation (Williams et al, 1955) is obtained ... 

Curves for the viscosity data, when displayed as a function of shear rate with temperature, show the same general shape with limiting viscosities at low shear rates and limiting slopes at high shear rates. These curves can be combined in a single master curve (for each asphalt) employing vertical and horizontal shift factors (77—79). Such data relate reduced viscosity (from the vertical shift) and reduced shear rate (from the horizontal shift). 

In order to allow for the effect of temperature on viscosity a shift factor, ar is often used. The Carreau equation then becomes... 

The viscosity flow curves for these materials are shown in Fig. 5.17. To obtain similar data at other temperatures then a shift factor of the type given in equation (5.27) would have to be used. The temperature effect for polypropylene is shown in Fig. 5.2. 

Investigation of the linear viscoelastic properties of SDIBS with branch MWs exceeding the critical entanglement MW of PIB (about -7000 g/mol ) revealed that both the viscosity and the length of the entanglement plateau scaled with B rather than with the length of the branches, a distinctively different behavior than that of star-branched PIBs. However, the magnitude of the plateau modulus and the temperature dependence of the terminal zone shift factors were found to... 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

Fig. 4.4 Scaling representation of the spin-echo data at Q ax- Different symbols correspond to different temperatures. Solid line is a KWW description (Eq. 4.8) of the master curve. The shift factors are taken from macroscopic viscosity measurements, a Polyisobutylene at Qmax l-0 A [125] measured on INI 1 (viscosity data from [34]). b Atactic polypropylene at Qmax=l-ll (viscosity data from [131]). (b Reprinted with permission from [126]. Copyright 2001 Elsevier)...

PPG (at higher temperatures) behaves like a typical pseudoplastic non-Newtonian fluid. The activation energy of the viscosity in dependence of shear rate (284-2846 Hz) and Mn was detected using a capillary rheometer in the temperature range of 150-180°C at 3.0-5.5 kJ/mol (28,900 Da) and 12-13 kJ/mol (117,700 Da) [15]. The temperature-dependent viscosity for a PPG of 46 kDa between 70 and 170°G was also determined by DMA (torsion mode). A master curve was constructed using the time-temperature superposition principle [62] at a reference temperature of 150°G (Fig. 5) (Borchardt and Luinstra, unpublished data). A plateau for G was not observed for this molecular weight. The temperature-dependent shift factors ax were used to determine the Arrhenius activation energy of about 25 kJ/mol (Borchardt and Luinstra, unpublished data). 

Fig. 2.5. Steady-state and dynamic oscillatory flow measurements on a 2 wt. per cent solution of polystyrene S 111 in Aroclor 1248 according to Philippoff (57). ( ) steady shear viscosity (a) dynamic viscosity tj, ( ) cot 1% from flow birefringence, (A) cot <5 from dynamic measurements, all at 25° C. (o) cot 8 from dynamic measurements at 5° C. Steady-state flow properties as functions of shear rate q, dynamic properties as functions of angular frequency m. Shift factor aT which is equal to unity for 25° C, is explained in the text, cot 2 % and cot 8 are expressed in terms of shear (see eqs. 2.11 and 2.22)...

Bueche-Ferry theory describes a very special second order fluid, the above statement means that a validity of this theory can only be expected at shear rates much lower than those, at which the measurements shown in Fig. 4.6 were possible. In fact, the course of the given experimental curves at low shear rates and frequencies is not known precisely enough. It is imaginable that the initial slope of these curves is, at extremely low shear rates or frequencies, still a factor two higher than the one estimated from the present measurements. This would be sufficient to explain the shift factor of Fig. 4.5, where has been calculated with the aid of the measured non-Newtonian viscosity of the melt. A similar argumentation may perhaps be valid with respect to the "too low /efi-values of the high molecular weight polystyrenes (Fig. 4.4). 

Simha and Utracki show that indeed all the experimental data for a given polymer solvent system fall on the same master curve. The master curve is different, however, for each different polymer-solvent system, while the shift factors aM v differ also. Therefore this method is less suited for predicting the viscosity of a new polymer-solvent system. 

The melt viscosity of SPS can be superposed on to a single curve according to the time-temperature law (Figure 18.11). The viscosity of SPS has a 3.4 power dependence on Mw (Figure 18.12). When the shift factor, aj, is plotted against... 

Figure 18.11 Superposed viscosity curve of SPS. Shear viscosities measured for several Mw and temperatures were superposed with reference to temperature (reference temperature 290 °C) and (reference 289000). ay = shift factor for temperature aM = shift factor for Mw...

From the vertical shift factor of the master curve, we are able to describe the mass dependence of the zero-shear viscosity in the iso-free volume state which is directly connected to the radius of gyration of the chains. In the molten state, it is generally assumed that the chains exhibit a Gaussian conformation and therefore the viscosity should be proportional to the molecular weight. 

When departures from the Cox-Merz rule are attributed to structure decay in the case of steady shear, the complex viscosity is usually larger than the steady viscosity (Mills and Kokini, 1984). Notwithstanding this feature, the relation between magnitudes of T a and T can be dependent on the strain amplitude used (Lopes da Silva et al., 1993). Doraiswamy et al. (1991) presented theoretical treatment for data on suspensions of synthetic polymers. They suggested that by using effective shear rates, the Cox-Merz rule can be applied to products exhibiting yield stresses. The shift factors discussed above can be used to calculate effective shear rates. 

Therefore, Or = 0.0012. This means that the longest relaxation time r of the polyisoprene at 100°C is 0.0012 times its value at 25°C. The viscosity tj changes roughly in proportion to the relaxation time if the small vertical shift factor is neglected. Thus, ... 

The viscosity of viscoelastic liquids at temperatures slightly above Tg can be determined by the procedures outUned above. Thus a step shear stress is imposed on the viscoelastic liquid at temperatures well above Tg, and once steady state is reached the sample is cooled to the temperature of interest then the straight line of J t) vs. t is recorded, and the viscosity is determined from the reciprocal of the slope of the straight line. By assuming that the elastic and viscous mechanisms have the same temperature dependence, the shift factor can be written in terms of the viscosity as (2,5)... 

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