Shear complex viscosity

A viscoelastic material also possesses a complex dynamic viscosity, rj = rj - - iv( and it can be shown that r = G jiuj-, rj = G juj and rj = G ju), where CO is the angular frequency. The parameter Tj is useful for many viscoelastic fluids in that a plot of its absolute value Tj vs angular frequency in radians/s is often numerically similar to a plot of shear viscosity Tj vs shear rate. This correspondence is known as the Cox-Merz empirical relationship. The parameter Tj is called the dynamic viscosity and is related to G the loss modulus the parameter Tj does not deal with viscosity, but is a measure of elasticity. 

There are commercially available in-line or on-line viscometer devices. In-line devices are installed directly in the process while on-line devices are used to analyze a side stream of the process. Most devices are based on measuring the pressure drop and flow rate through a capillary. The viscosity is either determined at a single shear rate or, at most, a few shear rates. Complex fluids, on the other hand, exhibit a viscosity that cannot be so easily characterized. In order to capture enough information that allows, for example, a molecular weight distribution to be inferred, it is necessary to determine the shear viscosity over reasonably wide ranges of shear rates. 

Figure 14.10 Master curves of steady shear viscosity, r (at lower shear rates) and complex viscosity, r (at higher frequencies) for the first seven generations of PAMAM dendrimers at 40°C in the bulk state...

ISO 6721-8 1997 Plastics - Determination of dynamic mechanical properties - Part 8 Longitudinal and shear vibration - Wave-propagation method ISO 6721-10 1999 Plastics - Determination of dynamic mechanical properties - Part 10 Complex shear viscosity using a parallel-plate oscillatory rheometer ISO 9311-2 2002 Adhesives for thermoplastic piping systems - Part 2 Determination of shear strength... 

Rheologists sometimes use an empirical functionality called the Cox-Mertz rule, which states that the complex viscosity value at a given frequency is equal to the steady shear viscosity at the same shear rate [14] ... 

Rheological properties of filled polymers can be characterised by the same parameters as any fluid medium, including shear viscosity and its interdependence with applied shear stress and shear rate elongational viscosity under conditions of uniaxial extension and real and imaginary components of a complex dynamic modulus which depend on applied frequency [1]. The presence of fillers in viscoelastic polymers is generally considered to reduce melt elasticity and hence influence dependent phenomena such as die swell [2]. 

Surface shear rheology at the oil-water interface is a sensitive probe of protein-polysaccharide interactions. In particular, there is considerable experimental evidence for a general increase in surface shear viscosity of protein adsorbed layers as a result of interfacial complexation with polysaccharides (Dickinson et al., 1998 Dickinson and Euston, 1991 Dickinson and Galazka, 1992 Semenova et al., 1999a Jourdain et al., 2009). One such example is the case of asi-casein + pectin at pH = 5.5 and ionic strength = 0.01 M (Ay = - 334 x 10 cm /mol) the interfacial viscosity after 24 hours was found to be five times larger in the presence of pectin (i.e., values of 820 80 and 160 20 mN m 1 with and without pectin, respectively) (Semenova et al., 1999a). 

Figure 8.12 illustrates the effect of complex formation between protein and polysaccharide on the time-dependent surface shear viscosity at the oil-water interface for the system BSA + dextran sulfate (DS) at pH = 7 and ionic strength = 50 mM. The film adsorbed from the 10 wt % solution of pure protein has a surface viscosity of t]s > 200 mPa s after 24 h. As the polysaccharide is not itself surface-active, it exhibited no measurable surface viscosity (t]s < 1 niPa s). But, when 10 wt% DS was introduced into the aqueous phase below the 24-hour-old BSA film, the surface viscosity showed an increase (after a further 24 h) to a value around twice that for the original protein film. Hence, in this case, the new protein-polysaccharide interactions induced at the oil-water interface were sufficiently strong to influence considerably the viscoelastic properties of the adsorbed biopolymer layer. 

Figure 8.13 Effect of the method of preparation of the complexes of sodium caseinate (CN) + dextian sulfate (DS) on the time dependence of interfacial shear viscosity, r s, and interfacial shear elasticity, Gs (frequency 0.1 s 1) ( ) CN + DS, mixed layer, freshly prepared complexes (O) CN + DS, mixed layer, 24-h-old complexes ( ) CN + DS, bilayer. Aqueous solutions contained 0.5 wt% CN and 1 wt% DS in 20 mM imidazole buffer at pH = 6. Reproduced from Jourdain et al. (2009) with permission.

If some or all of this curve is present, the models used to fit the data are more complex and are of two types. The first of these is the Carreau-Yasuda model, in which the viscosity at a given point (T ) as well as the zero-shear and infinite-shear viscosities are represented. A Power Law index (mi) is also present, but is not the same value as n in the linear Power Law model. A second type of model is the Cross model, which has essentially the same parameters, but can be broken down into submodels to fit partial data. If the zero-shear region and the power law region are present, then the Williamson model can be used. If the infinite shear plateau and the power law region are present, then the Sisko model can be used. Sometimes the central power law region is all that is available, and so the Power Law model is applied (Figure H. 1.1.5). 

This is the dynamic viscosity in small amplitude oscillatory shear which is the real component of the complex shear viscosity which is a function of the angular frequency of oscillation. 

The polyethenes prepared with catalyst 2 (Fig. 3a) have greatly elevated elastic modulus G values due to LCB compared to the linear polymers shown in Fig. 3b. LCB also shifts the crossover point to lower frequencies and modulus values. The measured complex viscosities of branched polymers (see also Table 2) are more than an order of magnitude higher than calculated zero shear viscosities of polymers having the same molecular weight but a linear structure. The linear polymers have, in turn, t] (0.02 radvs)... 

In the case of the oscillatory motion, equation (6.19) defines, in accordance with equation (6.8), the complex shear viscosity r](co) = rf I iif with components... 

Gortemaker (et al.), 1976). In Fig. 15.12, the dynamic moduli are plotted vs. reduced angular frequency. From these results the complex viscosity rf and its components // and rf were calculated. They were plotted vs angular frequency in Fig. 15.13, where also experimental values of the steady shear viscosity are shown. The agreement between rj q) and rf co) is clearly visible. This relationship between steady shear and sinusoidal experiments... 

FIG. 15.13 Non-Newtonian shear viscosity r/(q) at 170 °C vs. shear rate, q, for the polystyrene mentioned in Fig. 15.12, measured in a cone and plate rheometer (O) and in a capillary rheometer ( and ) and the dynamic and complex viscosities, rj (w) (dotted line), rj (w) (dashed line) and i (< ) (full line), respectively, as functions of angular frequency, as calculated from Fig. 15.12. From Gortemaker (1976) and Gortemaker et al. (1976). Courtesy Springer Verlag. 

Polymer melts are complex fluids. Their viscoelastic properties during flow depend not only on their molecular structure but also on the interactions they are likely to develop at the walls, depending on the physical and chemical features of the interface and the flow conditions. In addition, not all their properties can be determined and the constitutive equations used are in practice often limited to considerations on the shear viscosity. From a theoretical point of view, considerable difficulties are involved and the problem to be studied here has not been solved. In particular, even though the boundary conditions considered in... 

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