Steady shear viscosity, relationship

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... 

The relationship between steady shear and complex viscosity is fairly well established. Cox and Merz " found that an empirical relationship exists between complex viscosity and steady shear viscosity when the shear rates are the same. The Cox-Merz rule is stated as follows ... 

The relationship between steady-shear viscosity and dynamic-shear viscosity is also a common fundamental rheological relationship to be examined. The Cox-Merz empirical rule (Cox, 1958) showed for most materials that the steady-shear-viscosity-shear-rate relationship was numerically identical to the dynamic-viscosity-frequency profile, or r] y ) = r] m). Subsequently, modified Cox-Merz rules have been developed for more complex systems (Gleissle and Hochstein, 2003, Doraiswamy et al., 1991). For example Doriswamy et al. (1991) have shown that a modified Cox-Merz relationship holds for filled polymer systems for which r](y ) = t] (yco), where y is the strain amplitude in dynamic shear. 

An inherent assumption when using the above dynamic techniques is that the complex viscosity gives a good representation of the steady-shear viscosity during the curing reaction. This has been validated for many systems. However, care should be taken when relating the effects of cure on complex viscosity to the processing viscosity in other words the Cox Merz rule or a similar relationship must be validated. 

Figure 3 illustrates the relationship between steady shear viscosity and shear rate for PBTA homopolymer solutions in NMP/4% LiCl with various concentrations. This figure clearly revels the shear-thinning effect for isotropic (C C r) solutions and anisotropic (C > Ccj-) solutions with the most shear rate region. Meanwhile, a Newtonian plateau appears in a low shear rate region for anisotropic solutions, especially for C = 6 wt% and C = 6.5 wt%. Furthermore, the experimental data could be fitted with theoretical non-Newtonian fluid model. Among which, power-law model was applied for isotropic solutions and Carreau model (22) for anisotropic solutions, as shown below ... 

Dynamic viscosity data can be used to approximate the steady shear viscosity by taking advantage of an empirical relationship known as the Cox-Merz rule (Cox and Merz 1958), which relates the magnitude of the complex viscosity at frequency co to the steady shear viscosity at a shear rate y equal to co ... 

Using the inverse relationship between steady shear viscosity and MFI [99], Shenoy and Saini [51] write the modified form of E>oolittle s equation [100] in the following form ... 

There is a surprising but useful relationship between the steady shear viscosity t](y) and the amplitude of complex dynamic viscosity (or simply the complex... 

The longest relaxation time. t,. corresponds to p = 1. The important characteristics of the polymer are its steady-state viscosity > at zero rate of shear, molecular weight A/, and its density p at temperature 7" R is the gas constant, and N is the number of statistical segments in the polymer chain. For vinyl polymers N contains about 10 to 20 monomer units. This equation holds only for the longer relaxation times (i.e., in the terminal zone). In this region the stress-relaxation curve is now given by a sum of exponential terms just as in equation (10), but the number of terms in the sum and the relationship between the T S of each term is specified completely. Thus... 

Holden et al. (47) first noted the peculiar characteristics in the steady shear behavior of the SBS block copolymer melts. For a certain composition of styrene and butadiene, no limiting Newtonian viscosity was found at low shear rates. For some of the others, there exist two distinct viscosity vs. shear rate relationships (Figure 10). Arnold and Meier (73) carried out the experiments in oscillatory shear and found the same... 

For an unvulcanized polydimethylsiloxane, the biaxial viscosity was approximately six times the shear viscosity over the biaxial extensional rates from 0.003 to 1.0 s (Chatraei et al., 1981), a result expected for Newtonian fluids, that is, the relationship between the limiting value of biaxial extensional viscosity (j, ) at zero strain rate and the steady zero-shear viscosity (i o) of a non-Newtonian food is ... 

In addition to relationships between apparent viscosity and dynamic or complex viscosity, those between first normal stress coefficient versus dynamic viscosity or apparent viscosity are also of interest to predict one from another for food processing or product development applications. Such relationships were derived for the quasilinear co-rotational Goddard-Miller model (Abdel-Khalik et al., 1974 Bird et al., 1974, 1977). It should be noted that a first normal stress coefficient in a flow field, V i(y), and another in an oscillatory field, fri(ct>), can be determined. Further, as discussed below, (y) can be estimated from steady shear and dynamic rheological data. 

Section 4.2.1) gives a good description of the steady-shear-dynamic-shear relationship. They found that the following general relationship is a good representation of the chemo-viscosity profile ... 

We have seen that a simple apparent viscosity is used to characterize the stress-shear rate relationship for a non-Newtonian fluid. For a viscoelastic fluid, additional coefficients are required to determine the state of stress in any flow. For steady simple shear flow, the additional coefficients are given by the Criminale-Ericksen-Filbey equation... 

Experimental verification of Eqs 7.94 indicated that the scaling relationships are valid, but the shape of experimental transient stress curves, after step-change of shear rate, did not agree with Doi-Ohta s theory [Takahashi et al., 1994]. Similar conclusions were reported for PA-66 blends with 25 wt% PET [Guenther and Baird, 1996]. For steady shear flow the agreement was poor, even when the strain-rate dependence of the component viscosities was incorporated. Similarly, the... 

The rheological behavior of these materials is still far from being fully understood but relationships between their rheology and the degree of exfoliation of the nanoparticles have been reported [73]. An increase in the steady shear flow viscosity with the clay content has been reported for most systems [62, 74], while in some cases, viscosity decreases with low clay loading [46, 75]. Another important characteristic of exfoliated nanocomposites is the loss of the complex viscosity Newtonian plateau in oscillatory shear flow [76-80]. Transient experiments have also been used to study the rheological response of polymer nanocomposites. The degree of exfoliation is associated with the amplitude of stress overshoots in start-up experiment [81]. Two main modes of relaxation have been observed in the stress relaxation (step shear) test, namely, a fast mode associated with the polymer matrix and a slow mode associated with the polymer-clay network [60]. The presence of a clay-polymer network has also been evidenced by Cole-Cole plots [82]. 

Rameshwaram et al. [6] investigated the structure-property relationships and the effects of a viscosity ratio on the rheological properties of polymer blends using oscillatory and steady shear rheometry and optical microscopy. 

Fig. 6. Relationship between relative values of steady shear flow functions and concentration of silane coupling agnet for GF/PP and GF/mPP/PP composites (a) relative viscosity, (b) relative first normal stress difference...

Correlation Between Steady-Shear and Oscillatory Data. The viscosity function is by far the most widely used and the easiest viscometric function determined experimentally. For dilute polymer solutions dynamic measurements are often preferred over steady-shear normal stress measurements for the determination of fluid elasticity at low deformation rates. The relationship between viscous and elastic properties of polymer liquids is of great interest to polymer rheologists. In recent years, several models have been proposed to predict fluid elasticity from shear viscosity data. 

The capillary viscometer can only provide the viscosity-shear rate relationship for a polymer. It cannot give other viscometric functions. Viscometric functions associated with normal stress behaviour in steady shear... 

In steady state measurements one measures the shear stress (x)-shear rate (y) relationship using a rotational viscometer. A concentric cylinder or cone and plate geometry may be used depending on the emulsion consistency. Most cosmetic emulsions are non-Newtonian, usually pseudoplastic as illustrated in Fig. 1.11. In this case the viscosity decreases with applied shear rate (shear thinning behavior (Fig. 1.11)), but at very low shear rates the viscosity reaches a high limiting value (usually referred to as the residual or zero shear viscosity). 

The first of these relationships, eq. 4.2.4, follows from little more than the definitions of rj and t], while eq. 4.2.5 is less obvious (Coleman and Markovitz, 1964). Both relationships are of limited usefulness because they are relevant only for low shear rate properties. However, an empirical relationship, called the Cox-Merz rule, often holds fairly well at high shear rates. This rule states that the shear rate dependence of the steady state viscosity ij is equal to the frequency dependence of the linear viscoelastic viscosity ri that is. 

In this chapter, we have presented the rheological behavior of homopolymers, placing emphasis on the relationships between the molecular parameters and rheological behavior. We have presented a temperature-independent correlation for steady-state shear viscosity, namely, plots of log ri T, Y) r](jiT) versus log or log j.y, where Tq is a temperature-dependent empirical constant appearing in the Cross equation and a-Y is a shift factor that can be determined from the Arrhenius relation for crystalline polymers in the molten state or from the WLF relation for glassy polymers at temperatures between and + 100 °C. 

The corresponding relationships are noted for the storage and loss moduli, zero-shear viscosity, steady-state recoverable compliance, and average relaxation times (cf. eqns [32]-[35]) ... 

Steady-state shear stress-shear rate curves were used to obtain the relative viscosity (//,.)-volume fraction () relationship for the latex and emulsion. The results are shown in Figure 11.19 which also contains the theoretically predicted curve based on the Dougherty-Krieger equation [14],... 

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