Power law region

By measuring the scattered intensity as a function of q over multiple decades in q, information from multiple structural levels can be obtained. Two distinct scattering feamres distinguish each stmctural level, a power-law region and a knee region in a log-log plot of I q) versus q. An example of power-law scattering is shown in Figure 17.2. For particles with a smooth surface the... 

Polymers in solution or as melts exhibit a shear rate dependent viscosity above a critical shear rate, ycrit. The region in which the viscosity is a decreasing function of shear rate is called the non-Newtonian or power-law region. As the concentration increases, for constant molar mass, the value of ycrit is shifted to lower shear rates. Below ycrit the solution viscosity is independent of shear rate and is called the zero-shear viscosity, q0. Flow curves (plots of log q vs. log y) for a very high molar mass polystyrene in toluene at various concentrations are presented in Fig. 9. The transition from the shear-rate independent to the shear-rate dependent viscosity occurs over a relatively small region due to the narrow molar mass distribution of the PS sample. 

The fact that viscosity is independent of molar mass at high shear rates is of fundamental importance, since it follows that it is impossible to distinguish between different samples if the viscosity is measured in the power-law region. 

Fig. 13. Slope of the power-law region for narrowly distributed polystyrene in (O) trans-decalin (thermodynamically poor solvent) and ( ) toluene (thermodynamically good solvent). (-) theoretical course for a 9-solvent... 

Fig. 14. Slope in the power-law region of the flow curve as a function of the overlap parameter for narrowly distributed polystyrene in toluene...

Rubbery materials beyond the gel point have been studied extensively. A long time ago, Thirion and Chasset [9] recognized that the relaxation pattern of a stress r under static conditions can be approximated by the superposition of a power law region and a constant limiting stress rq at infinite time ... 

For the power law region at a shear rate of 20 1/s, the calculations lead to the temperature dependence of the power law viscosity function using the same method ... 

Finally, the concept of viscosity and resistance to flow of polymers as a function of shear rate will be discussed because there is often a misconception regarding the dissipation in an extruder at high shear rates when the viscosity is in the power law region. As previously discussed, the viscosity decreases as a function of increasing shear rate as shown in Fig. 3.23. Often this reduction in viscosity is misinterpreted as a reduction in the amount of power needed when the polymer is sheared at high rates. For an extruder, the misinterpretation would be that less motor power would be required to operate the machine at higher screw speeds. It... 

Relaxation dispersion data for water on Cab-O-Sil, which is a monodis-perse silica fine particulate, are shown in Fig. 2 (45). The data are analyzed in terms of the model summarized schematically in Fig. 3. The y process characterizes the high frequency local motions of the liquid in the surface phase and defines the high field relaxation dispersion. There is little field dependence because the local motions are rapid. The p process defines the power-law region of the relaxation dispersion in this model and characterizes the molecular reorientations mediated by translational displacements on the length scale of the order of the monomer size, or the particle size. The a process represents averaging of molecular orientations by translational displacements on the order of the particle cluster size, which is limited to the long time or low frequency end by exchange with bulk or free water. This model has been discussed in a number of contexts and extended studies have been conducted (34,41,43). 

Williamson (predicts first Newtonian plateau and power law region) q=il0-/ryn-1... 

Sisko (predicts power law region and second Newtonian plateau) Tl=TL+k yn"1... 

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). 

The complete flow curve allows for the response of the test sample to be predicted at rest (zero-shear behavior), when being poured out of a container (early Power Law region), when being pumped (middle Power Law region), or when being processed or consumed. Each part of the curve is important in assessing a sample s suitability. 

Fig. 13.3. Crossover diagram in the s — z plane (logarithmic scale). Shaded areas roughly correspond to power law regions. A dilute excluded volume region B dilute 0-region C semidilute 6-region D semidilute excluded volume region. Also drawn are lines z — 100 (long dashes)-, 5/z = 0.01 (short dashes) s = 0.03 (dotted)...

Fig. 3.5 Logarithmic plot of the shear rate-dependent viscosity of a narrow molecular weight distribution PS (A) at 180°C, showing the Newtonian plateau and the Power Law regions and a broad distribution PS( ). [Reprinted with permission from W. W. Graessley et al., Trans. Soc. Rheol., 14, 519 (1970).]...

The transition from the Newtonian plateau to the Power Law region is sharp for monodispersed polymer melts and broad for polydispersed melts (see Fig. 3.5). 

The slope of the viscosity curve in the Power Law region is not exactly constant. The flow index n decreases with increasing shear rate. Thus the Power Law equation holds exactly only for limited ranges of shear rate, for a given value of n. 

Figure 2-3 Plot of Shear Rate versus Apparent Viscosity for Shear Thinning Foods Identifying Three Separate Regions A Zero-Shear Viscosity at Low Shear Rates, a Power Law Region at Intermediate Shear Rates, and an Infinite-Shear Viscosity at High Shear Rates. Often, only data in the power law region are obtained.

Yoo et al. (1994) also found that the Cross and Carreau models were satisfactory for the most part in describing the data that covered the zero-shear and the power law regions of mesquite seed gum. Also in agreement with Lopes da Silva et al. (1992), the Carreau model described well the viscosity data shown in Figure 4-3 over a wide range of shear rates except in the power law region at high shear rates in concentrations > 1.4 g 100 mL . ... 

As the frequency is increased in a frequency scan, the Newtonian region is exceeded and a new relationship develops between the rate of strain, or the frequency, and the viscosity of the material. This region is often called the power law region and can be modeled by ... 

Several comments can be made about the nature of these approximate results. Note that Eqs. (6.23) and (6.24) give power-law region for both viscosity and normal stresses most experimental data for polymer solutions and melts do in fact exhibit a prominent power law region which extends over about three decades of shear rates. The values of the exponents — 2/3 for viscosity and 4/3 for the normal stress function - are not unreasonable values. In the power-law region it is often observed that the exponent for 6 is about twice that for tj. We must, however, note that comparison of Eqs. (6.21) and (6.22) with Eqs. (6.8) and (6.9) and a similar comparison of the power-law results shown in Table 2 indicate that the approximate procedure does not agree very well with the exact calculations. 

---