Shear rate vs. viscosity curve

Fig. 5. Shear rate vs viscosity for PDMS. Numbers on curves indicate molecular weights. To convert Pa-s to poise, multiply by 10.

Figure 1.2 Steady shear-rate (/) vs. viscosity (rj) curves of 1 wt% surfactant system at various temperatures (from Ref [37]).

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 value of the MW exponent for the low shear-rate melt viscosity, usually accepted as 3.4—3.5 for linear polymers, has been called in question for PE. Schreiber and Bagley (164) reported a value of 4.22 for linear PE Porter and Johnson (165) had earlier reported values up to 8 for LDPE. For branched polymers an effect of this kind could be due to the effect mentioned in Section 5, in which a cross-over point is found on a log-log plot of viscosity vs. MW when linear and slightly branched polymers are compared, since the curve for the branched polymers must be steeper than that for the linear ones near the cross-over point. 

It is common for industrial pumping and processing equipment to use shear rates that fall in the intermediate shear regime from about 10 to 1000 s 1 as illustrated in Table I. A convenient way to summarize the flow properties of fluids is by plotting flow curves of shear stress versus shear rate (r vs 7). These curves can be categorized into several rheological classifications (Figure 3). Suspensions are frequently pseudoplastic as shear rate increases viscosity decreases. This is also termed... 

Fig. 1. Apparent viscosity (q) vs. shear rate (y ) flow curves drawn for the suspensions investigated (below the dispersant optimum dosage).

Fig. 23. Viscosity vs shear rate curves for two fluids showing the fallacy of a single point measurement. Fluid Vl would appear to be more viscous than fluid B if measured only at point X, of the same viscosity if measured at point V, and less viscous if measured only at point Z.

The viscosity vs. shear rate data for PTF and BTF (Figure 9) were obtained in n-heptane solution at 25°C using a Contraves low shear rate instrument. n-Heptane was used as a nonpolar solvent, as it has a high enough boiling point to avoid losses due to evaporation during the measurement period. The curves presented here are reproducible in both directions of shear and are thus t ime-independent. 

Fig. 8.15. Viscosity vs shear rate in concentrated solutions of narrow distribution polystyrene The solvent in n-butyl benzene, the concentration is 0.300 gm/ml and the temperature is 30° C. The symbols are O for M = 860000 and for M = 411000 at low shear rates (155) and at high shear rates (346). The solid line for M= 860000 is the master curve for monodisperse systems from Graessley (227). The solid line for M=411000 is the master curve from Ree-Eyring (341). Either master curve fits data for both molecular weights...

The advantage of these models is that they predict a Newtonian plateau at low shear rates and thus at low shear stresses. We will see back these models in Chap. 16 where an extra term 7700 is added to the equations to account for the viscosity of polymer solutions at high shear rates. At high shear rates the limiting slopes at high shear rates in log r) vs. log y curves are for the Cross, the Carreau and the Yasuda et al. models —m, (n-1) and (n-1), respectively. 

FIGURE 11.12 Viscous behavior of complex fluids (i) shear stress vs. shear rate and (ii) viscosity vs. shear rate. The notation for the curves is (a) Newtonian, (b) shear thinning, (c) shear thickening, (d) Bingham plastic, and (e) pseudoplastic. 

The ratio of quantity of granulating liquid to batch size at the inflection point of power vs. time curve is constant irrespective of batch size and type of machine. Moreover, for a constant rate of low viscosity binder addition proportional to the batch size, the rate of change (slope or time derivative) of torque or power consumption curve is linearly related to the batch size for a wide spectrum of high shear and planetary mixers. In other words, the process end-point, as determined in a certain region of the curve, is a practically proven scale-up parameter for moving the product from laboratory to production mixers of different sizes and manufacturers. 

Figure 11. Plot of viscosity vs. shear rate for spring relaxation experiment. Job 5374. Key , fit curve Q, actual data.

Figure 12. Plot of viscosity vs. time for step shear rate experiment. Thixotropic recovery. Job 4373. Key , actual data Q, fit curve.

Aqueous pectin dispersions show flow behavior similar to many other polysaccharide solutions. Flow curves of specific viscosity rpp vs. shear rate have a Newtonian plateau (constant r sp) at low shear rates, followed by a shear thinning region at moderate shear rates (Morris et al., 1981). Most pectin solutions have relatively low viscosity compared to some other commercial polysaccharides, such as guar gum, mainly because of the lower MW. Consequently, pectin has limited use as a thickener. 

Fig. 6.19. Typical viscosity vs shear rate curve for a concentrated suspension.

Assuming that Ci, C2, and C3 are not affected by temperature, pressure, and dissolution of CO2, they can be determined from a viscosity-shear rate curve of the neat polymer. Namely, the coefficient, Ci, which is equivalent to n - 1, can be determined by the slope of the viscosity and shear rate curve. The values of C2 and C3 can be determined from data of viscosity vs. free volume fraction of the neat polymer. The data of free volume fraction required for determining C2 and C3 can be obtained from PVT data of the neat polymer at temperatures and pressures where the viscosity measurements of the neat polymer are performed. 

The effect of overall molecular weight or the number of blocks on rheological properties for the samples from the second fractionation can be illustrated as a plot of reduced viscosity vs. a function proportional to the principal molecular relaxation time (Figure 2). This function includes the variables of zero shear viscosity, shear rate, y, and absolute temperature, T, in addition to molecular weight, and allows the data to be expressed as a single master curve (10). All but one of the fractions from the copolymer containing 50% polystyrene fall on this... 

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