Shear viscosity function

This section describes two common experimental methods for evaluating i], Fj, and IG as functions of shear rate. The experiments involved are the steady capillary and the cone-and-plate viscometric flows. As noted in the previous section, in the former, only the steady shear viscosity function can be determined for shear rates greater than unity, while in the latter, all three viscometric functions can be determined, but only at very low shear rates. Capillary shear viscosity measurements are much better developed and understood, and certainly much more widely used for the analysis of polymer processing flows, than normal stress difference measurements. It must be emphasized that the results obtained by both viscometric experiments are independent of any constitutive equation. In fact, one reason to conduct viscometric experiments is to test the validity of any given constitutive equation, and clearly the same constitutive equation parameters have to fit the experimental results obtained with all viscometric flows. 

J m(t — t )y 1](t, t) dt. Consider a fluid with a single relaxation time, 20, and modulus, Go, and with h(y) = e y. Calculate the steady-state shear viscosity function... 

Fig. 1 Flow curve and shear viscosity function of Bingham- and Casson materials...

Cox and Merz [C20] have made the remarkable experimental observation that the non-Newtonian shear viscosity function of flexible chain polymers has the same form as the complex viscosity-frequency function, i.e.. 

Rheological models have been described for steady shear viscosity function, normal stress difference function, complex viscosity function, dynamic modulus function and the extensional viscosity function. The variation of viscosity with temperature and pressure is also discussed. 

With respect to the number of variables, it is fairly obvious that material functions are necessarily nonlinear but, of course, in well-selected asymptotic conditions of one of the parameters, with all the others constant, rnie may recover a linear behavior. For instance, at constant temperature, the shear viscosity function at vanishing shear rate of a pure, tmfilled polymer is the so-called pseudo-Newtonian viscosity, i.e. rio = limri(Y), and for (infinitesimally) low strain ampli-... 

F. 2 Shear viscosity function of an unfilled SBR1500 ctnnpound as measured at 100 °C with various instruments in the author s laboratory... 

In principle, the shear viscosity function in the linear viscoelastic range reduces to... 

As mentioned above, the Newtonian plateau is (or has been) rarely observed with gum rubbers so that po(T) must be obtained by extrapolating experimental data towards zero shear rate, by making use of an appropriate model for the shear viscosity function. In the author s experience, a most flexible model is the so-called Carreau-Yasuda equation, i.e. (at a given temperature T) ... 

Figure 4 illustrates how the Carreau-Yasuda model meets the shear viscosity data of Fig. 2. A non-linear fitting algorithm (i.e. Marquardt-Levenberg) was used to obtain the parameters given in the inset. As can be seen the fit curve provides a shear viscosity function that corresponds reasonably well with experimental data so that the high shear behavior is asymptotic to a power law and the very low shear behavior corresponds to the pseudo-Newtonian viscosity po- The characteristic time X (56.55 s) can be considered as the reverse of a critical shear rate (i.e. = Yc = 0.0177 s ) that corresponds to the intersection between the high shear power...

Fig. 4 Shear viscosity function of an unfilled SBR1500 compound at 100 °C as fitted with the Carreau-Yasuda model see Fig. 1 for symbols meaning...

Figure 5 shows steady shear viscosity data for a carbon black filled high cis-1.4 polybutadiene compound, as obtained using various rheometers. The Carreau-Yasuda equation was used to yield fit parameters given in the lower right inset the shear viscosity function q = f(y) is drawn in the left graph. As can be seen, a... 

Material Functions in the Nonlinear Viscoelastic Range 4.2.1 Shear Viscosity Function... 

The shear thinning behavior, as generally observed with polymer systems, is a typical nonlinear viscoelastic effect, so that by combining the Carreau-Yasuda and the Arrhenius equations a general model for the shear viscosity function can be written as follows ... 

Capillary rheometer experiments are tedious and time consuming. For instance, in the author s experience, a skilled operator has to work half-a-day to generate the shear viscosity function at one temperature, within the typical 10-10,000 s shear rate range. The interest of Eq. (8) is immediately obvious since, from the measured... 

Fig. 13 Shear viscosity function of a gum Ethylene-Propylene rubber as measured at three temperatures with a capillary rheometer (author s unpublished data)...

The method of using MFI as a ncmnalizing parameter to coalesce rheological parameter curves is not restricted to die shear viscosity function only. As a matter of fact, the unification tedmique can be extended to obtain coalesced curves of normal stress difference, conqdex viscosity, storage modulus, and ex-... 

Fig. 11 Left Influence of the wetting time of the silica particles in MMA (and correspondingly the Sauter mean diameter of the particle agglomerates Xpartide) on the Sauter mean diameter of the silica loaded MMA droplets. The droplet size distribution was measured with static light scattering (SLS). Right resulting shear viscosity functions...

Typical effects of carbon blacks on the shear viscosity function are illustrated in Figure 5.2, drawn using data published by Montes et al. Three types... 

Effects of carbon black on the shear viscosity function. (Data from S. Montes, J.L. White, N. Nakajima, /. Non-Newtonian Fluid Mech., 28,183-212,1988.)... 

Note that there are many (complex) fluids that do not exhibit a Newtonian plateau at low shear rate (stress) and whose shear viscosity function feeds the controversy on the existence of a yield stress. As noted by Barnes, when the flow is so slow than ages are necessary to detect it, at least one could consider that the yield stress is an engineering reality (H. Barnes. The yield stress—a review or "jtotvra pei" —everything flows /. Non-Newtonian Fluid Mech., 81,133-178,1999.)... 

Shear viscosity function of a carbon black filled BR compound. 

Thanks to their relatively lower viscosity, food and cosmetic products allow this type of behavior to be easily documented, when performing experiments with controlled stress rheometers. There are commercial versions of such instruments, essentially rotating systems (parallel disks or cone-and-plate), which have the capability to measure extremely small rotation rates (in the 10 rad/s, i.e., one revolution in 20 years ). Experiments performed in such conditions are called creep testing using controlled torques as low as 10 Nm with a resolution of 10 Nm. Figure 5.5 shows an example of the shear viscosity function (vs. shear stress) measured on a typical cosmetic product (body cream), using such a controlled stress rheometer. 

Typical shear viscosity function of a (complex) cosmetic material. 

As we will see below, nearly all proposed models follow this approach, but with respect to the definition of the shear viscosity function, i.e., T = o/y = /(y) = f (o), we consider that a criterion of coherence, if not validity, of such models is that they allow equations to be derived, either as t) = /(y) or as IT=F(o). 

---