Suspension viscosity-shear rate dependence

Krieger (1972) has argued that the shear-rate-dependent suspension viscosity t (0, y) is more appropriate than the solvent viscosity t]s to use in defining a dimensionless shear rate. Since r 4>, y)y is the shear stress a, Krieger s suggested dimensionless group is really a reduced shear stress ... 

The complex viscosity can be related to the steady-shear viscosity rf) via the empirical Cox-Merz rule, which notes the equivalence of steady-shear and dynamic-shear viscosities at given shearing rates ri y) = rj (co). The Cox-Merz rule has been confirmed to apply at low rates by Sundstrom and Burkett (1981) for a diallyl phthalate resin and by Pahl and Hesekamp (1993) for a filled epoxy resin. Malkin and Kulichikin (1991) state that for highly filled polymer systems the validity of the Cox-Merz rule is doubtful due to the strain dependence at very low strains and that the material may partially fracture. However, Doraiswamy et al. (1991) discussed a modified Cox-Merz rule for suspensions and yield-stress fluids that equates the steady viscosity with the complex viscosity at a modified shear rate dependent on the strain, ri(y) = rj yrap3), where y i is the maximum strain. This equation has been utilised by Nguyen (1993) and Peters et al. (1993) for the chemorheology of highly filled epoxy-resin systems. 

Figure la-c shows the dependence of the apparent viscosity of the enzyme hydrolysis suspension on biomass concentration. It clearly demonstrates a dramatic decrease of viscosity at the reloading point (i.e., after the initial 4 h). The experimental determination of the viscosity-shear rate and shear stress-shear rate relationships of the various formulation suspensions with different concentrations was performed with a variable speed rotational viscometer (2 to 200 rpm). 

An important distinction between polymeric liquids and suspensions arises from their different microstructures and is evidenced by the elastic recoil phenomena that polymers exhibit but suspensions do not. The polymeric or macromolecular system when deformed under stress will recover from very large strains because like an elastic material the restoring force increases with the deformation. With a suspension, however, the forces between the particles decrease with increasing separation so that there is limited mechanism for recovery. There are, however, a variety of rheological properties common to polymeric liquids that suspensions will exhibit including shear rate dependent viscosity and time-dependent behavior. We shall discuss these differences in more detail in the following section. 

Nicodemo et al [143] studied the shear rate dependence of the viscosity of suspensions in non-Newtonian liquids and compared it with that of suspensions in Newtonian liquids. The relative viscosity, when the suspending medium was Newtonian, decreased with increasing shear rate toward an asymptotic value which was a function of the filler content. The concentration dependence could be correlated by equations given in section 4.2.1. 

Lee, et al. measured viscosity as affected by shear rate for silica sphere suspensions, finding shear thinning at lower shear rates(57). In some but not all systems and volume fractions above 0.5, a reproducible abrupt transition to shear tbickening was found at elevated shear rates. The transition shear rate depended on concentration and temperature. In contrast, Jones, et al plot only a shear thinning region. A possible explanation for this difference is provided by the Peclet number Pe,... 

The typical viscous behavior for many non-Newtonian fluids (e.g., polymeric fluids, flocculated suspensions, colloids, foams, gels) is illustrated by the curves labeled structural in Figs. 3-5 and 3-6. These fluids exhibit Newtonian behavior at very low and very high shear rates, with shear thinning or pseudoplastic behavior at intermediate shear rates. In some materials this can be attributed to a reversible structure or network that forms in the rest or equilibrium state. When the material is sheared, the structure breaks down, resulting in a shear-dependent (shear thinning) behavior. Some real examples of this type of behavior are shown in Fig. 3-7. These show that structural viscosity behavior is exhibited by fluids as diverse as polymer solutions, blood, latex emulsions, and mud (sediment). Equations (i.e., models) that represent this type of behavior are described below. 

Thermal stabilities were assessed by the time-dependent change of melt viscosity at a constant temperature and shear rate (290°C, 50 s"1 respectively). Figure 6.4 shows that three ofthe six resins showed a significant drop in viscosity as a function of time at 290°C. The average decrease in viscosity for Kel-F 6050, Alcon 3000, and an experimental suspension is 37%. 

The flow properties of suspensions are complex. The apparent viscosity at a given shear rate increases with increasing solids concentration and rises extremely rapidly when the volume fraction of solids reaches about 50 per cent. The flow properties also depend on the particle size distribution and the particle shape, as well as the flow properties of the suspending liquid. 

Rheopectic Time-dependent dilatant flow. At constant applied shear rate, viscosity increases. In a flow curve, hysteresis occurs. Clay suspensions. A suspension which sets slowly on standing, but quickly when gently agitated due to time-dependent particle interference under flow. 

At very low shear rates (i.e., flow velocities), particles in a chemically stable suspension approximately follow the layers of constant velocities, as indicated in Fig. 2. But at higher shear rates hydro-dynamic forces drive particles out of layers of constant velocity. The competition between hydrodynamic forces that distort the microstructure of the suspension and drive particles together, and the Brownian motion and repulsive interparticle forces keeping particles apart, leads to a shear dependency of the viscosity of suspensions. These effects depend on the effective volume fraction of... 

The shear dependency of the viscosity of ceramic suspensions at low to intermediate shear rates can be measured with common rheometers. The results for a ceramic suspension with a high volume fraction of solids and a ceramic ink are shown in Fig. 6. 

Fig. 7. Qualitative sketch of the shear dependency of the viscosity of stable suspensions over a large range of shear rates.

The qualitative behavior of the viscosity of suspensions over a large range of shear rates is depicted in Fig. 7. This type of shear dependency is found, for example, for concentrated colloidal suspensions. These results cannot be immediately carried over to ceramic inks since the experiments are done with much higher concentrations of solids and at much lower shear rates than are applicable for the inkjet process. The suspended particles in these experiments are usually spherical at sizes of about 1 fim or smaller. But the shear dependency of the viscosity found for concentrated... 

This section draws heavily from two good books Colloidal Dispersions by Russel, Seville, and Schowalter [31] and Colloidal Hydrodynamics by Van de Ven [32] and a review paper by Jeffiey and Acrivos [33]. Concentrated suspensions exhibit rheological behavior which are time dependent. Time dependent rheological behavior is called thixotropy. This is because a particular shear rate creates a dynamic structure that is different than the structure of a suspension at rest. If a particular shear rate is imposed for a long period of time, a steady state stress can be measured, as shown in Figure 12.10 [34]. The time constant for structure reorganization is several times the shear rate, y, in flow reversal experiments [34] and depends on the volume fraction of solids. The viscosities discussed in Sections 12.42.2 to 12.42.9 are always the steady shear viscosity and not the transient ones. 

When departures from the Cox-Merz rule are attributed to structure decay in the case of steady shear, the complex viscosity is usually larger than the steady viscosity (Mills and Kokini, 1984). Notwithstanding this feature, the relation between magnitudes of T a and T can be dependent on the strain amplitude used (Lopes da Silva et al., 1993). Doraiswamy et al. (1991) presented theoretical treatment for data on suspensions of synthetic polymers. They suggested that by using effective shear rates, the Cox-Merz rule can be applied to products exhibiting yield stresses. The shift factors discussed above can be used to calculate effective shear rates. 

The viscous and elastic properties of orientable particles, especially of long, rod-like particles, are sensitive to particle orientation. Rods that are small enough to be Brownian are usually stiff molecules true particles or fibers are typically many microns long, and hence non-Brownian. The steady-state viscosity of a suspension of Brownian rods is very shear-rate- and concentration-dependent, much more so than non-Brownian fiber suspensions. The existence of significant normal stress differences in non-Brownian fiber suspensions is not yet well understood. 

Blood is a non-Newtonian suspension showing a shear-dependent viscosity. At low rates of shear, erythrocytes form cylindrical aggregates (rouleaux), which break up when the rate of shear is increased. Calculations show that the shear rate (D) associated with blood flow in large vessels such as the aorta is about 100 s b but for flow in capillaries it rises to about 1000 s b The flow characteristics of blood are similar to those of emulsions except that, while shear deformation of oil globules can occur with a consequent change in surface tension, no change in membrane tension... 

Keep in mind that the viscosity of the suspension and of the continuous phase liquid depend on the shear rate (shear stress) in most cases. 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

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