Suspension, relative viscosity

Concentration effects and interparticle forees are of rheological significance with injection molding mixes. Increasing the filler concentration (volume fraction) of a suspension increases the suspension relative viscosity tir = T)/rio, where ri is the suspension viscosity and Tjo is the continuous-phase viscosity. 

With this background of non-Newtonian behavior in hand, let us examine the viscous behavior of suspensions and slurries in ceramic systems. For dilute suspensions on noninteracting spheres in a Newtonian liquid, the viscosity of the suspension, r)s, is greater than the viscosity of the pure liquid medium, rjp. In such cases, a relative viscosity, rjr, is utilized, which is defined as rjs/rjL. For laminar flow, is given by the Einstein equation... 

Other biological fluids can be modeled as suspensions of particles in a solvent, such as was used for the description of suspensions and slurries in Section 4.1.2.2, namely, that the relative viscosity of the suspension is related to the hydrodynamic shape factor. 

Figure 1. Effects of the concentration of dextrans with various molecular weights on three indices of BBC aggregation (16). MAI indicates the average number of RBCs in each aggregation unit counted under the microscope. ESR is the maximum rate of sedimentation of erythrocytes in a calibrated tube, with corrections made for changes in viscosity and density of the suspending medium following the addition of dextrans. The relative viscosity (t)r) is the ratio of the viscosity of RBC suspension to that of the suspending medium at a shear rate fo 0.1 sec 1. The RBC concentration of the suspension was 1% for MAI and 45% for ESR and r)r measurements. The vertical bars represent SEM. (A), Dx 40 (O), Dx 80 (M), Dx 150 (A), Dx 500 ( ), Dx 2000.

In electrochemistry several equations are used that bear Einsteins name [viii-ix]. The relationship between electric mobility and diffusion coefficient is called Einstein relation. The relation between conductivity and diffusion coefficient is called - Nernst-Einstein equation. The expression concerns the relation between the diffusion coefficient and the viscosity and is known as the - Stokes-Einstein equation. The expression that shows the proportionality of the mean square distance of the random movements of a species to the diffusion coefficient and the duration of time is called - Einstein-Smoluchowski equation. A relationship between the relative viscosity of suspension and the volume fraction occupied by the suspended particles - which was derived by Einstein - is also called Einstein equation [ix]. 

Patzold (1980) compared the viscosities of suspensions of spheres in simple shear and extensional flows and obtained significant differences, which were qualitatively explained by invoking various flow-dependent sphere arrangements. Goto and Kuno (1982) measured the apparent relative viscosities of carefully controlled bidisperse particle mixtures. The larger particles, however, possessed a diameter nearly one-fourth that of the tube through which they flowed, suggesting the inadvertant intrusion of unwanted wall effects. 

With the bulk, deviatoric stress tensor denoted by a, Bossis and Brady define the relative viscosity of the suspension as... 

Normal stress differences do not exist in the absence of interparticle forces. Moreover, the relative viscosity of the suspension is a function of only particle densities approaching the maximum possible that still allow the suspension to flow, cluster size (and, as a result, the viscosity of the two-dimensional monolayer) appears to scale as... 

Analysing the rate of energy dissipation in concentrated suspensions of solid spheres, Frankel and Aerivos [69] also arrived at the conclusion that the relative viscosity of suspensions is a function of the relation a/h ... 

Fig. 4.8 Effect of volume fraction rigid spheres on the relative viscosity of the suspension. In these studies, the particle diameter ranged from 0.1 to 440 /xm. Data from Ref. 66.

Studies of the effect of particle size on viscosity suggest that Eqn. (1) is obeyed in the limit as cf> goes to zero. As the volume fraction of particles increases, the relative viscosity increases at a faster rate than linear rate, approaching infinity as the packing density of the suspension approaches that of densely packed solid particles Fig. 4.S.66 Regardless of particle size, all data in Fig. 4.8 scatter about a common curve. An empirical fit for this type of curve has been discussed by Kitano et al. 67... 

Fig. 4.9 Effect of particle asperity on the relative viscosity of molten polymer suspensions. Particles studied were as follows , glass spheres , natural calcium carbonate A, precipitated calcium carbonate o, glass fibers—aspect ratio = 18 ...

Einstein,2 for a suspension of rigid spheres of Tadius r which is small compared with the distance between them and the radius of a capillary, found for the relative viscosity (> o=viscosity of suspending liquid) ... 

Most food particles are not spherical in shape so that the empirical equation (Equation 2.25) that described well (Kitano et al., 1981 Metzner, 1985) the relative viscosity versus concentration behavior of suspensions of spheres and fibers... 

They reported values of A in the range 2.50-3.82 for dispersions of rigid solids, and 1.70-1.85 for red blood suspensions. For gelatinized 2.6% tapioca STDs, the relative viscosity was calculated using Equation 2.30 ... 

The viscosity (or apparent viscosity) of a suspension can be related to the viscosity of the continuous medium in terms of the relative viscosity, rjr. ... 

Figure 6.3 Relative viscosity as a function of the fraction of large spheres in a bimodal distribution of particle sizes with a 5 1 ratio of diameters, at various total volume percentages of particles. The arrow P Q illustrates the 50-fold reduction in viscosity that occurs when monosized particles in a 60 vol% suspension are replaced by a 50-50 mixture of large and small spheres. The arrow P S shows that if monosized spheres are replaced by a bimodal size distribution, the concentration of spheres can be increased from 60% to 75% without increasing the viscosity. (Reprinted from Barnes et al., An Introduction to Rheology (1989), with kind permission from Elsevier Science - NL, Sara Burger-hartstraat 25, 1055 KV Amsterdam, The Netherlands.)...

Thus, the relative shear viscosity r jr s of a concentrated suspension of monosized spherical particles should be a universal function of two dimensionless quantities and (7r (or Pe). Figure 6-4 shows that for three different particle radii 85, 141, and 310 nm, the relative viscosity tjr = V/Vs, is indeed a nearly universal function of for fixed d>. The dependence of tjr on can be fit by a simple expression... 

The tendency of particles to deform increases with increasing volume fraction. At effective volume fractions below about 0 0.4, suspensions of squishy spheres have viscosities similar to those of harder spheres (see Fig. 6-7). But at higher 0, there are substantial differences between the two, and layer deformability becomes important. Thus, for soft spheres at high 0, the dependence of relative viscosity on shear stress no longer obeys Eq. (6-14). Mewis et al. (1989) found that an equation of Cross (1965), which contains an extra parameter, works better ... 

Figure 6.29 Zero-shear relative viscosity versus particle volume fraction for aqueous suspensions of charged polystyrene spheres (a = 34 nm) in 5 x lO " M NaCl ( ) (Buscall et al. 1982a). The line is calculated by using Eq. (6-66) for the viscosity, with 0eff given by Eq. (6-64), and d ff by Eq. (6-67a) or (6-67b). The potential 1T(/ ) is given by Eq. (6-58) or (6-59) with k given by Eq. (6-61) the constant K is 0.10, and is in the range 50-90 mV. (From Buscall 1991, reproduced with permission of the Royal Society of Chemistry.)...

Disordered solutions of spherical micelles are not particularly viscoelastic, or even viscous, unless the volume fraction of micelles becomes high, greater than 30% by volume. Figure 12-7, for example, shows the relative viscosity (the viscosity divided by the solvent viscosity) as a function of micellar volume fraction for a solution of hydrated micelles of lithium dodecyl sulfate in water. Qualitatively, these data are reminiscent of the viscosity-volume-fraction relationship for suspensions of hard spheres, shown as a dashed line (see Section 6.2.1). The micellar viscosity is higher than that of hard-sphere suspensions because of micellar ellipsoidal shape fluctuations and electrostatic repulsions. 

For monodisperse or unimodal dispersion systems (emulsions or suspensions), some literature (28-30) indicates that the relative viscosity is independent of the particle size. These results are applicable as long as the hydrodynamic forces are dominant. In other words, forces due to the presence of an electrical double layer or a steric barrier (due to the adsorption of macromolecules onto the surface of the particles) are negligible. In general the hydrodynamic forces are dominant (hard-sphere interaction) when the solid particles are relatively large (diameter >10 (xm). For particles with diameters less than 1 (xm, the colloidal surface forces and Brownian motion can be dominant, and the viscosity of a unimodal dispersion is no longer a unique function of the solids volume fraction (30). 

Figure 11 shows the relative-viscosity-concentration behavior for a variety of hard-sphere suspensions of uniform-size glass beads. Even though the particle size was varied substantially (0.1 to 440 xm), the relative viscosity is independent of the particle size. However, when the particle diameter was small ( 1 fJLm), the relative viscosity was calculated at high shear rates, so that the effect of Brownian motion was negligible. Figure 8 shows that becomes independent of the particle size at high shear stress (or shear rate). 

Figure 13 illustrates another very interesting point. Here the relative viscosity of a bimodal suspension is plotted as a function of volume percent of small spheres in total solids. At any given total solids concentration, the relative viscosity decreases initially with the increase in volume percent of small spheres, and then it increases with further increase in small spheres. The minimum observed in the relative viscosity plots of a bimodal suspension is quite typical. There are no fundamental reasons why a similar behavior would not be true for emulsions. 

Figure 11. Relative viscosity vs. concentration behavior for suspensions of spheres having narrow size distributions. Particle diameters range from 0.1 to 440 pm. (Reproduced with permission from reference 30. Copyright 1965...

Figure 13. Dependence of relative viscosity upon particle size ratio for himodal suspensions of spheres. (Reproduced with permission from reference 28. Copyright 1971.)...

As the size ratio of the sand particle to the oil droplets d d increases to about 2, there is less dependence on the oil concentration, as shown in Figure 21b. When the size ratio increases to about 3, as shown in Figure 21c, the relative viscosity becomes independent of the oil concentration this result indicates that the emulsions act as a continuous phase toward the solids. Under this condition, the solids and the droplets behave independently, and no interparticle interaction occurs between the solids and the droplets. Yan et al. (64) showed that when the emulsions behave as a continuous phase toward the solids, the viscosity of the mixtures can be predicted quite accurately from the viscosity data of the pure emulsions and the pure solids suspensions. The viscosity of an emulsion-solids mixture having an oil concentration of Pq (solids-free basis) and a solids volume fraction of 0s (based on the total volume) can be calculated from the following equation ... 

In describing the viscosity of an emulsion, the volume fraction of the dispersed phase is the most important parameter. A model suggested by M. Mooney (15) in 1951 for solid suspensions and emulsions with highly viscous dispersed phase described the relative viscosity as a function of volume fraction and a coefficient, k, called the self-crowding factor. 

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