High shear limit

This model contains four rheological parameters the low shear limiting viscosity (rj0), the high shear limiting viscosity a time constant (X),... 

If r/0 (i], 1] ) and (ky)2 >S> 1, the Carreau model reduces to the equivalent of a power law model with a high shear limiting viscosity, called the Sisko model ... 

This deceptively simple expression is capable of describing the shear thinning response of monodisperse polymers with a high level of precision. In the high shear limit we obtain... 

Under shear flow, the minimum center-to-center separation, r, between the particles will be in the interval 2Rs < r < L i.e., at the high-shear limit r = 2R, and at quiescent conditions r = L. An analysis of the flow behavior in this case leads to the following expression for the zero shear rate (i.e., Pe < 1) limit of the relative viscosity ... 

A grafted layer of polymer of thickness L increases the effective size of a colloidal particle. In general, dispersions of these particles in good solvents behave as non-Newtonian fluids with low and high shear limiting relative viscosities (fj0 and rj ), and a dimensionless critical stress (a3aJkT) that depends on the effective volume fraction = (1 + L/a)3. The viscosities diverge at volume fractions m0 < for mo < fan < 4>moo> the dispersions yield and flow as pseudoplastic solids. 

Fig. 37. The ratio of the equivalent hard sphere volume fraction based on the measured intrinsic viscosity as a function of for polyfmethyl methacrylate) spheres with grafted poly( 12-hydroxy stearic add) layers such that a/L = 4.7 (Mewis et ai, 1989). Open and closed circles correspond to the low and high shear limits of suspension viscosity.

Beyond the percolation limit, the bridging network is more concentrated. Below the critical volume fraction, no continuously bridging networks are formed and the viscosity is low. As shown in Figure 12.7, this bridging network breaks up as the shear rate increases, giving different viscosities at different shear rates. As a result, this gives low and high shear limit viscosities observed at steady state for concentrated poljmier solutions and concentrated particulate suspensions (discussed later). 

This then gives a direct link between the thermodynamic stress and the osmotic pressure (or the compressibility) of the suspension. As a result of this stress, the viscosity will depend directly upon the structure, and the interpartide potential, V(ry). Using this interrelationship Batchelor has been able to evaluate the ensemble averages of both the mechanical and thermodynamic stresses by renormalizing the integrals. As a result, he has developed truncated series expressions for the low shear limit viscosity, and the high shear limit viscosity, t) , corresponding to... 

These theoretical expressions have been tested by experiments on monosized pol3oner lattices [42] and monosized silica [43] suspensions dispersed in water and other liquids. Both the low and the high shear-limiting viscosities are shown in Figure 12.13 to increase monotonically... 

The two rheological properties required—the yield stress, x , and the high-shear-limiting (or plastic) viscosity, —determine the flow behavior of a Bingham plastic. The viscosity fnnction for the... 

This chapter is an in-depth review on rheology of suspensions. The area covered includes steady shear viscosity, apparent yield stress, viscoelastic behavior, and compression yield stress. The suspensions have been classified by groups hard sphere, soft sphere, monodis-perse, poly disperse, flocculated, and stable systems. The particle shape effects are also discussed. The steady shear rheological behaviors discussed include low- and high-shear limit viscosity, shear thinning, shear thickening, and discontinuity. The steady shear rheology of ternary systems (i.e., oil-water-solid) is also discussed. 

In deriving equation 32 or equation 33, it is assumed that 0max is the solid volume fraction at which the suspended particles cease moving. Thus, the forces, such as shearing, that can disturb the suspension structure and hence improve the mobility of particles will have an effect on the value of max. This is confirmed by the fact that a value of kH = 6.0 is observed at low shear limit, that is, y 0 and at high shear limit, y - oo, kH = 7.1 is found. Typical values of max have been found with the use of Quemada s equation as 0max = 0.63 0.02 in the low shear limit and sterically stabilized silica spheres in cyclohexane (72, 85, 88). 

Figure 12 shows the variation of the high shear limit relative viscosity variation with particle volume fraction. One can observe that large discrepancies are present in the experimental data among different studies. This indicates the difficulty in measuring the viscosity of suspensions. Many factors can affect the experimental measurements. For instance, the uniformity of the particles, properties of the suspending medium, the wall effects of the viscometer, and even the time of the experiment (92). 

Jones and co-workers (72, 88) found that the suspension viscosity variation with shear rate can be fitted fairly well by the Cross equation, equation 12, with m = 0.5 — 0.84. Both the low and high shear limit relative viscosities, Mro, Mroo> can be expressed by the Quemada s equation with 0max = 0.63 and 0.71, respectively. 

Figure 12. High shear limit relative viscosity variation with volume fraction.

Figure 14 shows the variation of the steady shear relative viscosity at the high shear limit with the effective volume fraction as defined by equation 46 for poly(methyl methacrylate) (PMMA) suspensions of different sizes in decalin sterically stabilized by means of grafted poly (12-hydroxy stearic acid) chains with a degree of polymerization of 5. The stabilizing polymer layer thickness is 9 1 nm, in particular, A = 9 nm... 

The Einstein constant for a fiber suspension is also a function of shear rate. The ratio of low to high shear limit Einstein constant is rj 1.17 for rods and re/31 for discs (114). The low shear limit Einstein constant is given by... 

Figure 20. Universal low and high shear limit viscosity vs. solid volume fraction curve for monodisperse and hidisperse systems (130).

To numerically illustrate the effect of particle size on the dimensionless groups, we note that in water at standard temperature for a = 100 (im, a potential of 10 mV, and A = 10 °J, we have Pe = 10 y, N5R = 10 y, and = 10 y. Clearly for 100 /um particles, all of the dimensionless groups are very large compared to unity even at a shear rate of 1 s In the high shear limit, non-Newtonian behavior should vanish and the viscosity should attain a stationary value independent of the shear rate. We note here the analogue between the high shear limit for suspensions and the infinite-shear-rate-viscosity limit for polymers discussed above. 

For colloidal particles, the dimensionless parameters are generally small and non-Newtonian effects dominate. Considering the same example as above, but with particles of radius a = 1 /xm, the parameters take on the values Pe = y, N y = 10 y, and N = 10 y so that for shear rates of 0.1 s or less they are all small compared to unity. The limit where the values of the dimensionless forces groups are very small compared to unity is termed the low shear limit. Here the applied shear forces are unimportant and the structure of the suspension results from a competition between viscous forces. Brownian forces, and interparticle surface forces (Russel et al. 1989). If only equilibrium viscous forces and Brownian forces are important, then there is well defined stationary asymptotic limit. In this case, there is an analogue between suspensions and polymers which is similar to that for the high shear limit, wherein the low shear limit for suspensions is analogous to the zero-shear-rate viscosity limit for polymers. 

In the last section we introduced the concept of two asymptotic viscosity limits for shear thinning colloidal suspensions as a function of shear rate. One is the high shear limit which corresponds to high values of the Peclet number where viscous forces dominate over Brownian and interparticle surface forces. Generally this limit is attained with non-colloidal size particles since to achieve large Peclet numbers by increase in shear rate alone requires very large values for colloidal size particles. In this limit, non-Newtonian effects are negligible for colloidal as well as non-colloidal particles. 

Figure 9.3.1 High shear limit relative viscosity for monodisperse spherical particles as a function of solids volume fraction. Circles are data of Shapiro 8c Probstein (1992), squares are data of de Kruif et al. (1986), curves are semi-empirical equations.

Since the apparent viscosity 17—>00 as (f)— lk, it follows that l/k may be interpreted as the maximum packing fraction (f,- "f e high shear limit relative viscosity may therefore be written... 

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