Shear rate, colloidal suspensions

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. 

Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

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. 

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... 

Pe should control the onset of shear thickening in colloidal suspensions. This being the case, the shear rate at which shear thickening occurs is (approximately) given by... 

FIGURE 12.9 Viscosity (at low shear rate) of AI2O3 suspensions at different pH values adjusted by the addition of various amounts of 0.2 M AICI3. Near the isoelectric point the suspension is not colloidally stable, giving a high viscosity. Data taken from Reed [23]. 

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. 

As discussed in Sect. 4, in the fluid, MCT-ITT flnds a linear or Newtonian regime in the limit y 0, where it recovers the standard MCT approximation for Newtonian viscosity rio of a viscoelastic fluid [2, 38]. Hence a yrio holds for Pe 1, as shown in Fig. 13, where Pe calculated with the structural relaxation time T is included. As discussed, the growth of T (asymptotically) dominates all transport coefficients of the colloidal suspension and causes a proportional increase in the viscosity j]. For Pe > 1, the non-linear viscosity shear thins, and a increases sublin-early with y. The stress vs strain rate plot in Fig. 13 clearly exhibits a broad crossover between the linear Newtonian and a much weaker (asymptotically) y-independent variation of the stress. In the fluid, the flow curve takes a S-shape in double logarithmic representation, while in the glass it is bent upward only. 

Coupled controlled velocity, magnetic resonance imaging (MRI)/rheology measurements of thixotropic and yielding colloidal suspensions further demonstrate the importance of paired measurements [63], Shear rate profiles obtained in laminar tube flow for both Newtonian and non-Newtonian fluids from MRI... 

Here/is the shear stress (force per area), s is the rate of shear, (/ = 0, 1,2) is a quantity proportional to the relaxation time, is a constant proportional to the reciprocal of the shear modulus, X, is the fraction of a shear plane occupied by the ith flow unit, and / = 0, 1, and 2 indicate, respectively, the solvent, the Newtonian, and the non-Newtonian flow units. It is a well-known fact that (1) was applied with great success to various cases including colloidal suspensions and polymeric solutions. In this paper, we study the Newtonian terms [the first and second terms on the right of (1)] in more detail, and the nature of the Newtonian flow units of solutes or suspensoids will be considered. 

Time-dependent fluids are those for which the components of the stress tensor are a function of both the magnitude and the duration of the rate of deformation at constant temperature and pressure [4]. These fluids are usually classified into two groups—thixotropic fluids and rheopectic fluids—depending on whether the shear stress decreases or increases with time at a given shear rate. Thixotropic and rheopectic behavior are common to slurries and suspensions of solids or colloidal aggregates in liquids. Figure 10.2 shows the general behavior of these fluids. 

In many colloid systems particles are covered with adsorbed layers (see Chapter 5). These too influence the viscosity since the effective radius, and hence the effective volume fraction, is greater than that of the core particles. In attempting to fit experimental data on dispersions of spherical particles to theoretical equations the effective volume fractions must be employed. If measurements are made on very dilute suspensions and at low shear rates, equation (8.7) (retaining only the first two terms) may be used to calculate the effective volume fraction and hence the particle size, and the thickness of the adsorbed layer if the size of the core particles is known. This is not, however, a very precise method and generally other methods of finding the adsorbed layer thickness are to be preferred. 

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. 

Truly bimodal suspensions of colloidal and noncolloidal particles are of considerable practical interest. For such suspensions at low shear rates, the viscosity is high so that, for example, during storage, settling is reduced. On the other hand, because the mixture is shear thinning, at higher shear rates when the suspension is pumped the viscosity decreases, thereby enabling the mixture to be pumped at a lower pressure drop. 

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