Hard-sphere suspension viscosity

Fundamental theories of transport properties for systems of finite concentration are still rather tentative (24). The difficulties are accentuated by the still uncertain effects of concentration on equilibrium properties such as coil dimensions and the distribution of molecular centers. Such problems are by no means limited to polymer solutions however. Even for the supposedly simpler case of hard sphere suspensions the theories of concentration dependence for the viscosity are far from settled (119,120). 

Problems and Worked Examples 6.1 through 6.5, at the end of this chapter, will sharpen your skills in obtaining simple, practical estimations of the viscosity, modulus, and relaxation time of hard-sphere suspensions. 

Stable particle suspensions exhibit an extraordinarily broad range of rheological behavior. which depends on particle concentration, size, and shape, as well as on the presence and type of stabilizing surface layers or surface charges, and possible viscoelastic properties of the suspending fluid. Some of the properties of suspensions of spheres are now reasonably well understood, such as (a) the concentration-dependence of the zero-shear viscosity of hard-sphere suspensions and (b) the effects of deformability of the steric-stabilization layers on the particles. In addition, qualitative understanding and quantitative empirical equations... 

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. 

The flow properties of disordered micellar phases are now reasonably well understood. For spherical micelles the viscosity can be estimated from modified hard-sphere-suspension theories, while for disordered semidilute cylindrical micelles the Cates theory of entangled living polymers provides at least a good starting point, and in some cases nearly quantitative prediction of rheological properties. 

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). 

The viscosity of a hard sphere suspension is a function of the volume fraction of particles. Generally, the viscosity can be expressed by a virial expansion ... 

Figure 7.7. Relative viscosity of hard-sphere suspension in Newtonian fluid as a function of the volume fraction. Thomas curve represents the generalized behavior of suspensions as measured in 19 laboratories. The remaining curves were computed from Simha s, Mooney s and Krieger-Dougherty s relations assuming Einstein value for intrinsic viscosity of hard spheres, [T ] = 2.5, but different values for the maximum packing volume fraction, ([) = 0.78, 0.91, and 0.62 respectively.

To summarize, the dependence of relative viscosity on the volume fraction of suspended particles can be expressed by any of several theoretical or semi-empirical relations. These can be written in terms of the two parameters, [tj] and Thus = t1j.([ii], c )/( ) ). As it will be shown, the generality of this dependence extends beyond the monodispersed hard sphere suspensions. 

RELATIVE VISCOSITY vs. VOLUME FRACTION FOR HARD SPHERES SUSPENSIONS... 

At low concentration the relative viscosity of hard sphere suspension is usually approximated by a polynomial ... 

A number of factors affect the rheology of emulsions composition, the viscosity ratio of the dispersed-to-matrix phase (1 s 7/2/771), the droplet size and its distribution, rheology of the interphase, and so on. Often, well-stabilized emulsions follow the viscosity-concentration relationships developed for hard sphere suspensions, including the yield phenomena. In contrast, emulsions with deformable dispersed... 

In the limit of infinite dilution when all interactions between the particles are neglected, the relative viscosity of hard sphere suspensions is given by the Einstein equation... 

Figure 19 Relative zero shear viscosity of PNIPAM microgels obtained at different concentrations and different temperatures vs. the effective volume fraction. The line represents the mastercurve of hard sphere suspensions. Reproduced with permission from Senff, H. Richtering, W. J. Chem. Phys. 1999, fff, 1705. 2...

Figure 5.15 shows the good correlation of the Quemada model with 4>ma — 0-631 to the experimental results of low shear rate viscosity of Jones et al. (1991) for sihca hard sphere suspensions. 

Note that for hard spheres, surprisingly there is no influence of the particle size on the viscosity. The only concern about particle size in hard sphere suspensions is that if the particles are too big they will settle out. If the particles are neutrally buoyant sedimentation issues are not significant. 

It has been pointed out in the previous section that the rheological behaviour of hard sphere suspensions (non-interacting particles) was not influenced by the size of the particles. This is not true of attractive particle networks. When the particles in a suspension are attractive, smaller particle size results in increased rheological properties such as 5deld stress, viscosity and elastic modulus (described next). The influence of particle size can be determined by considering that the rheological properties of attractive particle networks depend upon the strength of the bond between particles and the number of bonds per unit volume that need to be broken. For example, consider the shear yield stress ... 

Hard spheres suspensions of narrow size distribution are Newtonian at low volume fraction and then shear thin between a zero shear rate viscosity ( jo) and a high shear rate plateau viscosity ( oo)-Numerous studies (Russel, 1989 Dhont, 1989 Vanderwerff, 1989 Bergenholz, 2001, 2002 Dekruif, 1985 Fuchs, 2002 Lionberger, 1998, 2000)... 

To summarize the results of this section, the relative viscosity of the hard-sphere suspension has been shown to depend on concentration via Eqs. 10.9 and 10.10. In most studies, there is not a significant crossover region there is no region in which... 

A. R. Altenberger andJ. S. Dahler. A renormalization group calculation of the viscosity of a hard-sphere suspension. J. Coll. Interf. ScL, 189 (1997), 379-381. 

The increase in viscosity with the volume fraction of solids in the suspension does not in itself imply any non-Newtonian behavior, as the stress can remain strictly linear in the shear rate (Guazzelli and Morris 2012). In fact, when we consider the hard-sphere suspension in a Newtonian liquid under Stokes flow conditions, the expectation is that the rheology should be Newtonian. This is a basic result of dimensional analysis since there is no intrinsic rate associated with either the fluid or the particles, the only rate is that set by the flow shear rate. 

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