Viscosity of Diluted Suspensions

Consider an extremely diluted suspension consisting of a viscous liquid and small spherical particles suspended in it. Small volume concentration of particles allows to neglect their interactions and to assume that each particle behaves as if it were surrounded by an infinite volume of pure hquid. It is obvious that with the increase of volume concentration of particles their mutual influence will become more and more prominent, and at a certain point it will become impossible to neglect it. Further on, for simplicity, the Brownian motion will be neglected. Besides, we assume that particles are small, so it is possible to neglect the influence of gravity and consider their motion as inertialess. It means that the velocity of particle motion is equal to the velocity of the flow, in other words, the particles are freely suspended in the liquid. 

The presence of particles in the liquid causes a disturbance of the velocity field that would establish in the liquid in the absence of particles. It is known [32] that for Stokes translational motion of an isolated solid sphere in an unbounded volume of viscous liquid, hydrodynamic disturbances of the velocity field attenuate with the increase of r as 1 /r. It is a sufficiently slow attenuation, which causes mathematical complications when we want to find disturbances caused by the presence of a large number of particles in the liquid. In particular, it results in slowly convergent (and sometimes even divergent) integrals. 

The addition of solid particles to a pure liquid results in an increase of viscous dissipation inasmuch as the total solid surface area increases. Therefore, if the suspension is to be considered as a Newtonian liquid, its viscosity should be greater than that of a pure liquid. Einstein [18] was the first to consider the problem of finding the viscosity of a suspension. He limited his analysis to the case of the Couette current of an infinitely dilute suspension. 

We mentioned in Section 4.2 that the Couette flow (shear flow) is characterized by the linear structure of the velocity field (Fig. 8.5), which in the Cartesian system of coordinates looks like 

Here h is the distance between the walls, one of which moves parallel relative to the other with the velocity Uq, which is taken to be constant. 

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This is the same as the expression for the viscosity of dilute suspensions (see for example Chapter 3). The expression is however limited to low concentrations of particles, i.e. for tp < 0.05 there are others that can be used throughout the range of volume fractions. Landel26 proposed that the elastic modulus can be described by... 

Viscosity of Diluted Suspensions 223 Fig. 8.5 The velocity profile in a shear flow. 

C. Lin, J.W. Shan, Electrically tunable viscosity of dilute suspensions of carbon nanotubes, Phys. Fluids, 2007,19,121702. 

According to Eq. (2.8) or (2.10) the concentration dependence of has two parameters, [rj and 0m- Both are measures of a specific physical quantity (respectively, shape and packing) and may be independently determined, for example, [rj] from viscosity of diluted suspensions and 0m from dry packing of solid particles. For anisometric particles, the magnitude of these parameters may also be theoretically predicted see the rheological summary in a quite recent monograph [64]. Once [rj] and 0m are known, Eq. (2.8) will correctly describe the rjr versus 0 dependence for complex industrial systems, for example, PVC (poly(vinyl chloride)) emulsions and plastisols, mica-reinforced polyolefins, and sealant formulations [44,65]. However, in some suspensions and blends, r] and 0m may vary with composition [66]. 

Einstein showed that the viscosity of dilute suspensions in the absence of interactions between the particles is proportional to the volume fraction of the dispersed phase, ( ), that is, the addition of particles to the dispersion medium results in energy dissipation due to the rotation of the particles in the shear force fiel ... 

Einstein [63-65] was the pioneer in the study of the viscosity of dilute suspensions of neutrally buoyant rigid spheres without Brownian motion in a Newtonian hquid. He proposed the following relationship between the relative viscosity of the suspension and the volume fraction of the suspended particles... 

Nawab, M.A. Mason, S.G. 1958. The Viscosity of Dilute Suspensions of Thread-Like Particles. /, Phys, Chem. 62 (10) 1248-1253... 

We can construct a useful and reasonably accurate theory of intrinsic viscosity of dilute polymer solutions, building directly on the Einstein result for viscosities of dilute suspensions of spheres (Chapter 10). Recalling this result in a slightly different form ... 

Einstein [73] has shown that the viscosity of dilute suspensions of rigid spherical particles is a function of the volume fraction of solids in the suspension and is independent of particle size, as shown in Eq. (4-3) ... 

Eq. (1) is valid for a particle settling without the interference of other particles, i.e., for diluted systems. The particle settling velocity decreases as particle concentration increases. This phenomenon is known as hindered settling. Several equations can be found in the literature to account for this phenomenon, but a simple method to calculate the hindered settling velocity is to use Eq. (1) replacing the liquid density and viscosity by the apparent density and viscosity of the suspension [24]. 

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... 

In order to overcome this difficulty, Rudin and Strathdee (1974) developed a semi empirical method for predicting the viscosity of dilute polymer solutions. The method is based on an empirical equation proposed by Ford (1960) for the viscosity of a suspension of solid spheres ... 

The simplest case to consider is steady flow of a dilute suspension of Newtonian drops or bubbles in a Newtonian medium. If the capillary number y a / F is small, so that the drops or bubbles do not deform under flow, then at steady state the viscosity of the suspension is given by Taylor s (1932) extension of the Einstein formula for solid spheres ... 

For dilute suspensions (10% or less phase volume) of spherical particles in a Newtonian fluid, the Einstein equation can be used to predict the viscosity of the suspension, rj, which also behaves as Newtonian ... 

Let us summarize the obtained results. From the expression for the viscosity of an infinite diluted suspension, it follows that the viscosity factor does not depend on the size distribution of particles. The physical explanation of this fact is that in an infinite diluted suspension W 1), particles are spaced far apart (in comparison with the particle size), and the mutual influence of particles may be ignored. Besides, under the condition a/h 1, we can neglect the interaction of particles with the walls. It is also possible to show that in an infinite diluted suspension containing spherical particles. Brownian motion of particles does not influence the viscosity of the suspension. However, if the shape of particles is not spherical, then Brownian motion can influence the viscosity of the suspension. It is explained by the primary orientation of non-spherical particles in the flow. For example, thin elongated cylinders in a shear flow have the preferential orientation parallel to the flow velocity, in spite of random fluctuations in their orientation caused by Brownian rotational motion. 

Jamieson, A. M., Simha, R., Newtonian viscosity of dilute, semi-dilute and concentrated polymer solutions. Chapter 1, in Polymer Physics From Suspensions to Nanocomposites and Beyond, Utracki L. A. and Jamieson A. M., Editors, J. Wiley Sons, New York (2010). 

Figure 12 Relative viscosity of different suspensions as a function of the volume frac-lion. Circles TiO, panicles 0.1 pm in diameter square. carbon black continuou,s line ideal dilute sphere result. (Taken from Ref. 63. Reprinted with permission of John Wiley and Suns. Inc.)...

Zero-Shear Viscosity The flow of dilute suspensions ( < 0.05) of rigid spheres in Newtonian liquid was described by Einstein in an article of 1906 on a new method for estimation of molecular dimension [37], and then corrected in 1911 [38] ... 

Consider a molecular liquid with Newtonian behaviour (see Chapter 4) such as water, benzene, alcohol, decane, etc. The addition of a spherical particle to the liquid will increase its viscosity due to the additional energy dissipation related to the hydrodynamic interaction between the liquid and the sphere. Further addition of spherical particles increases the viscosity of the suspension linearly. Einstein developed the relationship between the viscosity of a dilute suspension and the volume fraction of solid spherical particles as follows (Einstein, 1906) ... 

For dilute polymer solutions a simple approximate expression for specific viscosity can be derived from Einstein s equation for a dilute suspension of hard (incompressible) spheres in a liquid. The viscosity of a suspension of N spheres, each with a hydrodynamic volume Ve in a total volume V of a liquid, is given by... 

Though there is varied opinion about the relationship between the relative viscosity of a suspension and the volume concentration of the spheres for dilute suspension, one could get a reasonable estimate on using the simplest equation (4.1) of Einstein for 0 < 0.1. When 0.1 < 0 < 0.15, Thomas s [75] equation (4.2) or Ford s [77] equation (4.4) could be used for a reliable estimate. Of course for 0 < 0.1 too, these equations could be used and the result averaged out with the prediction from equation (4.1) to obtain a good conservative estimation. 

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