Viscosity dilute spheres

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.)...

Apart from chemical composition, an important variable in the description of emulsions is the volume fraction, outer phase. For spherical droplets, of radius a, the volume fraction is given by the number density, n, times the spherical volume, 0 = Ava nl2>. It is easy to show that the maximum packing fraction of spheres is 0 = 0.74 (see Problem XIV-2). Many physical properties of emulsions can be characterized by their volume fraction. The viscosity of a dilute suspension of rigid spheres is an example where the Einstein limiting law is [2]... 

For dilute dispersions of hard spheres, Einstein s viscosity equation predicts... 

In a very dilute solution, between the co-spheres of the ions the interstitial solvent is unmodified and has the same properties as in the pure. solvent,. The co-sphere of each positive ion and the co-sphere of each negative ion, however, may contribute toward a change in the viscosity. We should expect to find, in a very dilute solution, for each species of ion present, a total contribution proportional to the number of ions of that species present in unit volume. At the same time, we may anticipate that the electrostatic forces between the positively and the negatively charged ions must be taken into account. 

Theory presented earlier in this chapter led to the expectation that the frictional coefficient /o for a polymer molecule at infinite dilution should be proportional to its linear dimension. This result, embodied in Eq. (18) where P is regarded as a universal parameter which is the analog of of the viscosity treatment, is reminiscent of Stokes law for spheres. Recasting this equation by analogy with the formulation of Eqs. (26) and (27) for the intrinsic viscosity, we obtain ... 

For very dilute solutions, the motion of the ionic atmosphere in the direction of the coordinates can be represented by the movement of a sphere with a radius equal to the Debye length Lu = k 1 (see Eq. 1.3.15) through a medium of viscosity t] under the influence of an electric force ZieExy where Ex is the electric field strength and zf is the charge of the ion that the ionic atmosphere surrounds. Under these conditions, the velocity of the ionic atmosphere can be expressed in terms of the Stokes law (2.6.2) by the equation... 

The specific viscosity )jsp of a dilute solution of spheres is directly related to their hydrodynamic volume VV Nl denotes Avogadro s number. Typically the intrinsic viscosity [tj] follows a scaling law, the so-called Mark-Houwink-Sakurada equation ... 

In Chapter 3 (Section 3.5.2) the viscosity of a hard sphere model system was developed as a function of concentration. It was developed using an exact hydrodynamic solution developed by Einstein for the viscosity of dilute colloidal hard spheres dispersed in a solvent with a viscosity rj0. By using a mean field argument it is possible to show that the viscosity of a dispersion of hard spheres is given by... 

This value of kn is actually low by an order of magnitude for dilute suspensions of charged spheres of radius Rg. This is due to the neglect of interchain correlations for c < c in the structure factor used in the derivation of Eqs. (295)-(298). If the repulsive interaction between polyelectrolyte chains dominates, as expected in salt-free solutions, the virial expansion for viscosity may be valid over considerable range of concentrations where the average distance between chains scales as. This virial series may be approxi-... 

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 1906 Albert Einstein (Nobel Prize, 1921) published his first derivation of an expression for the viscosity of a dilute dispersion of solid spheres. The initial theory contained errors that were corrected in a subsequent paper that appeared in 1911. It would be no mistake to infer from the historical existence of this error that the theory is complex. Therefore we restrict our discussion to an abbreviated description of the assumptions of the theory and some highlights of the derivation. Before examining the Einstein theory, let us qualitatively consider what effect the presence of dispersed particles is expected to have on the viscosity of a fluid. 

The Einstein theory is based on a model of dilute, unsolvated spheres. In this section we examine the consequences on intrinsic viscosity of deviations from the Einstein model in each of the following areas ... 

It appears that one can develop a qualitative understanding of the simple flow properties at moderate concentration without going beyond concepts which are already inherent either in the dilute solution theory of polymers or in the properties of particulate suspensions. The dependence of viscosity on c[i ] is believed to reflect a particle-like or equivalent sphere (127) hydrodynamics in solutions of low to moderate concentration. In particular, the experimental facts do not force the consideration of effects which might arise from the permanent connectedness of the polymer backbones. Situations conducive to the entangling of molecules may be present, e.g., overlap of the coils, but either entanglement contributions are small, or else they are overwhelmed by the c[f ] interactions. 

Masumy and Tasomy 28) found that for regularly shaped resol PhFO microspheres it was best to use dilute aqueous solutions with viscosities of 15-50 Pa - sec, water contents of up to 50%, and free phenol contents of 6-9%. The resultant spheres included unexpanded monolithic particles, but these could be eliminated by using a chemical blowing agent such as dinitrosopentamethylenetetramine (DNPMTA) or azodiisobutyronitrile, or a surfactant (0.25 mass %). 

Our previous study (J 6) of self diffusion in compressed supercritical water compared the experimental results to the predictions of the dilute polar gas model of Monchick and Mason (39). The model, using a Stockmayer potential for the evaluation of the collision integrals and a temperature dependent hard sphere diameters, gave a good description of the temperature and pressure dependence of the diffusion. Unfortunately, a similar detailed analysis of the self diffusion of supercritical toluene is prevented by the lack of density data at supercritical conditions. Viscosities of toluene from 320°C to 470°C at constant volumes corresponding to densities from p/pQ - 0.5 to 1.8 have been reported ( 4 ). However, without PVT data, we cannot calculate the corresponding values of the pressure. 

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 ... 

Here, the sphere center is instantaneously situated at point 0 the sphere center translates with velocity U, while it rotates with angular velocity (a r is measured relative to 0 its magnitude r is denoted by r. Moreover, f = r/r is a unit radial vector. The latter solution is derivable in a variety of ways e.g., from Lamb s (1932) general solution (Brenner, 1970). [Equation (2.12) represents a superposition (Brenner, 1958) of three physically distinct solutions, corresponding, respectively, to (i) translation of a sphere through a fluid at rest at infinity (ii) rotation of a sphere in a fluid at rest at infinity (iii) motion of a neutrally buoyant sphere suspended in a linear shear flow. The latter was first obtained by Einstein (1906, 1911 cf. Einstein, 1956) in connection with his classic calculation of the viscosity of a dilute suspension of spheres, which formed part of his 1905 Ph.D. thesis.]... 

Einstein showed that the viscosity tj of a fluid in which small rigid spheres are present in dilute and uniform suspension is related to the viscosity tjq of the pure fluid (solvent) by the expression ... 

The simplest model to predict the viscosity of liquids containing solid particles is that derived for a dilute suspension of uniform, monodisperse, noninteracting hard spheres in a solvent of viscosity rj. ... 

The nice thing about the assumption of an equivalent hydrodynamic sphere is that it allows you to do two things first, use the Einstein relationship for the viscosity of a dilute solution of solid particles, given previously in Equation 12-43. Then we can use the definition of a volume fraction, (Equation 12-51) ... 

In this chapter, we first present a theory of the primary electroviscous effect in a dilute suspension of soft particles, that is, particles covered with an ion-penetrable surface layer of charged or uncharged polymers. We derive expressions for the effective viscosity and the primary electroviscous coefficient of a dilute suspension of soft particles [26]. We then derive an expression for the effective viscosity of uncharged porous spheres (i.e., spherical soft particles with no particle core) [27]. 

Following Einstein s work (Einstein, 1906, 1911) (Equation 2.24) on dilute rigid sphere dispersions, models for estimating viscosity of concentrated non-food dispersions of solids are based on volume fraction (p) of the suspended granules and the relative viscosity of the dispersion, % = (vIVs), where t] is the viscosity of the dispersion rjs is the viscosity of the continuous phase (Jinescu, 1974 Metzner, 1985). 

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