Dispersion viscosity Einstein equation

Hence, a dilute dispersion of rigid solid particles behaves as a Newtonian fluid with a volume fraction-dependent viscosity (Einstein equation) ... 

The relative viscosity of a dilute dispersion of rigid spherical particles is given by = 1 + ft0, where a is equal to [Tj], the limiting viscosity number (intrinsic viscosity) in terms of volume concentration, and ( ) is the volume fraction. Einstein has shown that, provided that the particle concentration is low enough and certain other conditions are met, [77] = 2.5, and the viscosity equation is then = 1 + 2.50. This expression is usually called the Einstein equation. 

Emulsions. Because emulsions are different from dispersions, different viscosity—concentration relationships must be used (71,87). In an emulsion the droplets are not rigid, and viscosity can vary over a wide range. Several equations have been proposed to account for this. An extension of the Einstein equation includes a factor that allows for the effect of variations in fluid circulation within the droplets and subsequent distortion of flow patterns (98,99). 

As the particle size of the disperse phase decreases, there is a corresponding increase in the number of particles and a concomitant increase in interparticulate and interfacial interactions. Thus, in general, the viscosity of a dispersion is greater than that of the dispersion medium. This is often characterized in accordance with the classical Einstein equation for the viscosity of a dispersion. 

The Einstein equation is one of those pleasant surprises that occasionally emerges from complex theories a remarkably simple relationship between variables, in this case the viscosity of a dispersion and the volume fraction of the dispersed spheres. A great many restrictive assumptions are made in the course of the derivation of this result, but the major ones are (a) that the particles are solid spheres and (b) that their concentration is small. 

The simplicity of the Einstein equation makes it relatively easy to test, but also limits its usefulness rather sharply. With so few variables involved, the quantities we may evaluate by Equation (41) are few. Viscosity is a measurable quantity from which we try to extract information about the dispersion. All that Equation (41) offers directly in this area is the evaluation of 0 from viscosity measurements, again provided 0 is small and the particles are spheres. 

To do this, we consider a dispersion of volume fraction 4> and examine the increment in viscosity dr) as a small amount of particles is added to the dispersion. If we take to be small enough that the Einstein equation, Equation (41), holds, the increment dr) that accompanies the addition of particles is then given by... 

List some of the conditions under which the Einstein equation for viscosity of dispersions fails and how one can correct the situation. 

Consider first the effect of a dispersed phase, of volume fraction continuous phase of viscosity D0 and dispersed particles (droplets) which do not attract. At low volume fractions the Einstein equation should apply to a suspension of solid particles at constant temperature,... 

In Section 5.1.2 the effect of solute molecules and particles on viscosity is briefly discussed. It follows that the intrinsic viscosity [t/] is a measure of the extent to which a certain solute can increase viscosity. (Remember that t] equals specific viscosity — 1) divided by concentration for infinitesimally small concentration.) According to the Einstein equation (5.6) the specific viscosity of a dispersion of spheres is 2.5q>, where

volume fraction. This means that [t/] = 2.5

0, where c is concentration in units of mass per unit volume. For a very dilute polymer solution the effective volume fraction can be given as the number of molecules per unit volume N times (4/3)jir, where ty, is the hydrodynamic radius see Eq. (6.5). Furthermore, N = c- M/Nav- For the amylose mentioned in the question just discussed, rh x 25 nm and M = 106 Da. It follows that [//] would equal... 

It is frequently desirable to be able to describe and/or predict dispersion viscosity in terms of the viscosity of the continuous phase (i/q) and the amount of dispersed material. A very large number of equations have been advanced for estimating emulsion, foam, suspension or aerosol viscosities. Most ofthese are empirical extensions of Einsteins equation for a dilute suspension of non-interacting spheres ... 

As a result, this equation is usually the only one needed for liquid or solid aerosols. Figure 6.18 shows several sets of experimental data compared with the Einstein equation. In practice once cp reaches between 0.1 and 0.5, dispersion viscosity increases significantly and can also become non-Newtonian (due to particle/droplet/bubble crowding or structural viscosity). The maximum volume fraction possible for an internal phase made up of uniform, incompressible spheres is 0.74, although emulsions and foams with an internal volume fraction of over 0.99 can exist as a consequence of droplet/bubble distortion. Figure 6.18 and Equation 6.33 illustrate why volume fraction is such a theoretically and experimentally favoured concentration unit in rheology. In the simplest case, a colloidal system can be considered Einsteinian, but in most cases the viscosity dependence is more complicated. 

The viscosity characteristics of liquids can be altered considerably by the presence of finely dispersed solid particles, especially of colloidal size. The viscosity of a suspension of rigid spherical particles in a liquid, when the distance between the spheres is much greater than their diameter, may be expressed by the Einstein equation ... 

One of the simplest ways to obtain the hydrodynamic radius of a particle is to measure the relative viscosity of dilute dispersions, and then apply the Einstein equation to obtain the effective volume fraction of the particles, 0eff- Assuming that the particles behave as hard spheres (when < h is small compared to the core radius... 

Several methods may be used to determine the adsorbed layer thickness, 8. Most of the methods depend on measuring the hydrodynamic radius of the particles with and without the adsorbed polymer layer. For example, one may measure the relative viscosity, of a dispersion with an adsorbed polymer layer. Assuming that the particles behave as hard spheres (when 8 is small compared with the particle radius R) of noninteracting units (low volume fraction of the disperse phase), can be related to the effective volume fraction, [Pg.355]

At an applied level, study of the rheology of emulsions is vital in many industrial applications of personal care products. It is useful to summarize the factors that affect emulsion rheology in a qualitative way. One of the most important factors is the volume fraction of the disperse phase, ( ). In very dilute emulsions (( )< 0.01), the relative viscosity, Tir, of the system may be related to ( ) using the simple Einstein equation (as for solid/ liquid dispersions) (15) i.e.. 

As pointed out by Kruyt (207) the viscosity of a sol depends on the volume fraction of the dispersed phase in accordance with the Einstein equation... 

A range of methods are available for making rheological measurements (qv) (39-42). A frequently encountered problem involves knowing the parti-cle/droplet/bubble size and concentration in a dispersion and the need to predict the suspension, emulsion, or foam viscosity. Many equations have been advanced for this purpose. In the simplest case, a colloidal system can be considered Einsteinian. Here, the viscosity of the colloidal system depends on that of the continuous phase, r]o, and the volume fraction of colloid, 0, according to the Einstein equation, which was derived for a dilute suspension of noninteracting spheres ... 

The viscosity of a very dilute dispersion of rigid spherical particles in a Newtonian fluid is described by the Einstein equation (Equation 8.2) [26]. Where q is the viscosity of the dispersion, qi is the viscosity of the fluid alone, ( > is the volume fraction of particles and kg is the Einstein coefficient, which is 2.5 for spherical particles, k depends upon both particle shape and orientation. 

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