Colloidal systems Einstein equation

The viscosity of colloidal systems depends upon the volume occupied by the colloidal particles. The simple equation of Einstein,... 

Colloidal systems, because of their large number of dispersed particles, show non-Newtonian flow behavior. For a highly dilute dispersion of spherical particles, the following equation has been proposed by Einstein ... 

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. 

In this chapter we examine some issues in mass transfer. The reader has already been introduced to some of the key aspects. In Chapter 3 (Section 7), flocculation kinetics of colloidal particles is considered. It shows the importance of diffusivity in the rate process, and in Equation 3.72, the Stokes-Einstein equation, the effect of particle size on diffusivity is observed, leading to the need to study sizes, shapes, and charges on colloidal particles, which is taken up in Chapter 3 (Section 4). Similarly some of the key studies in mass transfe in surfactant systems— dynamic surface tension, smface elasticity, contacting and solubilization kinetics—are considered in Chapter 6 (Sections 6, 7, 10, and 12 with some related issues considered in Sections 11 and 13). These emphasize the roles played by different phases, which are characterized by molecular aggregation of different kinds. In anticipation of this, the microstructures are discussed in detail in Chapter 4 (Sections 2,4, and 7). Section 2 also includes some discussion on micellization-demicellization kinetics. 

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 ... 

In combination with a general force balance, Einstein s diffiision law results in Equation 8.5b, which permits the estimation of the mass of each particle. Thus, upon combining diffiision experiments (for obtaining D) and sedimentation (gravitational) experiments (for obtaining ), we can estimate the mass of colloidal particles without any assumption about their shape. Finally, due to Einstein s diffiision law Df= kgT), the ratio f/fo is equal to DqID, where Do is the diffusion coefficient of a system containing the equivalent unsolvated spheres. 

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