Einstein equation for viscosity

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

Einstein Equation for Viscosity of Suspensions—Einstein derived a general equation for computation of the viscosity of a suspensoid in terms of viscosity of the medium, and the ratio of aggregate volume of solid particles to total volume of suspensoid. Einstein s equation is... 

Several size parameters can be used to describe the dimensions of polymer molecules radius of gyration, end-to-end distance, mean external length, and so forth. In the case of SEC analysis, it must be considered that the polymer molecular size is influenced by the interactions of chain segments with the solvent. As a consequence, polymer molecules in solution can be represented as equivalent hydrodynamic spheres [1], to which the Einstein equation for viscosity may be applied ... 

This was proposed according to an analogy to the following Einstein equation for viscosity r for colloidal solution ... 

The dynamic mechanical properties of a filled system, in the absence of interaction between components, can be described on the basis of a mechanical model proposed by Takayanagi for non-interacting polymer mixtures. This model is very useful for describing properties of filled systems with interfacial layers. Based on hydrodynamic considerations, an equation was proposed (analogous to the Einstein equation for viscosity of suspension) to calculate the modulus of composite, Ec ... 

For higher (0 > 0.05) concentrations where particle—particle interactions are noticeable, the viscosity is higher than predicted by the Einstein equation. The viscosity—concentration equation becomes equation 10, where b and c are additional constants (87). 

Using the Stokes-Einstein equation for the viscosity, which is unexpectedly useful for a range of liquids as an approximate relation between diffusion and viscosity, explains a resulting empirical expression for the rate of formation of nuclei of the critical size for metals... 

The Smoluchowski theory for diffusion-controlled reactions, when combined with the Stokes-Einstein equation for the diffusion coefficient, predicts that the rate constant for a diffusion-controlled reaction will be inversely proportional to the solution viscosity.16 Therefore, the literature values for the bimolecular electron transfer reactions (measured for a solution viscosity of r ) were adjusted by multiplying by the factor r 1/r 2 to obtain the adjusted value of the kinetic constant... 

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. 

We noted above that either solvation or ellipticity could cause the intrinsic viscosity to exceed the Einstein value. Simha and others have derived extensions of the Einstein equation for the case of ellipsoids of revolution. As we saw in Section 1.5a, such particles are characterized by their axial ratio. If the particles are too large, they will adopt a preferred orientation in the flowing liquid. However, if they are small enough to be swept through all orientations by Brownian motion, then they will increase [17] more than a spherical particle of the same mass would. Again, this is very reminiscent of the situation shown in Figure 2.4. 

The most important relationships used are - Fick s laws and - Einsteins equation for diffusion, Newtons viscosity law and Stokes s law (- Stokes s viscous force)... 

The effect of Co on the process of diamond powder compaction has been studied at the pressure of 8 GPa and temperatures between 1400-2000 °C. It is shown that the interaction between liquid Co and diamond particles speeds up the process if not changes the limiting value of shrinkage as compared with solid phase sintering. The dependences of the rate of diamond powder infiltration with cobalt and Co-WC, Co-Mo and Co-Ti melts on the temperature have been studied experimentally under high pressure. It is shown that the infiltration by pure cobalt occurs quicker as compared with that by cobalt-base alloys. Based on the Einstein equation for the viscosity of mixtures, an equation for the infiltration coefficient is derived which is in good agreement with the experimental data for Co-Ti and Co-WC alloys. 

The values of l) >,n — the diffusivity for the Brownian motion of aerosol — are calculated from the Stokes-Einstein equation. For spherical particulates with the effective radius rp, in a gas with the dynamic viscosity p2 (nearly constant for pressures about and less than one bar), the formula is ... 

The Stokes-Einstein equation for liquid-phase ordinary molecular diffusion coefficients in binary mixtures suggests that the product of Hab and the solvent viscosity /u-b should scale linearly with temperature T. Cite references (i.e., equations) from the literature and evaluate the product of Hab and /xb in terms of its scaling-law dependence on temperature for low-density gases. In other words ... 

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 ... 

Using the Thomas-Einstein equation for dynamic viscosity correction ... 

For droplets with low viscosity (comparable to that of the medium) the transmission of tangential stress across the 0/W interface from the continuous phase to the dispersed phase causes liquid circulation in the droplets. Energy dissipation is less than that for hard spheres and the relative viscosity is lower than that predicted by the Einstein equation. For an emulsion with viscosity qj for the disperse phase and q for the continuous phase,... 

The two properties that are most often studied in gelling systems are the viscosity and the elastic modulus. Unfortunately, the theoretical foundation for the critical exponents for such dynamic properties is much weaker than for the geometric properties, such as cluster size distribution. For example [24], to calculate the viscosity, one could start with the Einstein equation for the viscosity (//) of a sol. 

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