Einstein-Simha equation

The observation that a modified Einstein-Simha equation may be used to represent the viscosity data for poly (MPDMA AMPS) suggests... 

Lukhovitskii and Karpo [191] established that, in the concentration range accessible for viscometry, the efflux time of a polymer solution is a linear function of the polymer concentration. This is consistent with the Einstein-Simha equation. In the general case, the efflux time of a solution with a polymer concentration tending to zero, T, does not coincide with the efflux time of the pure solvent, T, When the efflux time of a polymer solution is reduced by rather than by (as is done in the standard method), the reduced viscosity becomes independent of the polymer concentration and equal to the intrinsic viscosity. The advantages of the proposed method are especially important for the determination of the intrinsic viscosity (M ) of ultra-high molecular weight polymers. 

This equation is analogous to Einstein-Simha viscosity equation for suspended particles of variable shapes (20,21). 

Many empirical and theoretical modifications have been made to Einstein s equations. A useful extension to dilute suspensions of anisotropic particles, such as clays, is given by the Simha Equation, which is approximately... 

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 majority of theories describing the concentration dependence of viscosity of diluted and moderately concentrated disperse systems is based on the hydrodynamic approach developed by Einstein [1]. Those theories were fairly thoroughly analyzed in the reviews written by Frish and Simha [28] and by Happel and Brenner [29], In a fairly large number of works describing the dependence of viscosity on concentration the final formulas are given in the form of a power series of the volume concentration of disperse phase particles — [Pg.111]

Figure 1 shows a comparison, published by Mori and Ototake [13], of the experimental dependences of viscosity on concentration of dispersions of solid particles based on the data of Vand [34], Robinson [12], Orr and Blocker [5], Dalla Valle and Orr [17] with the theoretical equations based on the hydrodynamic approach used by Einstein (1), Simha (30), Vand (31), Roscoe (44) and the phenomenological equation of Mori and Ototake (14). A more complicated form of the theoretical dependence, naturally makes it possible to describe experimental results over a wider range, but for concentrated dispersions most of theoretical equations remain inapplicable. 

Jeffrey [1923] extended Einstein s analysis to flow around an impermeable, rigid ellipsoid of revolution, and Simha [1940] further incorporated the effect of Brownian motion, deriving an equation of the form... 

Simha provided an equation for the viscosities of ellipsoids of revolution. The prolate ellipsoids of revolution are cigar-shaped while the oblate ellipsoids of revolution are disc-shaped (see Figure 5.3). According to derivations by Einstein and later by R. Simha,... 

Ai is the Einstein constant of 2.5 (from Equation 1-7), and has been found to be in the range of 10.05-14.1 according to Guth and Simha (1936). It is difficult to extrapolate the higher terms A3 and A4 in Equation 18. They are ignored with volumetric concentrations smaller than 20%. 

Derivations of the EiisfSTEiN equation that arc all rather complicated can be found in the papers by Einstein and Simha cited above, further in papers by Guth and Marks by Guth by Guth and Simha , by J, M. Burgers. We shall not reproduce these derivations here, but we wish to draw attention to a single point. 

Unfortunately, all these authors arrive at different results and it is none too clear which of their results, if any, should be trusted In all the papers the hydrodynamic interaction between pairs of particles is considered. Simha, VaND and, to a certain extent, De Bruyn, in addition to hydrodynamical also take mechanical interaction (encounters between particles, formation of pairs) into account. Usually the extension to higher concentrations is given in the form of a second term proportional to v to be added to the Einstein term 2.5 De Bruyn expects a simpler equation for the fluidity i/r, and expressed his results accordingly... 

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