Viscosity Einstein-Simha

Tn GPC the product [77] M has been widely accepted as a universal calibration parameter, where [77] is the intrinsic viscosity and M is the molecular weight. This product is defined by the Einstein-Simha viscosity expression (I) as... 

It can be shown following the arguments of Newman et al. (3), that only the number-average molecular weight and hydrodynamic radius are valid when the Einstein-Simha viscosity expression is applied to whole polymers. Higher moments require a polydispersity factor. Thus,... 

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

Benoit and coworkers demonstrated that it is possible to use a set of narrow polymer standards of one chemical type to provide absolute molecular weight calibration to a sample of a different chemical type (19,20). To understand how this is possible, one must first consider the relationship between molecular weight, intrinsic viscosity, and hydrodynamic volume, the volume of a random, freely jointed polymer chain in solution. This relationship has been described by both the Einstein-Simha viscosity law for spherical particles in suspension,... 

Data according to Eqs. (2-4) are reported in Table n. The v values (sixth column) were obtained from viscosity data and the Einstein-Simha... 

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

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. 

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 numerical correlations given by the direct BEM simulations are similar to the expressions given earlier. In fact, the first coefficient in the power expansion is close to the one predicted by Einstein [15]. The second coefficient in the power expansion is between the value suggested by Guth and Simha [25] and one suggested by Vand [65, 66], In Figure 10.29, the calculated BEM relative viscosity is collapsed for all cases. 

According to the Einstein theory, the intrinsic viscosity of a spherical particle suspension is 2.5. However, for a colloidal suspension of nonspherical particles, [r ] > 2.5. Jeffery [112] obtained the viscosity of an ellipsoidal particle suspension under shear. Incorporating Jeffery s results of velocity fields around the particle, Simha [113] obtained expressions for two explicit limiting cases of ellipsoids. Kuhn and Kuhn [114] also obtained an expression for intrinsic viscosity for the full range of particle aspect ratio (p) by taking an approach similar to Simha s method. 

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. 

Relative Viscosity of Suspensions One of the most interesting derivations of the T vs. (() dependence (covering the full range of concentration) was published by Simha [1952]. He considered the effects of concentration on the hydrodynamic interactions between suspended particles of finite size. (Note that previously the particles were simply considered point centers of force that decayed with cube of the distance.) Simha adopted a cage model, placing each solid, spherical particle of radius a inside a spherical enclosure of radius b. At distances x < b, the presence of other particles does not influence flow around the central sphere and the Stokes relation is satisfied. This assumption leads to a modified Einstein [1906, 1911] relation ... 

Figure 7.7. Relative viscosity of hard-sphere suspension in Newtonian fluid as a function of the volume fraction. Thomas curve represents the generalized behavior of suspensions as measured in 19 laboratories. The remaining curves were computed from Simha s, Mooney s and Krieger-Dougherty s relations assuming Einstein value for intrinsic viscosity of hard spheres, [T ] = 2.5, but different values for the maximum packing volume fraction, ([) = 0.78, 0.91, and 0.62 respectively.

Robert focused on two problems extension of Einstein s theory to higher concentrations and then to nonspherical particles. Armed with the tools provided by Lamb s hydrodynamic bible, Einstein s famous doctoral dissertation and a lengthy review by Guth and Mark, I started out. After an unsatisfactory beginning and a Black Sea vacation [Simha, 1999], he successfully extended the treatment of viscosity, t], to higher concentrations by including binary hydrodynamic interactions ... 

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,... 

Einstein s theoretical prediction(i) that the intrinsic viscosity of a dispersion of rigid spheres is precisely 5/2 (2.500. . . ) has been amply confirmed by experiment. Kuhn and Kuhn(2) and, independently, Simha ) showed how the intrinsic viscosities of ellipsoidal particles, both oblate and prolate, depend on their axial ratios. Their work also permits intrinsic viscosities to be estimated for cylindrical particles, and hence for rods and disks. Tayloi<" ) extended Einstein s theory to include fluid particles, specifically emulsions and foams. 

Attempts were also made to extend the Einstein law to higher concentration, both experimentally and theoretically. Eirich succeeded on the experimental side and Simha on the theoretical side. The work was extended to suspensions of elhpsoids and the dramatic increase in viscosity with elongation at constant volume fraction was derived by Simha and by Kuhn. The actual dependence of the intrinsic viscosity for a rod on length (and hence M) was as the square, not the linear dependence asserted by Staudinger. Another key figure in polymer science appears in this discussion Maurice L. Huggins (1897-1981). A more thorough... 

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