Viscosity Einstein equation

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). 

Mark and Houwink were the first to formulate the equation in the power form and to demonstrate its validity by means of empirical values. In reality, the Mark-Houwink Equation is simply the Einstein viscosity equation, which assumed spheres, transferred to particles with size dependent particle density. 

Einstein viscosity equation phys chem An equation which gives the viscosity of a sol in terms of the volume of dissolved particles divided by the total volume. Tn,stTn vis kas-od-e i.kwa-zhon ... 

The behavior of many fillers can be roughly treated as spheres. Current theories describing the action of these spherical-acting fillers in polymers are based on the Einstein equation (8.1). Einstein showed that the viscosity of a Newtonian fluid (t/J was increased when small, rigid, noninteracting spheres were suspended in a liquid. According to the Einstein equation, the viscosity of the mixture (17) is related to the fractional volume (c) occupied by the spheres, and was independent of the size of the spheres or the polarity of the liquid. 

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 ... 

In the case of a small heavy sphere falling through a suspension of large particles (hxed in space), we have A. > 1 the respective expansions, corresponding to Equation 5.265, were obtained by Fuentes et al. ° hi the opposite case, when A. 1, the suspension of small background spheres will reduce the mean velocity of a large heavy particle (as compared with its Stokes velocity ) because the suspension behaves as an effective fluid of larger viscosity as predicted by the Einstein viscosity formula. ... 

Problem 7-22. The Viscosity of a Multicomponent Membrane. An interesting generalization of the Einstein calculation of the effective viscosity of a dilute suspension of spheres is to consider the same problem in two dimensions. This is relevant to the effective viscosities of some types of multicomponent membranes. Obtain the Einstein viscosity correction at small Reynolds number for a dilute suspension of cylinders of radii a whose axes are all aligned. Although there is no solution to Stokes equations for a translating cylinder, there is a solution for a force- and torque-free cylinder in a 2D straining flow. The result is... 

For a solution (or suspension) whose solute molecules are rigid spheres the viscosity is represented by the Einstein viscosity equation ... 

The Einstein viscosity relation given by Equation 12.62 may be written as... 

Equation 12.67 predicts that the specific viscosity is proportional to the volume of the equivalent hydrodynamic sphere. The Einstein viscosity relation was derived for rigid spherical particles in solution. However, real polymer molecules are neither rigid nor spherical. Instead the spatial form of the polymer molecule in solution is regarded as a random coil. Theories based on this characteristic form of polymer molecules have resulted in the expression... 

The effect of filler loading on Mooney viscosity of the rubber compound is a good indication of the immobility and hydrodynamic effect caused by the filler in the unvulcanized rubber. Figure 3.8 shows a plot of Mooney viscosity against volume fraction of filler in the vulcanizate base for data shown in Table 3.3. The theoretical value was calculated using Equation (3.12). This equation is an extension of Einstein s equation. Einstein studied colloidal suspensions and emulsions by hydrodynamic analysis. The viscosity of the... 

The coefficient 2.5 of the first term on the right arises from the Einstein viscosity equation (see Section 3.8.2), the origin of the Guth-SmaUwood equation. The second term on the right is important for more concentrated dispersions of the fillers, taking into account their interaction. This equation also assumes the filler is dispersed in the polymer, and also predicts a lower bound increase in the modulus. 

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

Figure 12-14 shows the number-average molecular weight, Mn, as a function of the reduced time, f/%. Mn increases with the reaction time. Figure 12-15 shows plots of the reduced viscosity, Vsp/C, against the concentration, C, for solutions with r = 1.0 at reaction times, f/%, of 0.015, 0.71, 0.83, 0.91 and 0.96. The reduced viscosity, Vsp/C, of a solution containing spherical particles may be expressed by tbe following equation (Einstein, 1906) ... 

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 ... 

Vertically polarized DLS measurements (Xo= 514.5 nm) were performed at 25 C with a commercial BI-200SM instrument (from Brookhaven Instrument Corporation) equipped with a B1-9000AT correlator. The values of the hydrodynamic radius Rh were determined in NaCl solutions using the Stokes-Einstein equation. The viscosity of the polymer solutions was determined using a Rheometer Rheostress RS 100 (from Haake) at 25 C. 

The colloidal osmotic pressure is related to viscosity of a molecular solution, since both depend on the number of particles per unit volume and particle volume. If the hemoglobin contained in RBCs is dissolved in plasma, the osmotic pressure would be extraordinarily high, in the range of several atmospheres. Since this depends on the number of molecules in solutions, polymerization of the Hb is a practical way to maintain concentration and controls the number of particles. An additional advantage of polymerization is that as the number of particles decreases and the particle dimensions increase, the vasoactivity of hemoglobin decreases. Furthermore, since viscosity is a function of the total volume of the solute (Einstein s equation), the viscosity tends to remain constant. 

For dilute dispersions of hard spheres, Einstein s viscosity equation predicts... 

A somewhat similar problem arises in describing the viscosity of a suspension of spherical particles. This problem was analyzed by Einstein in 1906, with some corrections appearing in 1911. As we did with Stokes law, we shall only present qualitative arguments which give plausibility to the final form. The fact that it took Einstein 5 years to work out the bugs in this theory is an indication of the complexity of the formal analysis. Derivations of both the Stokes and Einstein equations which do not require vector calculus have been presented by Lauffer [Ref. 3]. The latter derivations are at about the same level of difficulty as most of the mathematics in this book. We shall only hint at the direction of Lauffer s derivation, however, since our interest in rigid spheres is marginal, at best. 

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