Viscosity Einstein expression

Perrin model and the Johansson and Elvingston model fall above the experimental data. Also shown in this figure is the prediction from the Stokes-Einstein-Smoluchowski expression, whereby the Stokes-Einstein expression is modified with the inclusion of the Ein-stein-Smoluchowski expression for the effect of solute on viscosity. Penke et al. [290] found that the Mackie-Meares equation fit the water diffusion data however, upon consideration of water interactions with the polymer gel, through measurements of longitudinal relaxation, adsorption interactions incorporated within the volume averaging theory also well described the experimental results. The volume averaging theory had the advantage that it could describe the effect of Bis on the relaxation within the same framework as the description of the diffusion coefficient. 

Self diffusion coefficients can be obtained from the rate of diffusion of isotopically labeled solvent molecules as well as from nuclear magnetic resonance band widths. The self-diffusion coefficient of water at 25°C is D= 2.27 x 10-5 cm2 s 1, and that of heavy water, D20, is 1.87 x 10-5 cm2 s 1. Values for many solvents at 25 °C, in 10-5 cm2 s 1, are shown in Table 3.9. The diffusion coefficient for all solvents depends strongly on the temperature, similarly to the viscosity, following an Arrhenius-type expression D=Ad exp( AEq/RT). In fact, for solvents that can be described as being globular (see above), the Stokes-Einstein expression holds ... 

Equation (44) can be transformed further by making a few more assumptions. Writing Rjj = r- + r-, and introducing the Stokes-Einstein expression for the diffusion constant of spheres in a continuous medium of viscosity rj... 

In the case of the intermolecular contribution to relaxation times of proton-containing molecules and for the effect of pressure on self-diffusion one can use the Stokes-Einstein expression, which relates the shear viscosity, rj, to the self-diffusion coefficient D ... 

Einstein expressed the diffusion coefficient D in terms of the particle radius a, fluid dynamic viscosity t, absolute temperature T, gas constant R, and Avogadro s Number Na (Einstein 1905) ... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

The relative viscosity of a dilute dispersion of rigid spherical particles is given by = 1 + ft0, where a is equal to [Tj], the limiting viscosity number (intrinsic viscosity) in terms of volume concentration, and ( ) is the volume fraction. Einstein has shown that, provided that the particle concentration is low enough and certain other conditions are met, [77] = 2.5, and the viscosity equation is then = 1 + 2.50. This expression is usually called the Einstein equation. 

The deviation from the Einstein equation at higher concentrations is represented in Figure 13, which is typical of many systems (88,89). The relative viscosity tends to infinity as the concentration approaches the limiting volume fraction of close packing ( ) (0 = - 0.7). Equation 10 has been modified (90,91) to take this into account, and the expression for becomes (eq. 11) ... 

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 same equation applies to other solvents. It is often easier to incorporate an expression for the diffusion coefficient than a numerical value, which may not be available. According to the Stokes-Einstein equation,6 diffusion coefficients can be estimated from the solvent viscosity by... 

A unified understanding of the viscosity behavior is lacking at present and subject of detailed discussions [17, 18]. The same statement holds for the diffusion that is important in our context, since the diffusion of oxygen into the molecular films is harmful for many photophysical and photochemical processes. However, it has been shown that in the viscous regime, the typical Stokes-Einstein relation between diffusion constant and viscosity is not valid and has to be replaced by an expression like... 

When solvated ions migrate within the electrolyte, the drag force applied by the surrounding solvent molecules is measured by solvent viscosity rj. Thus, in a solvent of lower viscosity, the solvated ions would move more easily in response to an applied electric field, as expressed by the Einstein—Stokes relation (eq 3). Solvents of low viscosity have always been considered the ideal candidates for electrolyte application however, their actual use was restricted because most of these solvents have low dielectric constants (Tables 1 and 2) and cannot dissociate ions effectively enough to prevent ion pairing. 

In 1906 Albert Einstein (Nobel Prize, 1921) published his first derivation of an expression for the viscosity of a dilute dispersion of solid spheres. The initial theory contained errors that were corrected in a subsequent paper that appeared in 1911. It would be no mistake to infer from the historical existence of this error that the theory is complex. Therefore we restrict our discussion to an abbreviated description of the assumptions of the theory and some highlights of the derivation. Before examining the Einstein theory, let us qualitatively consider what effect the presence of dispersed particles is expected to have on the viscosity of a fluid. 

The Einstein theory shows that volume fraction is the theoretically favored concentration unit in the expansion for viscosity, even though it is not a practical unit for unknown solutes. As was the case in the Flory-Huggins theory in Chapter 3, Section 3.4b, it is convenient to convert volume fractions into mass/volume concentration units for the colloidal solute. According to Equation (3.78), 0 = c(V2/M2), where c has units mass/volume and V2 and M2 are the partial molar volume and molecular weight, respectively, of the solute. In viscosity work, volumes are often expressed in deciliters —a testimonial to the convenience of the 100-ml volumetric flask In this case, V2 must be expressed in these units also. The reader is advised to be particularly attentive to the units of concentration in an actual problem since the units of intrinsic viscosity are concentration when the reduced viscosity is written as an expansion of powers of concentration c. (The intrinsic viscosity is dimensionless when the reduced viscosity is written as an expansion of powers of volume fraction 0.) With the substitution of Equation (3.78), Equation (42) becomes... 

EXAMPLE 13.5 Determination of the Thickness of Adsorbed Polymer Layer from the Intrinsic Viscosity of the Dispersion. An adsorbed layer of thickness 8RS on the surface of spherical particles of radius R increases the volume fraction occupied by the spheres and therefore makes the intrinsic viscosity of the dispersion greater than predicted by the Einstein theory. Derive an expression that allows the thickness of the adsorbed layer to be calculated from experimental values of intrinsic viscosity. 

We can determine from Eq. (8.2) the values of D for solvated metal ions. The value of D changes with changes in solvent or solvent composition. The viscosity (rj) of the solution also changes with solvent or solvent composition. However, the relation between D and tj can be expressed by the Stokes-Einstein relation ... 

For a suspension of solid spheres in a medium of viscosity rjo, Einstein (I) has derived the following expression for the relative vicosity of the suspension over that of the medium ... 

In order to evaluate this expression, we need to know the force v / that is responsible for producing the molecular flux. It could be an external force such as an electric field acting on ions. Then evaluation of Eq. 18-48 would lead to the relationship between electric conductivity, viscosity, and diflusivity known as the Nernst-Einstein relation. 

In this expression, the primary environmental determinant is the viscosity in the denominator. Note that the exponential is slightly larger than in the Stokes-Einstein relation (Eq. 18-52). Since viscosity decreases by about a factor of 2 between 0°C and 30°C, D,w should increase by about the same factor over this temperature range. Furthermore, the influence of the molecule s size is also stronger in Eq. 18-53 than in 18-52 (note r, = constant V 173). In Box 18.4 experimental information on the temperature dependence of D,w is compared with the theoretical prediction from Eqs. 18-52 and 18-53. 

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 type of chosen polymer and additives most strongly influences the rheological and processing properties of plastisols. Plastisols are normally prepared from emulsion and suspension PVC which differ by their molecular masses (by the Fickentcher constant), dimensions and porosity of particles. Dimensions and shape of particles are important not only due to the well-known properties of dispersed systems (given by the formulas of Einstein, Mooney, Kronecker, etc.), but also due to the fact that these factors (in view of the small viscosity of plasticizer as a composite matrix ) influence strongly the sedimental stability of the system. The joint solution of the equations of sedimentation (precipitation) of particles by the action of gravity and of thermal motion according to Einstein and Smoluchowski leads 37,39) to the expression for the radius of the particles, r, which can not be precipitated in the dispersed system of an ideal plastisol. This expression has the form ... 

The use of the Stokes-Einstein equation (2) relating the diffusion coefficient (D) of a spherical solute molecule to its radius (r), the viscosity of the medium (tj) and the Boltzmann constant (k) permits the rate coefficient ( en) to be expressed in (3) in terms of the viscosity of the medium. In this derivation, the... 

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. 

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