Einstein viscosity law

Filler particles such as talc or carbon black are routinely incorporated in rubber compounds to improve mechanical properties. Guth [51] related this reinforcement effect to the Einstein viscosity law for colloidal emulsions... 

The concept of universal calibration, as introduced by Benoit et al. (23,24), is based on the Einstein viscosity law,... 

Apart from chemical composition, an important variable in the description of emulsions is the volume fraction, outer phase. For spherical droplets, of radius a, the volume fraction is given by the number density, n, times the spherical volume, 0 = Ava nl2>. It is easy to show that the maximum packing fraction of spheres is 0 = 0.74 (see Problem XIV-2). Many physical properties of emulsions can be characterized by their volume fraction. The viscosity of a dilute suspension of rigid spheres is an example where the Einstein limiting law is [2]... 

Eimco High-Capacity thickener, 22 66 Einsteinium (Es), 1 463-491, 464t electronic configuration, l 474t ion type and color, l 477t metal properties of, l 482t Einstein relation, 22 238. See also Einstein s viscosity equation filled networks and, 22 571, 572 Einstein s coefficient, 14 662 Einstein s equation, 7 280 21 716 23 99 Einstein s law, 19 108 Einstein s viscosity equation, 22 54. 

FIG. 4.9 Experimental verification of Einstein s law of viscosity for spherical particles of several different sizes (Squares are yeast particles, Rs = 2.5 71m circles are fungus spores, Rs = 4.0 /xm triangles are glass spheres, R, = 80 /xm). Open symbols represent measurements in concentric-cylinder viscometers, and closed symbols represent measurements in capillary viscometers. (Data from F. Eirich, M. Bunzl, and H. Margaretha, Kolloid Z., 74, 276 (1936).)... 

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)... 

Being able to determine [r ] as a function of elution volume, one can now compare the hydrodynamic volumes Vh for different polymers. The hydrodynamic volume is, through Einstein s viscosity law, related to intrinsic viscosity and molar mass by Vh=[r ]M/2.5. Einstein s law is, strictly speaking, valid only for impenetrable spheres at infinitely low volume fractions of the solute (equivalent to concentration at very low values). However, it can be extended to particles of other shapes, defining the particle radius then as the radius of a hydrody-namically equivalent sphere. In this case Vjj is defined as the molar volume of impenetrable spheres which would have the same frictional properties or enhance viscosity to the same degree as the actual polymer in solution. 

The couple on the sphere vanishes unless it is restrained from rotating. If the sphere is also neutrally buoyant then F = 0, and only the last term in Eq. (147a) survives. By noting that the local rate of mechanical energy dissipation in the unperturbed flow is 2/iSjj Sjj, this ultimately leads to a simple proof of Einstein s law of suspension viscosity (Ela) for flow through cylinders (B17), provided that the spheres are randomly distributed over the duct cross section. 

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,... 

Einstein s viscosity law may therefore be expressed thus If a few rigid spheres are distributed in a liciuid, the coefficient of viscosity increases by a fraction which corresponds to 2.5 times the total volume of the suspended spheres. 

While few modern scientists will deny the direct relationship of molecular weight of polymers to their viscosity, this was not always tme. In the 1920 s, both Staudinger and Mark proposed such a relationship and in 1929 Staudinger derived his famous "viscosity law" based on the Einstein relationship of viscosity and concentration... 

Ahn [34] investigated the reorientation of di-terl-butylnitroxide (DTBN) in supercooled water at temperatures ranging from 15 to —33°C. The apparent Stokes hydrodynamic radius of DTBN in water was estimated to be about 0.35 nm. Good linear dependence of the reorientation time of the spin probe with the water viscosity is found according to the Debye-Stokes-Einstein (DSE) law. It is found that the ESR signal of DTBN is due to the supercooled liquid state, and not due to the signal from the rapid rotational motion of a spin probe in frozen water. Notice that the smaller spin probes PADS form clathrate cages in ice [18]. 

Surfactant solutions with globular micelles are generally Newtonian liquids with a low viscosity which increases linearly with the volume fraction of the particles according to Einstein s law... 

As an example, Figure 6.1 shows the mutual diffusivity of a mixmre of hexane and nitrobenzene at the critical composition as a function of T — Tc as measured by Taylor dispersion (Matos Lopes et al. 1992) and by light scattering (Wu et al. 1988) the solid curve represents the Stokes-Einstein diffusion law (6.21) with = 104 0.06. In Figure 6.2 a log-log plot of the viscosity ratio of a mixture of 3-methylpentane and nitroethane at the critical composition is shown as a function of the correlation length. The solid curve represents the power law (Q y with Q = 1.4nm and z = 0.063 (Burstyn et al. 1983). Experiments in fluids near the vapor-liquid critical point are consistent with these results (Guttinger Cannell 1980 Berg Moldover 1990). 

EDA complex formation, phase transitions 339 edge dislocation, Volterra process 419 effusity, phase transitions 320 Eglington-Glaser reaction, synthesis 99 eigenwaves, SLM 764 Einstein diffusivity-viscosity law 585... 

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 ... 

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. 

Einstein law linear viscosity growth versus concentration... 

One of the most popular applications of molecular rotors is the quantitative determination of solvent viscosity (for some examples, see references [18, 23-27] and Sect. 5). Viscosity refers to a bulk property, but molecular rotors change their behavior under the influence of the solvent on the molecular scale. Most commonly, the diffusivity of a fluorophore is related to bulk viscosity through the Debye-Stokes-Einstein relationship where the diffusion constant D is inversely proportional to bulk viscosity rj. Established techniques such as fluorescent recovery after photobleaching (FRAP) and fluorescence anisotropy build on the diffusivity of a fluorophore. However, the relationship between diffusivity on a molecular scale and bulk viscosity is always an approximation, because it does not consider molecular-scale effects such as size differences between fluorophore and solvent, electrostatic interactions, hydrogen bond formation, or a possible anisotropy of the environment. Nonetheless, approaches exist to resolve this conflict between bulk viscosity and apparent microviscosity at the molecular scale. Forster and Hoffmann examined some triphenylamine dyes with TICT characteristics. These dyes are characterized by radiationless relaxation from the TICT state. Forster and Hoffmann found a power-law relationship between quantum yield and solvent viscosity both analytically and experimentally [28]. For a quantitative derivation of the power-law relationship, Forster and Hoffmann define the solvent s microfriction k by applying the Debye-Stokes-Einstein diffusion model (2)... 

Loutfy and coworkers [29, 30] assumed a different mechanism of interaction between the molecular rotor molecule and the surrounding solvent. The basic assumption was a proportionality of the diffusion constant D of the rotor in a solvent and the rotational reorientation rate kOI. Deviations from the Debye-Stokes-Einstein hydrodynamic model were observed, and Loutfy and Arnold [29] found that the reorientation rate followed a behavior analogous to the Gierer-Wirtz model [31]. The Gierer-Wirtz model considers molecular free volume and leads to a power-law relationship between the reorientation rate and viscosity. The molecular free volume can be envisioned as the void space between the packed solvent molecules, and Doolittle found an empirical relationship between free volume and viscosity [32] (6),... 

At about the same time, Staudinger derived his well known "law of viscosity". His work was formulated in 1929 and published in 1930 (34, 35). Also based on the Einstein relationship, Staudinger s equation was a direct relationship between the specific viscosity and the polymer molecular weight. 

Another microscopic approach to the viscosity problem was developed by Gierer and Wirtz (1953) and it is worthwhile describing the main aspects of this theory, which is of interest because it takes account of the finite thickness of the solvent layers and the existence of holes in the solvent (free volume). The Stokes-Einstein law can be modified using a microscopic friction coefficient ci micro... 

The ions are regarded as rigid balls moving in a liquid bath. It is assumed that the macroscopic laws of motion in a viscous medium hold, and that the electrostatic interaction is determined by the theory of continuous dielectrics. This assumption implies that the moving particles are large compared to the molecular structure of the liquid. The most successful results of continuous theories can be found in any textbook of physical chemistry Stokes , law for viscous motion, Einstein s derivation of the dependence of viscosity on the concentration... 

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