Stokes-Einstein equation translational

Viscosity is a useful quantity, in that both rotational and translation mobility of molecules in solution are viscosity dependent and can be related to viscosity through the Stokes-Einstein equation ... 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

In dynamic light scattering (DLS), or photon correlation spectroscopy, temporal fluctuations of the intensity of scattered light are measured and this is related to the dynamics of the solution. In dilute micellar solutions, DLS provides the z-average of the translational diffusion coefficient. The hydrodynamic radius, Rh, of the scattering particles can then be obtained from the Stokes-Einstein equation (eqn 1.2).The intensity fraction as a function of apparent hydrodynamic radius is shown for a triblock solution in Fig. 3.4. The peak with the smaller value of apparent hydrodynamic radius, RH.aPP corresponds to molecules and that at large / Hs,Pp to micelles. 

One effect of diffusivity on the reaction rate constant of a bimolecular reaction has been proposed by Atkins (Equation 20.1) (Atkins, 1982). The Stokes-Einstein equation (Equation 20.2) (Edward, 1970) relates molecular size and local liquid phase viscosity to diffusion on very small scales. Eor the purposes of this review, we will refer to this type of diffusion as local translational diffusion. 

In a similar approach, double-stranded helicates of various lengths that were derived from copper and silver-based metallosupramolecular architectures have also been classified by their diffusion properties and estimates of the molecular sizes made [48]. Owing to the ellipsoidal structures, it was necessary to introduce appropriate shape factors to translate the hydrodynamic radii determined directly from the unmodified Stoke-Einstein equation into dimensions that were meaningful for these assemblies. Thus, knowledge of the width of the helicates (determined from the X-ray structure of a single complex in this case) allowed the determination of their lengths from the hydrodynamic radii. The results for a series of these helicates is summarised in Table 9.9. It was further shown that 2D DOSY spectra could be employed to differentiate the helicates of different lengths when present simultaneously in a mixture. 

In turn, the translational diffusion coefficient for homodispersed polymer molecules in a solvent at infinite dilution is related to an equivalent hydrodynamic diameter d through the Stokes-Einstein equation... 

The difiusion coefficient, in turn, is inversely proportional to the translational hydrodynamic diameter A h,t (Stokes-Einstein equation) ... 

The translational diffusion coefficient D of a polymer coil can be found from the time-dependent correlation function of scattered intensity measured in dynamic light-scattering experiments. Using the Stokes-Einstein equation, the translational diffusion coefficient D can be related to the apparent hydro-dynamic radius (the radius of equivalent hard sphere). In the limit of nondraining for the solvent coil formed by infinitely long chain, the hydrodynamic radius is given by... 

The fundamental rate expression to be considered is the Smoluchowski relation k = 4n iVDAB AB (Equation (2.1)). The derived expression ART/r] (Equation (2.3a)), is a useful approximation, but deviations from it are observed, because the Stokes-Einstein equation which is involved is derived by hydrodynamic theory for spherical particles moving in a continuous fluid, and does not accurately represent the measured values of translational diffusion coefficients in real systems. Although the proportionality Da 1 /rj is indeed a reasonable approximation for many solutes in common solvents, the numeral coefficient 1 /4 is subject to uncertainty. In the first place, this theoretical value derives from the assumption that in translational motion there is no friction between a solute molecule and the first layer of solvent molecules surrounding it, i.e., that slip conditions hold. If, however, one assumes instead that there is no slipping ( stick conditions), so that momentum is... 

The hydrodynamic radius of colloidal particles can be obtained from dynamic light scattering (DLS), also known as photon correlation spectroscopy (PCS). Here, the temporal fluctuations of scattered light intensity are measured to provide the autocorrelation function, analysis of which provides the translational diffusion coefficient. Then the Stokes-Einstein equation (Eq. 1.9) is used to determine a hydrodynamic radius. This method is described further in Section 1.9.2. 

In dynamic LLS, the Laplace inversion of each measured intensity-intensity time correlation function G q, t) in the self-beating mode can result in a line-width distribution G(L). G(7) can be converted into a translational diffusion coefficient distribution G(D) or further a hydrodynamic radius distribution /(Rh) via the Stokes-Einstein equation, Rh = (kBTI6nrio)/D, where kB, T and qo are the Boltzmann constant, the absolute temperature and the solvent viscosity, respectively. The time correlation functions were analyzed by both the cumulants and CONTIN analysis. 

It should be emphasized that D-5 and D-6 are general in effective diffusion coefficient (D), cage radius (p) and normalization radius (rj ). Analysis of the D=0 (f=0) limits was examined first since values for the yield, with no translational motion permitted, are Independent of the connection one chooses between the effective diffusion coefficient of the pair and the macroscopic viscosity of the medium. We have used the Stokes-Einstein equation (D-7) for this purpose where f is fluidity (1/n, cp.) and b is the effective radius for diffusive separation of the pair. 

A particle size is calculated from the translational diffusion coefficient Dj once a shape is assumed by far the most common choice is the sphere. For spheres the Stokes-Einstein equation is... 

Diffusion coefficients measured by the spin-echo technique provide a means of investigating the translational motion of molecules under the extremes of temperature and pressure. There have been numerous smdies of the self-diffusion coefficients of high-pressure liquids and supercritical fluids by NMR. As an illustration of the potential of these physicochemical measurements, we will focus on CO2 (3,28,33,38,39). The availability of a wide range of diffusion coefficients and viscosities allows one to test the Stokes-Einstein equation at the molecular level. From hydrodynamic theory,... 

For non-sphericd but symmetrical particles, there are two translational diffusion coefficients one parallel and one perpendicular to the symmetric axis However, only the average can be retrieved in most situations. This average translational diffusion coefficient Dt is related to both dimensions of the particle. It can generally he written in the same formula as the Stokes-Einstein equation ... 

The translational diffusion coefficient D (=DJ as a whole ((ji g< 1) for a sphere of radius R is expressed by the Stokes-Einstein equation ... 

As stated previously, additional information on the sample is obtainable from band broadening (plate height) measurements. If plate height is measured as a function of flow velocity at a fixed retention level, a linear relationship is obtained, as predicted by Equation 7. The slope of the line yields the diffusivity D of the sample, and the intercept provides the polydispersity a. . The D value translates into a value for the average particle diameter d via the Stokes-Einstein relationship... 

Thus, the combined SE and the DSE equations predict that the product Dtxc = (A Tc)sedse should equal 2r /9. Measurements of probe translational diffusion and rotational diffusion made in glass-formers have found that the product Dtr can be much larger than this value, revealing a breakdown of the Stokes-Einstein (SE) relation and the Debye-Stokes-Einstein (DSE) relation. There is an enhancement of probe translational diffusion in comparison with rotational diffusion. The time dependence of the probe rotational time correlation functions tit) is well-described by the KWW function,... 

The translational and rotational motion of a Brownian particle immersed in a fluid continuum is well described by the Stokes-Einstein and Debye equations, respectively. 

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