Stokes-Einstein formula

This result is often called the Stokes-Einstein formula for the difflision of a Brownian particle, and the Stokes law friction coefficient 6iiq is used for... 

In this section, we consider the advection and diffusion of a univariafe NDF n(t, x, ), where R+ is a passive scalar (i.e. the velocity u t, x) does not depend on ). For example, could denote the particle mass for fine particles in a liquid with very small Stokes number. The diffusion coefficient, F(4, is assumed to be a function of For example, using the Stokes-Einstein formula for diffusivity reported in Eq. (5.116) on page 187, F( ) = Fq/, where is the particle mass. The PBE for this case reads... 

Moreover, employing the Stokes-Einstein formula, we can write f f r(f)ndf = Fom/i-i, which is essentially the form that will be used in the numerical examples in Section 8.3.4. In summary, we will consider two variations of the moment-transport equations in the numerical examples in Section 8.3.4. The first example will use a closed moment system wherein the diffusivity does not depend on f ... 

Since the Stokes-Einstein formula is valid for finite-size particles, the NDF should be nonzero only for fo < f For convenience, we will allow 0 < f < oo and add fo to the denominator of the Stokes-Einstein formula in order to ensure that the diffusivity remains finite with f = 0. 

This constitutes the generalization of the Stokes-Einstein formulas (316). It applies for any choice of origin. The positive-definite character of the resistance matrix assures the existence of its inverse, as required by the preceding relation. 

Assuming sufficient added electrolyte in the as5unptotic limit of small c, no distinction is made between Do, Dtr, and Dm. In this limit, inserting/o for a sphere of radius R into the expression for Do, the Stokes-Einstein formula is obtained ... 

The Stokes-Einstein formula for diffusion coefficients is limited to cases in which the solute is larger than the solvent. Predictions for liquids are not as accurate as for gases. The Wilke and Chang [16] correlation for diffusion in liquids is an empirical correlation and is given by... 

The same expression describes the relaxation time of energy fluctuations with the thermal diffusivity replacing the diffusion coefficient. For a spherical Brownian particle, the diffusion coefficient is given by the Stokes-Einstein formula ... 

We can obtain a direct measurement of the diffusion constant for the particles in solution by curve fitting the time correlation function and therefore obtain the particle radius R using the Stokes-Einstein formula ... 

This expression is known as the Stokes-Einstein equation. This formula correctly relates diffusivity to molecular dimensions and viscosity for cases in which Stokes law is applicable. 

Here we show how the modified Kubo formula (187) for p(co) leads to a relation between the (Laplace transformed) mean-square displacement and the z-dependent mobility (z denotes the Laplace variable). This out-of-equilibrium generalized Stokes-Einstein relation makes explicit use of the function (go) involved in the modified Kubo formula (187), a quantity which is not identical to the effective temperature 7,eff(co) however re T (co) can be deduced from this using the identity (189). Interestingly, this way of obtaining the effective temperature is completely general (i.e., it is not restricted to large times and small frequencies). It is therefore well adapted to the analysis of the experimental results [12]. 

Formula (206), together with the out-of-equilibrium generalized Stokes-Einstein relation (203), is the central result of the present section. 

The main results of this sectionjire the out-of-equilibrium generalized Stokes-Einstein relation (203) between Ax2(z) and p(z), together with the formula (206) linking Teff(ffi) and the quantity, denoted as ( ), involved in the Stokes-Einstein relation. One thus has at hand an efficient way of deducing the effective temperature from the experimental results [12]. Indeed, the present method, which avoids completely the use of correlation functions and makes use only of one-time quantities (via their Laplace transforms), is particularly well-suited to the interpretation of numerical data. 

Stokes formula for e Stokes-Einstein equation - Einsteins general... 

Chang (W8) based on the Stokes-Einstein equation. In this formula the association parameter.r allows for differences in solvent behavior . r = 2.6 for water, 1.9 for methanol, 1.5 for ethanol, and 1.0 for benzene, ether, heptane, and other unassociated solvents. The average error for systems surveyed by the authors was about 10% the relationship cannot be used when a complex is formed between solute and solvent. For amyl alcohol. 

The values of l) >,n — the diffusivity for the Brownian motion of aerosol — are calculated from the Stokes-Einstein equation. For spherical particulates with the effective radius rp, in a gas with the dynamic viscosity p2 (nearly constant for pressures about and less than one bar), the formula is ... 

In 1908 and subsequent years, J.B. Perrin (1923) reported consistent values of Avogadro s number based on the Stokes-Einstein equation and experiment. Perrin determined experimentally values of (r ) for different colloidal particle sizes, temperatures, and liquid solutions, and substituted the measured values into the formula... 

This formula follows from the Stokes-Einstein equation, the relation (r ) = 6Dt, and the definition = R/k. 

Stokes formula for a Stokes-Einstein equation - Einsteins general laminar flow of spheres the basis of dynamic light formula for difFusion scattering particle sisdng... 

The diffusion coefficient of natural organics is related to the size of the molecules. Worch (1993) published a formula to relate MW with size for organic molecules (see Eqn. (2.1)). The size can then be related to diffusion coefficient by using the Stokes Einstein equation (see Appendix 5). While this method assumes spherical shapes, it allows the estimation as equivalent spheres. 

As mentioned earlier, the first signature of the influence of the protein surface on the dynamics of water came from the measurements of the rotational and translational diffusion coefficients of water in aqueous protein solutions. Analysis based on hydrodynamic formulas (such as Stokes-Einstein and Debye-Stokes-Einstein (DSE)) showed that an explanation of the observed values required a larger than actual radius of the protein to be used in the Stokes expression of the friction (from hydrodynamics). This indicated the presence of a substantially rigid water layer around the protein surface. However, the story turned out to be more complex. We have already discussed some of these aspects - we now turn to a more detailed discussion of several experimental results. 

The same procedure can be applied to the cuiiiulaiits or moiiieiils f7(r). With Stokes-Einstein s formula, the -average hydrodynamic riidius of the inacromolecules is determined from the z-average diffusion coefficient... 

The distribution formula (124b) has been used in deriving the above formula. The validity of (150) was shown by Perrin who showed that the Boltzmann constant k could be obtained by relating (150) with the Stokes-Einstein equation... 

This formula works well for solute molecules that are significantly larger than the solvent (r2 1.0 A) and are roughly spherical, with deviations from measured values of 20%. It is also useful for estimating the diffusivity of colloids. The Eyring theoretical approach treats diffusion as a rate process where movement from one point to another requires transition through an activated state. It yields the same dependence on temperature and viscosity as the Stokes-Einstein result 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 question about the difference between the macroscopic and microscopic values of the quantities characterizing the translational mobility (viscosity tj, diffusion coefficient D, etc.) has often been discussed in the literature. Numerous data on the kinetics of spin exchange testify to the fact that, with the comparable sizes of various molecules of which the liquid is composed, the microscopic translational mobility of these molecules is satisfactorily described by the simple Einstein-Stokes diffusion model with the diffusion coefficient determined by the formula... 

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. ... 

In order to define this accessibility curve clearly it is necessary to use a number of solute molecules which range in size over the whole range of pore sizes anticipated in the swollen structure. We have found the most suitable solutes to be the dextrans marketed by Pharmacia (Uppsala) Ltd. supplemented by a few low molecular weight sugars. Grotte (7) has reviewed evidence to show that these dextrans behave in solution as hydro-dynamic spheres, and that the diameters of these molecules in solution may be calculated from their diffusion coefficients according to the Einstein-Stokes formula ... 

Importantly, Yoshizaki and Yamakawa [25] found that, in contrast to /, [77] of a wormlike cylinder undergoes significant end surface effects until the axial ratio p reaches about 50, on the basis of numerical solutions to the Navier-Stokes equation with the no-slip boundary condition for spheroid cylinders, spheres, and prolate and oblate ellipsoids of rotation. They constructed an empirical interpolation formula for [ y] of a spheroid cylinder which reduces to eq 2.37 for p > 1 and to the Einstein value at p = 1. Then, with its aid, Yamakawa and Yoshizaki [4] formulated a modified theory of [77] for wormlike cylinders which agrees with the Yamakawa-Fujii theory [3] for Lj lq > 2.278 and with the Einstein value at Ljd = 1, regardless of dj2q smaller than 0.1. However, no formulation has as yet been made for L/2q < 2.278 and d/2q > 0.1, i.e., for short flexible cylinders. In what follows, the Yamakawa-Yoshizaki modification is referred to as the Yamakawa-Fujii-Yoshizaki theoiy. 

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