Stokes-Einstein equation generalized

In general, diffusion coefficients in gases can be often be predicted accurately. Predictions of diffusion coefficients in liquids are also possible using the Stokes-Einstein equation or its empirical parallels. On the contrary in solids and polymers, models allow coefficients to be correlated but predictions are rarely possible. 

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

For instance, dielectric relaxation times usually are close to, and scale simply with, the shear and bulk viscosity relaxation times, and t . Since the Stokes-Einstein equation connecting diffusivity with shear viscosity is generally valid, this implies that the structural relaxation time for a system could be estimated if the diffusivity were known. Indeed for a number of simple polar molecules the product seems experimentally to be roughly constant at 2X 10 cm, implying that perturbations of a liquid of diffusivity 1 X 10 cm /sec would require of the order 10 psec to be relaxed to 1/eth of their initial value. 

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

Kholodenko, A.L. and Douglas, J.F., Generalized Stokes-Einstein equation for spherical-particle suspensions, Phys. Rev. E 51,1081-1090 (1995). 

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. 

General properties single-chain dynamics approximation by a generalized Stokes-Einstein equation... 

The primary motivation for early studies of probe diffusion was the relationship between Dp and rj. In simple fluids. Dp follows the Stokes-Einstein equation with Dp /r. If the Stokes-Einstein equation remained valid in polymer solutions. Dp would accurately track the solution viscosity, so that measuring Dp would be a replacement for classical rheological measurements. However, in polymer solutions, it is often found experimentally that Dpt depends strongly on c and M. Probe particles generally diffuse faster than expected from rj. Obvious experimental artifacts such as probe aggregation and polymer adsorption by the probes all lead to probes that diffuse too slowly, not too swiftly, and therefore cannot lead to the observed non-Stokes-Einsteinian behavior. 

A systematic study of Dm (c) and related quantities was made by Koch, etal. 9) using polystyrene dioxane solutions. The solution inverse osmotic compressibility dfl/dc was obtained with static light scattering. A generalized Stokes-Einstein equation,... 

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 Stokes-Einstein equation [77] is generally applicable for molecular liquids and ILs, at least semi-quantitatively, for analyzing the diffusion coefficient D. Water is ubiquitous in the environment. It was shown that water induced accelerated ion diffusion compared to its effect on neutral species in ionic liquids [78]. We also observed that IL conductivity is significantly increased when [Bmim][BF4] is... 

Pa s) versus distance. The diSusion coefficient and the force are related by the generalized Stokes-Einstein equation Dj = PaksT. Here, pa is the mobility of the particle defined as Vp/F, where F is the driving force causing a drSt with velocity Vp. 

The theory for diffusion coefficients in liquids is not very well developed. Since the general theory for the calculation of binary diffusion coefficients in the liquid phase is missing, semi-empirical equations are often used. These equations describe the diffusion of a dissolved component in a solvent for the conditions when the concentration of solute is much lower than the solvent concentration. The treatment of the diffusion of a molecule A in a solvent B is based on the Stokes-Einstein equation, which correctly describes the trends in liquid-phase diffusion. 

The characteristics of diffusion coefficients described in this chapter are summarized in Table 5.7-1. In general, diffusion coefficients in gases and in liquids can often be accurately estimated, but coefficients in solids and in polymers cannot. In gases, estimates based on the Chapman-Enskog kinetic theory are accurate to around ten percent. In liquids, estimates are based on the Stokes-Einstein equation or its empirical parallels. These estimates, accurate to around twenty percent, can be supplemented by a good supply of experimental data. In solids and polymers, theories allow coefficients to be correlated but rarely predicted. 

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

Wilke and Chang (1955) developed an empirical relationship that was based on the temperature and viscosity characterization of the Stokes-Einstein relationship. It deviates from the equivalent diameter characterization by using another parameter, and incorporates the size of the solvent molecule and a parameter for polarized solvents. It is the most generally used of the available equations (Lyman et al., 1990) and is given as... 

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