Self Einstein equation

The dififiision time gives the same general picture. The bulk self-diffusion coefficient of copper is 10"" cm /sec at 725°C [12] the Einstein equation... 

Self-diffusion coefficients are dynamic properties that can be easily obtained by molecular dynamics simulation. The properties are obtained from mean-square displacement by the Einstein equation ... 

This equation, called the Kubo equation, is equivalent to the Einstein equation. However, it is easier to estimate self-diffusion coefficients from the slope of the mean-square displacements. 

Millet determined self-diffusion coefficients for Na and Cs+ ions in hydrated 1200 EW membranes using conductivity measurements and the Einstein equation, D+ = u+kT, where u+ is the absolute mobility of the given cation. u+ can be derived from the equivalent conductivity according to A = 0+IC+ = Fu+, where 0+ is the specific conductivity, C+ is the cation concentration (calculated on the basis of the dry membrane density, EW, and the water content), and F is the Faraday constant. The values of D+ determined via these conductivity measurements... 

The final section (Section 5.8) introduces dynamic light scattering with a particular focus on determination of diffusion coefficients (self-diffusion as well as mutual diffusion), particle size (using the Stokes-Einstein equation for the diffusion coefficient), and size distribution. 

What is the difference between the self-diffusion coefficient and the mutual diffusion coefficient Which of these two is described by the Stokes-Einstein equation ... 

Bueche et al. (33) determined chain dimensions indirectly, through measurements of the diffusion coefficient of C1 Magged polymers in concentrated solutions and melts.The self-diffusion coefficient is related to the molar frictional coefficient JVa 0 through the Einstein equation ... 

Diffusions NMR spectroscopy (e.g. PGSE = Pulsed Gradient Spin Echo STE = Stimulated Echo DOSY = Diffusion Ordered Spectroscopy) is a straightforward and accurate method for determination of the self-diffusion coefficient of a molecule. Its principal use in dendrimer chemistry is for size determination of dissolved dendrimers since the self-diffusion coefficient is directly correlated with the hydrodynamic radius of the molecule via the Stokes-Einstein equation [24]. Although one-dimensional and multidimensional diffusion NMR experiments can thus make an important contribution to structural characterisation of dendrimers, they have been used comparatively rarely until recently [25, 26]. 

Gottlieb MH and Sollner K (1968) Failure of the Nemst-Einstein equation to correlate electrical resistances and rates of ionic self-exchange across certain fixed charge membranes. Biophys J 8 515-35... 

The temperature dependencies of the viscosity (Figure 5.6) and the summation of the self-diffusion coefficient (Dcation + Oanion) (Figure 5.4) interestingly show the contrasted profiles with the indication of inverse relationship between viscosity and self-diffiision coefficient. This can be explained in terms of the Stokes-Einstein equation, which correlates the self-diffusion coefficient (Dcation Danion) with viscosity (q) by the following relationship ... 

The self-diffusion coefficients of CF and Na" in molten sodium chloride are, respectively, 33 x 10 exp(-8500// 7) and 8x10 exp(-4000// 7) cm s". (a) Use the Nernst-Einstein equation to calculate the equivalent conductivity of the molten liquid at 935°C. (b) Compare the value obtained with the value actually measured, 40% less. Insofar as the two values are significantly different, explain this by some kind of structural hypothesis. 

The determination of transport numbers in aqueous electrolytes is relatively easy (Chapter 3), but in molten salts it poses difficulties of concept, which in turn demand specialized apparatus. Explain why direct determination is difficult. Would it not be better to abandon the direct approach and use the approximate applicability of the Nernst-Einstein equation, relying on self-diffusion determinations Any counter considerations ... 

The limits of integration are the oxygen partial pressures maintained at the gas phase boundaries. Equation (10.10) has general validity for mixed conductors. To carry the derivation further, one needs to consider the defect chemistry of a specific material system. When electronic conductivity prevails, Eqs. (10.9) and (10.10) can be recast through the use of the Nemst-Einstein equation in a form that includes the oxygen self-diffusion coefficient Dg, which is accessible from ionic conductivity measurements. This is further exemplified for perovskite-type oxides in Section 10.6.4, assuming a vacancy diffusion mechcinism to hold in these materials. 

A precise determination of the frictional coefficient C in terms of the intermolecular potential and the radial distribution function at present constitutes the principal unresolved problem of the Brownian motion approach to liquid transport processes. It has been suggested by Kirkwood that an analysis of the molecular basis of self-diffusion might be a fruitful approach. The diffusion constant so calculated would be related to the frictional coefficient through the Einstein equation, Eq. 46. 

The theory presented here resolves itself into a generalization of the well-known Nemst-Einstein equation D = 22T/C>) to several components and optional concentration characteristics. The cases of two- and three-component mixtures are treated in detail. The latter case is also shown to be of interest in treating self diffusion in a binary mixture, a system which results from letting two components become diffusionally identical although still distinguishable. 

Electrical conductance of an ion-exchange membrane Xm in various ionic forms can be used for an approximate estimation of the ions self-diffusion coefficients Dj using the Nernst-Einstein equation ... 

The dendrimer size in solution can he obtained employing diffusion NMR spectroscopy [pulsed gradient spin-echo (PGSE), diffusion ordered spectroscopy (DOSY)] that allows the determination of the self-diffusion coefficient of a molecule, which is related to the hydrodynamic radius through the Stokes-Einstein equation. Using this technique, the flexible structure of dendrimers in solution was demonstrated in PAMAM derivatives, which swell or shrink with pH modification. This size variation in solution should be taken into account if interactions with nucleic acids are being studied. 

It is possible to estimate the self-diffusion coefficient at zero polymer concentration (iD2)a)i i) using the Stokes-Einstein equation as... 

Self-diffusion coefficients of distinct mobile species measured using PFG-NMR are based on spectral selectivity. In the context of IL selfdiffusion, PFG-NMR measures the time-averaged (miUisecond timescale) diffusion coefficients. Since ion-pair (cation-anion) interactions take place on a timescale faster than that, the measured diffusion coefficients are a weighted average over charged and neutral species. This is the reason for the variation between conductivity calculated from diffiasion coefficients determined by the Nemst-Einstein equation [15] (Eq. 1) and conductivity measured using impedance analysis ... 

The particle trajectories from the MD simulations can be used to determine the self-diffusion coefficient using the Einstein equation ... 

Since the response of PGSE-NMR (as well as PGSTE-NMR) measurements is insensitive to critical fluctuations, the diffusion data can profitably be used to detect the presence of miceUe-like aggregates in the water-rich region. According to the Stokes-Einstein equation (Eq. 1.2), the self-diffusion coefficient is expected to scale as the reciprocal of the viscosity ( > / ). However, as shown in Figure 1.2, this... 

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