Diffusion, Einstein

Let us first consider isothermal diffusion. Einstein s diffusion law can be written... 

For the sake of discussions in a later section we will summarize here the theoretical equations of two other important hydrodynamic properties, the translational and rotary frictional coefficients. The former, designated as / can be determined from either sedimentation (Svedberg and Pedersen, 1940) or diffusion (Einstein, 1905, 1906b Smoluchowski, 1906) measurements ... 

For diffusion in gases and colloidal systems, concentration is the origin (or driving force ) for diffusion. Einstein first showed in 1905 that from the concentration gradient, the diffusivity is (Kauzmann, 1966)... 

The Brownian motion of very small particles in a liquid is caused by collisions with the liquid molecules, l is mechanism can be described with an effective diffusivity. Einstein formulated the following equation for the diffusivity 0 of small P-particles with diameter d in a liquid with viscosity p. ... 

There is also a traffic between the surface region and the adjacent layers of liquid. For most liquids, diffusion coefficients at room temperature are on the order of 10 cm /sec, and the diffusion coefficient is related to the time r for a net displacement jc by an equation due to Einstein ... 

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... 

Reference 115 gives the diffusion coefficient of DTAB (dodecyltrimethylammo-nium bromide) as 1.07 x 10" cm /sec. Estimate the micelle radius (use the Einstein equation relating diffusion coefficient and friction factor and the Stokes equation for the friction factor of a sphere) and compare with the value given in the reference. Estimate also the number of monomer units in the micelle. Assume 25°C. 

Here, y is the friction coefficient of the solvent, in units of ps and Rj is the random force imparted to the solute atoms by the solvent. The friction coefficient is related to the diffusion constant D of the solvent by Einstein s relation y = kgT/mD. The random force is calculated as a random number, taken from a Gaussian distribu-... 

For particles of any shape at an absolute temperature T, Einstein showed that f is related to the experimental diffusion coefficient D by the expression... 

Figure 5 relates N j to collection efficiency particle diffusivity from Stokes-Einstein equation assumes Brownian motion same order of magnitude or greater than mean free path of gas molecules (0.1 pm at... 

Supercritical Mixtures Dehenedetti-Reid showed that conven-tionaf correlations based on the Stokes-Einstein relation (for hquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group D g l/T was constant. Thus, although no general correlation ap es, only one data point is necessaiy to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SFg and COg. 

Wilke-Chang This correlation for D°b is one of the most widely used, and it is an empirical modification of the Stokes-Einstein equation. It is not very accurate, however, for water as the solute. Otherwise, it apphes to diffusion of very dilute A in B. The average absolute error for 251 different systems is about 10 percent. ( )b is an association factor of solvent B that accounts for hydrogen bonding. 

The Stokes-Einstein equation has already been presented. It was noted that its vahdity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to Stokes-Einstein for molecules approximating spheroids is given by Tanford. Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. 

The oxide solid elecU olytes have elecuical conductivities ranging from lO Q cm to 10 cm at 1000°C and these can be converted into diffusion coefficient data, D, for die oxygen ions by the use of the Nernst-Einstein relation... 

The Einstein flux equation for surface diffusion in this situation is... 

A number of metals, such as copper, cobalt and h on, form a number of oxide layers during oxidation in air. Providing that interfacial thermodynamic equilibrium exists at the boundaries between the various oxide layers, the relative thicknesses of the oxides will depend on die relative diffusion coefficients of the mobile species as well as the oxygen potential gradients across each oxide layer. The flux of ions and electrons is given by Einstein s mobility equation for each diffusing species in each layer... 

The analysis of oxidation processes to which diffusion control and interfacial equilibrium applied has been analysed by Wagner (1933) who used the Einstein mobility equation as a starting point. To describe the oxidation for example of nickel to the monoxide NiO, consideration must be given to tire respective fluxes of cations, anions and positive holes. These fluxes must be balanced to preserve local electroneutrality tliroughout the growing oxide. The flux equation for each species includes a term due to a chemical potential gradient plus a term due to the elecuic potential gradient... 

In connection with the earlier consideration of diffusion in liquids using tire Stokes-Einstein equation, it can be concluded that the temperature dependence of the diffusion coefficient on the temperature should be T(exp(—Qvis/RT)) according to this equation, if the activation energy for viscous flow is included. 

Using the Stokes-Einstein equation for the viscosity, which is unexpectedly useful for a range of liquids as an approximate relation between diffusion and viscosity, explains a resulting empirical expression for the rate of formation of nuclei of the critical size for metals... 

Center-of-mass translational motion in MD simulations is often quantified in tenns of diffusion constants, D, computed from the Einstein relation. 

Analysis of neutron data in terms of models that include lipid center-of-mass diffusion in a cylinder has led to estimates of the amplitudes of the lateral and out-of-plane motion and their corresponding diffusion constants. It is important to keep in mind that these diffusion constants are not derived from a Brownian dynamics model and are therefore not comparable to diffusion constants computed from simulations via the Einstein relation. Our comparison in the previous section of the Lorentzian line widths from simulation and neutron data has provided a direct, model-independent assessment of the integrity of the time scales of the dynamic processes predicted by the simulation. We estimate the amplimdes within the cylindrical diffusion model, i.e., the length (twice the out-of-plane amplitude) L and the radius (in-plane amplitude) R of the cylinder, respectively, as follows ... 

By equating Fiek s seeond law and the Stokes-Einstein equation for diffusivity, Smoluehowski (1916,1917) showed that the eollision frequeney faetor takes the form... 

FIG. 2 Mean-square displacement (MSD) of helium atoms dissolved in polyisobutylene. There is a regime of anomalous diffusion (MSD a followed by a crossover at 100 ps to normal (Einstein) diffusion (MSD a r) [24],... 

Applying Einstein s formula, which links with the diffusion coefficient... 

N. C. Bartelt, T. L. Einstein, E. D. Williams. Measuring surface mass diffusion coefficients by observing step fluctuation. Surf Sci 572 411, 1994. 

Very commonly Eq. (4-5) is combined with Eq. (4-6), the Stokes-Einstein equation relating the diffusion coefficient to the viscosity -q. 

Equations (4-5) and (4-7) are alternative expressions for the estimation of the diffusion-limited rate constant, but these equations are not equivalent, because Eq. (4-7) includes the assumption that the Stokes-Einstein equation is applicable. Olea and Thomas" measured the kinetics of quenching of pyrene fluorescence in several solvents and also measured diffusion coefficients. The diffusion coefficients did not vary as t) [as predicted by Eq. (4-6)], but roughly as Tf. Thus Eq. (4-7) is not valid, in this system, whereas Eq. (4-5), used with the experimentally measured diffusion coefficients, gave reasonable agreement with measured rate constants. 

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