Einstein diffusion coefficient

Routh and Russel [10] proposed a dimensionless Peclet number to gauge the balance between the two dominant processes controlling the uniformity of drying of a colloidal dispersion layer evaporation of solvent from the air interface, which serves to concentrate particles at the surface, and particle diffusion which serves to equilibrate the concentration across the depth of the layer. The Peclet number, Pe is defined for a film of initial thickness H with an evaporation rate E (units of velocity) as HE/D0, where D0 = kBT/6jT ir- the Stokes-Einstein diffusion coefficient for the particles in the colloid. Here, r is the particle radius, p is the viscosity of the continuous phase, T is the absolute temperature and kB is the Boltzmann constant. When Pe 1, evaporation dominates and particles concentrate near the surface and a skin forms, Figure 2.3.5, lower left. Conversely, when Pe l, diffusion dominates and a more uniform distribution of particles is expected, Figure 2.3.5, upper left. 

Measurements of the free acid Ti vibrational lifetimes were also monitored as a function of base concentration (e.g., pyrrole with acetonitrile) to determine the effect of collisions and hydrogen-bond formation rates. Stern-Volmer plots of 1 /T1 rates versus base concentration enabled extraction of a bimolecular rate constant (kbm) for pyrrole acetonitrile of 2.5 0.2 x 1010 dm3/mol-s, which is slightly larger than the estimated Stokes-Einstein diffusion coefficient (0.73 x... 

Let us now briefly recall the main features of the behavior of D c °°(r) as a function of t for t > 0 [25]. First, the limiting value at infinite time of D ,c, 00(f) is, at any nonzero temperature, the usual Einstein diffusion coefficient kT/rp Above the crossover temperature Tc as defined above, ) e, 00(f) increases monotonously toward its limiting value. Below the crossover temperature (i.e., T < T( ).D"] (t) first increases, then passes through a maximum and finally slowly decreases toward its limiting value. Thus, in the quantum regime, I)" x(t) can exceed its stationary value, and the diffusive regime is only attained very slowly, namely after times t fth- At T - Q.If x(Vj can be expressed in terms of exponential integral functions ... 

The diffusion coefficients associated with translational motions when the radii of the diffusing radicals are not much larger than that of the solvent are expressed more accurately by D = kTI6nrr T (where r is the radius of the diffusing radical assuming a spherical shape and r (=yxr ) is the microviscosity. The value of /, the microfriction factor, can be calculated or taken equal to DsE/f exptb the ratio between the Stokes-Einstein diffusion coefficient (that considers van der Waals volumes, but not interstitial volumes) and the experimentally measured diffusion coefficient, Dexpti- As will be discussed later, these relationships appear to hold even in some polymer matrices. 

For low particle concentrations (unhindered diffusion of single particles) the diffusion coefficient D is given by the Stokes-Einstein diffusion coefficient... 

From the Taylor-Aris formulation for times t> a lD, where a is the capillary radius and D the Stokes-Einstein diffusion coefficient of the particle, the particle of radius will have had sufficient time to sample the full velocity profile. With the local particle velocity taken to be equal to that of the fluid (Eq. 4.2.14), the average particle velocity over the tube cross-section IJ is given by... 

Body force of component i Polymer mass concentration Macroscopic diffusion coefficient Cooperative diffusion coefficient Stokes-Einstein diffusion coefficient Rouse diffusion coefficient Self diffusion coefficient... 

Thus, Dcoop has the same structure as a simple Stokes-Einstein diffusion coefficient for a single blob. The essential feature is that >coop increases with concentration the restoring forces (due to the osmotic pressure gradient) ate stronger at high c. 

Notice that kTBj is the Nemst-Einstein diffusion coefficient D, which is the rate of swept area of cations [45]. Hence, the Nemst-Einstein equation is... 

In suspensions, it is common to consider particles whose sizes range down to the submicron scale, where Brownian motion and colloidal forces have pronounced effects (Russel et al. 1991). The influence of Brownian motion relative to shear flow is captured through a P clet number given by Pe = ( fl )/Do = (67tTioy )/fcT, where Do = kT/ 6ny Qa) is the Stokes-Einstein diffusion coefficient, k is Boltzmann s constant, and T is the temperature. The first form shows that Pe may be interpreted as a ratio of the hydrodynamic diffusion scaling with the shear rate and particle size ya ) as well as a dimensionless function of the volume fraction not shown. It is more common, however, to interpret Pe as the ratio of a diffusive timescale u IDq, relative to the flow timescale given by When Pe = 0, a Brownian suspension will approach a true equilibrium state through its thermal motions. Interparticle forces of many sorts are possible in a liquid medium. 

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. 

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... 

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... 

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... 

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. 

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. 

In a liquid that is in thermodynamic equilibrium and which contains only one chemical species, the particles are in translational motion due to thermal agitation. The term for this motion, which can be characterized as a random walk of the particles, is self-diffusion. It can be quantified by observing the molecular displacements of the single particles. The self-diffusion coefficient is introduced by the Einstein relationship... 

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]. 

T0 is a reference temperature which can be identified with T, and although the constant B is not related to any simple activation process, it has dimensions of energy. This form of the equation is derived by assuming an electrolyte to be fully dissociated in the solvent, so it can be related to the diffusion coefficient through the Stokes-Einstein equation. It suggests that thermal motion above T0 contributes to relaxation and transport processes and that... 

The same equation applies to other solvents. It is often easier to incorporate an expression for the diffusion coefficient than a numerical value, which may not be available. According to the Stokes-Einstein equation,6 diffusion coefficients can be estimated from the solvent viscosity by... 

---