Einstein-Smoluchowski relation

Diffusion occurs when there is a concentration gradient of one kind of molecule within a fluid. In terms of random walk model, the average distance, x, after an elapsed time, t, between molecule collisions in a diffusion movement is characterized by the Einstein-Smoluchowski relation,... 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

Figure 18.11 Diffusion distance, L, vs. diffusion time, t, for typical diffusivities calculated from the Einstein-Smoluchowski relation L = (2Dt)m, Eq. 18-8. The following diffusivities, D, are used (values in cm2s ) He in solid KC1 at 25°C KT10 molecular in water 1 O 5 molecular in air KT1 vertical (turbulent) in ocean 10° vertical (turbulent) in atmosphere 105 horizontal (turbulent) in ocean 106 to 108. Values adapted from Lerman (1979).

Fig. 4.14. Schematic diagram for the derivation of the Einstein-Smoluchowski relation, showing the transit plane Tin between and at a distance V from the left L and right fl planes. The concentrations in the left and right compartments are and respectively.

The dilemma may be resolved as follows. If in the Einstein-Smoluchowski relation pertains to the mean square distance traversed by a majority of the radioactive particles and if Geiger counters can—as is the case—detect a very small number of... 

The actual mechanism by which the ions constituting the ionic atmosphere are dispersed is none other than the random-walk process described in Section 4.2. Hence, the time taken by the ionic cloud to relax or disperse may be estimated by the use of the Einstein-Smoluchowski relation (Section 4.2.6)... 

From the Einstein—Smoluchowsky relation, if d is the diameter of the tube, then... 

As demonstrated in previous sections, Eq. 2.27 is out of the direct use due to absence of the data on the activity coefficients y in HASP. Instead of Eq. 2.27, another equation that contains DC is more helpful, namely the Einstein-Smoluchowski relation (Eq. 2.28) for mean square distance of a freely randomly walking particle during time t ... 

The difficulties mentioned can be resolved to some extent by recognizing that Eq. 1 holds only in the limit of extreme dilution - a situation wherein solute/solute interactions are negligible and the diffusion and mobility relate through solvent properties. In this limit, statistical treatments combine with force balances to yield an Einstein-Smoluchowski relation [9,10] (for electrolytic diffusion this is also sometimes called the Nemst-Einstein equation [11])... 

The reptation model, like the Rouse model, supposes that the friction involved in dragging the chain through its tube is proportional to the chain length, 5 = N i, Equation (33.33). The diffusion constant Dtubo for the chain moving through the tube is given by the Einstein-Smoluchowski relation, Equa-... 

Steep concentration profiles arise if the diffusion coefficient is low or if the length of the pore is high, or more precisely if the Thiele modulus 4> exceeds a value of about 1. As a criterion for the development of distinct concentration gradients we may also state that the characteristic time for diffusion, td, which equals the term I 1(2 D) according to the Einstein-Smoluchowski relation (see boxes), should be at least of the order of magnitude of the reaction time Tr ... 

The kinetics of repassivation of small pits in an early stage of their development seem to be related to the transport of aggressive anions, such as the chloride acciunulated locally during the intense dissolution process from the pit to the bulk [4,30]. If this transport is the rate-determining step, one expects the repassivation time to increase with the depth of a corrosion pit. If we simply apply the Einstein-Smoluchowski relation for the transport time out of a pit of radius r [Eq. (30)] and if the radius r is given by the local current density / and the lifetime of the pit by Eq. (31), we obtain for the repassivation time [Eq. (32)]... 

Emig and Klemm [19] also used the Einstein-Smoluchowski relation to estimate the diffusion time in microchannels. Nevertheless, for the basic considerations they set, in contrast to Pippel [20], the geometrical coefficient equal to 1. This causes a doubling of the estimated diffusion time ((d) ... 

In addition, the microscopic motion can be modeled as a random walk with step length, and step time, r, through the Einstein-Smoluchowski relation ... 

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