Einstein-Smoluchowski diffusion

The Einstein-Smoluchowski equation, = 2Dt, gives a measure of the mean-square displacements of a diffusing particle in a time t. There is the mean-square distance traveled by most of the ions. Common observation using dyes or scents shows that diffusion of some particles occurs far ahead of the diffusion front represented by the = 2Dt equation. Determine the distance of this Einstein-Smoluchowski diffusion front for a colored ion diffusing into a solution for 24 hr (D = 3.8 x 10 cm s ). Determine for the same solution how far the farthest 1% of the total diffused material diffused in the same time. Discuss how it is possible that one detects perfume across the space of a room in (say) 30 s. 

More recently an alternative explanation of the complex excimer behaviour observed in polymer systems has been proposed whereby the close proximity of some fluorophores leads to a time dependent rate of quenching analogous to that predicted by the Einstein-Smoluchowski diffusion theory for low molecular weight systems. This predicts a fluorescence response function of the form... 

Perrin model and the Johansson and Elvingston model fall above the experimental data. Also shown in this figure is the prediction from the Stokes-Einstein-Smoluchowski expression, whereby the Stokes-Einstein expression is modified with the inclusion of the Ein-stein-Smoluchowski expression for the effect of solute on viscosity. Penke et al. [290] found that the Mackie-Meares equation fit the water diffusion data however, upon consideration of water interactions with the polymer gel, through measurements of longitudinal relaxation, adsorption interactions incorporated within the volume averaging theory also well described the experimental results. The volume averaging theory had the advantage that it could describe the effect of Bis on the relaxation within the same framework as the description of the diffusion coefficient. 

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 Einstein-Smoluchowski equation, derived in Appendix 4.1, relates the mean thermal displacement, X, to the diffusion coefficient and mean lifetime. For a surface ... 

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).

The relation between length and time scales of diffusion, calculated from the Einstein-Smoluchowski law (Eq. 18-8), are shown in Fig. 18.11 for diffusivities between 10 10 cm2s 1 (helium in solid KC1) and 108 cm2s (horizontal turbulent diffusion in the atmosphere). Note that the relevant time scales extend from less than a millisecond to more than a million years while the spatial scales vary between 1 micrometer and a hundred kilometers. The fact that all these situations can be described by the same gradient-flux law (Eq. 18-6) demonstrates the great power of this concept. 

Does this sound familiar Yes, it should remind us of the Einstein-Smoluchowski law, Eq. 18-8. In fact, the analogy suggests that dispersion can be described by the same mathematical formalism as diffusion, that is, by the first and second Fickian laws (Eqs. 18-6 ad 18-14). We just have to replace diffusivity D by the dispersion... 

In electrochemistry several equations are used that bear Einsteins name [viii-ix]. The relationship between electric mobility and diffusion coefficient is called Einstein relation. The relation between conductivity and diffusion coefficient is called - Nernst-Einstein equation. The expression concerns the relation between the diffusion coefficient and the viscosity and is known as the - Stokes-Einstein equation. The expression that shows the proportionality of the mean square distance of the random movements of a species to the diffusion coefficient and the duration of time is called - Einstein-Smoluchowski equation. A relationship between the relative viscosity of suspension and the volume fraction occupied by the suspended particles - which was derived by Einstein - is also called Einstein equation [ix]. 

Einstein-Smoluchowski equation — Relationship between diffusion coefficient D, average width of a jump A of a microscopic species (atom, ion, molecule) involved in diffusion, and average time r between two jumps... 

The relaxation process may be accompanied by diffusion. Consequently, the mean relaxation time for such kinds of disordered systems is the time during which the relaxing microscopic structural unit would move a distance R. The Einstein-Smoluchowski theory [226,235] gives the relationship between x and R as... 

It is possible to calculate diffusion coefficients by computing the mean square displacement distance and dividing by 6t. [The basic relation here is the Einstein-Smoluchowski equation (Section 4.2.6)]. The values are surprisingly good and are shown in Table 2.26. 

The diffusion coefficient D has appeared in both the macroscopic (Section 4.2.2) and the atomistic (Section 4.2.6) views of diffusion. How does the diffusion coefficient depend on the structure of the medium and the interatomic forces that operate To answer this question, one should have a deeper understanding of this coefficient than that provided hy the empirical first law of Tick, in which D appeared simply as the proportionality constant relating the flux / and the concentration gradient dc/dx. Even the random-walk intapretation of the diffusion coefficient as embodied in the Einstein-Smoluchowski equation (4.27) is not fundamental enough because it is based on the mean square distance traversed by the ion after N stqis taken in a time t and does not probe into the laws governing each stq) taken by the random-walking ion. 

At first the results arising from the Einstein-Smoluchowski equation ( = 2Dt) may seem difficult to understand. Thus, the diffusion considered in the equation is random. Nevertheless, the equation tells us that there is net movement in one direction arising from this random motion. Furthermore, it allows us to calculate how far the diffusion front has traveled. Is there something curious about randomly moving particles covering distances in one direction Comment constructively on this apparently anomalous situation. 

The Einstein-Smoluchowski equation, = 2Dt, is a phenomenological equation derived for diffusion along one coordinate. (For example, after the release of a barrier, along a tube containing a liquid.) However, it also applies to any medium. Suppose, now, that metal ions, (e.g., Pt) are deposited on a Pd substrate. Calculate how far the Pt would diffuse into the Pd in 6 weeks. (The diffusion coefficient of Pt into Pd can be estimated from other data as 9 x 10- at295 K.)... 

Self-diffusion is the random translational motion of ensembles of particles (molecules or ions) originating from their thermal energy. It is well known that diffusion, which is closely related to the molecular size of the diffusing species, is given by the Einstein-Smoluchowski equation, Eq. (6.1) [8] ... 

Fig.V-4. A schematic representation of diffusion in the derivation of the Einstein-Smoluchowski equation...

Brownian motion theory was verified by many scientists (T. Svedberg, A. Westgren, J.Perrin, L.de Broglie and others), who both observed individual particles and followed the diffusion in disperse systems [5]. The influence of various factors, such as the temperature, dispersion medium viscosity, and particle size on the value of the Brownian displacement, was evaluated. It was shown that the Einstein-Smoluchowski theory describes the experimental data adequately and with high precision. 

E21.30(b) The Einstein-Smoluchowski equation related the diffusion constant to the unit jump distance and lime... 

The relative importance of migration and diffusion can be gauged by comparing Ud with the steady-state migrational velocity, u, for an ion of mobility Wj in an electric field (Section 2.3.3). By definition, v = where % is the electric field strength felt by the ion. From the Einstein-Smoluchowski equation, (4.2.2),... 

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