Diffusion distance, Einstein-Smoluchowsky

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).

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]. 

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, 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. 

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

The accumulation of corrosion products within the pits suggest that a high concentration of chloride is a necessary condition for a stable growth in their early stage of development. As a consequence, the kinetics of repassivation of small pits may be related to the transport of accumulated aggressive anions from the pit to the bulk electrolyte [19, 29]. If this transport is the rate-determining step, one expects the repassivation time to increase with the depth of a corrosion pit and thus to the distance the chloride has to travel by diffusion. If we simply apply the relation of Einstein-Smoluchowski for the transport time fr out of a pit of radius r (Eq. 14), and if the radius r is given by the local current density ic,p and the lifetime fp of the pit by Eq. (15), we obtain Eq. (16) for the repassivation time fr. 

Given the central role of the expulsion rate constant for micellar stability, formation, and dissociation, it is essential to determine the physical governing factors and functional form. Aniansson and Wall based their calculations [54] on a general diffusion in an external potential. In this approach, the diffusion coefficient, D(r) is dependent on the position, r, due to the potential V(r). In a sphero-symmetric system, we can imagine that the diffusion of a unimer only depends on the distance, r, from the origin and this problem can be summarized in a Einstein-Smoluchowski type equation ... 

The photoredox properties of porphyrin molecules at interfaces can be studied by time-resolved spectroscopic and photoelectrochemical techniques. A key difference between both approaches is that the latter only probes molecules located at distances from the interface not larger than the characteristic diffusion length of the excited state. According to the Einstein-Smoluchowski equation, the... 

The Peclet Number can be interpreted as the ratio between the transport time over the distance L by diffusion and by advection, respectively. Transport time by diffusion is expressed by the relation of Einstein and Smoluchowski (Eq. 18-8) ... 

Equation 6.33 states that the root-mean-square displacement is proportional to the square root of the number of jumps. For very large values of n, the net displacement of any one atom is extremely small compared to the total distance it travels. It turns out, that the diffusion coefficient is related to this root-mean-square displacement. It was shown independently by Albert Einstein (1879-1955) and Marian von Smoluchowski (1872-1917) that, for Brownian motion of small particles suspended in a liquid, the root-mean-square displacement, is equal to V(2Dt), where t is the time... 

The origin of both the diffusion and the migration is the random walk of ions. It was shown by Einstein and Smoluchowski that the diffusion coefficient is proportional to the mean square distance of the random movements of ions ... 

In order to apply the Smoluchowski equation (Equations (1.3), (2.1), (3.29)), we need values for the least distance of approach (rAn) and the diffusion coefficient (Dab)- The value of tab can be estimated from molecular volumes (Section 2.5.1.2). The diffusion coefficient can be determined by various methods, but experimental values are available only for a minority of the myriad possible situations. A common practice is to use the Stokes-Einstein relation (Section 1.2.3), which rests on the assumption that solute molecules in motion behave like macroscopic particles to which classical hydrodynamic theory can be applied. We shall first outline (a) the relation between the diffusion coefficient D and the mechanics of motion of particles in fluids, leading to the Stokes-Einstein equation relating D to solute size and solvent viscosity and (b) the direct experimental determination of D. We shall then (c) compare the results and note the reservations that are required in relying on the Stokes-Einstein estimates of D in various cases. 

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