Mean squared distance of diffusion

Brownian motion refers to the random thermal diffusion of a particle immersed in a fluid of interest due to continual collisions between the particle and the fluid molecules that are in continual, random motion. Such an effect is increasingly prevalent as the characteristic length scale of the particle decreases, meaning micron-size, or smaller, particles immersed in an otherwise quiescent fluid will display measurable diffusion owing to Brownian motion. Einstein [3] was the first to quantify the intensity of this diffusion for a spherical particle subjected to Stokes drag law in terms of the mean-square distance of diffusion defined as... 

It is found that the relaxation parameter T p as a function of temperature does not follow an increase with chain length, as the square of the number of methylene carbons. Nor is it linear with N, the number of methylene carbons, which should be true if relaxation to the lattice were rate controlling. Rather, it shows a temperature-induced increase of the minimum value of Tjp with about the 1.6 of N. So, both spin diffusion and spin lattice coupling are reflected. For a spin diffusion coefficient D of approximately 2 x 10 12 cm.2/sec., the mean square distance for diffusion of spin energy in a time t is the ft1 = 200/T A, or about 15A on a Tjp time scale. 

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 rapid diffusibUity of NO has critically important imphcations for its chemistry in the biological setting. The speed with which NO moves by random diffusion can be illustrated by consideration of its root mean square distance of displacement, which describes the distance a single NO molecule will move in any time interval based on its diffusion constant D (which is similar for aqueous solution and also tissue (brain) ) ... 

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 ... 

From a well-known result of calculus, the definite integral on the right-hand side is s/n so M is just equal to the quantity of diffusing substance. The present solution is therefore applicable to the case where M grams (or moles) per unit surface is deposited on the plane x=x at t=0. In terms of concentration, the initial distribution is an impulse function (point source) centered at x=x which evolves with time towards a gaussian distribution with standard deviation JlQit (Figure 8. 13). Since the standard deviation is the square-root of the second moment, it is often stated that the mean squared distance traveled by the diffusion species is 22t. 

The period of time t required for a molecule with a diffusion coefficient D (units of cm s ) to diffuse a mean square distance x, as given by the expression... 

In the pulmonary region, air velocities are too low to impact particles small enough to reach that region, and the mechanisms of deposition are sedimentation and Brownian diffusion. The efficiency of both processes depends on the length of the respiratory cycle, which determines the stay time in the lung. If the cycle is 15 breaths/min, the stay time is of the order of a second. Table 7.1 shows the distance fallen in one second and the root mean square distance travelled by Brownian diffusion in one second by unit density particles (Fuchs, 1964). Sedimentation velocity is proportional to particle density, but Brownian motion is independent of density. Table 7.1 shows that sedimentation of unit density particles is more effective in causing deposition than Brownian diffusion when dp exceeds 1 pm, whereas the reverse is true if dp is less than 0.5 pm. For this reason, it is appropriate to use the aerodynamic diameter dA equal to pj dp when this exceeds 1 pm, but the actual diameter for submicrometre particles. 

A more sophisticated method which has found wide application in the study of intracrystalline diffusion in zeolites is the nuclear magnetic resonance (NMR) pulsed field gradient self-diffusion method. The method, which is limited to hydrocarbons and other sorbates with a sufficient density of unpaired nuclear spins, depends on measuring directly the mean square distance traveled by molecules,... 

Diffusion is the random motion of molecules. The distance traveled between two positions is proportional to the square root of time. Einstein showed that the average of the square of the distance taken between the first position and the final position via many stops is related to the diffusion coefficient. The larger mean-square distance, the larger the diffusion coefficient expected. It can be seen that the mean-square displacement distance is related to the diffusion coefficient as given by ... 

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. 

The microscopic view of diffusion starts with the movements of individual ions. Ions dart about haphazardly, executing a random walk. By an analysis of one-dimensional random walk, a simple law can be derived (see Section 4.2.6) for the mean square distance-cc traversedby an ionin atimef. This is theEinstein-Smoluchowski equation... 

The Stokes-Einstein relation proved extremely useful in the classical work of Perrin. Using an ultramicroscope, he watched the random walk of a colloidal particle, and from the mean square distance traveled in a time t, he obtained the diffusion coefficient D from the relation (4.27)... 

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. 

Next, Fig. 5 shows a typical center of mass trajectory for the case of a full chaotic internal phase space. The eyecatching new feature is that the motion is no more restricted to some bounded volume of phase space. The trajectory of the CM motion of the hydrogen atom in the plane perpendicular to the magnetic field now closely resembles the random motion of a Brownian particle. In fact, the underlying equation of motion at Eq. (35) for the CM motion is a Langevin-type equation without friction. The corresponding stochastic Lan-gevin force is replaced by our intrinsic chaotic force — e B x r). A main characteristic of random Brownian motion is the diffusion law, i.e. the linear dependence of the travelled mean-square distance on time. We have plotted in Fig. 6 for our case of a chaotic force for 500 CM trajectories the mean-square distance as a function of time. Within statistical accuracy the plot shows a linear dependence. The mean square distance

of the CM after time t, therefore, obeys the diffusion equation... 

Simulations—isoergic and isothermal, by molecular dynamics and Monte Carlo—as well as analytic theory have been used to study this process. The diagnostics that have been used include study of mean nearest interparticle distances, kinetic energy distributions, pair distribution functions, angular distribution functions, mean square displacements and diffusion coefficients, velocity autocorrelation functions and their Fourier transforms, caloric curves, and snapshots. From the simulations it seems that some clusters, such as Ar, 3 and Ar, 9, exhibit the double-valued equation of state and bimodal kinetic energy distributions characteristic of the phase change just described, but others do not. Another kind of behavior seems to occur with Arss, which exhibits a heterogeneous equilibrium, with part of the cluster liquid and part solid. 

The randomizaton takes place by a a diffusion of atoms that is implicit in our earlier description of the initial randomization process as being akin to melting [1]. Later it was shown that the root-mean-square displacement of each atom must be of the order of the nearest-neighbor distance in order that the network lose all memory of the original crystal structure as measured by the structure factor S q) [21]. In this context, the melting point can be defined as that temperature for which the mean square displacement increases linearly with time. It appears, though, that a sequence of bond switches as illustrated in Fig. 1 is not the primary mechanism for self-diffusion in silicon [31,32]... 

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