Brownian motion and diffusion

All molecules are in constant thermal motion and are never completely at rest (except at absolute zero, 0 K). This random molecular motion is what we call heat , and objects will feel hot or cold depending on whether their molecular motions are more or less agitated than the surroundings. 

We cannot sensibly predict the exact motion of all the molecules in an object, but we can estimate average properties quite exactly. For example, a useful rule of thumb is that the average thermal energy (ignoring quantum effects) is available degree of freedom, so that the 

This IS a majoi pfirl of tht-subjuci known as Molecukir Thermodynamics or Slalistica Mechanics see Furihci Reading. 

Q Calculate the average thermal velocity of a water molecule in air at room temperature. 

In 1827 a Scoilish boianisi, Robert Brown, described (he perpetual random motion ol small particles inside grains oi pollen. Although he was not the lirst to observe this motion, his careiul experiments with dead pollen, particles of soot and other inert microscopic objects showed - contrary to popular belief that the grains were alive - that it was a universal properly of microscopic particles. 

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S. Chandrasekhar, Revs. Modern Phys., 16, 1 (1943), has summarized work on the general problem of diffusion and Brownian motion. Also T. R. Waite, J. Ch m. Phys., 28, 103 (1958). 

We begin by examining the rate of collision of suspended spherical particles in a static fluid due to Brownian motion. This theory was first put forward by the great Polish physicist M. Von Smoluchowski, to whom we have often referred. In consequence of the equivalence between diffusion and Brownian motion, we consider the relative motion between the particles as a diffusion process. The particles are assumed to be in sufficiently dilute concentration that only binary encounters need be considered. To further simplify the calculation, we consider the suspension to be made up of only two different-sized spherical particles, one of radius a, and the other of radius... 

Appreciate the thermal basis for molecular diffusion and Brownian motion... 

Nitsche, J.M., and Wei, J., Window effects in zeolite diffusion and Brownian motion over potential barriers, AlChE J., 37(5), 661-670 (1991). 

The results here presented are valid for particles falling freely in air. The settling rate of particles smaller than 10 m in diameter is insignificant (order of a few cm/ sec). Such particles cannot fall vertically they are only able to come into contact with the surface as a result of inertial, diffusion, and Brownian motion. In this case, not only vertical surfaces, but also plates inclined at ce > 90° may become dusty. 

Gravitational sedimentation techniques have limited worth for particles smaller than about a micron due to the long settling times required. In addition most sedimentation devices suffer from the effect of convection, diffusion and Brownian motion. These difficulties may be reduced by speeding up the settling process by centrifuging the suspension. 

Another example is that of the self-diffusion and Brownian motion of chains in solution which results in a fluctuation of their concentration on a time scale of about 10 s. As mentioned in Chapter 6, quasi-elastic light scattering is the ideal technique to determine the diffusion coefficient D from the self-correlation function of the scattered intensity. 

Plotting U as a function of L (or equivalently, to the end-to-end distance r of the modeled coil) permits us to predict the coil stretching behavior at different values of the parameter et, where t is the relaxation time of the dumbbell (Fig. 10). When et < 0.15, the only minimum in the potential curve is at r = 0 and all the dumbbell configurations are in the coil state. As et increases (to 0.20 in the Fig. 10), a second minimum appears which corresponds to a stretched state. Since the potential barrier (AU) between the two minima can be large compared to kBT, coiled molecules require a very long time, to the order of t exp (AU/kBT), to diffuse by Brownian motion over the barrier to the stretched state at any stage, there will be a distribution of long-lived metastable states with different chain conformations. With further increases in et, the second minimum deepens. The barrier decreases then disappears at et = 0.5. At this critical strain rate denoted by ecs, the transition from the coiled to the stretched state should occur instantaneously. 

Translational motion is the change in location of the entire molecule in three-dimensional space. Figure 11 illustrates the translational motion of a few water molecules. Translational motion is also referred to as self-diffusion or Brownian motion. Translational diffusion of a molecule can be described by a random walk, in which x is the net distance traveled by the molecule in time At (Figure 12). The mean-square displacement (x2) covered by a molecule in a given direction follows the Einstein-derived relationship (Eisenberg and Crothers, 1979) ... 

Since the random-walk approach is successful in molecular diffusion (K5) and Brownian motion studies (C14), it would seem that it might also be useful for the dispersion process. This has been considered by Baron (B2), Ranz (Rl), Reran (B5), Scheidegger (S6), Latinen (L4) and more recently by de Josselin de Jong (D14) and Salfman (SI, S2, S3). The latter two did not strictly use random-walk since a completely random process was not assumed. Methods based on statistical mechanics have been proposed by Evans et al. (E7), Prager (P8), and Scheidegger (S7). 

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... 

Diffusion can be considered as a stochastic or random process and described by the so-called Fokker-Planck equation adapted to Brownian motion. This equation is also known as the Smoluchowski equation. We consider the description of stochastic processes and Brownian motion in more detail in Section 11.1 and Appendix H. 

The foregoing clearly indicates that striation thickness is inversely proportional to total strain. We also note that the initial striation thickness is proportional to the size of the cube and inversely proportional to the volume fraction of the minor component. Hence, for any required final striation thickness, the larger the particles and the smaller the volume fraction of the minor component the more total strain is required. Therefore it is more difficult to mix a small amount of minor components into a major, than to make a 50-50 mixture, and the larger the individual particles of the minor, the more difficult it is to mix. By using Eq. E7.1-22, we can estimate the strain needed to reduce the striation thickness to a level where molecular diffusion or Brownian motion will randomize the mixture for a given strain rate and within the time (residence time) allotted for mixing. 

When aerosols are in a flow configuration, diffusion by Brownian motion can take place, causing deposition to surfaces, independent of inertial forces. The rate of deposition depends on the flow rate, the particle diffusivity, the gradient in particle concentration, and the geometry of the collecting obstacle. The diffusion processes are the key to the effectiveness of gas filters, as we shall see later. 

The process by which these particles migrate, either to a surface or to one another, is called diffusion, and their motion is described as brownian motion. Diffusion is important in aerosol studies because it represents the major dynamic effect acting on very small particles (d < 0.1 xm) and must be considered when the dynamics of these small particles are studied. 

The theories in this paper are first-principles statistical mechanics theories used to calculate static thermodynamic and molecular ordering properties (including solubilities of LCPs in various kinds of solvents) and dynamic properties (diffusion from Brownian motion). The diffusion of the LCP molecules constitutes a lower limit for the speed of processing of the LCPs. The static theory is used to calculate the packing of the bulky relatively rigid side chains of SS LCPs these calculations indicate that head-to-tail polymerization of the monomers of these SS LCPs will be very strongly favored. The intermolecular energies and forces calculated from the static theory are used in the dynamic theory. 

There is a striking contrast between our surface collision model in Ref. 7 and the formulas e.g. Ref. 22) used to relate t2 to the three-dimensional transport of spin label molecules. In the three-dimensional model, the spin labels are diffused by Brownian motion, in which case (18)... 

Fuoss and Kirkwood32 have obtained equations identical with Eqs. 31 and 32 without introducing, explicitly, the exponential decay function. Like Debye they reasoned as if the problem were mainly one of diffusion by Brownian motion under the influence of an external force. Treating this problem as a Sturm-Liouville equation, they developed /lt into a complete set of orthogonal functions yx. A relaxation time rx is associated with each of these functions. 

Diffusion Times. Brownian motion of molecules and particles is discussed in Section 5.2. The root-mean-square displacement of a particle is inversely proportional to the square root of its diameter. Examples are given in Table 9.4. The diffusion time for heat or matter into or out of a particle of diameter d is of the order of d2/ ()D where D is the diffusion coefficient. All this means that the length scale of a structural element, and the time scale needed for events to occur with or in such a structural element, generally are correlated. Such correlations are positive, but mostly not linear. 

PCS measures the diffusion coefficient of particles in the size range between 3 nm and a few micrometres. Particle size measurements for particles and/or aggregates smaller than 1 pm were performed on a Malvern Photon Correlation Spectrometer (PCS) Autosizer 4700 (633 nm, 5 mW, He-Ne laser). It is essentuial to use a red laser due to the fluorescence spectra of the humic substances (Goldberg and Weiner (1989)). A round quartz cell was used and temperature adjusted to 25 C. The method measures the diffusion coefficient (Brownian motion) of particles and is limited to about 3 nm... 

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