Brownian diffusion of particles

G.K. Batchelor, Brownian diffusion of particles with hydrodynamic interaction, J. Fluid Mech. 74 (1976) 1-29. 

Transport of particles by Brownian diffusion depends on the Schmidt number, Sc = v/D, where D is now the Brownian diffusivity of particles, given by (9.73) ... 

Thus the coefficient of Brownian diffusion of particles with small volume concentration W, suspended in a liquid that is at rest or undergoing translational motion with a constant velocity, has a constant value and is identical in all directions. 

Brownian diffusion (Brownian motion) The diffusion of particles due to the erratic random movement of microscopic particles in a disperse phase, such as smoke particles in air. 

The hydrodynamic drag experienced by the diffusing molecule is caused by interactions with the surrounding fluid and the surfaces of the gel fibers. This effect is expected to be significant for large and medium-size molecules. Einstein [108] used arguments from the random Brownian motion of particles to find that the diffusion coefficient for a single molecule in a fluid is proportional to the temperature and inversely proportional to the frictional coefficient by... 

The velocity, viscosity, density, and channel-height values are all similar to UF, but the diffusivity of large particles (MF) is orders-of-magnitude lower than the diffusivity of macromolecules (UF). It is thus quite surprising to find the fluxes of cross-flow MF processes to be similar to, and often higher than, UF fluxes. Two primary theories for the enhanced diffusion of particles in a shear field, the inertial-lift theory and the shear-induced theory, are explained by Davis [in Ho and Sirkar (eds.), op. cit., pp. 480-505], and Belfort, Davis, and Zydney [/. Membrane. Sci., 96, 1-58 (1994)]. While not clear-cut, shear-induced diffusion is quite large compared to Brownian diffusion except for those cases with very small particles or very low cross-flow velocity. The enhancement of mass transfer in turbulent-flow microfiltration, a major effect, remains completely empirical. 

Particulate diffusion does not play a significant role in the deposition of pharmaceutical aerosols. However, it is worth noting the mechanism by which diffusion of particles occurs in the lungs. The principle of Brownian motion is responsible for particle deposition under the influence of impaction with gas molecules in the airways. The amplitude of particle displacement is given by the following equation ... 

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

For dilute suspensions, particle-particle interactions can be neglected. The extent of transfer of particles by the gradient in the particle phase density or volume fraction of particles is proportional to the diffusivity of particles Dp. Here Dp accounts for the random motion of particles in the flow field induced by various factors, including the diffusivity of the fluid whether laminar or turbulent, the wake of the particles in their relative motion to the fluid, the Brownian motion of particles, the particle-wall interaction, and the perturbation of the flow field by the particles. 

The simple form in Eq. (7) can be maintained by replacing the Brownian diffusion coefficient in the expression kc = /-An /5 by the shear-induced hydrodynamic diffusion coefficient for the particles, Ds. Shear-induced hydrodynamic diffusion of particles is driven by random displacements from the streamlines in a shear flow as the particles interact with each other. For particle volume fractions between 20 and 45%, Ds has been related to... 

Particle capture occurs through an interception mechanism. Because of the strong electrostatic forces operating in the experimental system, the contribution of Brownian diffusion to particle capture is negligible. 

The particles of a ferrofluid rotate very rapidly in response to an imposed field or after the field is turned off. The rotary Brownian diffusivity of spherical particles is given by Eq. 

B(n .nbn chatigc 10 pf duc to Brownian diffusion of the aerosol integrated over all particle sizes... 

At one extreme, a suspension may be considered dilute if the thermal motion (Brownian diffusion) of the particles predominate over the imposed interparticle interaction [30-32]. In this case, the particle translational motion is large and only occasional contacts will occur between the particles that is, the particles do not see each other until a collision occurs, giving a random arrangement of particles. 

In practice, however there could be differences between the observed and estimated flux. The mass transfer coefficient is strongly dependent on diffusion coefficient and boundary layer thickness. Under turbulent flow conditions particle shear effects induce hydrodynamic diffusion of particles. Thus, for microfiltration, shear-induced difflisivity values correlate better with the observed filtration rates compared to Brownian difflisivity calculations.Further, concentration polarization effeets are more reliably predicted for MF than UF due to the fact diat macrosolutes diffusivities in gels are much lower than the Brownian difflisivity of micron-sized particles. As a result, the predicted flux for ultrafiltration is much lower than observed, whereas observed flux for microfilters may be eloser to the predicted value. 

The description of a number of meteorological phenomena is also based on the study of Brownian diffusion of aerosols to single solid and liquid particles. The increasing atmospheric pollution is a problem that requires understanding and description of the processes of atmospheric self-purification of chemical and mechanical pollutants and radioactive contaminants. The problem of settling aerosol particles on various collectors also arises in the analysis of filter efficiency. 

The electrostatic interaction between diffuse layers of ions surrounding particles is one of the most thoroughly theoretically developed factors of colloid stability. The theory of electrostatic factor is, essentially, the basis for the quantitative description of coagulation by electrolytes. This theory was developed in the Soviet Union by B.V. Derjaguin and L.D. Landau in 1935 -1941, and independently by the Dutch scientists E.Verwey and T. Overbeek, and is presently known by the initial letters of their names as the DL VO theory [44,45]. The DLVO theory is based on comparison of molecular interaction between the dispersed particles in dispersion medium and the electrostatic interaction between diffuse layers of ions, with Brownian motion of particles taken into account (in the simplest version of theory this is done on a qualitative level). 

Because of its large velocity, a freely rising bubble has a diffusion layer much thinner that in the described experiments. This effect can manifest itself only if the particles are small enough so that their thermal motion becomes significant. Thus, electro- and diffusiophoresis should be taken into account in describing the Brownian diffusion of sub-micron particles towards the bubble s mobile surface under the conditions of a sufficiently low electrolyte concentration. The influence of diffusiophoretic transport to the surface of a rising bubble through its diffusion layer is theoretically proved by Zholkovsky et al. (1983). 

If we relate the Brownian diffusivity D to the mean square displacements given by (9.66), then (9.67) can provide a convenient framework for describing aerosol diffusion. To do so, let us repeat the experiment above, namely, let us follow the Brownian diffusion of N0 particles placed at t = 0 at the y — z plane. To simplify our discussion we assume that N does not depend on y or z. Multiplying (9.67) by x2 and integrating the resulting... 

C At t = 0 a uniform concentration No of monodisperse particles exists between two horizontal plates separated by a distance h. Assuming that both plates are perfect absorbers of particles and the particles settle with a settling velocity vt, determine the number concentration of particles as a function of time and position. The Brownian diffusivity of the particles is D. 

Our analysis so far includes two major assumptions, that our particle is stationary and that all particles have the same radius. Let us relax these two assumptions by allowing our particle to undergo Brownian diffusion and also let it have a radius Rpi and the others in the fluid have radii Rp2. Our first challenge as we want to maintain the diffusion framework of (13.31) is to calculate the diffusion coefficient that characterizes the diffusion of particles of radius Rp2 relative to those of radius Rp. ... 

Characterizing the porous bed by means of a capillary model of the interstitial space, the physical basis of the size separation procedure can be demonstrated through examination of the convection and Brownian diffusion of the colloidal particles in a liquid flow through a circular capillary. Figure 5.7.1 shows two freely-rotating spherical Brownian particles of different size sampling a nonuniform Poiseuille flow. The center of the larger particle in its travel... 

Particles are transported across the quasi-laminar sublayer by Brownian motion analogous to gaseous molecular diffusion. The dependence of the particle Brownian diffusivity on particle size results in a transfer rate that depends on particle size (see (8.73)). Transfer is rapid, and hence resistance is low, for the very smallest particles. As particle size increases, the Brownian diffusivity decreases and transfer is less rapid (see Figure 8.8). The... 

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... 

A method for measuring the size of aggregates in aqueous environments is dynamic light scattering (DLS). This technique uses scattered light to measure diffusion rates (Brownian motion) of particles in stable suspensions to determine a size based on the Stokes-Einstein equation ... 

In 1935, Findeisen published his Uber das Absetzen kleiner in der Luft sus-pendierter Teilchen in der mensliche Lunge bei der Atmung, which was the first attempt to calculate the deposition of aerosols (3). The Findeisen model included four mechanisms for deposition of particles. These were (1) impaction, (2) sedimentation, (3) Brownian diffusion, and (4) interception. This review focuses only on the first two mechanisms proposed by Findeisen, because Brownian diffusion affects particles <1 jm these are nsnally too small for therapeutic purposes and interception is normally insignificant except for elongated particles such as fibers. For more information abont diffusion and interception, we refer the reader to Chap. 2. 

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