Particle Brownian diffusion

Figure 1.10 shows the velocities of deposition to a smooth surface of attached and unattached decay products, with a range of possible values of D. The scales are m s-1 for the attached and mm s 1 for the unattached decay products, illustrating the effect of attachment on diffusivity. For a nucleus of unit density with diameter, dp, equal to 0.17/um, the sedimentation velocity is 2 jum s 1, and deposition by Brownian diffusion and by sedimentation to upwards-facing surfaces are of comparable efficiency. For smaller particles, Brownian diffusion is always more effective. 

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... 

A) Submicron particles, Brownian diffusion > gravity. (B) Coarse particles (> 1 pm) with uniform size. (C) Coarse particles with size distribution. 

In the absence of hydraulic or wind forces, the water becomes quiescenf but natural or free convection processes remains operative. Driven by bottom residing thermal or concentration gradients. Equations 12.14 and 12.15 may be used for estimating these low-end MTCs. The chemical diffusion coefficient in the porewaters of the upper sediment layer is the key to quantifying the sediment-side MTC. Use Archie s law. Equation 12.18, to correct the aqueous chemical molecular diffusivity for the presence of the bed material. Bed porosity is the key independent variable that determines the magnitude of the correction factor. See Table 12.7 for typical porosity values in sedimentary materials. Eor colloids in porewaters. Equation 12.18 applies as well. The Stokes-Einstein equation (Equation 12.19) is recommended and some reported particle Brownian diffusion coefficients appear in Tables 12.9 and 12.10. Under quasisteady-state conditions, Equation 12.23 is appropriate for estimating the bed-side MTCs. 

The tendency for particles to settle is opposed by tlieir Brownian diffusion. The number density distribution of particles as a function of height z will tend to an equilibrium distribution. At low concentration, where van T Ftoff s law applies, tire barometric height distribution is given by... 

Photon Correlation Spectroscopy. Photon correlation spectroscopy (pcs), also commonly referred to as quasi-elastic light scattering (qels) or dynamic light scattering (dls), is a technique in which the size of submicrometer particles dispersed in a Hquid medium is deduced from the random movement caused by Brownian diffusion motion. This technique has been used for a wide variety of materials (60—62). 

In view of the facts that three-dimensional coUoids are common and that Brownian motion and gravity nearly always operate on them and the dispersiag medium, a comparison of the effects of particle size on the distance over which a particle translationaUy diffuses and that over which it settles elucidates the coUoidal size range. The distances traversed ia 1 h by spherical particles with specific gravity 2.0, and suspended ia a fluid with specific gravity 1.0, each at 293 K, are given ia Table 1. The dashed lines are arbitrary boundaries between which the particles are usuaUy deemed coUoidal because the... 

Table 1. Effect of Spherical Particle Size on the Relative Brownian Diffusion and the Sedimentation Distances After 1 h in Water at 293 K...

If 0 fluid streomline passes within one particle rodius of the collecting body, o particle traveling olong the streamline will touch the body and may be collected without the influence of inertia or brownian diffusion. 

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. 

From the image sequences, information on the velocities of nano-particles can be extracted. The statistical effect of Brownian motion on the flowing speed of the mixed liquid is found small enough to be ignored as shown in Fig. 37 where most of the particles trajectories in the liquid are straight lines and parallel with the wall basically. Therefore, Brownian diffusive motion is ignorable. 

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. 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

It is very interesting to note that in this case the factor e2P arises in the escape time instead of the Kramers factor associated with the good possibility for the Brownian particles to diffuse back to the potential well from a flat part of the potential profile, resulting in strong increasing of the escape time from the well (see, e.g., Ref. 83). 

We consider the process of Brownian diffusion in a potential cp(x). The probability density of a Brownian particle is governed by the FPE (5.72) with delta-function initial condition. The moments of transition time are given by (5.1). 

In the general case, when arbitrary interaction profiles prevail, the particle deposition rate must be obtained by solving the complete transport equations. The first numerical solution of the complete convective diffusional transport equations, including London-van der Waals attraction, gravity, Brownian diffusion and the complete hydrodynamical interactions, was obtained for a spherical collector [89]. Soon after, numerical solutions were obtained for a panoplea of other collector geometries... 

The polymer radius has to be larger than 80% of the particle radius to avoid adsorption limitation under orthokinetic conditions. As a rule of thumb a particle diameter of about 1 pm marks the transition between perikinetic and orthokinetic coagulation (and flocculation). The effective size of a polymeric flocculant must clearly be very large to avoid adsorption limitation. However, if the polymer is sufficiently small, the Brownian diffusion rate may be fast enough to prevent adsorption limitation. For example, if the particle radius is 0.535 pm and the shear rate is 1800 s-, then tAp due to Brownian motion will be shorter than t 0 for r < 0.001, i.e., for a polymer with a... 

FIGURE 2-10 Tracking a gold particle attached to a single molecule of phosphatidyl ethanolamine. What appears to be simple Brownian diffusion at a time resolution of 33 ms per video frame (A) is revealed to actually consist of fast hop diffusion by recording 300 times faster (B) at 110 ps per video frame. In (A) each color represents 60 frames = 2 seconds. In (B) each color indicates an apparent period of confinement within a compartment and black indicates intercompartmental hops. The residency time for each compartment is indicated. The hypothetical explanations are illustrated in part (C) and discussed in the text. Adapted from [29]. 

Heterodisperse Suspensions. The rate laws given above apply to monodisperse colloids. In polydisperse systems the particle size and the distribution of particle sizes have pronounced effects on the kinetics of agglomeration (O Melia, 1978). For the various transport mechanisms (Brownian diffusion, fluid shear, and differential settling), the rates at which particles come into contact are given in Table 7.2. 

According to Eq. (4) (Table 1.2), the time required to halve the concentration of the virus particles in the suspension containing the virus particles only would be almost 200 days. In the presence of bentonite (kb = 3.1 10 10 cm3 sec1 and Nd2 = 7.35 106 cm 3) we find after integrating that the free virus concentration after 1 hour of contact is only 2.6 particles cm 3. This example illustrates that the presence of larger particles may aid significantly in the removal of smaller ones, even when Brownian diffusion is the predominant transport mechanism. 

A rheological measurement is a useful tool for probing the microstructural properties of a sample. If we are able to perform experiments at low stresses or strains the spatial arrangement of the particles and molecules that make up the system are only slightly perturbed by the measurement. We can assume that the response is characteristic of the microstructure in quiescent conditions. Here our convective motion due to the applied deformation is less than that of Brownian diffusion. The ratio of these terms is the Peclet number and is much less than unity. In Equation (5.1) we have written the Peclet number in terms of stresses ... 

Fluorescence polarization. Emission anisotropy Brownian diffusion equation for a spherical particle... 

One of the important properties of particles that contributes to both the observed size distribution and the number concentration of aerosols in the atmosphere is the motion they undergo when suspended in air. This includes gravitational settling and Brownian diffusion. 

The rates of Brownian diffusion can be quantified by considering a box of cross-sectional area 1 cm2 having N particles cm 3 in one section (A) on the left, while the right section (B) is initially empty. Particles (or gas molecules) will always tend to diffuse from a region of higher concentrations to one of lower concentrations. 

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