Brownian diffusive motion

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). 

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. 

This is a modification of the Brownian diffusive motion of the ions in the solution. Because of non-ideality and its description in terms of the ionic atmosphere it has been shown (see Section... 

In the case of no restrictions imposed on the diain flexibility, the polymer chain represents a system without any directional memory and evokes a path of Brownian diffusive motion [9]. A suitable description of this self-similar process can be the random walk model rwm) depending on Euclidean dimensionality, d. The rwm-hased distribution function determines the probability of the root mean square chain end-to-end separation,... 

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. 

Diffusion The mixing of substances by molecular motion to equalize a concentration gradient. Applicable to gases, fine aerosols and vapors. (See Brownian diffusion.)... 

Various ligands bind to their protein sites in a diffusive motion. Similarly, the distance between different ends of a folded macromolecule changes in a way which can be described as a diffusive motion in the presence of a constraint potential (that keeps the parts of the molecule near their folded configurations). Brownian-type diffusive motion in the absence of a restrictive potential is characterized by a diffusion constant (Ref. 6)... 

Direct observation of molecular diffusion is the most powerful approach to evaluate the bilayer fluidity and molecular diffusivity. Recent advances in optics and CCD devices enable us to detect and track the diffusive motion of a single molecule with an optical microscope. Usually, a fluorescent dye, gold nanoparticle, or fluorescent microsphere is used to label the target molecule in order to visualize it in the microscope [31-33]. By tracking the diffusive motion of the labeled-molecule in an artificial lipid bilayer, random Brownian motion was clearly observed (Figure 13.3) [31]. As already mentioned, the artificial lipid bilayer can be treated as a two-dimensional fluid. Thus, an analysis for a two-dimensional random walk can be applied. Each trajectory observed on the microscope is then numerically analyzed by a simple relationship between the displacement, r, and time interval, T,... 

In the frame of the present review, we discussed different approaches for description of an overdamped Brownian motion based on the notion of integral relaxation time. As we have demonstrated, these approaches allow one to analytically derive exact time characteristics of one-dimensional Brownian diffusion for the case of time constant drift and diffusion coefficients in arbitrary potentials and for arbitrary noise intensity. The advantage of the use of integral relaxation times is that on one hand they may be calculated for a wide variety of desirable characteristics, such as transition probabilities, correlation functions, and different averages, and, on the other hand, they are naturally accessible from experiments. 

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... 

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 ... 

Diffusion here refers to the movement of ions and/or neutral species through the deposition bath or solution as a consequence of concentration gradients. It is primarily the result of random (Brownian) molecular motion, and it serves to produce more uniform distribution of the various component species in the bath. Depletion of ions next to the cathode will result in movement of the species from the (nearly unchanged) bulk of the bath toward the cathode. 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

Brownian elution mode occurs when the held-induced velocity of the analyte It/ l in the separation channel is constant and comparable to its diffusive motion. Under these conditions, that is, when the held-induced how and diffusive how are equal (see Figure 12.4a), at the equilibrium, one has... 

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. 

In general, particles of radius RSy2 will also be executing random Brownian motion (i.e., diffusion). In such a case, Dx should be replaced by Dn = (Dx + D2). The collision rate Z12 (where the second subscript now reminds us that particle 2 is also executing diffusive motion) is then... 

The second mechanism is capture by Brownian diffusion, which is more of a factor for smaller particles. Small particles are easily carried along by the moving fluid. However, because the particles are small, they are subject to random Brownian motion that periodically brings them into contact with the pore walls. When this happens, capture by surface adsorption occurs. 

Experiments on transfer of submicrometre radioactive particles to smooth surfaces (Wells Chamberlain, 1967 Chamberlain et al., 1984) have shown that the dependency of vg on D213 holds over many orders of magnitude of D. This means that the transport by Brownian diffusion becomes progressively less effective as the particle size increases. For example a particle of 0.1 pm diameter has a diffusivity of 6.8 x 10 10 m2 s 1, a factor 1.2 x 104 smaller than that of I2 vapour. Since D does not depend on the particle density, it is appropriate to discuss transport by Brownian motion in terms of the particle diameter. The aerodynamic diameter, dA, is equal to dppp2 where pp is the particle density in c.g.s. units (g cm-3) not SI units (kg m-3), and is the appropriate parameter for particles with dp> 1 pm, for which impaction and sedimentation are the mechanisms of deposition. 

The different mechanisms of deposition determine the shape of the curves in Fig. 6.9. For particles of diameter less than 0.2 /tm, Brownian diffusion is the dominant mechanism and vg varies according to Dm, as expected from equation (6.4). The minimum is at dp between 0.1 and 1 /un, where particles are too large to have appreciable Brownian motion but too small to impact. 

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. 

Brownian diffusion and enter another tube. If it repeats this action of small angular motions followed by translation into new tubes, it can work its way around a large circle of radius R. Through simple geometric arguments, this radius is R = 2 (L/ac) L. Since the motion... 

Once least squares values of the /3 s were obtained, it was desirable to extract from them as much information as possible about the original parameters. To do so, we make one further statement concerning the relations between the rate constants for mutual termination of polymeric radicals of different size. It has been shown (2) that termination rates in free radical polymerizations are determined by diffusion rates rather than chemical factors. The relative displacement of two radicals undergoing Brownian motion with diffusion coefficients D and D" also follows the laws of Brownian diffusion with diffusivity D = D -J- D" (11). It... 

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