Brownian particle diffusion coefficient

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... 

We now examine the connection between these computer simulations and the Smoluchowski equation. Consider the form of the kernel for aggregation of particles undergoing Brownian diffusion (see Fig. 17), where the diffusion coefficient of a particle of mass / is D,. In the frame of reference of particle /, the diffusion coefficient of particle j is (Z), + Dj). A collision occurs if the center of particle j enters a sphere of radius (r, + rj) around particle /, so [46]... 

WS7 Brownian diffusion coefficient for aqua sols (i.e., DOC, colloids, etc.) in the bed pore-waters WS8 Bioturbation driven bio-diffusion coefficient for particle and pore-water transport within the bed WS9 Particle settling and deposition into the deep ocean... 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

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... 

After the jump, the particle is taken to have reacted with a given probability if its distance from another particle is within the reaction radius. For fully diffusion-controlled reactions, this probability is unity for partially diffusion-controlled reactions, this reaction probability has to be consistent with the specific rate by a defined procedure. The probability that the particle may have reacted while executing the jump is approximated for binary encounters by a Brownian bridge—that is, it is assumed to be given by exp[—(x — a)(y — a)/D St], where a is the reaction radius, x andy are the interparticle separations before and after the jump, and D is the mutual diffusion coefficient of the reactants. After all... 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

A probabilistic kinetic model describing the rapid coagulation or aggregation of small spheres that make contact with each other as a consequence of Brownian motion. Smoluchowski recognized that the likelihood of a particle (radius = ri) hitting another particle (radius = T2 concentration = C2) within a time interval (dt) equals the diffusional flux (dC2ldp)p=R into a sphere of radius i i2, equal to (ri + r2). The effective diffusion coefficient Di2 was taken to be the sum of the diffusion coefficients... 

The above formulation can be readily extended to the two-dimensional diffusion of a Brownian particle in the presence of needle-like obstacles with a mean lifetime x, as has been made by Teraoka and Hayakawa [107], who obtained for the two-dimensional effective diffusion coefficient... 

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... 

Another derivation has been given by Resibois and De Leener. In principle, eqn. (287) can be applied to describe chemical reactions in solution and it should provide a better description than the diffusion (or Smoluchowski) equation [3]. Reaction would be described by a spatial- and velocity-dependent term on the right-hand side, — i(r, u) W Sitarski has followed such an analysis, but a major difficulty appears [446]. Not only is the spatial dependence of the reactive sink term unknown (see Chap. 8, Sect. 2,4), but the velocity dependence is also unknown. Nevertheless, small but significant effects are observed. Harris [523a] has developed a solution of the Fokker—Planck equation to describe reaction between Brownian particles. He found that the rate coefficient was substantially less than that predicted from the diffusion equation for aerosol particles, but substantially the same as predicted by the diffusion equation for molecular-scale reactive Brownian particles. 

Figure 5-4 illustrates Eqns. (5.40) and (5.44) by plotting the mean square displacement of many particles as a function of t. We can distinguish the Brownian from the pre-Brownian regime and correlate A with the diffusion coefficient D. 

The motion caused by thermal agitation and the random striking of particles in a liquid by the molecules of that liquid is called Brownian motion. This molecular striking results in a vibratory movement that causes suspended particles to diffuse throughout a liquid. If the colloidal particles can be assumed to be approximately spherical, then for a liquid of given viscosity (q), at a constant temperature (T), the rate of diffusion, or diffusion coefficient (D) is inversely related to the particle size according to the Stokes-Einstein relation (ref. 126) ... 

In addition to translational Brownian motion, suspended molecules or particles undergo random rotational motion about their axes, so that, in the absence of aligning forces, they are in a state of random orientation. Rotary diffusion coefficients can be defined (ellipsoids of revolution have two such coefficients representing rotation about each principal axis) which depend on the size and shape of the molecules or particles in question28. 

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... 

Many excellent introductions to quasi-elastic light scattering can be found in the literature describing the theory and experimental technique (e.g. 3-6). The use of QELS to determine particle size is based on the measurement, via the autocorrelation of the time dependence of the scattered light, of the diffusion coefficients of suspended particles undergoing Brownian motion. The measured autocorrelation function, G<2>(t), is given by... 

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