Sphere Brownian diffusion coefficients

For rigid spheres the rotary Brownian diffusion coefficient is... 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

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. 

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... 

Table 2.1 Diffusion coefficients and Brownian displacements calculated for uncharged spheres in water at 20°C...

Consider a sphere of radius a held fixed in a creeping flow field with approach velocity U. The fluid contains Brownian particles having a diffusion coefficient D. Should the Peclet number 2aUiDUj have a value much greater than unity, the diffusion boundary layer will be sufficiently thin so that curvature effects and tangential diffusion are negligible. Under these conditions the convective-diffusion equation assumes the following form ... 

The r-average translational diffusion coefficient l> is calculated from the equation Dj = V/q2. For a collection of identical spheres undergoing ordinary Brownian motion in solution. 

The diffusion coefficient. Dp, that appears in the transport equations in the diffusion boundary layer, was defined by treating the disentangling chains in the boundary layer as Brownian spheres. Thus, a Stokes-Einstein type diffusivity... 

The effect of free volume on penetrant diffusion coefficients in polymers is often described using concepts from the Cohen and Turnbull model (7. This statistical mechanics model provides a simplistic description of diffusion in a liquid of hard spheres. A hard sphere penetrant is considered to be trapped in a virtual cage created by its neighbors. Free volume is defined as the volume of the cage less the volume of the penetrant. Free volume fluctuations, which occur randomly due to thermally-stimulated Brownian motion of neighboring hard spheres, provide opportunities for the penetrant to execute a diffusion step if the gap (i,e, free volume fluctuation) occurs sufficiently close to the i netrant to be accessible and is of sufficient size to acconunodate it. The diffusion coefficient of a penetrant is given by ... 

Carter JM, Phillies GDJ. Second-order concentration correction to the mutual diffusion coefficient of a suspension of hard Brownian spheres. J Phys Chem 1985 89 5118-5124. 

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]... 

Total internal reflection microscopy (TIRM) was introduced in 1987 by Prieve et al. [343]. TIRM allows to probe the interaction of a single microsphere with a transparent flat plate. In a TIRM experiment, a microsphere is allowed to sediment toward the plate. The technique relies on repulsive forces between sphere and plate. This repulsion will typically result from electric double layer or steric forces. They keep the sphere from getting into contact with the plate. Thermal fluctuations will constantly change the precise distance. The distance between sphere and plate is monitored by the light intensity scattered from the particle when illuminated by an evanescent wave and can be determined with a resolution of w 1 nm. By recording the fluctuations in vertical position of the sphere due to Brownian motion, the potential energy of interaction and the diffusion coefficient of the sphere can be deduced. For overviews of the technique, see Refs [344, 345]. 

Consider a dilute suspension of spherical particles A in a stationary liquid B. If the spheres are sufficiently small, yet large with respect to the molecules of stationary liquid, the collisions between the spheres and the liquid molecules B lead to a random motion of the spheres. This motion is called the Brownian motion. Dilute diffusion of suspended spherical colloid particles is related to the temperature and the friction coefficient by... 

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