Brownian diffusivity calculations

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. 

Another approach we can use to describe the stress relaxation behaviour and all the linear viscoelastic responses is to calculate the relaxation spectrum H. Ideally we would like to model or measure the microstructure in the dispersion and include the role of Brownian diffusion in the loss of structural order. The intermediate scattering... 

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

MIRAGE2 Bulk equilibrium with RH based on Kohler theory. Hysteresis is treated Mechanistic, parameterized activation based on Kohler theory bulk CCN only Modal activation. Brownian diffusion, autoconversion, precip. rate independent of aerosols Calculated modal scaveng. coeff using a parameterization of the collective efficiency of aerosol particles by rain drops with size dependence Two-moment sedimentation for aerosols, nosedimentation for cloud droplets/ices... 

The physics behind this relation is the fluctuation-dissipation theorem the same random kicks of the surrounding molecules cause both Brownian diffusion and the viscous dissipation leading to the frictional force. It is -instructive to calculate the time scale t required for the particle to move a... 

Figure 3.13 CompEirison of experiment and theory for the deposition of monodisperse latex particles on a free-slanding wafer 4 in. in diameter. The air mainstream velocity normal to the wafer was 30 cm/sec, typical of microelectronics clean room operations. The diffu-sion equation wa.s solved numerically using calculated velocity and temperature distributions. The curves show that a small increase in surface temperature eHeelivcly suppresses deposition over a wide intermediate particle size range. Larger particles deposit by sedimentation smaller ones break through the thermal barrier by Brownian diffusion. (After Ye et aL, 1991.)...

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 collision rate is initially extremely fast (actually it starts at infinity) but for t 4Rp/nD, it approaches a steady-state value of /coi = 8nRp DNq. Physically, at t — 0, other particles in the vicinity of the absorbing one collide with it, immediately resulting in a mathematically infinite collision rate. However, these particles are soon absorbed by the stationary particle and the concentration profile around our particle relaxes to its steady-state profile with a steady-state collision rate. One can easily calculate, given the Brownian diffusivities in Table 9.5, that such a system reaches steady state in 10-4 s for particles of diameter 0.1 pm and in roughly 0.1 s for 1 pm particles. Therefore neglecting the transition to this steady state is a good assumption for atmospheric applications. 

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. ... 

Figure 20.6 shows E calculated from (20.53) as a function of the collected particle radius (dp/2), for raindrop radius of 0.1 mm and 1 mm. As expected, Brownian diffusion dominates for dp <0.1 pm, whereas impaction and interception control removal for large dp. The characteristic minimum in E occurs in the regime where the particles are too large to have an appreciable Brownian diffusivity yet too small to be collected effectively by either impaction or interception. 

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. 

The diffusion of NPs in blood flow can be due to (1) Brownian diffusion caused by the bombardment of fluid molecules and (2) shear-induced diffusion due to the presence of red blood cells (RBCs) in shear flow. The Brownian diffusion coefficient, Dpr, can be calculated using the Stokes-Einstein equation ... 

Similar agreement with free-molecular momentum calculations for spherical particles has been obtained through study of Brownian diffusion in nonequilibrium gases as described by the Fokker-Planck equation [2.23,24]. 

The electrolyte induced rapid coagulation of polydisperse mixtures of polystyrene microspheres, titania and kaolinite was studied by Photon Correlation Spectroscopy (PCS). A method has been developed that enables the rate constant to be calculated without precise knowledge of the particle size distribution (7). For Brownian Diffusion-controlled Aggregation ... 

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... 

Further support for this approach is provided by modern computer studies of molecular dynamics, which show that much smaller translations than the average inter-nuclear distance play an important role in liquid state atom movement. These observations have conhrmed Swalin s approach to liquid state diffusion as being very similar to the calculation of the Brownian motion of suspended particles in a liquid. The classical analysis for this phenomenon was based on the assumption that the resistance to movement of suspended particles in a liquid could be calculated by using the viscosity as the frictional force in the Stokes equation... 

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