Diffusion Fuchs

When the radius of an aerosol particle, r, is of the order of the mean free path, i, of gas molecules, neither the diffusion nor the kinetic theory can be considered to be strictly valid. Arendt and Kallman (1926), Lassen and Rau (1960) and Fuchs (1964) have derived attachment theories for the transition region, r, which, for very small particles, reduce to the gas kinetic theory, and, for large particles, reduce to the classical diffusion theory. The underlying assumptions of the hybrid theories are summarized by Van Pelt (1971) as follows 1. the diffusion theory applies to the transport of unattached radon progeny across an imaginary sphere of radius r + i centred on the aerosol particle and 2. kinetic theory predicts the attachment of radon progeny to the particle based on a uniform concentration of radon atoms corresponding to the concentration at a radius of r + L... 

Diffusion of the gas to the surface (rg). As described by Fuchs and Sutugin (1970, 1971) in their comprehensive treatment of highly dispersed aerosols, the rate of transfer of mass to the surface of a spherical particle by diffusion of a gas is described by... 

For the other extreme of the free molecular regime where Kn - oc, the particle radius is small compared to the mean free path. In this case, the thermal velocity distribution of the gas is not distorted by uptake at the surface. In effect, the gas molecules do not see the small particles. For this case, Fuchs and Sutugin (1970, 1971) show that for diffusion to a spherical particle of radius a... 

A more quantitative measure of stability, known as the stability ratio, can be obtained by setting up and solving the equation for diffusive collisions between the particles. Quantitative formulations of stability, known as the Smoluchowski and Fuchs theories of colloid stability, are the centerpieces of classical colloid science. These and related issues are covered in Section 13.4. 

Attachment is mainly by diffusion, although, if the decay products are ions, electrostatic attraction to charged nuclei of opposite sign makes a small additional effect (Bricard Pradel, 1966). The rate constant for attachment XA, is given by an equation originally applied to evaporation from small droplets (Fuchs, 1959). 

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. 

As mentioned earlier, k may be reduced by an electrostatic interaction, whose magnitude may be given the Fuchs symbol W1, but also because of reversibility (30). Because the radical, upon collision with a particle, may add monomer units and/or terminate with a radical already present in the particle, Hansen and Ugelstad applied the theory of Dankwerts for diffusion with reaction to determine the overall capture rate. Under these conditions the diffusion equation of Fick must be modified to read ... 

The rate of gas uptake is dominated by diffusion for 7 greater than 4D /Ca. At atmospheric pressure and 10 /un radius droplets this limitation occur for 7 < 0.02. The limitations of gas diffusion are discussed in greater detail by Fuchs and Sutugin (fi) and by Schwartz (2). 

The collection efficiency of a fiber is defined as the ratio of the number of particles striking the fiber to the number which would strike it if the streamlines were not diverted [Dorman, 1966]. For particles with dp > 1 pm, the particles are collected primarily by inertial impaction and interception. Collection by diffusion is only important for submicron particles. In the following, we only discuss the methodology of collection efficiency by interception. Details about the filtration of fine and ultrafine powders or granular filtration in general can be found in Davies (1973), Fuchs (1964), Dickey (1961), Matteson and Orr (1987), and Tien (1989). 

Bammer R, Fazekas F, Augustin M, Simbrunner J, Strasser-Fuchs S, Seifert T, Stollberger R, and Hartung HP (2000) Diffusion-weighted MR imaging of the spinal cord. AJNR Am J Neuroradiol 21 (3) 587-591 Behrens S, Thron A (1999) Long-term follow-up and outcome in patients treated for spinal dural arteriovenous fistula. J Neurol 246 181-185... 

Table 1 compares the dimensionless coagulation coefficient predicted by the present model with other models. Since the Hamaker constant for most of the aerosol systems is of the order of 10"12 eig, this value is used in the calculation of the lower bound. Particle diffusion coefficients based on Philips slip correction factor for an accommodation coefficient of unity are used for the calculation of the coagulation coefficients ft (the Fuchs interpolation formula) and fts (the Sitarski... 

It is possible to extend the useful range of this equation to somewhat smaller drops. Fuchs (1959) initially pointed out that Langmuir s equation could not be correct for small particles having diameters approaching the mean free path of the gas since the equation predicted rates of molecular escape which exceeded the evaporation rate into a vacuum. To correct this difficulty, Fuchs considered the diffusion process to start a distance of approximately one mean free path from the droplet surface. 

Frictional electrification, 183-184 Frictional resistance, 50 Friedlander, S. K, 312, 316 Fuchs, N. A., 61 and brownian rotation, 138 and coagulation, 313-314 and diffusion charging, 188 and diffusion and collisions, 304 and equilibrium charge distributions, 204-205... 

Bai, X. S., and Fuchs, L., Numerical model for turbulent diffusion flames with applications. In Computational Fluid Dynamics (C. Hirsch, J. Periaux and W. Kordulla, eds.). Elsevier, Amsterdam, 1992, vol. l,p. 169. 

Thus the root mean square displacement in 1 s for a 1 pm particle settling in water, viscosity 0.001 Pa s, at an absolute temperature 300 K is 0.938 pm this is almost the same as the distance settled under gravity by a quartz particle (density 2650 kg m" ) in 1 s (0.90 pm). A comparison of Brownian movement displacement and gravitational settling displacement is given by Fuchs [8]. For a size determination to be meaningful the displacement of the particles due to Brownian diffusion must be much smaller than their displacement due to gravity, hence the condition ... 

In subsequent analyses it was shown that if diffusion in the presence of an energy barrier is considered (Fuchs, 1934 Overbeek, 1952) then the initial rate of disappearance of the primary particles could be written as... 

This result collapses to that of Fuchs and Debye for diffusion in a field of force when is a constant. When K(/ ) = 0 for / > the Smoluchowski result of Eq. (1.2) is obtained. It is useful to denote deviations from Eq. (1.2) by a dimensionless stability factor 1F(0) so that... 

In the continuum regime dp 0.1 /xm), the ion fluxes ft and ft can be estimated from steady-state solutions to the ion diffusion equation (2.43) in the presence of a Coulomb force field. surrounding the particles image forces are neglected (Fuchs and Sutugin, 1971) ... 

Unlike diffusion, which is a stochastic process, particle motion in the inertial range is deterministic, except for the very important case of turbulent transport. The calculation of inertial deposition rates Is usually based either on a force balance on a particle or on a direct analysis of the equations of fluid motion in the case of colli Jing spheres. Few simple, exact solutions of the fundamental equations are available, and it is usually necessary to resort to dimensional analysis and/or numerical compulations. For a detailed review of earlier experimental and theoretical studies of the behavior of particles in the inertial range, the reader is referred to Fuchs (1964). 

The process of evaporation during dip coating was studied only recently (2, 22a), although the essential physics of evaporation has been well known for over a century. According to Fuchs (23), James Clerk Maxwell wrote an article on diffusion for the Encyclopedia Britannica in which he considered the stationary evaporation of a spherical droplet in an... 

Table 7-4. Mobilities b(r), Diffusion Coefficients D(r), Average Thermal Velocities v(r), Mean Free Paths A (r), and Fuchs s Correction Factor f (r) for Like Particles, as a Function of Radius for Spherical Panicles in Aira...

The rate of coagulation is affected by interaction between particles separated by distances greater than twice their radius. According to N.A. Fuchs [56], one has to look at the diffusion of particles of radius r in their interaction field, described as a function of energy (negative ) on distance, u(R). Fick s equation in this case can be written as... 

From the theoretical point of view a similarity exists between electrostatic retardation of ion transport and coagulation retardation, known as slow coagulation (Fuchs, 1934). Both phenomena arise from electrostatic repulsion caused by the existence of the diffuse part of the DL. In the slow coagulation theory, the electric field if the DL is derived from the Gouy-Chapman model (cf. Chapter 2). This model does not consider a deviation of the diffuse layer from equilibrium. Initially, the same simplification was used by Dukhin et al. (1973) in describing the DL effect on the electrostatic retardation of adsorption. 

Robbins PE, Jenkins WJ (1998) Observations of temporal changes of tritium-He-3 age in the eastern North Atlantic thermocline Evidence for changes in ventilation J Mar Res 56 1125-1161 Robbins PE, Price JF, Owens WB, Jenkins WJ (2000) The importance of lateral diffusion for the ventilation of the lower thermocline in the subtropical North Atlantic. J Phys Oceanogr 30 67-89 Roether W, Fuchs G (1988) Water mass transport and ventilation in the Northeast Atlantic derived from tracer data. Philos Trans R Soc London, Ser. A 325 63-69 Roether W, Kromer B (1978) Field determination of air-sea gas-exchange by continuous measurement of Rn-222. Pure Appl Geophysics 116 476-485... 

Fuchs Theory The matching of continuum and free molecule fluxes dates back to Fuchs (1964), who suggested that by matching the two fluxes at r = A + Rp, one may obtain a boundary condition on the continuum diffusion equation. This condition is, assuming unity accommodation coefficient... 

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