Spheres diffusion through

It was suggested by Sutherland [363] that for a sphere diffusing through a medium consisting of molecules of comparable size to the diffusant, the friction coefficient is equal to... 

Kops-Werkhoven, et al. measured tracer diffusion of dilute 38 nm radius silica spheres diffusing through suspensions of 33 nm silica spheres, all in cyclo-heptane(12). The experimental method was dynamic light scattering particle motions were observed over distances large compared to their radius. They found ku =-2.7 0.3. 

Numerous models have been proposed to interpret pore diffusion through polymer networks. The most successful and most widely used model has been that of Yasuda and coworkers [191,192], This theory has its roots in the free volume theory of Cohen and Turnbull [193] for the diffusion of hard spheres in a liquid. According to Yasuda and coworkers, the diffusion coefficient is proportional to exp(-Vj/Vf), where Vs is the characteristic volume of the solute and Vf is the free volume within the gel. Since Vf is assumed to be linearly related to the volume fraction of solvent inside the gel, the following expression is derived ... 

A catalyst bed contains a uniform mixture of equivolume porous spheres and cubes with two pairs of opposite faces sealed to diffusion. Accordingly the cubes may be regarded as slabs with diffusion through two faces. The spheres have a diameter of 0.02 ft, so cubes of the same volume have edges of 0.01614 ft. 

Another important example of a solid transformation where growth requires diffusion through the product film is the transformation of solid spheres. The principles of this process are similar to those for planar films, but now the concentration profile is not linear, and the expression one obtains for the transformation and the solid conversion is more complex. 

The strict solution for the problem of the resistance to the motion of a small sphere moving through gas has been obtained by Baines et al. (1965). They considered both specular and diffuse reflection of the molecules at the surface of the sphere mass of which is large in comparison with the mean mass of gas molecules and the radius to be small compared with the mean free path of gas molecules. All these assumptions are applicable for circumstellar outflows. Fadeyev and Henning (1987) used these solutions for calculation of momentum transfer from silicate dust grains to gas molecules in cool 0-rich red giants... 

The present models describe the extraction of porous particles completely filled with a liquid solute. The dissolution of liquid is followed by intraparticle diffusion up to the particle surface and then by external diffusion through a solvent boundary layer into the flowing solvent bulk. Three different particle geometries were modeled spheres, cylinders with ends mechanically sealed and cylinders with open ends available for extraction. 

The theory described above is applied to the binary hard-sphere system with the size ratio (Jijci = 0.2 and the concentration of small particles ci = 0.5. Figure 1 shows the diffusion constants D, of small (s = 1) and big (s = 2) particles versus the total packing fraction 9. The dotted lines show the power-law fit D = f o 9 — fj with 7=1.31 and 2.36 for small and big particles, respectively. The diffusion constant of the big particles is found to vanish at 95 =0.52, which is close to the liquid-glass transition point (9=0.516) of a one-component hard-sphere system, while Di becomes zero at 9 =0.53 (> 9b). This means that for 9b < 9 < 9x there exists a new phase (delocalized phase) with mobile small particles diffusing through the voids of a glassy structure formed by the immobile big particles. 

The diffusion coefficients in solids are typically very low (on the order of 10 to 10" mVs), and thus the diffusion process usually affects a thin layer at the surface. A solid can conveniently be treated as a semi-infinite medium during transient mass diffusion regardless of its size and shape when the penetration depth is small relative to the thickness of the solid. When this is not the case, solutions for one dimensional transient mass diffusion through a plane wall, cylinder, and sphere can be obtained from the solution.s of analogous heat conduction problems using the Heisler charts or one term solutions pieseiited in Chapter 4. 

In the first approximation the collision phenomena are described in terms of hard sphere molecular diameters, which are independent of temperature. Actually, the diameters decrease with higher temperature, approaching individual limits [1]. Let us consider a single molecular entity with the mass m and the diameter afmj, which diffuses through a gas consisting mainly of more abundant dissimilar molecules of the mass m2, the diameter dm.2 and the concentration no. If the collision diameter is mi.2 = (dm, 1 + <7m.2)/2, the tracer molecule must collide each second with the host molecules contained in a volume of about nm 2um. Because the host molecules also move, the mean relative speed u 1,2 is... 

Use To form a protecting layer of the tiny spheres over liquid surfaces, such as oils in big tanks, to reduce evaporation to separate helium from natural gas because of the wide difference in relative rates of diffusion through the spheres as an extender in plastics to achieve low density. 

In another example, Beveridge studied the oxidation of zinc sulfide spheres and reported that the global rate was, in turn, controlled by the surface chemical reaction at low temperatures, diffusion through the zinc oxide product layer at intermediate temperatures, and external mass transfer resistance at higher temperatures. 

The recombination rate in such a microspace can be estimated according to Goselle et al. (1979). When two reactants, one of which is much smaller and more diffusive (H+) than the other (0-), are locked in a space with internal diameter R, we can regard the heavier reactant (0-) as practically immobile target in the center of a reaction sphere. The proton can diffuse through the reaction space, but wherever it penetrates the Coulomb cage of the proton emitter, protonation takes place. The rate constant of the reaction is thus controlled by two radii, the Debye radius (RD), where electrostatic interaction dominates, and the radius of the reaction sphere, out of which the proton cannot escape. 

In these expressions, p is the porosity of the SiC, i.e. the volume fraction of empty space. In the asymmetric MG model, we have chosen the coating to be the solution, since the opposite choice of SiC-encapsulated liquid spheres will not permit diffusion through the medium. With this choice, the SiC does not percolate and hence there is no structural support. The selectivity of the membrane is based in part on the size and shape of the protein molecules. The expressions for (pD)eff in the effective medium models [Equations (12.2) and (12.3)] do not contain a size scale, but it is necessary to introduce a scale in order to account for the size of a protein molecule. For simplicity, we assume that the proteins are spherical with effective (hydration) radius r. The excluded volume within the pores due to nonzero size is taken into account by replacing the porosity p with an effective porosity p. For the columnar... 

A model of the actual immobilization process with intact spherical catalyst particles was developed using the experimentally determined binding kinetics (48). The system was treated as a group of porous spheres suspended in a well-mixed solution of heparinase. The enzyme diffused through the porous network, where it reacted with the surface cyanate esters to produce the bound enzyme. 

It follows from Eq. (25) that any periodicity (/) in the propagator should give rise to a coherence peak, that is, to a local maximum for q = ybg = Iti/I in the representation of the NMR spin echo intensity versus the intensity of the field gradient pulses (78 ). Such behavior has indeed been observed in recent PEG NMR studies of water diffusion through the free space within an array of loosely packed monodisperse polysterene spheres [78],... 

Analytical expressions for effective diffusion coefficients in complex media can be obtained only when the geometry is simple. Consider, for example, the diffusion of solute through a periodic array of spherical obstructions, in which the solute diffuses through the continuous interstitial space with a diffusion coefficient i A.pore and through the spheres with a diffusion coefficient Z>a,s-The effective diffusion coefficient for such a composite material can be determined exactly [94] ... 

Radial Diffusion through the Section of a Cylinder or a Sphere... 

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