Diffusion physical mechanism

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

In general, both physical and weak chemical bonds are responsible for mucoadhesion. Physical/mechanical bond formation can be explained as the entanglement of the adhesive polymer and the extended mucin chains. When this diffusion is mutual, it leads to maximum bioadhesive strength. 

Supposing, on the other hand, that the physical mechanism is that some elements of fresh liquid are swept into the surface at a velocity Vn which is so rapid that the diffusion path remains negligibly short, then the mass-transfer coefficient of such elements of liquid is given by Eq. (11). At moderate turbulence, other elements of liquid may, however, approach the surface obliquely, so that the fresh liquid resides for appreciable times in the surface, and to such elements Eq. (7) applies. The total resistance to mass transfer is then the sum of the two resistances acting in parallel, i.e.,... 

Another unique attribute of polymerizations of multifunctional monomers is the dominance of reaction diffusion as a termination mechanism [134,136, 143-146]. Reaction diffusion involves the mobility of radicals by propagation through unreacted functional groups. This termination mechanism is physically different from translation and segmental diffusion termination mechanisms which involve the diffusion of polymer macroradicals and chain segments to bring radicals within a reaction zone before terminating. Whereas normal termination mechanisms are related to the diffusion coefficient of the polymer, reaction diffusion must be considered differently. In essence, reaction diffusion is... 

One important aspect of method development in SFE is that the extractability of analytes from sample matrices is strongly dependent upon the nature or type of the sample matrix. The efficiency of the extraction of a discrete set of analytes needs to be optimized for that set of analytes from a standard matrix and also optimized for the extractability of those specific analytes from a particular sample matrix. However, the optimum set of parameters for both of these cases may be distinctly different. This is due to the analytes, depending on the nature of the sample matrix, having an affinity for the physical outside surface of the matrix or actually being present within the sample matrix particles. This affinity results in either the predominance of a solubility mechanism, a diffusion type mechanism or a physical adsorption type mechanism. Therefore, the optimum extraction conditions will vary if the sample of interest is a polymer, soil or vegetable matrix. 

Although different physical mechanisms can cause diffusive mixing, they all cause a net transport of a chemical from areas of higher concentration to areas of lower concentration. All diffusive processes are also referred to as Fickian transport because they all can be described mathematically by Fick s first law, which states that the flow (or flux) of a chemical (N, g/h) is proportional to its concentration gradient (dC/dx) ... 

The physical mechanism described by this equation can be understood by starting at time zero with a velocity distribution sharply peaked at v = vo- As time passes, the maximum of this distribution is shifted toward smaller velocities, as a result of a systematic friction undergone by the particles (first term on the right-hand side of the equation). Furthermore, the peak broadens progressively as a result of diffusion in velocity space (second term on the right-hand side, which is the velocity space equivalent of the similar coordinate space term in Fick s law of diffusion). The final time-independent distribution reached by the Brownian particle is nothing more than the familiar Maxwell distribution ... 

More recent investigations suggest that many different mechanisms (for example path windiness, pore geometry, or concentration dependant diffusion) may be important (2A). By casting the diffusion equation in the appropriate geometry, the physical mechanisms of release can be elucidated without resort to any simple empirical factors. 

As explained in Chapter 5, the transport mechanism in dense crystalline materials is generally made up of incessant displacements of mobile atoms because of the so-called vacancy or interstitial mechanisms. In this sense, the solution-diffusion mechanism is the most commonly used physical model to describe gas transport through dense membranes. The solution-diffusion separation mechanism is based on both solubility and mobility of one species in an effective solid barrier [23-25], This mechanism can be described as follows first, a gas molecule is adsorbed, and in some cases dissociated, on the surface of one side of the membrane, it then dissolves in the membrane material, and thereafter diffuses through the membrane. Finally, in some cases it is associated and desorbs, and in other cases, it only desorbs on the other side of the membrane. For example, for hydrogen transport through a dense metal such as Pd, the H2 molecule has to split up after adsorption, and, thereafter, recombine after diffusing through the membrane on the other side (see Section 5.6.1). 

A related matter concerns the physical mechanism by which radicals (primary or oligomeric) are acquired by the reaction loci. One possibility, first proposed by Garden (1968) and subsequently developed by Fitch and Tsai (1971), is that capture occurs by a collision mechanism. In this case, the rate of capture is proportional to, inter alia, the surface area of the particle. Thus, if the size of the reaction locus in a compartmentalized free-radical polymerization varies, then a should be proportional to r, where r is the radius of the locus. A second possibility (Fitch, I973) is that capture occurs by a diffusion mechanism. In this case, the rate of capture is approximatdy proportional to r rather than to r. A fairly extensive literature now exists concerning this matter (see, e.g., Ugelstad and Hansen, 1976, 1978. 1979a, b). The consensus of present opinion seems to favor the diffusion theory rather than the collision theory. The nature of the capture mechanism is not. however, relevant to the theory discussed in this chapter. It is merely necessary to note that both mechanisms predict that the rate of capture will depend on the size of the reaction locus constancy of a therefore implies that the size of the locus does not change much as a consequence of polymerization. 

The assumption has been made in deriving Eq. (2.32) that the diffusion coefficient is independent of the separation R. As discussed above, there are a host of physical mechanisms that give rise to a spatially dependent diffusion coefficient, and a formulation of the encounter problem is required which accounts for these. Also, there is reason to believe that Eq. (2.32) needs to be modified for reasons associated with the relationship between D and discussed in Section IIB. The reduction of the FPE in Section IIB gives the relative friction as... 

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