Other Diffusion Paths

Atomic migration may also occur along dislocations, grain boundaries, and external surfaces. These are sometimes called short-circuit diffusion paths inasmuch as rates are much faster than for bulk diffusion. However, in most situations, short-circuit contributions to the overall diffusion flux are insignificant because the cross-sectional areas of these paths are extremely small. 

Vacancy diffusion occurs via the exchange of an atom residing on a normal lattice site with an adjacent vacancy. 

For interstitial diffusion, an atom migrates from one interstitial position to an empty adjacent one. 

Pick s First Law Diffusion flux is defined in terms of mass of diffusing species, cross-sectional area, and time according to Equation 5.1. 

For nonsteady-state diffnsion, there is a net accmnulation or depletion of diffusing species, and the flux is dependent on time. 

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The development of composite micro/mesoporous materials opens new perspectives for the improvement of zeolytic catalysts. These materials combine the advantages of both zeolites and mesoporous molecular sieves, in particular, strong acidity, high thermal and hydrothermal stability and improved diffusivity of bulky molecules due to reduction of the intracrystalline diffusion path length, resulting from creation of secondary mesoporous structure. It can be expected that the creation of secondary mesoporous structure in zeolitic crystals, on the one hand, will result in the improvement of the effectiveness factor in hydroisomerization process and, on the other hand, will lead to the decrease of the residence time of products and minimization of secondary reactions, such as cracking. This will result in an increase of both the conversion and the selectivity to isomerization products. 

For a flat-plate porous particle of diffusion-path length L (and infinite extent in other directions), and with only one face permeable to diffusing reactant gas A, obtain an expression for tj, the particle effectiveness factor defined by equation 8.5-5, based on the following... 

The product cystine is presumably formed in the recombination of two thiyl radicals. This free-radical model is suitable for formal treatment of the kinetic data however, it does not account for all possible reactions of the RS radical (68). The rate constants for the reactions of this species with RS-, 02 and Cu L, (n = 2, 3) are comparable, and on the order of 109-10loM-1s-1 (70-72). Because all of these reaction partners are present in relatively high and competitive concentrations, the recombination of the thiyl radical must be a relatively minor reaction compared to the other reaction paths even though it has a diffusion controlled rate constant. It follows that the RS radical is most likely involved in a series of side reactions producing various intermediates. In order to comply with the noted chemoselectivity, at some point these transient species should produce a common intermediate leading to the formation of cystine. 

While microscopic techniques like PFG NMR and QENS measure diffusion paths that are no longer than dimensions of individual crystallites, macroscopic measurements like zero length column (ZLC) and Fourrier Transform infrared (FTIR) cover beds of zeolite crystals [18, 23]. In the case of the popular ZLC technique, desorption rate is measured from a small sample (thin layer, placed between two porous sinter discs) of previously equilibrated adsorbent subjected to a step change in the partial pressure of the sorbate. The slope of the semi-log plot of sorbate concentration versus time under an inert carrier stream then gives D/R. Provided micropore resistance dominates all other mass transfer resistances, D becomes equal to intracrystalline diffusivity while R is the crystal radius. It has been reported that the presence of other mass transfer resistances have been the most common cause of the discrepancies among intracrystaUine diffusivities measured by various techniques [18]. 

Dayanada 1979) and Fig. 4.14 shows the measured composition profiles of the various elements in a diffusion couple from Al-Nb-Ti at 1200°C (Hellwig 1990). In the latter case the diffusion path crosses three two-phase fields, (Nb, Ti) AI3 + TiAl, TiAl + NbzAl, and /3 + Nb2Al. Other good examples of this technique in practice can be found in the work of van Loo and co-workers (1978,1980,1981). 

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.,... 

Brilman et al. [42] and Lin et al. [44] using a numerical method, Nagy [48] by using an analytical method, investigated the effect of the second, third, etc. particles (perpendicular to the gas-liquid interface) on the absorption rate. They obtained that, in most cases, the first particle determines the absorption rate. However, in special cases, the effect of these particles can also be important. Nagy solved the mass transfer problem analytically for the number of particles in the diffusion path [48]. For the sake of completeness we will give the absorption rate for that case, as well (for details see [48] ). The mass transfer is accompanied here by a first-order chemical reaction. This situation is illustrated in Fig. 1 where three particles are located behind each other. The absorption rate... 

Other factors which may also be responsible to some extent for the small coefficient obtained for the coconut carbon include the approximation of spherical particle geometry, the assumption of an isotropic medium, and the assumption of a radial diffusion path (13). 

Other types of diffusive sampler have been less widely applied for indoor air studies. These include tube type samplers that are solvent desorbed and radial type samplers consisting of a cylindrical adsorbent surface that has a short diffusive path resulting in an effective uptake rate that is typically 100 times that of the tube type sampler (Cocheo, Boaretto and Sacco, 1996). One type of radial sampler developed for measuring carbonyl compounds in indoor air comprises silica gel coated with 2,4-DNPH as the adsorbent within a sintered polyethylene tube that acts as a diffusive membrane (Uchiyamaa, Aoyagi and Ando, 2004). 

Figure 12.4 The pulse scheme of the three-pulse echo sequence to determine Xe diffusion coefficients in polymers and other porous systems. The shaded areas are magnetic field gradient pulses with amplitude g and length 8. The time between the two gradient pulses A determines the time during which the diffusion path length is...

It is easy to notice that during the same nine seconds three B atoms could have displaced from substance B across the AB layer to interface 1, if the chemical transformations at interface 1 (including also external diffusion of the B atoms to phase A through interface 1) would occur instantaneously or if the excessive B atoms (there are two such atoms in the case under consideration) would be used in the formation of the layers of other compounds of the same binary system. In the examined case of a single compound, this possibility of diffusion of excessive B atoms from interface 2 to interface 1 is not realised because the diffusion path is closed up until the full completion of chemical transformations at interface 1. However, the existence of such a possibility must be borne in mind when analysing the multiple layer growth. 

In regards to the stationarity of the reaction-diffusion process, it should be emphasised that the number of the B atoms diffusing across the ApBq layer is always equal to their number combined by the A surface into the ApBq compound at interface 1, if the growth of this layer is not accompanied by the formation of other compounds or solid solutions. The case under consideration is characterised by a kind of forced stationarity due to (z) the impossibility of any build-up of atoms at interfaces between the solids, (z z) the limited number of diffusion paths in the ApBq layer for the B atoms to travel from interface 2 to interface 1 and (Hi) the finite value of the reactivity of the A surface towards the B atoms. The stationarity is only... 

Mesostructures prepared through S°I° or N°F pathways have either wormhole6,7 or lamellar framework structures8,9. The wormhole structures possess a three-dimensional channel structure. Moreover, the framework domain size can be made very small (e.g., 20-200 nm), which introduces an intraparticle textural porosity that is complementary to the framework porosity. The combination of wormhole framework pores and textural pores can greatly facilitate access to catalytic centers in the framework walls. Lamellar frameworks, on the other hand, can be folded into vesicular particles with very thin mesostructured shells and hollow cores. These hierarchical structures also can facilitate access to reactive catalytic centers in the framework walls by minimizing the diffusion path length. 

Inequality 11 was substituted into Equation 8, together with reasonable values of other parameters and 0.1 cm < a < 0.2 cm as a was found to be in this study. This leads to the conclusion that, for systems in which r is less than 0.1 cm, the local equilibrium assumption is applicable (i.e., q is sufficiently small) when D is nearly equal to as observed in the experiments. In soils in which the exchanging particles are not spherical, r would represent approximately the mean diffusion path within clay aggregates or within clay coatings on coarse particles. 

The diffusion of small molecules into the skin from the external world is limited by the stratum comeum (SC). There is now considerable evidence that a major pathway for such diffusion through the stratum comeum itself consists of the intercellular spaces and, in particular, the lipid component of the intercellular spaces. Such lipids are arranged in lamellae that may well be bilayers and that in other respects also resemble biological membranes. In addition, there are other components within the intercellular spaces (e.g., proteins and an aqueous phase) that, although not well understood, mean that the intercellular diffusion path is a heterogeneous material. Within such material, lipids appear to play a very important role (Potts and Guy, 1992), and... 

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