Mechanisms of Diffusion

Table II.1 which depends on the pellet size, so the familiar plot of effectiveness factor versus Thiele modulus shows how t varies with pellet radius. A slightly more interesting case arises if it is desired to exhibit the variation of the effectiveness factor with pressure as the mechanism of diffusion changes from Knudsen streaming to bulk diffusion control [66,...

This analysis makes possible the determination of a chemical diffusion coefficient from experimental data having made no use of a model, and which takes no account of tire atomic mechanism of diffusion, and assumes tlrat the same chemical diffusion coefficient applies to each component of the alloy. 

The diffusion coefficient corresponding to the measured values of /ch (D = kn/4nRn, is the reaction diameter, supposed to be equal to 2 A) equals 2.7 x 10 cm s at 4.2K and 1.9K. The self-diffusion in H2 crystals at 11-14 K is thermally activated with = 0.4 kcal/mol [Weinhaus and Meyer 1972]. At T < 11 K self-diffusion in the H2 crystal involves tunneling of a molecule from the lattice node to the vacancy, formation of the latter requiring 0.22 kcal/mol [Silvera 1980], so that the Arrhenius behavior is preserved. Were the mechanism of diffusion of the H atom the same, the diffusion coefficient at 1.9 K would be ten orders smaller than that at 4.2 K, while the measured values coincide. The diffusion coefficient of the D atoms in the D2 crystal is also the same for 1.9 and 4.2 K. It is 4 orders of magnitude smaller (3 x 10 cm /s) than the diffusion coefficient for H in H2 [Lee et al. 1987]. 

Miyazaki et al. [1984] and Itskovskii et al. [1986] have supposed that the mechanism of diffusion consists in the exchange reactions... 

Some rubber base adhesives need vulcanization to produce adequate ultimate strength. The adhesion is mainly due to chemical interactions at the interface. Other rubber base adhesives (contact adhesives) do not necessarily need vulcanization but rather adequate formulation to produce adhesive joints, mainly with porous substrates. In this case, the mechanism of diffusion dominates their adhesion properties. Consequently, the properties of the elastomeric adhesives depend on both the variety of intrinsic properties in natural and synthetic elastomers, and the modifying additives which may be incorporated into the adhesive formulation (tackifiers, reinforcing resins, fillers, plasticizers, curing agents, etc.). 

When this system was studied over time, it was found that the marker wires move toward each other. This shows that the most extensive diffusion is zinc from the brass (an alloy of zinc and copper) outward into the copper. If the mechanism of diffusion involved an interchange of copper and zinc, the wires would not move. The diffusion in this case takes place by the vacancy mechanism described later, as zinc moves from the brass into the surrounding copper. As the zinc moves outward, vacancies are produced in the... 

One type of diffusion mechanism is known as the interstitial mechanism because it involves movement of a lattice member from one interstitial position to another. When diffusion involves the motion of a particle from a regular lattice site into a vacancy, the vacancy then is located where the site was vacated by the moving species. Therefore, the vacancy moves in the opposite direction to that of the moving lattice member. This type of diffusion is referred to as the vacancy mechanism. In some instances, it is possible for a lattice member to vacate a lattice site and for that site to be filled simultaneously by another unit. In effect, there is a "rotation" of two lattice members, so this mechanism is referred to as the rotation mechanism of diffusion. 

In this equation, N is the number of ions per cm3, q is the charge on the ion, and a is a factor that varies from about 1 to 3 depending on the mechanism of diffusion. Because conductivity of a crystal depends on the presence of defects, studying conductivity gives information about the presence of defects. The conductivity of alkali halides by ions has been investigated in an experiment illustrated in Figure 8.11. 

The aim of many of the studies of diffusion is to relate the measured diffusion coefficient to a mechanism of diffusion. By this is meant a model of atomic jumps that accurately reproduces the diffusion coefficient and the measured concentration profile over a wide range of temperatures. This objective has been most pronounced... 

The mechanism of diffusion of these permeant molecules in these membranes is an issue that must be explored in detail. We have shown [71] that the R = -C2H4-OH-derivatized nanotubules flood when immersed in water. In contrast, permeation experiments with inorganic salts suggest that the R = -C16H33 nanotubules do not flood with water. Hence, in these membranes the permeate molecule is partitioned into and diffuses through the Cig phase within the tubes. 

To obtain a more complete description, we need to find an analytic expression for the pre-exponential factor Dq of the diffusion coefficient by considering the microscopic mechanism of diffusion. The most straightforward approach, which neglects correlated motion between the ions, is given by the random-walk theory. In this model, an individual ion of charge q reacts to a uniform electric field along the x-axis supplied, in this case, by reversible nonblocking electrodes such that dCj(x)/dx = 0. Since two... 

Figure 18.3. Graphical illustration of the mechanisms of diffusion a) interchange by rotation b) migration through interstitials (c) atoms exchange position with vacancy. (From C. Kittel, Introduction to Solid State Physics, 7th ed., Wiley, New York, 1996, with permission from Wiley.)...

By comparing the measurable zone with a standard response line, the concentration of the dilution can be determined and the potency of the sample may be calculated. For a complete discussion of the mechanics of diffusion, the formation of the zone edge, and the relationships between concentration and zone size, the reader should refer to Kavanagh s classic text ( ). 

Polymer network structure is important in describing the transport through biomedical membranes [139, 140]. The mechanism of diffusion in membranes may be that of pure diffusion or convective transport depending on the mesh size of the polymer network. With this in mind, polymer membranes are typically divided into three major types described below [141]. 

Macroporous membranes - these are membranes containing large pores. The pore size is usually between 0.1 and 1 pm. Convective transport through the pore space is the mechanism of diffusion in this case. 

Nonporous gel membranes - these membranes do not contain a porous structure and thus diffusion occurs through the space between the polymer chains (the mesh). Obviously in this case, molecular diffusion rather than convective transport is the dominant mechanism of diffusion in these membranes. 

In the case of macroporous membranes, the pores are of large enough size and the mechanism of diffusion is such that the diffusion coefficient of a solute through the membrane may be described as the diffusion coefficient through the solvent-filled pores of the membrane. The macroporous membrane may be characterized by a porosity, e, and a tortuosity, x, as well as a partition coefficient, Kp, which describes how the solute distributes itself in the membrane. These parameters are usually included in the description of transport in macroporous membranes by incorporating them into the diffusion coefficient as... 

As with thermal conductivity, we see in this section that disorder can greatly affect the mechanism of diffusion and the magnitude of diffusivities, so that crystalline ceramics and oxide glasses will be treated separately. Finally, we will briefly describe an important topic relevant to all material classes, but especially appropriate for ceramics such as catalyst supports—namely, diffusion in porous solids. 

Inert markers have been used to obtain additional information regarding the mechanism of spinel formation. A thin platinum wire is placed at the boundary between the two reactants before the reaction starts. The location of the marker after the reaction has proceeded to a considerable extent is supposed to throw light on the mechanism of diffusion. While the interpretation of marker experiments is straightforward in metallic systems, giving the desired information, in ionic systems the interpretation is more complicated because the diffusion is restricted mainly to the cation sublattice and it is not clear to which sublattice the markers are attached. The use of natural markers such as pores in the reactants has supported the counterdiffusion of cations in oxide spinel formation reactions. A treatment of the kinetics of solid-solid reactions becomes more complicated when the product is partly soluble in the reactants and also when there is more than one product. 

In reading the literature on the fluid mechanics of diffusion, one encounters numerous difficulties because of the diversity of reference frames and definitions used by authors in various fields. Frequently more time is spent in translating from one system of notation to another than is spent in the actual study of the physics of the problem. It is hoped that the glossaries of terminology and symbols given here will be of use to those whose fields of research require a familiarity with the literature on diffusion. This exposition should emphasize the extreme importance of giving lucid definitions in any discussion of diffusion and mass transfer. 

For liquid-phase catalytic or enzymatic reactions, catalysts or enzymes are used as homogeneous solutes in the hquid, or as sohd particles suspended in the hquid phase. In the latter case, (i) the particles per se may be catalysts (ii) the catalysts or enzymes are uniformly distributed within inert particles or (hi) the catalysts or enzymes exist at the surface of pores, inside the particles. In such heterogeneous catalytic or enzymatic systems, a variety of factors that include the mass transfer of reactants and products, heat effects accompanying the reactions, and/or some surface phenomena, may affect the apparent reaction rates. For example, in situation (iii) above, the reactants must move to the catalytic reaction sites within catalyst particles by various mechanisms of diffusion through the pores. In general, the apparent rates of reactions with catalyst or enzymatic particles are lower than the intrinsic reaction rates this is due to the various mass transfer resistances, as is discussed below. 

The decay of the end-to-end vector correlation function was used to gain insight into the mechanism of diffusion. As in the simulations of June et... 

The picture is less clear for molecular crystals when the molecules deviate strongly from a globular form. NMR data and tracer diffusion data are then often in disagreement. Diffusion profiles (In c, vs. distance) are found to be curved, which is usually attributed to additional heterogeneous and fast diffusion pathways. For plastic crystals, this could indicate that many of them possess a highly defective structure. Even for the aromatic ring molecule benzene, which forms a non-plastic crystal, one finds a D (NMR)/D (tracer) ratio on the order of 103. This cannot be understood unless one invokes other than bulk lattice mechanisms of diffusion. 

Because different driving forces can arise for a chemical species and because the mechanisms of diffusion comprising the microscopic basis for D are essentially independent of the driving force, all the driving forces can be collected and attributed to the generalized diffusion potential, , introduced in Chapter 2. 

Figure 10.3 Mechanism of diffusion in amorphous glasses by thermally activated...

H.I. Aaronson. Atomic mechanisms of diffusions nucleation and growth and comparisons with their counterparts in shear transformations. Metall. Trans., 24A(2) 241-276, 1993. 

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