Jump diffusion mechanism

In the case of H-SSZ-24, the values of the pre-exponential factor experimentally obtained (see Table 5.4) do not agree with the values theoretically predicted by the equation for a jump diffusion mechanism of transport in zeolites with linear channels, in the case of mobile adsorption [6,26], Furthermore, the values obtained for the activation energies are not representative of the jump diffusion mechanism. As a result, the jump diffusion mechanism is not established for H-SSZ-24. This affirmation is related to the fact that in the H-SSZ-24 zeolite Bronsted acid sites were not clearly found (see Figure 4.4.) consequently p- and o-xylene do not experience a strong acid-base interaction with acid sites during the diffusion process in the H-SSZ-24 channels, and, therefore, the hopping between sites is not produced. 

The dependence of the inverse of the characteristic times for water translation in 12% water-DP12 at T = 1.4Tg = 475 K (circles) shows the characteristic shape of jump-diffusion mechanism (Equation 3.2) without any contribution from a continuous diffusion mechanism. When the segmental mobility of the oligomer is increased by eliminating the torsional barriers between the monomeric residues, water mobility acquires a continuous diffusion component (circles). The data for water in the monomer at the same reduced temperature T = l-4Tg = 335 K and water content is also shown for comparison (squares). The existence of a continuous component in water diffusion in these low water content mixtures requires a continuous component in the mobility of the sugar. 

Experimental Results on Hydrogen Jump Diffusion Mechanisms 801... 

These features may give rise to a coexistence of several types of H motion with different characteristic jump rates. Here we shall discuss the experimental results on hydrogen jump diffusion mechanisms in a number of representative interme-tallic compounds. A comprehensive review of the experimental studies in this field before 1992 can be found in Ref [11]. 

Another example of an interesting H jump diffusion mechanism has been reported for hydrogen dissolved in the cubic A15-type compound Nb3Al [87]. In this compound H atoms occupy the tetrahedral 6d sites coordinated by four Nb atoms. The 6d sites form three sets of nonintersecting chains in the <100>, <010> and <001> directions. The distance between the nearest-neighbor d sites in the... 

Studies of the effect of permeant s size on the translational diffusion in membranes suggest that a free-volume model is appropriate for the description of diffusion processes in the bilayers [93]. The dynamic motion of the chains of the membrane lipids and proteins may result in the formation of transient pockets of free volume or cavities into which a permeant molecule can enter. Diffusion occurs when a permeant jumps from a donor to an acceptor cavity. Results from recent molecular dynamics simulations suggest that the free volume transport mechanism is more likely to be operative in the core of the bilayer [84]. In the more ordered region of the bilayer, a kink shift diffusion mechanism is more likely to occur [84,94]. Kinks may be pictured as dynamic structural defects representing small, mobile free volumes in the hydrocarbon phase of the membrane, i.e., conformational kink g tg ) isomers of the hydrocarbon chains resulting from thermal motion [52] (Fig. 8). Small molecules can enter the small free volumes of the kinks and migrate across the membrane together with the kinks. 

In the case of interstitials—self-interstitials, impurities, or dopants—two diffusion mechanisms can be envisaged. In the simplest case, an interstitial can jump to a neighboring interstitial position (Fig. 5.8a). This is called interstitial diffusion and is sometimes referred to as direct diffusion to distinguish it from vacancy diffusion (indirect diffusion). 

The vacancy will follow a random-walk diffusion route, while the diffusion of the tracer by a vacancy diffusion mechanism will be constrained. When these processes are considered over many jumps, the mean square displacement of the tracer will be less than that of the vacancy, even though both have taken the same number of jumps. Therefore, it is expected that the observed diffusion coefficient of the tracer will be less than that of the vacancy. In these circumstances, the random-walk diffusion equations need to be modified for the tracer. This is done by ascribing a different probability to each of the various jumps that the tracer may make. The result is that the random-walk diffusion expression must be multiplied by a correlation factor, / which takes the diffusion mechanism into account. 

The high conductivity of (3-alumina is attributed to the correlated diffusion of pairs of ions in the conduction plane. The sodium excess is accommodated by the displacement of pairs of ions onto mO sites, and these can be considered to be associated defects consisting of pairs of Na+ ions on mO sites plus a V N l on a BR site (Fig. 6.12a and 6.12b). A series of atom jumps will then allow the defect to reorient and diffuse through the crystal (Fig. 6.12c and 6.12d). Calculations suggest that this diffusion mechanism has a low activation energy, which would lead to high Na+ ion conductivity. A similar, but not identical, mechanism can be described for (3"-alumina. 

For a random walk, f = 1 because the double sum in Eq. 7.49 is zero and Eq. 7.50 reduces to the form of Eq. 7.47. In principle, f can have a wide range of values corresponding to physical processes relating to specific diffusion mechanisms. This is readily apparent in extreme cases of perfectly correlated one-dimensional diffusion on a lattice via nearest-neighbor jumps. When each jump is identical to its predecessor, Eq. 7.49 shows that the correlation factor f equals NT.6 Another extreme is the case of f = 0, which occurs if each individual jump is exactly opposite the previous jump. However, there are many real diffusion processes that are nearly ideal random walks and have values of f 1, which are described in more detail in Chapter 8. 

At more elevated temperatures, the diffusion mechanisms become more complex and jumps to more distant sites occur, as do collective jumps via multiple defects. At still higher temperatures, adatoms apparently become delocalized and spend significant fractions of their time in flight rather than in normal localized states. In many cases, the Arrhenius plot becomes curved at these temperatures (as in Fig. 9.1), due to the onset of these new mechanisms. Also, the diffusion becomes more isotropic and less dependent on the surface orientation. 

Molecular reorientations at Bjerrum fault sites are responsible for the dielectric properties of ice. A second type of fault (proton jumps from one molecule to a neighbor) accounts for the electrical conductivity of ice, but cannot account for the high dielectric constant of ice. Further discussion of such ice faults is provided by Franks (1973), Franks and Reid (1973), Onsager and Runnels (1969), and Geil et al. (2005), who note that interstitial migration is a likely self-diffusion mechanism. 

A number of diffusion mechanisms in crystalline solids are possible. Atoms vibrate in their equilibrium sites after that, periodically, these oscillations turn out to be large enough to give rise to a jump from one site to the other. The order of magnitude of the frequency of these oscillations is about 1012-1013 Hz. In this regard, it has been shown that the jump rate at which an atom jumps into an empty neighboring site is given by [30]... 

In the case of H-ZSM-11, the results reported in Table 5.4 indicate that the mechanism for p- and o-xylene diffusion is by hopping between sites or jump diffusion, since the values reported for D0 are within the limits 1 x 10 4 < < 5 x 10" cm2/s. Hence, the values of the pre-exponential... 

The first qualitative observation of vacancy-induced motion of embedded atoms was published in 1997 by Flores et al. [20], Using STM, an unusual, low mobility of embedded Mn atoms in Cu(0 0 1) was observed. Flores et al. argued that this could only be consistent with a vacancy-mediated diffusion mechanism. Upper and lower limits for the jump rate were established in the low-coverage limit and reasonable agreement was obtained between the experimentally observed diffusion coefficient and a theoretical estimate based on vacancy-mediated diffusion. That same year it was proposed that the diffusion of vacancies is the dominant mechanism in the decay of adatom islands on Cu(00 1) [36], which was also backed up by ab initio calculations [37]. After that, studies were performed on the vacancy-mediated diffusion of embedded In atoms [21-23] and Pd atoms [24] in the same surface. The deployment of a high-speed variable temperature STM in the case of embedded In and an atom-tracker STM in the case of Pd, allowed for a detailed quantitative investigation of the vacancy-mediated diffusion process by examining in detail both the jump frequency as well as the displacement statistics. Experimental details of both setups have been published elsewhere [34,35]. A review of the quantitative results from these studies is presented in the next subsections. 

For exchange with a copper adatom the diffusion mechanism is illustrated in Fig. 3. In the measurements, embedded indium atoms are observed to make jumps of several atomic spacings. For the adatom mechanism, this would imply that the indium adatoms reinsert themselves into the terrace after making... 

In Section 3 we derive that for the vacancy-mediated diffusion mechanism, one expects the shape of the jump length distribution to be that of a modified Bessel function of order zero. Both distributions can be fit very well with the modified Bessel function, again confirming the vacancy-mediated diffusion mechanism for both cases. The only free parameter used in the fits is the probability prec for vacancies to recombine at steps, between subsequent encounters with the same embedded atom [33]. This probability is directly related to the average terrace width and variations in this number can be ascribed to the proximity of steps. The effect of steps will be discussed in more detail in Section 4. 

It appears that the short time dynamics of water molecules at or near the hydrophilic model surface and at a soluble protein surface is much slower than that of the bulk water. It is important to note that the more significant slow dynamics of interfacial water is reflected in the long residence time for jump diffusion. This suggests that there may be a common underlying mechanism for the slowing down of the single-particle dynamics of interfacial water. 

The Haven ratio may deviate from unity when correlation effects and possibly different jump distances and jump frequencies can not be neglected [51]. For a vacancy diffusion mechanism Hr equals the well-known tracer correlation factor /. 

The dependence of the inverse of the characteristic times for water translation does not show exclusively the linear behavior of continuous diffusion (Equation 3.1) or exclusively the shape of a simple jump diffusion model (Equation 3.2), but is very well represented (lines) by considering that both mechanisms contribute to the relaxation (Equation 3.3). The symbols correspond to the simulation data at different temperatures 365 K (circles), 335 K (squares), and 310 K (triangles). 

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