Jump diffusion

There is not sufficient experimental evidence to continue this discussion quantitatively at the present time, but the sparse experimental data suggests that for a given compound, the Dq value is significantly lower than is the case in simple metals. This decrease may be attributed to a low value in the conelation factor which measures the probability drat an atom may either move forward or return to its original site in its next diffusive jump. In simple metals this coefficient has a value around 0.8. 

In tire transition-metal monocarbides, such as TiCi j , the metal-rich compound has a large fraction of vacairt octahedral interstitial sites and the diffusion jump for carbon atoms is tlrerefore similar to tlrat for the dilute solution of carbon in the metal. The diffusion coefficient of carbon in the monocarbide shows a relatively constairt activation energy but a decreasing value of the pre-exponential... 

In addition to the adsorption and desorption explained in Sec. II A, we can also include the diffusion process within our master-equation formahsm (10) [47]. For this purpose, we must only include the supplementary channel of the diffusive transition into the right-hand side of Eq. (10). A diffusion jump... 

Here the lattice positions i and j should be adjacent and the -function assures that one of the two lattice positions is occupied and the other one is free, r/j is a characteristic time scale for a diffusion jump. The time-dependence of the average si) is calculated by approximating the higher moments (siSj) [49]. In practice the analysis is rather involved, so we do not give further details here. An important result, for example, is the correction to the Wilson-Frenkel rate (33) at high temperatures ... 

A formalism similar to that used for partially adiabatic proton transfer reactions was applied in the calculation of the transition probability. This model of the diffusion jump is similar to the model of the diffusion of light defects in solids which was first considered in Ref. 62. 

The diffusion-jump model for SBA and VML binding to Tn-PSM and the other PSM analogues can be envisioned as occurring with one subunit of SBA or VML bound to one aGalNAc residue of Tn-PSM at a time (Fig. 3A). (Two subunits of individual SBA or VML molecules simultaneously binding to a single Tn-PSM chain is not supported by the enhanced affinities of both lectins to 38/40-mer Tn-PSM relative to aGalNAcl-G-Ser (Table I). If two subunits of an SBA tetramer were bound to 38/40-mer Tn-PSM, the... 

The form of h(f) is discussed further in Chap. 8, Sect. 3.2. Noyes [269] argued that h(t) is related to the diffusion jump size and frequency. While not disputing this suggestion, recombination probability experiments are probably not the means to study the general details of diffusive motion, especially when there are several other unknown parameters (e.g. feact and r0) to be determined from experimental studies (see Sect. 3). 

In some cases, particularly in the growth of aerosol particles, the assumption of equilibrium at the interface must be modified. Frisch and Collins (F8) consider the diffusion equation, neglecting the convective term, and the form of the boundary condition when the diffusional jump length (mean free path) becomes comparable to the radius of the particle. One limiting case is the boundary condition proposed by Smoluchowski (S7), C(R, t) = 0, which presumes that all molecules colliding with the interface are absorbed there (equivalent to zero vapor pressure). A more realistic boundary condition for the case when the diffusion jump length, (z) R, has been shown by Collins and Kimball (Cll) and Collins (CIO) to be... 

An argument in favour of the correctness of the values of R calculated with the help of eqn. (26) is a correlation detected in ref. 28 between the values of R found in liquid solutions and the values of Rtatt 10 1 s found in solid solutions for the same acceptors (Fig. 25). Such a correlation must exist if the mechanism of electron transfer in either case is tunneling. The lower values of R, as compared with those of Rt, for the reactions of eaq and e(j with the same acceptor are accounted for in a natural way in terms of the tunneling mechanism by the difference of the characteristic times during which there occurs a tunneling. In liquids the characteristic time is the time of a diffusion jump at T = 300 K, 10 10 s, in solids, it is the time between the end of irradiation and the measurement of the radiation yield of et r, t 103s. 

To estimate the activation energy of diffusion at lower temperatures let us assume the pre-exponential factor Du to be independent of T and, by the order of magnitude, to be equal to D = A2vk % 10 4cm2s 1 (A % 10 8 cm is the characteristic value of a diffusion jump, vk 1012s 1 is the characteristic frequency of atomic oscillations in a solid). Then from the experimental values of D one can find the activation energy of diffusion Ea = 9 kcalmol 1... 

We note, finally, that with the help of equations (12) with a concrete form of the function F [e.g., (26)], it is also possible to solve the very interesting problem of the diffusion jump of fuel across the flame zone as is shown in Fig. 3, the concentration of the mutually penetrating substances in the transition across the reaction zone falls sharply, but does not become zero. Since the temperature and reaction rate also fall on both sides of the reaction zone, the concentration of fuel which has already reached a certain distance from the flame in the oxidation zone no longer changes. 

The high frequency relaxation is attributed in part to the modulation of intermolecular dipolar interactions by the translational diffusion. The cutoff frequency (60 MHz at 55°C) corresponds to the local diffusive jump frequency that is estimated from measurements of the diffusion coefficient (D 10"6 cm2/sec at 55°) (19, 21). This cutoff frequency also varies in temperature with the same activation energy (Eact 0.25 eV) as the diffusion frequency. 

Discussion. We can now propose a coarse description of the paraffinic medium in a lamellar lyotropic mesophase (potassium laurate-water). Fast translational diffusion, with D 10"6 at 90 °C, occurs while the chain conformation changes. The characteristic times of the chain deformations are distributed up to 3.10"6 sec at 90 °C. Presence of the soap-water interface and of neighboring molecules limits the number of conformations accessible to the chains. These findings confirm the concept of the paraffinic medium as an anisotropic liquid. One must also compare the frequencies of the slowest deformation mode (106 Hz) and of the local diffusive jump (109 Hz). When one molecule wants to slip by the side of another, the way has to be free. If the swinging motions of the molecules, or their slowest deformation modes, were uncorrelated, the molecules would have to wait about 10"6 sec between two diffusive jumps. The rapid diffusion could then be understood if the slow motions were collective motions in the lamellae. In this respect, the slow motions could depend on the macroscopic structure (lamellar or cylindrical, for example)... 

Transport, A companion transport model that also acknowledges the fact that penetrant may execute diffusive jumps into and out of the two sorption environments expresses the local flux, N, at any point in the polymer in terms of a two part contribution (17-20) ... 

Of course, this equation can also be expressed in an equivalent form in terms of Q, the activation energy per mole of diffusion jumps ... 

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