Jump diffusion coefficient

The measurement of relaxation times 7j and T2 and the subsequent application of the theory formulated by Bloembergen et al. (236), and extended by Kubo and Tomita (272) and Torrey (288), leads to the determination of motional and thermodynamic parameters such as mean times between molecular jumps, diffusion coefficients, and activation enthalpies for translation. For example, Resing and Thompson (289, 290) used this... 

Garcia-Belmonte, G. et al. 2006. Jump diffusion coefficient of different cations intercalated into amorphous WO3. Solid State Ionics, TI, 1635-1637. 

First, the jump diffusion coefficient Dj is introduced, which is proportional to the tracer diffusion coefficient, D, that reflects random walks of a particle ... 

Consequently, Dj can be calculated by Monte Carlo simulation [13, 160, 161]. In the multiple trapping framework the jump diffusion coefficient is given by [49]... 

Combining the general form of the chemical and jump diffusion coefficient, (86) and (87), we have, for the exponential distribution. 

It is important to note that (97) indicates that the conductivity is determined exclusively by the transport level and is completely independent of the presence and distribution of traps, in the context of the multiple trapping model that we have used herein. The steady-state conduction is not affected by the trapping process because the traps remain in equilibrium. Alternatively, one can view conduction as the result of the displacement of the whole electron density, n, with a smaller jump diffusion coefficient see (96). 

Adsorption Kinetics. In zeoHte adsorption processes the adsorbates migrate into the zeoHte crystals. First, transport must occur between crystals contained in a compact or peUet, and second, diffusion must occur within the crystals. Diffusion coefficients are measured by various methods, including the measurement of adsorption rates and the deterniination of jump times as derived from nmr results. Factors affecting kinetics and diffusion include channel geometry and dimensions molecular size, shape, and polarity zeoHte cation distribution and charge temperature adsorbate concentration impurity molecules and crystal-surface defects. 

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... 

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

Diffusion has often been measured in metals by the use of radioactive tracers. The resulting parameter, DT, is related to the self-diffusion coefficient by a correlation factor/that is dependent upon the details of the crystal structure and jump geometry. The relation between DT and the self-diffusion coefficient Dsclf is thus simply... 

Referring to the Pick Equations of diffusion, let us now reexamine 1/T, the jump frequency, so as to relate diffusion processes to lattice vibration processes. The reciinrocal of the time of stay is the jump frequency and is related to the diffusion coefficient by ... 

The values derived in this way for the diffusion coefficients exhibit surprising agreement with the experimental values, even for small ions, better than a coincidence in the order of magnitude. More detailed theoretical analysis indicates that the formation of holes required for particle jump is analogous to the formation of holes necessary for viscous flow of a liquid. Consequently, the activation energy for diffusion is similar to that for viscous flow. 

When (DEB), is much smaller than unity, the polymer relaxation is relatively rapid compared to diffusion. In this case, conformational changes take place instantaneously and equilibrium is attained after each diffusional jump. This is the type of diffusion encountered ordinarily and is called viscous diffusion. Therefore, the transport will obey classical theories of diffusion. When (DEB), is much larger than unity, the molecular relaxation is very slow compared to diffusion and there are no conformational changes of the medium within the diffusion time scale. In this case, Fick s law is generally valid, but no concentration dependence of the diffusion coefficient is expected. This is termed elastic diffusion. When (DEB), is in the neighborhood of unity, molecular rearrangment... 

After the jump, the particle is taken to have reacted with a given probability if its distance from another particle is within the reaction radius. For fully diffusion-controlled reactions, this probability is unity for partially diffusion-controlled reactions, this reaction probability has to be consistent with the specific rate by a defined procedure. The probability that the particle may have reacted while executing the jump is approximated for binary encounters by a Brownian bridge—that is, it is assumed to be given by exp[—(x — a)(y — a)/D St], where a is the reaction radius, x andy are the interparticle separations before and after the jump, and D is the mutual diffusion coefficient of the reactants. After all... 

One of earliest approaches of estimating the diffusion coefficient through a polymer carrier is that of Eyring (1936). In this theory, diffusion of a solute through a medium is presented as a series of jumps instead of a continuous process. Therefore, in Eq. (18) in Table I, which comes from the Eyring analysis, X is the diffusional jump of the drug in the polymer and v is the frequency of jumping. 

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 simplest and most basic model for the diffusion of atoms across the bulk of a solid is to assume that they move by a series of random jumps, due to the fact that all the atoms are being continually jostled by thermal energy. The path followed is called a random (or drunkard s) walk. It is, at first sight, surprising that any diffusion will take place under these circumstances because, intuitively, the distance that an atom will move via random jumps in one direction would be balanced by jumps in the opposite direction, so that the overall displacement would be expected to average out to zero. Nevertheless, this is not so, and a diffusion coefficient for this model can be defined (see Supplementary Material Section S5). 

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 list below shows the last position reached, in units of the jump step a, during a random walk for 100 atoms, each of which makes 200 jumps. If the jump time is 10-3 s and the jump distance, a, is 0.3 nm, estimate the diffusion coefficient (a) in units of a2 s-1 and (b) in units of m2 s-1 ... 

Fick s (continuum) laws of diffusion can be related to the discrete atomic processes of the random walk, and the diffusion coefficient defined in terms of Fick s law can be equated to the random-walk displacement of the atoms. Again it is easiest to use a one-dimensional random walk in which an atom is constrained to jump from one... 

Pulsed field gradient (PFG)-NMR experiments have been employed in the groups of Zawodzinski and Kreuer to measure the self-diffusivity of water in the membrane as a function of the water content. From QENS, the typical time and length scales of the molecular motions can be evaluated. It was observed that water mobility increases with water content up to almost bulk-like values above T 10, where the water content A = nn o/ nsojH is defined as the ratio of the number of moles of water molecules per moles of acid head groups (-SO3H). In Perrin et al., QENS data for hydrated Nation were analyzed with a Gaussian model for localized translational diffusion. Typical sizes of confining domains and diffusion coefficients, as well as characteristic times for the elementary jump processes, were obtained as functions of A the results were discussed with respect to membrane structure and sorption characteristics. ... 

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