Molecular jump length

Peppas and Reinhart have also proposed a model to describe the transport of solutes through highly swollen nonporous polymer membranes [155], In highly swollen networks, one may assume that the diffusional jump length of a solute molecule in the membrane is approximately the same as that in pure solvent. Their model relates the diffusion coefficient in the membrane to solute size as well as to structural parameters such as the degree of swelling and the molecular weight between crosslinks. The final form of the equation by Peppas and Reinhart is... 

The process of molecular diffusion may be viewed conceptionally as a sequence of jumps with statistically varying jump lengths and residence times. Information about the mean jump length /(P and the mean residence time t, which might be of particular interest for a deeper understanding of the elementary steps of catalysis, may be provided by spectroscopic methods, in particular by quasielastic neutron scattering (see next Section) and nuclear magnetic resonance (NMR). 

The broadening values obtained for methane diffusion can be fitted to a molecular jump model with a Gaussian distribution of jump lengths (25,62). Thus, the scattering law becomes a Lorentzian,... 

In contrast, for zeolite NaX the reduction of the molecular translational mobility with increasing sorbate concentration is mainly due to the reduction of the jump lengths, with jump rates practically unaffected by concentration. For hydrocarbons in NaX, the jump lengths are found to be corre-... 

Simulations of C NMR lineshapes have shown that experimental spectra that appear to result from a superposition of two different lines (cf. Fig. 15) can be explained by the above-mentioned molecular jump model. Analogous conclusions were drawn from macroscopic sorption kinetic data (82). From the experimental C NMR lineshapes, a mean residence time tj of 20 and 150 p-s for a concentration of six molecules per u.c. at 250 and 200 K, respectively, was derived. Provided that these jumps detected in C NMR spectroscopy are accompanied by a translational motion of the molecules, it is possible to derive self-diffusivities D from the mean residence times. Assuming the diffusion path of a migrating molecule as a sum of individual activated jumps, for isotropic systems the relation (P) = 6Dtj is valid, where (P) denotes the mean square jump length. Following experimental and theoretical studies on the preferential sorption sites of benzene molecules in the MFI framework (83-90), in our estimate the mean distance between adjacent sorption sites is assumed to be 1 nm. 

It has been demonstrated that the combined application of various NMR techniques for observing molecular rotations and migrations on different time scales can contribute to a deeper understanding of the elementary steps of molecular diffusion in zeolite catalysts. The NMR results (self-diffusion coefficients, anisotropic diffiisivities, jump lengths, and residence times) can be correlated with corresponding neutron scattering data and sorption kinetics as well as molecular dynamics calculations, thus giving a comprehensive picture of molecular motions in porous solids. 

These last two equations are derived on the basis of the Eyring theory of holes in liquids. The assumptions here, in contrast to those of the Stokes-Einstein equations, are that the diffusing molecules are of the same order of size as those of the solvent. The discontinuity of the liquid medium thus plays an essential part in the Eyring theory, the fundamental length X being the distance between successive positions of the diffusing solute or solvent molecule as it jumps between the molecules of the liquid. The quantities D and )/, however, refer to the diffusion constant and the viscosity of the system measured in the usual way. They represent the observed effect of very large numbers of such molecular jumps. 

FC data set at 30 Larmor frequencies and two director orientations makes use of more than 50 Ti v pairs at one constant temperature to find, by iterative steps, up to 11-14 unknown quantities. It is evident that not all the model characteristics could be evaluated reliably because of very strong correlations in the fitting process. In particular the fitting of both OF-mode cut-off frequencies of the anisotropy ratios trand e, and of the jump length parameter a, proved rather insensitive near physically plausible constraints. Some results obtained in this way are listed in Table 1 to illustrate differences between 5CB and 8CB. These data have been interpreted with astonishing success by means of the molecular geometry, i.e. by spin-pair orientations and separations, and available visco-elastic material constants, i.e. 

The main interest of the QNS technique lies, however, in the S (Q, to) behaviour at higher Q values, i.e. when the translational process is followed on molecular distances. A departure from Pick s law is generally observed ". This is indicative of the microscopic aspect of the motion and its interpretation needs the elaboration of more sophisticated models involving characteristic jump lengths, jump times, etc. 

In the earlier NMR studies the self-diffusivity was derived indirectly from measurements of the spin-lattice relaxation time over a range of temperatures. The correlation time which is roughly equivalent to the average time between successive molecular jumps, may be derived from such information and the self-diffusivity is then estimated from the Einstein relation using an assumed mean square jump length A. For an isotropic cubic lattice... 

The jump length a which has to be inserted in the equations to account for the experimental data turns out to be abnormally large (orders of magnitude larger than the molecular size). 

Upon employment of refined instrumentation, it was possible to estimate the average distance between nanopores in a 60% cross-linked rigid resin and thereby the jump length and the jump frequencies of the molecular water random walk process. These turned out to have the approximate value of 12-13 A and 1.5 x 10 -135x10 Hz for temperature varying between 5 and 90°C (Soles and Yee 2000). ... 

In a second set of experiments [63], the same measurements were carried out on the isotopic molecule HD. The same jump diffusion model holds for this molecule, with Thd = 0 33 nm. As can be expected from simple arguments, the mean jump lengths are found to be in the ratio of the square root of their molecular masses ( Th2/Thd = V 3/2 ). On the other hand the temperature dependence of the diffusion coefficients (which are of the order of 10 m s ) points out an unexpected feature. As a matter of fact, one can represent each result on an Arrhenius coordinate system Ln(D) = f(l/T) but this leads to different Dq values and different activation energies, namely Eh2 = 240 K and Ehd = 337 K and one... 

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