Jump lengths

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

The deviations from Gaussian behaviour were successfully interpreted as due to the existence of a distribution of finite jump lengths underlying the sublinear diffusion of the proton motion [9,149,154]. A most probable jump distance of A was found for PI main-chain hydrogens. With the model... 

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... 

As mentioned, the Peppas-Reinhart theory is valid in the case of highly swollen membranes. Additional work by Peppas and Moynihan [158] resulted in a theory for the case of moderately swollen networks. This theory was derived much like the Peppas-Reinhart theory with the exceptions that in a moderately swollen network, one may not assume that the diffusional jump length of the solute in the membrane, X2, i3, is equal to the diffusional jump length of the solute in pure solvent, X2, i and, also, one may not assume that the free volume of the polymer/solvent system is equal to the free volume of the solvent. The initial... 

A one-dimensional random walk is not necessarily symmetric with respect to jumps toward the right and toward the left. If the chemical potential gradient is sufficiently weak we may still approximate the jump length distribution by an exponentially decaying function, but distinguish that toward the right from that toward the left. 

This jump length distribution corresponds to the nearest neighbor, discrete random walk of jump length /. 

In a one-dimensional random walk with excursion from the origin, the mth moment of the jump length distribution can be derived from... 

As in (VIII.2.1) we write IT as a function of starting point and jump length r = X-X... 

The dependence on r describes the relative probability of various jump lengths, while the dependence on X determines the overall transition prob-... 

In Table 16-2, the time scale for elementary activated motion is given in the first place. It is converted into an energy scale by virtue of the E = (2n-h/t) relation, If we assume that the atomic jump length a is 2 A, the time scale may be converted into a diffusion coefficient scale by D = az/(2-t). One notes that (with the exception of /J-NMR) nuclear spectroscopies monitor the atomic jump behavior of relatively fast diffusing species. 

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... 

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