Diffusion Jump parameters

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). 

Equation 7.48 relates the macroscopic diffusivity and microscopic jump parameters for uncorrelated diffusion in one dimension. 

Somewhat closer to the designation of a microscopic model are those diffusion theories which model the transport processes by stochastic rate equations. In the most simple of these models an unique transition rate of penetrant molecules between smaller cells of the same energy is determined as function of gross thermodynamic properties and molecular structure characteristics of the penetrant polymer system. Unfortunately, until now the diffusion models developed on this basis also require a number of adjustable parameters without precise physical meaning. Moreover, the problem of these later models is that in order to predict the absolute value of the diffusion coefficient at least a most probable average length of the elementary diffusion jump must be known. But in the framework of this type of microscopic model, it is not possible to determine this parameter from first principles . 

It is interesting to attempt a simple theory of the parameter Tq. Relaxation of p occurs at the cluster surfaces via a diffusive jump across the interface. Using the free-volume model of self-diffusion (Section VII), we obtain the following for Tq,... 

Even if the jump parameters, obtained by fitting the different models to the data, can vary, the diffusion coefficient should stay constant if there are enough experimental points at small Q values, since all models share the same broadening behavior in this Q range of the form DQ. ... 

Transport Properties. Sorption and transport properties are highly dependent on the post-vitrification history of glassy polymers (77) hence one would expect parameters such as physical aging, antiplasticization and amorphous orientation to affect transport properties. The reduction in diffusivity and permeability due to aging, orientation, and antiplasticization can be modeled via entropy or fi ee volume arguments (77). In addition, diffusive jumps of penetrant molecules in glassy polymers can be affected by (facilitated by) the segmental mobility that is manifested in sub-Tg relaxations 78),... 

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]. 

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... 

The perturbation theory presented in Chapter 2 implies that orientational relaxation is slower than rotational relaxation and considers the angular displacement during a free rotation to be a small parameter. Considering J(t) as a random time-dependent perturbation, it describes the orientational relaxation as a molecular response to it. Frequent and small chaotic turns constitute the rotational diffusion which is shown to be an equivalent representation of the process. The turns may proceed via free paths or via sudden jumps from one orientation to another. The phenomenological picture of rotational diffusion is compatible with both... 

Fig. 4.16 Time evolution of the mean squared displacement (r ) (empty circle) at 363 K and the non-Gaussian parameter 2 obtained from the simulations at 363 K (filled circle) for the main chain protons of PL The solid vertical arrow indicates the position of the maximum of 2> At times r>r(Qinax)> the crossover time, a2 assumes small values, as in the example shown by the dotted arrows. The corresponding functions (r ) and a2 are deduced from the analysis of the experimental data at 320 K in terms of the jump anomalous diffusion model and are displayed as solid lines for (r )and dashed-dotted lines for a2- (Reprinted with permission from [9]. Copyright 2003 The American Physical Society)...

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... 

These equations can also be used to derive diffusion parameters. For example, within the nearest neighbor jump approximation, we have from the last equation,... 

An Arrhenius analysis, when properly applied, can provide diffusion parameters and clarifies the energetics of surface diffusion, but does not give any information about the geometrical aspect of these atomic jumps and how these jumps are related to the atomic structure of the substrate. Such information can be derived from displacement distributions. A... 

Auerbach et al. (101) used a variant of the TST model of diffusion to characterize the motion of benzene in NaY zeolite. The computational efficiency of this method, as already discussed for the diffusion of Xe in NaY zeolite (72), means that long-time-scale motions such as intercage jumps can be investigated. Auerbach et al. used a zeolite-hydrocarbon potential energy surface that they recently developed themselves. A Si/Al ratio of 3.0 was assumed and the potential parameters were fitted to reproduce crystallographic and thermodynamic data for the benzene-NaY zeolite system. The functional form of the potential was similar to all others, including a Lennard-Jones function to describe the short-range interactions and a Coulombic repulsion term calculated by Ewald summation. 

Fig. 32a, b. Temperature variation of a. the geometrical parameters jump distance r and distance L between neighbouring chains, b. Residence time x and diffusion constant D... 

Solutions for this type of kinetics can only be achieved numerically. Model calculations with constant kinetic parameters have been made [H. Wiedersich, et al. (1979)], however, the modeling of realistic transport (diffusion) coefficients which enter into the flux equations is most difficult since the jump rate vA vB. Also, the individual point defects have limited lifetimes which determine the magnitude of correlation factors (see Section 5.2.2). Explicit modeling for dilute or non-dilute alloys can be found in [A.R. Allnatt, A.B. Lidiard (1993)]. 

Equation 7.35 is a fundamental relationship between the diffusivity and the mean-square displacement of a particle diffusing for a time r. Because diffusion processes in condensed matter are comprised of a sequence of jumps, the mean-square displacement in Eq. 7.31 should be equivalent to Eq. 7.35. This equivalence, as demonstrated below, results in relations between macroscopic and microscopic diffusion parameters. 

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