Single-file systems with

Under the assumption that molecular propagation in single-file systems proceeds by activated jumps of step length I with a mean life time r between succeeding jump attempts, and that jump attempts are only successful if they are directed to a vacant site, the mobility factor may be shown to be given by the relation [10]... 

With this more general equation, representation of the effectiveness factor in terms of the Thiele modulus also becomes possible for single-file diffusion. As an example. Fig. 16 shows the result of a computer simulation of diffusion and reaction within a single-file system consisting of A = 100 sites for different occupation numbers in comparison with the dependence... 

Since normal and single-file diffusion are described by the same propagator, Eq. 5, due to the analogy of Eq. 2 and Eq. 3 the propagation pattern of a given particle in a single-file system coincides with that of normal diffusion... 

Fig. 2 Various time regimes of molecular propagation in single-file systems as resulting from MD simulations. The inset indicates the channel diameters considered in the different simulations. In all cases, the diameter of the diffusants was assumed to be equal to 0.383 nm. From [35] with permission...

Experiments with single-file systems of finite extension are thus found to be easily affected by the influence of the boundary conditions. Therefore, one should be aware of the fact that the observed displacements are sufficiently below the hmiting values for which the boimdary conditions start to become relevant. 

Eq. 21 with Eq. 23 results as the solution of the corresponding differential equation of normal diffusion with the appropriate initial and boundary conditions. These relations hold with the adequate interpretation of D as a self-diffusivity or a transport diffusivity, respectively, for both tracer exchange between the initially adsorbed species A by species B and the relative uptake in an adsorption experiment. It should be noted that Eq. 21 also describes the molecular uptake by single-file systems, since with respect to adsorption/desorption there are no differences between single-file systems and systems which permit normal diffusion. 

Fig. 3 The normalized tracer exchange curves in single-file systems obtained by dynamic Monte Carlo (DMC) simulations for various file lengths (=L in the figiu e) and loadings (0) (points). The dashed and solid lines show the best fit lines for 0 = 0.5 with the slope of 1/2 and 1/4 expected for the mechanism of normal and single-file diffusion, respectively, in the limit of short times. From [57] with permission...

Fig.4 Comparison of the concentration profiles of tagged particles obtained by DMC simulations for tracer exchange in single-file systems of length L (oscillating solid lines) with the concentration profiles for normal diffusion, with Dsim and N given in Table 1 (solid lines) at times ti = 0.93 x 10 r, t2 = 2.1 x 10 r, t = 3.7 x 10 , and t4 = 7.6 x 10 r (r is the duration of the elementary diffusion step). From [57] with permission...

Fig. 5 Concentration profiles of the molecules of species B within the single-file system under stationary conditions and comparison with the dependence to be expected for ordinary diffusion (broken line, Eq. 29). The quantity 2L(klDy (the Thiele modulus 0) in Eq. 29 has been chosen to coincide with the generalized Thiele modulus (cf. Eq. 30) of the single-file reaction for k = 1.27 x 10 (0 = 2.77). z denotes the distance from the middle of the file and L = NX its length. From [1] with permission...

Our understanding of diffusion and reaction in single-file systems is impaired by the lack of a comprehensive analytical theory. The traditional way of analytically treating the evolution of particle distributions by differential equations is prevented by the correlation of the movement of distant particles. One may respond to this restriction by considering joint probabilities covering the occupancy and further suitable quantities with respect to each individual site. These joint probabilities may be shown to be subject to master equations. 

Fig. 8 Average residence time profile (in units of the time r between two jump attempts) of the particles in the single-file systems considered in Fig. 7. From [72] with permission...

The benefit of the analytical treatment presented thus far for the calculation of the characteristic functions of the single-file system is only limited by the increasing complexity of the joint probabilities and the related master equations. This treatment, however, has suggested a most informative access to the treatment of systems subjected to particle exchange with the surroundings and to internal transport and reaction mechanisms [74,75]. Summing over all values (Ji = 0 and 1 and, subsequently, over all sites i, Eq. 31 may be transferred to the relation Eq. 34... 

It should be emphasized, however, that likely in none of these studies was the zeolite material of such an ideal structure as implied in data analysis. In this respect, experimental studies with artificially created single-file systems [127-129] may provide a much higher rehability of the pre-supposed structural features. A treatise on the substantial deviations of the real structure, with particular emphasis on the consequences for ideal host systems for single-file diffusion as evidenced by optical techniques, is given in [95]. Irrespective of these limitations, however, a number of peculiarities of catalytic reactions in zeolites with one-dimensional channel systems are most Ukely to be attributed to the special conditions of molecular transport and molecular arrangement imder single-file conditions. 

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