Single-file systems

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]... 

For an estimate of the correlation between t he intracrystalline mean life time and the system properties, the exchange curve between the (labelled) molecules of a single-file system and the (unlabelled) surroundings ( tracer exchange curve ) may be assumed to be determined by a single dimensionless parameter... 

Combining eqs. 2, 5, 7, and 9, for sufficiently large Thiele moduli (i.e. in the transport-limited case), the effectiveness factor of single-file systems is found to obey the relation... 

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... 

Dsim Modified (center-of-mass) diffusivity in a single-file system... 

Mean time between two jump attempts, residence time Mean exchange time between two adjacent molecules Average time that a particle which is foimd at site i, has already spent in the single-file system... 

The mean square displacement in single-file systems may quite generally be shown to be related to the movement of a sole molecule by the expression [8,10]... 

Combining Eqs. 3, 6 and 7, the mobility factor F of a single-file system and the diffusivity of a sole molecule in this system may be shown to be related to... 

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... 

Molecular dynamics (MD) simulations in single-file systems are additionally comphcated by the requirement that in the absence of external forces the center of mass must be preserved. This comphcation results from the fact that, as a consequence of the correlated motion in a single-file system, the shift of a particular molecule must be accompanied by shifts of other molecules in the same direction. Depending on the total amoimt of molecules under consideration, the conservation of the center of mass therefore prohibits arbitrarily large molecular shifts. The maximum mean square displacement may be shown to obey the relation [22]... 

The great expense in calculation time due to the inevitably large particle numbers in single-file systems calls for the application of simplified potentials. Figure 1 shows the results obtained for spherical molecules diffusing in an unstructured tube [22]. Particle-particle and particle-wall interactions have been simulated by a shifted-force Lennard-Jones potential [26] and an... 

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...

The concept of single-file diffusion has most successfully been applied for MD simulations in carbon nanotubes [36-39], yielding both the square-root time dependence of the molecular mean square displacement and a remarkably high mobility of the individual, isolated diffusants. In [40-42], the astonishingly high single-particle mobilities in single-file systems have been attributed by MD simulations to a concerted motion of clusters of the adsorbed molecules. 

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. 

Since the two limiting cases of open and closed ends have been shown to lead, respectively, to an enhancement and a reduction of the mean square displacement in comparison to an infinite single-file system, it may be anticipated that, imder the influence of boundary conditions intermediate between these two hmiting cases, molecular propagation in a finite single-file system may even proceed as in a single-file system of infinite extension. 

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. 

As soon as the movements of different species have to be distinguished from each other, however, the mutual correlation of the molecules in singlefile systems makes it impossible to predict the evolution of the particle distributions by differential equations. Eor this reason, the time dependence of the tracer exchange in single-file systems has thus far only been investigated by Monte Carlo simulations [1,55-57]. 

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...

As in the case of tracer exchange, quantitative information about the correlated effect of transport and catalytic reactions in single-file systems has... 

SO far only been attained by Monte Carlo simulations. Figure 5 illustrates the situation due to the combined effect of diffusion and catalytic reaction in a single-file system for the case of a monomolecular reaction A B [1]. For the sake of simplicity it is assumed that the molecular species A and B are completely equivalent in their microdynamic properties. Moreover, it is assumed that in the gas phase A is in abimdance and that, therefore, only molecules of type A are captured by the marginal sites of the file. Figure 5 shows the concentration profile of the reaction product B within the singlefile system imder stationary conditions. A parameter of the representation is the probabiUty k that during the mean time between two jump attempts (t), a molecule of type A is converted to B. It is related to the intrinsic reactivity k by the equation... 

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