Single-file systems displacements

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]... 

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 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. 

Another deviation from the pattern of ordinary diffusion must be expected if the reactant and product molecules are subjected to single-file conditions, i.e. if (i) the zeolite pore system consists of an array of parallel channels and if (ii) the molecules are too big to pass each other. In this case, the molecular mean-square displacement z t)) is found to be proportional to the square root of the observation time, rather than to the observation time itself. First PFG NMR studies of such systems are in agreement with this prediction [8]. By introducing a mobility factor F, in analogy to the Einstein relation for ordinary diffusion. 

Anomalous diffusion is also possible in microporous solids. For instance, it is possible for molecules to be confined in a channel system in which they cannot pass each other, and this will obviously affect molecular displacement in a time interval. This case is termed single-file diffusion , and the mean square displacement in a time t is then given... 

There has recently been much interest in the phenomenon of single file diffusion, which occurs in a unidimensional pane system when the diffusing molecules are too large to pass one another. In this situation the mean square displacement increases in proportion to the square root of elapsed... 

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