Molecular 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]... 

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]... 

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. 

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. 

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... 

This dependence is represented by the dotted line in Fig. 6. In a first attempt to systematize the simulation results of molecular reaction and diffusion in single-file systems, a generalized Thiele modulus has been introduced [1]. Combining Eq. 25 and Eq. 23, the Thiele modulus may be expressed in the alternative notation... 

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. 

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... 

Before the introduction of measuring techniques such as pulsed field gradient (PEG) NMR ([14,16,45], pp. 168-206) and quasielastic neutron scattering (QENS) [49,50], which are able to trace the diffusion path of the individual molecules, molecular diffusion in adsorbate-adsorbent systems has mainly been studied by adsorption/desorption techniques [ 16]. In the case of singlefile systems, adsorption/desorption techniques cannot be expected to provide new features in comparison to the case of normal diffusion [51,52]. In adsorption/desorption measurements it is irrelevant whether or not two adjacent molecules have exchanged their positions. But it is this effect which makes the difference between normal and single-file diffusion. 

Several other authors have reported similar systems. For example, Balaram et al. studied closely related pentapeptides with enantiomorphic sequences, Boc-D-Pro-Aib-Xxx-Aib-Val-OCHs and Boc-Pro-Aib-D-Xxx-Aib-D-Val-OCHs, where Xxx refers to Leu, Val, Ala, or Phe, and observed that they yielded molecular structures with a similar backbone conformation but different packing patterns in crystals. Some of the peptides form entrapped water wires that are in the hydrophobic channel, a situation in contrast to the single-file arrangements in amphipathic channels formed by aquaporins. 

---