Single-file case

In complete agreement with the fact that the product molecules in a singlefile system are prevented from leaving the system by their file neighbors, the concentration profiles in the single-file cases show a much more pronounced tendency of accumulation of the reaction products in the file center than in the case of normal diffusion. Under stationary conditions, the effective reactivity k is related to the intrinsic reactivity k by the equation... 

Before compound selection can begin, it is necessary to preprocess the set of vendor compound files and the Pharmacia Compound Collection (PCC) to ensure that they both represent the most current compound information. In the former case, it is necessary to generate a single file for all vendor compounds free of redundancies, that is, each compound should appear only once in the file. The set of vendors supplying each... 

Business continuity procedures, including disaster recovery procedures, should ensure minimal disruption in the case of loss of data or any part of the system. It is necessary to ensure that the integrity of the data is not compromised during the return to normal operation. At the lowest level, this may mean the accidental deletion of a single file, in which case a procedure should be in place for restoring the most recently backed-up copy. At the other extreme, a disaster such as a fire could result in loss of the entire system. For this situation a procedure addressing the following should be in place ... 

Until now we have used the database for a very simple purpose, namely to extract information from a single file. However, it is also possible to connect several files. Let us suppose that we want to use dBASE for the following problem. In atomic absorption spectroscopy (AAS), one has to choose between the flame and the (flameless) graphite tube methods. The flame methods does not have such a low detection limit as the graphite tube, but it is easier to handle, less prone to interferences and more robust. For that reason the user s strategy will often be to apply the flame method above a certain concentration limit and the flameless method below it. The flame method has its own experimental characteristics and we suppose that we have another database file in which the characteristics for flame methods are given per element. In that case, we would like the consultation to go like this ... 

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. 

It is evident that, in the cases of single-file diffusion and barrier-limited molecular exchange, a suitable definition of the Thiele modulus cannot be based on eq. 1. These cases may also be covered, however, if the Thiele modulus is expressed in terms of the... 

It has been demonstrated by numerical simulations [9] that, with this definition, eq. 2 provides a reasonable order-of-magnitude estimate of the effectiveness factor also in the case of single-file diffusion. While in the case of ordinary diffnsion the intracrystalline mean life time may be easily correlated with the crystal size and the internal mobility [11], similar analytical expressions for single-file diffusion have not been established. The rule-of-thumb given in Ref. [10] on the basis of a few first numerical simulations turned out to be of rather limited validity in recent more refined considerations [12]. 

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... 

In this case, the matrix of Pick diffusivities depends on the surface coverage and is given by the single file Maxwell-Stefan relationship. 

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... 

It should be noted that in the literature the term single-file diffusion has sometimes been used to indicate only the mutual steric hindrance of the molecules in zeolite channels and not the additional constraint of an absolute exclusion of their mutual passage. Such terminology is clearly misleading since in this case molecular propagation would also be subjected to Pick s laws. 

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...

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. 

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. 

Most importantly, combining the expressions for the intracrystalline mean life time Eq. 23 and the effective self-diffusivity (Eq. 19), in the case of singlefile diffusion the mean time of molecular exchange is found to scale with L, rather than with the dependence typical of normal diffusion. Therefore, under the conditions of single-file diffusion, the exchange rate with zeoUte crystallites decreases even more pronounced with increasing crystal sizes as in the case of normal diffusion. 

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

---