Diffusivity single-file diffusion

It is known that the displacement of single particle in single-file diffusion follows the time dependence r2(t) °c t n. Hence the tracer diffusion coefficient vanishes at long times for diffusion on an infinitely long lattice. However, MC simulations show that the collective... 

Since the fraction of empty sites in a zeolite channel determines the correlation factor (Section 5.2.2), as is well known from single-file diffusion in the pores of a membrane, the strong dependence of the diffusion coefficients on concentration can be understood. This is why a simple Nernst-Planck type coupling of the diffusive fluxes (see, for example, [H, Schmalzried (1981)]) is also not adequate. Therefore, we should not expect that sorption and desorption are symmetric processes having identical kinetics. Surveys on zeolite kinetics can be found in [A. Dyer (1988) J. Karger, D.M. Ruthven (1992)]. 

These formulations are applicable only to normal systems in which diffusing molecules can pass each other in the channels. When molecules cannot pass we have single file or string of pearls diffusion which has entirely different features. This topic has been well reviewed by K rger [9] and will not be considered here. 

J. KSrger, Single File Diffusion, Ch. 7 in Molecular Sieves Science and Technology, H.G. Karge and J. Weitkamp eds., Wiley-VCH (2002). 

In zeolites Na X and ZSM-5, the self-diffusion coefficients were found to decrease with increasing concentration while for zeolite NaCa A they are essentially constant The highest diffusivities were observed in zeolite Na X. This is in agreement with the fact that due to the internal pore structure the steric restrictions of molecular propagation in zeolite Na X are smaller Aan those in Na Ca A and ZSM-5 (94). Mass transfer and chemical reaction in zeolite channels in which the individual molecules cannot pass each other (single-file... 

The first term on the right-hand side of the equation denotes the friction between species the second term represents friction between a species and the surface. and )Ji+i are the Maxwell-Stefan diffusivities. The first term on the right side is often referred to as an exchange term that represents the probability of molecules exchanging places on the surface. Since this exchange is not likely to occur in narrow zeolite channels, it is commonly neglected. This is called single-file diffusion. 

FIGURE 10.24 Model of the prevention of secondary reactions by single-file diffusion in tnicroporous membranes. (Adapted from Lange, C., Storck, S Tesche, B and Maier, W.F., J. Catal, 175, 280, 1998.)... 

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]. 

For single file diffusion mechanism, with no possible counter-exchange between adsorbed species, these equations simplify to... 

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

Por single file diffusion mechanism, it is assumed that no coimter-exchange is possible between the adsorbed solutes i and j. For single file diffusion mechanism the Pick surface diffusivity matrix is given by ... 

Kaczmarski et al. used a similar model for the calculation of the band profiles of the enantiomers of 1-indanol on a chiral phase in HPLC [29,57]. These authors ignored the external mass transfer and assumed that local equilibrium takes place for each component between the pore surface and the stagnant fluid phase in the macropores (infinite fast kinetics of adsorption-desorption). They also assumed that surface diffusion contribution is much faster than pore diffusion and neglected pore diffusion entirely. Instead of the single file Maxwell-Stefan diffusion, these authors used the generalized Maxwell-Stefan diffusion (see Chapter 5).The calculation (see below) requires first the selection of equations to calculate the surface molecular flux [29,57,58],... 

For small por s with a pore diameter of about the same size as the molecular diameter (situation b2 in Fig. 9.20) counter-exchange will not take place and terms with D12 are absent. This is the so-called single file diffusion for which can be written [89] ... 

If single file diffusion prevails the single component permeation data (which determine D,) and single component adsorption data are sufficient to predict the binary fluxes via Eq. (9.52). 

Channel proteins transport water or specific types of Ions and hydrophilic small molecules down their concentration or electric potential gradients. Such protein-assisted transport sometimes is referred to as facilitated diffusion. Channel proteins form a hydrophilic passageway across the membrane through which multiple water molecules or ions move simultaneously, single file at a very rapid rate. Some ion chan-... 

As iff 1 increases so should the degree of saturation at which Z>ab begins to deviate significantly from D. The phenomenon of single-file diffusion is, of course, excluded if Ifi > 1. 

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