Modelling micropore filling

There has been fierce debate (see Refs. 232, 235-237) over the usefulness of the preceding methods and the matter is far from resolved. On the one hand, the use of algebraic models such as modified DR equations imposes artificial constraints, while on the other hand, the assumption of the validity of the /-plot in the MP method is least tenable just in the relatively low region where micropore filling should occur. 

Measuring the specific surface area, A, related to the mass of PS does not require a textural model (a morpho-independent parameter, i.e., one can apply an approach of partitioning and, correspondingly, the second statement of texturology, as we have already done for volume-related parameters). Let us consider the most widespread adsorption method based on proportionality of adsorption, Q. and the specific surface area in the absence of volumetric effects (capillaiy condensation, micropore filling, etc.) ... 

Other -more complicated- models to evaluate the microporous volume exist. The Dubinin-Radushkevich model46,47,48,49 is based on thermodynamical considerations concerning the process of micropore filling. Full discussion of this model is beyond the scope of this book. The reader is referred to the standard work of Gregg and Sing50 on adsorption for a detailed treatment. 

There is a growing interest in the presentation of physisorption isotherms in a generalized integral form. This approach was first applied to physisorption in the submonolayer region (Adamson et al., 1961), but much of the current interest is centred on the analysis of micropore filling isotherms. An apparent advantage is that it provides a means of constructing a series of model isotherms by systematically... 

Various other aspects of fractal analysis have been discussed by Van Damme and Fripiat and their co-workers. For example, by extending the BET model to fractal surfaces, Fripiat et al. (1986) were able to show that the apparent fractal dimension is reduced by the progressive smoothing of a molecularly rough surface. Alternatively, the effect of a micropore filling contribution is to enhance the fractal dimension. 

Because of its diatomic nature and permanent quadrupole moment, the physisorp-tion of nitrogen at 77 K presents special problems. The application of DFT is facilitated if the molecules are assumed to be spherical, which was the approach originally adopted by Seaton et al. (1989) and also by Lastoskie et al. (1993). The analytical procedures already outlined in Chapter 7 (Section 7.6) do not depend on the meniscus curvature and are in principle applicable to both capillary condensation and micropore filling. The non-local version of the mean field theory (NLDFT), which was used by Lastoskie, gave excellent agreement with computer simulation when applied to the carbon slit pore model. However, as pointed out earlier, these computational procedures are not entirely independent since they involve the same model parameters. 

The Ne adsorption isotherms on model AIPO4-5 micropores were calculated from the Tarazona s version of the nonlocal density functional theory [34,35] which has beer actually applied to the study on micropore filling [36,37]. The necessary parameters were obtained fram the adsorption isotherms of Ne on AIPO4-S at 27K and 30K in a lov pressure range. 

MCM-41 and HMS materials show adsorption at a pressure lower than the threshold at 0.43 p/p°. In this region it is difficult to evaluate the pore size with classical method based on the Kelvin equation, because both micropore filling and capillary condensation can occur. Instead DFT (silica model) permits a better evaluation of pore size distribution in this region, observing a very narrow pore size distribution for lVICM-41 (Figure 6, curve b)... 

Basic adsorption isotherms have been described in this chapter. For micro-porous membranes, the use of the DR equation to describe micropore filling has been shown to be quite adequate. Techniques for the determination of surface area and pore size distribution have ben presented. The use of potential functions for the determination of pore size distribution in microporous materials has been described. Although the potential function techniques give consistent and satisfactory results, caution must be exerted in using these techniques for the calculation of the pore size distribution, due to the uncertainty involved in the values of the parameters used in the calculation and the simplifying assumptions employed in the derivation of the model equations. 

The hydrogen adsorption on SWNTs can be described using the Dubinin-Astakhov (DA) model by a traditional theory of micropore filling. The... 

A microkinetics model has been constracted that explicitly describes the dependence of overall reaction rate on micropore filling. The model can be illustrated with figure 1. Molecules from the gas phase adsorb to the zeolitic micropore sites. Transport steps are introduced between molecules adsorbed in the micropore with the metal sites and with the acidic protons, also located in the micropores. Communication of molecules between catalytically active sites again is only possible via the micropore sites. 

These simple model calculations illustrate the dramatic effect of micropore filling on catalytic performance. Since dilution by pentane does not increase the rate of hexane in Mordenite. The experiments by Spivey and Bryant indicate that diffusion has to be explicitly considered. It is likely that not only the higher heat of adsorption of alkanes in Mordenite compared to Faujasite, but ultimately the one dimensional stracture and resulting single file diffusion are responsible for the unique behavior of the Mordenite catalyst. 

However, the decrease in diffusion rate with increasing micropore filling is much faster than found experimentally or with molecular dynamics studies. A molecular dynamics model study, using an idealization of the one-dimensional zeolite micropore, shows that in open channels single-file diffusion behavior occurs only for particular time regimes (see Fig. 4.42). 

The surface areas determined by the BET method are apparent surface areas only since the BET adsorption equation is, in principle, not valid when micropore filling occurs. The determination of the true surface area in the micropores depends on the method used for the evaluation of the adsorption isotherms arul on the model used for the shape of the micropores (cylindrical, slit-shaped or other). 

The calculations in ref. 25 for model micropores only consider interactions between a single adsorptive molecule and the walls of the model micropore. They do not account for interactions between adsorptive molecules and so cannot model the process of micropore filling. Recently (ref. 43) results from molecular modelling studies were reported for the adsorption of nitrogen on porous carbons in which both adsorptive-adsorbent and inter-adsorptive interactions were considered. Using an approximate theory of inhomogeneous fluids known as mean-field theory, a function p(p, w) was derived (ref. 43) which relates the... 

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