Micropore size distribution description

Practical solids are generally heterogeneous, and this subject of heterogeneity is the topic of Chapter 6, where the concept of distribution of the interaction energy between adsorbate molecules and solid atoms is discussed. For systems, such as non-polar hydrocarbons on activated carbon, where the adsorption force is dispersive by nature, the role of micropore size distribution is important in the description of solid heterogeneity. The concept of distribution is not restricted to the interaction energy between adsorbate molecules and solid atoms, it can be applied to the Henry constant, the approach of which has been used by Sircar, and it can be applied to free energy, which was put forward by Aharoni and Evans. 

The DA equation is corresponding to the choice of arbitrary value in the WeibulTs distribution function (eq. 4.2-1). With this additional parameter in the adsorption isotherm equation, the DA equation provides flexibility in the description of adsorption data of many microporous solids ranging from a narrow to wide micropore size distribution. The following table shows the degree of filling when the adsorption potential is equal to some fraction of the characteristic energy. 

Because n =3 is found to describe well data of solids having narrow pore size distribution, the DA equation with n = 3 is generally used as the local isotherm for the description of micropore size distribution as we shall discuss later in Section 4.4. 

Since adsorption of many adsorbates in micropores of carbonaceous solids is due to the dispersion force, the micropore size is therefore playing a major role in the attraction of adsorbate molecules. In this sense, a distribution of the micropore size is a more fundamental description of heterogeneity than the distribution in characteristic energy as done in the previous section. If we let the micropore size distribution as f(x) such that Wo f(x) is the micropore volume having micropore size between x and x + dx, then the volume of micropore occupied by adsorbate at a given adsorption potential A is ... 

In this section, we show another approach but very similar to what we did in Section 6.10, that of Hovarth and Kawazoe (1983), to obtain the average potential energy for a slit shape pore, from which a method is derived to determine the micropore size distribution from the information of experimental isotherm data. What to follow is the brief description of the theory due to Horvarth and Kawazoe. 

The micropore size-induced energy distribution is then fitted by a polynomial for the subsequent use in the dynamics calculation. The model equations for the dynamics studies are the same as those presented in Section 11.6. The only difference is that the energy distribution is deduced from the micropore size distribution, instead of the uniform energy distribution. Hu and Do (1995) have studied this and they have shown that the approach using the micropore size distribution as the source of system heterogeneity seems to provide a better description of the desorption data than the approach using the uniform energy distribution. 

Swiatkowski, A, Trznadel, B.J., and Zietek, S., Description of active carbon micropore size distribution based on the Horvath-Kawazoe equation adapted to benzene adsorption isotherms, Adsorpt. Sci. Technol., 14(1), 59-68(1996). 

No current theory is capable of providing a general mathematical description of micropore fiUirig and caution should be exercised in the interpretation of values derived from simple equations. Apart from the empirical methods described above for the assessment of the micropore volume, semi-empirical methods exist for the determination of the pore size distributions for micropores. Common approaches are the Dubinin-Radushkevich method, the Dubinin-Astakhov analysis and the Horvath-Kawazoe equation [79]. 

As we noted in Sec. II. B. the use of the statistical polymer method alone does not allow the complete description of microporous systems and could be completed by fractal method. We have noted in Sec. IV that the statistical polymer method, though seems very effective for the estimation of various (first of all additive) parameters of macromolecules (even branched cross-linked ones), does not allow the obtainment of enough information about the empty space between them, i.e., micropores. Sometimes that is not important (e.g., if one is interested mainly in the energy distribution of micropores which is directly determined by that of macromolecules), but sometimes the direct information about micropores (especially their size distribution) is indispensable, and then the combined use of the statistical polymer and fractal... 

The objective of this chapter is to present the fundamental theories of adsorption followed by the description and discussion of experimental techniques for the measurements of adsorption isotherms and for the determination of surface area and pore size distribution. The adsorption of gases on microporous membranes and the inter-relation between adsorption and permeation are then discussed. The adsorption in liquid phase is briefly presented. The chapter concludes with a brief summary. 

The pore size distribution in the carbon support is an important factor for a well performance of the catalyst. Pores in the nanometric scale are classified by lUPAC in three groups the micropores are those with diameters lower than 2 nm, the mesopores with diameters between 2 and 50 nm and the macropores with diameters larger than 50 run. Each pores size offers different benefits, the micropores produce materials with high surfaces area but could be inaccessible to liquid solutions or have slow mass transport. A material with mesopores has a lower surface area but better accessibility than those with micropores. FinaUy, materials with macropores show the lowest surface area, but they are easily accessible to liquid fuel. For this reason, the structured carbons, principally mesoporous carbon, have attracted considerable attention due to their potential application in the catalyst area, where the challenging is to favour the dispersion of catalyst and allow the accessibility of liquids that feed the anode side of a DMFC. In the following sections a description of different carbons support wdl be discussed stressing on the effect of the porous structure. 

In Chapter 2, we discussed the fundamentals of adsorption equilibria for pure component, and in Chapter 3 we presented various empirical equations, practical for the calculation of adsorption kinetics and adsorber design, the BET theory and its varieties for the description of multilayer adsorption used as the yardstick for the surface area determination, and the capillary condensation for the pore size distribution determination. Here, we present another important adsorption mechanism applicable for microporous solids only, called micropore filling. In this class of solids, micropore walls are in proximity to each other, providing an enhanced adsorption potential within the micropores. This strong potential is due to the dispersive forces. Theories based on this force include that of Polanyi and particularly that of Dubinin, who coined the term micropore filling. This Dubinin theory forms the basis for many equations which are currently used for the description of equilibria in microporous solids. 

Thus, this work contains the first report of the lifetimes and FV (hole) size distributions in polymers distinguished by an unusually loose structure (high free volume) PTMSP and microporous PPO membranes. It was shown that PATFTT and CONTIN descriptions of the annihilation data are in reasonable agreement maxima of CONTIN peaks coincide with Xi conq>onents in the PATFIT finite-term spectra. The o-... 

Seaton and co-workers [9] state that because of the unphysical nature of the underlying assumpUon, this method does not provide a reliable means of determining micropore pore size distributions. Indeed they state that there is no current analysis method that is based on a realistic description of micropore filling, although several semi-empirical approaches have been presented [89-91]. 

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