Adsorption cylindrical pore

The variant of the cylindrical model which has played a prominent part in the development of the subject is the ink-bottle , composed of a cylindrical pore closed one end and with a narrow neck at the other (Fig. 3.12(a)). The course of events is different according as the core radius r of the body is greater or less than twice the core radius r of the neck. Nucleation to give a hemispherical meniscus, can occur at the base B at the relative pressure p/p°)i = exp( —2K/r ) but a meniscus originating in the neck is necessarily cylindrical so that its formation would need the pressure (P/P°)n = exp(-K/r ). If now r /r, < 2, (p/p ), is lower than p/p°)n, so that condensation will commence at the base B and will All the whole pore, neck as well as body, at the relative pressure exp( —2K/r ). Evaporation from the full pore will commence from the hemispherical meniscus in the neck at the relative pressure p/p°) = cxp(-2K/r ) and will continue till the core of the body is also empty, since the pressure is already lower than the equilibrium value (p/p°)i) for evaporation from the body. Thus the adsorption branch of the loop leads to values of the core radius of the body, and the desorption branch to values of the core radius of the neck. 

These calculations lend theoretical support to the view arrived at earlier on phenomenological grounds, that adsorption in pores of molecular dimensions is sufficiently different from that in coarser pores to justify their assignment to a separate category as micropores. The calculations further indicate that the upper limit of size at which a pore begins to function as a micropore depends on the diameter a of the adsorbate molecule for slit-like pores this limit will lie at a width around I-So, but for pores which approximate to the cylindrical model it lies at a pore diameter around 2 5(t. The exact value of the limit will of course depend on the actual shape of the pore, and may well be raised by cooperative effects. 

Figure 9,16 Comparison of theory with experiment for rg/a versus K. The solid line is drawn according to the theory for flexible chains in a cylindrical pore. Experimental points show some data, with pore dimensions determined by mercury penetration (circles, a = 21 nm) and gas adsorption (squares, a= 41 nm). [From W. W. Yau and C. P. yidXont, Polym. Prepr. 12 797 (1971), used with permission.]...

Pore size has been evaluated through the DFT method, using the NLDFT adsorption branch model for cylindrical pore [10],... 

Physical properties of calcined catalysts were investigated by N2 adsorption at 77 K with an AUTOSORB-l-C analyzer (Quantachrome Instruments). Before the measurements, the samples were degassed at 523 K for 5 h. Specific surface areas (,S BEX) of the samples were calculated by multiplot BET method. Total pore volume (Vtot) was calculated by the Barrett-Joyner-Halenda (BJH) method from the desorption isotherm. The average pore diameter (Dave) was then calculated by assuming cylindrical pore structure. Nonlocal density functional theory (NL-DFT) analysis was also carried out to evaluate the distribution of micro- and mesopores. 

The left-hand side of equation 17.42 is a characteristic dimension of the pores in which condensation or evaporation occurs at an adsorbate pressure Pm. This depends on the geometry of the pores, as well as whether adsorption or desorption is occurring. Figure 17.10 illustrates the conditions in a cylindrical pore. For desorption occurring from the free surface of condensate ... 

Figure 5.15 Comparison of the hydrogen adsorption in a slit and cylindrical pore [18].The amountofabsorbed hydrogen correlates with the specific surface area of the sample the maximum is at 0.6 mass% (p — 6 MPa, T— 300 K). No significant difference was found in the calculated amount of hydrogen between the slit and cylindrical pores. The calculation was verified experimentally with excellent agreement.

The shape of the hysteresis loop in the adsorption/desorption isotherms provides information about the nature of the pores. The loops have been classified according to shape as A, B and E (De Boer, 1958) or as HI - H4 by lUPAC (Sing et al, 1985). Ideally, the different loop shapes correspond to cylindrical, slit shaped and ink-bottle pores the loops in the isotherm IV and V of Figure 5.3 correspond to cylindrical pores. Wide loops indicate a broad pore size distribution (for an example see Fig. 14.9). The absence of such a loop may mean that the sample is either nonporous or microporous. These generalizations provide some initial assistance in assessing the porosity of a sample. In fact the adsorption/desorption isotherms are often more complicated than those shown in Figure 5.3 owing to a mixture of pore types and/or to a wide pore size distribution. 

Specific surface areas of the materials under study were calculated using the BET method [22, 23]. Their pore size distributions were evaluated from adsorption branches of nitrogen isotherms using the BJH method [24] with the corrected form of the Kelvin equation for capillary condensation in cylindrical pores [25, 26]. In addition, adsorption energy distributions (AED) were evaluated from submonolayer parts of nitrogen adsorption isotherms using the algorithm reported in Ref. [27],... 

Application of MMSs as model adsorbents for development and verification of adsorption methods to calculate PSDs rests upon availability of reliable independent estimates of the MMS pore size. This fundamental problem was already solved for MCM-41 [1], silica with honeycomb arrays of approximately cylindrical pores. Taking advantage of its simple geometry, the following relation between the pore diameter, wd, the primary mesopore volume, Vp, and the XRD (100) interplanar spacing, d, was derived [10, 20] ... 

Recently, the Horvath-Kawazoe (HK) method for slit-like pores [40] and its later modifications for cylindrical pores, such as the Saito-Foley (SF) method [41] have been applied in calculations of the mesopore size distributions. These methods are based on the condensation approximation (CA), that is on the assumption that as pressure is increased, the pores of a given size are completely empty until the condensation pressure corresponding to their size is reached and they become completely filled with the adsorbate. This is a poor approximation even in the micropore range [42], and is even worse for mesoporous solids, since it attributes adsorption on the pore surface to the presence of non-existent pores smaller than those actually present (see Fig. 2a) [43]. It is easy to verify that the area under the HK PSD peak corresponding to actually existing pores does not provide their correct volume, so the HK-based PSD is not only excessively broad, but also provides underestimated volume of the actual pores. This is a fundamental problem with the HK-based methods. An additional problem is that the HK method for slit-like pores provides better estimates of the pore size of MCM-41 with cylindrical pores than the SF method for cylindrical pores. This shows the lack of consistency [32,43]. Since the HK-based methods use CA, one can replace the HK or SF relations between the pore size and pore filling pressure by the properly calibrated ones, which would lead to dramatic improvement of accuracy of the pore size determination [43] (see Fig. 2a). However, this will not eliminate the problem of artificial tailing of PSDs, since the latter results from the very nature of HK-based methods. 

According to the classical treatment of Cohan [8], which is the basis of the conventional BJH method [14], capillary condensation in an infinite cylindrical pore is described by the Kelvin equation using cylindrical meniscus, while desorption is associated with spherical meniscus. In large pores the following asymptotic equation is expected to be valid [8] Pd/Po = (PA/P0)2, where Pd/Po and Pa/Po are the relative pressures of the desorption and adsorption, respectively. An improved treatment [9-11, 13], originated from Deijaguin [9], takes into... 

The adsorption isotherm of N, on FSM-16 at 77 K had an explicit hysteresis. As to the adsorption hysteresis of N-, on regular mesoporous silica, the dependencies of adsorption hysteresis on the pore width and adsorbate were observed the adsorption hysteresis can be observed for pores of w 4.0nm. The reason has been studied by several approaches [5-8]. The adsorption isotherm of acetonitrile on FSM-16 at 303K is shown in Fig. 1. The adsorption isotherm has a clear hysteresis the adsorption and desorption branches close at PIP, = 0.38. The presence of the adsorption hysteresis coincides with the anticipation of the classical capillary condensation theory for the cylindrical pores whose both ends are open. The value of the BET monolayer capacity, nm, for acetonitrile was 3.9 mmol g. By assuming the surface area from the nitrogen isotherm to be available for the adsorption of acetonitrile, the apparent molecular area, am, of adsorbed acetonitrile can be obtained from nm. The value of am for adsorbed acetonitrile (0.35 nnr) was quite different from the value (0.22 nm2) from the liquid density under the assumption of the close packing. Acetonitrile molecules on the mesopore surface are packed more loosely than the close packing. The later IR data will show that acetonitrile molecules are adsorbed on the surface hydroxyls in... 

It is widely accepted that two mechanisms contribute to the observed hysteresis. The first mechanism is thermodynamic in origin [391,392], It is illustrated in Fig. 9.14 for a cylindrical pore of radius rc. The adsorption cycle starts at a low pressure. A thin layer of vapor condenses onto the walls of the pore (1). With increasing pressure the thickness of the layer increases. This leads to a reduced radius of curvature for the liquid cylinder a. Once a critical radius ac is reached (2), capillary condensation sets in and the whole pore fills with liquid (3). When decreasing the pressure again, at some point the liquid evaporates. This point corresponds to a radius am which is larger than ac. Accordingly, the pressure is lower. For a detailed discussion see Ref. [393],... 

An example of the adsorption to one such material is shown in Fig. 9.16. The siliceous material, called MCM-41, contains cylindrical pores [397], With increasing pressure first a layer is adsorbed to the surface. Up until a pressure of P/Po 0.45 is reached, this could be described by a BET adsorption isotherm equation. Then capillary condensation sets in. At a pressure of P/Po 0.75, all pores are filled. This leads to a very much reduced accessible surface and practically to saturation. When reducing the pressure the pores remain filled until the pressure is reduced to P/Pq rs 0.6. The hysteresis between adsorption and desorption is obvious. At P/Po 0.45 all pores are empty and are only coated with roughly a monolayer. Adsorption and desorption isotherms are indistinguishable again below P/Po 0.45. 

Figure 9.16 Adsorption isotherms of Argon at 87 K in siliceous material MCM-41 with cylindrical pores of 6.5 nm diameter. Redrawn after Ref. [398].

The pore volume and the pore size distribution can be estimated from gas adsorption [83], while the hysteresis of the adsorption isotherms can give an idea as to the pore shape. In the pores, because of the confined space, a gas will condense to a liquid at pressures below its saturated vapor pressure. The Kelvin equation (Eq. (4.5)) gives this pressure ratio for cylindrical pores of radius r, where y is the liquid surface tension, V is the molar volume of the liquid, R is the gas constant ( 2 cal mol-1 K-1), and T is the temperature. This equation forms the basis of several methods for obtaining pore-size distributions [84,85]. 

For many of activated microporous carbons,80"84 the isotherms exhibit prominent adsorption at low relative pressures and then level off, i.e., the isotherms exhibit Type I behavior. Type I isotherms may be also observed in the mesoporous materials with pore sizes close to the micropore range. In particular, in the case of gas adsorption on highly uniform cylindrical pores, the adsorption isotherms exhibit discernible steps at relative pressures down to 0.1 or perhaps even lower.85"87 Such Type I behavior can be indicative of some degree of broadening of the mesopore size distribution. 

According to the IUPAC,79 the hysteresis loops are classified into four types from Type HI to Type H4. The Type HI loop in Figure 3 is characteristic of the mesoporous materials consisting of the pores with cylindrical pore geometry or the pores with high degree of pore size uniformity.88,89 Hence, the appearance of the HI loop on the adsorption isotherms for the porous solids generally indicates facile pore connectivity and relative narrow PSD. 

Because of the conceptual and mathematical simplicity of these models, they have been used recently to describe adsorption in zeolite and clay structures,53,54,55,56 based on either a cylindrical pore model or a packed sphere model. 

The adsorption process is, in this case, described with the help of a potential in between a perfect cylindrical pore of infinite length but finite radius, rp [18]. The calculation is made with the help of a model similar to those developed by Horvath-Kawazoe for determining the MPSD [18], which includes only the van der Waals interactions, calculated with the help of the L-J potential. In order to calculate the contribution of the dispersion and repulsion energies, Everett and Powl [45] applied the L-J potential to the case of the interaction of one adsorbate molecule with an infinite cylindrical pore consisting of adsorbent molecules (see Figure 6.20), and obtained the following expression for the interaction of a molecule at a distance r to the pore wall [18]... 

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