Slit-like pores

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. 

To provide an approximate quantitative framework, the following limits of width of slit-like pores, for nitrogen at 77 K are suggested for the respective processes ... 

R. Evans, U. Marini Bettolo Marconi, P. Tarazona. Capillary condensation and adsorption in cyhndrical and slit-like pores. J Chem Soc Faraday Trans 2 52 1763-1787, 1986. 

Sec. Ill is concerned with the description of models with directional associative forces, introduced by Wertheim. Singlet and pair theories for these models are presented. However, the main part of this section describes the density functional methodology and shows its application in the studies of adsorption of associating fluids on partially permeable walls. In addition, the application of the density functional method in investigations of wettability of associating fluids on solid surfaces and of capillary condensation in slit-like pores is presented. 

The singlet-level theories have also been applied to more sophisticated models of the fluid-solid interactions. In particular, the structure of associating fluids near partially permeable surfaces has been studied in Ref. 70. On the other hand, extensive studies of adsorption of associating fluids in a slit-like [71-74] and in spherical pores [75], as well as on the surface of spherical colloidal particles [29], have been undertaken. We proceed with the application of the theory to more sophisticated impermeable surfaces, such as those of crystalline solids. 

Following Ref. 122, we consider the adsorption of associating hard spheres, Eq. (60), with d = 0.45, a = 0.09 in a slit-like pore with crystalline walls. A weakly associated fluid and a highly associated fluid have been studied. The weakly associated fluid was characterized by the coefficient A (cf. Eq. (79)) equal to 1, whereas in the case of the highly associated fluid A = 100. 

The density functional approach has also been used to study capillary condensation in slit-like pores [148,149]. As in the previous section, a simple model of the Lennard-Jones associating fluid with a single associative site is considered. All the parameters of the interparticle potentials are chosen the same as in the previous section. Our attention has been focused on the influence of association on capillary condensation and the evaluation of the phase diagram [42]. 

The fluid is confined to a slit-like pore of width H. Each of the pore walls is the source of the Lennard-Jones (9-3) potential and the total adsorbing potential, thus... 

First we are looking for the adsorption of a fluid consisting of particles of species m, in a slit-like pore of width H. The pore walls are chosen normal to the z axis and the pore is centered at z = 0. Adsorption of the fluid m, i.e., the matrix, occurs at equihbrium with its bulk counterpart at the chemical potential The matrix fluid is then characterized by the density profile, p (z) and by the inhomogeneous pair correlation function A (l,2). The structure of that fluid is considered... 

A. Milchev, K. Binder. Dynamics of polymer chains confined in slit-like pores. J Physique 7/5 21-31, 1996. 

Yethiraj and Hall [94] studied the density profiles, surface forces, and partition coefficient of freely jointed tangent hard-sphere chains between hard walls. The theory was able to capture the depletion of chain sites at the surface at low densities and the enhancement of chain sites at the surface at high densities. This theory is in qualitative agreement with simulations for the density profiles and partitioning of 4 and 20 bead chains, although several quantitative deficiencies are present. At low densities the theory overestimates the value of the density profile near the surface. Furthermore, it predicts a quadratic variation of density with distance near the surface, whereas in reality the density profile should be linear in distance, for long chains. At high densities the theory underestimates the value of the density near the surface. The theory is quite accurate, however, for the partition coefficient for hard chains in slit-like pores. 

Skvortsov and Gorbunov [44,45] showed that it holds for the slit-like pores and for the (Gaussian-type) coiled macromolecules... 

Let us consider a slit-like pore of width D along whose walls the ip(x) potential is localized (Fig. 4). We shall regard the interaction of monomers with the walls as a short-range interaction and the characteristic radius of interaction as being of the order of the segment size a. The exact assignment of the form of the potential is immaterial for our purposes, since it describes the effective interaction of units with the pore walls, renormalized by the solvent molecules. Conditions are to be as follows ... 

Fig. 4a and b. Distribution of the segment density at different values of the energy 0 (a) and schematic picture of lattice-like chain of length N in a slit-like pore of width D (b). 0cis the critical energy characteristic of the case when the entropy losses of the macromolecule in the pore are compensated by the energy of interaction with the wall. attractive potential of a depth 0 and with a characteristic radius of interaction r0 of the order of the segment size a... 

A detailed study of the chromatographic behaviour of the lattic-like models of chains in slit-like pores has been performed in Refs. 59 6I), for all chain length N and pore size D ratios and all energies of the interaction of units with pore walls, 0. In Ref. 62 a strict analytical theory has been elaborated for the separation according to the functionality of macromolecules interacting with the surface only by their terminal groups. 

We have calculated the energy 0 in this way for some polymers and separation conditions (Table 2) and, using the lattice-like model and a slit-like pore, we have found the distribution coefficients, K 1, for these macromolecules as a function of N, D, 0 and 0f 65). It turned out that for such a crude model not only the calculated KJj 1 values were close to the experimental ones, but also, which is especially important, that the chemical nature of the macromolecule, the functional groups and the separation conditions (the mobile phase composition) were correctly accounted for. Two examples of such calculations are given in Figs. 8 and 9. 

We have performed such calculations for samples of non-functional polybutadienes 66) (Fig. 11) and, using the found Ax and X0 values, we calculated Kd0) for a cubic lattice model and a slit-like pore within the whole experimentally accessible eab range using Eq. (3.16). The result presented in Fig. 12 shows a good agreement of the experimental data with the calculated curves. Even such a crude model as the lattice-like model and a slit-like pore can be successfully applied to assess the change in the retention volume as a function of the composition of the mobile phase. 

Let us now consider the behaviour of cyclic macromolecules at critical conditions. The macrocycle at critical conditions be transferred from the liquid phase into the pore in the following way (a) transform a cycle of length N into a linear molecule (AF = —3/2 In N) (b) transfer the linear molecule into the pore since this occurs at critical conditions, AF = 0 (c) form again a cycle in a slit-like (two-dimensional) pore (AF = In N). In going from (a) to (c), it is seen that the total free energy change for the transfer of the macrocycle into the pore at critical conditions is not equal to zero but depends on the size of the ring... 

Both types of molecular sieves, MCM-36 and MCM-41, demonstrate large BET surface area and high static sorption capacity (see Table 2). Considerable qualitative differences are observed in N2 isotherms, which are shown in Figure 3. The nitrogen isotherm for MCM-41, prepared with cetyltrimethylammonium cation, is type IV [9] and shows the characteristic reversible steep capillary condensation at p/p0 = -0.4 corresponding to the pore opening -40 A [1]. MCM-36 also shows the type IV isotherm with almost linear and reversible uptake increase up to - p/p0 = 0.5, followed by a hysteresis loop. This profile of adsorption/desorption is typical for layered materials with slit-like porosity generated between layers [9],... 

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. 

Type H3 hysteresis loop, which does not level off near the saturation vapor pressure, is characteristic of the mesoporous materials being comprised of agglomerates of plate-like particles with slit-shaped pores.79,86 Type H4 loop, which features parallel and almost horizontal branches, is attributable to the adsorption/ desorption in narrow slit-like pores. However, Type H4 loop was recently reported for MCM-41 being comprised of particles with internal voids of irregular shape and broad PSD,90 and also... 

The formalism of nonlocal functional density theory provides an attractive way to describe the physical adsorption process at the fluid - solid interface.65 In particular, the ability to model adsorption in a pore of slit - like or cylindrical geometry has led to useful methods for extracting pore size distribution information from experimental adsorption isotherms. At the moment the model has only been tested for microporous carbons and slit - shaped materials.66,67 It is expected that the model will soon be implemented for silica surfaces. 

Kowalczyk P, Tanaka H, Kaneko K, Terzyk AP, and Do DD. Grand canonical Monte Carlo simulation study of methane adsorption at an open graphite surface and in slit like carbon pores at 273 K. Langmuir, 2005 21(12) 5639-5646. 

In Figure 6.15, the adsorption isotherm of N2 at 77 K on the silica 68bslE [42], where the capillary condensation effect is obvious, is shown. Capillary condensation is normally characterized by a step in the adsorption isotherm. In materials with a uniform PSD, the capillary condensation step is remarkably sharp [20], However, in practice, the hysteresis loop is seen in materials consisting of slit-like pores, cylindrical-like pores, and spherical pores, that is, ink-bottle pores [2,41], The... 

The Type H3 loop, which docs not exhibit any limiting adsorption at high p/p°, is observed with aggregates of plate-like particles giving rise to slit-shaped pores. Similarly, the Type H4 loop is often associated with narrow slit-like pores, but in this case the Type I isotherm character is indicative of microporosity. 

Everett and Powl (1976) applied both the 9-3 and the 10-4 expressions in their theoretical treatment of potential energy profiles for the adsorption of small molecules in slit-like and cylindrical micropores. As one would expect, the two corresponding potential energy curves were of a similar shape, but the differences between them became greater as the pore size was reduced. Strictly, the replacement of the summation by integration is dependent on the distance between the molecule and the surface plane, becoming more accurate as the distance is increased (Steele, 1974). 

Results from SAXS analysis and GA (using COp gas) for the same coal samples are shown below in Table I. From the results of SAXS analysis it may be concluded that the investigated Victorian brown coal possesses an extensive micropore system containing between 10 to 10 pores per gram. Additionally, according to the shape hypothesis these pores may be either slit-like (thin discs with large diameters) or filament-like (long narrow cylinders) and are... 

Normally, carbon materials are characterised using the DFT which assumes a slit-like pore geometry [12]. However, these zeolite templated carbons are best described by the DFT-Hybrid... 

Kwong-Soave type). Epi is calculated from geometric considerations involving r, and H. Fa(H) can be related to the pore volume distribution fimction Fy(H) if the pore shape is known. In this work, we consider classical slit-like pores to describe the porous structure of the activated carbon and we take a bimodal gaussian for the mathematical expression of Fv(H). As a consequence ... 

Additionally, the PSDs were calculated using Micromeritics software with the Density Functional Theory (DFT) for the slit-like pore model. 

Figure 2. Pore size distribution in (a) MN200, (b) MN500, and (c) Carboxen 1003 and Carboxen 1010, calculated using a model of slit-like pores and the regularisation procedure CONTIN (a, b, c) and Micromeritics DFT (a, b).

Textural Parameters. Adsorption-desorption isotherms of N2 at 77K were determined in a Micromeritics ASAP 2010 with a micropore system. Prior to measurement, the samples were outgassed at 140 C for at least 16 h. The specific surface area was determined by the BET method, assuming that the area of a nitrogen molecule is 0.162 nm [12]. Micropore volume was calculated by the t-plot method using the Harkins and Jura [13] thickness. We used model isotherms calculated from density functional theory (DFT) to determine the pore size distributions and cumulative pore volume of the pillared samples by taking the adsorption branch of the experimental nitrogen isotherm, assuming slit-like pores [14]. 

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