Horvath-Kawazoe

This equation describes the additional amount of gas adsorbed into the pores due to capillary action. In this case, V is the molar volume of the gas, y its surface tension, R the gas constant, T absolute temperature and r the Kelvin radius. The distribution in the sizes of micropores may be detenninated using the Horvath-Kawazoe method [19]. If the sample has both micropores and mesopores, then the J-plot calculation may be used [20]. The J-plot is obtained by plotting the volume adsorbed against the statistical thickness of adsorbate. This thickness is derived from the surface area of a non-porous sample, and the volume of the liquified gas. 

Figure 6. Pore size distribution ( ) and cumulative pore volumes ( ) Microporous domain, Horvath-Kawazoe equation.

A number of models have been developed for the analysis of the adsorption data, including the most common Langmuir [49] and BET (Brunauer, Emmet, and Teller) [50] equations, and others such as t-plot [51], H-K (Horvath-Kawazoe) [52], and BJH (Barrett, Joyner, and Halenda) [53] methods. The BET model is often the method of choice, and is usually used for the measurement of total surface areas. In contrast, t-plots and the BJH method are best employed to calculate total micropore and mesopore volume, respectively [46], A combination of isothermal adsorption measurements can provide a fairly complete picture of the pore size distribution in sohd catalysts. Mary surface area analyzers and software based on this methodology are commercially available nowadays. 

Thus, either type I or type IV isotherms are obtained in sorption experiments on microporous or mesoporous materials. Of course, a material may contain both types of pores. In this case, a convolution of a type I and type IV isotherm is observed. From the amount of gas that is adsorbed in the micropores of a material, the micropore volume is directly accessible (e.g., from t plot of as plot [1]). The low-pressure part of the isotherm also contains information on the pore size distribution of a given material. Several methods have been proposed for this purpose (e.g., Horvath-Kawazoe method) but most of them give only rough estimates of the real pore sizes. Recently, nonlocal density functional theory (NLDFT) was employed to calculate model isotherms for specific materials with defined pore geometries. From such model isotherms, the calculation of more realistic pore size distributions seems to be feasible provided that appropriate model isotherms are available. The mesopore volume of a mesoporous material is also rather easy accessible. Barrett, Joyner, and Halenda (BJH) developed a method based on the Kelvin equation which allows the calculation of the mesopore size distribution and respective pore volume. Unfortunately, the BJH algorithm underestimates pore diameters, especially at... 

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]. 

Pore size obtained by Horvath-Kawazoe analysis of N2 adsorption data. 

The N2 adsorption-desorption isotherm at -196°C and the micro- and mesopore size distributions are presented in figure 2. In the partial pressure range -0.02-0.3 the upward deviation indicates the presence of supermicropores (15-20A) or small mesopores (20-25A). From the De Boer t-plot the presence of an important microporosity can be deduced, so a unique combined micro- and mesoporosity is present for this type of material. Indeed, this combined pore system is confirmed when considering the micropore (Horvath-Kawazoe) and mesopore (Barrett-Joyner-Halenda) size distributions with maxima at respectively 6A and 17.5 A pore diameter (figure 5). An overview of the surface area, micro- and mesoporosity data of the unmodified PCH can be found in table 1. 

Figure 5. Micropore (Horvath-Kawazoe) and mesopore (Barrett-Joyner-Halenda) size distributions of A) the unmodified PCH B) Al-PCH-75% C) AI-PCH-200%...

Deficiencies of the Horvath-Kawazoe method and other similar procedures... 

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. 

Figure 1. N2 adsorption isotherms and (inset) Horvath-Kawazoe pore size distribution for HMS and its functionalized analogs prepared by grafting and direct incorporation.

In order to determine the PSD of the micropores, Horvath-Kawazoe (H-K) method has been generally used. In 1983, Horvath and Kawazoe" developed a model for calculating the effective PSD of slit-shaped pores in molecular-sieve carbon from the adsorption isotherms. It is assumed that the micropores are either full or empty according to whether the adsorption pressure of the gas is greater or less than the characteristic value for particular micropore size. In H-K model, it is also assumed that the adsorbed phase thermodynamically behaves as a two-dimensional ideal gas. 

HRTEM High resolution transmission electron microscopy H-K Horvath-Kawazoe... 

The Horvath-Kawazoe pore size model (or one of its modifications) has become recently available as a fully implemented software package. 

Figure 4.6 shows the PSDs obtained from the high-resolution N2 adsorption isotherms at 77 K (Figure 4.5) by applying the Horvath-Kawazoe method (Figure 4.6a), Dubinin-Astakhov method...

FIGU RE 4.9 Comparison of the PSD obtained for different samples by applying different methods (a) Sample ACF1, (b) sample AC2, and (c) sample AC1. DR-C02 is the PSD obtained by applying the Dubinin-based method proposed by Cazorla-Amoros et al. [10] to C02 at 273 K. HK, DFT, and DA are the PSDs obtained by applying Horvath-Kawazoe, DFT, and Dubinin-Astakhov methods to the N2 adsorption isotherm at 77 K, respectively. 

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]... 

Figure 5 presents the N2 adsorption/desorption isotherms for calcined (650 °C) MSU-G silicas assembled from C H2 +,NH(CH2)2NH2 surfactants with n = 10, 12, and 14. The inset to the figure provides the framework pore distributions. The maxima in the Horvath-Kawazoe pore size distributions increase in the order 2.7, 3.2, 4.0 nm as the surfactant chain length increases. The textural porosity evident from the hysteresis loop at P/P0... 

The characterization of porous materials exhibiting a composite pore structure encompassing micro-meso-and perhaps macro-pore sizes, is of particular significance for the development of separation and reaction processes. Among the characterization methods for materials exhibiting ultramicropore structures, DpDubinin-Radushkevich (DR)[2], Dubinin-Astakov (DA) [3], Dubinin-Stoeckli (DS) [4], as well as the Horvath-Kawazoe (HK), [5] methods are routinely used for the evaluation of the micropore capacity and the pore size distriburion (PSD). 

A Two-Stage Horvath-Kawazoe Adsorption Model for Pore Size Distribution Analysis... 

The Horvath-Kawazoe (HK) method is capable of generating model isotherms more efficiently than either molecular simulation (MS) or density functional theory (DFT) to characterize the pore size distribution (PSD) of microporous solids. A two-stage HK method is introduced that accounts for monolayer adsorption in mesopores prior to capillary condensation. PSD analysis results from the original and two-stage HK models are evaluated. 

Two kinetic (CMS-Kl, CMS-K2) and one equilibrium (CMS-R) carbon molecular sieves, used originally for separation of gaseous mixtures, were investigated. The adsorption Nj isotherms at 77 K, in static conditions where obtained. In the case of the two first sieves the adsorption was so low that the calculation of parameters characterizing the texture was impossible. The volume of nitrogen adsorbed on the sieve CMS-R is remarkable From obtained results parameters characterizing micropore structure according to Dubinin -Radushkevich equation and Horvath - Kawazoe method were determined. 

From obtained isotherm were determined parameters characterizing micropore structure according to Dubinin - Radushkevich equation [6] and Horvath - Kawazoe method [7] which are presented below ... 

The methods depend on the theoretical treatment which is used. A majority of them are based on the Generalised Adsorption Isotherm (GAI) also called the Integral Adsorption Equation (LAE). The more recent approaches use the Monte Carlo simulations or the density functional theory to calculate the local adsorption isotherm. The analytical form of the pore size distribution function (PSD) is not a priori assumed. It is determined using the regularization method [1,2,3]. Older methods use the Dubinin-Radushkevich or the Dubinin-Astakhov models as kernel with a gaussian or a gamma-type function for the pore size distribution. In some cases, the generalised adsorption equation can be solved analytically and the parameters of the PSD appear directly in the isotherm equation [4,5,6]. Other methods which do not rely on the GAI concept are sometimes used the MP and the Horvath-Kawazoe (H-K) methods are the most well known [7,8]. 

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