Horvath-Kawazoe model

Nitrogen isotherms were measured by using an ASAP (Micromeritics) at 77K. Prior to each analysis, the samples were outgassed at S73K for 10 - 12 h to obtain a residual pressure of less than 10 torr. The amount on nitrogen adsorbed was used to calculate specific surface area, and the micro pore volumes determined from the BET equation [14] and t-plot method [15], respectively. Also, the Horvath-Kawazoe model [16] was applied to the experimental nitrogen isotherms for pore size distribution. 

The DFT method is such a complex in nature compared to other simple methods such as the Horvath-Kawazoe model, despite the fact that its PSD predictions for mesoporous sorbents are more accurate. Elaborate computer codes, available as software packages commercially, are needed for determination of PSD. The extension of the model to cylindrical [24] or spherical pores would further add to the complexity of the model. 

The corrected Horvath-Kawazoe models discussed in this work provide a simple pictiue for adsorbate interactions in a micropore and are reasonably fast and accurate methods provided the desired pore size is within the microporous range (<2.0nm). Some distinct advantages of these methods is the faeile extension to different pore geometries compared to other methods and the ease of availabiUty of required physical parameters from Uterature. However, these methods are also known to be very sensitive to certain input parameters and introduce artifacts in the PSD results for large-pore materials. 

Pore Size Averages and Pore Volume Using die Horvath-Kawazoe Model with Nitrogen at 77K (0.3 cc step with 8... 

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... 

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. 

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

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]... 

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. 

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]. 

A two-stage Horvath-Kawazoe adsorption model for pore size distribution analysis... 

The calculation methods for pore distribution in the microporous domain are still the subject of numerous disputes with various opposing schools of thought , particularly with regard to the nature of the adsorbed phase in micropores. In fact, the adsorbate-adsorbent interactions in these types of solids are such that the adsorbate no longer has the properties of the liquid phase, particularly in terms of density, rendering the capillary condensation theory and Kelvin s equation inadequate. The micropore domain (0.1 to several nm) corresponds to molecular sizes and is thus especially important for current preoccupations (zeolites, new specialised aluminas). Unfortunately, current routine techniques are insufficient to cover this domain both in terms of the accuracy of measurement (very low pressure and temperature gas-solid isotherms) and their geometrical interpretation (insufficiency of semi-empirical models such as BET, BJH, Horvath-Kawazoe, Dubinin Radushkevich. etc.). 

In this paper, a modified HK method is presented which accounts for spatial variations in the density profile of a fluid (argon) adsorbed within a carbon slit pore. We compare the pore width/filling pressure correlations predicted by the original HK method, the modified HK method, and methods based upon statistical thermodynamics (density functional theory and Monte Carlo molecular simulation). The inclusion of the density profile weighting in the HK adsorption energy calculation improves the agreement between the HK model and the predictions of the statistical thermodynamics methods. Although the modified Horvath-Kawazoe adsorption model lacks the quantitative accuracy of the statistical thermodynamics approaches, it is numerically convenient for ease of application, and it has a sounder molecular basis than analytic adsorption models derived from the Kelvin equation. 

The principal drawback of the DFT method is that it is computationally intensive relative to the classical adsorption models, although it is still much less compute-intensive than full Monte Carlo molecular simulation. A semianalytic adsorption model that retains computational efficiency while accounting for gas-solid potential interactions in micropores was originally proposed by Horvath and Kawazoe [12], In the Horvath-Kawazoe or HK method, a pore filling correlation is obtained by calculating the mean heat of adsorption (/> required to transfer an adsorbate molecule from the gas phase to the condensed phase in a slit pore of width // ... 

Semiempirical models such as the Horvath-Kawazoe (HK) method [19] and the Dubinin model [20] and their derivatives These models generally make specihc assumptions regarding the shape of the pores and/or the distribution function that describes the pore sizes within the adsorbent. To varying degrees, the semiempirical methods incorporate adsorbate-adsorbent interaction energies into the calculation of the theoretical isotherm. 

Two principal semiempirical adsorption models have enjoyed widespread use for adsorbent PSD characterization the Horvath-Kawazoe (HK) method [19] and its derivatives, and approaches based upon the ideas of Dubinin [20] for modeling micropore distributions. Each of these methodologies is considered in turn. 

Fig. 16. Comparison of PSDs obtained using the Dubinin-Stoeckli (DS), Horvath-Kawazoe (HK), and density functional theory (DFT) methods to interpret an isotherm generated from molecular simulation of nitrogen adsorption in a model carbon that has a Gaussian distribution of slit pore widths [120]. Results are shown for mean pore widths of 8.9 A (left) and 16.9 A (right).

Cheng, L.S. and Yang, R.T. (1994). Improved Horvath—Kawazoe equations including spherical pore models for calculating micropore size distribution. Chem. Eng. Sci., 49, 2599-609. 

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