Horvath-Kawazoe approach

N2 isotherms at 77 K are used for practical reasons (e.g., simultaneous determination of the BET surface area). The use of the Kelvin equation was a popular approach for estimating the pore size distribution. Many procedures were proposed for calculating the pore size distribution from the N2 isotherms over the period between 1945 and 1970 (Rouquerol et al., 1999). The method proposed by Barrett, Joyner, and Halenda (1951), known as the BJH method, continues to be used today. In the BJH method, the desorption branch of the isotherm is used, which is the desorption branch of the usual hysteresis loop of the isotherm for the mesoporous sorbent. The underlying assumptions for this method are 

Many questions have been raised concerning the validity of the Kelvin equation, in particular, the lower limit of the pore size that one could use with this approach. Clearly, this method does not apply when the pore size approaches molecular dimensions, or a few molecular sizes. Molecular simulations by Jes-sop et al. (1991) showed that the Kelvin equation fails to account for the effects of fluid-wall interactions. The density functional theory study of Lastoskie et al. (1993), as well as other work, indicated that the Kelvin equation would underestimate the pore size and should not be extended below a pore size of 7.5 nm. 

Over the past two decades, different types of HK models have been developed depending upon the pore geometry. The original HK model discussed slit-shaped pores (Horvath and Kawazoe, 1983), whereas models for cylindrical pores (Saito and Foley, 1991), and spherical pores (Cheng and Yang, 1994) have also been proposed. The basic framework for all the different HK models is the same  

The original HK model will be given first, followed by the corrected model by Rege and Yang (2000). 

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According to the lUPAC classification of pores, the size ranges are micropoies (<2 nm), tnesopores (2-50 nm), and macropores (>50 mn) (lUPAC, 1972). All useful sorbents have micropores. The quantitative estimation of pore size distribution (PSD), particularly for the micropores, is a crucial problem in the characterization of sorbents. Numerous methods exist, of which three main methods will be described Kelvin equation (and the BJH method), Horvath-Kawazoe approach, and the integral equation approach. 

IV. MODELS BASED ON THE HORVATH-KAWAZOE APPROACH A. HK Methods... 

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]. 

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]. 

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. 

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. 

Alternatively, the method of Horvath and Kawazoe [33] can be applied which relies on high resolution adsorption measurements at very low pressures. It is certainly a very attractive approach as it can be easily combined with conventional measurements of N2 and Ar adsorption isotherms. It is strongly suggested, however, that the method needs to be calibrated against well-known and well-defined microporous materials before using it for unknown samples. 

Sorption in micropores can be described by the Dubinin-Radushkevic formalism that has been adapted by Stoeckli et al. This is a largely empirical approach and it should be emphasized that the use of a combination of Langmuir types isotherm leads to similar quantitative results. For evaluation of the distribution of micropores, one can either rely on high-resolution measurements of mostly nitrogen adsorption as suggested by Horvath and Kawazoe or use a combination of probe molecules of different minimum kinetic diameter. More recently, approaches based on density functional theory are put forward. 

Horvath and Kawazoe s Approach on the Micropore Size Distribution... 

Pore Size Distribution. As mentioned earlier, NLDFT and GCMC are considered to be the most accurate methods for micro- and mesopore size analysis. Their application is limited by the availability of kernel isotherms. Although semiempirical analytical methods may fill the gap, it is however worthwhile to recall that these methods often underestimate the pore sizes for a given equilibrium pressure. The approach of Horvath and Kawazoe (HK) is based on a fundamental statistical analysis of a fluid confined to a slit-shaped pore (24). An extension of this method to cylindrical and spherical pore models was made by Saito and Foley (SF, (25)) and Cheng and Yang, respectively (26). 

From the equation, the relation between p and / is uniquely defined and thus the relation between the effective pore size (/—J.) and p is also obtained as shown in Table 2 2 The relation is also plotted in Fig. 2 17 In the table and figure, the pore size calculated by Dollimore s method is included Both methods approach pore size of around 13 4 A and plPi=Q 05, and Horvath and Kawazoe suggested that their method should be applied to measurements below plp,=0 5 and that Dollimore s method should be used above this relative pressure... 

In the microporous region the pore width is approaching molecular dimensions, for instance the effective width of the nitrogen molecule is given as 0.364nm. It is clear that the pore wall adsorption fields are almost overlapping and adsorbate molecules are in close proximity to them. This results in enhanced uptake of adsorbate at very low relative pressures. Horvath and Kawazoe (2) calculated that, for pores in carbons, pores filled at the low relative pressure shown in Table 1. 

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