Pore size distributions Horvath-Kawazoe method

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. 

Horvath G and Kawazoe K 1983 Method for oaloulation of effeotive pore size distribution in moleoular sieve oarbon J. Chem. Eng. Japan 16 470-5... 

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]. 

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]. 

Specific surface area was calculated from the Brunauer-Emmett-Teller (BET) equation for N2 adsorption at 77 K (Micromeritics, ASAP 2010) [10], The t-method of de Boer was used to determine the micropore volume [11]. The pore size distribution curves of micropores were obtained by the Horvath-Kawazoe (H-K) method [12]. 

G. Horvath and K. Kawazoe, Method for the calculation of effective pore size distribution is molecular sieve carbon, J. Chem. Eng. Jpn. 16 (1983) pp. 470-475. 

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. 

A series of good quality MCM-41 samples of known pore sizes was used to examine the applicability of the Horvath-Kawazoe (HK) method for the pore size analysis of mesoporous silicas. It is shown that the HK-type equation, which relates the pore width with the condensation pressure for cylindrical oxide-type pores, underestimates their size by about 20-40%. The replacement of this equation by the relation established experimentally for a series of well-defined MCM-41 samples allows for a correct prediction of the pore size of siliceous materials but does not improve the shape of the pore size distribution (PSD). Both these versions of the HK method significantly underestimate the height of PSD. In addition, PSD exhibits an artificial tail in direction of fine pores, ended with a small peak, which may be interpreted as indicator of non-existing microporosity. 

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). 

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. 

For the purpose of describing pore size distribution in this range, Horvath and Kawazoe (1983) proposed a method based on potential func-... 

Fig. 2. 18. Effective pore size distributions of carbon molecular sieves calculated by Horvath-Kawazoe method.

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