Horvath-Kawazoe equation

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

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]. 

Classical methods, like DS and HK, show shortcomings in the determination of MSD due to the assumptions involved in their formulation of the adsorption process Dubinin equation does not show linearity in the Dubinin plot for single slit pores and Horvath-Kawazoe equation assumes that at a given pressure a pore is either completely filled or completely empty, which is contrary to the behavior observed in computer simulations Resulting MSD are shifted respect to those obtained by Monte Carlo simulations, by amounts that vary with the actual distribution, and too small micropores are predicted... 

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. 

Du et al. [75] investigated the surface area of purified and pristine HiPco nanotubes by performing N2 and Ar adsorption isotherms. Interestingly, this study found that there were significant differences in the specific surface areas of the pristine HiPco samples, even when their reported impurity levels were similar. These authors analyzed their data using the Horvath—Kawazoe equation... 

Cheng, L.S., and Yang, R.T., Predicting isotherms in micropores for different molecules and temperatures from a known isotherm by improved Horvath-Kawazoe equations. Adsorption, 1(3). 187-196 (1995). 

Swiatkowski, A, Trznadel, B.J., and Zietek, S., Description of active carbon micropore size distribution based on the Horvath-Kawazoe equation adapted to benzene adsorption isotherms, Adsorpt. Sci. Technol., 14(1), 59-68(1996). 

Predicting Isotherms in Micropores for Different Molecules and Temperatures from a Known Isotherm by Improved Horvath-Kawazoe Equations, Adsorption 1 (1995), p. 187-196. 

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. 

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... 

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]. 

Specific micropore volumes derived from the Horvath-Kawazoe (HK) and Dubinin-Astakhov (DA) methods. Characteristic energies from the Dubinin-Astakhov equation. 

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]. 

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 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.). 

Figure I Relation between filling pressure and pore width predicted by the modified Kelvin equation (MK), the Horvath-Kawazoe method (HK), density Junctional theory (DFT), and molecidar simulation (points) for nitrogen adsorption in carbon slits at 77 K [8].

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. 

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. 

Hgure It.l Pore-filling pressure dependence on the pore width for nitrogen adsorption in carbon slit pore at 77.35 K. (Solid line) NLDFT. (Dashed line) Horvath-Kawazoe method. (Dash-dot line) Kelvin equation. 

In 1983 Horvath and Kawazoe [143] proposed a method to derive analytical equations for the average potential in a micropore of a given geometry, which in fact relate the adsorption potential with the pore size x. These equations are used to express the amount adsorbed in micropores as a function of the pore width and subsequently to calculate the micropore volume distributioa Thus, the Horvath-Kawazoe (HK) procedure is a logical extension of the metliod based on the Kelvin equation to the micropore range, and can be considered as an extension of the condensation approximation method to the region of fine pores [4]. Further improvements and modifications of this method are reported elsewhere [144, 153-157]. 

The Horvath-Kawazoe (H-K) equation was developed by Horvath and Kawazoe [111] to characterize the microporous structure of an adsorbent. They used the 10-4 potential to describe the potential energy of adsorbate molecules in a slit pore and... 

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. 

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