Horvath-Kawazoe method width

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].

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. 

The Horvath-Kawazoe method can also be appUed to adsorption of carbon dioxide at 273 K. This gives information about smaller micropores than does nitrogen adsorption. A partial micropoie distribution for PM-1 from CO2 adsorption is compared with the result from N2 adsorption in Figure 2.3. Extension of the distribution to higher pore widths requires higher pressures than were possible on the instrument used to obtain these data. The apparent distribution from CO2 adsorption may be affected by the presence of specific adsorption sites. Nevertheless, the data support the idea that the material is essentially microporous. 

The micropores method is frequently used to determine pore size distributions below 20 A in inorganic membranes. For example, Larbot et al. [77] analyzed the pore size distribution of a Ti02 membrane with a mean pore size of 8 A. They also studied how pore size distribution width changes due to small variations of the sol— gel synthesis method. Other methods to analyze micropores are sometimes used too for instance, Kumar et al. [116] used the Horvath—Kawazoe method to select the appropriate sinterizing (calcination temperature) technique to prepare zeolite membranes. 

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. 

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 // ... 

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).

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 and Kawazoe (HK) method [39] was developed to determine the PSD of active carbons from nitrogen adsorption isotherm. All pores are assumed to have slit shape. This method rests on the assumption that the adsorption state of a pore is either empty or completely fiUed. The demarcation pressure between these two states is called the pore-filling pressure, and it is a function of pore width. The equilibrium of a pore exposed to a bulk phase of constant chemical potential is obtained from the minimization of the following grand thermodynamic potential ... 

---