Dubinin-Astakhov method

Figure 4.6 shows the PSDs obtained from the high-resolution N2 adsorption isotherms at 77 K (Figure 4.5) by applying the Horvath-Kawazoe method (Figure 4.6a), Dubinin-Astakhov method...

FIGU RE 4.9 Comparison of the PSD obtained for different samples by applying different methods (a) Sample ACF1, (b) sample AC2, and (c) sample AC1. DR-C02 is the PSD obtained by applying the Dubinin-based method proposed by Cazorla-Amoros et al. [10] to C02 at 273 K. HK, DFT, and DA are the PSDs obtained by applying Horvath-Kawazoe, DFT, and Dubinin-Astakhov methods to the N2 adsorption isotherm at 77 K, respectively. 

The pore size distribution is displayed in Fig. 5 for the samples activated under reflux conditions during 6 h, calculated by the Dubinin-Astakhov method [28]. These samples have a similar pore diameter, with a value between 17 and 19 A but the intensity of the curves is different. This difference shows the influence of the starting metakaolin over the formation of porosity, where MK-900 shows again the worse properties. The samples activated at longer time did not show internal surface. 

Textural characterisation of the samples was carried out by measuring apparent density (mercury at 0.1 MPa), mercury porosimetry and N2 and CO2 adsorption isotherms, at -196 and 0 °C, respectively. The apparent surface areas of the samples were obtained by using the BET equation [5]. The micropore size analysis was performed by means of the t-plot and the Dubinin-Astakhov methods [6]. 

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]. 

On the other hand, for the microporous carbons with pore size distribution (PSD) with pore fractality, the pore fractal dimensions56,59,62 which represent the size distribution irregularity can be theoretically calculated by non-linear fitting of experimental adsorption isotherm with Dubinin-Astakhov (D-A) equation in consideration of PSD with pore fractality.143"149 The image analysis method54,151"153 has proven to be also effective for the estimation of the surface fractal dimension of the porous materials using perimeter-area method.154"159... 

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]. 

However a recent study has shown that the Stoeckli method (based on the Dubinin-Astakhov theory) [6] gave results similar to those obtained from the molecular simulation methods [9]. On the other hand, the H-K and the MP methods are known to be rather inconsistent. 

Several methods have been proposed for the characterisation of the Micropore Size Distribution (MPSD) that take into account the energetic heterogeneity of solid surfaces [9,10]. The Dubinin-Radushkevich (DR) and Dubinin-Astakhov (DA) equations have been used to describe the adsorption process on structurally heterogeneous solids [11,12]. From these equations, the adsorption isotherm can be expressed as follows ... 

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

The pore volumes of the obtained hard carbons were measured using the molecular probe method [3]. Adsorption isotherms of the probe molecules were measured at 298 K using an adsorption apparatus (Bel Japan, Belsorp 28). The employed probe molecules were CO2, C2H6> n-C4H o and (-C4H10 (minimum molecular dimensions 0.33, 0.40, 0.43 and 0.50 nm, respectively). By applying the Dubinin-Astakhov equation (n=2) [1] to the measured isotherms, the limiting micropore volumes corresponding to the minimum size of the adsorbed molecules were determined. 

The most relevant characteristic of porous materials is the disposal of a high effective surface/volume relationship, usually expressed in terms of their specific surface area (area per mass unit), which can be determined from nitrogen adsorption/desorption data. Different methods are available for determining the specific surface area (Brunauer-Emmett-Teller, Langmuir, and Kaganer), micropore volume (f-plot, ttj, and Dubinin-Astakhov), and mesopore diameter (Barrett-Joyner-Halenda Leroux et al., 2006). Table 1.1 summarizes the values of specific surface area for selected porous materials. 

A brief review of methods based on the integral adsorption Eq, (73) showed that they are attractive to evaluate the pore volume distribution. The analytical solution of this integral for sub-integral functions represented by the Dubinin-Astakhov equation and gamma-type... 

The following table 5.3-3 shows the various formula for the spreading pressure and the pure component hypothetical pressure for various commonly used isotherms. Some isotherms such as Langmuir, Freundlich, LRC have analytical expressions for the spreading pressure as well as the pure component hypothetical pressure. Other isotherms, such as O Brien Myers, Ruthven, Toth and Nitta have analytical expression for the spreading pressure, but the pure component hypothetical pressure expressed in terms of the reduced pressure must be determined from a numerical method. For other general isotherms, such as Unilan, Aranovich, Dubinin-Radushkevich, Dubinin-Astakhov, Dubinin-Stoeckli, Dubinin-Jaroniec, one must resort to a numerical method to obtain the spreading pressure as well as the pure component hypothetical pressure. 

The difficulty in the case of microporous materials stems from the porefilling mechanism. For this reason, the surface area of such materials is often determined by other methods than BET, which is based on layer formation. From the Dubinin equation the micropore volume Wo can be converted to the surface area. The as isotherm comparison method is an independent method for estimating the micropore volume and the surface area (20). The reference isotherm is a plot of the measured isotherm normalized by the amount of gas adsorbed at a fixed relative pressure, typically at p/po = 0.4. High resolution as analysis (21) yields more information about the characteristic texture of the adsorbent. Further methods (MP (22), -plot (23), Dubinin-Astakhov (11), Dubinin-Stockli (12), and so on) are also available for more reliable estimates of the micropore volume and surface area. 

A further criticism of the methods for single parameter estimates of micropore size is that the DR equation frequently does not result in a good fit to equilibrium, microporous adsorption data. To represent such data the Dubinin-Astakhov (DA) equation (ref. 30) has been proposed. This may be written as... 

Dubinin, Polanyi, and Radushkevich proposed about 1947 a simple but very useful empirical theory allowing one to calculate the amount of gas adsorbed in a microporous sorbent. The theory was based on a pore filling model. Today it is used for both characterization of porous solids and also for engineering purposes. It has been extended by several authors among them predominantly Astakhov (1970). The theory is still the subject of further investigations, mainly by statistical mechanics and computational methods (DFT) [7.1-7.3, 7.48-7.55],... 

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