Dubinin-Astakhov

The original DR equation is thus a special case of the Dubinin-Astakhov equation, with m = 2 parameter of Equation (4.18) for m = 2 is related to the structural constant B of the DR treatment through the simple expression... 

Dubinin-Kadushkevich. This model (29) is the same as the more general Dubinin-Astakhov equation (30) (see below), with n = 2. Dubinin-Astakhov ... 

EOS, LRC (loading ratio correlation), Dubinin-Radushkevich (D-R), and Dubinin-Astakhov (D-A) are more suitable. 

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]. 

Semiernpirical Isotherm Models. Some of these models have been shown to have some thermodynamic inconsistencies and should be used with due care. They include models based on the Polanyi adsorption potential (Dubinin-Radushkevich, Dubinin-Astakhov, Radke-Prausnitz, Toth, UNI LAN. and BET). 

A more general expression, called the Dubinin-Astakhov (D-A) equation98 is written as... 

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... 

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. 

Stoeckli (1993) has pointed out that the Dubinin-Astakhov equation (Equation (4.45)) can be derived from Equation (4.52), but McEnaney (1988) and others (e.g. Jaroniec et al. 1997) have drawn attention to the difficulty in arriving at an unambiguous interpretation of the energy distribution function. Indeed, Stoeckli et al. (1998) have now pointed out that Equation (4.45) can be usefully applied to a number of adsorption isotherms on non-porous solids. A comprehensive review of the significance and application of Equation (4.52) is given by Rudzinski and Everett (1992). 

Over the past 30 years many organic molecules of different size, shape and polarity have been used as molecular probes. A high proportion of the experimental isotherms on porous carbons have been analysed by application of the Dubinin-Radushkevich (DR) equation or, in a few cases, by the Dubinin-Astakhov (DA) equation. So far, the more sophisticated Dubinin-Stoeckli (DS) treatment (Stoeckli, 1993) has been applied by very few other investigators. 

Various attempts were made by Dubinin and his co-workers to apply the fractional volume filling principle and thereby obtain a characteristic curve for the correlation of a series of physisorption isotherms on a zeolite (Dubinin, 1975). As was noted in Chapter 4, the original Dubinin-Radushkevich (DR) equation (i.e. Equation (4.39)) was found to be inadequate and in its place the more general Dubinin-Astakhov (DA) equation was applied (i.e. Equation (4.45)). 

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. 

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]. 

The Dubinin-Astakhov (D-A) [6] equation was applied to the N2 adsorption isotherms. The accessible pore width, L, was calculated fi om the expression proposed by Stoeckli and Ballerini [9]. 

Textural parameters derived from the Dubinin-Astakhov equation applied to CO2... 

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. 

Ozawa S., Kusumi S. and Ogino Y., Physical adsorption of gases at high pressure, IV. An improvement of the Dubinin-Astakhov adsorption equation. J. Colloid Interface Sci. 56 (1976) pp.83-91. 

Since lAST-based calculations require a precise description of pure component adsorption data, especially in the low coverage range, Freundlich, Langmuir, Toth and Dubinin-Astakhov models were compared. 

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