Dubinin-Stoeckli

As described earlier, one of the first methods used to obtain PSD from the Dubinin equation is the so-called Dubinin-Stoeckli method [38-43], For strongly activated carbons with a heterogeneous collection of micropores, the overall adsorption isotherm is considered as a convolution of contributions from individual pore groups. Integrating the summation and assuming a normal Gaussian equation for the distribution of MPV with respect to the K parameter (Equation 4.19), Stoeckli obtained an equation useful to estimate the micro-PSD. 

The mathematical elegance of the Dubinin-Stoeckli approach is impressive, but it remains to be seen whether the basic DR equation is strictly applicable to a narrow... 

The first procedure is based on the Dubinin-Stoeckli principles of volume filling (see Section 4.4.4). The energy of immersion A l/ is related to the micropore volume W0(d) and the characteristic energy Eq for a given micropore size and immersion liquid (Bansal et al., 1988) by the expression ... 

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. 

The characterization of porous materials exhibiting a composite pore structure encompassing micro-meso-and perhaps macro-pore sizes, is of particular significance for the development of separation and reaction processes. Among the characterization methods for materials exhibiting ultramicropore structures, DpDubinin-Radushkevich (DR)[2], Dubinin-Astakov (DA) [3], Dubinin-Stoeckli (DS) [4], as well as the Horvath-Kawazoe (HK), [5] methods are routinely used for the evaluation of the micropore capacity and the pore size distriburion (PSD). 

Three commercial activated carbons were used BPL, CAL and GAe, manufactured by Chemviron, Calgon and CECA respectively. In addition, sample GAe-oxl was prepared by oxidation of GAe in aqueous solution of (NH4)2S20g and further pyrolysis in N2 flow at 773 K [5]. The specific surface areas were obtained applying the BET and Dubinin-Asthakov equations to the adsorption of N2 at 77 K and CO2 at 273 K respectively. Moreover, the C02 adsorption data permitted the evaluation of the micropore size distributions and the mean value of pore width using the Dubinin-Stoeckli equation [6] which supposes a gaussian distribution of pore sizes. 

The micropore size distribution curves obtained by applying the Dubinin-Stoeckli equation to the CO2 adsorption data are shown in Fig. 1. CAL has the narrowest micropore size distribution, BPL and GAe show very similar curves and GAe-ox 1 has the most open distribution that extends beyond the microporosity limit (2 nm). The mean value of the micropore size distribution ranges between 1.47 and 1.77 nm for CAL and GAe-oxl respectively. 

Figure 1 Micropore size distributions obtained from Dubinin-Stoeckli equation...

Sbet is the specific surface area calculated using standard BET method Vp is the total pore volume computed from adsorption at the maximal p/po value parameters with the DS subscripts were computed using the modified Dubinin-Stoeckli equation [7]. 

Based on the Dubinin s theory of volume filling of micropores, a modified Dubinin-Stoeckli (DS) equation was used to estimate micropore contribution with correction related to adsorption in mesopores [7]. The Sps, Vds and other parameters with the DS subscript (Table 1) were calculated over the pore range at the half-widA xds = 0.2-1.0 nm. 

In another investigation [ 18], the PSDs obtained using the Dubinin-Stoeckli (DS)... 

Figure 3 Comparison of PSDs obtained using the Dubinin-Stoeckli (DS), Horvalh-Kawazoe (HK), and density fitnctional theory (DFT) methods to interpret an isotherm generated from molecular simulation of nitrogen adsorption in a model carbon that has an Gaussian distribution of slit pore widths (18]. Results are shown for mean pore widths of 8.9 A (left) and 16.9 A (right).

In the Dubinin-Stoeckli (DS) method, a Gaussian pore size distribution is assumed for 7(B) in Eq. (39), based on the premise that for heterogeneous carbons, the original DR equation holds only for those carbons that have a narrow distribution of micropore sizes. This assumption enables Eq. (39) to be integrated into an analytical form involving the error function [119] that relates the structure parameter B to the relative pressure A = -RT ln(P/Po)-The structure parameter B is proportional to the square of the pore halfwidth, for carbon adsorbents that have slit-shaped micropores. 

Figure 17.9 Pore size distributions obtained for the sample CFS50 applying the Dubinin-Stoeckli (DS) equation to the CO2 and Nj adsorption data.

Comparison of the parameters obtained by means of the Dubinin-Stoeckli (DS) and Jaroniec-Choma (JC) equations for nitrogen adsorption on the active carbons studied... 

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. 

According to these results, and many others published in the literature [20,23,35,38-44,49-97], to increase the adsorption capacity of an AC high hydroxide/carbon ratios need to be used. However, in addition to the increase in surface area and micropore volume, it is also important to analyze the effect on the MPSD. Eigure 1.11 presents the MPSD calculated by applying the Dubinin-Stoeckli (DS) equation [10,11] to the N2 adsorption data. The higher the KOH/anthracite ratio, the wider the pore size distribution and the higher the mean pore size. These MPSD curves agree with what can be deduced from the difference in the micropore volumes calculated from N2 and CO2 adsorption data. 

FIGURE 1.11 MPSD calculated by applying the Dubinin-Stoeckli (DS) equation to the N2 adsorption data for samples prepared with different KOH/anthracite ratios (redrawn from Lozano-Castello, D., Cazorla-Amoros, D., Linares-Solano, A., and Quinn, D.F. Carbon 40(7) 989-1002, 2002. With permission). 

FIGURE 1.15 Mean pore size (L calculated from Dubinin-Stoeckli [DS] equation) of AC versus the maximum HTT. 

The low partial pressure regions of an isotherm are also where micropore filling occurs. Micropores, with widths less than 2 nm, are easily filled by a few monolayers of most adsorbents, and the second group of equations or theories attempt to extract micropore characteristics from the initial stages of the isotherm. These are similar to the surface area techniques and include Henry s Law based interpretations, the Langmuir-Brunauer equation [IS], and the Dubinin-Stoeckli based theories [16,17]. 

Dubinin-Stoeckli, DS). These methods are based on Dubinin s theory of the volxmie filling of micropores (TVFM), the density functional theory (DFT) and the Horvath-Kawazoe method. However, CO2 provides a complement to N2 adsorption for the assessment of the narrow microporosity [19]. A frequently observed disagreement between the PSD obtained finm adsorption isotherms of different gases is mostly attributed to molecular sieving and networking effects [20], and to specific adsorbate-carbon interactions [9, 21]. Although these factors are important, possible inconsistencies in the PSD may also be caused by the choice of parameters for intermolecular interactions [9]. 

II. MODELS BASED ON THEORY OF VOLUME FILLING A. Dubinin-Stoeckli Method... 

The DA equation forms the basis for the Dubinin-Stoeckli (DS) method for PSD determination. As in other TVF methods, it is assumed that microporosity is composed of different pore groups and the local adsorption in each can be described by the DA equation. The exponent n is chosen to be 3, which gives sufficient flexibiUty to fit the isotherm equation. It must be noted, however, that DS equations derived by using n = 2 also fi equently occin in literature [22,50]. 

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