Dubinin-Stoeckli method

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. 

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

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). 

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. 

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]. 

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. 

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. 

FIGURE 1.20 (a) N2 adsorption isotherms at 77 K and (b) MPSD by applying the Dubinin-Stoeckli (DS) method corresponding to an AC, and the same material after KOH post-heat treatment (redrawn from Martin-Gulldn, I., Marco-Lozar, J.P., Cazorla-Amoros, D., and Linares-Solano, A. Carbon 42(7) 1339-1343, 2004. With permission). 

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]. 

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]. 

These procedures proposed by Dubinin and by Stoeckli arc, as yet, in the pioneer stage. Before they can be regarded as established as a means of evaluating pore size distribution, a wide-ranging study is needed, involving model micropore systems contained in a variety of chemical substances. The relationship between the structural constant B and the actual dimensions of the micropores, together with their distribution, would have to be demonstrated. The micropore volume would need to be evaluated independently from the known structure of the solid, or by the nonane pre-adsorption method, or with the aid of a range of molecular probes. 

The first part of this paper focuses on the structural study of TS deposits using scatming and transmission electron microscopy in order to characterize the mesopore and macropore network, as well as the microstructure. In the second part, we present a comparative study of adsorption properties of TS deposits and some reference samples. We have measured volumetry adsorption isotherms of various probe molecules like CH4, N2, CsHg and have determined parameters like energy adsorption, micropore volume and pore size distribution (PSD), using empirical methods and both Dubinin-Asthakov and Stoeckli models [3]. 

NTR suggest that these mesopores are probably closed. In order to confort this hypothesis, it could be interesting to p orm experimental methods such as liquid intrusion and thermoporosity [17], Gas adsorption study has revealed a TS deposits microporosity and its quantitative characterization has been done using Dubinin-Asthakov and Stoeckli theory. To valid these results, we will extend the number of probe molecules and compare our experimental results with DFT a statistical method [18]. Finally, this observed multiscale porosity can play a role in diffusion and retention of hydrogen, studies are on progress to put in evidence this effect. 

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