Pore size distributions Dubinin equation

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... 

It is apparent that any limited range of linearity of a simple DR plot cannot be used to give a reliable evaluation of the pore size distribution. In order to describe a bixnodal micropore size distribution, Dubinin (1975) applied a two-term equation, which we may write in the form... 

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]. 

In order to estimate the pore size distributions in microporous materials several methods have been developed, which are all controversial. Brunauer has developed the MP method [52] using the de Boer t-curve. This pore shape modelless method gives a pore hydraulic radius r, which represents the ratio porous volume/surface (it should be realised that the BET specific surface area used in this method has no meaning for the case of micropores ). Other methods like the Dubinin-Radushkevich or Dubinin-Astakov equations (involving slitshaped pores) continue to attract extensive attention and discussion concerning their validity. This method is essentially empirical in nature and supposes a Gaussian pore size distribution. 

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.

TSA - total surface area determined by N2 adsorption (BET method) V(N2) and V(C02) - micropore volumes calculated by the application of the Dubinin-Radushkevitch equation to N2 adsorption at 77 K and CO2 adsorption at 273 K pore size determined according to the BJH method - maximum value of the BJH pore size distribution peak calculated from the adsorption branch of the N2 isotherm capacitance values... 

In Chapter 2, we discussed the fundamentals of adsorption equilibria for pure component, and in Chapter 3 we presented various empirical equations, practical for the calculation of adsorption kinetics and adsorber design, the BET theory and its varieties for the description of multilayer adsorption used as the yardstick for the surface area determination, and the capillary condensation for the pore size distribution determination. Here, we present another important adsorption mechanism applicable for microporous solids only, called micropore filling. In this class of solids, micropore walls are in proximity to each other, providing an enhanced adsorption potential within the micropores. This strong potential is due to the dispersive forces. Theories based on this force include that of Polanyi and particularly that of Dubinin, who coined the term micropore filling. This Dubinin theory forms the basis for many equations which are currently used for the description of equilibria in microporous solids. 

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. 

More national and international standardization procedures for mercury porosimetry and the derivation of pore size distributions from adsorption isotherms are in preparation. Regarding the weakness of the two-parameter BET model for surface area determination in addition the three-parameter BET equation or improved approximations [26] should be considered. Competitive evaluation methods, like the method of Dubinin, Horwath-Kawazoe, Kaganer and Radushkevich are being discussed. 

Porosity and pore-size distributions were determined by gas adsorption and immersion calorimetry, with the measurement of helium and bulk densities. Volumes of micropores were calculated using the Dubinin-Radushkevich (DR) equation (Section 4.2.3) to interpret the adsorption isotherms of N2 (77 K), CO2 (273 K) and n-C4H o (273 K). Volumes of mesopores were evaluated by subtracting the total volume of micropores from the amount of nitrogen adsorbed at p/p° = 0.95. The two density values for each carbon were used to calculate the volume of the carbon skeleton and the total volume of pores (including the inter-particle space in monolithic disks). Immersion calorimetry of the carbon into liquids with different molecular dimensions (dichloromethane 0.33 run benzene 0.37 nm and 2,2-dimethylbutane 0.56 nm) permits the calculation of the surface area accessible to such liquids and subsequent micropore size distributions. The adsorption of methane has been carried out at 298 K in a VTI high-pressure volumetric adsorption system. Additional techniques such as mercury porosimetry and scanning electron microscopy (SEM) have also been used for the characterization of the carbons. 

Pores are classified by the International Union of Pure and Applied Chemistry (lUPAC) by pore size as micropores (<2 nm), mesopores (2-50 nm) and macropores (>50 nm). Micropores are sometimes divided into ultramicropores (<0.7 nm) and supermicropores (1.4—2.0 nm). The terms nanopore and nanoporosity are not defined precisely but refer to nanometre-sized pores. Characterisation of the porous structures of materials is difficult because some MOF materials are flexible. A variety of isotherm equations and adsorptives have been used to characterise porous structures using gas adsorption techniques. Porous structures are characterised by surface areas [determined using Langmuir, Bmnauer-Emmett-Teller (BET), Dubinin-Radushkevich (DR), etc., equations], pore volumes [total, micropore (DR), etc.] and pore size distributions. 

The typical example of pore size distribution is presented on Fig. 3 for a sample of silica gel of medium porosity exhiting the classical hysteresis loop of type HI. Curve 1 shows the volume differential pore size distribtuion calculated from the adsorption and desorption isotherms of nitrogen by using equation (6) by Dubinin s method from the adsorption isotherm (curve 2) and the desorption isotherm (curve 3) using the model of unrelated pores. 

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 isotherms developed by Dubinin and coworkers employ a power to which the adsorption potential is raised that indicates the prevalent type of pores. The Dubinin-Radnshkevich eqnation [1 in Table 14.3] was intended for microporons adsorbents, since the exponent is 2. The Dnbi-nin-Astakhov equation (m) allows the exponent B to vary, bnt a reasonable lower limit is unity (for macroporous adsorbents). The Dnbinin-Stoeckli equation (n) allows a distribution of pore sizes, which is a feature of many adsorbents. 

The calculation methods for pore distribution in the microporous domain are still the subject of numerous disputes with various opposing schools of thought , particularly with regard to the nature of the adsorbed phase in micropores. In fact, the adsorbate-adsorbent interactions in these types of solids are such that the adsorbate no longer has the properties of the liquid phase, particularly in terms of density, rendering the capillary condensation theory and Kelvin s equation inadequate. The micropore domain (0.1 to several nm) corresponds to molecular sizes and is thus especially important for current preoccupations (zeolites, new specialised aluminas). Unfortunately, current routine techniques are insufficient to cover this domain both in terms of the accuracy of measurement (very low pressure and temperature gas-solid isotherms) and their geometrical interpretation (insufficiency of semi-empirical models such as BET, BJH, Horvath-Kawazoe, Dubinin Radushkevich. etc.). 

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