Broekhoff-de Boer method

Table 2 shows that the specific surface Sbet and the Dubinin micropore volume Vp increase with the EDAS/TEOS ratio. Figure 2a presents the evolution of the cumulative surface distribution over the micro- and the mesopore range obtained by applying the Brunauer and the Broekhoff-de-Boer methods in their respective domains. For low EDAS content, 0.025 < EDAS/TEOS < 0.06, the structure is mainly microporous, whereas for high EDAS content, 0.1 < EDAS/TEOS < 0.2, mesopores are also present. Indeed, a large mesopores distribution... 

Figure 2. (a) Cumulative surface area as function of pore diameter obtained by applying Brunauer and Broekhoff-de-Boer methods, (b) FHH plots (Equation 1). The exponents m and the linear range A(p/p ) used for their extraction are presented in Table 2. 

Lukens, W.W., Jr, Schmidt-Winkel, P., Zhao, D., et al. (1999). Evaluating pore sizes in mesoporous materials a simplified standard adsorption method and a simplified BroekhofF-de Boer method. Langmuir, 15, 5403-9. 

SBA-15 samples with diameters from 5 to 10 nm have been prepared by tuning the temperature of the first step of the synthesis [5], MCM-41 has been prepared in the presence of hexadecyl trimethyl ammonium by using methylamine as pH-controlling agent [6], The pore size from N2 adsorption at 77 K has been evaluated by the Broekhoff and de Boer method, shown to correctly evaluate the pore size of ordered mesoporous silicas [7]. 

Comparison of the pore size distribution determined by the present method with that from the classical methods such as the BJH, the Broekhoff-de Boer and the Saito-Foley methods is shown in Figure 4. Figure 5 shows a close resemblance of the results of our method with those from the recent NLDFT of Niemark et al. [16], and XRD pore diameter for their sample AMI. The results clearly indicate the utility of our method and accuracy comparable to the much more computationally demanding density functional theory. There are several other methods published recently (e. g. [21]), however space limitations do not permit comparison with these results here. It is hoped to discuss these in a future publication. 

The Broekhoff-de Boer t method for the determination of surface areas is not based on a new theory. It is an empirical method in which the adsorption isotherm of a material of unknown surface area is compared with a standard isotherm, the common t curve, valid for f materials with a surface area of 1 m2. 

Important trends in N2 isotherm when the PS beads are used as a physical template are shown in Table 1 and Fig. 2. In Table 1, PI is the alumina prepared without any templates, P2 is prepared without ]4iysical template (PS bead), P3 is prepared without chemical template (stearic acid), and P4 is prepared with all templates. For above 10 nm of pore size and spherical pore system, the Barrett-Joyner-Halenda (BJH) method underestimates the characteristics for spherical pores, while the Broekhoff-de Boer-Frenkel-Halsey-Hill (BdB-FHH) model is more accurate than the BJH model at the range 10-100 nm [13]. Therefore, the pore size distribution between 1 and 10 nm and between 10 and 100 nm obtained from the BJH model and BdB-FHH model on the desorption branch of nitrogen isotherm, respectively. N2 isotherm of P2 has typical type IV and hysteresis loop, while that of P3 shows reduced hysteresis loop at P/Po ca. 0.5 and sharp lifting-up hysteresis loop at P/Po > 0.8. This sharp inflection implies a change in the texture, namely, textural macro-porosity [4,14]. It should be noted that P3 shows only macropore due to the PS bead-free from alumina framework. 

On dry gels, standard characterization techniques for porous media are used, several of which have been described in Volume 2 of this series helium pycnometry for pore volume determination (Section 6.3.1.2) as well as nitrogen adsorption at 77 K for surface area (Section 6.3.2.2, BET method), for microporosity (Section 6.3.3.2, Dubinin-Radushkevich method), for pore size distribution (Section 6.3.3.3, BJFl method), and for total pore volume (Section 6.3.3.4). When characterizing gels by nitrogen adsorption, other methods are also used for data interpretation, for example, the t-plot method for microporosity (Lippens and de Boer, 1965) and the Dollimore-Heal method (Dollimore and Heal, 1964) or Broekhoff-de Boer theory for mesoporosity (Lecloux, 1981). 

Complete details about the method of construction of 3-D porous networks through Monte Carlo simulation can be found elsewhere [10] similarly, the precise algorithm employed to replicate sorption processes in porous networks has been reported somewhere else [11]. Here, we will only mention several key aspects regarding these porous network and sorption simulations. First, the critical conditions required for cavities (hollow spheres) and necks (hollow cylinders open at both ends) to be fully occupied by either condensate or vapor have been calculated by means of the Broekhoff-de Boer (BdB) equation [12], while the thickness of the adsorbed film has been approximated via the Harkins-Jura equation [13]. Some other important assumptions that are made in this work are (i) the pore volume is exclusively due to sites (ii) bonds are considered as volumeless windows that communicate neighboring sites (ili) bonds can merge into a site without suffering of any geometrical interference with adjacent throats. 

In order to correct these discrepancies, Broekhoff and de Boer [5] generalized Kelvin s equation by taking pore shape into account, as well as the influence of surface curvature on the thickness of the adsorbed layer. The BdB method can be applied to both adsorption and desorption isotherms using four different pore models defined by a shape factor. Unfortunately, owing to computational difficulties, this last method, although more general, has been far less applied than the first two. 

The assessment of mesopores and their size distribution is only sensible with type rv isotherms. In that case the Kelvin equation can be used and can be well adapted to various pore shapes. The modification of the Kelvin equation by Broekhoff and de Boer allows for a more realistic assessment than the classic method. Using the t method data treatment, it is possible to assess the micropore volume and, from its deviations from ideality, the geometry of possible micro and meso pores. 

Among many classical approaches available in the literature, a method developed by Broekhoff and de Boer (BdB) [47—53] for description of vapor adsorption and desorption in cylindrical pores and slit pores is the most thermodynamically rigorous and elegant for more than 35 years. This method relies on a reference system, which is a flat surface having the same structure and surface chemistry as that of the adsorbent. The pores of the adsorbent can have either... 

All these assumptions do not exactly agree with results obtained from molecular simulations. However, erroR resulting from these assumptions in the case of cylindrical pore and in the case of the reference flat surface may partly compensate each other. The advantage of the BdB method is that in the framework of their model all thermodynamic derivations are strictly correct. Details of this method can be found in the excellent papeR by Broekhoff and de Boer. 

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