NLDFT theory

Thus, either type I or type IV isotherms are obtained in sorption experiments on microporous or mesoporous materials. Of course, a material may contain both types of pores. In this case, a convolution of a type I and type IV isotherm is observed. From the amount of gas that is adsorbed in the micropores of a material, the micropore volume is directly accessible (e.g., from t plot of as plot [1]). The low-pressure part of the isotherm also contains information on the pore size distribution of a given material. Several methods have been proposed for this purpose (e.g., Horvath-Kawazoe method) but most of them give only rough estimates of the real pore sizes. Recently, nonlocal density functional theory (NLDFT) was employed to calculate model isotherms for specific materials with defined pore geometries. From such model isotherms, the calculation of more realistic pore size distributions seems to be feasible provided that appropriate model isotherms are available. The mesopore volume of a mesoporous material is also rather easy accessible. Barrett, Joyner, and Halenda (BJH) developed a method based on the Kelvin equation which allows the calculation of the mesopore size distribution and respective pore volume. Unfortunately, the BJH algorithm underestimates pore diameters, especially at... 

Much better agreement has been found with the results of the Deijaguin-Broekhoff-de Boer theory [10,11] (Figure 3, top). For pores wider than ca. 6 nm the equilibrium and spontaneous capillary condensation transitions predicted by the NLDFT are well approximated by the semi-empirical equations of Broekhoff and de Boer [10,11] In smaller pores, the deviations are substantial (Figure 3, bottom)... 

The NLDFT predicts the critical point for capillary condensation phase transition (capillary critical pore size) at ca. 2 nm, which is approximately the minimum pore size in which capillary condensation is experimentally observed [21,27], However, the theory fails to predict the disappearance of the hysteresis loop for pores smaller than ca. 4 nm (hysteresis critical point) [20,15], It should be noted that the theory of Broekhoff and de Boer fails to predict both critical points unless some additional semi-empirical corrections are made [16]... 

Recent Monte Carlo simulations of N2 in cylindrical pores fully support the results of the NLDFT calculations [28]. Thus, it appears that the failure of the NLDFT to predict the disappearance of the hysteresis loop at relative pressures below ca, 0.4 and pores smaller than ca, 4 nm is of a fundamental nature and cannot be explained by approximations made in the theory. 

Figure 3. Capillary hysteresis of nitrogen in cylindrical pores at 77 K Equilibrium desorption (black squares) and spontaneous condensation (open squares) pressures predicted by the NLDFT in comparison with the results of the Broekhoff and de Boer theory [10, 11],...

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. 

Because of its diatomic nature and permanent quadrupole moment, the physisorp-tion of nitrogen at 77 K presents special problems. The application of DFT is facilitated if the molecules are assumed to be spherical, which was the approach originally adopted by Seaton et al. (1989) and also by Lastoskie et al. (1993). The analytical procedures already outlined in Chapter 7 (Section 7.6) do not depend on the meniscus curvature and are in principle applicable to both capillary condensation and micropore filling. The non-local version of the mean field theory (NLDFT), which was used by Lastoskie, gave excellent agreement with computer simulation when applied to the carbon slit pore model. However, as pointed out earlier, these computational procedures are not entirely independent since they involve the same model parameters. 

Essential progress has been made recently in the area of molecular level modeling of capillary condensation. The methods of grand canonical Monte Carlo (GCMC) simulations [4], molecular dynamics (MD) [5], and density functional theory (DFT) [6] are capable of generating hysteresis loops for sorption of simple fluids in model pores. In our previous publications (see [7] and references therein), we have shown that the non-local density functional theory (NLDFT) with properly chosen parameters of fluid-fluid and fluid-solid intermolecular interactions quantitatively predicts desorption branches of hysteretic isotherms of nitrogen and argon on reference MCM-41 samples with pore channels narrower than 5 nm. 

The adsorption and desorption isotherms have been calculated for the Nj sorption at 77K in cylindrical pores of MCM-41 materials in the range 1-12 nm. The points of spinodal and equilibrium transitions are plotted in Fig. 2. There are several features worth noticing. As the pore size increases, the line of spinodal desorption saturates at the value corresponding to the spinodal decomposition of the bulk liquid. The line of equilibrium capillary condensation asymptotically approaches the Kelvin equation for the spherical meniscus and the line of spontaneous capillary condensation asymptotically approaches the Kelvin equation for the cylindrical meniscus. This asymptotic behavior is in agreement with the classical scenario of capillary hysteresis [12] capillary condensation occurs spontaneously after the formation of the cylindrical adsorption film on the pore walls while evaporation occurs after the formation of the equilibrium meniscus at the pore end. Most interestingly, the NLDFT predictions of equilibrium and spontaneous capillary condensation transitions for pores wider than 6 nm are approximated by the semi-empirical equations of the Deijaguin-Broekhoff-de Boer theory [13]. 

The non-local density functional theory (NLDFT) with properly chosen parameters of fluid-fluid and fluid-solid intermolecular interactions quantitatively predicts both adsorption and desorption branches of capillary condensation isotherms on MCM-41 materials with the pore sizes from 5 to 10 nm. Both experimental branches can be used for calculating the pore size distributions in this pore size range. However for the samples with smaller pores, the desorption branch has an advantage of being theoretically accurate. Thus, we recommend to use the desorption isotherms for estimating the pore size distributions in mesoporous materials of MCM-41 type, provided that the pore networking effects are absent. 

The nonlocal density functional theory (NLDFT) has been used to characterize mesocaged structures.[164] NLDFT analysis gives accurate information about the cage size, the total meso- and micropore volumes and surface area, and the pore-wall thickness in combination with XRD measurements. Argon- and nitrogen-desorption data on FDU-1 provided evidence that there are two major populations of pore entrances. Argon desorption was superior in providing information about pore connectivity in FDU-1 samples. 

Because of the present availability of commercial software, the nonlocal version of density functional theory (i.e., NLDFT) is now widely used for pore... 

Figure 7.3 A comparison of experimental data for nitrogen adsorbed at 77 K on Sterling FT-G(2700) graphite with the modified nonlocal density fimctional theory [10] (MNLDFT) and unmodified [17] nonlocal density fimctional theory (NLDFT).

G activated carbon (points) with the fit given by the nonlocal density functional theory (NLDFT) models (line), (b) The pore width distribution for the carbon. 

Figure 7.10 The average pore fluid density in pores of various widths as calculated hy nonlocal density functional theory (NLDFT). Note the periodic nature of the density, with density maxima near the positions of the minima in the distrihutions shown in Figs 7.8(h) and 7.9(h).

The solid line is calculated by the nonlocal density functional theory (NLDFT), which will be described to some detail in Section 11.4. As seen in the figure, the curve obtained with the HK method (dashed fine) correlates with NLDFT much better than that calculated with the Kelvin equation (dash-dotted line). 

Density functional theory is a powerful tool to study many phenomena in physical chemistry and chemical engineering. It was popularized in the early 1960s by a number of authors [72-74]. But it is not until the 1980s that this theory had found widespread appHcations in many interfacial problems. Capillary condensation in pore was systematically studied [75], and the first paper [76] applying this technique to the problem of PSD determination of carbon particle appeared in 1989. This work used a local DFT, and it is now superseded by the NLDFT, which was developed by Tarazona and Evans [77-79]. This is the method that is now widely used in the characterization of pore size distribution. 

This would allow performing accurate PSD calculations using these simple algorithms. Theoretical considerations [13], nonlocal density functional theory (NLDFT) calculations [62, 146], computer simulations [147], and studies of the model adsorbents [63, 88] strongly suggested that the Kelvin equation commonly used to provide a relation between the capillary condensation or evaporation pressure and the pore size underestimates the pore size. 

Beside classical methods of pore size analysis, there are many advanced methods. Seaton et al. [161] proposed a method based on the mean field theory. Initially this method was less accurate in the range of small pore sizes, but even so it g ve a more realistic -way for evaluation of the pore size distribution than the classical methods based on the Kelvin equation [162]. More rigorous methods based on molecular approaches such as grand canonical Monte Carlo (GCMC) simulations [147, 163-165] and nonlocal density functional theory (NLDFT) [86, 146, 147, 161, 163-169] have been developed and their use for pore size analysis of active carbons is continuously growing. 

The non-local density functional theory (NLDFT) is well established and widely presented in the literature. The distribution of density in a confined pore can be obtained for an open system in which a pore is allowed to exchange mass with the surroundings. From the thermodynamic principle, the density distribution is obtained by minimization of the following grand potential written below for the one-dimensional case [170] ... 

Finn and Monson [139] first tested the predictability of IAS theory for binary systems using the isothermal isobaric Monte Carlo simulation on a single surface. However, this system does not represent real adsorption systems. Tan and Gubbins [140,141] conducted detailed studies on the binary equilibria of the methane-ethane system in slit-shaped micropores using the nonlocal density function theory (NLDFT). The selectivity of ethane to methane was studied in terms of pore width, temperature, pressure, and molar fractions. 

HCP-l,4-benzenedimethanol (HCP-BDM) and HCP-ben l alcohol (HCP-BA) networks synthesized by the Friedel-Crafts self-condensation method have shown high selectivity for CO2 over N2, measured by nitrogen adsorption isotherm at two different temperatures, 273 and 298 Nonlocal density functional theory (NLDFT) calculations confirmed the pore sizes to be below 2 nm for both of the networks. Such small pore sizes, as well as the high ojq gen content in both the polymers, is most probably tbe reason for the strong interactions with polar CO2 rather than N2, making these good candidates for the selective separation of CO2 from N2. 

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