NLDFT

Pore size has been evaluated through the DFT method, using the NLDFT adsorption branch model for cylindrical pore [10],... 

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... 

Calculate the average thickness thp of the pore walls of the SBA-15 from Problem 6 by using the calculated lattice constant a and an average mesopore diameter of 8.10 nm (derived from N2 sorption isotherms using the NLDFT method). The scheme illustrating the ordering of the pores given below will help you do this. 

Fig. 7 (a) Nitrogen adsorption (closed symbols) and desorption (open symbols) isotherms at 77 K of mesoporous NU-Ge-1. (b) NLDFT pore size distribution calculated from the adsorption branch... 

Fig. 4.4 NLDFT Pore size distribution (NLDFT-PSD) in porous carbons with high voiume of uitra- and super-micropores (V = 0.4-0.7 crn g" ), and mesopores 1.2-1.4 cm g ).Li/Ci...

Two kernels of theoretical isotherms in cylindrical channels have been constructed corresponding to the adsorption and desorption branches. For a series of samples [2-4], we show that the pore size distributions calculated from the experimental desorption branches by means of the desorption kernel satisfactory coincide with those calculated from the experimental adsorption branches by means of the adsorption kernel This provides a convincing argument in favor of using the NLDFT model for pore size characterization of nanoporous materials provided that the adsorption and desorption data are processed consistently,... 

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 1. (a) Comparison of the NLDFT isotherm in a 107 nm diameter cylindrical pore with the standard nitrogen isotherm on nonporous oxides [26], (b) corresponding statistical film thickness plot. 

Figure 2. Capillary hysteresis of nitrogen in cylindrical pores at 77 K. Equilibrium desorption (black squares) and spinodal condensation (open squares) pressures predicted by the NLDFT in comparison with the results of Cohan s equation (the BJH method) for spherical (crosses and line) and cylindrical (line) meniscus.

Structural parameters of the MCM-41 materials calculated by means of the NLDFT method are listed in Table 1. We note very good agreement between the results obtained from the desorption and adsorption branches of the isotherms, especially for samples 1 - 3. It is worth noting that the pore wall thickness (1.2-1,8 nm) of wide-pore MCM-41 materials is larger than that usually obtained for conventional MCM-41, and tends to increase with the pore diameter. 

Figure 4. Comparison of the NLDFT N2 isotherm in 5.1 nm cylindrical pore at 77 K with the isotherm on enlarged MCM-41 material [2, 3] (sample 1 in Table 1).

Figure 6. The pore size distributions of enlarged MCM-41 materials [2-3] calculated from adsorption (dotted lines) and desorption (solid lines) branches of nitrogen isotherms by the NLDFT method.

Figure 7. The pore size distribution of wide-pore material [4] (sample 4 in Table 1) calculated from adsorption and desorption branches of nitrogen isotherm by the NLDFT method.

VpDfT and SPDFT are the pore volume and the pore surface area, respectively, calculated by the NLDFT method. 

Dpdft is the mean pore diameter calculated from the NLDFT pore size distribution ctpdft is the standard deviation calculated from the NLDFT pore size distribution dwaii = ao - DPDFT- the pore wall thickness assuming cylindrical pores... 

In this paper we have presented a new model for determining the pore size distribution of microporous and mesoporous materials. The model has been tested using the adsorption isotherms on pure as well as mixtures of MCM-41 materials. The experimental data of adsorption of nitrogen at 77.4 has been inverted using regularization technique. The results of PSD by the present model are compared with the pore size obtained from other classical methods, NLDFT [16] as well as the that obtained by X-ray diffraction methods. 

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. 

Figure 5. Comparison of pore size distribution of sample AMI of Niemark et al. [16] determined by current method with that from regularization with NLDFT, and the XRD pore diameter.

The NLDFT and GCMC methods can be carried out to obtain a set of local isotherms for pores of different widths. The PSD is obtained by solving the GAI equation (Equation 4.28) in its discrete form ... 

Figure 4.6b), and NLDFT method (Figure 4.6c). All these PSDs were obtained using the software provided by the manufacturer of the commercial volumetric analyzer, where the N2 adsorption isotherms were measured. Note that CMS1, for the reason commented before (lack of N2 adsorption at 77 K due to diffusional problems), cannot be analyzed. The three methods qualitatively show the same evolution of the PSD, that is, the expected one deduced from the N2 adsorption isotherm analysis. The micropore size and the width of the distribution increase in the order ACF1 < AC2 < AC1. 

In Figure 6.17a, the adsorption isotherm of N2 at 77 K on a Fisher powdered active carbon is shown. In this isotherm, the capillary condensation effects are not pronounced, and the material, from an adsorption point of view, behaves similar to a zeolite. This is due to the highly developed micropore network present in this carbon and the few mesopores present in it. Additionally, in Figure 6.17b, the adsorption isotherm of N2 at 77 K on a SWCNT is shown [80], In the following sections, the data reported in Figure 6.17b are applied to illustrate the application of the BET method for the determination of the specific surface area, and the Saito-Foley and NLDFT methods for the determination of the PSD. 

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