Accurate Relations between Pore Size and the Pressure of Capillary

Department of Chemistry, Okayama University of Science, 1-1 Ridai-cho, Okayama 700-0005, Japan ... Helmut Kaiser , Roger Pynn , William J. DeSisto , M...
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Langmuir 2006, 22, 4165-4169

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Accurate Relations between Pore Size and the Pressure of Capillary Condensation and the Evaporation of Nitrogen in Cylindrical Pores Kunimitsu Morishige* and Masayoshi Tateishi Department of Chemistry, Okayama UniVersity of Science, 1-1 Ridai-cho, Okayama 700-0005, Japan ReceiVed NoVember 17, 2005. In Final Form: March 1, 2006 To examine the theoretical and semiempirical relations between pore size and the pressure of capillary condensation or evaporation proposed so far, we constructed an accurate relation between the pore radius and the capillary condensation and evaporation pressure of nitrogen at 77 K for the cylindrical pores of the ordered mesoporous MCM-41 and SBA-15 silicas. Here, the pore size was determined from a comparison between the experimental and calculated X-ray diffraction patterns due to X-ray structural modeling recently developed. Among the many theoretical relations that differ from each other in the degree of theoretical improvements, a macroscopic thermodynamic approach based on Broekhoff-de Boer equations was found to be in fair agreement with the experimental relation obtained in the present study.

I. Introduction Capillary condensation of vapors within mesoporous materials is characterized by a distinct step in an adsorption isotherm often accompanied by a hysteresis loop.1 This phenomenon is a shifted gas-liquid phase transition resulting from the confinement of a fluid and has been extensively studied to elucidate the behavior of the fluid in the pores.2 The condensation pressure depends on the pore size and shape and also on the strength of the interaction between the fluid and pore walls. Knowing the relation between the pore size and the pressure of the condensation or evaporation for the pores of a given shape and surface chemistry and a statistical film thickness curve appropriate to the pores, we can determine a pore size distribution (PSD) in the sample with pores of the same (or similar) shape and surface chemistry from the adsorption isotherm that represents the amount of adsorbed fluid as a function of the vapor pressure. The determination of PSDs is one of the most important applications of physical adsorption. Conventional mesoporous materials consist of an interconnected network of pores of varying shape, curvature, and size. To obtain PSDs of these solids, we must assume that the solid is made up of a collection of independent, noninterconnected pores of some simple geometry (usually a slit or cylindrical shape).1 Therefore, knowledge of an accurate relation between the pore size and the pressure of the capillary condensation or evaporation for the pores of a simple geometry is very important. There have been many studies3-20 concerning the relation between the pore size and the pressure of the condensation or evaporation of nitrogen at 77 K in the cylindrical pores of silicas (1) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: New York, 1982. (2) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska-Bartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573. (3) Barrett, E. P.; Joyner, L. G.; Halenda, P. P. J. Am. Chem. Soc. 1951, 73, 373. (4) Cranston, R. W.; Inkley, F. A. AdV. Catal. 1957, 9, 143. (5) Dollimore, D.; Heal, G. R. J. Appl. Chem. 1964, 14, 109. (6) Broekhoff, J. C. P.; de Boer, J. H. J. Catal. 1967, 9, 15. (7) Broekhoff, J. C. P.; de Boer, J. H. J. Catal. 1968, 10, 377. (8) Celestini, F. Phys. Lett. A 1997, 228, 84. (9) Naono, H.; Hakuman, M.; Shiono, T. J. Colloid Interface Sci. 1997, 186, 360. (10) Maddox, M. W.; Olivier, J. P.; Gubbins, K. E. Langmuir 1997, 13, 1737. (11) Kruk, M.; Jaroniec, M.; Sayari, A. Langmuir 1997, 13, 6267. (12) Zhu, H. Y.; Lu, G. Q.; Zhao, X. S. J. Phys. Chem. B 1998, 102, 7371. (13) Sonwane, C. G.; Bhatia, S. K. Chem. Eng. Sci. 1998, 53, 3143. (14) Lukens, W. W.; Schmidt-Winkel, P.; Zhao, D.; Feng, J.; Stucky, G. D. Langmuir 1999, 15, 5403. (15) Miyahara, M.; Kanda, H.; Yoshioka, T.; Okazaki, M. Langmuir 2000, 16, 4293.

because the measurement of a nitrogen adsorption isotherm at a liquid-nitrogen temperature is a standard technique for characterizing porous materials by means of gas adsorption, and a cylindrical pore geometry is often assumed in the methods for the PSD calculation. Most of them are theoretical. For many years, direct experimental verification of the relation was exceptionally difficult because of the lack of well-defined porous solids. The discovery of ordered mesoporous materials,21,22 however, gave us an opportunity for such verification. To this end, Naono et al.,9 Kruk et al.,11 Zhu et al.,12 Sonwane and Bhatia,13 Lukens et al.,14 Qia et al.,18 Ustinov et al.,19 and Kowalczyk et al.20 have compared the experimental relation between the pore size and the pressure of capillary condensation obtained for FSM-1621 and MCM-4122 ordered mesoporous silicas with various forms of the theoretical relation. MCM-41 and FSM-16 materials give several Bragg peaks in their diffraction patterns due to 2D ordering in the arrangement of cylindrical pores of uniform size. In these studies, the pore sizes of the solids were estimated by a simple geometrical relation among pore size, pore volume, and geometrical surface area,9,14 a geometrical relation among pore size, pore volume, and (100) interplanar spacing,11,13,18-20 and/or a simple subtraction of an assumed pore wall thickness from a lattice parameter.12 The pore sizes thus obtained still lack the accuracy required for a strict verification of the theoretical or semiempirical relations between the pore size and the pressure of the capillary condensation or evaporation. In recent years, X-ray structural modeling has been developed to be able to obtain the structural characteristics of the ordered mesoporous materials from a comparison between the experimental and calculated diffraction patterns.22-29 The method is (16) Neimark, A. V.; Ravikovitch, P. I. Microporous Mesoporous Mater. 2001, 44-45, 697. (17) Kruk, M.; Jaroniec, M. Chem. Mater. 2001, 13, 3169. (18) Qiao, S. Z.; Bhatia, S. K.; Zhao, X. S. Microporous Mesoporous Mater. 2003, 65, 287. (19) Ustinov, E. A.; Do, D. D.; Jaroniec, M. J. Phys. Chem. B 2005, 109, 1947. (20) Kowalczyk, P.; Jaroniec, M.; Terzyk, A. P.; Kaneko, K.; Do, D. D. Langmuir 2005, 21, 1827. (21) Yanagisawa, T.; Shimizu, T.; Kuroda, K.; Kato, C. Bull. Chem. Soc. Jpn. 1990, 63, 988. (22) Beck, J. S.; Vartuli, J. C.; Roth, W. J.; Leonowicz, M. E.; Kresge, C. T.; Schmitt, K. D.; Chu, C. T.-W.; Olson, D. H.; Sheppard, E. W.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. J. Am. Chem. Soc. 1992, 114, 10834. (23) Feuston, B. P.; Higgins, J. B. J. Phys. Chem. 1994, 98, 4459. (24) Edler, K. J.; Reynolds, P. A.; White, J. W.; Cookson, D. J. Chem. Soc., Faraday Trans. 1997, 93, 199. (25) Hammond, W.; Prouzet, E.; Mahanti, S. D.; Pinnavaia, T. J. Microporous Mesoporous Mater. 1999, 27, 19. (26) Tun, Z.; Mason, P. C. Acta Crystallogr., Sect. A 2000, 56, 536.

10.1021/la053105u CCC: $33.50 © 2006 American Chemical Society Published on Web 03/25/2006

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based on the established principle of X-ray diffraction by a periodically ordered distribution of electron density and sound compared to the methods due to the geometrical relations mentioned above. The present study aims at obtaining an accurate relation between the pore size and the pressure of capillary condensation and the evaporation of nitrogen at 77 K in the cylindrical pores of porous silicas from the analysis of the diffraction patterns for MCM-41 and SBA-15 having a 2D hexagonal arrangement of cylindrical pores. To examine the theoretical and semiempirical relations so far reported, the results obtained will be compared with these relations. 2. Experiment 2.1. Materials. The MCM-41 samples were synthesized using alkyltrimethylammonium bromide surfactants with alkyl chain lengths of 10, 12, 14, 16, and 18 carbons as structure-directing agents. Sample preparations have been given elsewhere.30 In the present study, the samples are referred to as MCM-41(Cn), where n represents the carbon chain lengths of surfactants used in the synthesis. The SBA-15 samples were prepared using Pluronic 123 triblock copolymer as a structure-directing agent, according to the procedure of Kruk et al.31 Three kinds of SBA-15 samples were obtained by changing the aging temperature. The samples are referred to as SBA15(T), where T represents the aging temperature. 2.2. Measurements. Adsorption isotherms of nitrogen at 77 K were measured volumetrically on a homemade semiautomated instrument equipped with a Baratron capacitance manometer (type 127) with a full scale of 1000 Torr. Small-angle X-ray diffraction patterns were obtained on a Rigaku NANO-Viewer using Cu KR radiation. The incident beam was point focused on the sample using confocal max-flux optics, and the diffracted beam was detected by an imaging plate. Vacuum flight tubes were inserted between the X-ray sourse and the sample and between the sample and the detector to eliminate parasitic air scattering. The exposure time was 10 min. The diffraction patterns consisted of several concentric circles so that the scattered intensity was azimuthally integrated and plotted as a function of the scattering angle, 2θ. 2.3. X-ray Data Analysis. Several approaches have been reported for X-ray diffraction (XRD) structural investigations of the ordered mesoporous materials.22-29 Among them, we adopted the continuous density function technique developed by Solovyov et al.29 because of its flexibility with respect to changing the structural model. In this approach, the averaged density distribution in the material is modeled by flexible analytical function F with adjustable parameters. The intensities of the diffraction reflections are calculated by the Fourier integral of the model function. The adjustable parameters of F are refined by minimizing the difference between calculated and experimental powder diffraction profiles using the Rietveld technique.32 The structure characteristics are determined from the refined model parameters. The intensities of the diffraction peaks of MCM-41 and SBA-15 having a 2D hexagonal arrangement of cylindrical pores are known to decrease rapidly with their order. This is due to both the form factor of the cylinders and to the existence of an appreciable disorder similar to frozen thermal fluctuations, the effect of which may be described by a Debye-Waller factor. A smooth, continuous transition of the electron density from wall to pore was taken into account by incorporating a wall density slope of 0.3 Å in the model. A hexagonality parameter was also introduced into the refined models for MCM41. Therefore, adjustable parameters are the lattice parameter a, the (27) Impe´ror-Clerc, M.; Davidson, P.; Davidson, A. J. Am. Chem. Soc. 2000, 122, 11925. (28) Sauer, J.; Marlow, F.; Schuth, F. Phys. Chem. Chem. Phys. 2001, 3, 5579. (29) Solovyov, L. A.; Kirik, S. D.; Shmakov, A. N.; Romannikov, V. N. Microporous Mesoporous Mater. 2001, 44-45, 17. (30) Morishige, K.; Fujii, H.; Uga, M.; Kinukawa, D. Langmuir 1997, 13, 3494. (31) Kruk, M.; Jaroniec, M.; Ko, C, H,; Ryoo, R. Chem. Mater. 2000, 12, 1961. (32) Rietveld, H. M. J. Appl. Crstallogr. 1969, 2, 65.

Figure 1. Adsorption-desorption isotherms of nitrogen at 77 K on MCM-41 samples. (Volumes adsorbed for C12, C14, C16, and C18 were incremented by 200, 400, 600, and 800 mL(STP)/g, respectively.) Desorption points are represented by closed symbols. mesopore radius R, the root-mean-square displacement of the cylinders u, a common scale factor, and the hexagonality parameter p. The profile shape function was represented as the product of a symmetrical profile function and an asymmetrical correction to it.32 As a symmetrical profile function, a modified pseudo-Voight function was adopted in which the Gauss and Lorentz functions may have unequal peak heights and half-width-at-half-maximum intensities.33 Allowance for the background contributions to the overall diffraction pattern was made using a background function that is linear in six refinable background parameters.33

3. Results 3.1. Adsorption Isotherm. Figures 1 and 2 show the adsorption-desorption isotherms of nitrogen at liquid-nitrogen temperature on five kinds of MCM-41 and three kinds of SBA15 samples, respectively. In accord with previous studies,9,13,30 none of the isotherms for MCM-41 samples showed appreciable hysteresis, although a distinct step due to capillary condensation was observed in each case. An increase in the pore size with the use of surfactants of longer alkyl chain lengths led to a shift in this step to higher relative pressures. However, all of the isotherms for SBA-15 showed clear hysteresis loops of type H1 in the IUPAC classification.34 The difference in the appearance of the adsorption hysteresis between MCM-41 and SBA-15 samples is concerned with the larger pore sizes of the SBA-15 sample.35 The pore size of SBA-15 increased with an increase in the aging temperature, as revealed by a gradual shift of the adsorption hysteresis to higher relative pressures. The pressure of capillary condensation or evaporation was determined at the midpoint of the adsorption step. The pressures of the condensation and evaporation on each sample of MCM-41 were almost identical, indicating that the observed pressure represents an equilibrium transition point. In hysteretic isotherms such as those for SBA(33) Izumi, F. In The RietVeld Method; Young, R. A., Ed.; IUCr Monographs on Crystallography; Oxford University Press: Oxford, U.K., 1993; Vol. 5, p 236. (34) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Pure Appl. Chem. 1985, 57, 603. (35) Morishige, K.; Ito, M. J. Chem. Phys. 2002, 117, 8036.

Pore Size Vs Condensation/EVaporation in Pores

Figure 2. Adsorption-desorption isotherms of nitrogen at 77 K on SBA-15 samples. (Volumes adsorbed for SBA-15(353 K) and SBA-15 (373 K) were incremented by 200 and 400 mL(STP)/g, respectively.) Desorption points are represented by closed symbols.

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Figure 4. X-ray diffraction patterns of SBA-15 samples.

Figure 5. Comparison between the experimental and calculated XRD patterns for MCM-41 (C16). The experimental profile is shown by circles, and the calculated profile is shown by a solid line.

Figure 3. X-ray diffraction patterns of MCM-41 samples.

15, the experimental condensation and evaporation pressures correspond to some pressure points between the equilibrium transition and spinodal condensation pressures and the equilibrium transition and spinodal evaporation pressures, respectively. 3.2. Diffraction Pattern. Figures 3 and 4 show the powder X-ray diffraction patterns of MCM-41 and SBA-15 samples, respectively. The four diffraction peaks indexed to the (100), (110), (200), and (210) reflections of a 2D hexagonal lattice were observed for MCM-41(C18) and MCM-41(C16), whereas the five peaks (100), (110), (200), (210), and (300) were observed for SBA-15 prepared at an aging temperature of 373 K. The lattice parameter increased with an increase in the alkyl chain length of the surfactant for MCM-41 and with an increase in the

aging temperature for SBA-15. The diffraction peaks for MCM-41(C14) were evidently broader than for other MCM-41 samples. This indicates that the crystallographic quality of MCM-41(C14) is poor. The MCM-41(C14) sample seems to be composed of at least two different phases that have slightly different unit cell parameters but almost the same pore size. The relative intensities of these diffraction peaks changed with the alkyl chain length of the surfactant for MCM-41 and with the aging temperature for SBA-15. The relative intensities of the ordered mesoporous solids with the 2D hexagonal array of cylindrical pores depend not only on the ratio between wall thickness and pore diameter and on the unit cell size but also on the hexagonality of the pore shape and on the roughness of the pore wall. Structural information is obtained by X-ray structural modeling. Figures 5 and 6 show comparisons between the experimental and calculated XRD patterns for MCM-41(C16) and SBA-15(373 K), respectively. The quality of the fit is reasonable. Table 1 summarizes the refined parameters. The errors in the unit cell parameter and mesopore radius are due to the standard deviation of the parameters refined in nonlinear least-squares fitting. These errors are comparable to those reported in previous studies.29,36,38 The wall thickness (hw) along a line from one pore center to the

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Figure 6. Comparison between the experimental and calculated XRD patterns for SBA-15 (373 K). The experimental profile is shown by circles, and the calculated profile is shown a solid line.

next was obtained by subtraction (a - 2R). The profile shape function employed in the present study was not able to reproduce the diffraction profile for MCM-41(C14). Therefore, X-ray structural modeling was not tried for this sample. Hexagonality in pore geometry was noticeable for MCM-41(C16) and MCM41(C18), whereas the mesopores of other MCM-41 as well as SBA-15 samples were almost cylindrical. The wall thickness of MCM-41 increased only slightly with an increase in pore size, although their values were in fair agreement with those so far reported for MCM-41.23,29,36 The wall thickness of SBA-15 decreased with an increase in the aging temperature. The values of R, hw, and u for SBA-15 are close to the reported values of SBA-15 samples prepared in a similar way.27,37,38 It is known that the pore wall of SBA-15 is rougher than that of MCM-41 and actually has spongelike morphology with many micropores.27,37,38 The wall roughness could be taken into account by incorporating a wall density slope of long distance. A wall density slope of 10 Å, however, did not appreciably change the values of the refined mesopore radius. Physisorption of a gas in the micropores, which exist in the pore walls of the main channel, takes place at relative pressures lower than for the occurrence of capillary condensation of the same gas in the main channels. Hofmann et al.38 have revealed in their latest work that the mesopore radius of SBA-15 obtained in X-ray structural modeling does not change appreciably by the adsorption of krypton in the micropores and starts to decrease with the formation of the adsorbed film on the pore walls of the main channels. This clearly indicates that the mesopore radius of SBA-15 determined here represents the real pore radius of the main channels responsible for the occurrence of capillary condensation.

4. Discussion 4.1. Examination of Reported Experimental Relations. Kruk, Jaroniec, and Sayari (KJS)11 have proposed a modified Kelvin equation that is able to reproduce satisfactorily the experimental relation between the pore size and the capillary condensation pressure of nitrogen at 77 K obtained by them. In their work, the pore size was estimated on the basis of the geometrical relation among the interplanar spacing, pore volume, and pore diameter for the honeycomb structure. This relation (36) Solovyov, L. A.; Belousov, O. V.; Shmakov, A. N.; Zaikovskii, V. I.; Joo, S. H.; Ryoo, R.; Haddard, E.; Gedeon, A.; Kirik, S. D. Stud. Surf. Sci. Catal. 2003, 146, 299. (37) Solovyov, L. A.; Fenelonov, V. B.; Derevyankin, A. Yu.; Shmakov, A. N.; Haddard, E.; Geden, A.; Kirik, S. D.; Romannikov, V. N. Stud. Surf. Sci. Catal. 2001, 135, 287. (38) Hofmann, T.; Wallacher, D.; Huber, P.; Birringer, R.; Knorr, K.; Schreiber, A.; Findenegg, G. H. Phys. ReV. B 2005, 72, 064122.

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relies on the assumptions that the material takes an ideal porous structure over whole particles of the powder sample and the pore wall density is the same as that of a nonporous amorphous silica (2.2 g/cm3). These assumptions do not seem to be strictly held for MCM-41 ordered mesoporous silicas. To examine the validity of the semiempirical relation reported by KJS, we compared our data with the KJS relation. The adsorption isotherms of nitrogen at 77 K on the mesoporous silicas with pore radii of less than 2 nm are reversible, and thus we should compare the experimental relation with the KJS curve in this pore size region. As Figure 7 shows, our experimental data do not fit the semiempirical relation of KJS well. Evidently, the difference comes from the inaccuracy of the pore sizes estimated on the basis of the geometrical relation. Similarly, many researchers9,11-14,18-20 have tested the validity of various theoretical relations between the pore size and the pressure of the capillary condensation or evaporation using the pore sizes estimated on the basis of the same or similar geometrical relations for the 2D hexagonal arrangement of uniform cylindrical pores. All of these methods of pore size determination are based on several assumptions that are acceptable only in an approximate way. Therefore, these experimental relations between the pore size and the capillary condensation or evaporation pressure cannot strictly test the validity of the theoretical equations concerning capillary condensation. X-ray structural modeling for the evaluation of pore size is based on the established principle of X-ray diffraction by a periodically ordered distribution of electron density. The geometrical methods are prone to lead to inaccurate pore sizes because of the small particle size, inaccurate value of the pore wall density, and sometimes the presence of micropores in the real materials. However, the effect of small particle size results in only line broadening of the diffraction peak and does not affect the relative intensities. The change in pore wall density and the presence of micropores in mesopore channels do not affect the relative intensities. XRD modeling is able to give relatively accurate pore size values that are not influenced by these factors. The presence of disorders such as pore wall roughness and fluctuations in the positions of cylindrical pores may nevertheless result in systematic errors in refined pore size values, although some of these disorders were accounted for in the present XRD modeling. The effect of sample quality on the relation between pore size and condensation/evaporation pressures is very interesting in this respect. 4.2. Examination of Theoretical Relations. In hysteretic isotherms, we have two branches of adsorption (capillary condensation) and desorption (capillary evaporation) in the pressure region where capillary condensation occurs. Correspondingly, two different equations have been theoretically derived for capillary condensation and capillary evaporation. At this point in the argument, however, readers should notice that the pressures calculated from the equations for capillary condensation and evaporation do not necessarily correspond to the experimental adsorption and desorption pressures, respectively. A theoretical capillary condensation pressure, in fact, corresponds to a spinodal capillary condensation transition.16 When the pressure is increased beyond the equilibrium transition point, the energy barrier for the transition from a metastable multilayer to a stable condensedliquid state decreases and eventually vanishes at the limit of stability of the adsorbed film (vaporlike spinodal).39,40 In real adsorption experiments, the spinodal is never reached because of finite thermal energy, temperature fluctuations, and structural inhomogeneity of samples. However, a theoretical equation for (39) Vishnyakov, A.; Neimark, A. V. J. Phys. Chem. B 2001, 105, 7009. (40) Ustinov, E. A.; Do, D. D. J. Phys. Chem. B 2005, 109, 11653.

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Table 1. Structural Parameters of the Calculated Forms of the Samples Studied sample

unit cell parameter a(Å)

mesopore radius R(Å)

rms displacement u(Å)

hexagonality

wall thickness hw(Å)

MCM-41(C10) MCM-41(C12) MCM-41(C16) MCM-41(C18) SBA-15(323 K) SBA-15(353 K) SBA-15(373 K)

33.5 ( 0.1 36.3 ( 0.1 44.3 ( 0.1 48.2 ( 0.1 98.1 ( 0.3 108.0 ( 0.3 111.3 ( 0.3

12.1 ( 0.1 13.4 ( 0.1 16.9 ( 0.1 18.8 ( 0.1 32.1 ( 0.3 38.1 ( 0.3 42.4 ( 0.3

3.6 3.5 4.5 4.8 9.3 9.3 9.9

0.0 0.0 0.53 0.32 0.0 0.0 0.0

9.5 9.9 10.7 10.8 33.9 31.8 26.5

capillary evaporation gives a relation between the pore size and the equilibrium capillary condensation pressure because it has long been thought that as the vapor pressure decreases, evaporation occurs near the equilibrium via formation of the hemispherical meniscus at the pore ends. As our previous studies35,41 and other studies11,18 have shown, such a theoretical view is not necessarily supported by direct experimental evidence. In principle, the experimental capillary condensation and evaporation pressures correspond to some pressure points between the equilibrium transition and spinodal capillary condensation pressures and the equilibrium transition and spinodal capillary evaporation pressures, respectively. When there is no adsorption-desorption hysteresis, the equilibrium transition corresponds to both adsorption and desorption branches of experimental isotherms. In the hysteretic isotherms, the position of the adsorption/desorption branch is controlled by the height of the energy barrier between the full liquid pore and the vapor coexisting with the liquid film. When the pressure is increased beyond some critical value, capillary condensation takes place in the cylindrical pores, the walls of which are already covered with the adsorbed film. Capillary evaporation leaves the adsorbed film on the pore walls. Therefore, the correction for the adsorbed film thickness must be introduced into theoretical treatments based on the Kelvin equation and related equations. Many equations3-8,10,12-16,18-20 have been proposed that differ from each other in the degree of theoretical improvement. We compared our experimental results with those theoretically derived relations between the pore size and the pressure of capillary condensation and capillary evaporation. Among them, a macroscopic thermodynamic approach based on Broekhoff-de Boer (BdB) equations6,7 was found to be in fair agreement with the experimental relation obtained by us. Figure 7 also shows a comparison of the experimental relation between the pore size and the pressure of capillary condensation and evaporation with the theoretical relations of BdB. Here, the results of the BdB relation have been cited from the numerical values tabulated in

Figure 7. Comparison of the experimentally obtained relation between the pore size and the pressure of capillary condensation (filled circles) and capillary evaporation (hollow circles) with the semiempirical relation proposed by KJS11 and the theoretical relations of BdB.6,7

their original papers.6,7 The equilibrium transition pressures obtained experimentally for MCM-41 samples almost coincide with the theoretical relation for equilibrium capillary evaporation. In addition, the experimental data points obtained for SBA-15 samples seem to be compatible with the principle for the occurrence of adsorption hysteresis. The experimental data points for capillary condensation are located between the equilibrium transition and spinodal capillary condensation pressures, and those for capillary evaporation are located below the equilibrium transition pressures. Very recently, Ustinov et al.19 and Kowalczyk et al.20 have reported the modification of the BdB approach by accounting for the effects of pore size on the potential exerted by the pore walls and the surface tension of the liquid, respectively. These modified approaches have shown distinct deviations from the original BdB relation in the region of smaller pore sizes. However, the experimental relation showed reasonable agreement with the BdB relation for the equilibrium transition in this region. Unfortunately, the theoretical improvements19,20 for the BdB approach resulted in worse agreement with the experimental relation. Neimark and Ravikowitch16 and Ustinov et al.19 have compared the relation between pore size and pore condensation/ evaporation pressures obtained by the nonlocal density functional theory (NLDFT) with the BdB relation. Although the results of the NLDFT method showed deviations from those of the classical BdB method in the region of smaller pore sizes, the deviations observed for the equilibrium transition were relatively small. The experimental isotherms of nitrogen at 77 K on the mesoporous silicas with pore radii of less than 2 nm are reversible, and thus the experimental relation should be compared with the theoretical relation for the equilibrium transition in this region of pore size. In this respect, the deviation of the experimental relation from the NLDFT relation is relatively small, although the agreement of the experimental relation with the BdB relation is better than for the NLDFT theory. Better agreement with the BdB relation, however, does not necessarily validate the BdB approach for small pores. The concepts of surface tension and meniscus inherent to macroscopic thermodynamic approaches such as the BdB become obscure. In these theoretical studies, the pores are always assumed to be cylindrical in shape, infinite in length, and smooth in pore wall structure. The pores of ordered mesoporous silicas such as MCM-41 are not like this. Further theoretical and experimental investigations are needed to solve this point. Acknowledgment. We express our sincere thanks to K. Hoshino of Rigaku Corporation for measurement of the smallangle X-ray diffraction patterns for the ordered mesoporous silicas. This work was supported by the High-Tech Research Center Project for Private Universities through a matching fund subsidy from MEXT (Ministry of Education, Culture, Sports, Science and Technology), 2001-2005. LA053105U (41) Morishige, K.; Nakamura, Y. Langmuir 2004, 20, 4503.