Determination of Mesopore Size Distributions from Argon Adsorption

Apr 17, 2002 - The common t-curve and the relation between the diameter of cylindrical pores and the capillary condensation pressure were determined a...
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J. Phys. Chem. B 2002, 106, 4732-4739

Determination of Mesopore Size Distributions from Argon Adsorption Data at 77 K Michal Kruk and Mietek Jaroniec* Department of Chemistry, Kent State UniVersity, Kent, Ohio 44242 ReceiVed: October 8, 2001

Argon adsorption at 77 K on ordered mesoporous silicas (MCM-41, SBA-15) with pore diameters up to 9 nm and on disordered mesoporous silicas with larger pore sizes was investigated, allowing us to successfully develop a method to calculate pore size distributions (PSDs) for cylindrical pores with diameters below 15 nm. It was found that for argon at 77 K the capillary condensation pressure tends to gradually increase as the pore diameter increases up to about 15 nm, whereas larger mesopores do not exhibit capillary condensation. The capillary evaporation pressure was much less clearly related to the pore size, and the steepness of the desorption branch was not always correlated with the degree of structural ordering of a given adsorbent. A good correlation was found between the positions of capillary condensation steps on nitrogen and argon adsorption isotherms at 77 K, whereas the correlation between the positions of capillary evaporation steps was much worse. Therefore, for argon adsorption at 77 K, the PSD calculations from adsorption branches of isotherms are feasible, but calculations from desorption branches are expected to be inherently difficult. The changes in the statistical film thickness as the pore size decreases were examined, and it was concluded that the common t-curve provides a satisfactory description of multilayer adsorption for pore sizes > 2 nm. The common t-curve and the relation between the diameter of cylindrical pores and the capillary condensation pressure were determined and subsequently used in PSD calculations based on a well-known algorithm. The resulting PSDs were in good agreement with those calculated from nitrogen data (77 K) and argon data (87 K), and thus the present work on argon adsorption at 77 K provided a firm foundation for the pore size analysis of pore sizes < 15 nm.

1. Introduction materials1,2

After the discovery of ordered mesoporous and a subsequent rapid growth in this area3,4 fueled by the prospects of applications in catalysis, separations, and nanotechnology, there is a strong incentive to develop more accurate, reliable, and informative approaches for the characterization of porous materials. Gas adsorption holds much promise for the characterization of porous solids.5,6 However, there is much disagreement in the literature as to how to interpret gas adsorption data and as to what information can actually be inferred from them. There is also a need to identify gases suitable for pore size analysis, to establish the pore size ranges that can be probed by particular gases, and to ascertain whether adsorption or desorption or both branches of isotherms are suitable for pore size determination.5,7,8 Nitrogen is the most commonly used adsorbate for the calculation of pore size distributions (PSDs),5,7 but the opportunities in application of other adsorbates have also been explored.9 In particular, argon adsorption at 87 K10-14 and 77 K15-17 was found to be highly useful for the characterization of microporous materials. Argon adsorption in micropores was also extensively studied using computer simulations18,19 and advanced theoretical approaches, such as nonlocal density functional theory.9,12 Argon adsorption on ordered mesoporous solids20-38 has also attracted much attention, both from experimental and theoretical points of view. This resulted in the development of three new methods for pore size analysis on the basis of argon adsorption: (i) a method based on nonlocal density functional theory (NLDFT);28 (ii) an empirical method * Telephone: (330) 672-3790. Fax: (330) 672-3816. E-mail: jaroniec@ columbo.kent.edu.

derived and validated using MCM-41 silicas as model adsorbents (developed before only for argon adsorption at 87 K);32 and (iii) a theoretical approach.33 The first method is based on the arbitrary assignment of the NLDFT equilibrium transition to the desorption branch of the hysteresis loop of an experimental isotherm and the NLDFT metastable adsorption branch to the experimental adsorption branch.28,29 This assignment leads to apparently consistent PSDs from both adsorption and desorption branches of isotherms for good-quality large-pore MCM-41 and SBA-15 silicas.36,37 However, an extensive experimental work32 on argon adsorption at 87 K on MCM-41 silicas with uniform, approximately cylindrical pores of size 2-6.5 nm (as determined using an independent method) provided strong evidence that, in the region of adsorptiondesorption hysteresis, desorption data are not particularly suitable for the pore size analysis. This is because the steepness of the desorption branch is not necessarily correlated with the degree of structural ordering of samples, and an increase in the capillary evaporation pressure as the pore diameter increases is somewhat irregular. However, it was possible to find a good relation between the capillary condensation pressure and the pore diameter, and this allowed us to obtain consistent PSDs from nitrogen adsorption data at 77 K39 and argon adsorption data at 87 K32 using a well-known simple algorithm proposed by Barrett, Joyner, and Halenda (BJH).40 Argon adsorption isotherms at 87 K reported in this work32 were already extensively used in theoretical and computer-simulation studies of adsorption in porous media and in the development of methods for pore size analysis.33,35-37 The aim of the current study was fourfold: (i) the measurement of argon adsorption isotherms at 77 K on ordered

10.1021/jp0137423 CCC: $22.00 © 2002 American Chemical Society Published on Web 04/17/2002

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mesoporous silicas with a wide range of pore diameters in order to obtain model adsorption isotherms suitable for the development and testing of methods for PSD calculation and theories of adsorption in porous media; (ii) the determination of whether adsorption or desorption or both branches of argon isotherms at 77 K are suitable for PSD calculations; (iii) the estimation of the limiting pore diameter above which argon at 77 K does not exhibit capillary condensation; (iv) the development of a useful method to calculate PSDs from argon adsorption data at 77 K and the comparison of PSDs obtained from these data with those assessed from argon isotherms at 87 K and nitrogen isotherms at 77 K. 2. Experimental Section 2.1. Materials. The synthesis and characterization of MCM41 silicas used herein were reported in our earlier studies39,41-50 (see Supporting Information). Silica gels LiChrospher Si-100, Si-1000, and Si-4000 were acquired from EM Separations, Gibbstown, NJ. Samples are often denoted in the text and figures using their pore diameter in nanometers. 2.2. Measurements. Argon adsorption measurements at 77 K were carried out on a Micromeritics ASAP 2010 volumetric adsorption analyzer. During the adsorption measurements, the saturation vapor pressure was acquired periodically and later used to calculate the relative pressure. Before the measurements, the samples were outgassed under vacuum at 473 K for 2 or more hours to reach the residual pressure below or equal to 6 µmHg. 2.3. Calculations. The BET specific surface area51 was calculated using adsorption data in the relative pressure range from 0.04 to 0.2, except for the smaller-pore samples, for which intervals at somewhat lower pressures were used in order to avoid the inclusion of data grossly affected by the capillary condensation. The argon atom cross-sectional area 0.138 nm2 was used in the BET analysis.5 The total pore volume51 was calculated from the amount adsorbed at a relative pressure of about 0.99. Bulk argon at 77 K is in a solid or gas state, but it was suggested that argon adsorbed in the MCM-41 mesopores at this temperature is in a liquidlike state.23 Therefore, the argon density in the pores at 77 K was assumed to be the same as that used in our earlier study of argon adsorption at 87 K,32 at which temperature the bulk argon is a liquid. The external surface area and the primary mesopore volume were determined from the Rs plot analysis5,42,51 performed in the Rs range from 1.6 to 1.95, except for the 6.5 nm MCM-41, 7.4 nm SBA-15, and 8.9 nm SBA-15 samples, for which the ranges from 1.6 to 2.0, 1.7 to 2.0, and 1.8 to 2.0, respectively, were used. The pore diameter of the MCM-41 samples that exhibit a 2-dimensional (2-D) hexagonal structure of parallel, nonintersecting, disconnected pores has been previously determined using the following geometrical equation:41,52

wd ) cd

(

)

FVp 1 + FVp

1/2

(1)

where c is a constant equal to 1.213 for the cylindrical pore geometry, d is XRD (100) interplanar spacing, F is the pore wall density (assumed to be 2.2 g cm-3), and Vp is the primary pore volume. The pore diameter for the SBA-15 silicas was assessed using a modified eq 153 that takes into account the existence of micropores in the SBA-15 pore walls, which was proven recently.54 The resultant equation for the 2-D hexagonallyordered porous material with microporosity in the pore walls

Figure 1. Argon adsorption isotherms measured at 77 K for highlyordered MCM-41 silicas with pore diameters from 2.4 to 6.5 nm (the isotherms for samples from 6.0 to 2.4 are offset vertically by 150, 300, 450, 600, 750, 900, 1050, 1200, 1350, and 1500 cm3 STP g-1, respectively).

assumes the following form:53,55

wd ) cd

(

Vp 1/F + Vp + Vmi

)

1/2

(2)

where Vmi is the volume of micropores in the pore walls. The pore diameter of one of the disordered large-pore silicas45 was estimated by comparison with reported data for an ordered largepore MSU-H silica (pore diameter of 10.2 nm, as evaluated using eq 2)55 that exhibited the nitrogen capillary condensation step centered at the same relative pressure. The statistical film thickness of argon in the MCM-41 pores was evaluated as discussed in detail elsewhere.32,39 3. Results and Discussion 3.1. Argon Adsorption Isotherms of MCM-41. Argon adsorption isotherms at 77 K on highly-ordered MCM-41 silicas with pore diameters from 2.4 to 6.5 nm are shown in Figure 1 (data for 2.4 and 2.6 nm MCM-41 were taken from ref 49), and the corresponding structural data are listed in Table 1. Although some argon adsorption isotherms at 77 K on MCM41 have already been reported,20,21,23,24,26,28-31,49 the samples studied therein exhibited a much narrower range of pore diameters and usually a lower degree of structural uniformity. In particular, complete adsorption-desorption isotherms for MCM-41 with pore diameters of 5.5 nm or larger are reported for the first time, since the only previous study of MCM-41 with such large pores provided only adsorption branches of isotherms.24 All the isotherms shown in Figure 1 featured clearly pronounced capillary condensation steps, whose positions shifted to higher pressures as the pore diameter increased. Adsorption in pores of diameters lower than or equal to 3.2 nm was reversible, whereas adsorption-desorption hysteresis is observed in 3.5 nm or larger pores. In terms of the relative pressure, this corresponds to the reversibility of adsorption-desorption isotherms below about 0.27 and to the adsorption-desorption hysteresis above this limit (see data in Table 1 of the Supporting Information). In all cases shown in Figure 1, the hysteresis loops were of H1 type (types of hysteresis loops are discussed in ref

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TABLE 1: Structural Properties of Selected MCM-41 Silicasa Vt Vp Sex SBET sample (m2 g-1) (cm3 g-1) (cm3 g-1) (m2 g-1) 2.4 2.6 3.2 3.5 3.9 4.2 4.5 4.8 5.5 6.0 6.5

640b 620b 860c 870d 840d 890d 880d 800d 710d 700d 610d

0.54 0.59 0.71 0.80 0.88 1.04 1.04 1.00 1.03 1.10 1.03

0.51 0.52 0.67 0.75 0.79 0.95 0.93 0.88 0.98 1.05 0.95e

40 70 40 60 80 90 110 120 50 50 80

wAr,77K (nm)

wd (nm)

2.37 2.63 3.10 3.46 3.82 4.29 4.50 4.88 5.59 5.93 6.32; 6.86f

2.36 2.59 3.16 3.51 3.87 4.20 4.51 4.80 5.53 5.96 6.53

a

Notation (for additional information, see Table 1 in the Supporting Information): SBET, BET specific surface area; Vt, total pore volume; Vp, primary mesopore volume; Sex, external surface area; wAr,77K, position of the maximum on the PSD calculated as proposed in the current work; wd, pore diameter evaluated using eq 1. b,c,d Relative pressure intervals 0.01-0.02, 0.04-0.14, and 0.04-0.2, respectively, were used in the BET analysis. e Includes the micropore volume of about 0.02 cm3 g-1. f Two peaks on the PSD.

7), that is, with parallel adsorption and desorption branches. However, for less well-ordered MCM-41 samples, hysteresis loops of H2 type or even more complicated hysteresis loop shapes were observed (see Figure 1S of the Supporting Information). In general, the width of the hysteresis loop increased as the pore diameter increased when samples with a similarly high degree of structural ordering were compared (see 3.5-4.8 nm MCM-41 samples, Figure 1). This trend does not extend to 5.5-6.5 nm samples that exhibited quite broad hysteresis loops of similar width, most likely because their structural ordering is not as perfect as that of the aforementioned smaller-pore materials. In the case of argon at 77 K, different shapes of hysteresis loops were observed for the MCM-41 silicas with similar pore diameters, largely depending on the degree of structural ordering of the materials. As can be seen in Figures 1 and 1S, highlyordered MCM-41 with pore diameters of 4.1-4.2 nm exhibited a narrow H1 hysteresis loop, but even in this case, the desorption branch exhibited some tailing for one sample (Figure 1S, the uppermost isotherm). A less-well-ordered MCM-41 sample39 synthesized using the postsynthesis hydrothermal restructuring method42 exhibited a triangular hysteresis loop (H2 type). Similar behavior was observed for the 4.6 nm sample42 synthesized under similar conditions. Another less-ordered MCM-41 silica, which was synthesized at relatively low temperature using hexadecyldimethylamine as an expander,45 exhibited a desorption branch with two steeper regions. A similar shape of the hysteresis loop was observed for another sample prepared under similar conditions45 as well as for a well-ordered MCM-41 with 3.9 nm pores.43 One of the large-pore MCM-41 samples42 exhibited an extremely broad hysteresis loop, which was much broader than the loops observed for other samples of similar pore diameters. For a given sample, a similar steepness of adsorption branches of isotherms was observed in the case of argon adsorption at 77 K (data reported herein) and 87 K32 and nitrogen adsorption at 77 K.39,41-50 However, the steepness of the desorption branches often differed considerably. For instance, 5.8 nm MCM-41, whose argon isotherm at 77 K shown in Figure 1S features a rather broad desorption branch, had a very steep desorption branch in the case of nitrogen adsorption at 77 K.42 It is clear that the steepness of a desorption branch does not always correspond to the degree of structural ordering of MCM-41 silicas, and for the same sample, this steepness

Figure 2. Higher pressure parts of argon and nitrogen adsorption isotherms at 77 K for disordered silicas (amount adsorbed was converted to the corresponding volume of liquid adsorbate).

Figure 3. Experimental relations for the pore diameter versus (i) the capillary condensation pressure (hollow circles) and (ii) the capillary evaporation pressure (filled circles) for argon at 77 K in cylindrical siliceous pores. The line shows an empirical description of relation (i) using eq 4.

may be considerably different for different adsorbates and for the same adsorbate at different temperatures. 3.2. Argon Adsorption Isotherms for Silicas with Larger Mesopores. As reported elsewhere,54 SBA-15 silicas of pore diameters below 9 nm (more specifically, 7.4 and 8.9 nm, as calculated using eq 2) exhibited argon adsorption isotherms at 77 K with H1 hysteresis loops that are similar to those observed on nitrogen adsorption isotherms. Disordered large-pore silica with an average pore diameter of about 10 nm also exhibited argon adsorption isotherms with a hysteresis loop shape that is similar to that for the corresponding nitrogen adsorption isotherm (Figure 2).45 However, the similarity did not extend over larger pore diameters, as LiChrospher Si-100 silica with an average pore diameter of about 19 nm (estimated from nitrogen adsorption data using a method described in ref 39) exhibited an adsorption isotherm intermediate between types II and IV in the case of argon at 77 K, whereas the corresponding nitrogen adsorption isotherm was of type IV (Figure 2). Moreover, the amount adsorbed expressed as the volume of liquid adsorbate was much smaller in the former case. This suggests that for argon at 77 K capillary condensation does not take place in pores of a diameter that is above about 19 nm (this limit will be further elucidated hereafter). 3.3. Relation between the Pore Size and Capillary Condensation/Evaporation Pressure. The experimental relation between the pore diameter of siliceous cylindrical pores and the capillary condensation/evaporation pressures for argon at 77 K is shown in Figure 3. The capillary condensation pressure for argon at 77 K gradually increased as the pore diameter

Determination of Mesopore Size Distributions

Figure 4. Experimental relations for (i) nitrogen capillary condensation pressure versus argon capillary condensation pressure and (ii) nitrogen capillary evaporation pressure versus argon capillary evaporation pressure, observed at 77 K on silica samples with a wide range of average pore diameters.

increased. The scatter of data was relatively small. For very similar pore diameters (estimated using eq 1), the capillary condensation relative pressures for different samples did not differ by more than 0.05, whereas, for very similar capillary condensation pressure values, the pore diameters were not more than 0.5 nm apart. In most cases, the agreement was much better than these extreme cases. The scatter of the experimental relation between the capillary evaporation pressure and the pore diameter was much larger. In particular, the same capillary evaporation pressure was recorded for two samples with pore diameters that were more than 1.2 nm apart (see data for 4.5 nm MCM-41 in Figure 1 and for 5.8 nm MCM-41 in Figure 1S). Moreover, for very similar pore diameters, the capillary evaporation relative pressure was found to differ by 0.08 in some cases (see Figure 3). This large scatter can largely be traced back to differences in the structural perfection of the samples. Less-ordered materials tended to exhibit lower capillary evaporation pressures for a given pore diameter. The results discussed above indicate that, in the case of argon adsorption at 77 K in cylindrical pores, the pore diameter is correlated well with the capillary condensation pressure and much less correlated with the capillary evaporation pressure. This provides good prospects for pore size analysis based on adsorption branches of isotherms and questions the feasibility of the pore size assessment from desorption branches. To further investigate the prospects for the pore size assessment from argon adsorption data at 77 K, we studied the correlation between the capillary condensation pressure of argon for a given sample and that for nitrogen at 77 K for the same sample (Figure 4) and found it to be very good. However, the correlation between the capillary evaporation pressure of argon for a given sample and that of nitrogen for the same sample was much less satisfactory in the region of adsorption-desorption hysteresis. This suggests that a consistent pore size analysis from adsorption branches of argon and nitrogen isotherms at 77 K can be achieved, whereas prospects for consistency in the analysis from desorption branches are not good. 3.4. Implications for Assignment of Branches of Hysteresis Loops Generated by DFT and Computer Simulation. The results presented above have major implications for the elucidation of the relation between experimental adsorption isotherms and adsorption isotherm data generated using computer simulations and theoretical methods, such as NLDFT. The computationally-generated adsorption data often exhibit wide hysteresis loops with determinable points of (i) spinodal capillary condensation, (ii) spinodal capillary evaporation, and (iii) equilib-

J. Phys. Chem. B, Vol. 106, No. 18, 2002 4735 rium transition.28,29,35-37 In principle, the experimental capillary condensation pressure corresponds to some pressure point between the equilibrium transition and the spinodal capillary condensation pressures, whereas the experimental capillary evaporation pressure corresponds to some point between the equilibrium transition and spinodal capillary evaporation pressures. When there is no adsorption-desorption hysteresis, the equilibrium transition has to correspond to both adsorption and desorption branches of experimental isotherms. However, in the case of hysteresis, the assignment is certainly less clear. It was often arbitrarily assumed that spinodal capillary condensation corresponds to the experimental capillary condensation, whereas the equilibrium transition corresponds to the experimental capillary evaporation.28,29,35-37 Although this interpretation has some theoretical rationale, it is inconsistent with the experimental data reported herein. The lack of a clear relation between the pore diameter and the experimental capillary evaporation pressure, and the dependence of the latter on the ordering of the adsorbents studied suggest that the capillary evaporation in real systems is delayed because of the presence of constrictions in pore channels56 and, in general, other pore network effects.5,7,8 In practice, porous solids are usually much less ordered than any MCM-41 sample considered herein, which makes a general assignment of the equilibrium transition to the experimental desorption branch even more questionable and less acceptable in practical PSD calculations. In the context of the aforementioned identification, it was noted that the occurrence of pore connectivity effects would result in capillary evaporation at pressures that do not correspond to the equilibrium transition.36 The current study of well-ordered materials suggests that it is hard to find a system in which there are no pore connectivity effects of any sort, whether they be single pore effects56 or pore network effects.5,7,8 Thus, one should not attribute the experimental capillary evaporation pressure to the equilibrium transition, except perhaps for some highly-ordered materials with channel-like pores. But even in these cases, the validity of this interpretation is uncertain. As for the experimental capillary condensation pressure, it is not likely to correspond in general to the spinodal capillary condensation. It was recently acknowledged that, in the region of transition between reversible and irreversible (that is, accompanied with hysteresis) adsorption behavior, the experimental capillary condensation corresponds to some point between the equilibrium transition and spinodal capillary condensation.57 Our data suggest that this is the case for the entire hysteresis region, and moreover, there is a chance that the capillary condensation is delayed more for highlyordered materials than for less-perfectly-ordered ones (see Supporting Information). 3.5. Determination of the t-curve. Because of the fact that adsorption in mesopores involves the monolayer-multilayer adsorption before the onset of capillary condensation (and desorption from the multilayer after the capillary evaporation), the development of a useful method for the PSD calculation requires knowledge of the multilayer formation process in pores of different sizes. Therefore, the statistical film thickness data for MCM-41 pores of different diameters (see Figures 5 and 2S) were examined. The statistical film thickness before the onset of capillary condensation was similar for the pores of diameters above about 4 nm, but as the pore size decreased further, the increased statistical film thickness was observed well before the onset of the capillary condensation. This increase gradually became more pronounced as the pore diameter decreased, but the departures from the statistical film thickness for larger pores were still moderate even for pore sizes down

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TABLE 2: Reduced Adsorption Isotherm of Argon at 77 K on LiChrospher Si-1000 Silicaa p/p0

Rs

p/p0

Rs

p/p0

Rs

p/p0

Rs

1.68 × 10-5 2.47 × 10-5 3.84 × 10-5 5.60 × 10-5 7.73 × 10-5 1.022 × 10-4 1.309 × 10-4 1.630 × 10-4 1.990 × 10-4 2.389 × 10-4 2.824 × 10-4 3.295 × 10-4 3.80 × 10-4 4.35 × 10-4 4.93 × 10-4 5.55 × 10-4 6.20 × 10-4 6.94 × 10-4 7.66 × 10-4 8.40 × 10-4 9.18 × 10-4 9.99 × 10-4 1.086 × 10-3 1.176 × 10-3 1.268 × 10-3 1.363 × 10-3 1.461 × 10-3 1.560 ×10-3 1.663 ×10-3 1.766 ×10-3

0.0092 0.0136 0.0180 0.0224 0.0267 0.0310 0.0353 0.0396 0.0438 0.0480 0.0522 0.0563 0.0603 0.0643 0.0683 0.0722 0.0760 0.0798 0.0836 0.0872 0.0909 0.0944 0.0981 0.1016 0.1052 0.1086 0.1120 0.1153 0.1186 0.1217

1.872 × 10-3 1.979 × 10-3 2.088 × 10-3 2.198 × 10-3 2.446 × 10-3 2.702 × 10-3 2.965 × 10-3 3.234 × 10-3 3.51 × 10-3 3.88 × 10-3 4.26 × 10-3 4.66 × 10-3 5.08 × 10-3 5.52 × 10-3 5.97 × 10-3 6.41 × 10-3 6.87 × 10-3 7.35 × 10-3 7.83 × 10-3 8.33 × 10-3 8.85 × 10-3 9.38 × 10-3 9.92 × 10-3 0.01048 0.01133 0.01260 0.01491 0.01764 0.02006 0.02516

0.1248 0.1279 0.1309 0.1338 0.1401 0.1463 0.1522 0.1580 0.1636 0.1706 0.1775 0.1843 0.1910 0.1976 0.2040 0.2101 0.2160 0.2219 0.2277 0.2334 0.2390 0.2446 0.2500 0.2554 0.2634 0.2746 0.2932 0.3128 0.3285 0.358

0.02983 0.0352 0.0401 0.0450 0.0502 0.0574 0.0698 0.0798 0.0895 0.0995 0.1191 0.1390 0.1601 0.1803 0.2007 0.2213 0.2419 0.2625 0.2829 0.3034 0.3239 0.344 0.362 0.383 0.400 0.421 0.441 0.461 0.481 0.501

0.382 0.406 0.425 0.444 0.462 0.485 0.520 0.546 0.569 0.591 0.631 0.667 0.701 0.732 0.761 0.789 0.816 0.842 0.867 0.891 0.915 0.938 0.958 0.981 1.000 1.023 1.045 1.067 1.090 1.112

0.521 0.541 0.561 0.581 0.601 0.621 0.641 0.660 0.680 0.699 0.719 0.739 0.758 0.778 0.797 0.817 0.836 0.856 0.876 0.901 0.912 0.922 0.933 0.943 0.954 0.964 0.975 0.985 0.996 0.998

1.135 1.159 1.182 1.207 1.232 1.257 1.283 1.308 1.336 1.363 1.392 1.423 1.456 1.492 1.528 1.564 1.601 1.643 1.687 1.746 1.772 1.798 1.825 1.853 1.883 1.915 1.947 1.980 2.017 2.051

a Notation: p, equilibrium vapor pressure; p , saturation vapor pressure; R , standard reduced adsorption, that is the amount adsorbed divided by 0 s the amount adsorbed at p/p0 ) 0.40 (equal to 7.689 cm3 STP g-1). The t-curve can be obtained by multiplying the Rs values by 0.522 nm. The saturation vapor pressure values recorded during the measurement were between 191.1 and 192.5 Torr.

Figure 5. Statistical film thickness for argon at 77 K in cylindrical pores of different sizes. The solid line shows the fit of the statistical film thickness with argon adsorption isotherm data for LiChrospher Si-1000 macroporous silica.

to about 2.4 nm. The departures became large within the entire relative pressure range only for 2.0 nm pores. These results suggest that it is possible to derive a common statistical film thickness curve (t-curve) that would provide a satisfactory description of multilayer adsorption in pores of different sizes down to 2.4 nm or perhaps even smaller. Such a t-curve can be based on an adsorption isotherm for a suitable macroporous silica gel.32,39 Therefore, we carefully measured argon adsorption isotherms at 77 K on two macroporous silicas, LiChrospher Si1000 and Si-4000. The argon isotherm at 77 K for the macroporous silica with a higher BET specific surface area (19.3 m2 g-1 calculated from argon data at 77 K) is shown in Figure 3S. The isotherm for the other silica is essentially superimposable after a normalization that is required to compare adsorptions for samples of different specific surface areas. In contrast to

the nitrogen adsorption isotherm at 77 K and the argon adsorption isotherm at 87 K, the argon adsorption isotherm at 77 K did not exhibit a steep increase related to the buildup of subsequent adsorbed layers and/or to the capillary condensation as the saturation vapor pressure is approached. Thus, the argon isotherm at 77 K looks as if it were truncated,26 in other words, as if the saturation vapor pressure were not reached.29,34 The argon adsorption isotherm data for LiChrospher Si-1000 are provided in Table 2. These data were used in the Rs plot analysis to determine the primary mesopore volume and the external surface area for the samples under study (see Table 1). In general, these two quantities assessed from argon adsorption data at 77 K were similar to those evaluated from nitrogen and argon adsorption data at 77 K and 87 K, respectively, although the primary mesopore volume assessed from argon data at 87 K was usually slightly lower than those of the other assessments. Since the total pore volume determined from the argon adsorption at 77 K does not account correctly for the secondary mesopore volume (these pores are often too large to exhibit capillary condensation under these conditions), it is in some cases considerably smaller than that determined using argon at 87 K or nitrogen at 77 K (compare Table 1 with data from ref 46). For all the ordered mesoporous materials studied, the Rs plot analysis from argon data at 77 K suggested the presence of micropores, which is similar to the results from the argon adsorption at 87 K32 but is in contrast to the assessment from the nitrogen adsorption at 77 K.39,42,43,46,47 Some possible explanations of this behavior were discussed elsewhere.32 The BET specific surface area assessed from the argon adsorption at 77 K was similar to that determined from the argon data at 87 K, but was about 20% smaller than the nitrogen BET surface area (compare data in Table 1 with those from refs 39, 42, 46, and 50).

Determination of Mesopore Size Distributions

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To obtain the t-curve valid within the entire relative pressure range, the statistical film thickness data for four large-pore MCM-41 silicas were fitted with the data for LiChrospher Si1000.32,39 In doing so, it was assumed that the statistical film thickness for the macroporous silica is equal to that for the MCM-41 silicas at a relative pressure of 0.2. This assumption is somewhat arbitrary but guarantees that there is no relative pressure value for which the fitted t-curve has a larger value than that for the average statistical film thickness for the largepore MCM-41 samples. The resultant t-curve can be obtained from the data listed in Table 2 simply by multiplying the standard reduced adsorption values by the constant provided in the caption to the table. The fit was satisfactory for moderate values of the relative pressure (see Figure 5) but was less satisfactory at lower relative pressures (see Figure 2S), similar to the results for argon adsorption at 87 K32 and in contrast to those for nitrogen adsorption at 77 K, in which case a good fit at low pressures was obtained.39 3.6. Pore Size-Capillary Condensation Pressure Relation. In addition to the knowledge of the statistical film thickness for pores of different sizes, PSD calculations require an expression for the relation between the pore diameter and the capillary condensation (or evaporation) pressure. It is convenient to express this relation as a sum of two terms, one being the statistical film thickness and the second one being a relation between the pore core radius (the radius of void space confined by the film adsorbed on the pore walls) and the capillary condensation/evaporation relative pressure. It was shown above that the capillary condensation pressure is much better correlated with the pore diameter than the capillary evaporation pressure is, and therefore only the relation between the capillary condensation pressure and pore core radius was established. To do that, the statistical film thickness values corresponding to the experimentally-observed capillary condensation pressures were calculated by interpolating the t-curve (Table 2) and were subtracted from the pore radii to obtain the core radii. This allowed us to obtain the experimental relation for the core radius as a function of capillary condensation pressure. To use such a relation in the pore size analysis, one needs to find its approximated and extrapolated form. One can attempt to use an expression in the form of the Kelvin equation with a constant correction term (rc ) -a/log(p/p0) + b, where rc is the pore core radius; p/p0 is the relative pressure; a is a constant that is in principle related to the surface tension, molar volume, and temperature; and b is the constant correction term), which was a good approximation formula for the experimental argon adsorption data at 87 K and nitrogen adsorption data at 77 K.32,39 However, such an expression does not exhibit proper behavior for pressures close to the saturation vapor pressure, because it predicts that the pore core radius for which the capillary condensation takes place rapidly increases as the pressure approaches saturation vapor pressure. This is in contrast with the experimental data that show the lack of capillary condensation for pore sizes above about 20 nm. To obtain a satisfactory fit with a reasonable behavior at pressures close to saturation, it was necessary to introduce an additional constant that modifies the relative pressure value in the Kelvin-equation-type relation:

rc(p/p0)[nm] ) -

0.5393 + 0.343 log(0.8259p/p0)

(3)

Equation 3 combined with the t-curve provides an empirical equation for the relation between the capillary condensation pressure and the diameter of cylindrical pores for Ar adsorption

Figure 6. Comparison of pore size distributions calculated from argon adsorption data at 77 (hollow triangles) and 87 K (hollow circles) and nitrogen adsorption data at 77 K (filled circles) for MCM-41 silicas discussed in ref 32.

at 77 K:

r(p/p0)[nm] ) -

0.5393 + 0.343 + t(p/p0) (4) log(0.8259p/p0)

Equation 4 is in excellent agreement with the experimental data (see Figure 3) for pore sizes up to 10 nm, which corresponds to relative pressures up to 0.87 and behaves well for relative pressures closer to saturation, predicting that pores 15 nm in diameter would exhibit capillary condensation at a relative pressure of 0.99. This is reasonable in light of the available data, but further studies will be required to refine our current knowledge of the pore diameter limit at which capillary condensation can be observed for argon at 77 K. It is interesting to note here that the form of eq 4 that describes the pore diameter-capillary condensation pressure relation for argon at 77 K is different from that of the empirical equation derived for argon adsorption at 87 K. This confirms earlier expectations33 that the form of the empirical relation derived using our approach32,39 may be different for the same adsorbate at different temperatures. This is not a major limitation from the utilitarian point of view, because practically useful temperatures of lowpressure adsorption measurements are largely restricted to 77 and 87 K (boiling points of liquid nitrogen and argon), which are conditions that we have already explored32,39 or describe herein. In addition, empirical relations in the form of eq 4 can readily be adjusted for differences in the surface properties of materials to be characterized simply by employing a t-curve characteristic of a given surface type.58 3.7. Calculation of Mesopore Size Distributions. Equation 4 and the t-curve reported herein (data provided in Table 2) can readily be used to calculate mesopore size distributions for pore sizes below about 15 nm for argon adsorption data at 77 K. For this purpose, an algorithm based on the concept outlined in the work of Barrett, Joyner, and Halenda (BJH)40 was used, but no simplifying assumptions proposed in the original BJH work were employed.8,32,39 The calculations were carried out from adsorption branches of the isotherms. Shown in Figure 6 is a comparison of PSDs calculated from argon adsorption data at 77 and 87 K,32 as well as nitrogen adsorption data at 77 K39,59 for eleven MCM-41 samples with pore diameters from 2 to 6.5 nm used in ref 32. For all samples, the peaks on PSDs calculated

4738 J. Phys. Chem. B, Vol. 106, No. 18, 2002

Figure 7. Comparison of pore size distributions calculated from argon and nitrogen adsorption data at 77 K for periodic mesoporous organosilicas and SBA-15 silicas.

from argon data at 77 K were centered within 0.2 nm and were often even within 0.1 nm from the pore diameter determined using eq 1. In general, a very good agreement in the position of peaks, their width, and height was observed, but argon at 77 K tended to produce somewhat sharper peaks. PSDs were also calculated from argon adsorption data at 77 K reported earlier for periodic mesoporous ogranosilicas HMM-1 and HMM-260 (see Figure 4S) and SBA-15 silicas,54 and compared (see Figure 7) with the corresponding PSDs calculated from nitrogen adsorption data at 77 K. HMM-1 is a 2-D hexagonal material with a porous structure akin to that of MCM-41, so it is not surprising that the PSD calculation methods calibrated on MCM-41 silicas provided essentially identical results from both nitrogen and argon data. HMM-2 is a 3-D hexagonal structure with cagelike pores, which is conspicuously different from that of MCM-41 with channellike pores. Differences in the pore shape are likely to influence adsorption properties, such as the pore volume-surface area ratio, capillary condensation pressure for a given pore diameter, and so forth.61 So, in general, methods calibrated on pores of a certain shape may be less accurate for pores of a different shape, and moreover, no perfect consistency in PSD calculations from data for different adsorbates or temperatures is expected. Nonetheless, the PSDs calculated for HMM-2 from nitrogen and argon adsorption data were in very good agreement with one another, although the pore diameter for this material with cagelike pores was most likely underestimated by about 1 nm.61 The case of the HMM-2 hybrid material is particularly instructive for the assessment of the feasibility of PSD calculations from adsorption and desorption branches of isotherms. One can notice that the adsorption branches of nitrogen and argon isotherms at 77 K (Figure 4S) were of similar shape and steepness at pressures corresponding to the capillary condensation in primary mesopores, whereas the desorption branches for these two adsorbates exhibited markedly different shapes and steepnesses at pressures of capillary evaporation from primary mesopores. Consequently, PSDs calculated for this material from argon and nitrogen adsorption branches of isotherms were in excellent agreement, whereas one can hardly envision that any consistency can be achieved for PSDs calculated from the desorption branches. As seen in Figure 7, the PSDs calculated from argon adsorption data at 77 K as proposed herein are also in good agreement with the nitrogen adsorption PSDs in the case of SBA-15 silicas. It is clear that in general the PDS calculations from adsorption branches of isotherms provide a similar, if not essentially identical, picture of the PSD independent of the selection of nitrogen or argon as an adsorbate and of the choice of the

Kruk and Jaroniec measurement temperature. The possibility of such similarity in the PSD assessment can be predicted by comparing the shapes of adsorption isotherms for different gases or temperatures, but the realization of this possibility requires a proper calibration, for instance that proposed in our earlier studies32,39 and employed herein. As was extensively discussed in our earlier work and herein, a similar consistency cannot be generally expected for calculations from desorption branches, although it may be possible in special cases. The contemplation of these special cases may make an impression that desorption data are in general suitable for pore size analysis and capable of providing similar PSDs in the case of different gases or the same gas at different temperatures,29 or that consistent PSDs can be evaluated from adsorption and desorption branches of isotherms.36,37 However, the experimental data for ordered mesoporous materials with gradually increasing pore diameters spanning over a broad range of values (including the samples with a similar pore diameter but a wide variation of degree of structural perfection) indicate that such promising results of PSD calculations from desorption branches of isotherms are merely special cases, whereas such calculations in general may be highly inconsistent and may provide a misleading picture of PSDs. Finally, it is important to note that the reference adsorption isotherm and eq 4 provided herein can readily be used to generate local adsorption isotherms for pores of different sizes (all pore sizes above 15 nm would be represented by the standard adsorption isotherm data) and to solve the integral equation for overall adsorption6 in order to get the PSD. This way would be analogous to that used in PSD calculations based on local isotherms generated using DFT9,28 or computer simulations.9 4. Conclusions In the case of argon adsorption at 77 K in cylindrical siliceous pores, there exists a good correlation between the pore diameter and the capillary condensation pressure, whereas the capillary evaporation pressure is much less correlated with the pore diameter. Moreover, for some MCM-41 silicas, there may be no correlation between the steepness of a desorption branch and the degree of structural ordering (judged from XRD), whereas the steepness of an adsorption branch is in general wellcorrelated with the structural ordering of MCM-41. This suggests that the assessment of PSDs from adsorption branches of isotherms is feasible, but that from desorption branches is questionable. The calculation of PSDs from argon data at 77 K is restricted to pore diameters below about 15 nm, because no capillary condensation is observed in wider pores. Using the methodology developed during our earlier work, we obtained consistent PSDs from adsorption branches of isotherms in cases of argon at 77 and 87 K, and nitrogen at 77 K. Argon at 77 K is known to be suitable for micropore analysis, and thus the current work suggests that this gas at 77 K is suitable for studies of microporous and mesoporous materials when the information about pores wider than about 15 nm is not necessary. Acknowledgment. The donors of the Petroleum Research Fund administrated by the American Chemical Society are gratefully acknowledged for support of this research. Drs. A. Sayari (U. Ottawa, Canada) and R. Ryoo (KAIST, Korea) are acknowledged for providing MCM-41, SBA-15, and large-pore silica samples. Supporting Information Available: Information about samples, methods used, and additional discussion and explana-

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