Experimental and Theoretical Investigations of Adsorption Hysteresis

Ind. Eng. Chem. Res. 1998, 37, 2271-2283

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Experimental and Theoretical Investigations of Adsorption Hysteresis and Criticality in MCM-41: Studies with O2, Ar, and CO2† C. G. Sonwane, S. K. Bhatia,* and N. Calos‡ Department of Chemical Engineering, The University of Queensland, St. Lucia, Brisbane QLD 4072, Australia

MCM-41 materials of six different pore diameters were prepared and characterized using X-ray diffraction, transmission electron microscopy, helium pycnometry, small-angle neutron scattering, and gas adsorption (argon at 77.4 and 87.4 K, nitrogen and oxygen at 77.4 K, and carbon dioxide at 194.6 K). A recent molecular continuum model of the authors, previously used for adsorption of nitrogen at 77.4 K, was applied here for adsorption of argon, oxygen, and carbon dioxide. While model predictions of single-pore adsorption isotherms for argon and oxygen are in satisfactory agreement with experimental data, significant deviation was found for carbon dioxide, most likely due to its high quadrupole moment. Predictions of critical pore diameter, below which reversible condensation occurs, were possible by the model and found to be consistent with experimental estimates, for the adsorption of the various gases. On the other hand, existing models such as the Barrett-Joyner-Halenda (BJH), Saito-Foley, and Dubinin-Astakhov models were found to be inadequate, either predicting an incorrect pore diameter or not correlating the isotherms adequately. The wall structure of MCM-41 appears to be close to that of amorphous silica, as inferred from our skeletal density measurements. 1. Introduction Considerable effort has been made in recent years to develop porous materials of well-defined pore geometry. Such materials, in addition to their fundamental applications, have potential for use as catalyst and supports/shape-selective adsorbents. The syntheses and applications of ordered microporous solids, such as zeolites, are long established with numerous patents and papers, but it was not until recently that interest in preparing mesoporous materials (diameter g 20 Å) arose, principally catalyzed by the discoveries of Yanagisawa et al. (1990) and Kresge et al. (1992). MCM-41, a member of the M41S family of materials, is prepared by surfactant liquid-crystal templating. It has been confirmed from transmission electron microscopy and X-ray diffraction studies that these materials have uniform-sized parallel hexagonal pores (Alfredsson et al., 1994), arranged in a honeycomb type lattice with a nonintersecting pore system. The pore diameter can be tailored between 20 and 200 Å (with a very narrow distribution) either by varying the surfactant chain length, or by adding auxiliary organic molecules, such as trimethylbenzene, during synthesis or by thermally restructuring (Khushalani et al., 1995). They have very high surface areas in the range of 600-1300 m2/g (usually calculated from the BET equation), and from R-s plots it has been shown that there are no significant micropores present in MCM-41. From a model describing X-ray diffraction pattern intensities, it has been shown that the most favored pore shape of MCM-41 is hexagonal (Alfredsson et al., 1994). Surface properties of MCM-41 can be modified by incorporating heteroatoms such as boron, titanium, vanadium, etc. Moreover, * To whom correspondence should be addressed. E-mail: [email protected]. Fax: +61 7 3365 4199. Telephone: +61 7 3365 4263. † This paper is dedicated to Professor L. K. Doraiswamy on the occasion of his 70th birthday. ‡ Department of Chemistry.

the silanol groups present on the surface are suitable for chemical bonding of organic ligands or anchoring inorganic species. The ease of surface modification promises wide application of MCM-41 in the field of catalysis (Zhao et al., 1996; Casci, 1994). The adsorption of gases in mesoporous materials is generally characterized by a distinct step in the isotherm accompanied by a hysteresis loop (Gregg and Sing, 1982). However, adsorption on MCM-41 materials of pore size less than about 40 Å appears to be unusual, in that the isotherm of nitrogen at 77.4 K is hysteresisfree. A similar phenomenon has been reported in the literature for the adsorption of argon, oxygen, carbon dioxide, ethylene, benzene, carbon monoxide, and cyclopentane (cf. Table 1). This has not been reported in the past for any other mesoporous material. Although many models have been proposed to describe the phenomenon of hysteresis in mesopores, the most widely accepted has been the Kelvin-Cohan theory. According to this, during adsorption, as a consequence of cooperative adsorbate-adsorbate interactions together with the influence of the curvature of the wall, at a certain pressure the pore is abruptly filled with saturated adsorbate following multilayer adsorption on the walls. During desorption the filled pore empties to an adsorbate layer by evaporation of the core. Thus, during condensation the shape of the adsorbate meniscus passes from cylindrical to hemispherical, while during desorption the reverse occurs. Hysteresis has been considered to be a consequence of this difference in the shapes of the adsorbate/vapor interface. This is one of the classical explanations and assumes the state of the fluid in the pores to be the same as the bulk saturated fluid, which may not be valid when the pore diameter is on the order of a few molecular diameters. It is not surprising, therefore, that the Kelvin-Cohan theory fails to explain the absence of hysteresis observed in MCM-41 materials of pore size e 40 Å. While pore networking, constrictions, and different pore shapes

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2272 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 Table 1. Region of Reversible Adsorption (i.e., No Hysteresis) for MCM-41 adsorbate nitrogen

argon

cyclopentane

CO H2O CO2

C2H4 O2

SO2 benzene methanol ethanol propanol butanol

diameter (Å)

temp (K)

refs

303 g173 g195 > 197 (>)195 g148 g163 g76 g91 > 92 (>)77.4 (>)273 298 (>)303 (>)303 (>)303 (>)303

Morishige et al., 1997 Morishige et al., 1997 Kruk et al., 1997 Ravikovitch et al., 1995 Ravikovitch et al., 1995 Sonwane and Bhatia, 1998 Branton et al., 1993 Inagaki et al., 1996 Schmidt et al., 1995 Llewellyn et al., 1994 Schulz-Ekloff et al., 1997 Naona et al., 1997 Morishige et al., 1997 Morishige et al., 1997 Morishige et al., 1997 Branton et al., 1994 Llewellyn et al., 1994 Rathousky et al., 1994, 1995 Rathousky et al., 1994, 1995 Rathousky et al., 1994, 1995 Franke et al., 1993 Llewellyn et al., 1996 Inagaki et al., 1996 Branton et al., 1995a Morishige et al., 1997 Morishige et al., 1997 Morishige et al., 1997 Branton et al., 1995b Morishige et al., 1997 Morishige et al., 1997 Morishige et al., 1997 Morishige et al., 1997 Morishige et al., 1997 Branton et al., 1994 Branton et al., 1995b Inagaki et al., 1996 Branton et al., 1995a Branton et al., 1995a Branton et al., 1995a Branton et al., 1995a

could also cause the hysteresis, these factors may not apply to the MCM-41 structure. Among the few models developed for prediction of adsorption in MCM-41 materials, the nonlocal density functional theory (NLDFT) of Ravikovitch et al. (1995) predicts a capillary critical size of about 18 Å for nitrogen adsorption at 77.4 K, considerably lower than the experimental value of 40 Å. The authors concluded that the absence of hysteresis could be because of the metastability of the adsorption branch of the isotherm, while they also questioned some of the fundamental assumptions of the NLDFT model such as the applicability of the Carnahan-Starling equation of state and the mean-field spherical model of fluid-fluid interactions for nitrogen adsorption. They also concluded that there is no direct correlation between the experimental lower closure point of the hysteresis loop and the limit of metastability of the liquid phase. Another promising theory is the one proposed recently by Evans et al. (1986), which suggests that the hysteresis is an intrinsic property of the phase transition in a single idealized pore. In addition to the adsorbate layer, there is an enhanced concentration of gas in the pore cavity. The amount of this gas depends on temperature, relative pressure, pore size, and the properties of the pore wall. At some point along the adsorption path, possibly after the metastable state has been reached, there is an irreversible transition to a stable liquid state. Desorption from this state may then proceed by an irreversible

transition from a metastable liquid state to a stable state comprising the pore gas and adsorbed layer. The hysteresis due to a transition between a capillary gas and a capillary liquid appears at a temperature that is lower than the critical temperature of the bulk fluid. The theory predicts that the smaller the pore diameter, the lower the capillary critical temperature. Although a shift of the gas-liquid critical point of the fluid in the porous material to lower temperatures relative to that of the bulk fluid has been experimentally verified, there have been only a few experimental studies to try to establish a quantitative relationship between the pore size and the critical point shift. On the basis of the above grounds, Branton et al. (1994) and Rathousky et al. (1995) suggested that the absence of hysteresis in MCM-41 materials for some of the adsorbates is actually a fundamental property of the system, being a function of the type of adsorptive, the temperature, and the pore size. On the basis of the proposed model of Evans et al. (1986) and on their studies with the adsorption of cyclopentane on MCM41 materials, Rathousky et al. (1995) suggested the relation

1-

TPC d ≈ TC R

(1)

where TC is the bulk critical temperature of the adsorptive, TPC is the critical temperature of the adsorptive in the pores, d is the molecular diameter, and R is the radius of the pore. However, this expression fails to explain the absence of hysteresis of nitrogen adsorption in MCM-41 materials of pore diameter ≈ 40 Å at 77.4 K. In recent papers (Bhatia and Sonwane, 1998; Sonwane and Bhatia, 1998) we have presented a molecularcontinuum model which incorporates molecular concepts in the classical framework. While applying the model to the adsorption of nitrogen at 77.4 K in MCM-41 materials and incorporating the concept of tensile stress of fluids confined in the pores, we estimated (Sonwane and Bhatia, 1998) the limiting pore diameter to be 33 Å which is closer to the experimental value of 40 Å. This result also compares favorably with that of Maddox et al. (1997), who obtained a value between 28 and 32 Å from molecular simulations, while considering a heterogeneous potential model. The latter approach also predicts that heterogeneity leads to the condensation transition being manifested instead as a continuous pore filling. This is consistent with the prior finding of Karykowski et al. (1994) based on simulation of argon adsorption in energetically heterogeneous slitlike pores. The adsorption of nitrogen, argon, carbon dioxide, ethylene, oxygen (Morishige et al., 1997), benzene (Inagaki et al., 1996), cyclopentane (Rathousky et al., 1994; 1995), and carbon monoxide (Llewellyn et al., 1996) on MCM-41 materials has been found to be of type IV without hysteresis. Table 1 provides an overview of the hysteresis-related work done on various MCM-41 materials using different adsorbates at different temperatures. We have previously presented a theory (Bhatia and Sonwane, 1998; Sonwane and Bhatia, 1998) for the adsorption in MCM-41 materials, which incorporates pore-size-dependent potential fields within the framework of the classical approach. We also provided the comparison between model-predicted and experimental adsorption isotherms of nitrogen at 77.4 K for

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MCM-41 materials of different pore sizes (Sonwane and Bhatia, 1998). Here we report our findings for the adsorption of argon, oxygen, and carbon dioxide in these materials and compare experimental and model-predicted isotherms for these cases. Characterization of the structure has been done by helium pycnometry, X-ray diffraction, transmission electron microscopy, and small-angle neutron scattering and the results are presented here. In the work discussed here we first present the detailed procedure used in the synthesis and characterization of MCM-41 materials of six different pores sizes by various techniques. Subsequently, in the Results and Discussion section the wall structure of MCM-41 is discussed by using the small-angle neutron scattering and skeletal density data. The estimates of lattice spacing and pore diameter and their comparison with micelle diameter are then provided followed by the characterization of the MCM-41 materials by an adsorption isotherm of various gases along with the estimates of pore size distribution. The failure of the conventional models to predict either the pore size or the isotherm is discussed followed by model predictions, using our models (Bhatia and Sonwane, 1998; Sonwane and Bhatia, 1998) for pore size, adsorption isotherm, and isotherm reversibility criteria. 2. Experimental Section 2.1. Synthesis. A series of MCM-41 materials of varying pore size were synthesized hydrothermally as described in the literature (Beck et al., 1992). The samples of MCM-41 are designated as Cn where n stands for the number of carbon atoms present in the alkyl chain of the alkyltrimethylammonium halide surfactant used in the synthesis. Surfactants with C8 and C10 chain lengths (purity >98% and >99%, respectively) were obtained from Tokyo Kasei Organic Chemicals, Tokyo, Japan. Surfactants C12 (99%) and C14 (99%) were supplied by Aldrich, Sydney, Australia. C16 surfactant (99%) was supplied by Merck Pty Ltd., Kilsyth (Victoria) Australia, and C18 (>98%) was supplied by Fluka Chemicals, Sydney, Australia. Sodium silicate (27.5% w/w) and sulfuric acid (98%) were procured from Ajax Chemicals, Brisbane, Australia and BDH Chemicals, Kilsyth (Victoria) Australia, respectively. All the materials were used as received. Water purified by reverse osmosis was used for all the work. A typical synthesis of C16-MCM-41 involves the following procedure. A total of 9.11 g of cetyltrimethylammonium bromide (C16 surfactant) was dissolved in 30 g of water. As the surfactant is not soluble completely at room temperature, the solution was heated gently, until it formed a colorless solution. Then 0.70 g of sulfuric acid (98%) was added to it, and the mixture was stirred for 20 min. In a separate flask, 10.93 g of sodium silicate was dissolved in 20 g of water and the mixture was stirred for 30 min. This solution was then added to the stirring solution in the previous flask. This initially formed a thick gel, which after adding a small amount water and stirring for 30 min formed a viscous solution. This solution was transferred to a stainless steel reactor and heated to 100 °C for 150 h under autogenous pressure conditions. After the reactor was cooled to room temperature, the white product that settled at the bottom of the reactor was recovered by filtration, washed with water using a vacuum filtration apparatus, and dried under ambient conditions over-

night. This as-synthesized product was then calcined at 550 °C for 10 h with an initial ramp of 1 °C/min in air. Subsequently, the product was cooled and stored in dried glass bottles. 2.2. Characterization. X-ray powder diffraction data were obtained on a Philips (Almelo, Holland) PW1840 diffractometer using Co KR radiation (40 kV, 25 mA) with a Fe filter. It was equipped with a special variable-receiving slit, which was fixed to 0.2 mm. The wavelength of the radiation was 1.790 26 nm. MCM41 materials have large d spacings, and it has been reported that the error in measurement of lattice spacing increases rapidly for lattice spacing greater than about 10 Å and diffraction angles (2θ) less than about 10°. This is because of several sources of error, some of which arise in conventional use of diffractometer and others of which are related to the nature of the diffracting materials and the diffraction process itself. Longchain organic compounds such as normal alcohol are often used for calibration of lattice spacing in the range of 10-50 Å. In the current work, we used tetradecanol as the standard material for calibration because it has higher order reflections in the same range as that of quartz and lower reflections closer to that of MCM-41 materials (Brindley and Wan, 1974; Brindley, 1981). The calibration was performed according to the procedure reported by Brindley (1981). The d spacings of the MCM-41 materials were corrected using the above calibration. The adsorption isotherms of CO2, Ar, and O2 were measured on a Micromeritics ASAP 2010 analyzer using standard volumetric techniques. Before the analysis, the samples were outgassed for 250 °C for at least 12 h at about 6 × 10-6 Torr. Transmission electron micrographs of the sample C18 were obtained for particles dispersed with acetone on a holey carbon grid, using a JEOL 2010 TEM operating at 200 kV. The sample C8 was likewise prepared; however, the micrographs were obtained on a JEOL JEM 4010 HRTEM operating at 400 kV, in an attempt to better image the nature of the silica framework. The skeletal density of the MCM-41 samples was measured on a fully automatic gas displacement helium pycnometer (AccuPyc 1330 pycnometer, Micromeritics, Inc.). The reported density is the mean of 30 different runs. The samples were outgassed at 300 °C for at least 12 h before the measurement was performed. For some samples, an additional analysis was done, in which the material was vacuum degassed for 10 h at about 6 × 10-6 Torr using a molecular drag pump. Small-angle neutron scattering (SANS) patterns of samples C8, C10, C16, and C18 were recorded on the LOQ instrument at the ISIS facility Rutherford-Appleton Laboratory, Chilton (Oxon), U.K. 3. Results and Discussion 3.1. Wall Structure of MCM-41. Although there is little conclusive information on the wall structure of MCM-41, some recent evidence (Maddox et al., 1997) would suggest highly crystalline, quartz-like walls. In that work skeletal densities of about 2.7-3.0 gm/cm3 were found, which is consistent with the known value of 2.7 gm/cm3 for quartz and much higher than that of about 2.2 gm/cm3 for amorphous silica. However, while MCM-41 materials are considered as crystalline materials because of the regular arrangement of pores in a two-dimensional hexagonal array, no clear evidence for highly crystalline walls has been found and the broad

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Figure 1. Relation between the skeletal density and the number of carbon atoms present in the aliphatic chain of the surfactant used in synthesis.

range of Si-O-Si bond angles seen in solid-state NMR studies suggests an amorphous or partially crystalline structure, more closely related to amorphous silica. NMR also suggests that 20% ((10%) of the silicons in MCM-41 are present as silanol species (Beck et al., 1992). It has been reported by many researchers, based on X-ray diffraction studies, that walls of MCM-41 resemble amorphous silica. It was suggested by Maddox et al. (1997) that the high value of skeletal density they obtained could be because of the presence of heavy cations and/or functional groups attached to the surface of MCM-41. However, the sample used by them was purely siliceous, with little chance of the presence of heavy cations. Therefore, for additional estimates of skeletal densities, which provide useful information about the wall structure of MCM-41, we measured the density of all the MCM-41 samples prepared for this work, and the results are shown in Figure 1. In this and all subsequent figures dashed lines are drawn through the data only to guide the eye. Although these materials were found to be X-ray amorphous, some indications of the local structures and conformations of the silicate groups could be given by the true density of the material, using the volume per [SiO4] unit, by analogy to known crystalline silica structures. The densest silica structure is found in the mineral stishovite, a high-pressure phase with rutiletype packing of octahedral silicate groups. However, (under standard conditions) the more favorable tetrahedral arrangement of [SiO4] may pack in a variety of corner-shared frameworks, with variable density according to the relative conformations of these tetrahedra. Although in R-quartz each [SiO4] tetrahedron in the structure connects to four other [SiO4] tetrahedra, as in cristobalite, the corner-sharing arrangements and tetrahedral tilts differ to occupy less volume. Thus, the density of the silica is an indication of the silicate conformation. There is a limit on the expansion of each of these framework types, so complete Q4 coordination of silica has a minimum density of about 2.2 g/cm3. Indeed, the theoretical minimum for the cristobalite structure type is 2.18 g/cm3 (Taylor, 1972). Lower densities may be found in another class of silica framework structures with different conformations of

Figure 2. Porod plots from small-angle neutron scattering data.

corner-shared silica tetrahedra, namely, the ZSM zeolites, in which the [SiO4] units are no longer closepacked but define nanopores. White et al. (1997) have suggested a two-phase structure for the walls of these MCM-41 zeolites, with an open type of silica lining the void and a more dense framework support. However, the skeletal densities of between 2.15 and 2.3 g/cm3 observed for the present materials suggest that the cell walls are of the topologically more compact forms of silica (i.e., cristobalite-type) rather than a significantly different framework. Our experimental densities of different MCM-41 samples obtained by a helium pycnometer are shown in Figure 1 along with a comparison with some of the standard silica reported in the literature. The density of C8, C10, and C12 is around 2.10-2.12 g/cm3 which is close to tridymite-2H (2.197 g/cm3), high cristobalite (2.202 g/cm3), Pmmn (1.995 g/cm3), Cmcm (1.98 g/cm3), and Cmc21 (1.972 g/cm3). The density of samples C14, C16, and C18 was found be in the range of 2.25-2.29 g/cm3, which is closer to the density of low cristobalite (2.331 g/cm3), high cristobalite (2.202 g/cm3) and tridymite-2H (2.197 g/cm3). These densities were taken from PDF-2 Database Sets 1-46 of Powder Diffraction File 1996 (CDD, Swarthmore, PA). Each sample was analyzed at least two times with different times of outgassing (>12 h) or temperatures (250-300 °C), but the sample skeletal densities were unchanged (within 0.5%). The scattering from an isotropic system is described by its differential scattering cross section I(Q) measured as a function of its scattering vector Q. The scattering vector Q is related to the wavelength of the incident radiation, λ, and the scattering angle θ by Q ) (4π/λ) sin(θ/2). The results of neutron scattering are presented in the form of Porod plots in Figure 2. The sample C10, C16, and C18 SANS patterns show distinct peaks corresponding to the pore separations in each of these materials. The d spacing (2π/Q) obtained from the first peak of these SANS curves was found to be very close to that obtained from the XRD peaks. The peak resulting from the porosity of the sample C8 is very diffuse, with an ill-defined maximum at 27.0 Å, while those of C16 and C18 were at 39.8 and 40.8 Å, respectively. These values are close to the values obtained from X-ray diffraction patterns 28.9, 37.85, and

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Figure 3. Powder diffraction pattern of some calcined MCM-41 samples.

43.27 Å for C8, C16, and C18, respectively (to be discussed subsequently). Porod plots of each of these materials yield their fractal dimensions. Fractal dimensions of the samples C8, C10, C16, and C18 show a Porod slope of 2.63, 3.79, 3.81, and 3.62, respectively, to the resolution limit of about 200-300 Å. The C8 and C18 sample SANS patterns were also recorded for materials soaked in 63% D2O-37% H2O. In this porefilling experiment the sharp peaks and the extraneous small-angle scatter disappeared, confirming that the scattering arose from the ordered pore structures. In an analysis of the SANS data, to a resolution limit of 209 Å, C8 appears as an ideal mass fractal, according to the Porod slope of 2.63 (Keefer and Schaefer, 1986). At finer resolutions (i.e., larger q values), the nature of the C8 material becomes more like a surface fractal, with a Porod slope of 1.59. On the other hand, C18 behaves as a Euclidean scatterer (i.e., smooth surface) to a resolution limit of 184 Å, according to the observed Porod slope of 3.62 (Keefer and Schaefer, 1986). Thus, C10, C16, and C18 appear to be euclidean scatterers for a resolution limit of about 200 Å. The estimates of particle size and surface area of different samples from neutron scattering results will be given in detail in a subsequent paper. 3.2. Lattice Spacing and Pore Size. X-ray diffraction patterns (not on the same scale) of samples C10-C16 are shown in Figure 3. Around 3-4 peaks of each sample were observed and were indexed on a hexagonal lattice, typical of MCM-41 materials. Co KR radiation, having wavelength of 1.790 26 Å, was used for X-ray diffraction measurements. For the C18 sample the 100 peak could not be observed, but we observed other peaks, which were indexed on a hexagonal lattice

Figure 4. High-resolution transmission electron micrograph of calcined (a) C8 and (b) C18 samples.

as 110, 200, and 210 as seen in Figure 3. The absence of the 100 peak was attributed to its proximity to the detector limit. The d spacing for C18 samples was estimated from the 110 peak after indexing its 110, 200, and 210 peaks on a hexagonal index pattern typical of MCM-41. The XRD powder diffraction peak of the C8 sample was very weak as compared to the peaks of C10-C18 samples, indicating a low crystallinity of the former. A very sharp peak in XRD represents a highly crystalline material with a very narrow pore size distribution. Transmission electron microscopy of samples C8 and C18 was carried out, partly prompted by the XRD characterization. Parts a and b of Figure 4 depict the transmission electron micrographs of the samples C8 and C18, respectively. Figure 4 a suggests an amorphous fractal nature for the C8 sample, with probably a broader pore size distribution. For the C18 sample Figure 4 b shows the honeycomb structure of the pores in the product. As is seen in Figure 4 b, not all the pores in the figure are arranged in the same hexagonal lattice plane, i.e., the arrays of pores on the right-hand side do not match the arrays of pores on the left-hand side. On the macroscale, the structure of the C18 sample was seen to consist of lumps of small particles/crystallites. The XRD d spacing of these samples is consistent with the literature (Beck et al., 1992). The values of the lattice parameter, which is more appropriate as an indicator of pore size, are shown in Figure 5 for samples prepared with different surfactant chain lengths and may be seen to compare well with the literature values (Beck et al., 1992; Morishige et al., 1997; Kruk et al., 1997). Our method of synthesis

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Figure 5. Relation between the lattice parameter and the number of carbon atoms present in the aliphatic chain of the surfactant, alkyltrimethylammonium halide, used in the synthesis of MCM41 samples.

Figure 6. Relation between the pore diameter (obtained by argon, oxygen, and carbon dioxide adsorption) and the number of carbon atoms present in the aliphatic chain of the surfactant in the synthesis of MCM-41 samples.

of MCM-41 is similar to that of Beck et al. (1992) while different from that of others. The difference in the lattice parameter may be attributed to small variations in pH and calcining conditions. It can also be concluded that the different procedures of synthesis using the same surfactant can give samples of different lattice parameters. As seen from Figure 5, the lattice spacing increases gradually with an increase in the chain length of the surfactant used in the synthesis. The lattice parameters of C8 and C10 are very close, possibly because the 100 peak obtained in the C8 sample is broader. Except for the two samples C8 and C10, where the lattice parameter was very close, the increase of one -(CH2)2- group causes an average increase of 3.3 Å in the lattice parameter (for C8-C16 samples), which is higher than 2.5 Å reported by Kruk et al. (1997), probably due to the difference in the synthesis procedure. From sample C16 to sample C18 the increase in lattice spacing of about 6.3 Å is much larger than 3.3 Å. This increase is probably because of a relative increase in the pore diameter or a decrease in the wall thickness. As an aid to our interpretations, the expected micelle diameter was also estimated by using the equation of Schulz-Ekloff et al. (1997)

the volume of the hydrophilic group. So, with the selection of a ) 0.64 nm2, the value of the packing parameter was obtained as 0.33, which satisfies the condition. As the values of the cross section of the headgroups of all the surfactants were difficult to obtain from the literature, we assumed the cross-sectional area of the head of each surfactant is the same as that of the C16 surfactant. This assumption may be valid for the present case, as each surfactant has same hydrophilic headgroup “trimethylammonium halide”. It was also observed that for the lattice spacing obtained by this method, assuming a constant wall thickness, the addition of one -(CH2)2- group gives an increase of 3.6 Å in the lattice spacing, which is very close to the experimental value of 3.3 Å seen in our results but further removed from the value of 2.5 Å reported by Kruk et al. (1997). Nevertheless, we do not know the reason for the substantial increase of the lattice spacing in going from C16 to C18 in the present results. The pore diameter was calculated from the combination of gas adsorption and XRD determined lattice spacing, assuming that MCM-41 is made of infinitely long cylindrical pores arranged on a two-dimensional hexagonal lattice. The values of pore diameters obtained by various gas isotherms are as shown in Table 4 and are plotted in Figure 6 along with the predicted diameter of the micelles. The calculation needs the skeletal density, which was taken from the experimental results with helium pycnometry. The following equation was used:

Dm )

0.04(27.4 + 26.9n) a

(2)

and the values are plotted in Figure 6 along with the other estimates of the diameter of the samples. Here n is the number of carbon atoms in the alkyl chain, and a is the cross-sectional area of the headgroup. For the C16 surfactant micelle, the value of a has been reported as between 0.6 and 0.7 nm2. We found that with these extremes the value of the packing parameter (v/alc) was obtained as 0.35 and 0.30, respectively. The critical value of a for the packing parameter to be in the range of 1/3-1/2 is 0.64 nm2. Schulz-Ekloff selected a value of 0.63 nm2, but we chose 0.64 nm2, which is the maximum permissible value. For a stable cylindrical micelle, the packing parameter has to satisfy the condition 1/3 < v/alc < 1/2, where lc is the critical chain length of the surfactant given by lc ≈ (0.154 + 0.1265n) nm and v is

x x

D ) dXRD

8 πx3

FtVP 1 + FtVP

(3)

where dXRD is the d spacing, VP is the pore volume, and Ft is the true density. The values of pore diameter obtained from adsorption of different gases are very close but significantly higher than the micelle diameter. This suggests that the pore diameter increases either during hydrothermal treatment or during the calcining step. This may also indicate that the wall thickness decreases during that step. Although the diameter

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Figure 7. Adsorption isotherms of oxygen at 77.4 K on MCM-41 samples of different pore sizes (adsorption, b; desorption, O). The solid lines are model predictions.

Figure 9. Adsorption isotherms of argon at 87.46 K on MCM-41 samples of different pore sizes (adsorption, b; desorption, O). The solid lines are model predictions.

Figure 8. Adsorption isotherms of argon at 77.35 K on MCM-41 samples of different pore sizes (adsorption, b; desorption, O). The solid lines are model predictions.

Figure 10. Adsorption isotherms of carbon dioxide at 194.66 K on MCM-41 samples of different pore sizes (adsorption, b; desorption, O).

obtained from argon adsorption data at 87.4 K is the highest while that from oxygen adsorption at 77.4 K is the lowest, the results follow a consistent trend. One more estimate of the pore diameter was obtained by subtracting a constant wall thickness of 10 Å from the lattice parameter, consistent with the report of Chen et al. (1993). The diameter obtained by this approximation is very close but lower than other estimates discussed above. 3.3. Adsorption Isotherms and Characterization. The experimental adsorption isotherms of oxygen (at 77.4 K), argon (at 77.4 and 87.46 K), and carbon dioxide (at 194.66 K) are shown as the symbols in Figures 7-10, respectively. In these figures the amount adsorbed has been normalized by using the t-plot volume. The results of experimental and theoretical predictions of adsorption of nitrogen at 77.4 K on MCM41 of different pore sizes have already been presented (Sonwane and Bhatia, 1998). In the case of nitrogen adsorption, except for sample C8, in which the adsorption isotherm did not have any pronounced step which is a characteristic of mesoporous materials, all the

samples showed a distinct rise in the amount adsorbed over a very small pressure range, indicating a very narrow pore size distribution. In the present case the rise in the amount adsorbed was very sharp for samples C16 and C18. Theoretical predictions of the isotherm by nonlocal density functional theory (Ravikovitch et al., 1995), molecular simulations (Maddox and Gubbins, 1994; Maddox et al., 1997), and the molecular-continuum approach (Sonwane and Bhatia, 1998) also show a distinct rise in the amount adsorbed. However, the experimental rise is continuous as opposed to the sudden jump anticipated for conventional capillary condensation, reflecting surface heterogeneity or the presence of a pore size distribution. These features are most evident for the C8 sample, for which the rise is much more gradual. By assuming the pore wall of MCM-41 to be made up of eight equal sectors with different interactions for each sector, Maddox et al. (1997) have found that the critical diameter for reversible adsorption of nitrogen adsorption at 77.4 K is between 28 and 32 Å, which is closer to the experimental results and more realistic than the DFT value of around

2278 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 Table 2. Surface Area, Pore Volume, and Diameter of Various Samples by Adsorption of Different Gases sample and d spacing (Å) molecule

temp (K)

property

C8 (28.90)

C10 (29.15)

C12 (31.21)

C14 (34.74)

C16 (37.85)

C18 (43.27)

Ar

77.35

Ar

87.46

O2

77.35

SBET (m2/g) VP (cm3/g) SBET (m2/g) VP (cm3/g) SBET (m2/g) VP (cm3/g) SBET (m2/g) VP (cm3/g) SBET (m2/g) VP (cm3/g)

1100 0.34 873 0.38 1063 0.32 946 0.35 937 0.36

973 0.50 965 0.52 1150 0.42 868 0.39 1318 0.70

1027 0.54 963 0.61 1163 0.53 1123 0.59 1280 0.75

1015 0.88 954 0.65 888 0.52 1018 0.65 1162 0.79

895 0.79 1125 0.92 909 0.63 1013 0.72 1240 0.93

920 1.07 942 0.82 1043 0.88 1338 1.14 1123 0.98

CO2 N2

194.66 77.35

15-20 Å (Ravikovitch et al., 1995) obtained by assuming the surface to be homogeneous. As suggested earlier (Kruk et al., 1997), the interplanar spacing was found to be well-correlated with the condensation pressure in mesopores for nitrogen, argon, oxygen, and carbon dioxide. However, sample C8, where the lattice parameter is very close to that of C10, shows an inconsistent trend. There is a sharp decrease in the condensation pressure in going from C10 to C8, even though the lattice parameter has not been changed significantly (lattice parameters of C8 and C10 are 33.4 and 33.7 Å, respectively, and relative condensation pressures for nitrogen are 0.06 and 0.16, respectively). This behavior has been observed for all the gases studied here. This may indicate that the wall thickness of the sample C8 is large, yielding a small pore diameter for the same lattice parameter as C10. This suggests that the wall thickness changes regularly with the lattice parameter or remains constant. This is consistent with the molecular dynamics computer simulations of MCM-41 (Feuston and Higgins, 1994), which showed that, in order for the porous structure to be stable, the wall thickness has to increase when the distance between pores decreases. In general, the order of condensation pressure is CO2 (at 194.66 K) > N2 (at 77.4 K) > Ar (at 87.46 k) > O2 (at 77.4 K) with the exception of the sample C16 where the order is CO2 > Ar > N2 > O2 and C8 where the order is CO2 > Ar > N2 > O2. It was suggested (Llewellyn et al., 1994) that argon forms a meniscus in sample C12 more readily than nitrogen because of its smaller size and lack of quadrupole moment, avoiding a preferential orientation of the molecules within the adsorbate phase. The pore volume obtained by nitrogen adsorption in the present work is consistent with the literature. The values of pore volume for each gas are reported in Table 2. Our method of synthesis is similar to that of Beck et al. (1992) but different from those of Morishige et al. (1997) and Kruk et al. (1997). The pore volumes obtained in the current work are close to those of Morishige et al. (1997) but slightly different from those of Beck et al. (1997). The samples follow a consistent trend for pore volume; i.e., the pore volume increases with an increase in the number of carbon atoms used in the alkyl chain of the surfactant used in the synthesis of the samples with few exceptions. The slight difference in the pore volume of the current work as compared to the literature may be attributed to the difference in crystallinity because of slight differences in the synthesis/ calcining condition. The surface areas obtained by the BET method for all the samples using different adsorptives are shown in Table 2. Although the surface areas for all the samples using different adsorptives are in the range of

Table 3. Lennard-Jones Parameters and Other Properties of Various Adsorbates Used O2 Ar Ar CO2 (77.35 K) (77.35 K) (87.46 K) (194.66 K) σFF (Å) FF (K) σFS (Å) FS (K) PC (N/m2) × 103 TC (K) VC (m3/kmol) P0 (mmHg) Vsl (m3/kmol) γ∞ (N/m) area of cross section (Å2/molecule)

3.467 106.7 3.7167 51.8705 5043 154.6 0.0734 160 0.026531 0.015535 14.3

3.418 124 3.6 61.28 4898 150.9 0.0746 200 0.02757 0.01407 16.6

3.418 124 4.52 25.06 4898 150.9 0.0746 760 0.028728 0.011772 16.6

3.941 195.2 3.90 75.99 7380 304.1 0.0939 1434 0.035288 0.02554 22.2

900-1000 m2/g, there is no trend observed. It should be noted that the BET surface area is the total surface area rather than the mesopore surface area. We observed that the area by the t-plot method, which is actually a mesopore area, is close to the BET area, indicating the presence of insignificant secondary mesopores. The values of cross-sectional area of the different molecules used for finding the BET area are shown in Table 3. Good correlation of the BET equation could only be obtained for sample C8 over the entire range of 0.05-0.35 relative pressures, with some deviations for the other samples. Therefore, only the initial linear range of the BET plot was chosen for calculation of the surface area. As indicated by Kruk et al. (1997), the BET method may be very inaccurate for estimating the surface area for the small pore MCM-41 materials, as the primary mesopore volume is very close to the monolayer capacity. One of the interesting features of adsorption of argon at 77.4 and 87.6 K is the fact that the isotherm is very sharp at 77.4 K but is broader at 87.6 K. To verify if this is explained by the presence of a pore size distribution, we differentiated the modified Kelvin equation to yield

∆R )

2γ∞vsl ∆P*

0.6832 + 3.54 × 10-10 RgTP*(ln(P*)) (-ln(P*))1.4453P* (4) 2

where γ is the surface tension, vM is the molar volume, P* is the relative pressure, R is the gas constant, and T is the temperature. The first term in the above equation is obtained by differentiating the Kelvin equation, while the second term is obtained by differentiating the Halsey equation fitted to our experimental data for adsorption of nitrogen on a standard nonporous silica at 77.4 K. The values of ∆P* were obtained from the difference between the two sharp knees on the desorp-

Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2279 Table 4. Comparison of Diameters (Å) of the MCM-41 Samples Estimated by Various Methods Using Different Gases adsorbate

temp (K)

nitrogen

77.35

argon

77.35

argon

87.6

oxygen

77.35

carbon dioxide

194.6

sample

Saitofoley

C8 C10 C12 C14 C16 C18 C8 C10 C12 C14 C16 C18 C8 C10 C12 C14 C16 C18 C8 C10 C12 C14 C16 C18 C8 C10 C12 C14 C16 C18

18.8 25.3 28.1 34.8 41.2 50.8 18.7 21.5 23.6 33.8 39.0 51.3 19.1 21.4 24.6 28.3 35.4 37.8 18.6 22.6 25.6 31.8 36.5 46.0 17.5 20.5 22.7 26.7 29.4 34.5

BJH ads des

XRD

11.3 14.6 16.6 20.3 24.2 29.5 13.2 15.3 17.2 25.2 29.0 37.7 12.6 14.3 16.5 18.8 23.7 25.5 12.9 14.7 19.0 19.8 26.9 33.2 15.6 18.6 20.8 24.8 27.6 33.2

23.14 27.30 29.63 33.80 37.78 43.64 22.70 25.29 27.62 34.46 36.73 44.17 23.38 25.57 28.32 32.59 37.71 42.38 22.29 24.18 27.50 31.08 35.21 42.82 22.97 23.7 28.19 32.54 36.08 44.59

14.7 16.7 17.5 21.7 25.4 29.6 14.5 16.0 17.8 24.9 28.2 36.6 13.8 14.7 17.0 19.3 24.1 25.0 14.3 15.4 19.7 19.8 26.3 30.5 15.6 18.6 20.2 23.4 26.8 31.4

tion branch of the isotherm in the capillary condensation region, signifying the beginning and the end of the phase transition. For the six samples (C8, C10, C12, C14, C16, and C18), the values of ∆RP were obtained as 1.66, 1.45, 1.72, 1.45, 1.56, and 1.34 Å, respectively, from the 77.4 K data. On the other hand, at 87.6 K the values were 1.44, 1.06, 1.33, 2.10, 1.45, and 1.00 Å, respectively, in reasonable agreement with the 77.4 K estimates, indicating that the broadening of the capillary condensation region at 87.6 K as compared to 77.4 K is attributable to pore size distribution and temperature effects. 3.4. Failure of Conventional Models. The BJH method has been shown to underestimate the pore size of MCM-41 materials using nitrogen adsorption data (Ravikovitch et al., 1995; Kruk et al., 1997), but there has been no evidence of its applicability using other adsorbates. Another method, the extension of the Horvath-Kawazoe method to cylindrical pore geometry (Saito and Foley, 1991), was also used here. The results of application of the BJH method (using adsorption as well as desorption data), and the Saito-Foley method to MCM-41 materials in the range of 10-200 Å for adsorption of argon at 77.4 and 87.4 K, nitrogen and oxygen at 77.4 K, and carbon dioxide at 194.6 K, and a comparison with the experimental values of the pore diameter are shown in Table 4. Clearly, the BJH method uniformly underpredicts the diameter of MCM41, a discrepancy reconciled by incorporating the interaction potential (Bhatia and Sonwane, 1998; Sonwane and Bhatia, 1998). In general, the Saito-Foley method slightly overpredicts the diameter in the case of adsorption of argon, oxygen, and nitrogen at 77.4 K, even though the predictions for argon at 87.6 K are quite close to the experimental values. This method was found to significantly underpredict the diameter when

Figure 11. DA plots for various MCM-41 samples for adsorption of nitrogen at 77.4 K.

applied for adsorption of carbon dioxide at 194.6 K. Carbon dioxide is known for its chemisorption properties and has a high quadrupole moment, suggesting that its adsorption is very sensitive to the presence of polar groups. Therefore, a usual Lennard-Jones (LJ) type of potential model may not be applicable. The Dubinin-Astakhov (DA) equation is widely used as a model for micropore filling. The DA plots with optimum n for all the MCM-41 materials are shown in Figure 11. The values of the optimized exponent n were 4.25, 3.11, 4.34, 3.99, 4.69, and 2.88 for the samples C8, C10, C12, C14, C16, and C18, respectively. For optimization, the adsorption data of nitrogen up to P/P0 of 0.1 was used. In the case of samples whose condensation pressure is below or in the range of 0.1, the pressure up to the first knee of the isotherm was used for the optimization. It can be seen from Figure 11 that while the fit is not satisfactory for C8 and C14 samples, it is surprisingly good for sample C18 in spite of its large mesopores. This raises the possibility of the existence of micropores in the C18 sample, a feature already suggested by Maddox and Gubbins (1994). 3.5. Mathematical Modeling. In recent papers (Bhatia and Sonwane, 1998; Sonwane and Bhatia, 1998) we have proposed a new model which incorporates the molecular approach within the classical framework. In this model the equilibrium thickness t of the adsorbed layer at pressure Pg is given by

φ˜ (t,R) +

γ vs(R - t)

∫PP vg dP ) (R ∞-l t - λ/2)2 g

0

(5)

where R is radius of the pore and φ˜ (t,R) is the positiondependent incremental local potential due to the solid. The integral (second term) was obtained using the Benedict-Webb-Rubin (BWR) equation of state. The fluid-solid interaction potential parameters were obtained by fitting the condensation pressures satisfying the stability boundary (Sonwane and Bhatia, 1998) s dφ˜ γ∞vl (R - t + λ/2) ) dt (R - t - λ/2)2

(6)

to the MCM-41 data. The results of the fitting for

2280 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998

a

b

c

d

Figure 12. Experimental and fitted variation of condensation pressure with pore diameter on MCM-41 for (a) oxygen adsorption at 77.35 K, (b) argon adsorption at 77.35 K, (c) argon adsorption at 87.46 K, and (d) carbon dioxide adsorption at 194.66 K.

adsorption of oxygen at 77.4 K, argon at 77.4 and 87.4 K, and carbon dioxide at 194.6 K are shown in Figure 12a-d, with parameters used in the calculations included in Table 3. For the calculations the surface tension and molar volume of the adsorbate in saturated conditions were estimated from the corresponding states correlation and the Hankinson-Brobst-Thomson (HBT) equation (Reid et al., 1987), respectively. The LJ parameters for oxygen, carbon dioxide, and argon were taken from Reid et al. (1987). It can be seen that the fit for carbon dioxide is less satisfactory compared to those for oxygen and argon, which may be attributed to its high quadrupole moment and the possibility of chemisorption, as discussed earlier. The isotherm predictions are shown in Figures 7-10. The predictions for carbon dioxide adsorption deviated significantly from the data, with much better success for oxygen and argon adsorption at 77.4 K. Some deviation is also noted for argon adsorption at 87 K for extreme cases. This may perhaps be accounted for by consideration of the pore size distribution, a feature open for further studies. 3.6. Adsorption Hysteresis and Criticality. As seen from Figure 9, the adsorption isotherm of argon at 87.46 K on samples C8-C16 is hysteresis-free, but that of C18 shows a small hysteresis. The helium pycnometry results show that the density of sample C18 is very close to those of C14 and C16, indicating that in going from C16 to C18 there is some other factor which causes the hysteresis. The saturation pressure of argon at 87.46 K was taken as 760 mmHg. At this temperature the isotherms are not sharp as compared to adsorption at 77.35 K. Interestingly, samples C14 and C16, which did not clearly show hysteresis for argon adsorption at 87.4 K, show very well-defined hysteresis at 77.4 K for the same adsorbate. Our results of argon adsorption at 77.4 are consistent with those of Llewellyn et al. (1994). As the temperature 77.4 K is below the triple point of argon (88.8 K), there is some doubt as to the choice of an appropriate standard state, with the choice of P0 as the saturation pressure of solid found to yield inaccurate results with adsorption on nonporous

materials (Gregg and Sing, 1982), and the alternative choice as the vapor pressure of the supercooled liquid, which is 220 mmHg at 77.2 K, being questioned in recent years. Therefore, in the present calculations, the value of P0 was measured by the adsorption instrument (ASAP 2010) and found to be 200 mmHg. For the adsorption of carbon dioxide at 194.7 K, we could observe hysteresis even for C8, but the extent was very small. Surprisingly, this hysteresis extended down to very low pressure. In Figure 10 the effective saturation pressure, P0, is taken as that of supercooled liquid with P0 ) 1434 mmHg (Branton et al., 1995b). Unlike the results reported by Branton et al. (1995b) for CO2 adsorption on the C16 sample at 194.7 K, we observed a complete isotherm with plateau, consistent with the results of Morishige et al. (1997). Interestingly, for the C16 sample, Morishige et al. (1997) did not observe any hysteresis at 194.6 K, which is very well-defined in our results. The results of oxygen adsorption at 77.4 K shows that samples C14, C16, and C18 show very welldefined hysteresis. The saturated pressure of oxygen at 77.4 K was taken as 160 mmHg. Interestingly, in all the above isotherms it can be observed that the hysteresis loop expands and becomes well-defined (with almost vertical lines for the isotherm in the capillary condensation region) as the pore size increases. It has been reported (Schulz-Ekloff et al., 1997) that some of the samples of MCM-41 show a large hysteresis due to condensation in external (or intercrystalline) pores, with the hysteresis closing before desorption from the intercrystalline pores. Kruk et al. (1997) also reported that significant secondary mesopores are present in the C16 and C8 samples. In the present work, out of the six samples studied only C8 and C18 showed hysteresis due to condensation in external pores. Results of these findings along with the macrostructure characterization will be discussed in future publications. It has been reported that the surface of MCM-41 is heterogeneous due to 20-30% of silanol coverage and/ or other unknown factors (Beck et al., 1992; Branton et al., 1995a,b; Franke et al., 1993). Also by high-resolution electron microscopy, it has been reported that the shape of the pores is hexagonal (Alfredsson et al., 1994). By matching the small-angle X-ray scattering data, it has been shown that the walls of MCM-41 materials are made up of two different kinds of materials: highdensity and low-density silica phases (Edler et al., 1997). All these factors suggest that the surface of MCM-41 may be highly heterogeneous. Recently, it was suggested that surface heterogeneity may lead to the condensation transition being manifested instead as a continuous pore filling, based on the simulation of argon adsorption in slitlike pores (Karykowski et al., 1994). On the basis of a heterogeneous potential, Maddox et al. (1997) have reported a value of critical pore diameter of 28-32 Å, which is closer to the experimental value of about 40 Å than the DFT value of 15-20 Å based on a homogeneous potential model (Ravikovitch et al., 1995). In our recent article (Sonwane and Bhatia, 1998) we showed that there are three kinds of criticalities existing during adsorption-desorption in the pores. The first is the condensation criticality, which occurs because of the instability of the meniscus during adsorption. We derived eq 6 as a modification of the classical Cohan relationship, with a first-order condensation phase transition occurring at the corresponding solution tc

Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2281 Table 5. Limiting Pore Diameter (Å) for Different Adsorptives O2 Ar Ar CO2 (77.35 K) (77.35 K) (87.46 K) (194.66 K)

mechanism capillary coexistence multilayer formation mechanical stability of the hemispherical meniscus

24.0 17.6 25.0

22.7 20.0 25.0

23.0 18.7 30.2

34.3 21.4 40.0

when it exists. Under the condition of t < tc, the interface is stable. With eq 6 we found that there exists a critical pore size below which the phenomenon of multilayer adsorption followed by phase transition does not occur. That pore size satisfies s d2φ˜ 2γ∞vl (R - t + λ/2) ) dt2 (R - t - λ/2)4

(7)

The above relation, in combination with eqs 5 and 6, yields the critical values shown in Table 5. The values for different gases in MCM-41 are consistently in the range of 17-20 Å pore diameter and represent the critical pore diameter below which adsorption proceeds by volume filling. The second criticality predicted by our model was for the desorption branch of the isotherm. There exists a pore diameter below which the phenomenon of capillary evaporation (which involves changing of the hemispherical meniscus to a cylindrical meniscus) does not occur. The following relation was obtained (Sonwane and Bhatia, 1998) for the capillary coexistence curve which is to be solved in conjunction with eq 5:

φ˜ (R,R) +

∫P

Pg 0

vg dP -

2γ∞vsl (R - t)2

) [(R - t)(R - t - λ) + λσFF/4] 0 (8)

The critical diameters predicted by the above equation were larger than those obtained from the adsorption branch. The critical diameter obtained for carbon dioxide adsorption was exceptionally high (34 Å). The values for other adsorbates are reported in Table 5. A third criticality is obtained by using the YoungLaplace equation

Pg - Pl )

2γ∞ R - t - λ/2

(9)

in conjunction with our model and using the tensile stress hypothesis (Burgess and Everett, 1970) by virtue of which Pl > -τ0, where is τ0 is the maximum tensile stress obtained from the metastable region. Through this approach, it was found in our earlier paper that the limiting pore size for hysteresis to occur for nitrogen adsorption was around 33 Å. Similar calculations were done for oxygen, argon, and carbon dioxide. The results are shown in Table 5, with all the values consistent with our experimental findings. The predicted limiting pore diameter as a function of temperature is plotted in Figure 13. For this curve, the fitted Lennard-Jones potential parameters obtained for the adsorption of argon at 77.4 K were used. The dotted straight line represents the semiempirical equation proposed by Rathousky et al. (1995) based on the adsorption of cyclopentane at different temperatures. It can be seen that predictions by the current model are more satisfactory, based on the extensive experimental data on

Figure 13. Variation of critical temperature for isotherm reversibility with pore diameter (s, current work; - -, Rathousky et al., 1995; NP, nonporous; M, Morishige et al. (1997); C, current work).

nonporous materials as well as the MCM-41 materials for different adsorbates. The interpretation of this critical size is that for smaller pores the hemispherical meniscus is mechanically unstable during desorption and the latter therefore occurs through the reversal of the adsorption branch. This critical size is always larger than that for thermodynamic coexistence, so in practice the latter criticality does not affect the desorption. 4. Conclusions MCM-41 materials of six different pore sizes are successfully prepared and characterized by XRD, TEM, helium pycnometry, SANS, and gas adsorption (argon at 77.4 and 87.4 K, nitrogen and oxygen at 77.4 K, and carbon dioxide at 194.6 K). Based on the skeletal density, it was concluded that the wall structure of MCM-41 materials may be close to that of amorphous silica. A recent molecular-continuum model, utilized earlier by the authors for nitrogen adsorption, was successfully extended here for other gases and even for different temperatures and found to predict the isotherms satisfactorily. However, the predictions for carbon dioxide were unsatisfactory, possibly due to the high quadrupole moment and chemisorption on the surface of MCM-41. The model was also found to predict the critical diameters close to the experimental values obtained for MCM-41 materials. Estimates of pore diameter by various gas adsorption data were found to be very close to the diameter obtained from the XRD lattice spacing using 10 Å wall thickness but significantly different from the diameter of the micelles and predictions by other widely used models (e.g., BJH, Saito-Foley). It is found that the Dubinin-Astakhov model fails even for the smallest diameter MCM-41 materials. Acknowledgment The research has been supported by a grant (No. 97/ UQ NSRG004G) from The University of Queensland. The award of an overseas Postgraduate Research Scholarship to one of us (C.G.S.) by the University of

2282 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998

Queensland is also gratefully acknowledged. Dr. Richard Heenan and the LOQ facility of ISIS, RutherfordAppleton Laboratory, Chilton (Oxon), U.K., and the access to Major Facilities Research Program of the Australian Nuclear Science and Technology Organization are acknowledged for their essential role in collection and processing of SANS data. Notation a ) across-sectional area of the head of the surfactant d ) diameter of the molecule dXRD ) d spacing D ) diameter of the pore Dm ) diameter of the micelle lc ) length of the aliphatic chain n ) number of carbon atoms in the surfactant alkyl chain P*) relative pressure (Pg/P0) Pg ) gas-phase pressure P0 ) vapor pressure Rg ) universal gas constant R ) pore radius t ) thickness of the adsorbate T ) temperature TPC ) pore critical temperature TC ) bulk critical temperature v ) volume of headgroup of the surfactant vg ) volume of the fluid vsl ) saturated molar volume VP ) mesopore volume Greek Letters Ft ) skeletal density λ ) interlayer spacing τ0 ) maximum tensile stress γ∞ ) surface tension for infinite radius of curvature φ˜ (t,R) ) position dependent incremental potential

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Received for review December 1, 1997 Revised manuscript received March 20, 1998 Accepted March 25, 1998 IE970883B