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Structure of the Mesoporous Silica SBA-2, Determined by a Percolation Analysis of Adsorption Manuel Pe´rez-Mendoza,†,‡ Jorge Gonzalez,§ Paul A. Wright,§ and Nigel A. Seaton*,† School of Chemistry, University of St Andrews, North Haugh, St Andrews KY16 9ST, U.K., and Institute for Materials and Processes, School of Engineering and Electronics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, U.K. Received March 16, 2004. In Final Form: July 26, 2004 We have carried out a percolation analysis of the adsorption of ethane and nitrogen in SBA-2, a structured mesoporous silica consisting of a hexagonal close-packed (hcp) array of spherical cavities connected by cylindrical channels. Our analysis explains the different uptakes of nitrogen and ethane in terms of the greater accessibility of the network to the smaller nitrogen molecule. The analysis also allows us to quantify the connectivity of the SBA-2 pore network. The effective coordination number of the cavities, defined as the average number of channels per cavity that are large enough to allow nitrogen to pass, is 4.9, much less than the theoretical maximum value of 12. Taking into account only the smaller set of channels large enough to admit ethane, the effective coordination number is 1.8, just above the percolation threshold of the network.
1. Introduction Periodic mesoporous silica materials, such as the MCM and SBA families, are obtained by forming silica in the aqueous phase of a templating micellar solution. Once the surfactant forming the micellar phase is removed, by either calcination or extraction, the resultant solid has a long-range-ordered porous structure reflecting the regular arrangement of the micelles in the micellar solution.1-3 Depending on the surfactant and the synthesis conditions, solids with different pore networks are obtained. The pore network of SBA-2 consists of spherical cavities, arranged in a hexagonal close-packed (hcp) array (space group P63/ mmc); the cavities are connected in three dimensions by small microporous channels.4-6 Such an apparently well connected pore structure makes SBA-2 of interest for applications in adsorption and catalysis.7 However, the way in which the channels connect the neighboring cavities is as yet unclear. These channels seem to be created during the removal of the surfactant at the calcination stage of the process, and thus, it is unlikely that all the connections will be of the same size and that the cavities will be connected following a regular pattern. The connectivity of a porous solid has a strong effect on diffusion rates8 * Corresponding author. † University of Edinburgh. ‡ Current address: Instituto de Carboquı´mica-CSIC. C/ Miguel Luesma Casta´n,4. 50018 Zaragoza, Spain. § University of St Andrews. (1) Huo, Q. S.; Margolese, D. I.; Stucky, G. D. Chem. Mater. 1996, 8 (5), 1147-1160. (2) Sakamoto, Y.; Diaz, I.; Terasaki, O.; Zhao, D. Y.; Perez-Pariente, J.; Kim, J. M.; Stucky, G. D. J. Phys. Chem. B 2002, 106 (12), 31183123. (3) Sakamoto, Y. H.; Kaneda, M.; Terasaki, O.; Zhao, D. Y.; Kim, J. M.; Stucky, G.; Shim, H. J.; Ryoo, R. Nature 2000, 408 (6811), 449-453. (4) Hunter, H. M. A.; Garcia-Bennett, A. E.; Shannon, I. J.; Zhou, W. Z.; Wright, P. A. J. Mater. Chem. 2002, 12 (1), 20-23. (5) Zhou, W. Z.; Hunter, H. M. A.; Wright, P. A.; Ge, Q. F.; Thomas, J. M. J. Phys. Chem. B 1998, 102 (36), 6933-6936. (6) Zhou, W. Z.; Garcia-Bennett, A. E.; Hunter, H. M. A.; Wright, P. A. Stud. Surf. Sci. Catal. 2002, 141, 379. (7) Hunter, H. M.; Wright, P. A. Microporous Mesoporous Mater. 2001, 43 (3), 361-373. (8) Liu, H. L.; Zhang, L.; Seaton, N. A. Langmuir 1993, 9 (10), 25762582.
Figure 1. PSD obtained from the analysis of ethane adsorption at 263 K.
and, hence, on the performance of the solid as an adsorbent or catalyst support. It is thus important to be able to characterize the pore-network connectivity of SBA-2 and other ordered silicas. In a previous communication,9 we described the adsorption of ethane and methane on SBA-2 using a model in which both the cavities and the interconnecting channels are described by a pore size distribution (PSD) with the cavities and channels represented as spheres and cylinders, respectively. The PSD f(d) ()dV/dd, where V is the pore volume and d is the diameter of the channel or cavity) was obtained by using a Monte Carlo simulationbased method; this PSD is reproduced in Figure 1. The smaller pores (3-35 Å in diameter) are the cylindrical channels, while the larger pores (36-50 Å in diameter) are the cavities. The pores were considered to be independent, in the sense that the interfaces between the pores were ignored; the spheres have complete surfaces, without openings to neighboring channels, and the channels are modeled using periodic boundary conditions along the axis of the channel. No account was taken of the effect of porenetwork connectivity: all pores were considered to be (9) Perez-Mendoza, M.; Gonzalez, J.; Wright, P. A.; Seaton, N. A. Langmuir 2004, 20, 7653.
10.1021/la0493159 CCC: $27.50 © 2004 American Chemical Society Published on Web 09/28/2004
Structure of the Mesoporous Silica SBA-2
accessible to all adsorptives. These simulations of adsorption in pores of different sizes, combined with experimental data for ethane adsorption at 264 K up to high pressures, were used to obtain the PSD shown in Figure 1 using the method of Davies and Seaton.10,11 This model is capable of accurately predicting the adsorption of ethane and methane at different temperatures, again using grand canonical Monte Carlo (GCMC) simulation to model adsorption in individual pores, using only ethane adsorption data at a single temperature to obtain the PSD. Despite this success, evidence of a pore-network connectivity effect was seen in the comparison between methane and ethane adsorption, where the PSD obtained from ethane adsorption underpredicted methane adsorption, suggesting that the network has a different accessibility to the two adsorptives, due to the presence of constrictions in the pore network that allow the passage of the smaller species but not the larger one. This effect has also been observed in the comparison of nitrogen adsorption with the adsorption of the organic species n-hexane, cyclopentane, and mesitylene on SBA-2,12 where the adsorbent had much lower capacities for the organic species than for nitrogen. Therefore, it is of interest to carry out a more thorough pore-geometry and network-connectivity analysis of adsorption on SBA-2. To this end, we have made further experimental measurements of nitrogen adsorption at 77 K, the standard temperature for this type of experiment. As the nitrogen molecule is much smaller than the ethane molecule, the comparison between nitrogen and ethane adsorption would be expected to show a much larger connectivity effect (with the nitrogen penetrating the pore network more completely than the ethane) than that in the methane/ethane comparison. An appropriate way to address such an analysis is by considering adsorption as a percolation process and using percolation theory (see, e.g., ref 13) to analyze the connectivity of the network. We begin by defining the pore network in terms of the variables of percolation theory. SBA-2 has a three-dimensional network consisting of cavities (“sites” in the language of percolation) linked by channels (“bonds”). In the adsorption problem we are studying, some cavities and channels are large enough to accommodate a particular adsorptive species, while others are not; in the corresponding percolation problem, some sites and bonds are “occupied”, while others are not. When the fractions of cavities and channels that are large enough to accommodate a particular species (the “site occupation probability” and the “bond occupation probability”, respectively) are sufficiently large, the network “percolates”, which is to say that the adsorptive can penetrate from one side of the network to the other or, in adsorption terms, that all macroscopic regions of the adsorbent are accessible to the adsorptive species. (However, this does not imply that all the pores that are large enough to accommodate the adsorptive actually do so, since except at very high site and bond occupation probabilities some pores are isolated from the percolating cluster of accessible pores.) 2. Monte Carlo Simulation in Individual Pores As in our earlier simulation study of adsorption in SBA2,9 the GCMC simulation method was used to simulate (10) Davies, G. M.; Seaton, N. A.; Vassiliadis, V. S. Langmuir 1999, 15 (23), 8235-8245. (11) Davies, G. M.; Seaton, N. A. Langmuir 1999, 15 (19), 62636276. (12) Garcia-Bennett, A. E.; Williamson, S.; Wright, P. A.; Shannon, I. J. J. Mater. Chem. 2002, 12 (12), 3533-3540. (13) Stauffer, D.; Aharony, A. Introduction to Percolation Theory, 2nd ed.; Taylor and Francis: London, 1992.
Langmuir, Vol. 20, No. 22, 2004 9857 Table 1. Interaction Parameters Used in GCMC Simulationsa molecule
/κB, K
σ, Å
CH4 C2H6 (2CLJ) N2 (2CLJ) O
149.92 139.8 33.4 185
3.7327 3.512 3.383 2.708
quad. moment, C‚m2
bond length, Å
-3.712
2.353 0.923
a and σ are the Lennard-Jones parameters for the individual sites (one site for methane and oxygen and two sites for ethane and nitrogen), and the bond length is the distance between the centers of the sites in the two-site molecules.
adsorption, in this case of nitrogen at 77 K. A complete description of the simulation method is given in our earlier paper. In summary, both the spherical cavities and cylindrical channels consist of a regular array of three concentric layers of oxygen atoms. The adsorptive molecules have pairwise interactions with the atoms in the pore wall. The contribution of silicon atoms toward the adsorption is known to be small for silicas,14 so the silicon atoms are neglected, with the small contribution made by the silicon atoms to the adsorption potential implicitly included in the potential parameters of the oxygen atoms in the solid. The oxygen atoms were represented by Lennard-Jones sites, with the same potential parameters as those used previously for the simulation of adsorption on similar silica materials.15 Nitrogen was modeled as a two-site LennardJones molecule with a quadrupole, following Kjems and Dolling.16 The Lorentz-Berthelot rules were applied to calculate the mixed Lennard-Jones parameters. All parameters employed in the nitrogen simulation together with those previously used to simulate ethane adsorption are given in Table 1. The cutoff distance was 18.6 Å. 3. Percolation Analysis of the Accessibility of the Pore Network The cavities in SBA-2 are arranged in a hcp array, so this is also the structure of the network in which percolation occurs; the network has an underlying coordination number of Z ) 12. This is thus the maximum number of connections a site in the network can have to its neighbors. In practice, some of the channels might not be present and some of those that are present might be too small to admit nitrogen (as nitrogen is the smallest probe molecule we used, we cannot in any case distinguish between these two categories). For the moment, we assume that all the channels are present; we will return to this assumption later. As the cavities are much larger than the probe molecules, it is clear that the channels control access to the pore space of the material, so in percolation terms, it is the bonds that determine the accessibility of the network. Thus, for this particular percolation problem, all the sites are occupied, while only some of the bonds (those corresponding to channels that are large enough to accommodate the adsorptive under consideration) are occupied. In our case, ethane is the larger molecule, which experimentally “sees” a smaller proportion of the pore network than nitrogen. Ethane is excluded from both cavities and channels because it is too large to pass through some of the channels. (Although we are modeling the channels as straight cylinders, the real shape is certainly more disordered, so the increased length of the ethane (14) Bezus, A. G.; Kiselev, A. V.; Lopatkin, A. A.; Du, P. Q. J. Chem. Soc., Faraday Trans. 2 1978, 74, 367-379. (15) Yun, J. H.; Duren, T.; Keil, F. J.; Sexton, N. A. Langmuir 2002, 18 (7), 2693-2701. (16) Kjems, J. K.; Dolling, G. Phys. Rev. B 1975, 11 (4), 1639-1647.
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Figure 2. Schematic diagram showing the connectivity in a 2D lattice for large (ethane) and small (nitrogen) molecules. The relative sizes of the cavities, channels, and molecules have been modified for the sake of clarity.
molecule compared with nitrogen is likely to have a significant impact on its ability to penetrate narrow channels.) Some of the channels that are inaccessible to ethane are nevertheless large enough to accommodate ethane molecules but simply isolated from the bulk gas by channels that are too small to allow the ethane to pass. The cavities, in contrast, are all large enough to accommodate ethane, but some will be isolated by channels that are smaller than the ethane molecules. This effect is illustrated in Figure 2. As nitrogen is the smallest adsorptive used in this study (Table 1), we assume for the moment that it probes the whole pore network, that is, that all cavities and channels are accessible to it. Reformulating this physical description in terms of percolation theory, we have (i) pore volume is associated with both bonds and sites, (ii) all the sites are occupied (the site occupation probability, ps ) 1), (iii) only some of the bonds are occupied (the bond occupation probability, pb < 1). Clearly, the accessibility of both the bonds and the sites depends on the bond occupation probability. Thus, the percolation analysis of the SBA-2 lattice comprises two different problems depending on whether attention is focused on the sites or on the bonds: (i) Adsorption in the channels corresponds to the bond percolation problem.13 The fraction of accessible bonds, Pb (corresponding to the fraction of pores in which ethane can occur) is a function of pb. This function has been computed by Yanuka17 for the face-centered cubic (fcc) lattice, which has the same coordination number (Z ) 12) as the hcp lattice. It is known that the percolation properties of a network depend to a very good approximation on the dimensionality of the network and on the coordination number of the network, and only to a small degree on other aspects of the topology of the network.18 This is confirmed by the very similar values of the percolation threshold for the hcp and fcc networks.18 Exploiting this principle of “dimensional invariance”, we can use the fcc results of Yanuka as a substitute for the corresponding hcp results. Yanuka’s results are shown in Figure 3a. (17) Yanuka, M. Percolation theory and capillarity in relation to pore geometry and topology. Ph.D. Thesis, University of Guelph, Guelph, Canada, 1984. (18) Lorenz, C. D.; May, R.; Ziff, R. M. J. Stat. Phys. 2000, 98 (3-4), 961-970.
Figure 3. Percolation functions for (a) the bond problem and (b) site percolation in the bond problem in fcc lattices.
(ii) The connection between adsorption in the cavities and percolation theory is more complex. Here, the accessibility of the sites, Ps, depends not on the site occupation probability, ps, but on the bond occupation probability, pb. This is “site percolation in the bond problem”, which has also been studied by Yanuka; this function is shown in Figure 3b. (Again, we apply fcc results to our hcp network.) The accessibility of the bonds, Pb, is related to the PSD of the channels, fch(w) ()dVc/dd, where Vc is the channel volume) by19 b
P
∫0∞f ech(d) dd ) ∞ ∫0 f nch(d) dd
(1)
where f ech(d) is the channel PSD detected by ethane and f nch(d) is the channel PSD detected by nitrogen. The accessibility of the sites, Ps, is related to the PSD of the cavities by s
P
∫0∞f eca(d) dd ) ∞ ∫0 f nca(d) dd
(2)
where f eca(d) is the spherical cavity PSD detected by ethane and f nca(d) is the cavity PSD detected by nitrogen. In writing these equations, we are assuming, for the moment, that all the channels and cavities are accessible to nitrogen molecules. Strictly speaking, these integrals should be over the pore number distribution (i.e., dN/dd, where N is the number of pores), rather than the pore volume distribution (i.e., the PSD as we have defined it, dV/dd). It can be (19) Lopez-Ramon, M. V.; Jagiello, J.; Bandosz, T. J.; Seaton, N. A. Langmuir 1997, 13 (16), 4435-4445.
Structure of the Mesoporous Silica SBA-2
Figure 4. Experimental (symbols) and simulated (lines) adsorption of nitrogen at 77 K. Dashed line: prediction from the PSD obtained from the analysis of ethane adsorption. Solid line: isotherm obtained by fitting the PSD to nitrogen adsorption data.
Figure 5. Comparison of the PSDs obtained by analyzing ethane and nitrogen adsorption data.
shown19 that the two distributions are equivalent if, in the pore size range in which the pores are big enough to accommodate both species (i.e., that a PSD is detected by both species), the pore diameters are uncorrelated with their accessibility; this is equivalent to the reasonable assumption that the sizes of the channels and spheres are uncorrelated with their position in the network.
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Using eqs 1 and 2 coupled with the PSDs shown in Figure 5, we obtain the following values for the bond and site accessibilities: Pb ) 0.32 and Ps ) 0.93. We are now in a position to fit the bond occupation probability, pb, to the accessibility results of Yanuka,17 shown in Figure 3. In principle, either Pb or Ps could be used to determine the value of pb. However, as the bulk of the pore volume is in the cavities, it is likely that the error in the channel PSD is relatively high (as a big change in the PSD has only a small effect on the absolute adsorption in the material as a whole) and therefore that the cavity PSD is the more reliable one. Thus, from Figure 3b, Ps ) 0.93 corresponds to pb ) 0.16. In adsorption terms, this means that only 16% of the channels linking the cavities will allow passage of ethane, while the rest will be blocked for this adsorptive. Put another way, the “effective coordination number” of the portion of the network that is accessible to ethane is not 12 for this adsorptive, but instead, it is 0.16 × 12 ) 1.8. As the accessibility function is very steep in this region, any uncertainty in the value of Ps has only a small effect on the value of pb. The percolation threshold, below which the network is disconnected, is pb ) 0.120,18 so the network is just above its percolation threshold for the passage of ethane. We now use this value of pb to calculate the corresponding value of Pb from Yanuka’s accessibility data for the bond problem (Figure 3a). This gives Pb ) 0.13 instead of the value of 0.32 calculated from the comparison of the ethane and nitrogen channel PSDs (f ech(d) and f nca(d)) using eq 1. The most natural explanation for this is that the denominator in eq 1 is too large, suggesting that (contrary to our assumption) not all the channels are accessible even to nitrogen or, in other words, that, as far as can be detected by the smaller of our probe molecules, some of the channels are missing. (This does not imply, of course, that they would necessarily be detected as missing by a smaller probe molecule, e.g., hydrogen.) This can be quantified as follows. The mean number of channels per cavity accessible to species i is
nich ) ZPb,i
(3)
where Z ) 12 for our hcp network. For ethane, Pb,e ) 0.13 and nech ) 1.56. The ratio of the mean number of channels accessible to the two species is
4. Results and Discussion
nech
We begin by using the PSD obtained earlier9 from the analysis of ethane adsorption at 263 K (Figure 1) to predict nitrogen adsorption at 77 K. (The fit between the simulated and experimental ethane isotherms is shown in ref 9.) The predicted adsorption is shown in Figure 4 (dashed line) along with the experimental data (symbols). Although the shape of the isotherm is correct, the amount adsorbed is too low everywhere. To investigate the reason for this, a new PSD based on the analysis of nitrogen, rather than ethane, adsorption was obtained. A comparison of these PSDs is shown in Figure 5. The maxima of the PSDs are coincident, but the PSD from the nitrogen adsorption data is higher at all values of the pore size, indicating a higher accessibility of the pore network to the smaller nitrogen molecules. The simulated nitrogen adsorption generated using the new PSD is given by the solid line in Figure 4, showing a very good fit between simulation and experiment. This comparison between the PSDs obtained from ethane adsorption at 263 K and nitrogen adsorption at 77 K provides strong evidence that adsorption in this material should indeed be treated as a percolation process.
nnch
)
Pb,e Pb,n
(4)
The accessibility is related to the PSD by eq 1, giving
∫0∞f ech(d) dd ) 0.32 ) nnch ∫∞f nch(d) dd 0
nech
(5)
Substituting for nech ) 1.56, we get nnch ) 4.9 and, from eq 3, Pb,n ) 0.41. Figure 3 shows that the network is far above its percolation threshold at this value of the bond accessibility, so that the corresponding bond occupation probability is pb,n ) 0.41. Returning to the language of adsorption, 41% of the channels are large enough to accommodate nitrogen, while as we found above 16% of the channels are large enough to accommodate ethane (pb,e ) 0.16). We conclude with a speculative comment relating to the synthesis of the SBA-2 material. We recall that the channels seem to be formed by the removal of the
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surfactant molecules from the cavities, which in the case of the sample we studied is by calcination. During calcination, the surfactant molecules are broken down into smaller units, which must then force at least one channel to a neighboring cavity. The main reaction products that evolved during the calcination of our SBA-2 material are trimethylamine, propane diamine, and fragments of the hydrocarbon tail of the surfactant molecules.20 The reaction products then diffuse from cavity to cavity, within the percolating cluster of cavities, until they reach the surface of the particle. We can thus see that the removal of the surfactant is a percolation process, in which the role of accessibility is the same as that in adsorption. (The dynamics of the processes are, however, completely different: the calcination process creates the topology, while the adsorption process obeys it.) The effective coordination number of the network accessible to ethane was found to be 1.8, just above the percolation threshold. As the reaction products are either of a similar size to the ethane molecules (in the case of the hydrocarbon fragments) or of a slightly larger size, the accessibility of the network to the reaction products and to ethane should be similar. This suggests that during the calcination the reaction products create a network which is only slightly above the percolation threshold for ethane adsorption. Provided the calcination process is sufficiently slow (that (20) Hunter, H. M. A.; Wright, P. A. Microporous Mesoporous Mater. 2001, 43, 361-273.
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is, that the characteristic time for reaction is less than the characteristic time for diffusion through the developing pore network), this is exactly what would be expected on physical grounds. 5. Conclusions Our percolation analysis of adsorption in the SBA-2 pore network, coupled with GCMC simulations of the adsorption of nitrogen and ethane, has provided new information about the connectivity of the SBA-2 pore network. The effective coordination number of the cavities, taking into account channels that are large enough to permit the passage of nitrogen, is 4.9. Taking into account only the smaller set of channels large enough to admit ethane, we found that the effective coordination number is 1.8. Combined with our earlier paper,9 this work gives a rather full picture of the geometry and connectivity of the SBA-2 pore network. This approach is generalizable to other adsorbents with a regular pore structure consisting of cavities linked by channels. Acknowledgment. The authors acknowledge the financial support of the U.K. Engineering and Physical Sciences Research Council. M.P.-M. gratefully acknowledges the financial support from the MECyD (Spain). LA0493159
