J. Phys. Chem. B 2000, 104, 11435-11439
11435
Sizing of Cylindrical Pores by Nitrogen and Benzene Vapor Adsorption C. Nguyen and D. D. Do* Department of Chemical Engineering, The UniVersity of Queensland, St. Lucia, 4072 QLD, Australia ReceiVed: February 24, 2000; In Final Form: August 2, 2000
A new method for sizing the diameter of cylindrical mesopores is presented in this paper. The effects of the pore dimension and the cylindrical shape on adsorption in mesopores are accounted for by using the concepts of enhanced pore pressure and enhanced surface layering. In conjunction with the conventional Kelvin equation, this treatment is capable of modeling the capillary condensation process in mesopores. The model is tested against adsorption data of benzene and nitrogen onto MCM-41 samples. The isotherm model fitting is satisfactory. The sizing of the pore in MCM-41 using this approach does not suffer from the pore size underestimation problem experienced with the traditional Kelvin equation.
Introduction
Adsorption in Porous Media
Capillary condensation is one of the characteristic features of adsorption in mesopores. This process is usually described using the Kelvin and Cohan equations for desorption and adsorption, respectively.1 Either of the equations in conjunction with an equation for the statistical thickness of the adsorbed layer t is commonly used for mesopore size characterization.2 The combination of the classical Kelvin equation with an equation for the t thickness is usually referred to as the modified Kelvin equation. It is, however, known that this combined equation suffers from some inherent incapability. It underestimates the diameter of narrower pores.3,4 The narrower the pore, the larger the error. It has been established that the lower limit of application range of the modified Kelvin equation for nitrogen adsorption at 77 K is about 7.5 nm.5 There are numerous explanations for the limitation of the modified Kelvin equation. They are based on arguments such as that the meniscus existence in smaller pores is questionable, and that the liquid properties of the adsorbed phase in narrow mesopores are different from those of the bulk liquid at the same temperature.6 There are also arguments concerning the layering process in mesopores,4,7 for example at a given pressure the thickness of the adsorbed layer in narrower mesopores is greater than that occurring on a flat surface. Many attempts were made to “correct” the Kelvin equation.6,8 Recently we have proposed a model of adsorption in carbonaceous adsorbents,9,10 that is, materials with slitlike pores. The adsorption process is modeled as a combination of various processes, that are affected by the interference of the proximity of the pore walls. The model was applied successfully on the adsorption of nitrogen at 77 K as well as that of other adsorbates at near-ambient temperatures.10 The above model is now extended to cylindrical pores. The subject porous material in this study is MCM-41, a silica-based adsorbent known for its narrow mesopore size distribution.11 This adsorbent has recently been subjected to many studies.12 In this work, benzene and nitrogen are used as the probe molecules to investigate the pore structure of MCM-41.
Enhancement of Adsorption in Pores. It is well-known that adsorption in pores, especially in micropores, is enhanced by the proximity of the pore walls.2 This is because the force fields of the pore walls overlap when they are sufficiently close to each other. Theoretically, the enhanced potential of a molecule in a pore can be several (up to ∼3.5) times higher than its potential with a flat surface of the same solid structure. In our previous papers, we argued that the potential energy enhancement can be translated into the adsorption modeling by allowing parameters of different adsorption processes to bear some dependence on the adsorbent-adsorbate interaction, which is pore size dependent. The potential enhancing effects on the gas and adsorbed phases can be accounted for by introducing concepts of the pore-enhanced pressure and the enhanced surface layering, respectively. The former implies that gas-phase molecules are attracted and literally “compressed” against each other by the pore walls, whereas the latter means that adsorbed molecules are held more strongly in pores than onto a flat surface. These concepts are presented in details elsewhere.9,10 Adsorption in Porous Media. On the basis of the concept of the enhanced pore pressure, adsorption in pores can be pictured as a process whereby gas-phase molecules are drawn into the pore interior. Once inside, they are further pressed against each other as a result of the overlapped potential forces. If the enhanced pressure is beyond the corresponding vapor pressure, the adsorbed phase turns into liquid (albeit compressed liquid). It is, therefore, understood that in some narrow pores, because of the very large enhanced pressures, the adsorbed phase exists in the liquid form even at low bulk pressures. This liquid filling process progresses to pores larger in size as the bulk pressure increases. This resembles the micropore filling process described in the Dubinin theory,13,14 and to some extent even the Horvath-Kawazoe approach.15 Parallel to this, in larger pores, adsorption occurs as a surface-layering process until a stage when the enhanced pressure reaches the vapor pressure value or the conditions for the pore filling (for mesopores this is termed the capillary condensation) to take place are satisfied. Model Equations. The pore-enhanced pressure is calculated using the following equation:
* Corresponding author.
10.1021/jp0007282 CCC: $19.00 © 2000 American Chemical Society Published on Web 10/26/2000
11436 J. Phys. Chem. B, Vol. 104, No. 47, 2000
pp(r) ) pb exp
[
]
-〈Egp(r)〉 RT
Nguyen and Do
(1)
with pb being the bulk gas pressure, R the gas constant, T the system temperature, and 〈Egp〉 the average potential of the pore gas-phase molecules (which is pore size dependent). This potential energy is estimated in this work using the potential energy at the pore center Egp. The statistical thickness of the adsorbed layer is described by means of a Brunauer-Emmett-Teller (BET) equation, and the enhancement effect is accounted for via the BET coefficient Cp as follows:
[
]
Qp(r) - Qs Cp(r) ) Cs exp RT
(2)
with Cs being the BET coefficient for adsorption on a flat surface, and Qp and Qs are the heats of adsorption at zero loading in a pore and on a flat surface, respectively. Equation 2 shows that the enhanced adsorption affinity coefficient is a function of the difference between the heats of adsorption. This difference is contributed mainly by the difference between the adsorption potential energy in a pore and that on a surface, because other energy-contributing factors such as the vibrational and rotational energies, etc., are assumed to be similar for those two situations. Therefore, for simplicity, we will calculate the heats of adsorption at zero loading as the absolute values of the minimum potential energies at corresponding locations. It is known that the thickness of the adsorbed layer on a curved surface is higher than that on a flat surface. Inferring to the case of adsorption in pores of slit and cylindrical shapes, the thickness in the latter must be higher, that is, the calculated statistical thickness needs to be corrected for the curvature of the pore wall. In a series of papers, Broekhoff and De Boer16,17 introduced two empirical equations for the calculation of the t thickness in pores of two different ranges. The use of two equations means that there may be a kink in the t thickness when plotted as a function of the pore size. The equation for the narrower pore range has an extra term to account for the effect of the pore curvature. This is, in our opinion, an empirical way to compensate for the enhancement of adsorption. Because this effect has been already accounted for in our model, we only need to account for the geometrical curvature of the pore wall in the calculation of the thickness of the adsorbed layer. The thickness of adsorption in a pore t can be calculated from the thickness of adsorption on a flat surface ts using the following equation:18
ts(p) t(p,r) t(p,r)2 ) r r 2r2
(3)
with r being the radius of the empty pore. The Kelvin equation written for adsorption in a cylindrical pore with allowance for the adsorption-enhancing effects is as follows:
2γνm cos θ r - t(pp,r) ) RT ln(pp/po)
(4)
with θ being the contact angle (which is usually taken to be zero) and νm and γ the molar volume and surface tension of the adsorbate liquid, respectively. Results and Discussion Potential Calculation. The heat of adsorption at zero loading can be calculated as the absolute value of the minimum potential
of the molecule residing in a pore. A potential energy equation such as Σ12-6, 9-3, etc. can be used to calculate the molecule potential if a rigid and structureless configuration can be assumed for the solid surfaces. For adsorption occurring in a pore, the minimum of these potentials is dependent on the pore dimension. Similarly, the potential at the pore center is also a function of the pore dimension. In this paper, we assume a cylindrical slab structure for the silica pore wall so that the 9-3 potential equation can be used. This potential equation written for a cylindrical pore of radius r is as follows:19
φ ) 3πnsfσ3sf
[ ( )∑ () 21 σsf
32 r
9 ∞
i)0
Ri
a
2i
9 + 2i r σsf
-
( )∑ ( )] r
3 ∞
i)0
βi
a
3 + 2i r
2i
(5)
with
R1/2 i )
Γ(-4.5) Γ(-4.5 - i)Γ(i + 1)
β1/2 i )
Γ(-1.5) Γ(-1.5 - i)Γ(i + 1)
and
where n is the solid mass center density and a is the distance of the gas molecule from the axis. This equation gives the pore center potential φcenter when the distance a is set to be 0. The equation after rearrangement is as follows:20
( )[
σsf 5 φcenter ) - π2nsfσ3sf 2 r
4
1-
( )]
21 σsf 32 r
6
(6)
Potentials in MCM-41 Pores. The solid wall configuration for MCM-41 presented by Maddox et al.21 is adopted in this paper. Here, the solid is assumed to consist solely of oxygen molecules with a solid mass center density of 20/nm3. The collision diameter σss is 0.265 nm. The solid atom interaction energy ss/k is less defined in the original paper. It is reported to vary in the range from 152 to 1313 K. The interaction coefficients σff and ff/k for nitrogen and benzene are 0.375 nm and 95.2 K,21 and 0.535 nm and 412.3 K, respectively.22 The minimum potential energy and the potential energy at the center of MCM-41 pores are calculated using eqs 5 and 6. Figure 1 displays the nitrogen potential curves for two values of ss/k, 329 and 1313 K, to show the strong effect of the solid atom interaction energy on the potential energy calculation. The continuous curves are for minimum potential energy, whereas the dotted curves are for potential energy at the center. The potential curves of benzene when ss/k ) 329 K are also shown in the same figure for comparison. The deeper well depth of benzene, compared with nitrogen, is due to a significantly higher ff/k of benzene. In general, the minimum and pore center potentials are the same for smaller pores. They both increase rapidly as the pore radius gets below the value of ∼0.9 σsf because of strong short-range repulsive forces exerted by the solid atoms. When the pore radius is greater than σsf, the potential curves deviate from each other to approach two different asymptotes. The asymptote for the pore center potential curve is zero, which is the potential of the bulk gas phase (reference state), whereas that for the minimum pore potential curve is the minimum potential of adsorption on a flat surface
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J. Phys. Chem. B, Vol. 104, No. 47, 2000 11437
Figure 3. Model fitting of nitrogen isotherm at 77 K (solid line) and the extracted PSD for the sample C16R.
Figure 1. Pore center (solid) and pore minimum (dotted) potentials as functions of pore size. Figure 4. Model fitting of nitrogen isotherm and the extracted PSD for the sample C16K.
Figure 2. Nitrogen pore-filling pressure curves for MCM-41 materials.
(8.7 kJ for benzene when ss/k ) 329 K). This potential value is dependent on the choice of the ss/k value. Nitrogen Adsorption. The Pore-Filling Pressure CurVes. The pressure at which a pore is filled with the adsorbate liquid is called the pore-filling pressure corresponding to the size of that pore. It can be calculated using the concepts presented before. Figure 2 shows the nitrogen pore-filling pressure as a function of the pore size for two values of ss/k (continuous lines). A curve calculated using the traditional modified Kelvin equation is also included in the figure (dotted line). Furthermore, the shifted curve of the Kelvin equation, translated horizontally to the right by 0.3 nm, is also plotted in the same figure. This more-or-less “empirical” curve was introduced to match the Kelvin equation with the X-ray diffraction (XRD) measurements.23 We can see that the curves calculated using our approach are closer to this “shifted” curve than to the original Kelvin curve. It is understood that if the solid-adsorbate interaction is taken to be zero, our pore-filling pressure curve will reduce to the Kelvin curve. On the other hand, by increasing the solid-fluid interaction parameters, we can produce the pressure-filling curve resembling the “shifted” Kelvin curves. It is important to note that the addition of 0.3 nm to the pore radius calculated using the Kelvin equation is an empirical correction. Therefore, it may not be valid for pores having diameters outside the range indicated in the figure. Our results
are further compared with the density functional theory (DFT) results taken from the work of Jaroniec et al.24 A general agreement between the DFT curve and that obtained with a solid-fluid interaction energy of 1313 K is noticeable, especially for larger pores. It is again worth noting that a better agreement could be expected by varying the solid-fluid interaction energy. PSD of MCM-41 Materials. We will use the experimental data of MCM-41 samples prepared using C16 bromide surfactants taken from the work of Kruk et al.23 and Ravikovitch et al.,25 where characterizing methods such as the BJH, XRD, and DFT techniques were used. In this paper, these MCM-41 samples are coded C16K and C16R, respectively. Their PSDs in the form of a histogram are calculated using a nonlinear optimization procedure, the principle of which is presented elsewhere.26 From the discussion above, we can expect that different sets of solid-fluid interaction parameters give different PSD results. The goodness of fit is, nevertheless, always satisfactory as the examples shown in Figures 3 and 4. In Figure 3, the primary pore diameter of the sample C16R is estimated to be ∼3.5 nm when ss/k ) 329 K. This is compared with the values of 2.4 and 3.6 nm calculated using the BJH method (Kelvin equation based) and the DFT technique, respectively.25 It shows that the method presented here does not suffer from the same problem experienced with methods based on the traditional Kelvin equation. The dependence of the primary pore diameter on the solid atom interaction energy (ss/k) is demonstrated by comparing this diameter with that calculated using a value of 1313 K for ss/k. It is found that the primary pore diameter changes very little (less than 0.2 nm) when the interaction energy increases by a factor of 4 (from 329 to 1313 K). In principle, it is reasonable to say that the approach presented above is applicable for adsorption in cylindrical pores of the MCM-41 materials. The problem of determining the relevant interaction parameters is of less importance, and can be resolved once we have a reference material having a defined pore structure. For the purpose of this paper, we choose to use ss/k ) 329 K. This value is found to be suitable for one of the models presented in the work of Maddox et al.21 Figure 4 shows the results of the model fitting against the data of the sample C16K. The fitting is again satisfactory. The
11438 J. Phys. Chem. B, Vol. 104, No. 47, 2000
Nguyen and Do
Figure 6. Benzene isotherm fitting at 303 K and PSD for the sample C16N.
Figure 5. Filling pressure curves by benzene vapor for MCM-41 pores.
resulting PSD is comparable with that of the sample C16R (Figure 3). Besides the shape, the primary diameters of the samples are also similar. This is interesting because, as mentioned before, these two samples are prepared by two different groups using surfactants of a similar molecule length. Benzene Adsorption. Adsorption of benzene, one of the frequent probe molecules in adsorption studies, is used to further validate the technique presented here. We will test the model against the benzene adsorption data of two MCM-41 samples, coded C16N and C18N, which are prepared using C16 and C18 surfactants, respectively. These samples were investigated in a previous work.27 Benzene adsorption data are measured using volumetric technique. Pore-Filling Pressure CurVe. The coefficient Cs of the BET equation for benzene is determined to be 6.1 using the adsorption data of a nonporous aerosol sample at 303 K measured in our laboratory. The benzene pore-filling pressure curves for two temperatures, 298 and 303 K, calculated using the above procedure are presented in Figure 5. Here again, the solid atom interaction energy is taken to be 329 K. The filling pressure curve for 303 K calculated using the traditional Kelvin equation is also included in the figure for comparison purposes. The curves in the figure show that, similar to the case of slit pores,9 the filling pressure in cylindrical pores is a very strong function of the pore size when it is small. For larger pores, where the enhancement is less significant, our filling pressure result approaches that calculated using the modified Kelvin equation at the same temperature. It is observed that for pores of radius >3 nm, the curve generated using our model and that using the modified Kelvin equation at the same temperature are very close to each other. This is comparable with the mentioned limit pore diameter of 7.5 nm set for the applicability of the Kelvin equation by Lastoskie et al.5 It is worth noting that this limit was proposed for nitrogen adsorption at its liquid temperature, whereas the calculation here is for benzene at 303 K. The effects of temperature on the pore-filling pressure can be investigated by comparing the curves in the figure. As seen, a temperature change of 15 degrees does not have a significant effect on the pore-filling process. In general, a decreasing temperature makes the pore-filling process occur at lower relative pressures. Model Fitting. A set of pore radii is selected to represent the pore spectrum with an emphasis on the range immediate to the mean pore size. The benzene isotherms are generated for those pores using the model equations. These isotherms are used as the local isotherms in the calculation of the overall adsorption
Figure 7. Model fitting of benzene isotherms at 303 K and the extracted PSD for the sample C18N.
isotherm. Matching this with the experimental data will result in the derivation of the PSD of the solid. The model fitting to the benzene isotherm at 303 K for the samples C16N and C18N are shown in Figures 6 and 7, respectively. The fitting results shown in the normal and logarithmic pressure scales are excellent for the whole range of pressure. The PSD of the solid samples are also included in the figure. The results show that most pores of the sample concentrate within a narrow interval. Similar to the case of PSD by nitrogen data, there are some additional mesopores of sizes in the range immediate to the micropore region. The contribution of these pores is clearly small. The presence of these “unwanted” pores (some of which could be classified as micropores) is also observed in other work.12,21 Here, we argue that this pore volume to some extent could be an artifact if we consider the following: (a) The PSD is obtained via an optimization procedure, which always has some residual. (b) There are always some errors in the experimental measurement, and the working principles of our optimization procedure26 are such that any overestimation of adsorption at low or underestimation at high pressure ranges will result in a biased (positive) micropore volume. The mean primary pore size of the sample C16N, which is ∼3.7 nm, is 0.2 nm larger than those of the samples C16K and C16R. A possible explanation is that the diameter of the sample C16N is calculated using benzene as the probing molecule, whereas nitrogen was used in the case of C16K and C16R. This could also be a result of the uncertainties in the fluid-solid interaction paramters. Pore Size by Different Methods. The regular structure of MCM-41 allows for the application of more direct methods for pore size measurement such as XRD. By assuming that the pores are arranged in a hexagonal pattern in an infinite array, a simple equation to calculate the pore size from the lattice spacing d100 obtained from XRD is available:12
d ) 1.2128
(
)
FVp 1 + FVp
1/2
d100
(7)
The solid density can be taken to be the same as the density of the amorphous silica (2.2 g/cm3). This solid density also gives
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J. Phys. Chem. B, Vol. 104, No. 47, 2000 11439
TABLE 1: Mean Pore Diameter (Å) of Sample C18N Using Benzene Adsorption Data mean variance
eq 8
Kelvin eq
eq 7
new model
35.2 -
37 2.8
43.1 -
43.5 ∼2
the mass center density of 20/nm3 used in the potential calculation.21 In a simpler way, the diameter of cylindrical pores can also be calculated if the pore volume Vp and the total surface area A are known (for example, from the nitrogen adsorption analysis) as follows:
d)
4Vp A
(8)
The pore-sizing results for the sample C18N by different methods are displayed in Table 1. The results using the traditional Kelvin approach and XRD are also shown in the table. As seen, our method yields a result that is very comparable with the XRD results, whereas the Kelvin and eq 8 approaches underpredict the pore size. Equations 7 and 8 provide means to estimate the average pore diameter only. On the other hand, the PSD of the material can be calculated using the traditional Kelvin or the present approach. Results in Table 1 indicate that the sample C18N has quite a narrow PSD. The newly presented model gives a narrower PSD than the traditional Kelvin approach. Isotherm Simulation. Data of benzene adsorption onto the sample C18N at 298 K are now used to test the isotherm simulation using our technique. The isotherm is simulated assuming the PSD shown in Figure 7. The results are shown in Figure 8. As seen, the model predicts benzene adsorption quite accurately. This is evidence supporting our approach to address the problem of adsorption in cylindrical pores. Conclusions Adsorption in mesopores, especially those at the lower end of the range, can be described using the technique presented above. The allowance for the adsorption enhancement renders the method the flexibility to avoid the problem of pore size underestimation experienced with the modified Kelvin equation. As expected, the calculated PSD is dependent on the solidfluid interaction parameters. This uncertainty can be eliminated if the PSD of a “reference” sample can be estimated independently. We found that the solid atom interaction energy of 329 K is suitable for benzene data. On the other hand, when applied to nitrogen data, the median pore diameter is smaller than that calculated from XRD measurement. Acknowledgment. Support from the Australian Research Council (ARC) is gratefully acknowledged.
Figure 8. Simulated benzene adsorption onto the sample C18N at 298K.
References and Notes (1) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; John Wiley & Sons: New York, 1984. (2) Greg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. (3) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids; Academic Press: London, 1999. (4) Evans, R.; Marconi, U. M. B. Chem. Phys. Lett. 1985, 114, 415. (5) Lastoskie C., Gubbins K. E., Quirke N. J Phys. Chem. 1993, 97, 4786. (6) Maglara, E.; Kaminsky, R.; Conner, W. C. In Characterization of Porous Solids IV; McEnaney, B., Mays, T. J., Rouquerol, J., RodriguezReinoso, F., Sing, K. S. W., Unger, K. K., Eds.; The Royal Society of Chemistry: Cambridge, 1997. (7) Everret, D. H. In Characterization of Porous Solids I; Unger, K. K., Rouquerol, J., Sing, K. S. W., Kral, H., Eds.; Society of Chemical Industry: London, 1988. (8) Brunauer, S. The Adsorption of Gases and Vapors; Oxford University Press: Oxford, 1945. (9) Nguyen, C.; Do, D. D. Langmuir 1999, 15, 3608. (10) Nguyen, C.; Do, D. D. Proceedings of the 2nd Pacific Basin Conference on Adsorption Science and Technology; World Scientific: Singapore, 2000. (11) Davis, M. E.; Saldarriaga, C.; Montes, C.; Garees, J.; Crowder, C. Nature (London) 1988, 331, 698. (12) Kruk, M.; Jaroniec, M.; Sayari, A. J. Phys. Chem. B 1997, 101, 583. (13) Dubinin, M. M. Chem. ReV. 1960, 60, 235. (14) Dubinin, M. M. Carbon 1978, 25, 593. (15) Horvath, G.; Kawazoe, K. J. Chem. Eng. Jpn. 1983, 16, 470. (16) Broekhoff, J. C. P.; De Boer, J. H. J. Catal. 1968, 10, 386. (17) Broekhoff, J. C. P.; De Boer, J. H. J. Catal. 1967, 9, 15. (18) Karnaukhov, A. P.; Kiselev, A. V. Z. Fiz. Khim. 1960, 34, 1019. (19) Everett, D. H.; Powl, J. C. J. Chem. Soc., Faraday Trans. 1976, 72, 619. (20) Do, D. D. Adsorption Analysis: Equilibria and Kinetics; Imperial College Press: London, 1998. (21) Maddox, M. W.; Olivier, J. P.; Gubbins, K. E. Langmuir 1997, 13, 1737. (22) Reid, R. C.; Prausnitz, J. M.; Polling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987. (23) Kruk, M.; Jaroniec, M.; Sayari, A. Langmuir 1997, 13, 6267. (24) Jaroniec, M.; Kruk, M.; Olivier, J. P.; Koch, S. Stud. Surf. Sci. Catal. 2000, 128, 71. (25) Ravikovitch, P. I.; Wei, D.; Chueh, W. T.; Haller, G. L.; Neimark, A. V. J. Phys. Chem. B 1997, 101, 3671. (26) Nguyen, C.; Do, D. D. Langmuir 2000, 16, 1319. (27) Nguyen, C.; Sonwane, C. G.; Bhatia, S. K.; Do, D. D. Langmuir 1998, 14, 4950.
