6622
Langmuir 2000, 16, 6622-6627
Verification of the Condensation Model for Cylindrical Nanopores. Analysis of the Nitrogen Isotherm for FSM-16 Hideki Kanda,† Minoru Miyahara,* Tomohisa Yoshioka,‡ and Morio Okazaki§ Department of Chemical Engineering, Kyoto University, Kyoto 606-8501, Japan Received December 1, 1999. In Final Form: April 25, 2000 An improved condensation model for the estimation of pore-size distribution in the range of nanometers is examined. In the authors’ previous paper, the model proved its reliability in computer experiments employing the molecular dynamics technique. The model is tested here in a real experimental system with a MCM-41-like ordered mesoporous silicate, FSM-16. The study includes how to find the adsorbate-solid interaction strength, which is taken into account as an additional contribution for condensation other than the Kelvin effect. The true pore size of the material is separately determined to be 3.2 ( 0.2 nm by high-resolution transmission electron microscopy observation, and by a “colloidal particle adsorption method”. The conventional model for condensation, the Kelvin model, underestimates the pore size of FSM-16 to be 2.5 nm from the nitrogen isotherm. The present model successfully predicts the pore size to be 3.4 nm, and proves its reliability in the real experimental system. The effect of the pore wall’s potential on the capillary coexistence relation is further discussed comparing ordered mesoporous silicates and the usual silica materials.
1. Introduction Quantitative evaluation of the microstructure of porous solids using physisorption is of crucial importance for proper application of, e.g., adsorbents and catalysts. The pore-size evaluation, however, suffers from a problem in the range of nanometers. Models for adsorption in micropores cannot directly be applied, and the Kelvin model fails to describe the pore size though condensation itself occurs in this range of pores.1-6 A straightforward application of statistical thermodynamics, such as the density functional theory and molecular simulation technique,1-4 to this problem could be a solution at the expense of simplicity. Even after these preceding works, however, the Kelvin model is still often, and widely, used for nanopore characterization, which is a direct reflection of the lack of an accurate and handy model. What we are aiming at is to give a simple concept and a model for the condensation phenomena in nanopores, which was successfully done for slit pores.5,6 Further, a model for cylindrical nanopores, taking into account the attractive potential from the pore wall and the dependency of the surface tension on curvature, was proposed in the authors’ previous paper. The model was compared with * To whom correspondence should be addressed. Phone: +81-75-753-5582. Fax: +81-75-753-5913. E-mail: miyahara@ cheme.kyoto-u.ac.jp. † Present address: Department of Chemical Energy Engineering, Central Research Institute of Electric Power Industry, Yokosuka, Kanagawa 240-0196, Japan. ‡ Present address: Department of Chemical Engineering, Hiroshima University, Higashi-Hiroshima 793-0046, Japan. § Present address: Polytech College Kinki, Kishiwada, Osaka 624-0912, Japan. (1) Evans, R.; Marconi, U. M. B.; Tarazona, P. J. Chem. Phys. 1986, 84, 2376. (2) Heffelfinger, G. S.; van Swol, F.; Gubbins, K. E. Mol. Phys. 1987, 61, 1381. (3) Peterson, B. K.; Gubbins, K. E. Mol. Phys. 1987, 62, 215. (4) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (5) Miyahara, M.; Yoshioka, T.; Okazaki, M. J. Chem. Phys. 1997, 106, 8124. (6) Yoshioka, T.; Miyahara, M.; Okazaki, M. J. Chem. Eng. Jpn. 1997, 30, 274.
molecular dynamics (MD) simulations of a Lennard-Jones (LJ) fluid as an ideal experimental system, and proved its reliability.7 However, the MD simulations do not, in a strict sense, reproduce a real system, in which many other factors such as the atomic structure of the adsorbate, that of the solid surface, and the third body force may affect the condensation. Therefore, the condensation model must be tested in a real system as the last stage of the series of verifications. For testing the model, porous media of known pore size must be used. Typical porous materials such as silica gels, controlled pore glasses, and porous carbons are difficult to use for this purpose because observation by, e.g., electron micrography, cannot reach their pore structure developed within solid bodies, and because their pores have, more or less, distributed sizes. Recently, highly ordered porous silicates have been developed. MCM-418 is synthesized with rodlike micelles of cationic surfactant as a template, and exhibits a hexagonal arrangement of cylindrical nanopores. Separately, Inagaki et al. developed a family of mesoporous materials of similarly ordered structure made from kanemite, which are called FSMs.9,10 Because of the unique feature of these ordered mesoporous materials, they have been attracting much attention as model porous materials for various purposes, because the true diameter can be specified using an electron microscope.8 Branton et al., for example, examined the nitrogen isotherm of MCM-41 and indicated the usefulness of MCM41 as a reference material for verification of porecharacterization methods.11 More recently, Naono et al. reported pore sizes of ordered mesoporous silicates prepared with a similar method, on the basis of geometrical (7) Miyahara, M.; Kanda, H.; Yoshioka, T.; Okazaki, M. Langmuir 2000, 16, 4293. (8) Beck, J. S.; Vartuli, J. C.; Roth, W. J.; Leonowicz, M. E.; Kresge, C. T.; Schmitt, K. D.; Chu, C. T-W.; Olson, D. H.; Sheppard, E. W.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. J. Am. Chem. Soc. 1992, 114, 10834. (9) Inagaki, S.; Fukushima, Y.; Kuroda, K. J. Chem. Soc., Chem. Commun. 1993, 22, 680. (10) Inagaki, S.; Yamada, Y.; Fukushima, Y.; Kuroda, K. Science and Technology in Catalysis 1994; Kodansha: Tokyo, 1994; p 143. (11) Branton, P. J.; Hall, P. G.; Sing, K. S. W. J. Chem. Soc., Chem. Commun. 1993, 1257.
10.1021/la991575g CCC: $19.00 © 2000 American Chemical Society Published on Web 07/14/2000
Condensation Model for Cylindrical Nanopores
Langmuir, Vol. 16, No. 16, 2000 6623
the single term of 2γ/r0 seen in the Kelvin model: The meniscus is not hemispherical in the proposed model. Equation 1 gives the local curvature of the condensation phase, and a geometrical integration of eq 1 will determine the shape of the interface, which will give the pore size if summed with the thickness of the adsorbed film on the interior surface of the pore, t ()R - r0). The thickness of the adsorbed film is given by eq 2 since the radii of the
( )
kT ln
Figure 1. Schematic figure of the gas-condensate interface and surface adsorption phase.
consideration of cylindrical adsorption/condensation space.12 These results demonstrate a high degree of uniformity in pore sizes, which is thought to be suitable for our purpose. In this paper the authors’ model is tested with the nitrogen adsorption isotherm on the ordered mesoporous material FSM-16, to prove its validity. 2. Condensation Model The details of the model are given elsewhere for slit pores6 and for cylindrical pores.7 The latter, which is applied here, is briefly explained below. The feature of the model is nonuniformity in the curvature of the gascondensate interface, which results from two factors included in the model: (i) the external force field from the pore wall and (ii) the dependency of the surface tension on the curvature. Referring to Figure 1, the basic equation to describe the critical condensation condition at temperature T is
kT ln
( )
pg γ(F) ) ∆ψ(r) - Vp psat F(r)
(1)
where
1 1 1 ) + F(r) F1(r) F2(r) k is Boltzmann’s constant, p is pressure, and the subscript g indicates the gas phase and sat indicates saturation at T. Vp is the volume per molecule of liquid, γ is the surface tension of the liquid, which is expressed by the GibbsTolman-Koenig-Buff equation13 to account for the dependency of the surface tension on the curvature.14 F1(r) and F2(r) are the radii of the two principal curvatures of the gas-condensate interface, which varies depending on the distance from the center of the pore r. ∆ψ(r) is the contribution of the attractive potential energy from the pore wall, which must be expressed as an “excess” amount compared with a potential energy that a molecule would feel if the pore wall consists of a liquid of the adsorbate molecules. Any potential function can be chosen that may be suitable to express the interaction between the adsorbate and pore walls for a given adsorption system. The contribution of the potential energy, together with the curvature-dependent surface tension, gives the two different principal radii of curvature in eq 1, instead of (12) Naono, H.; Hukuman, M.; Shiono, T. J. Colloid Interface Sci. 1997, 186, 360. (13) Melrose, J. C. Ind. Eng. Chem. 1968, 60, 53. (14) Tolman, R. C. J. Chem. Phys. 1949, 17, 333.
pg γ(r0) ) ∆ψ(r0) - Vp psat r0
(2)
two curvatures will be zero and r0, respectively, at the surface of the adsorption film in a cylindrical pore. Since the potential function ∆ψ(r) itself depends on the pore radius R, one needs iterative calculation to obtain the relation between pg/psat and R, but it can be done quite easily. Note that the convention for definition of the potential function brings a difference in t ()R - r0) given by the above equation from those obtained in experiments. As a result, the pore radius R also has a similar difference. This point will be considered in section 4. 3. Determination of the True Pore Diameter of FSM-16 The material employed for the test was a mesoporous silicate, FSM-16.9 FSM-16 is derived from kanemite and has highly ordered mesopores of hexagonal regularity,10 which resemble those of MCM-41. Our intention here is to obtain the true pore size of this material by some means other than physisorption, with which the model prediction can be compared for verification. The true pore size of FSM-16 can be determined with the following two methods. First, observation of FSM-16 by transmission electron microscopy (TEM) was made with an acceleration voltage of 200 kV and a magnification of 50 000, which was decided not to destroy its thin pore wall. An example of the micrograph is shown in Figure 2. Unlike the usual porous media, FSM-16 is able to show a TEM image of the “pore” because of its thin pore walls, uniform pore size, and straight pore structure. A regular honeycomb-like arrangement of channels with uniform size is recognizable. The average distance between the edges of the shadow is 3.2 nm with a possible error of (0.2 nm. The thickness of its pore walls of 0.9 (0.1 nm is also determined from the picture. Its pore walls are much thinner than those of regular porous materials, which exhibit a weaker attractive potential field in the pore space. The resultant feature in the capillary coexistence relation, in comparison with the usual silica materials, is discussed in section 4. Note here that the honeycomb array, though it is not typically shown in Figure 2, exposes only a limited amount of “outer” surface available for surface adsorption. This is a big difference compared with the carbon nanotubes, which form individual tubes with both interior and outer surfaces available for adsorption. Thus, in the case of the FSM, the effect of adsorption on the outer surface is considered to be negligibly small. The validity of the above pore size was supported by another trial made with nanoparticles, or colloidal dispersion of rhodium (Rh) as a probe for detecting the pore size. Rh nanoparticles were made by the reduction of rhodium(III) chloride using methanol as a reducing agent and poly(vinylpyrrolidone) (PVP) as a dispersing agent
6624
Langmuir, Vol. 16, No. 16, 2000
Kanda et al.
Figure 2. Example of the TEM image of FSM-16.
according to Hirai et al.15 PVP (150 mg, 360 000 in average molecular weight) and rhodium(III) chloride (8.8 mg, 0.033 mmol) were dissolved in a methanol (25 mL)-water (25 mL) mixed solvent to form a rose-pink solution. Refluxing the solution at 370 K under air in a heating bath for 4 h gave a homogeneous light brown to dark brown solution of a Rh colloidal dispersion.15 Then, FSM-16 powder was contacted with the colloidal dispersion for 10 days in a batchwise operation thermostated at 298 K. The diameter of individual Rh particles, before and after contact, was determined from TEM photographs with magnification of 157 000 and 188 000. The samples for TEM observation were prepared by evaporating a small amount of the solution on a carbon film supported on copper grids. An example of the picture is shown in Figure 3. The obtained particle size was distributed between 1.5 and 5.5 nm in diameter. For each sample of before and after contact, we measured diameters of ca. 750 particles to obtain the size distribution. The distribution was normalized so as to coincide with the range of diameter larger than 4.5 nm. As shown in Figure 4, the particle of size less than 3.0-3.2 nm shows a clear decrease in the particle size distribution after contact with FSM-16, which is in good agreement with the former result. This colloid-probe method alone may not be sufficient to determine the pore size precisely, but at least it supports the pore size from the TEM observation. Thus, we take 3.2 ( 0.2 nm as the true pore diameter of FSM16. 4. Application of the Proposed Model to FSM-16
Figure 3. Example of the TEM image of Rh particles.
Figure 4. Normalized size distributions of Rh particles before and after contact with FSM-16.
4.1. Determination of Potential from Pore Walls. Potential Function and Interaction Parameter. Adsorption of nitrogen on silicate is a typical physisorption phenomenon attributed to dispersion force. Among possible choices in potential function, the most simple concept of a
structureless cylindrical wall made of a Lennard-Jones solid given by Peterson et al.16 is employed here, which is analogous to the LJ 9-3 potential for a planar surface. The function ∆ψ should be obtained from subtraction of a corresponding potential energy for the adsorbate’s liquid
(15) Hirai, H.; Nakao, Y.; Toshima, N. J. Macromol. Sci., Chem. 1978, A12 (8), 1117.
(16) Peterson, B. K.; Walton, J. P. R. B.; Gubbins, K. E. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1789.
Condensation Model for Cylindrical Nanopores
Langmuir, Vol. 16, No. 16, 2000 6625
state from the potential of the pore wall. Further, only the attractive term of the potential is enough to be considered: The repulsive term can be negligible because the effect of the repulsive potential decreases rapidly with the distance from a pore wall, and it hardly influences the condensation phenomena in the inner portion of the pore. Thus, the function is
∆ψ ) ψgs - ψgg ) -π(gsFsσgs6 - ggFgσgg6)K3(r,R) ) 3C K (r,R) (3) 2 3 where Figure 5. Nitrogen adsorption isotherm for FSM-16.
2 C ) π(gsFsσ6gs - ggFgσ6gg) 3
∫0πdθ (- Rr cos θ + (1 - (Rr )
K3(r,R) ) R-3
2
1/2 -3
) )
sin2 θ
ψgs is the potential field from the pore wall and ψgg is the one that a molecule would feel if the pore wall consisted of a liquid of adsorbate molecules. and σ are the energy and size parameters, for which the subscript gs indicates gas-solid interaction and gg means gas-gas interaction. Fs is the number density of interaction sites in the solid wall, Fg is the number density of the adsorbate particle in the liquid state, and r is the distance from the center of the pore. The interaction parameter C in eq 3 is the representative for the relative strength of the wall’s potential, for which the factor 2/3 is introduced for ease of referring to those for a planar surface. Equation 3 indicates that if the potential energy from the pore wall is equal to that for the adsorbate liquid, the influence of the potential on condensation will vanish. Another possibility for the potential function may first be the one proposed by Tjatjopoulos et al.17 for a cylindrical tube, which is analogous to the LJ 10-4 potential for a planar sheet. If the wall consisted of a single atomic layer as the case for the buckeytube, this potential would be most appropriate. On the other hand if the wall has, say, more than four or five atomic layers, Peterson’s function probably can stand. The FSM, whose wall is considered to have ca. three atomic layers, is just the intermediate of the two functional formulas. Then we tried a potential function for expressing a cylindrical wall with finite thickness δ: ψgs was set as ψgs(r,R) - ψgs(r,R+δ) to express an annulus of solid. Both this setting and Tjatjopoulos’s function were, however, finally not adopted in the present study because of the mathematical difficulty in determining unknown parameters included. Note that, however, the difficulty is only for the determination of the potential parameters for the unique material FSM, and that the pore size calculation itself would be made straightforward as long as the parameters are known in some way. This compromise to use the functional formula for an infinite solid, instead of that for a finite thickness, would bring some disadvantage: Estimated pore sizes would inevitably have some overestimation even with appropriately evaluated potential parameters, because Peterson’s function has a longer decay length than the reality. This point will be considered again in section 4.2. Interaction Parameter for FSM-16. If there exists a “nonporous ultrathin silica solid with the same thickness as that of the wall of FSM-16”, its standard isotherm could give the C parameter for FSM-16. This is not, however, (17) Tjatjopoulos, G. J.; Feke, D. L.; Man, J. A. J. Phys. Chem. 1988, 92, 4006.
Figure 6. Schematic representation of the difference between the experimentally obtained thickness of the adsorbed film and that defined in eqs 2 and 5.
the case at present. We thus decided to use the noncondensed portion of the adsorption isotherm of FSM-16 itself to determinate the C parameter for FSM-16, CFSM-16, using eq 2. Figure 5 shows the nitrogen adsorption/desorption isotherms for the FSM-16 at the normal boiling temperature of nitrogen measured with an automated volumetric gas adsorption instrument, BELSORP 28 (Bel Japan, Inc.). The steep rise, which is a reflection of a highly uniform pore structure, occurs at a relative pressure of ca. 0.3. The BET surface area was calculated to be 920 m2/g using adsorption data in the range 0.05 < pg/psat < 0.15. In this range, no condensation was assumed to occur because of the high uniformity in the pore size. With the BET surface area, the amount adsorbed can be converted to the statistical thickness of the adsorbed film. As noted in section 2, correction relating to the definition of the potential function must be made for comparison of the experimental thickness and that in eq 2. Figure 6 illustrates the difference. The experimental thickness includes merely the adsorbed volume. On the other hand, t includes the radius of surface atoms of the pore wall because the origin of the potential function is placed at the center of them, and t excludes the radius of adsorbate molecules because the location of the center of the adsorbate molecules decides their behavior. The correction then will be
t ) R - r0 )
VAds σgg - σss ) ABET 2 VAds 0.354 nm - 0.300 nm (4) ABET 2
where VAds is the adsorbate’s volume obtained using the density for the bulk liquid and ABET is the BET surface area. The thickness of nitrogen’s monolayer, which is given by the liquid-phase density and the molecular cross-
6626
Langmuir, Vol. 16, No. 16, 2000
Kanda et al.
Figure 7. Fitting of eq 2 to the nitrogen adsorption isotherm of FSM-16 for determination of CFSM-16.
Figure 8. Fitting of eq 5 to the nitrogen standard isotherm on nonporous silica for determination of CS.
sectional area,18 is employed for σgg. The van der Waals diameter of the bridged oxygen atom of 0.30 nm19 is employed for σss. We fitted eq 2 to the adsorption data of the noncondensed portion. Calculation of eq 2 needs the pore diameter to determine CFSM-16. Thus, the calculation was made iteratively to obtain convergence with the peak pore radius of the pore size distribution, which is given in the next section: the sensitivity of CFSM-16 to the pore radius is rather low, and the convergence was easily reached. The fitted result is shown in Figure 7. The determined CFSM-16 is 6.5 × 10-23 J nm3. The fitting was made for the range of surface coverage from 0.9 to 1.1: For a smaller amount adsorbed the repulsive potential, which is not included in eq 2, affects the adsorption behavior; the upper limit was set to be free from condensation. Strictly speaking, the use of the BET surface area is inconsistent, because two different models of surface adsorption (BET and eq 2) are applied to the same phenomenon. The consistent way would be to calculate the surface area from the pore size distribution (PSD) based on the present model as described in section 4.2, to find convergence in the surface area. We examined this way of calculation, and found the following. The surface area determined from the PSD calculation suffered from relatively large uncertainty depending on what pore size the calculation goes down. For example, the cumulative surface area was 570 m2/g for 2R > 3 nm and 870 m2/g for 2R > 2 nm. The latter may be understood as a tolerable consistency with the assumed surface area of 920 m2/g, but we cannot be sure about the appropriate pore size at which the cumulative calculation should be stopped. A possible guide could be the calculation step at which the amount adsorbed becomes negative, but the successive calculation tends to cumulate errors in the residual amount adsorbed as is often the case for the PSD calculation. Because of the above uncertainty, we finally quit employing the cumulative surface area, and decided to use the BET one. Note that the surface area, in this case, was needed only to determine the C parameter for this unique material FSM: For the usual porous material the parameter should separately be determined from a standard isotherm on a corresponding nonporous solid. Further, the C parameter does not need a high degree of accuracy. Rather, 20-30% uncertainty can still be satisfactory to ensure possible deviation of less than (0.2 nm in the estimated pore size. This tolerance is another reason for using the BET surface area.
Interaction Parameter for the Usual Silica. For a usual porous material that has thick pore walls and a broad distribution in pore size, the constant C should be obtained from a standard adsorption isotherm on a nonporous material consisting of the same chemical composition. We proposed a method for the determination of the C parameter20 based on the Frenkel equation,21 which is given by eq 5. Note that the potential ψ here is for the
(18) Emmett, P. H.; Brunauer, S. J. Am. Chem. Soc. 1937, 59, 1553. (19) Bondi, A. Physical Properties of Molecular Crystals, Liquids, and Glasses; Wiley: New York, 1968.
kT ln
( )
pg ) ψgs - ψgg ) psat 2 1 C - π(gsFsσgs6 - ggFgσgg6) 3 ) - 3 (5) 3 t t
planar surface as a function of t. Comparing eqs 3 and 5, one may know that the constant C determined from a standard isotherm on a planar surface can describe the effective strength of the potential within a cylindrical pore. We tried to determine the parameter for silica, termed CS, by fitting eq 5 to a standard adsorption isotherm on nonporous silica. A standard adsorption isotherm of nitrogen on nonporous silica provided by Bel Japan, Inc., which is almost the same as that by Pierce,22 is used for determining the constant CS. The result of fitting is shown in Figure 8. CS is found to be 1.3 × 10-22 J nm3 from a portion of the isotherm with surface a coverage from 0.9 to 1.8, which was found to be suitable for finding a proper value of the C parameter by our research result:20 For a larger amount than this range, a strongly structured first layer of adsorption would bring an additional contribution to the amount adsorbed, which cannot be described by the Frenkel equation, while the effect of a repulsive potential arises for a lower amount as described before. Eventually the thus determined value of CS does not apply to FSM-16 because of its thin pore walls, but the determined CS is expected to provide a capillary coexistence relation for the usual porous silica, as described later. CFSM-16 is about half of CS. This difference must come from the unique thin pore walls of 0.9 nm, because dispersion force by solid atoms far from 0.9 nm up to infinity must be effective in the latter case. 4.2. Prediction of the Pore Size with the Proposed Model. The relation between the critical relative pressure for condensation and the pore diameter was calculated by solving eqs 1 and 2 for the two cases with CFSM-16 or CS, which is shown in Figure 9 together with those by the (20) Miyahara, M.; Yoshioka, T.; Nakamura, J.; Okazaki, M. J. Chem. Eng. Jpn. 2000, 33, 103 (21) Halsey, G. J. Chem. Phys. 1948, 16, 931. (22) Pierce, C. J. Phys. Chem. 1968, 72, 3637.
Condensation Model for Cylindrical Nanopores
Langmuir, Vol. 16, No. 16, 2000 6627
Figure 10. Comparison of the pore-size distributions for FSM16. The shaded range gives the “true” pore size. Figure 9. Comparison of capillary coexistence curves. The pore diameter corresponds to experimentally observable ones.
Kelvin model. The raw value of the pore diameter predicted by the proposed model expresses the distance between the center of the surface atoms of the pore walls. Then the experimentally defined pore diameter is smaller than that of the model by about the size of the surface atoms. The van der Waals diameter of the bridged oxygen atom of 0.30 nm19 was subtracted from the raw value: Thus, the obtained value, in Figure 9, expresses the size of the accessible pore space, which in general means “pore size”. The influence of the difference of the two values of the C parameters is rather small, but one may recognize ca. 20-30% of underestimation by the Kelvin model compared with the proposed model in the case of FSM. For a usual porous silicate the difference would be greater. Roughly about half of the difference in pore size between the Kelvin model and the proposed model comes from the potential effect and the other half from the stronger surface tension of the curved surface. The higher the pressure, the smaller the difference between the two models, because of the decrease of the influence of the potential from the pore walls, and the decrease of the surface tension with weaker curvature. Using these coexistence curves, we calculated the corresponding pore-size distribution for FSM-16. The scheme of calculation was similar to that by Dollimore and Heal,23 but the thickness of the adsorbed film and capillary coexistence employed were those of our model. Figure 10 shows the results, where the shaded range gives the “true” pore size obtained by TEM observation. The Kelvin model predicts the peak pore size to be 2.5 nm, and underestimated the pore size of FSM-16 by about 20%. The present model successfully predicted the peak pore size to be 3.4 nm, and proved its reliability in a real system. The possible error of +0.2 nm may have been caused by the use of the LJ 9-3 analogue function. The potential function in reality should decrease more rapidly with the distance from the wall within a pore of FSM-16 because of the quite thin pore walls. We examined this effect by employing the potential function derived by Tjatjopoulos17 for a cylindrical tube, which is analogous to the LJ 10-4 potential for a planar sheet, as ψgs. Contrary to the case with Peterson’s function, this function is thought to have a shorter decay length than the reality in which at least (23) Dollimore, D.; Heal, G. R. J. Appl. Chem. 1964, 14, 109.
a few of the atomic layers exist within the wall. Note that the examination here is not strict in the definition of ∆ψ: ψgg was also expressed by the same functional form to avoid the mathematical difficulty in determining parameters, as explained in section 4.1. The calculated peak pore size based on this function was 3.1 nm, which is smaller, as expected, than that obtained with Peterson’s function, and may include some degree of underestimation. Accordingly, it is strongly suggested that the possible overestimation for the FSM’s pore size originates from its unique character of the extremely thin walls, and that the present model is expected to predict pore sizes even more accurately, for normal porous media, than this success for FSM-16. 5. Conclusion The condensation model proposed by the authors is tested on a MCM-41-like mesoporous silicate, FSM-16. Because of its unique feature, the true pore size was measured with TEM and the “colloidal particle adsorption method” to be 3.2 ( 0.2 nm. The present model successfully predicts the peak pore size of 3.4 nm from the nitrogen isotherm, and proves its reliability in a real experimental system, in addition to previously reported success in computer experiments. The possible error of +0.2 nm is thought to have resulted from the employment of a potential function for a semiinfinite solid: one could use another potential function with, e.g., finite thickness, but it would need determination of additional potential parameters with mathematical difficulty. Considering the simple procedure and sufficient accuracy, the potential function employed in this study is recommended for poresize estimation in general. The present model is expected to give more accurate estimation for normal porous silica made of thicker pore walls, employing the potential parameter for silica, CS, determined in this work. Acknowledgment. It is a pleasure to thank Dr. S. Inagaki, Toyota Central R&D Laboratories, Inc., for the generosity in supplying FSM-16. We are deeply indebted to Dr. M. Tsuji, Institute for Chemical Research, Kyoto University, for the TEM observation of FSM-16 and Rh nanoparticles. This research was supported in part by the Grant-in-Aid for Scientific Research, No. 09750827, provided by the Ministry of Education of Japan. LA991575G
