A Method for the Estimation of Pore Anisotropy in Porous Solids

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A Method for the Estimation of Pore Anisotropy in Porous Solids P. J. Pomonis* and G. S. Armatas Department of Chemistry, University of Ioannina, Ioannina 45 110, Greece Received March 1, 2004. In Final Form: June 2, 2004 In this work a method for the estimation of pore anisotropy, b, in porous solids is suggested. The methodology is based on the pore size distribution and the surface area distribution, both calculated from trivial N2 adsorption-desorption isotherms. The materials used for testing the method were six MCM-Alx solids in which the ordered pore structure (for x ) 0) was gradually destroyed by the introduction of Al atoms (x ) 5, 10, 15, 20, 50) into the solids. Additionally, four silicas having random porosity were examined, in which the surface of the parent material SiO2 (pure silica) was gradually functionalized with organosilicate groups of various lengths (tSisH, tSisCH2OH, tSis(CH2)3OH) in order to block a variable amount of pores. As pore anisotropy, the ratio bi ) Li/Di is defined where Li and Di are the length and the diameter of each group of pores i filled at a particular partial pressure (Pi/P0). The ratio of the surface area Si over the pore volume Vi, at each particular pressure (Pi/P0), is then expressed as Si3/Vi2 ) 16π(Nibi) ) 16πλi, where Nibi is the number of pores having anisotropy bi which are filled at each pressure (Pi/P0) and λi is the total anisotropy of all the pores Ni belonging to the group i of pores. Then plot of λi vs (Pi/P0) provides a clear picture of the variation of the total pore anisotropy λi as the partial pressure (Pi/P0) increases. For the functionalized silicas there appears a continuous drop of λi as partial pressure (Pi/P0) increases, a fact indicating that both Ni and bi are continuously diminished. In contrast, for the MCM-Alx materials a sudden kink of λi appears at the partial pressure where the well-defined mesopores are filled up, a fact indicating that at this point Ni and/or bi is large. The kink disappears as the ordered porosity is destroyed by increasing the x doping in MCM-Alx. The pore anisotropy bi of each group i of pores is then estimated using the expression (Si3/Vi2) ) 8πNiriSi and plotting log(λi) vs log ri. From those plots, the values of si can be found and therefore the values of bi ) 0.5riSi are next defined. In the MCM-Alx materials the maximum pore anisotropy b is very high (bi ∼ 250) for x ) 0. Then as mesoporosity is destroyed by increasing x, the maximum b values drop gradually to b ∼ 11 (x ) 5), b ∼ 8 (x ) 10), and b ∼ 3 (x ) 15). For x ) 20 and x ) 50, the maximum b obtains values equal to unity. The same phenomena, although less profound, are also observed for the functionalized silicas, where the anisotropy b is altered by the process of functionalization and from bi ∼ 0.5 for the nonfunctionalized or bi ∼ 0.9 for the solid functionalized with tSisH groups drops to b ) 0.3 and b ) 0.2 for the solid functionalized with tSis(CH2)OH and tSis(CH2)3OH, respectively. A correlation factor F is suggested in cases where the pore model departs from the cylindrical geometry.

Introduction The pivotal assumption behind all the calculations of porosity and the pore size distribution (PSD) in porous solids1-3 is that the pores and the capillaries in such semicontinuous media assume a cylindrical shape.4,5 Then application of the Kelvin equation, in the form of various algorithms embedded into the commercial or homemade porosimeters, provides, among other things, the diameter of the pores which, as mentioned, were considered cylinders. But clearly, a typical cylinder is characterized by two geometrical parameters, its diameter D as well as its length L. Now, a question naturally occurring in the mind is what is the ratio of b ) L/D of those pores or capillaries? Perhaps up to 1992, before the invention of MCM materials6-8 which clearly possess long cylindrical pores, this question was somehow irrelevant: The ordinary porous solids such as silicas, aluminas, aluminosilicates, * To whom correspondence may be addressed: Department of Chemistry, University of Ioannina; tel. +32651098350; fax. +32651098795; e-mail [email protected]. (1) Gregg, J. S.; Sing, K. S. W. In Adsorption, Surface Area and Prosity, 2nd ed.; Academic Press: New York, 1982. (2) Rouquerol, F.; Rouquerol, J.; Sing, K. S. W. Adsorption by Powders and Porous Solids; Academic Press: New York, 1999. (3) Thomas, J. M.; Thomas, W. J. In Indroduction to the Principles of Heterogeneous Catalysis; Academic Press: London, 1967. (4) Lowell, S. In Introduction to Power Surface Area; John Willey and Sons: Chichestre, Brinstone, Toronto, 1979. (5) Thomas, J. M.; Thomas, W. J. In Principles and Practice of Heterogeneous Catalysis; VCH: New York, 1997.

aluminophosphates, etc., possess a more or less random porous network and it was difficult to talk about distinct geometrical characteristics such as dimeter and length of openingssand which was what. Besides, various observations made by elecron microscopy have shown that many traditional porous materials appear as a random collection of packed spherical and/or semispherical particles9,10 as well as of sticks.11,12 Clearly with this kind of space-to-pore relationship it is not easy to attribute distinguished Euclidean geometrical parameters such as D and L. Nevertheless Dullien13 from various microphotographic figures of solids with random porosity showed that that in many cases the length L of pores is very close to their diameter D, i.e., L ∼ D. But with the arrival of MCM materials such questionssi.e., the distinction between L and Dshave a new and clear meaning. So the purpose of this paper is (6) (a) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Nature 1992, 359, 710. (b) Beck, J. C.; Vartuli, J. C.; Roth, W. J.; Leonowicz, C. T.; Kresge, C. T.; Schmitt, K. D.; C. Chu, T.-W.; Olson, D. H.; Sheppard, E. W.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. J. Am. Chem. Soc. 1992, 114, 10834. (7) Ciesla, U.; Schuth, F. Micropor. Mesopor. Mater. 1999, 27, 131. (8) Øye, G.; Sjo¨blom, J.; Sto¨cker, M. Adv. Colloid Interface Sci. 2001, 89-90, 439. (9) Reyes, S. C.; Iglesia, E. J. Catal. 1991, 129, 457. (10) Drewry, H.; Seaton, N. A. AIChE J. 1995, 41, 880. (11) Mace, O.; Wei, J. Ind. Eng. Chem. Res. 1991, 30, 909. (12) Smith, B. J.; Wei, J. J. Catal. 1991, 132, 41. (13) Dullien, F. A. L. In Porous Media: Fluid Transport and Pore Structure, 2nd ed.; Academic Press: New York, 1992.

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to propose a method for the estimation of the anisotropy parameter b ) L/D of pores in porous solids based on standard N2 adsorption-desorption isotherms. Clearly this work would not be feasible without the use of MCMtype materials. So we shall proceed based on the assumption that in a collection of pores, for each pore i, or rather for each group of pores in a small interval of pore diameter Di ( ∆Di, the length Li is related to the diameter Di via a relation of the form

bi ) Li/Di

(1)

where bi is a dimensionless positive number expressing the anisotropy of those pores in the group i. If bi is larger than unity (bi . 1), those pores have a large anisotropy and such a situation is expected in MCM-type solids. If bi ∼ 1 (Li ∼ Di), the pores are isotropic and according to Dullien13 this should be the case in ordinary porous solids with a random porous network. If on the other hand bi < 1, the pores are again anisotropic as to their diameter Di and length Li, but in this last case it is better to imagine a shallow cavity rather than a typical pore. For the simple case of nonintersecting capillaries the ratio of the volume Vp contained by the pores to the surface area Sp created by these same pores is 4Vp/Sp ) D h p, which is the average pore diameter.1-5 For other geometric configurations of pore assemblies, one may write5,14

D hp )

( )

1 4Vp F Sp

(2)

where F is a factor characteristic of the particular pore geometry. This factor F obtains values equal to unity for nonintersecting capillaries and is equal to F ) 0.613 for cubic packing of spheres, F ) 0.433 for orthorombic packing of spheres, F ) 0.229 for rombohedral packing of spheres, and finally F ) 0.293 for tetragonal-spheroidal packing of spheres. So F can be considered as a measure of deviation from the cylindrical geometry assumed in trivial N2 porosimetry.14 Wheeler in a much quoted paper15 tried to provide an answer to the fact that the ratio of the experimentally determined pore volume and surface area does not characterize a porous system completely. He reached the conclusion that in a catalyst pellet, the average pore diameter D h p is given by

D hp )

( )

4Vp τ(1 - ψ) Sp

(3)

where Vp is the specific pore volume, Sp is the specific pore surface area, τ is a roughness factor, and ψ is the pellet porosity. This model is obviously equivalent to the geometric nonintersecting pore model when the product τ(1 - ψ) is equal to unity. Wheeler further assumed that in a practical porous pellet, some pores are parallel to the gas flow while others are at right angles, and the average orientation of pores in the direction of gas flow is 45°.5,15 In this case he estimated that the average pore length L equals

L ) 21/2

Vx Sx

(4)

(14) Thomas, J. M.; Thomas, W. J. in Principles and Practice of Heterogeneous Catalysis; VCH: New York, 1997; p 283. (15) (a) Wheeler, A. Adv. Catal. 1951, 3, 249. (b) Wheeler, A. Catalysis 1955, 2, 118.

where Vx is the volume of the pellet and Sx is the external surface area of it. Combination of eqs 3 and 4 provides a measure of pore anisotropy bi as defined above by eq 1

bi )

Li 21/2(Vx/Sx) ) Di 4(Vp/Sp)τ(1 - ψ)

(5)

The roughness factor τ in (5) was assumed to be 2 for practical purposes. On the other hand for a pellet made of particles that do not possess any internal porosity, the ratio Vx/Sx was equal to dparticle/6 for spheres and cubes.15 Thus eq 5, according to that treatment, obtains the form

bi )

Li 0.058dparticle ) Di (Vp/Sp)(1 - ψ)

(6)

Equation 6 could be used to estimate the anisotropy factor bi provided that the values for dparticle and ψ are known and besides the solid particles do not possess any internal porosity. For a uniform collection of nonporous particles, for which dparticle and ψ are well defined, it is possible in principle to estimate a unique value of bi. But for a collection of particles with various dparticle values and variable ψ this method seems intractable and much more where internal porosity is introduced into the problem. This is probably one of the reasons that up until today it has not been extensively applied for the estimation of the anisotropy parameter bi of pores. Nevertheless such a parameter bi is important since it should play a central role in controlling mass transport properties in porous solids. The purpose of this paper is to provide a method for estimating the values of the anisotropy parameter bi of pores. The method is based on a particular treatment of N2 adsorption data, namely, of the specific surface area Spi and the specific pore volume Vpi eastimated at each point i, i.e., (Pi/P0) of the adsorption isotherm. The materials used for testing were six MCM type solids modified with the addition of Al (MCM-Alx x ) 0, 5, 10, 15, 20, 50) and four silicas SiO2-X which suffered gradual functionalization with functional groups X of increasing length (X ) tSisH, tSisCH2OH, tSis (CH2)3OH). Experimental Section Synthesis of Materials MCM-Alx (x ) 0, 5, 10, 15, 20, 50). Aluminum-containing MCM-48 samples were synthesized by a pathway similar to the procedure described by the Unger group.16 Namely n-hexadecyltrimethylammonium bromide template (CTAB) (6.6 mmol) was dissolved in deionized water (2.8 mol), ethanol (0.87 mol), and TEOS (98%). Different amounts of aluminum precursors (AlNO3‚18H2O) (Si + Al ) 16 mmol, Al/(Si + Al) ) 0, 5, 10, 15, 20, 50) were added to the surfactant solution. The solution was stirred for 10 min (400 revolutions min-1), and then aqueous ammonia (32%, 0.20 mol) was added. The molar composition of the gel was 1 M metallic ions (Si + Al)/12.5 M NH3/54 M EtOH/0.4 M CTAB/174 M H2O. After the mixture was stirred for 2 h at room temperature, the resulting solid was recovered by filtration, washed with distilled water, and dried in air at ambient temperature and then at 110 °C for 12 h. The template was removed by calcination at 823 K for 6 h. The obtained materials will be designated in the text as MCM-Alx, where x ) 0, 5, 10, 15, 20, 50 indicates the ratio of Al/(Si + Al) into the synthesis bath. Synthesis of Materials SiO2-X (X ) tSisH, tSisCH2OH, tSis(CH2)3OH). The inorganic support SiO2 was synthesized as follows: Commercial tetraethoxysilane (TEOS), oxalic acid (OA), and hexanol were used without purification. An amount (16) Schumacher, K.; Grun, M.; Unger, K. K. Micropor. Mesopor. Mater. 1999, 27, 201.

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Figure 1. N2 adsorption-desorption isotherms and the corresponding PSD according to the Horvath-Kawazoe method for the MCM-Alx solids. of OA in 0.66 mol of hexanol was mixed with 0.40 mol of TEOS in a ratio of OA/4TEOS ) 7.5 in a 500-mL two-necked flask. The reaction mixture was kept at 70 °C for 2 h with mechanical stirring. Then, 4.0 mol of deionized water containing TEOS in molar ratio H2O/TEOS ) 10 was added. After 24 h at 70 °C under mechanical stirring the resulting solid was recovered by filtration, washed with ethanol and distilled water, and dried in air at 100 °C for 12 h. The organic part was removed by calcination at 873 K for 6 h. Three samples, based of the original porous solid SiO2, were prepared with gradual functionalization of its acid sites with organosilicate groups (X ) tSisH, tSisCH2OH, tSis(CH2)3OH). The functionalization was carried out as follows: The inorganic support SiO2 was dried in an oven at 150 °C for 2 h and refluxed in toluene for 12 h with the estimated amount of the corresponding organic alkoxy silicates (CH3O)3SiH (trimethoxysilane), (CH3O)3SiCH2Cl ((chloromethyl)triethoxysilane), and (CH3O)3Si(CH2)3Cl ((chloropropyl)trimethoxysilane) with carbon chains of different length. The resulting solids were recovered by filtration, washed with toluene and ethanol, and dried in air at 110 °C for 24 h. N2 Porosimetry. A Fisons Sorptomatic 1900 instrument was used to carry out the pore size distribution measurements both for the six MCM-Alx as well as for the four SiO2-X. The characterization included the determination of nitrogen adsorption-desorption isotherms at 77 K. Prior to each experiment the samples were degassed at 250 °C in a vacuum of 5 × 10-2 mbar for 12 h for the MCM-Alx and at 60 °C for 24 h for the SiO2-X.

The desorption branch of the isotherms was used for the calculation of the pore size distribution (PSD). The specific surface area of the sample was calculated by applying the BET equation using the linear part (0.05 < P/P0 < 0.30) of adsorption isotherm and assuming a closely packed BET monolayer. X-ray Diffractometry. The MCM-Alx and the SiO2-X materials were examined by X-ray diffractometry in order to examine the degree of order in their structure. The measurements took place in a Bruker Advance D8 system using Cu KR radiation (λ ) 1.15418 Å) with a resolution of 0.01°.

Results The N2 adsorption-desorption isotherms are shown in Figure 1 for the MCM-Alx materials and in Figure 2 for the hybrid organic-inorganic SiO2-X solids. In the same Figures 1 and 2 the PSDs are shown, calculated according to the Horvart-Kawazoe (HK) method for the MCM-Alx and according to the BJH methodology for the SiO2-X porous solids. In Table 1 the estimated values of specific surface area Sp (m2 g-1), of specific pore volume Vp (cm3 g-1), and of the mean hydraulic pore diameter Dp ()4Vp/Sp) of pores are included for all the solids, MCM-Alx as well as SiO2-X. The results from the X-ray diffraction analysis for the samples MCM-Alx are shown in Figure 3. For the samples

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Figure 2. N2 adsorption-desorption isotherms and the corresponding PSD according to the BJH method for the SiO2-X solids. Table 1. Specific Surface Areas, Sp, Specific Pore Volumes, Vp, and the Mean Hydraylic Pore Diameter, D h p, of the PSD (See Figures 1 and 2) for the Solids MCM-Alx and SiO2-Xa materials

Sp(BET) (m2 g-1)

Vp (at P/P0 ) 0.99) (cm3 g-1)

D h p ) 4Vp/Sp (nm)

bmax

L h ) bD hp (nm)

Lmax ) bDmax (nm)

MCM-Al0 MCM-Al5 MCM-Al10 MCM-Al15 MCM-Al20 MCM-Al50 SiO2 SiO2-SiH SiO2-SiCH2OH SiO2-Si(CH2)3OH

1304 1383 1378 1237 1180 785 897 790 776 675

0.74 0.86 0.84 0.73 0.70 0.47 1.57 1.41 1.34 1.24

2.48 2.52 2.53 2.52 2.40 2.38 7.40 6.98 6.90 6.77

250 11 8 6 0.13 0.03 0.5 0.9 0.25 0.15

500 21 15 2.6 0.2 0.05 3.7 6.3 1.7 1.0

500 28 20 7.5 0.3 0.07 3.1 4.5 1.3 1.4

a

The anisotropy b ()bmax) as well as the length of the pores given either by Lmean ) bDmean and Lmax ) bDmax are also given.

Discussion

Figure 3. X-ray diffractogram for the solids MCM-Alx.

SiO2-X, the X-ray diffractograms indicated an amorphous background and lack of any crystalline or porous order and therefore are not shown here.

For the first group of materials (MCM-Alx) tested, we observe from Table 1 that low substitution of Si by Al in the MCM-Alx solids (MCM-Al5 and MCM-Al10) results in a small increase by almost 5-6% of the specific surface area Sp from 1304 to 1383 m2 g-1 (MCM-Al5) and 1378 m2 g-1 (MCM-Al10). But further addition of Al (samples MCM-Al15, MCM-Al20, and MCM-Al50) results in a gradual drop of specific surface areas to 1237, 1180, and 785 m2 g-1, respectively. The initial increase of the specifc surface area should be a attributed to various imperfections, defects, cracks, etc. created in the initial silicious structure by the addition of Al without destroying the organized mesoporous structure and thus forming some new cavities for adsorption. But further addition of Al (x g 15) results in extensive destruction of organized porosity, as corroborated also by the X-ray diffraction (XRD) data shown below. As a result, the specific surface area drops. In a similar fashion, the specific pore volume Vp increases slightly from 0.74 cm3 g-1 (MCM-Al0) to 0.86 and 0.84 cm3 g-1 for the samples MCM-Al5 and MCMAl10, respectively. Thereafter further addition of Al results in a drop of Vp to 0.73, 0.70, and 0.47 cm3 g-1 for the MCMAl15, MCM-Al20, and MCM-Al50 samples. The reasons should be similar to the ones discussed abovesinitial

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creation of small defects in the structure but finally extensive destruction of the MCM parent structure. The hydraulic mean pore diameter Dp ()4Vp/Sp) is slightly affected and initially increases from 2.48 to 2.52-2.53 nm, but further on it decreases to 2.40 and 2.38 nm for the solid MCM-Al50. From Figure 1 we also observe that the introduction of Al atoms into MCM-Alx solids affects the knee of the adsorption isotherms at P/P0 ∼ 0.1-0.3, a fact showing a gradual destruction of organized and ordered mesoporosity: In the final sample, MCM-Al50, the features of mesoporosity can be hardly distinguished. So this gradual destruction of the organized/ordered mesoporosity is expected to be expressed by a gradual drop of anisotropy factor bi along this series of solids MCM-Alx, x ) 0 f 50. The gradual destruction of the organized porosity can be also seen from the X-ray results shown in Figure 3. Those diffractograms show typical cubic structure (IR3 h d) similar to the ones observed for MCM-48 solids17 or AlMCM-48 solids.18-20 We observe that as the amount of Al increases, the d211 diffraction peak moves to higher 2θ angles, a fact indicating that the interpore distance decreases. Besides and more important, the sharpness of the d211 lines decreases as the Al content increases, a fact showing the deterioration of the ordered porous structure. Those results are in line with observations by other authors.21,22 The second group of materials, which were amorphous according to the XRD experiments, is based on typical silica SiO2, which possesses a random porous network. Its initial porosity is blocked by functionalization with a functional group of increasing size, tSisH, tSisCH2OH, and tSis(CH2)3OH. As expected this procedure results in a gradual drop of Sp (from 897 to 675 m2 g-1) and Vp (from 1.57 to 1.24 cm3 g-1) values (Table 1). The mean hydraulic pore diameter D h p also decreases (from 7.40 to 6.77 nm). Those results are certainly due to the gradual blocking of pores by the functional groups. This effect of gradual blocking is expected to be expressed by a gradual drop of anisotropy value bi, which in any case should not be very far away from unity for the parent material SiO2 according to Dullien.13 Let us see now how these qualitative and somehow intuitive predictions as to the variation of b in MCM-Alx and SiO2-X, can be expressed in a more quantitative way. From the N2 porosimetry measurements we can easily calculate at each Pi ()P/P0) the values corresponding to Spi (m2 g-1) and to Vpi (cm3 g-1). After expressing those values in similar units, we can carry out the following calculation: At each Pi we assume that a number Ni of cylindrical pores is filled up. Those pores have a diameter Di and a lengh Li. The ratio

bi )

Li Li ) Di 2ri

reasons. Then we assume that

Spi ) Ni(2πri)Li ) Ni(2πri)(2biri) ) 4πNibiri2 (8) Vpi ) Ni(2πri2)Li ) Ni(πri2)(2biri) ) 2πNibiri3 (9) and easily

Spi3/Vpi2 ) [(4π)3/(2π)2](Nibi) ) 16π(Nibi) ) 16πλi (10) where λi ) Nibi. This product (Nibi) ) λi provides a measure of the total anisotropy of the pores i filled at that particular pressure Pi. Although the particular values of Ni and bi are not known, their product (Nibi) ) λi is very useful as such, since it provides the variation of total anisotropy of the porous network as the pores are filled up. In Figure 4 there are two such examples of variation of λi ) Nibi as a function of Pi ()P/P0) as well as a function of rp ) Dp/2, for one material with ordered porosity, MCM-Al0, and another material with a random porous network, SiO2SiH. We observe that the total pore anisotropy λi at both and each solid is initially high. This is expected since Ni and/ or bi should be large. Then as the partial pressure Pi ()P/ P0) of N2 increases, the total anisotropy λi decreases, as the pores are filled up. This behavior is common both to MCM-Al0 as well as to SiO2-SiH material as perhaps expected. Nevertheless there is a critical difference in MCM-Al0: At the point where the filling of mesopores is completed, i.e., at the knee in the adsorption isotherm, a kink of the λi is observed. This is due to the fact that suddenly, exactly at this point, a large number of pores Ni with large anisotropy bi are filled up. On the contrary in the silicate material with the random porous network, there is a continuous drop of λi ) biNi, which reflects the continuous smooth flooding of larger and larger pores with liquid N2. Nevertheless the problem of estimating the particular values of bi at each particular Pi ()P/P0) or Ni remains. A way out of this dilemma is to consider that

Li ) riRi

Then following a procedure similar to that applied above for eqs 8, 9, and 10, we can easily obtain

riRi-1 riRi-1 ) 16πNi ) [(4π) /(2π) ]Ni 0 2 2 V2 Si3

(17) Chen, F. X.; Huang, L.; Li, R. Chem. Mater. 1997, 9, 2685. (18) Schmidt, R.; Akporiage, D.; Stocker, M.; Ellested, O. H. J. Chem. Soc., Chem. Commun. 1994, 1494. (19) Romero, A. A.; Alba, M. D.; Klinowski, J. J. Phys. Chem. B 1998, 102, 123. (20) Chem., F. X.; Song, F.; Li, Q. Micropor. Mesopor. Mater. 1999, 29, 305. (21) Reddy, K. M.; Song, C. Catal. Lett. 1996, 36, 103. (22) Tanev, P. T.; Chibwe, M.; Pinnavaia, T. J. Science 1994, 368, 321.

3

2

(12)

i

Since (Si3/Vi2)/(16π) ) λi (see eq 10), after rearranging and taking logarithms we have

(7)

is a factor we shall call the anisotropy factor for obvious

(11)

log(Si3/Vi2) ) log(16πNi) + log

riRi-1 2

(13a)

or

log[(Si3/Vi2)/(16π)] ) log(λi) ) (log Ni - log 2) + Ni + si log ri (13b) (Ri - 1)log ri ) log 2 Then plots of log(λi) vs log(ri) should provide lines with slope si ) Ri - 1 at each point i. Two typical such plots for the materials MCM-Al0 with ordered porosity and SiO2SiH with random porosity are shown in Figure 5.

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Figure 4. Variation of the parameter λi ) biNi as a function of P ()Pi/P0) (left) and ri (right) for the ordered porous solid MCM-Al0 (upper part) and the disorder porous material SiO2tSiH (lower part).

Figure 5. Presentation of relationship (13b) in the form log(λi) ) f(log ri).

The slope at each point i of the plots in Figure 5 is equal to si ) Ri - 1, and since Li ) biDi ) 2ribi ) riRi, it follows that

bi ) 0.5riRi-1 ) 0.5risi

(14)

Therefore we can estimate the values of the anisotropy bi of each group of pores i from eq 14 at each ri. This is shown in Figure 6 for all the materials MCM-Alx as well as in Figure 7 for all the materials SiO2-X. The corresponding values of bmax as well as of the pore length, or rather the maximum pore length Lmax ) bDmax, as well as the mean pore length Lmean ) bDmean are cited in Table 1. We mention that Dmax was estimated from the maxima of PSD while the Dmean was estimated from the relationship Dp ) 4Vp/ S p. The results in Figure 6 as well as in Table 2 verify the intuitive assumption that the pores in the material MCMAlo are long and ordered and characterized by high values of anisotropy parameter bi. At the pressure Pi ()P/P0), where the mesopores are filled, bi ∼ 250. Since the mean D h p ) 2.48 nm (see Table 1) the length of those pores is around 500-600 nm or 0.5-0.6 µm. Addition of Al in the solids MCM-Al5 and MCM-Al10 results in a gradual destruction of ordered mesoporosity, as shown clearly in the corresponding N2 adsorption-desorption isotherms in Figure 1. This is expressed by a drop of anisotropy to values bi ∼ 11 for MCM-Al5 and bi ∼ 8 for MCM-Al10. The mean diameter D h p of those mesopores is around 2.522.53 nm; therefore their length should be around 28 nm for MCM-Al5 and 20 nm for MCM-Al10. For the material MCM-Al15, bi ∼ 6, which means that the length of pores is 2.52 × 6 ∼ 15 nm. Up to this point we can speak about organized mesoporosity and pores having cylindrical shape. Then for sample MCM-Al20, the anisotropy bi obtains values around 0.13. Since in this case D h p ∼ 2.4 nm (see Table 1) it follows that the length of pores is around 0.3 nm. In this case we can imagine that the pores are more similar to shallow cavities rather than cylinders. Finally for the sample MCM-Al50, the anisotropy bi is only ∼0.03. Since D h p ∼ 2.38 nm in this sample, it follows that the apparent pore length is only ∼0.07 nm. This should correspond to a multifractured structure without any conventional pores extended in a straight way.

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Figure 6. The pore anisotropy bi for the materials MCM-Alx estimated for each group of pores with radius ri.

As far as the materials SiO2-X are concerned (Figure 7), the anisotropy of the parent silica SiO2 is around bi ∼ 0.5. Then functionalization with tSisH groups increases the anisotropy to bi ∼ 0.9, which means that Li ≈ Di, in agreement with previous observations by Dullien based on microphotography.13 This effect is understood if we imagine that the tSisH groups are small enough and do not block completely the pores but make them narrower; therefore the ratio bi ) Li/Di increases. But functionalization with tSisCH2OH and tSis(CH2)3OH groups results in a drop of maximum anisotropy to bi ) 0.25-0.3 and further to bi ) ∼0.15. These results can be appreciated if we assume that the narrower parts of pores are gradually blocked by the groups added; therefore their length is restricted and the bi ) Li/Di values decrease too. As noticed above, those pores correspond actually to shallow cavities. We emphasize the fact that the values of bi obey actually some kind of distribution and the values of bi used in the

above discussion for MCM-Alx and SiO2-X materials refer to the maximum value of that distribution. An important point is that the above methodology for the estimation of anisotropy parameter b, and consequently of an apparent pore length Lp ) bDp, is valid mainly for pores having a more or less cylindrical structure like MCM solids. On the contrary, in cases where the pore structure deviates from cylindrical geometry, the results are expected to exhibit large deviations from reality. This is probably the reason that for samples with disordered structure like MCM-Al20 and MCM-Al50 as well as the SiO2-X solids the values of anisotropy b appear less than unity so the length of pores appears less than their diameter which is difficult to understand. A correction factor F could be introduced in this case according to relationship (2) which obtains values between unity (for nonintersecting capillaries) to 0.22-0.23 for more complex pore geometries.14 Then, eq 1 should obey

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Langmuir, Vol. 20, No. 16, 2004

Pomonis and Armatas

Figure 7. The pore anisotropy bi for the materials SiO2-X estimated for each group of pores with radius ri.

the form bi ) Li/FDi and therefore

Fbi ) Li/Di ) bapparent

(15)

Therefore in such cases, where we are away from the cylindrical geometry, the values of estimated/apparent anisotropy factor bapp contain the correction factor F and

bi,true ) bi,apparent/F

(16)

Since F obtains values as low as 0.22-0.23, then the bi,true should be four or five times larger compared to bi,apparent. So the values of bmax in Table 1 which appear abnormally low (b ) 0.13 and 0.03 for MCM-Al20 and MCM-Al50 and b ) 0.25 and 0.5 for SiO2-SiCH2OH and SiO2-S(CH2)3OH) correspond actually to values of b higher by four or five times. Those corrected values approach unity and are much more realistic. But the exact value of correction factors F cannot be easily estimated for each particular solid. So the method provides accurate results for cylindrical pores but as we move to more and more disorder porosities some correction factor F is needed. But by the same token, if we accept the experimental observation of Dullien13 that L ∼ D, the proposed methodology provides values of b which give us a kind of “stick” to estimate the

departure of pores, in random porous systems, from the pure cylindrical geometry. Conclusions A simple method for the estimation of pore anisotropy factor bi ) Li/Di is proposed, based on N2 adsorptiondesorption isotherms. The method is based on the estimation of surface area Spi, the pore volume Vpi, and pore radius ri ) Di/2 corresponding to each partial pressure Pi ()P/P0) of adsorption isotherm. Then plots of log(Spi3/Vpi2) vs log ri provide lines whose slope at each point of them equals si. Then the anisotropy factor bi can be estimated from the simple relationship bi ) 0.5risi. This methodology applies better to pores with pure cylindrical geometry, while for random porous systems deviating from this geometry some kind of correction is needed for the estimation of bi. Acknowledgment. We acknowledge financial support from EU under the project GROWTH-INORGPORE (G5RD-CT-2000-00317). LA049470N