Relationship between Pore Connectivity and Mean Pore Size in

corresponding specific surface areas Sp (m2 g-1), specific pore volumes (cm3 g-1), the mean hydraulic pore diameter dp (4Vp/Sp) (nm), the maximum of t...
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Langmuir 2001, 17, 8397-8404

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Relationship between Pore Connectivity and Mean Pore Size in Modulated Mesoporous Vanado-Phosphoro-Aluminates and Some Similarities with the Branching of Trees P. J. Pomonis,* K. M. Kolonia, and G. S. Armatas Department of Chemistry, University of Ioannina, Ioannina 45110, Greece Received July 19, 2001. In Final Form: October 10, 2001 Sixteen mesoporous vanado-phosphoro-aluminate solids, of the general formula Al100PXVY (X, Y ) 0, 5, 10, 20), have been prepared and characterized by their N2 adsorption/desorption isotherms. The materials were truly mesoporous, and the addition of P and/or V affects in a very precise and profound way the corresponding specific surface areas Sp (m2 g-1), specific pore volumes (cm3 g-1), the mean hydraulic pore diameter dp (4Vp/Sp) (nm), the maximum of the pore size distribution (psd) dmax (nm), and the full width at half-maximum (fwhm) (nm) of the psd. The pore connectivity c was also determined and the dimensionality Dcc of adsorption was estimated in the pressure range where capillary condensation of N2 in the pores takes place. The dimensionality Dcc was found to be related to the dmax, reduced over the fwhm, via the relation Dcc ≈ 3.7(dmax/fwhm). The two parameters, dimensionality Dcc and connectivity c, are also related to each other via the simple relation log c ∝ -1.83 log Dcc, which in turn results in the relation log c ∝ -1.83 log(dmax/fwhm). A parallelism is drawn between the connectivity of pores in solids and its relation to the pore size, on one hand, and the degree of branching b of trees and its relation with the branch diameter dk, on the other. Using allometric relations already established for such hierarchically developed natural systems, that is, trees, we reach a relation between b and dk of the form log b ∝ -2.34 log(dk,max/2dmean) where dk,max is the average diameter of the higher order (thinner) branches in a typical tree and dmean is the arithmetic mean diameter of all branches of that tree. The reasons for these similarities are discussed.

Introduction The N2 adsorption/desorption isotherms at 77 K are the most usual and valuable technique for characterization of mesoporous solids. From such isotherms, the specific surface area Sp (m2 g-1), the specific pore volumes Vp (cm3 g-1), and the hydraulic mean pore diameters dp ) 4Vp/Sp (nm) are routinely found.1 Another parameter of porous solids which can be also calculated from such N2 isotherms is the dimensionality D of adsorption. Indeed, various surface phenomena related to heterogeneous chemistry, surface science, and materials processing display statistical or random self-similarity in many orders of magnitude.2-4 Therefore, determining the fractal surface dimension of porous media is an important step toward understanding reaction and diffusion mechanisms in porous reactants or catalysts,3-7 percolation of fluids,8-10 * Corresponding author. Tel: +30651 98350. Fax: +30651 98795. E-mail: [email protected]. (1) (a) Gregg, J. S.; Sing, W. S. K. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1991. (b) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids; Academic Press: San Diego, 1999. (c) Lowell, S. Introduction to Powder Surface Area; John Wiley and Sons: New York, 1979. (2) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: New York, 1983. (3) The Fractal Approach to Heterogeneous Chemistry; Avnir, D., Ed.; Wiley: New York, 1989. (4) Fractals in the Natural and Applied Sciences; Novak, M. M., Ed.; North-Holland: Amsterdam, 1994. (5) (a) Coppens, O. M.; Froment, F. G. Chem. Eng. Sci. 1995, 50, 1013. (b) Coppens, O. M.; Froment, F. G. Chem. Eng. Sci. 1995, 50, 1027. (6) Coppens, O. M.; Froment, F. G. Chem. Eng. Sci. 1994, 49, 4897. (7) Coppens, O. M.; Froment, F. G. Chem. Eng. Sci. 1996, 51, 2283. (8) Sahini, M. Applications of Percolation Theory; Taylor and Francis: London, 1994. (9) Stauffer, D.; Aharony, A. Introduction to Percolation Theory, 2nd ed.; Taylor and Francis: London, 1994. (10) Oxaal, V. et al. Nature 1987, 329, 32.

the fracture of materials,10,11 and so on. Experimental determinations of fractal dimensions can be obtained by employing adsorption techniques,13-18 scanning electron microscopy,19-20 small-angle X-ray scattering,21-25 nuclear magnetic relaxation,26,27 and thermal gravimetric methods.28 For materials with extensive micro- and/or macroporosity,1 which are routinely characterized by N2 porosimetry, this kind of data can be used to estimate the surface fractal dimension D by employing two methods, either the modified Frenkel-Hill-Halsey (FHH) equation29-34 or the (11) Lung, W. C.; Zhang, Z. S. Physica D 1989, 38, 242. (12) Ali, M.; Gennert, A. M.; Glavkson, G. T. In Applications of Fractals and Chaos; Crilly, J. A., Earnshaw, A. R., Jones, H., Eds.; Springer Verlag: Berlin, 1993. (13) Avnir, D.; Farin, D.; Pfeifer, P. J. Chem. Phys. 1983, 79, 3566. (14) Avnir, D.; Farin, D.; Pfeifer, P. Nature 1984, 308, 261. (15) Avnir, D.; Farin, D.; Pfeifer, P. J. Colloid Interface Sci. 1985, 103, 112. (16) Avnir, D.; Farin, D.; Pfeifer, P. Phys. Rev. Lett. 1989, 62, 1997. (17) Rolle-Kampczyk et al. J. Colloid Interface Sci. 1993, 159, 366. (18) Wang, F.; Li, S. Ind. Eng. Chem. Res. 1997, 36, 1598. (19) Krohn, E. C.; Thompson, H. A. Phys. Rev. B 1986, 33, 6366. (20) Nakabayashi, H.; Murataki, T.; Tanigaki, M. Colloids Surf. 1998, 139, 163. (21) Schaefer, W. D.; Martin, E. J.; Wiltzius, P.; Cannell, D. Phys. Rev. Lett. 1984, 52, 2371. (22) Stermer, L. D.; Smith, M. D.; Hurd, J. A. J. Colloid Interface Sci. 1989, 131, 592. (23) Hurd, J. A.; Schaefer, W. D.; Smith, M. D.; Ross, B. S.; Mehaute, L. A.; Spooner, S. Phys. Rev. B 1989, 39, 9742. (24) Weidler, G. P.; Degovics, C.; Laggner, P. J. Colloid Interface Sci. 1998, 197, 1. (25) Froehlich, J.; Kreitmeier, S.; Goentz, D. Gummi Kunstst. 1998, 51, 370. (26) Mandelson, S. K. Phys. Rev. B 1986, 34, 6503. (27) Rigpy, P. S.; Gladden, F. L. Chem. Eng. Sci. 1996, 51, 2263. (28) Tao, P. D. Thermochim. Acta 1999, 338, 125. (29) De Gennes, J. P. In Physics of Disordered Materials; Adler, D., Fritzsche, H., Ovhinsky, R. S., Eds.; Plenum: New York, 1985. (30) Avnir, D.; Jaroniec, M. Langmuir 1989, 5, 1431. (31) Pfeifer, P.; Cole, W. M. New J. Chem. 1990, 14, 221.

10.1021/la011126s CCC: $20.00 © 2001 American Chemical Society Published on Web 11/30/2001

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thermodynamic method.35-38 These two methods are eventually equivalent to each other as shown recently by various groups.39,40 The calculation of D values, using the modified FHH equation, takes place using plots of the form

ln V ) constant + (D - 3) lnln(P0/P)

(1)

where V is the adsorbed volume of nitrogen at 77 K and P/P0 is the partial pressure. The calculation of D using the thermodynamic method takes place using the following kind of plots:

ln S ) constant + (D - 2) ln rp

(2)

where S is the calculated surface area and rp is the corresponding pore radius. We emphasize that the application of the above equations shows some differentiation in the calculation of D values with the range of P/P0, which is due to the limits of capillary or multilayer condensation. To be more precise, according to eq 1 two values of D can be estimated, one lower (D ) 2.20-2.40) at lower ranges of P/P0 and another higher (D ) 2.40-2.50) at higher ranges of P/P0.41 Intuitively, those values could be ascribed to a 2D-like adsorption at low P/P0 and a rather 3D-like dimensional adsorption, which approaches gradually condensation, at higher values of P/P0. Equation 2 provides much richer information provided that we extend its application to the range of P/P0 where condensation takes place. In a previous relevant work,42 it was shown that plots of ln S versus ln rp in such an extended range of P/P0 exhibit at least two distinct slopes corresponding to two (D - 2) values, or two distinct values of exponent 1/m in the FHH equation,

N ) K/(P0/P)1/m

(3)

At low P/P0 values, the D values are around 2 and correspond practically to the dimensionality of the surface as “seen” by the adsorbing species. This value will be designated in the following as Dsa, meaning dimensionality of surface adsorption. But at higher P/P0 values, where pore condensation takes place, the log S versus log rp plots result in slopes which give D values between 4 and 12 (!). These values will be designated in the following as Dcc meaning dimensionality of capillary condensation. It was suggested there that such high Dcc values correspond actually to the clustering of N2 molecules during their supercritical condensation into the pores.42 It is exactly this exponent, in the critical condensation region, which (32) Pfeifer, P.; Odert, M.; Cole, W. M. Proc. R. Soc. London, Ser. A 1989, 423, 169. (33) Pfeifer, P.; Wu, J. Y.; Cole, W. M.; Krim, J. J. Phys. Rev. Lett. 1989, 62, 1997. (34) Yin, Y. Langmuir 1991, 7, 216. (35) Neimark, A. V. JETP Lett. 1990, 51, 607. (36) Neimark, A. V.; Hanson, M.; Unger, K. K. J. Phys. Chem. 1993, 97, 6011. (37) Neimark, A. V.; Unger, K. K. J. Colloid Interface Sci. 1993, 158, 412. (38) Neimark, A. V. Adsorpt. Sci. Technol. 1990, 7 (4), 210. (39) Jaroniec, M. Langmuir 1995, 11, 2316. (40) Sahouli, B.; Blackev, S.; Browery, F. Langmuir 1996, 12, 2872. (41) (a) Stathopoulos, N. V.; Petrakis, E. D.; Hudson, J. M.; Falaras, P.; Neofytides, G. S.; Pomonis, P. J. Stud. Surf. Area Catal. 2000, 128, 593-602. (b) Pomonis, J. P.; Ladavos, K. A. Adsorption of Gases in Porous Solid Surfaces. In Encyclopedia of Surface and Colloid Science; Marcel Dekker: New York, 2001. (42) Petrakis, E. D.; Pashalidis, I.; Theoharis, R. E.; Hudson, J. M.; Pomonis, J. P. J. Colloid Interface Sci. 1997, 185, 104-110.

will be examined in this paper and which will be calculated using eq 2 suggested originally by Neimark.35-38 Additional information which can be extracted from N2 adsorption/desorption measurements and the corresponding pore size distribution (psd) is related to percolation effects.8,9 Percolation analysis pays special attention to the connectivity of the pore system and its relation to the adsorption-desorption hysteresis loop.43 Since the observed hysteresis in nitrogen isotherms may be related to the topology of the pore network, the natural language to describe this phenomenon is percolation theory. Taking the desorption isotherm as a bond percolation phenomenon and the assumption that pore network effects dominate, Seaton and co-workers43-47 have proposed a method to determine the mean coordination number of the pore network, c, and the characteristic size of the particles, L, expressed as the number of the pore lengths. The procedures for the determination of c and L from the experimental adsorption data can be summarized as follows:43 First, the psd is obtained using the method of Barrett et al.48 with a cylindrical pore model. Then, the bond occupation probability f is obtained as a function of percolation probability F from the adsorption and desorption isotherms, using the psd obtained above as an input. Finally, the best c and L values are obtained by fitting the experimental scaling data (F, f) obtained above to a generalized scaling relation between F and f. For a detailed description, one may refer to the work of Seaton.43 In our analysis, the generalized scaling relation

Lβ/νcΦ ) G[(cf - 3/2)L1/ν]

(4)

was constructed using the simulation data of Kirkpatrick.49 The critical exponents β and ν have values of 0.41 and 0.88, respectively.49 As far as we know, it is unknown to what extent those two properties, dimensionality of condensation Dcc and pore connectivity c, might be related. Although intuitively this might not be impossible, such a kind of scrutinization is difficult because of the lack of an extended range of data, which is necessary for a meaningful comparison. The purpose of this work is testing to what extent Dcc and c are interrelated in a class of mesoporous vanadophosphoro-aluminates possessing modulated porosity.50-52 Experimental Section Preparation of Specimens. The samples prepared have the general formula Al100PXVY-600 where X, Y ) 0, 5, 10, and 20 and 600 °C is the final firing temperature. The preparation took place as follows: The calculated amounts of Al(NO3)3‚9H2O (Merck p.a.) and H3PO4 (Ferak p.a.) were dissolved in 250 mL of distilled water, and V2O5 which was dissolved in 10 mL of NH4OH was added to the first solution. Then, NH4OH (Ferak p.a.) was added gradually under stirring up to pH ) 9.5. The formed gel was dried at 110 °C for 24 h. Since relevant thermogravimetric studies51 had shown that such gels lose weight around 300-350 (43) Seaton, A. N. Chem. Eng. Sci. 1991, 46, 1895. (44) Liu, H.; Zhang, L.; Seaton, A. N. Chem. Eng. Sci. 1992, 47, 4393. (45) Liu, H.; Zhang, L.; Seaton, A. N. J. Colloid Interface Sci. 1993, 156, 285. (46) Liu, H.; Zhang, L.; Seaton, A. N. Langmuir 1993, 9, 2576. (47) Liu, H.; Seaton, A. N. Chem. Eng. Sci. 1994, 49, 1869. (48) Barrett, P. E.; Joyner, G. L.; Halenda, H. P. J. Am. Chem. Soc. 1951, 73, 373. (49) Kirkpatrick, S. In III-Condensed Matter; Ballian, R., Mayward, R., Toulouse, G., Eds.; North-Holland: Amsterdam, 1979. (50) Kolonia, M. K.; Petrakis, E. D.; Angelidis, N. T.; Trikalitis, N. P.; Pomonis, J. P. J. Mater. Chem. 1997, 7 (9), 1925. (51) Gougeon, D. R.; Bodart, R. P.; Harris, K. R.; Kolonia, M. D.; Petrakis, E. D.; Pomonis, J. P. Phys. Chem. Chem. Phys. 2000, 2, 5286. (52) Kolonia, M. K.; Petrakis, E. D.; Vaimakis, C. T.; Economou, D. E.; Pomonis, J. P. Thermochim. Acta 1997, 293, 93.

Modulated Mesoporous Vanado-Phosphoro-Aluminates Table 1. Specific Surface Areas Sp, Specific Pore Volumes Vp, and Mean Pore Sizes dp of the Al100PXVY Solidsa sample Al100P0V0 Al100P5V0 Al100P10V0 Al100P20V0 Al100P0V5 Al100P5V5 Al100P10V5 Al100P20V5 Al100P0V10 Al100P5V10 Al100P10V10 Al100P20V10 Al100P0V20 Al100P5V20 Al100P10V20 Al100P20V20

Sp (BET) Vp (BET) dp (4Vp/Sp) dmax of fwhm (m2 g-1) (cc g-1) (nm) psd (nm) (nm) 201.3 245.3 319.8 239.5 270.5 352.6 386.1 336.0 301.9 359.0 321.5 257.9 176.7 286.3 305.2 199.9

0.430 0.518 1.192 0.928 0.446 0.703 1.146 0.475 0.497 0.833 1.109 0.358 0.456 0.743 1.020 0.245

8.5 8.4 14.0 15.5 6.6 8.0 11.9 5.7 6.6 9.3 13.8 5.6 10.3 10.4 13.4 4.9

6.54 6.34 9.14 10.74 5.44 6.24 9.24 10.34 5.24 6.94 10.24 12.44 8.14 7.34 9.24 12.64

2.5 2.4 4.4 6.3 2.1 2.9 4.6 5.9 2.1 3.4 5.4 10.2 5.2 4.1 6.0 10.3

a

The maximum dmax and the fwhm of the pore size distribution are also shown. Table 2. Dimensionality of Adsorption Calculated According to the Thermodynamic Method (Equation 2)a thermodynamic method (eq 2) sample

Das (low P/P0, surface adsorption)

Dcc (high P/P0, capillary condensation)

Al100P0V0 Al100P5V0 Al100P10V0 Al100P20V0 Al100P0V5 Al100P5V5 Al100P10V5 Al100P20V5 Al100P0V10 Al100P5V10 Al100P10V10 Al100P20V10 Al100P0V20 Al100P5V20 Al100P10V20 Al100P20V20

2.25 2.15 2.10 2.16 2.22 2.10 2.10 2.16 2.20 2.09 2.21 2.74 2.78 2.04 2.18 2.53

10.48 8.18 7.27 6.02 10.05 8.24 7.20 6.15 10.90 7.98 6.71 5.02 6.12 6.09 5.57 5.01

a The D and D values correspond to different D values at low sa cc (Dsa, surface adsorption) and high pressures (Dcc, capillary condensation) according to the corresponding plots in Figure 3.

°C and stabilize their weight above this temperature, the heating to this temperature took place very slowly (∼1° min-1) in a tubular furnace under atmospheric conditions. The final firing temperature was set to 600 °C for a 6 h period. The 16 samples prepared and some of their properties are in Table 1. Surface Area and Porosity. The pore size distribution measurements were carried out by using a Fisons Sorptomatic 1900 instrument. The characterization techniques included the determination of nitrogen adsorption-desorption isotherms at 77 K from which the pore size distributions were also found and shown in Figure 1. The calculated specific surface areas and pore volumes are cited in Table 1.

Langmuir, Vol. 17, No. 26, 2001 8399 Table 3. Connectivities c and Characteristic Size L of the Particle sample

c

L

Al100P0V0 Al100P5V0 Al100P10V0 Al100P20V0 Al100P0V5 Al100P5V5 Al100P10V5 Al100P20V5 Al100P0V10 Al100P5V10 Al100P10V10 Al100P20V10 Al100P0V20 Al100P5V20 Al100P10V20 Al100P20V20

6.55 ( 0.28 12.28 ( 0.26 11.21 ( 0.22 13.95 ( 0.42 5.85 ( 0.05 6.99 ( 0.06 9.06 ( 0.43 9.56 ( 0.69 5.85 ( 0.06 7.80 ( 0.11 10.55 ( 0.44 14.97 ( 0.22 10.63 ( 0.17 8.94 ( 0.24 10.86 ( 0.17 15.17 ( 0.21

1.94 ( 0.10 1.64 ( 0.06 2.04 ( 0.08 1.62 ( 0.06 2.37 ( 0.05 2.03 ( 0.03 1.57 ( 0.04 1.36 (0.03 2.51 ( 0.06 1.84 ( 0.03 1.45 ( 0.04 1.25 ( 0.07 1.58 ( 0.04 1.74 ( 0.07 1.43 ( 0.02 1.29 ( 0.01

Dsa and Dcc, are included in Table 2 and shown in Figure 3 as a function of sample composition. The Calculation of Connectivities. The procedure for the calculation of connectivities c, closely following Seaton,43 can be summarized as follows: The bond occupation probability f was obtained as a function of percolation probability F from the adsorption isotherms (Figure 1) using the pore size distribution obtained as follows:

∫r*r)∞ nr dr f ) r)∞ ∫0 nr dr

(5)

Vflat max - Vdes f ) F Vflat max - Vads

(6)

where nr is the corresponding psd using the BJH method for cylindrical pores, Vflat max is the part of the desorption curve before the start of desorption, and Vdes and Vads are the corresponding volumes in the desorption and adsorption curves. Then, the best c and L values are obtained by fitting the experimental scaling data (F, f) obtained above to the generalized scaling relation (eq 4) between F and f.49 For a detailed description, one may refer to the original work of Seaton.43 Typical fitting results are shown in Figure 4. The calculated values of c and L are in Table 3. The variation of connectivities c is shown as a function of sample composition in Figure 3. Before going to the discussion, we mentioned that the information from the above analysis is restricted to microand mesopores, while no information relevant to macropores is obtained. Besides, the connectivity values correspond to a mean coordination number and no information about its distribution in the pore network is provided. Discussion

Results The Estimation of Fractal Dimensionality of Adsorption. The calculation of dimensionality D of adsorption took place using the thermodynamic approach (eq 2). For the calculation of D values according to eq 2, the drawn lines are in Figure 2. Those lines show at least two distinct slopes, one at low P/P0 and another at higher P/P0. From these slopes, two D values were calculated and are designated in the following as Dsa (meaning dimensionality of surface adsorption) and Dcc (meaning dimensionality of capillary condensation). Those values,

The Correlation between Dimensionality of Condensation Dcc, Connectivity c, and Maximum Pore Size dmax. As can be seen from the data shown in Figure 3, between the connectivities c and the dimensionalities of adsorption Dcc in the capillary condensation range there exists an inverse relationship. Namely, when c increases, Dcc appears to drop and vice versa. After some trials, we reached the conclusion that the best linear fitting between those two quantities is achieved when a log-log plot is used. So, in Figure 5 we have plotted the log c values versus the log Dcc quantities.

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Figure 1. Nitrogen adsorption-desorption isotherms (77 K) in Al100PXVY mesoporous solids and the corresponding pore size distributions (BJH).

In this figure, an additional point corresponding to a typical MCM-48 sample is shown. For this specific sample, the adsorption-desorption isotherms, the fitting of eq 4 for the estimation of connectivity c, and the plot according to eq 2 for the estimation of Dcc values are shown in Figure 6. We mention that the used pore size distribution plot, necessary for the estimation of c and L values in MCM48, was that of Howarth-Kawazoe and not the BJH. This is because the BJH theory does not apply in the MCM materials and instead it is the HK theory/plots which provide more realistic results.53,54 The fitting data shown in Figure 5 obey the relationship

log c ) 2.56 - 1.83 log Dcc

(7)

with a correlation coefficient r ) 0.9233, which is better than 95% confidence limits. This relation means that when Dcc f 2, which is the minimum possible value for surface adsorption (Dcc f Dsa), then c f 2, in other words, the connectivity tends to exactly two: This is a remarkable result corresponding to the lowest possible connectivity, or branching, of pores. Naturally, connectivity equal to two corresponds to a lack of branching of pores and to a rather continuous, more or less ideal, tubular pore. This situation corresponds to ideal MCM structures. In our case the calculated value for c was found to be equal to ∼2.5, which shows some deviation from an ideal tubular MCM model, as probably expected in real systems. Another important point is that the dimensionality of adsorption Dcc in the capillary condensation region is related to the maximum dmax of the psd (see Table 1) reduced over the fwhm of the distribution of pores (also (53) (a) Horvath, G.; Kawazoe, K. J. Chem. Eng. Jpn. 1983, 16, 470. (b) Coustel, N.; Renzo, D. F.; Fajula, F. J. Chem. Soc., Chem. Commun. 1994, 967. (54) Kim, S. S.; Zhang, W.; Pinnavaia, J. T. Science 1998, 282, 1302.

in Table 1). This relationship is shown in Figure 7 and has the form

Dcc ≈ 3.7(dmax/fwhm)

(8)

with a correlation coefficient r ) 0.9229, which is better than 95% confidence limits. The dmax corresponds to the most probable size of pores in the solid, while the fwhm corresponds practically to 2σ where σ is the variance of the distribution.55 The higher the values of variance (or fwhm), the wider the distribution and vice versa. So the meaning of eq 8 is that the dimensionality of adsorption Dcc, when critical condensation of N2 takes place into the pores, is directly proportional to the most probable size of pores, reduced over the variance of distribution. In other words, Dcc is proportional to the slenderness of the distribution, but especially when applied to large pores. We draw attention to the fact that the Dcc values do not actually correspond to the fractal surface dimension; this parameter is in fact given by the Das values in Table 2. The Dcc is actually a scaling parameter of the critical condensation in pores providing some sense of the supercritical clustering of nitrogen molecules in the confined space. Equation 8 can be substituted into the relation in eq 7 and the result will be

log c ) 1.52 - 1.83 log[dmax/fwhm] ) 1.52 - 1.83 log(dmax/2σ)

(9)

Equation 9 clearly implies that the pore connectivity, for the materials tested, decreases when the dmax values increase but drops when the distribution of the pores is narrow. (55) Boas, L. M. Mathematical Methods in the Physical Sciences; J. Wiley and Sons: New York, 1966.

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Figure 2. Variation of log Sp as a function of ln rp for the calculation of dimensionalities Dsa and Dcc of adsorption, according to the thermodynamic method (eq 2).

Figure 3. Variation of Dsa and Dcc for the Al100PXVY solids as a function of their composition. The variation of connectivity c is also shown in the lower part.

biomimetic model, namely, the branching of trees, as a paradigm and example. In trees, we encountered branches which are branched, and the branching ratio b will be used as a parallel to the pore connectivity in solids. Besides, the new branches are thinner than the old ones, exactly as smaller pores are started out of larger ones. A Parallelism between Pore Connectivity in Solids and the Branching in Trees. As shown above, the connectivity between the pores, or the branching of pores, is high when their predominant size is small (eq 9). By the same token, it seems that porosity made up of large pores cannot be highly branched or connected. Intuitively, this situation represents exactly what we often do when we draw pores in published papers, textbooks, or the blackboard: we start with a large pore (first generation, macropore?) and then draw smaller branches out of it (second generation, mesopores?) and even smaller pores (third generation, micropores?) connected to the second generation and so on. Indeed, we cannot easily imagine a system of pores of similar size but highly branched. If we attempt to draw a highly connected/branched porous system, by necessity we are forced to move to smaller sizes. This is exactly the way the trees have developed: a high degree of branching results in thinner branches as all the children who have climbed on trees know very well. So this experience is rather common, the scientific subject is quite old, going back to Leonardo da Vinci,56

We consider that this last result has a deeper physical and natural meaning as we shall try to show using a

(56) The Notebooks of Leonardo da Vinci; Richter, P. J., Ed.; Dover: New York, 1883, Reprint 1970; Vol. I.

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Figure 4. Typical fitting results of eq 4 for the indicated solids and the corresponding c and L values.

and the relevant literature is huge. The reader can find a wealth of references in recent textbooks and collected volumes.57 (57) (a) Scaling in Biology; Brown, H. J., West, B. G., Eds; Oxford University Press: Oxford, 2000. (b) Niklas, J. K. Plant Biomechanics: An Engineering Approach to Plant Form and Function; Chicago University Press: Chicago, 1992.

The various models, based on different physical principles, which describe the development and the construction of branches in the trees, are usually categorized as follows: pipe models, hydraulic models, resistancecapacitance models, and plant-architectural models.58 Now, the interrelations between the various structural and functional variables of plants are often described by

Modulated Mesoporous Vanado-Phosphoro-Aluminates

Figure 5. Relationship between the connectivity c and the dimensionality Dcc of adsorption in the condensation part of adsorption isotherms shown in Figure 1. The triangle (2) corresponds to a typical MCM sample (see Figure 6).

the so-called allometric equations. There are literally hundreds of such equations, which have the form

y ) axb

(10)

where a and b are constants. Such an allometric relationship correlates the number of branches Nk to their radius rk.59

Nk ∝ rk-b

(11)

where k is the generation of branching as explained in Figure 8c. In the same figure, the schematic branching of a tree is exemplified according to the so-called pipe model (Figure 8a,b). Finally, in Figure 8d the experimental application of eq 11 is shown for a small poplar tree, the data read from an often-cited source59 and reproduced in Table 4.

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Figure 7. Relationship between Dcc and (dmax/fwhm) for the Al100PXVY solids.

The equation relating the total number of branches Nk of order k and the corresponding mean branch diameter dk of the same order of branching, in the cited example/ data, is

log Nk ∝ -R log dk

(12)

with the scaling constant R ≈ 2.34. For other trees, of different kind and/or age, the scaling constant R might be different but the allometric relation (eq 12) holds.58-61 Now, it can be easily appreciated that the ratio b ) Nk/Nk-1 corresponds to the branching number b and from the relation in eq 12 easily follows

log b ) log(Nk/Nk-1) ∝ -R log(dk/dk-1)

(13)

Choosing for Nk ) Nk,max and dk ) dk,max from the data cited in Table 4, it can be easily calculated that dk-1 ≈ 2dmean. Then, substitution of those values in eq 13 provides

Figure 6. Left: Typical adsorption-desorption isotherms of a MCM-48 mesoporous solid. Middle: The fitting of eq 4 for the calculation of connectivity. Right: The plot according to eq 2 for the estimation of dimensionality of adsorption. The pore size distribution employed for the estimation of c and L values was the Hovarth-Kawazoe one (see text).

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Pomonis et al. Table 4. Branching Analysis in a Small Poplar Treea order of branch (ki)

number of branches Nk

branch diameter (dk, cm)

k6 k5 k4 k3 k2 k1

1 2 15 68 256 960

14 8.0 3.5 2.0 1.1 0.7

a

Figure 8. (a) Macroscopic architecture of part of a plant branching network; (b) pipe model representation of the microscopic architecture of such a network illustrating a tightly bundled plant vascular system composed of diverging vessel elements; (c) topological representation of such a network, where k specifies the order of the level, beginning with the trunk or main stem (k ) 1) and ending with the petiole (k ) N); (d) application of the relationship in eq 11 for a small poplar tree, data from ref 59 cited in Table 4. Parts a, b, and c are adapted from ref 58.

log b ∝ -R log(dk,max/2dmean)

(14)

where R ≈ 2.34, for a small poplar tree. This equation should be compared with eq 9 describing similar quantities in a porous system.

log c ∝ -R′ log(dmax/2σ)

(15)

where R′ ≈ 1.83, for pores in solids. We consider that the similarity between the last two scaling equations provides a biomimetic model for understanding the structure of porous systems. We are very much aware of the fact that we have used just one such biomimetic example, the small poplar tree, to draw conclusions. But selection of another tree will not affect the qualitative argument, except perhaps for the scaling parameter R: It is well documented that trees in dried areas show shorter, thicker branches and a higher degree of branching b. On the contrary, trees in wet areas show longer, thinner branches and a lower degree of branching.57 That means that the scaling constant R in eq 14 will be larger in the first case as compared to the second. Perhaps in the vast diversity of trees in Nature there is somewhere a tree, or a family of trees, for which R ) R′. But actually the purpose of this work is not to trace it but to show that we can look into characteristics of porous solids from the point of view of biomimicry and hierarchical development encountered in natural systems, like trees, which are evolved according to the physical laws controlling the mass and heat transfer of liquids and gases and the strength

Data from McMahon (ref 59).

of materials.57 As far as we know, there is no such description for the development of porous solids based on fundamental physical laws except perhaps the MCM case. There are simulations of course, but those attempts describe the result but do not explain it. So perhaps a search to this direction might be fruitful. Another point which might be considered as a weakness of this work is that actually all the critical parameters c, Dcc, and dmax for the tested Al100PXVY solids were extracted from a single experiment, namely, N2 adsorption. It would be useful to try to obtain data for one of those parameters, preferably c, using a quite different technique, that is, optical, to cross-check eq 9. Finally, we are fully aware that the proposed parallelism between the pore connectivity and the tree branching is nothing more than a conjecture for the moment and more data from various sources is needed to establish the extent of its validity. The conjecture that smaller pore sizes appear to solids with higher connectivity and vice versa, as expressed by eqs 9 and 15, is not just an experimental observation but has been also predicted recently theoretically by Monte Carlo simulations in a 3D porous network by F. Rojas and co-workers.64 Namely, those authors state that (i) the smallest sites are linked to the biggest possible bonds (throats) thus acquiring a low connectivity and (ii) the biggest sites adopt the maximum possible connectivity and allocate small and medium size bonds (throats) rather than large ones. Those observations are in agreement with the results of this work. What might be the physical reason for the similarity between eq 14, which refers to the branching of trees, and eq 15 referring to the connectivity of pores? We suggest that this is due to the fractal heterosimilarity between the two systems: If the skeleton of a tree is scaled down from the centimeter size to the nanometer size and cut in random pieces and those pieces are reshuffled in any possible ways, the resulting system can be considered to represent, under a 10-7 scale reduction, the porous system of a solid, self-similar to the original one, that is, the two systems being heterosimilar. LA011126S (58) Enquist, J. B.; West, B. G.; Brown, H. J. In Scaling in Biology; Brown, H. J., West, B. G., Eds.; Oxford University Press: Oxford, 2000. (59) McMahon, A. T. Sci. Am. 1975, 231 (1), 92. (60) (a) Niklas, J. K. Am. J. Bot. 1993, 80, 461. (b) Niklas, J. K. Plant Allometry: The Scaling of form and Process; Chicago University Press: Chicago, 1994. (61) Shinozaki, O. K.; Yoda, K.; Hozumi, K.; Kira, T. Jpn. J. Ecol. 1964, 14, 97. (62) Thomas, C. S. Evol. Ecol. 1966, 10, 517. (63) (a) McMahon, A. T.; Bonner, T. J. On Size and Life; Scientific American Library: New York, 1983. (b) McMahon, A. T.; Kronauer, R. E. J. Theor. Biol. 1976, 59, 443. (64) Cordero, S.; Rojas, F.; Riccardo, J. L. Colloids Surf., A 2001, 187-188, 425.