Morphometry of Porous Solids: Lacunarity, Fractal Dimensions

Langmuir 2002, 18, 10421-10429

10421

Morphometry of Porous Solids: Lacunarity, Fractal Dimensions, Connectivity, and Some Topological Similarities with Neurons G. S. Armatas, K. M. Kolonia, and P. J. Pomonis* Department of Chemistry, University of Ioannina, Ioannina 45110, Greece Received July 10, 2002. In Final Form: October 18, 2002 The topological morphometry of 16 mesoporous phosphoro-vanado-aluminate solids possessing random porous network and 5 Al-modified MCM 48 materials, possessing ordered porosity, were investigated using the parameters lacunarity (L), fractal dimension (Dv) and connectivity (c) of their porous structure. The phosphoro-vanado-aluminate group of materials employed had the general formula Al100PXVY, with X, Y ) 0, 5, 10, 20, the balance being oxygen. The Al-MCM 48 solids contained Al in 0, 5, 10, 15, and 20%. The porosities of the solids were determined using standard N2 adsorption-desorption measurements. The BJH methodology was employed for the determination of the pore size distribution (psd) curves for the Al100PXVY solids while for the Al-MCM 48 materials the psd was estimated using the Howarth-Kawazoe method. From the psd curves, the L of the solids was determined using the formula L ) M(2)/[M(1)]2, suggested previously by Allain and Cloitre, where M(1) and M(2) are the firstand second-order momenta of distribution. The Dv of the solids was estimated from plots of the form ln Vp ) f[ln(ln(Po/P))] suggested originally by Avnir and Jaroniec, where Vp (cm3 g-1) corresponds to the specific pore volume of the porous materials. Finally the connectivity c of the porous network was determined according to the method of Seaton. The quantities L and c for all materials were found to be interrelated via the relationship: ln L ) 0.24 - 1.62 ln c, while the quantities Dv and L are interrelated via the equation Dv ) 2.47 - 1.41L. Of interest is the fact that a relationship between D and L, similar to that described above for porous networks, has also been observed previously by Smith and Lange for neurons. The physical meaning underlying this kind of heterosimilarity is discussed from the point of view of natural necessity imposed on the development of such seemingly dissimilar systems that is pores and neurons.

Introduction The necessity for understanding the characteristic features and the invisible intricacies of the form of porous solids, as well as the internal logic behind their development, has led the workers/researchers in the field to define, categorize, and investigate various properties of their porosity. Then, those properties are measured or calculated and, if the physicochemical reasons controlling them are eventually well understood, they can be manipulated accordingly for practical/technological applications, let us say adsorption and separation. Such topological properties of porous solids we shall deal with in this paper are the lacunarity (L), the fractal dimension (D), and the connectivity (c) of the porous network. It is worth noticing that all those properties are eventually estimated from a single type of experiment, namely N2 adsorption-desorption porosimetry. Trivial data obtained from such adsorption-desorption measurements concern the specific surface area Sp (m2 g-1), the specific pore volume Vp (cm3 g-1), the mean hydraulic pore diameter dp ) 4Vp/Sp (nm), and the pore size distribution (psd) from which its maximum dmax (nm) as well as its variance 2σ (nm) corresponding roughly to the full width at the half-maximum (fwhm) of distribution can be easily determined. The questions naturally occurring in such cases are (i) which of the above-mentioned properties are interrelated, (ii) how are they related, i.e., which is the mathematical/ physical form of their relationship, and (iii) why this interdependence, if any, is observed. * Corresponding author: P. J. Pomonis, Tel. +30651 0 98350, Fax +30651 0 98795, e-mail: [email protected].

One of the purposes of this paper is to trace answers to the above three questionsswhich, how, and whysfor the properties L, D, c, dmax, and 2σ which have an intensive physicochemical character. However, before going into experimental details and results, let us introduce the meaning of the less common property of L, explain what we shall mean with the term of D, and recall the meaning ofc and the method of its estimation. The Lacunarity (L). The term lacunarity was cited for the first time by Benoit Mandelbrodt1 in order to describe the gaps between various features of fractal objects. The origin of this term is from the Latin word Lacuna, for gap, and Mandelbrodt invented this property in his effort to describe objects with similar D but otherwise different texture details. In terms of fractal geometry, the term arises from the relationship

property ) (prefactor) × (quantity B)exponent (1) where the exponent corresponds to D. L corresponds to the prefactor.2,3 In the case of adsorption of various molecules having volume υ in a sample of porous solid, the number n adsorbed from each species is given by2,3 n ) βυ-R where β and R are constants for the tested system. Since the total volume V of the sample is given by V ) nυ, we can (1) Mandelbrodt, B. In The Fractal Geometry of Nature; W. H. Freeman and Co.: New York, 1977. (2) Farin, D.; Avnir, D. In The Fractal Approach to Heterogeneous Chemistry-Surfaces, Colloids, Polymers; Avnir, D., Ed.; J. Wiley and Sons: Chichester, New York, Brisbane, Toronto, Singapore, 1989. (3) Harrison, A. In Fractals in Chemistry; Oxford Science Publications; Oxford University Press: Oxford, New York, Tokyo, 1995.

10.1021/la026213e CCC: $22.00 © 2002 American Chemical Society Published on Web 11/22/2002

10422

Langmuir, Vol. 18, No. 26, 2002

Armatas et al.

write V ) β υ1-R. We may also express this in terms of a linear dimension r, the radius of the probe molecule such that υ ) kr,3 where k depends on the shape of the probe molecule. Then the last equation becomes V ) βk1-Rr3-3R ) β′r-D. The exponent D ) 3R - 3 is characteristic of the (volume) fractal dimensionality of the system and can be easily determined from relevant plots of the form ln V ) ln β′ - D ln r. The literature on this subject is very large and an extensive collection of relevant D values can be found in a collective volume edited by Farin and Avnir.2 The constant β or β′ in the above equations corresponds to the lacunarity of the system. So actually, eq 1 can be written in the form

(property i of a porous solid) ) (lacunarity) × (size of stick used to probe property i)-fractal dimentionality (2) However, while D is found on a routine basis for various systems from logarithmic plots of eq 2, on the contrary L has received little attention, if any at all. So it is rather scarce to trace any relevant data in the literature referred to L from adsorption data on porous solids. The reason is due, probably, to the fact that this pre-exponential term is complex and its interpretation is not well-developed, known, and understood. Nevertheless, Russ4 observes that fractal time profiles (fractal lines) generated by a Gaussian distribution of random number generators, of similar mean value h, but with different standard deviation σ, possesses different texture, and the parameter which differentiates them is the parameter of Lacunarity proposed by Mandelbrodt, but no interpretation has yet been developed. The only relevant hint known to us in the literature of porous materials is one made by Neimark et al.5 that lacunarity is indeed a structural characteristic of porous fractal objects. Nevertheless, despite the rarity of data on lacunarity referred to adsorption data on porous solids, there exists a very sophisticated method by Allain and Cloitre6,7 for its quantitative measurement. Without going into details, we mention that this is done using the formula

L ) M(2)/[M(1)]2

(3)

where M(1) and M(2) are the first- and second-order momenta of distribution of gaps. In the case of porous materials, the gaps actually correspond to the parts of the one-dimensional space, which do not correspond to pores. In other words, a porous solid, which contains the whole spectrum of pore sizes, does not have gaps, i.e., poreless spaces in the psd, and its L should be small. On the contrary, a porous solid with only one size of pores, like zeolites or MCM, should show large L. The Fractal Dimension (D). The fractal dimensionality of the materials is a much more elaborated property as compared to lacunarity, although they both appear in the same formula (2). For materials with extensive microand/or mesoporosity, which are routinely characterized by N2 porosimetry, this kind of data can be used to estimate either the volume Dv by employing the modified Frenkel(4) Russ, J. C. In Fractal Surfaces; Plenum Press: New York and London, 1994. (5) Neimark, A. V., Koylu, U. O.; Rosner, E. D. J. Colloid Interface Sci. 1996, 180, 590. (6) (a) Allain, C.; Cloitre, M. Nature 1991, 297, 47-49. (b) Allain, C.; Cloitre, M. Phys. Rev. A: At., Mol., Opt. Phys. 1991, 44, 35523558. (7) Turcotte, D. L. In Fractals and Chaos in Geology and Geophysics, 2nd ed.; Cambridge University Press: Cambridge, 1997.

Table 1. Specific Surface Areas Sp, Specific Pore Volumes Vp, and Mean Pore Sizes dp of the Al100PXVY Solidsa

sample

Sp (BET) (m2 g-1)

Vp (BET) (cm3 g-1)

dp (4Vp/Sp) (nm)

dmax of psd (nm)

fwhm ≈ 2σ (nm)

Al100P0V0 Al100P5V0 Al100P10V0 Al100P20V0

201.3 245.3 319.8 239.5

0.430 0.518 1.192 0.928

8.5 8.4 14.0 15.5

6.54 6.34 9.14 10.74

2.5 2.4 4.4 6.3


270.5 352.6 386.1 336.0

0.446 0.703 1.146 0.475

6.6 8.0 11.9 5.7

5.44 6.24 9.24 10.34

2.1 2.9 4.6 5.9


301.9 359.0 321.5 257.9

0.497 0.833 1.109 0.358

6.6 9.3 13.8 5.6

5.24 6.94 10.24 12.44

2.1 3.4 5.4 10.2


176.7 286.3 305.2 199.9

0.456 0.743 1.020 0.245

10.3 10.4 13.4 4.9

8.14 7.34 9.24 12.64

5.2 4.1 6.0 10.3

MCM-Al0 MCM-Al5 MCM-Al10 MCM-Al15 MCM-Al20

1304 1383 1378 1237 1180

0.74 0.86 0.84 0.73 0.70

2.27 2.49 2.44 2.36 2.37

2.48 2.52 2.53 2.52 2.40

0.74 0.77 0.97 1.15 1.10

a The maximum (d max) and the values 2σ ≈ fwhm of the pore size distribution are also shown.

Hill-Halsey (FHH) eq 48 or the surface Ds employing the so-called thermodynamic method elaborated by Neimark.9 These two methods are not completely equivalent to each other as proposed by various groups10 but provide quite differentiated information. Namely, the calculation of Dv values, using the modified FHH equation, takes place using plots of the form10,11

ln Vp ) constant - (Dv - 3)[ln ln(Po/P)]

(4)

where Vp is the adsorbed volume of nitrogen at 77 K at each P/Po and P/Po is the partial pressure. The calculation of Ds using the thermodynamic method takes place using the following kind of plots9,12

ln Sp ) constant - (Ds - 2) ln rp

(5)

where Sp is the calculated surface area at each rp and rp is the corresponding pore radius. It is important to emphasize that the two equations provide different parameters. The first (eq 4) provides the fractal dimensionality of the pore volume Dv while the second (eq 5) the fractal dimensionality of the surface Ds. (8) (a) De Gennes, J. P. In Physics of Disordered Materials; Adler, D., Fritzsche, H., Ovhinsky, R. S., Eds.; Plenum: New York, 1985. (b) Avnir, D.; Jaroniec, M.; Langmuir 1989, 5, 1431. (c) Pfeifer, P.; Cole, W. M. New J. Chem. 1990, 14, 8221. (d) Pfeifer, P.; Odert, M.; Cole, W. M. Proc. R. Soc. London, Ser. A 1989, 423, 169. (e) Pfeifer, P.; Wu, J. Y.; Cole, W. M.; Krim, J. J. Phys. Rev. Lett. 1989, 62, 1997. (f) Yin, Y. Langmuir 1991, 7, 216. (9) (a) Neimark, A. V. JETP Lett. 1990, 51, 607. (b) Neimark, A. V.; Hanson, M.; Unger, K. K. J. Phys. Chem. 1993, 97, 6011. (c) Neimark, A. V.; Unger, K. K. J. Colloid Interface Sci. 1993, 158, 412. (d) Neimark, A. V. Adsorpt. Sci. Technol. 1990, 7 (4), 210. (10) (a) Jaroniec, M. Langmuir 1995, 11, 2316. (b) Sahouli, B.; Blackev, S.; Browery, F. Langmuir 1996, 12, 2872. (11) (a) Stathopoulos, N. V.; Petrakis, E. D.; Hudson, J. M.; Falaras, P.; Neofytides, G. S. 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. (12) Petrakis, E. D.; Pashalidis, I.; Theoharis, R. E.; Hudson, J. M.; Pomonis J. P. J. Colloid Interface Sci. 1997, 185, 104-110.

Morphometry of Porous Solids

Langmuir, Vol. 18, No. 26, 2002 10423

Figure 1. Adsorption-desorption isotherms of N2 (77 K) for the Al100PXVY (X, Y ) 0, 5, 10, 20) porous solids and the corresponding pore size distribution curves calculated from the desorption branch according to the BJH methodology.

In this paper, we shall deal only with eq 4 and the Dv values calculated from it. The Connectivity (c). Another additional property, which can be calculated from N2 adsorption/desorption measurements, and the corresponding pore size distribution, is connectivity. The calculation of connectivity is based on the percolation analysis of the porous network according to the ground-breaking work of Seaton and coworkers.13 Taking the desorption isotherm as a bond percolation phenomenon and using the assumption that pore network effects dominate, they 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. It is unknown to what extent those properties,L, Dv of the porous solid, and c, might be related. Although intuitively this might not be impossible, such a kind of scrutinization is difficult because of the lack of extended range of data, which is necessary for a meaningful comparison. Since all those parameters are calculated from (13) (a) Seaton, A. N. Chem. Eng. Sci. 1991, 46, 1895. (b) Liu, H.; Zhang, L.; Seaton, A. N. Chem. Eng. Sci. 1992, 47, 4393. (c) Liu, H.; Zhang, L.; Seaton, A. N. J. Colloid Interface Sci. 1993, 156, 285. (d) Liu, H.; Zhang, L.; Seaton, A. N. Langmuir 1993, 9, 2576. (e) Liu, H.; Seaton, A. N. Chem. Eng. Sci. 1994, 49, 1869.

a simple nitrogen adsorption measurement, they are also expected to be related to the pore size distribution of the solid and, namely, to the parameters characterizing it, i.e., dmax, 2σ, and/or the ratio dmax/2σ. The purpose of this work is to test to what extent those quantities are interrelated in a class of mesoporous vanado-phosphoro-aluminates possessing modulated but random porosity14 and a class of MCM 48 solids modified with the addition of Al, which possess ordered mesoporosity. Experimental Section Preparation of Specimens. The first group of solids employed is 16 mesoporous vanado-phosphoro-aluminates possessing random porous network.13 The samples prepared have the general formula Al100PXVY-600 where X, Y ) 0, 5, 10, 20, and 600 °C, 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 (14) (a) Kolonia, M. K.; Petrakis, E. D.; Angelidis, N. T.; Trikalitis, N. P.; Pomonis, J. P. J. Mater. Chem. 1997, 7 (9), 1925-1931. (b) Gougeon, D. R.; Bodart, R. P.; Harris, K. R.; Kolonia, M. D.; Petrakis, E. D.; Pomonis, J. P. Phys. Chem. Chem. Phys. 2000, 2, 5286-5292. (c) Kolonia, M. K.; Petrakis, E. D.; Vaimakis, C. T.; Economou, D. E.; Pomonis, J. P. Thermochim. Acta 1997, 293, 93-100.

10424


Armatas et al.

Figure 2. Adsorption-desorption isotherms of N2 (77 K) for the MCM-AlX solids. The corresponding psd curves have been calculated according to the Howarth-Kawazoe method. 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 studies14c have shown that such gels lose weight around 300-500 °C and stabilize their weight above this temperature, the heating to this temperature took place very slowly (∼1 deg/min) in a tubular furnace under atmospheric conditions. The final firing temperature was set to 873 K. The samples with some of their properties are in Table 1. The MCM 48 solids possessing ordered porosity and containing 0, 5, 10, 15, and 20% Al atoms will be designated next as MCMAlX where X ) 0, 5, 10, 15, and 20. They were prepared as follows: The started materials were cetyltrimethylammonium bromide (C16TAB), water, NH3 (32% extra pure), and tetraethyl orthosilicate (TEOS, 98%). The surfactant (0.048 mol dm-3) was first dissolved in water at 303 K followed by addition of ammonia (1.03 mol dm-3) and TEOS (0.325 mol dm-3). The white product was filtrated, washed, dried at 363 K for 24 h, and then calcined slowly at a heating rate of 1 deg min-1 up to 823 K where it remained for 5 h. The five samples thus prepared, with 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. They are shown in Figure 1 for the Al100PXVY solids and in Figure 2 for the MCM-AlX materials. The calculated specific surface areas and pore volumes are cited in Table 1.

Results The Estimation of Fractal Dimensions. As explained in the Introduction, the Dv of the pore volume can be calculated from N2 adsorption data according to eq 4. Typical plots for the materials in Table 1 are shown in Figure 3. The values of Dv found form the plots in Figure 3 are collected in Table 2.

The Calculation of Connectivities. The procedure for the calculation of c, following closely that of Seaton,13 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 as follows:

∫r*r)∞nr dr f ) r)∞ ∫0 nr dr

(6)

Vflat max - Vdes f ) F V -V

(7)

flat max

ads

where nr is the corresponding psd using the BJH method for cylindrical pores, Vflat max is the part of 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 generalized scaling relation (8) between F and f.15

Lβ/νcf ) G[cf - 3/2 L1/ν]

(8)

For a detailed description, one may refer to the original work of Seaton.13 Typical fitting results are shown in Figure 4. The c and L values calculated from the hysteresis loop of the adsorption-desorption isotherms at the range 0.3 < p/p0 < 0.6 are in Table 2. (15) Kirkpatrick, S., In III-Condensed Matter.; Ballian, R., Mayward, R., Toulouse, G., Eds.; North-Holland: Amsterdam, 1979.


Langmuir, Vol. 18, No. 26, 2002 10425

Figure 3. Typical plots of the form ln V ) f[ln ln(P0/P)], for some of the samples in Table 1, according to eq 4 for the estimation of Dv values. Table 2. Dimensionality Dv of the Pore Volume, Connectivities c, Characteristic Size L of Particles and Lacunarities L for the Samples in Table 1 sample

Dv

c

L

L


2.51 2.47 2.42 2.38

6.55 ( 0.28 12.28 ( 0.26 11.21 ( 0.22 13.95 ( 0.42

1.94 ( 0.10 1.64 ( 0.06 2.04 ( 0.08 1.62 ( 0.06

0.0142 0.0143 0.0212 0.0216


2.40 2.38 2.40 2.41

5.85 ( 0.05 6.99 ( 0.06 9.06 ( 0.43 9.56 ( 0.69

2.37 ( 0.05 2.03 ( 0.03 1.57 ( 0.04 1.36 (0.03

0.0245 0.0259 0.0265 0.0316


2.40 2.43 2.35 2.44

5.85 ( 0.06 7.80 ( 0.11 10.55 ( 0.44 14.97 ( 0.22

2.51 ( 0.06 1.84 ( 0.03 1.45 ( 0.04 1.25 ( 0.07

0.0319 0.0341 0.0447 0.0545


2.46 2.36 2.44 2.42

10.63 ( 0.17 8.94 ( 0.24 10.86 ( 0.17 15.17 ( 0.21

1.58 ( 0.04 1.74 ( 0.07 1.43 ( 0.02 1.29 ( 0.01

0.0560 0.0781 0.0840 0.0142

MCM-Al0 MCM-Al5 MCM-Al10 MCM-Al15 MCM-Al20

2.15 2.26 2.10 2.03 2.12

2.6 ( 0.2 2.7 ( 0.3 2.9 ( 0.1 3.2 ( 0.2 4.0 ( 0.1

1.02 ( 0.02 0.89 ( 0.01 0.96 ( 0.04 0.86 ( 0.02 0.74 ( 0.02

0.18 0.21 0.23 0.25 0.27

The Calculation of Lacunarities. The L for each of the prepared materials, 16 Al100PXVY and 5 MCM-AlX, were calculated according to the relationship (3) proposed originally by Allain and Cloitre.6 To be more precise, the calculation of L took place as follows: We consider a network of pores with known pore size distribu-

tion, according to the Barrett-Joyner-Halenda (BJH) method for the AlPV samples and the Howarth-Kowazoe method for the MCM samples. Then, we consider that the minimum size of pores equals 0.5 nm (rmin ) 0.5 nm) and that the maximum size of pores equals 25.0 nm (rmax ) 25.0 nm). Then a segment is chosen with a unit length of r0 ) 0.5 nm which corresponds roughly to be the size of one N2 molecule (π(d/2)2 ) 0.162 nm2, d0 ≈ 0.46 nm) used as a probe. Next, we consider a density function of pore sizes, which describes an infinite number of pore groups with a particular mean pore size. The total number of pore groups considered is given by

N(r0) )

rmax - rmin r0

(9)

The number of steps of size r0 that contain s pores is defined as n(s,r0), and the sequence of probability is given by

p(s,r0) )

n(s,r0) N(r0)

(10)

while the normalized probability is

p′(s,r0) )

p(s,r0) rmax

∑ p(s,r0)

rmin

(11)

10426


Armatas et al.

Figure 4. Typical fitting results of eq 8 for the indicated solids from Table 1 and the corresponding c and L values. The c and L values for all the samples of Table 1 are collected in Table 2.

Finally, the first- and second-order momenta of this distribution are defined according to r

M1(r0) )

∑sp(s,r0)

(12)

s)1 r

M2(r0) )

s2p(s,r0) ∑ s)1

(13)

The L is defined in terms of the moments by

L(r0) )

M2(r0)

(14)

[M1(r0)]2

The calculated L values are collected in Table 2. Discussion (m2 g-1)), specific

The values of specific surface area (SP pore volume (VP (cm3 g-1)), the maximum of the psd (dmax (nm)), the variance of the distribution 2σ (nm), the ratio dmax/2σ, the connectivities (c), the fractal dimension Dv of pore volumes, and finally the lacunarity (L) are shown in Figure 5 for the Al100PXVY solids and in Figure 6 for the MCM-AlX samples.

As mentioned in the Introduction, one of the main purposes of this work is to compare the inter-relationships between the intensive parameters L, Dv, and c on one hand and the psd parameters dmax, 2σ, and their ratio dmax/2σ on the other. We should mention that the extensive parameters Sp (m2 g-1) and Vp (cm3 g-1) do not show any meaningful relationship with the intensive ones. Nevertheless, they are depicted in Figures 5 and 6 with the purpose of providing the reader with a clearer picture of the materials at hand. The relationship between the L of the 16 Al100PXVY solids and the 5 MCM-AlX materials on one hand and the fwhm of their pore size distribution on the other is shown in Figure 7. In Figure 8 , the corresponding relationship is shown between the L values and the 2σ of the psd. Those parameters (dmax and 2σ) are clearly related to L, and the corresponding equations are

L ) 0.19(2σ)-1.19

(15)

L ) 1.07(dmax)-1.67

(16)

Both equations/relationships exhibit a very high level of significance. It is worthwhile to emphasize that these relationships include both the alumino-phosphorovanadate materials with a random porosity as well as the


Langmuir, Vol. 18, No. 26, 2002 10427

Figure 5. Variation of the parameters (a) Sp (m2 g-1), (b) Vp (cm3 g-1), (c) dmax (nm), (d) 2σ ≈ fwhm (nm), (e)dmax/2σ, (f) Dv, (i)c, and (j) L as a function of sample composition for the Al100PXVY porous solids.

Figure 6. Variation of the parameters (a) Sp (m2 g-1), (b) Vp (cm3 g-1), (c) dmax (nm), (d) 2σ ≈ fwhm (nm), (e)dmax/2σ, (f) Dv (i)c, and (j) L as a function of sample composition for the MCM-AlX materials.

MCM-AlX porous solids with order porosity. The physical meaning of relationship 15 is that the wider the psd is, the smaller the L of the gaps in the pore size distribution is, as actually expected. The physical meaning of eq 16 is somehow similar showing that L of pore distribution decreases as we move to higher values of dmax. This similar kind of dependence

is routed on the well-known fact that it is difficult to develop porous solids with narrow psd at high dmax values. Nevertheless, the relationship between dmax/2σ and L is much poorer, as shown in Figure 9. Clearly, the two groups of materials Al100PXVY and MCM-AlX show somehow differentiated behavior, and as a result the correlation coefficient of the relationship ln(L) ) f[ln(dmax/

10428


Figure 7. The relationship between L and the fwhm of the pore size distribution for the 16 Al100PXVY and the 5 MCM-AlX porous solids.

Armatas et al.

Figure 10. The relationship between L and c of the porous solids.

This relationship (Figure 10) is described by the equation

L ) 1.27 c-1.62

Figure 8. The relationship between L and the dmax of the psd for all the materials tested.

(17)

with a correlation coefficient of R ≈ 0.96. High values of c are related to low L values and vice versa. This implies that the samples which appear with large L in their psd, or in other words possess narrow psd and small dmax (see Figures 7 and 8), also possess low average number of connections between their pores. We emphasize the fact that those observations are valid for the two groups of systems studied (phosphoro-vanado-aluminates with random porous networks and MCM with ordered but not interconnected porosity). Finally, in Figure 11 (left-hand part) we have plotted the values of L of the pore distribution versus the Dv of the porous materials. The figure shows that high values of L are, by necessity, accompanied by low values of Dv and vice versa. The two quantities are related via the equation

Dv ) 2.47 - 1.4L (for porous solids)

(18)

with a correlation coefficient R ) 0.91. Is such an interdependence restricted to the networks of porous solids or it is also apparent in other networks? Also, if it is apparent, in what kind of networks is it so? An answer to this short list of questions is shown in the right-hand part o Figure 11, which refers to neurons and has been taken from a recent publication by Smith and Lange.16 These researchers examined a very large number of neurons of different physiological origin and with different external features such as nerve cells (astrocytes and oligodendrocytes) and brain cells (also astrocytes and oligodendrocytes). Their purpose was to establish some kind of morphometric relationships between the L and D of neurons with their normal or abnormal functioning and/or development. Those studies resulted, among others, in the relationship shown in

Figure 9. The relationship between L and dmax/2σ.

2σ)] is much lower (R ) 0.90) compared to the R ≈ 0.98 values observed in (15) and (16). One of the most interesting results of this work is the close interrelation between L and the mean c of the pores.

(16) Smith, T. G.; Lange, D. G., Biological Cellular MophometryFractal Dimesnions, Lacunarity, and Nultifractals. In Fractals in Biology and Medicine; Losa, A. G., Merlini, D., Nonnenmacher, T., Weibel, R. E., Eds.; Birkhauser: Basel, Boston, Berlin, 1997. (b) Smith, T. G., Jr.; Lange, G. D.; Marks, W. B. J. Neurosci. Methods 1996, 69, 123-136. (c) Smith, T. G.; Lange, G. D., Jr. Fractal Studies of Neuronal and Glial Morphology. In Fractal Geometry in Biological Systems: an Analytical Approach; Iannaconne, P. M., Khoka, M., Eds.; C. R. C. Press: Boca Raton, FL, 1996, pp 173-186. (d) Smith T. G., Jr.; Marks, W. B.; Lange, G. D.; Sheriff, W. H., Jr.; Neale, E. A. J. Neurosci. Methods 1998, 26, 75-81.


Langmuir, Vol. 18, No. 26, 2002 10429

Figure 11. Left-hand: The interdependence between Dv of the porous solids and L of their pore size distribution. Right-hand: Similar for neurons after ref 15a.

Figure 11 (right-hand part), which is given by the equation

D ) 2.3 - 2.9L (for neurons)

(19)

There is a remarkable similarity between eqs 18 and 19 describing the relationship between L and D of the groups of structures (pores and neurons) which looks and sounds dissimilar in a first approach. However, in a second thought, it can be certainly appreciated that the morphometric characteristics and features of such systems, and probably other similar ones such as trees,17 obey some common laws such as the ones described by eqs 18 and 19. The DV or D of systems such as those above describes and corresponds to the density of space occupation by the pores or the neurons or by the branches in the case of trees:17 More dense occupation necessitates a higher degree of branching of neurons, trees, or pores. In the last case, that means higher pore connectivity. The higher order of branchings/connections should be, by necessity, thinner. In porous materials, this means that we move from macropores to mesopores and next to micropores. For physiological systems such as neurons and trees, this architecture is needed in order to interact better with all the parts of their environment, to make use of all the substances (water, light, minerals, air) available in it and affecting this environment in the most effective way. For porous materials developed randomly and under no specific conditions (like the Al100PXVY in this work) the development of channels (pores), from which the volatile components of the precursor escape during heating and firing, follows exactly the same logic; small escaping routes

(transformed eventually to micropores) are joined to larger escaping routes (eventually mesopores) and finally to relative large channels (macropores). This seems to be the most effective way to remove the volatiles, and the resulting porous solid bears the stamp of this process. Now if gaps of space occupation exist in the vicinity of neurons or trees, that means that this particular space does not interact with its environment, so it is practically useless for life. In pore systems, the existence of gaps in their psd means that there are no pores of this particular size for some “unnatural” reasons such as the specific method of synthesis applied, for example, in the synthesis of MCM in the present case. So, such systems are not good for operations necessitating such spaces, for example, sieving operations. So, although the concept of L may play a secondary role in the description of scaling patterns compared to D, it conveys some very important properties of the systems under study, like its stiffness in rocks,18 the efficiency of systems such as neurons16 and trees17 in interacting with their environment, and finally the efficiency of porous systems to interact with various molecules. The use of L might be considered a complementary one to that of D in describing scaling systems. Acknowledgment. Parts of this work took place with financial assistance from EU projects ERBFMRRXCT960084 and G1RD-2000-25043. LA026213E (17) Pomonis, J. P.; Kolonia, M. K.; Armatas, S. G. Langmuir 2001, 17, 8397-8404. (18) Pyrak-Nolte, J. L., Myer, R. L.; Nolte, D. D. Pure Appl. Geophys. 1992, 138, 680-706.

Morphometry of Porous Solids: Lacunarity, Fractal Dimensions

Recommend Documents