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Phosphate Alumina Process by Sol-Gel: Textural and Fractal Properties A. Balankin,*,† T. Lo´pez,‡ R. Alexander-Katz,‡ A. Co´rdova,‡ O. Susarrey,† and R. Montiel‡ Universidad Auto´ noma Metropolitana-Iztapalapa, A.P. 55-534, 09340 Me´ xico, Distrito Federal, Me´ xico, and SEPI-ESIME, Instituto Polite´ cnico Nacional, 07738 Me´ xico, Distrito Federal, Me´ xico Received September 30, 2002. In Final Form: February 6, 2003 In the present work, we report an alternative way to prepare inorganic particles of alumina by sol-gel, starting with a metal alkoxide (aluminum butoxide) with controlled pore sizes to obtain a maximum amount of air hollows occluded inside the particle to produce a favorable change in the effective refraction index of the clusters. Furthermore, we characterized the texture and fractal nature of the synthesized materials by means of transmission electron microscopy, thermodynamic analysis of adsorption-desorption isotherms, and small-angle X-ray scattering. The results of this study give a deeper insight into the mechanisms of gel formation.
Introduction There has been a considerable effort to synthesize new light-scattering particles that can replace TiO21-3 in specific applications where opaque materials are needed. Examples of this are (i) latex synthesis with hollow polymer particles, such as the Rhopaque case of Rohm and Haas addressed to applications for the paper industry; (ii) organic-inorganic hybrid synthesis,4 where structured inorganic particles were synthesized with closed pores; and (iii) inorganic hollow- or porous-particle synthesis.5-8 In all these cases, the holes or pores were on the order of 1 /2 of a wavelength of light to achieve maximum scattering. A different approach is to include a great number of micropores (or mesopores) such that the effective dielectric constant of the material is close to that of air. Here, the particles, and not the pores, are on the order of 1/2 of a wavelength. Following this idea,9 several materials were prepared by means of the sol-gel technique.10-15 To establish a relationship between the porosity and the lightscattering properties of such particles, it is necessary to have additional information about their structures. It is well-known that the structure of a particulate gel is highly disordered, but at least on some length scales, they are * Author to whom correspondence should be addressed. † Instituto Polite ´ cnico Nacional. ‡ Universidad Auto ´ noma Metropolitana-Iztapalapa. (1) Mun˜oz, E.; Boldu, J. L.; Andrade, E.; Novaro, O.; Lo´pez, T.; Gomez, R. J. Am. Ceram. Soc. 2001, 84, 392. (2) Ramanurti, M.; Leong, K. W. J. Aerosol Sci. 1987, 18, 17. (3) Pfeifer, P.; Avnir, D. J. Chem. Phys. 1983, 79, 3558. Avnir, D.; Farin, D.; Pfeifer, P. J. Chem. Phys. 1983, 79, 3566. Avnir, D.; Farin, D.; Pfeifer, P. Nature (London) 1984, 308, 261. (4) Kawahashi, N.; Matiejevic, E. J. Colloid Interface Sci.1991, 143, 103. (5) Abreu Filho, P. P.; Galembeck, F. Langmuir 1990, 6, 1013. (6) Lima, E. C. O.; Galembeck, F. Colloids Surf. 1993, 75, 65. (7) Beppu, M. M.; de Oliveira, L. E. C.; Galembeck, F. J. Colloid Interface Sci. 1996, 178, 93. (8) Beppu, M. M.; de Oliveira, L. E. C.; Sassaki, R. M.; Galembeck, F. J. Coat. Technol. 1997, 69 (No. 867), 81. (9) Digne, M.; Sautet, P.; Raybaud, P.; Euzen, P.; Toulhoat, H. J. Catal. 2002, 211, 1. (10) Yoldas, B. E. Am. Ceram. Soc. Bull. 1975, 54, 286. (11) Mizushima, Y.; Hori, M. J. Non-Cryst. Solids 1994, 167, 1. (12) Gardes, G. G.; Pajonk, G. M.; Teichner, S. J. Bull. Soc. Chim. Fr. 1976, 1321. (13) Mizushima, Y.; Hori, M. J. Appl. Catal. 1992, 88, 137. (14) Pierre, A. C.; Uhlman, D. R. J. Non-Cryst. Solids 1986, 82, 271. (15) Brinker, C. J.; Scherer, G. W. Sol-gel Science; The physics and chemistry of Sol-gel processing; Academic Press: New York, 1990; p 196.
often statistically self-similar and can be described in terms of fractal geometry, or defined by an average cluster size and a fractal dimension.16 Statistical self-similarity implies that a finer structure is revealed as the object is magnified; similarly, morphological complexity means that a finer structure (increased resolution and detail) is revealed with increasing magnification. The fractal dimension measures the rate of addition of structural detail with increasing magnification, scale, or resolution. The fractal dimension, therefore, serves as a quantifier of the complexity.17 It has been established that during the sol-gel transition the aggregates grow with a fractal structure.18 To gain a deeper insight into the complexity of the internal structures of gels and, hence, a better understanding of the mechanisms of their formations, it is desirable to probe the structures using several independent techniques. In this paper, we analyze the fractal structures of two sol-gel materials with metal alkoxides as precursors. Aluminas were modified with glycerin and phosphoric acid as additives to obtain different mesoporosities. Three methods were used to establish the fractal character of the materials: direct methods of image analysis of transmission electron microscopy (TEM) micrographs, small-angle X-ray scattering (SAXS), and the analysis of adsorption-desorption isotherms [Brunauer-Emmett-Teller (BET) analysis]. These techniques portray the gel structure from different perspectives on different length scales. Methods of Fractal Measurement. Fractal geometry represents a major advantage over previous methods for quantifying and simulating the complex structures of gels. Perhaps the most important practical aspect of fractal analysis may be the use of the fractal dimension as a quantitative variable that morphologists can study as a dependent variable in the context of many independent variables. The fractal dimension governs the chemical,18 thermodynamic,19,20 optical,17,18 and mechanical21-23 properties of gels. Moreover, to characterize the structures of gels, the most commonly used techniques are those based (16) Mandelbrot, B. B. The Fractal Geometry of Nature, 3rd ed.; Freeman: New York, 1983. (17) Gouyet, J. F. Physics and Fractal Structures; Springer: New York, 1996. (18) Bunde, A., Havlin, S., Eds. Fractals in Science; SpringerVerlag: New York, 1994. (19) Neimark, A. Physica A 1992, 191, 258; JETP Lett. 1990, 51, 607. (20) Balankin, A. S. Phys. Lett. A 1996, 210, 51.
10.1021/la026630r CCC: $25.00 © 2003 American Chemical Society Published on Web 04/02/2003
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on rheological measurements.23 In these cases, one needs a model, named the scaling theory, relating the structures of gels to rheological properties.21 Among the methods of fractal characterization of gel structures are scattering methods17,18,24,25 (light, X-ray, and neutron scattering), adsorption-desorption experiments,19,26,27 rheological measurements,23 and the fractal analysis of gel images by, for example, electron micrographs.16-18 Image analysis is the only method that permits the determination of three fractal dimensions that completely characterize the gel structure from the fractalgeometry point of view:27,28 the mass dimension of the particle aggregates (clusters), fractal dimension of the pore space, and fractal dimension of the pore-clusters interface. Moreover, the fractal-image analysis permits the detection of the loss of statistical self-similarity, employing multifractal analysis.29,30 There are numerous methods available to estimate the fractal dimensions of particle clusters, their boundaries, and pore spaces from digitized electron micrographs.16-18,31 These are based on different definitions of the metric or fractal dimension (Hausdorf-Besicovich and MinkowskiBouligand dimensions, Kolmogorov capacity, etc.)32 and may lead to different values of measured fractal dimensions.29,31 This uncertainty, as well as the uncertainty associated with converting the gray-scale micrograph into the black-and-white image, commonly used for fractal analysis, leads to differences in the data reported by authors using different techniques. So, to avoid confusion, it is important to specify the method used in image conversion and fractal-dimension estimation. In this work, we first analyze the histogram of the grayscale image (obtained by a computer-assisted electron microscope) using the Scion Image software.33 Then, the gray-scale image was converted to a black-and-white image using the histogram maximum as a threshold value. The fractal dimensions of the clusters (black) and pores (white) were found by the box-counting, DB,16,32 and masscounting, DM,16 methods using the commercial BENOIT 1.2 software.34 The fractal dimension of the cluster boundaries (pore-cluster interface) was determined by the divider, DD,16,32 and perimeter-area, DP,16,33 methods, also adopted in the BENOIT 1.2 software. At least four images of the gel obtained at each synthesis condition were analyzed. The box dimension is defined as the exponent DB in the relationship
N(∆) ∝ ∆-DB
(1)
where N(∆) is the number of boxes of linear size ∆ (21) Shih, W. H.; Shih, W. Y.; Kim, S. I.; Lu, J.; Aksay, I. A. Phys. Rev. A: At., Mol., Opt. Phys. 1990, 42, 4772-4778. (22) Balankin, A. S. Phys. Rev. B 1996, 53, 5438. (23) Hagiwara, T.; Kumagai, H.; Nakamura, K. Food Hydrocolloids 1998, 12, 29. (24) Vijayakumar, R.; Wihitby, K. T. Aerosol Sci. Technol. 1984, 3, 17. (25) Harrison, A. Fractals in Chemistry; Oxford University Press: Oxford, U.K., 1995. (26) Pfeifer, P.; Wu, Y. J.; Cole, M. W.; Krim, J. Phys. Rev. Lett. 1989, 62, 1997. (27) Sahimi, M. Flow and Transport in Porous Media and Fractured Rocks; VCH: New York, 1995. (28) Korvin, G. Fractal Models in the Earth Sciences; Elsevier: London, 1992. (29) Balankin, A. Eng. Fract. Mech. 1997, 57, 135. (30) Balankin, A. S.; Izotov, A. D.; Novikov, V. U. Inorg. Mater. 1999, 35, 1047. (31) Balankin, A. S.; Sandoval, F. Rev. Mex. Fis. 1997, 43, 545. (32) Falconer, K. Techniques in Fractal Geometry; Wiley: New York, 1997. (33) Scion Image. http://www.scioncorp.com. (34) BENOIT 1.2. http://www.trusoft-international.com.
necessary to cover a data set of points distributed in a two-dimensional plane. In the definition of the box dimension, a box is counted as occupied and enters the calculation of N(∆) regardless of whether it contains one point or a relatively large number of points (to avoid the remainder problem,31,33 N(∆) is averaged through a set of 72 covered networks with box size ∆ rotated with angle increments of 5°).34 Furthermore, to confirm the statistical autosimilarity of cluster and pore structures,31 we also have determined their information dimensions using the same images. The information dimensions effectively assign weights to the boxes in such a way that the boxes containing a greater number of points count more than the boxes with fewer points. The information entropy I(∆) for a set of N(∆) boxes of linear size ∆ is defined as N(∆)
I(∆) ) -
pi ln pi ∑ i)1
(2)
Here, pi ) ni/n, where ni is the number of pixels in the ith box and n is the total number of pixels in the image. The information dimension is determined from the scaling relationship
I(∆) ∝ -DI log ∆
(3)
It is easy to see that, in general, DI e DB and DI ) DB only for (statistically) self-similar fractals. In this work, the statistical self-similarities of cluster and pore structures were verified using the following criterion:31
DB - DI < σ[0.5(DB + DI)]
(4)
where DI is the information dimension of the structure,33 measured using the BENOIT 1.2 software and σ[...] is the standard deviations of the data for a set of studied images. If criterion 4 fails, the structure should be considered a multifractal,31 and in this way, it is characterized by an infinite spectrum of generalized dimensions, Dq, also called the Re´nyi dimension spectrum;33 otherwise, it can be treated as a statistically self-similar fractal. The divider (or ruler) dimension, DD, of the cluster boundary is determined from the scaling relationship
NR(δ) ∝ δ-DD
(5)
where NR(δ) is the number of steps taken by walking a divider (ruler) of length δ on the cluster boundary.33,34 It should be noted that, in general, a ruler of length δ will not cover the line exactly and we will be left with a remainder. The BENOIT 1.2 software keeps this remainder and, therefore, has noninteger values of NR(δ).34 The perimeter-area method is based on the relationship between the area, S, and the perimeter, P, of the cluster with a fractal boundary:16
SDP ∝ P2
(6)
where both the perimeter and the area are measured by the box-counting method.34 The criterion of statistical selfsimilarity of the cluster boundaries is31
DD - DP < σ[0.5(DP + DD)]
(7)
Otherwise, the boundary can be statistically autoaffined if31
DP ) 1/(2 - DD)
(8)
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for (statistically) self-similar boundaries DP ) DD.17 If criterion 7 fails, the boundary should be treated as a selfaffined fractal31 that possesses anisotropic scale invariance.16,33 The mass-counting or mass dimension of the structure (cluster or pore-space) is determined from the scaling relation34
M(R) ∝ RDM
(9)
where M(R) is the number of black (or white) pixels in the circle of radius R whose center is located at the image center. The mass dimension equals the box-counting dimension only for isotropic fractals. In this way, the difference DB - DM may be used as a characteristic of the structure anisotropy.31 As we mentioned before, a different method to obtain DM is given by scattering methods. In many of the applications of aggregate materials, the scattered intensity I(q) of an aggregate, relative to the incident intensity, can be factored into two terms:
I(q) ) (n/V)P(q) S(q)
(10)
where P(q) and S(q) are the form and structure factors, respectively; n is the number of individual particles in the aggregate particle; V is the volume of the sample illuminated by the beam; and the vector q is the difference between the scattered-wave vector and the incident one, and its magnitude is given by
q ) (4π/λ) sin(θ/2)
(11)
where θ is the scattering angle relative to the incident direction and λ is the wavelength in the medium. If a is the radius of the individual particles forming the aggregate and ξ is the characteristic length of the size of the aggregate particle, then for 1/ξ , q , 1/a, P(q) ) 1 and the scattered intensity will be proportional to S(q). For centrosymmetric particles, the structure factor can be written in terms of the static pair correlation function g(r) as
S(q) ) 1 +
sin(qr) dr qr
∫0∞g(r) r2
4πn V
(12)
The function g(r) describes the probability that some other particle is at a distance r from any other particle in the aggregate. For a fractal object with a fractal dimension DM, g(r) can be written as
g(r) ) (V/n)p(DM/4π)aDMrDM-3 exp(-r/ξ)
(13)
where p is the number density of the primary particles. When g(r) is substituted in eq 12, for 1/ξ , q , 1/a, S(q) can be reduced to
S(q) ) 1 + C(qa)-DM
(14)
where C is a constant. Equation 14 constitutes the basic expression to determine DM by SAXS. The fractal dimension of the pore-cluster interface may also be determined by a thermodynamic method19 based on the analysis of adsorption-desorption isotherms within a model of equicurvature surfaces, according to which the interface area, S, scales with the mean radius of curvature, r, as
S ≈ r2-Dfs where
(15)
r)
2σv RT(-ln X)
(16)
and
S)
N(Xf1) -ln X dN ∫N(X)
RT σ
(17)
where X ) P/P0, P is the current nitrogen pressure and P0 is the saturation pressure, R is the universal gas constant, T is the temperature, σ is the surface tension of the interface, v is the molar volume of the adsorbate, and N(X) is the amount of adsorbed nitrogen at the relative pressure X. Note that the relationship between X and N(X) can be obtained directly from the adsorptiondesorption isotherms without a model or any additional assumptions.19 In this way, the fractal dimension of the pore-cluster interface is defined as Dfs ) 2 + R, where R is the slope of the graph of ln(S) versus ln[ln(1/X)]. It should be noted that only in the case of ideal pore surfaces do the fractal dimensions defined from the adsorption-desorption isotherms have the same values. In practice, in the region of capillary condensation, the adsorption and desorption isotherms generally do not coincide. In such a case, preference should apparently be given to the desorption branch because polymolecular adsorption is manifested to a lesser extent during desorption than during adsorption.19 Experimental Section Sample Preparation. The samples were prepared in such a way that the phosphate was occluded in the final dry gel. In these kinds of sol-gel compounds, several species can be trapped or occluded and can remain stable until high temperatures are reached. In this particular case, a fraction of the phosphates is inside the alumina pores and another anchored to the surface through the hydroxyl groups. Because a number of surface OH groups depend on the pretreatment temperature, the gels were thermally treated under soft conditions. The substitution of OH ions by PO4 ions is an endothermic and easy process: -Aln(OH-) + H3PO4 f Aln(PO4-) + H2O, where n is the coordination of the surface OH. The endothermicity can be explained by steric constraints: the optimal Al-PO4 distance (3.2-3.4 Å) is larger than the optimal Al-OH one (1.8-2.0 Å). As a consequence, it is difficult to simultaneously accommodate two- or threefoldcoordinated PO4 groups without strong local deformations, which are energetically unfavorable. A study of the surface phosphation will be made, but the results of vibrational analysis9 are consistent with the hypothesis. The hydroxyl surface density and vibrational properties of the hydroxyl groups clearly show that the selected models are valid representations of the active surfaces of alumina materials. Materials: phosphoric acid, H3PO4 (Baker, 85-87%); distillated water; ethyl alcohol absolute grade, CH3CH2OH (Baker, 99.9%); aluminum tri-sec-butoxide, ATB (Aldrich, 97%); ammonium hydroxide, NH4OH (Baker, 30%); glycerin reactive grade (Aldrich, 99.5%); monobasic sodium phosphate, NaH2PO4 (Mallinckrodt, 99.8%); and tert-butyl alcohol, (CH3)3COH (Baker, 99.6%). Phosphate Alumina. An aqueous solution of 0.01 M H3PO4 was added drop by drop to the reaction flask (with 70 mL of absolute CH3CH2OH and water, in reflux), and simultaneously, 15.76 mL of ATB was added during 1 h at 65 °C under constant stirring. The pH was adjusted to 9 with NH4OH. Hydrolysis and condensation occurred at 65 °C for 4 days. Finally, the solution was heated at 120 °C for 24 h. The sample was named Al2O3-pH 9-NH4OH (Table 1), and we shall use the abbreviation ALP to refer to this sample in the paper. Phosphated Alumina with Glycerin. An aqueous solution of 0.045 M NaH2PO4 and, simultaneously, 15.76 mL of ATB were added drop by drop to the reaction flask (with 70 mL of (CH3)3COH and 4.4 mL of glycerin) following the same procedures and conditions as before. The pH was adjusted to 10 with NH4OH.
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Figure 2. Adsorption-desorption isotherms for ALP. Figure 1. General flow diagram for alumina phosphate synthesis. Table 1. Synthesis Conditions of Alumina Phosphates sample ALP
added catalyst
H3PO4 (0.01 M) ALPG NaH2PO4 (0.045 M)
final ATB/glycerin ATB/alcohol ATB/H2O pH molar ratio molar ratio molar ratio 9 10
1:1
1:20
1:30
1:20; (CH3)3COH
1:50
Additionally, the dried gel was subjected to a heat treatment of 500 °C for 4 h. The sample was named Al2O3-pH 10-glycerinNaH2PO4-tert-butyl alcohol and abbreviated ALPG. Table 1 lists and Figure 1 shows a diagram that describes the process conditions and methods used to prepare the samples.
Figure 3. Adsorption-desorption isotherms for ALPG.
Characterization Surface Areas and Porosities of the Gels. Physical nitrogen adsorption at -196 °C occurred at a relative pressure (P/P0) between 0.05 and 0.998 with an Autosorb3B from Quantachrome. The samples were prepared as follows: the gel samples of alumina were dried for 24 h at 110 °C, and the solids were subjected to controlled heating at 350 °C under vacuum until degasification was achieved. The area measurements were taken using nitrogen as the adsorbent. The surface area was determined by the BET method and the pore size distribution by the Barret-Joyner-Halenda (BJH) method. TEM. The alumina particles were embedded into epoxy resin (Epon 12) and dried at 60 °C for 24 h in order to strengthen the resin. Sections of 80 nm were cut by means of a RMC-MT7000 microtome operating at cryogenic conditions (-120 °C) and then were deposited on a copper grid and carbon-coated with a Blazers evaporator. Finally, the samples were observed by TEM in a Leo EM-910 (120 kV) electron microscope. SAXS. SAXS was performed using a Siemens 5000 X-ray generator with a copper target operating at 30 kV and 20 mA, and Cu KR X-rays were selected by means of a Ni-foil filter. The X-rays were collimated with a Kratky camera from Anton Paar with a slit of 20 µm, and the scattered X-rays were detected using a linear position-sensitive detector with a mixture of argon and methane gases. Aluminum filters were used as attenuators. The sample was heated at 70 °C and held at 3 × 10-8 Torr in situ for 2 h before the measurements were taken. The Pair Distribution of Heterogeneity program by Glatter35 was used to desmear and analyze the scattering zones. (35) Glatter, O. J. Appl. Crystallogr. 1977, 10, 415.
Figure 4. Pore size distribution for ALP and ALPG by the BJH method.
Results and Discussion The adsorption-desorption isotherms of these samples are shown in Figures 2 and 3. Both have a hysteresis loop due to capillary condensation, indicating mesoporous-type materials. These isotherms are a combination of the type H2-IV isotherm, usually found in materials with a crosslinked porous system, and type H3-IV, normally assigned to slitlike pores. Adsorption at a low relative pressure increases rapidly, suggesting that a small amount of micropores is present. The pore size distribution was calculated by applying the BJH method to the adsorption branches of the isotherms. These distributions are shown in Figure 4. Both samples showed monomodal distributions. ALPG has a narrower pore size distribution and a higher surface area and volume fraction relative to those of ALP, as indicated in Table 2. Figure 5 presents the
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Figure 7. Representative TEM electromicrographs of (a) ALP and (b) ALPG.
Figure 5. Neimark analysis of the BET isotherms.
Figure 6. SAXS diffractograms for ALP and ALPG. Table 2. Alumina Phosphate: Textural Analysis Data material
BET superficial area (As, m2 g-1)
pore diameter (Dp, Å)
ALP ALPG
308 433.4
39.6 58.1
results of the analysis by the method proposed by Neimark19 of the adsorption-desorption isotherms of both the ALP and the ALPG samples. As Figure 5 shows, when ln Sc is plotted against ln[ln(P0/P)], there is a linear region for both the desorption and the adsorption isotherms, indicating that in that region Sc follows the same scaling law as the radius of curvature r (from eq 16, -ln[ln(P0/P)] ) ln r + constant). The fractal dimensions Dfs for ALPG and ALP, as defined by eq 15, are 2.46 and 2.58,
respectively, for the desorption branch, pointing out that ALP has a slightly rougher surface than ALPG. SAXS diffractograms are shown in Figure 6 for both of the samples. From these, one sees that there is a linear region for the q values between 1.2 and 7.0 Å-1. Using eq 14, we find that DM ) 1.816 ( 0.013 and 1.695 ( 0.015 for ALPG and ALP, respectively. In the case of ALPG, this corresponds closely to a diffusion-controlled aggregation; however, for ALP, its fractal dimension implies that the gel is less packed than ALPG. Representative TEM images of the two gels are shown in Figure 7. Typical fractal graphs obtained using the BENOIT 1.2 software are shown in Figures 8 and 9. It should be noted that at least four images of the gel obtained at each synthesis condition were analyzed. The fractal dimensions obtained from a set of images of the same gel vary in conformity with a normal distribution with standard deviations on the order of 15% of the mean value (see Table 3). One can see from the graphs in Figure 8 that the gels possess statistical scale invariance within a wide range of length scales (up to 2 orders of magnitude) and, according to criterion 4, they can be treated as self-similar fractals (see also Table 3). Moreover, the gel clusters and pores for ALP are characterized by almost the same fractal dimension. For ALPG, the difference between the fractal dimensions of the pores and those of the solids is slightly larger, yet their uncertainty intervals overlap. For both ALP and ALPG, all the fractal dimensions are close to the theoretical value of D ) 1.8 for a cluster aggregated under a diffusion-controlled process.36 Table 3 also shows that the clusters of ALP are rather isotropic because DM is almost equal to DB. Furthermore, on a small scale (from ∼3 to 30 nm), the cluster boundaries of ALP are characterized by the fractal dimension DD ) 1.45 ( 0.02, close to that of ALPG with DD ) 1.46 ( 0.01, indicating that on that scale the cluster boundaries of both are equally tortuous (according to the adsorption-desorption measurements, ALP was slightly more tortuous than ALPG). On a larger scale, ALP shows
Table 3. Summary of the Fractal Dimensions cluster
pore space
cluster boundary
DB
DI
DM
DB
DI
DM
ALP
TEM
1.85 ( 0.03
1.85 ( 0.02
1.84 ( 0.02
1.83 ( 0.03
1.83 ( 0.02
1.84 ( 0.02
ALPG
BET X-ray TEM
1.45 ( 0.02a; 1.29 ( 0.03 1.44 ( 0.02 b 2.58 ( 0.01
1.83 ( 0.03
1.87 ( 0.02
1.85 ( 0.02
not found
1.46 ( 0.01a; 1.31 ( 0.02c; 1.75 ( 0.05 b 1.30 ( 0.02d 2.46 ( 0.01
BET X-ray a
1.695 ( 0.015 1.83 ( 0.01 not found 1.816 ( 0.015
4-30 nm. b 30-300 nm. c 40-260 nm.
d
290-8100 nm.
DD
DP
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Figure 8. Representative BENOIT runs on ALP and ALPG samples for (a) the box-counting of the cluster, (b) the box-counting of the pore space, (c) mass versus radius of the solid particles, and (d) the information entropy of the solid particles.
more or less the same cluster-boundary scaling properties, yet surprisingly, ALPG shows a much higher value for DD. The latter is not in agreement with the results for the cluster boundary given by the perimeter-area analysis, which shows that the cluster-boundary scaling properties are very much the same for ALP and ALPG no matter the scale. Even more, criterion 7 for the statistical selfsimilarity of the cluster boundary is not satisfied. It is not clear if the origin of this discrepancy is in the clusters having a more complex structure or is inherent to the algorithm used, and it should be further investigated. As shown in Table 3, there are some differences between the results obtained by TEM, BET, and SAXS; however, it should be noted that there is a difference between the three methods scalewise. With respect to the clusters and pore spaces, SAXS points out a fractal character of the gel from about 0.17 to 1.1 nm, whereas the TEM measurements go from about 3 to 600 nm. Notice that SAXS, in principle, should have a larger upper-scale limit that in some cases can reach 100 nm with the corresponding experimental setup, and it will be interesting to overlap both techniques. The nature of the gel cluster generated by the aggregation of colloids or polymerization of alkoxides depends on the diffusion of the elemental species and of the growing clusters (effects of dilution, viscosity, and temperature). It depends also on the probability of aggregation during contact between two species (reactivity). Various models were proposed that are associated with a limiting mech(36) Meakin, P. Fractals, Scaling and Growth far from Equilibrium; Cambridge University Press: Cambridge, U.K., 1998.
anism for the growth: diffusion-limited monomer-cluster or cluster-cluster growth and reaction-limited monomercluster or cluster-cluster growth. The resulting clusters usually exhibit a fractal structure for scales above that of the elemental unit size. The data presented in Table 3 indicate the difference in the mechanisms of ALP and ALPG gel formation. Namely, for the ALPG gel we determine that in all the studied length scales the fractal dimension of the clusters of 1.82 is consistent with diffusion-limited cluster-cluster aggregation. On the other hand, we find that the mass fractal dimension obtained by TEM for ALP is larger than that obtained by SAXS on a completely different scale, indicating that self-similarity breaks down below 3.0 nm. This observation suggests that on the microscopic (below 3 nm) and mesoscopic length scales the structure of the ALP gel clusters is very different. This difference indicates the difference in the mechanism of cluster formation at the early times versus the final stage. Namely, at the earlier time the small clusters with the fractal dimension of about 1.65 are formed, probably because of the ballistic deposition of elementary particles, while the spanning cluster is built mainly by the aggregation of the diffusing, nucleating clusters. Now, using eq 16 it is observed that the length scale of the region where the desorption measurements indicate a fractal character of the gel goes from about a 0.25- to a 1.0-nm pore radius. In the case of the adsorptiondesorption method, only the region well before saturation can give us information about the interfacial surface; once saturation is reached, the pores are filled. If, for example,
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Figure 9. Representative BENOIT runs on ALP and ALPG samples for (a, b) the number of ruler steps along the cluster boundary versus the ruler length for ALP and ALPG, respectively, and (c, d) the area of the cluster versus its perimeter for ALP and ALPG, respectively.
one takes the desorption isotherm of ALPG, as shown in Figure 3, with a saturation plateau from about P/P0 ≈ 0.65, the maximum Kelvin radius will be on the order of 2.2 nm and, in principle, this will represent the upper limit at which this technique could give us information about the interfacial area. Actually, long before saturation the BET isotherms present multilayer adsorption, which occurs at about the relative pressure where the hysteresis begins, which, for the case of ALPG, is approximately P/P0 ≈ 0.4 and, when translated into a radius of curvature, gives a radius of about 1.0 nm, corresponding to our experimental upper limit. This means that even if the fractal character extends to larger scales, the thermodynamic method used cannot give us information about the scaling at larger scales. In this sense, adsorptiondesorption, SAXS, and TEM are complementary tech-
niques. The study presented in this work clearly demonstrates the great potentials of the available fractalcharacterization techniques, the significance of an appropriate interpretation of data, and the abundance of information contained here. The fractal analysis of solgel systems permits the gain of deeper insight into the complexity of gel structures and, hence, a better understanding of the mechanisms of their formation. In this way, the improvement of the fractal-characterization methods must also contribute to the optimization of the synthesis conditions to prepare better materials. Acknowledgment. We express our gratitude to Vı´ctor Hugo Lara and Dr. Pedro Bosch for their support in the X-ray work included in the present work. LA026630R
