Mass and Surface Fractal in Supercritical Dried Silica Aerogels

Dec 16, 2014 - E-mail: [email protected]. ... The aerogels presented most of the mass-fractal characteristics of the original wet gels at large lengt...
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Mass and Surface Fractal in Supercritical Dried Silica Aerogels Prepared with Additions of Sodium Dodecyl Sulfate Amanda P. Perissinotto, Carlos M. Awano, Dario A. Donatti, Fabio S. de Vicente, and Dimas R. Vollet* IGCE, Departamento de Física, Unesp − Univ Estadual Paulista, Cx.P. 178, 13500-970 Rio Claro, SP, Brazil ABSTRACT: Silica wet gels were prepared from hydrolysis of tetraethoxysilane (TEOS) with additions of sodium dodecyl sulfate (SDS). The surfactant was removed after gelation. Wet gels exhibited mass-fractal structure with mass-fractal dimension D (typically around 2.25) in a length scale extending from a characteristic size ξ (typically about 10 nm) of the mass-fractal domains to a characteristic size a0 (typically between 0.3 and 0.4 nm) of the primary particles building up the fractal domains. ξ increased while D and a0 diminished slightly as the SDS quantity increased. Aerogels with typical specific surface of 1000 m2/g and density of 0.20 g/cm3 were obtained by supercritical drying of the wet gels after washing with ethanol and n-hexane. The pore volume and the mean pore size increased with the increase of the SDS quantity. The aerogels presented most of the mass-fractal characteristics of the original wet gels at large length scales and exhibited at a higher resolution level at about 0.7 nm a crossover to a mass-surface fractal structure, with apparent mass-fractal dimension Dm ∼ 2.4 and surface-fractal dimension Ds ∼ 2.6, as inferred from small-angle X-ray scattering (SAXS) and nitrogen adsorption data.

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work,21 a fixed concentration of SDS was used with varied quantities of an oil phase to produce hydrophobic ambient pressure dried silica aerogels. It was concluded that the typical values of the specific surface of the ambient pressure dried aerogels were comparable with those obtained by supercritical drying, and the pore volume and the mean pore size increased with increasing the oil phase. In this work, silica wet gels were prepared from hydrolysis of TEOS with different quantities of SDS. The surfactant was removed after gelation. Monolithic supercritical dried aerogels were obtained after washing the wet gels with water, ethanol and n-hexane. Interesting modifications on the surface of the aerogels could be inferred from small-angle X-ray scattering (SAXS) and nitrogen adsorption data. To study adequately the characteristics of the surface of the aerogels by SAXS is necessary to get SAXS data with enough quality in the Porod region, which is not always possible due to mainly the background contribution. In this work, the SAXS data were measured with high precision using synchrotron radiation and a two-dimensional (2D) position sensitive X-ray detector, which allowed us to infer interesting characteristics of the surface of the aerogels and correlate them with nitrogen adsorption data. This study has scientific and technological interest because of the inherent importance of the system for applications in several areas of knowledge and also because of the original method employed to describe the whole structural characteristics of the aerogels, which with certainty will be of interest for

INTRODUCTION Silica aerogels are an interesting class of materials because of their often low density and high specific surface area.1 Aerogels have been considered for scientific and technological applications in several areas, such as thermal isolation,2 catalysis,3 separation,4 adsorption,5 sensing,6 enzyme immobilization,7 controlled drug delivery,8−10 and nanotechnology.11 Silica wet gels prepared from hydrolysis of tetraethoxysilane (TEOS) can be described as a continuous solid network embedded in a large volume liquid phase. Such a structure is very often described in terms of a mass-fractal structure, since it presents primary particles forming domains in which the mass m scales in a power law with the probe length r as m ∝ rD, where D is the mass-fractal dimension. The supercritical extraction of the liquid phase of the wet gels (supercritical drying) often yields monolithic aerogels with structure not so far from that of the original wet gels, at least at large length scales. Additional structural changes are expected to occur with the supercritical drying at a higher resolution level, where the primary particles are often modified with respect to the size and the surface characteristics, depending on the conditions of preparation and on the subsequent treatments to the supercritical drying. Surfactants can form micelles in a variety of forms (spherical, cylindrical, lamellae) and structural organization such as lamellar, cubic, hexagonal, or three-dimensional wormhole,12 so they have been used as appropriate matrices to prepare and modify a large variety of mesoporous silica with interesting structural properties.12−16 Sodium dodecyl sulfate (SDS) is a very popular anionic surfactant, although one could find only few cases in the literature in which it has been used as a modifier for silica mesoporous structure.15,17−21 In a previous © 2014 American Chemical Society

Received: October 29, 2014 Revised: December 15, 2014 Published: December 16, 2014 562

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Langmuir several researchers dealing with complex and ill-defined geometries as those found in the gel structures.

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EXPERIMENTAL SECTION

Monolithic silica aerogels were prepared from acid hydrolysis of TEOS with additions of SDS as a surfactant agent. Different quantities of SDS were dissolved in 48.0 mL of distillated water under mechanical stirring at 45 °C to obtain solutions with SDS concentrations of 1, 25, 50, 75, and 100 times the SDS critical micelle concentration (∼8.2 × 10−3 M). Volumes of 5.0 mL of ethanol and 5.0 mL of 1.0 N HCl were added to each one of the different SDS solutions under stirring at 45 °C, and, after homogenization for 30 min, 25 mL of TEOS was finally added to the mixtures. The hydrolysis was then promoted at 45 °C for 60 min under mechanical stirring and reflux conditions. Thus, the TEOS/water/ethanol/HCl molar ratio in the final sols was 1:26.5:0.78:0.070, while the SDS/TEOS molar ratios were varied to 0, 0.0035, 0.0875, 0.175, 0.26, and 0.35. The hydrolyzed solutions were cast in sealed plastic containers and kept at 40 °C for gelation, which occurred in a few hours, and aging for 5 days to obtain monolithic wet gels. The wet gels were washed with distilled water, ethanol, and n-hexane to remove most of the SDS constituents, nonreacted residues, and byproducts of hydrolysis. The wet gels were named U0, U1, U25, U50, U75, and U100, the numbers meaning the SDS concentration relative to the critical micelle concentration of the SDS solutions used in the preparations of the gels. Samples of this set of wet gels were kept in sealed container and studied by SAXS. For the production of aerogels, the liquid phase of wet gels was exchange by ethanol at room conditions. The ethanol was then exchanged by liquid CO2 in an autoclave followed by supercritical CO2 extraction to yield monolithic aerogels samples named A0, A1, A25, A50, A75, and A100, derived correspondingly from the wet gels U0, U1, U25, U50, U75, and U100. The aerogels were yet degassed at 120 °C in vacuum conditions (∼3 × 10−3 mmHg) for about 24 h before they were studied by nitrogen adsorption and SAXS. The SAXS experiments were carried out using synchrotron radiation with a wavelength λ = 0.1608 nm at the SAXS beamline of the National Synchrotron Light Laboratory (LNLS), Campinas, Brazil. The beam was monochromatized by a silicon monochromator and collimated by a set of slits defining a pinhole geometry. A 2D position sensitive X-ray detector was used to obtain isotropic SAXS intensity I(q) as a function of the modulus of the scattering vector q = (4π/ λ) sin(θ/2), where θ is the scattering angle. The experimental setup allowed us to obtain SAXS data from q0 = 0.1379 nm−1 up to qm = 3.3480 nm−1 with resolution of about 4 × 10−4 nm−1. The data were corrected by sample attenuation and parasitic scattering and normalized with respect to the beam intensity. Nitrogen adsorption isotherms were obtained at liquid nitrogen temperature (77 K) by using an ASAP 2010 Micromeritics apparatus. The data were analyzed for the BET specific surface (SBET), the total pore volume (Vp), as the volume of nitrogen adsorbed at a point close to the nitrogen saturation pressure, and the mean pore size (lp) as lp = 4Vp/SBET.

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Figure 1. SAXS patterns for the wet gels prepared with different additions of SDS. The curves were vertically shifted for the sake of clarity. The full lines (red lines) are fittings of eq 1, through eqs 2 and 3, to the experimental data (points). A straight line with slope equal to −2.25 was drawn as a reference.

I(q) = AP(q) S(q)

(1)

where A is a constant, P(q) the form factor of an individual scatter, and S(q) an effective structure factor. An approach for S(q) accounting for the cutoff ξ at low-q can be cast as23 S(q) = 1 + BΓ(D + 1)sin[(D − 1)arctan(qξ)] /(1 + q2ξ 2)(D − 1)/2 (D − 1)qξ

(2)

where B = (ξ/ra)D represents an average number of primary particles in the fractal cluster, with ra being the gauge of measurement (a characteristic dimension of the primary particle), and Γ(D+1) being the gamma function of the argument (D+1). P(q) is often well approximated by the Debye−Bueche form25,26 P(q) = 1/(1 + a0 2q2)2

(3)

For systems with primary particles with characteristic size a0 small enough, P(q) is fairly a constant value at low- and intermediary-q while the structure factor S(q) is dominated by the second term in eq 2, which is much larger than the unit term 1. The second term of S(q) can be approximated to a Guinier’s law type at very low-q and exhibits a crossover at q ∼ ξ −1 to a power law on q as S(q) ∝ q−D. So, the intensity I(q) at low- and intermediary-q is determined mainly by the parameters ξ and D according to the second term of S(q) in eq 2. In the Porod law region (q → ∞), the second term of S(q) in eq 2 is much smaller than the unit term 1, while P(q) tends either to a Porod’s law scattering as P(q) ∝ q−4 or to a so small primary particle scattering according to Guinier’s law as P(q) ∝ ∼(1−2a02q2), and so does the intensity I(q). In the case of a Guinier’s law contribution for P(q), we should have r0′ = 101/2a0, where r0′ is the radius of an equivalent spherical primary particle; and in the case of a Porod’s law contribution, we should have r0′ = (3/4ϕ)a0, where ϕ is pore volume fraction (typically about 0.9 in the present system). Due to this

RESULTS AND DISCUSSION

Wet Gels. Figure 1 shows the SAXS intensity I(q) of the ethanol- and n-hexane-washed wet gels prepared with different SDS quantities. The curves can be described as the scattering from a mass-fractal structure, since the intensity is a power law on q as I(q) ∝ q−D, where D is the mass-fractal dimension (1 < D < 3),22 typically equal to 2.25, in the most part of the central q domain. The curves depart from the power-law behavior at low-q, due to the finite size ξ of the mass-fractal domains, and slightly at very high-q, due to the finite characteristic size a0 of the primary particle building up the mass-fractal domain.23,24 The intensity from such a system of fractal-like aggregates of primary particles can be decomposed as23,24 563

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Langmuir ambiguity on the real contribution of P(q) at high-q, we preferred not to assign the parameter B directly to (ξ/a0)D, but instead to keep B as an independent parameter to be fitted, leaving an implicit factor f in B so that B = (ξ/ra)D = ( fξ/a0)D. Figure 1 shows fittings of eq 1, through eqs 2 and 3, to the experimental I(q) of the wet gels. The fitting process was carried out by finding the set of parameters A, B, ξ, D, and a0 which best fit the experimental data in a Kratky q2I(q) versus q plot using a nonlinear least-squares routine (Levenberg− Marquardt algorithm). Table 1 shows the values of the fitted parameters ξ, D, a0, and B of the wet gels. Table 1. SAXS Mass-Fractal Properties of the Wet Gels ξ (nm) U0 U1 U25 U50 U75 U100

8.17 (8) 9.64 (9) 9.78 (9) 10.3 (1) 11.0 (1) 11.3 (1)

D 2.22 2.32 2.27 2.24 2.23 2.26

(1) (1) (1) (1) (1) (1)

a0 (nm)

B (102)

0.32 0.43 0.44 0.38 0.27 0.23

1.7 1.8 1.7 3.1 8.7 12

(1) (1) (1) (1) (2) (3)

(1) (1) (1) (2) (9) (1)

Figure 2. Nitrogen adsorption isotherms for the aerogels prepared with different additions of SDS. Open symbols correspond to the desorption branch. Isotherms were shifted 1000 cm3/g with respect to the others for the sake of clarity (except A100 which was shifted 1500 cm3/g with respect to A75).

The characteristic size ξ of the mass-fractal domains (typically of about 10 nm) increased slightly as the quantity of SDS increased. The fractal dimension D and the characteristic size a0 of the primary particles increased at first with the first addition of SDS (with respect to the aerogel prepared without SDS) and decreased fairly regularly afterward with increasing the SDS quantity. The increase of the mass-fractal size ξ should be due to the increase of the number of micelles of SDS, as the SDS concentration is increased during the step of preparation of the wet gels. The greater concentration of the SDS micelles allows the interparticle spacing to be adequately filled up to higher length scales during the clusters growing, allowing the mass-fractal law m ∝ rD to hold up to higher values of ξ. This effect of the micelle concentration on ξ is particularly in agreement with the increase of the parameter B (number of primary particle per cluster), and the minor decrease of the mass-fractal dimension D and diminution of the characteristic size a0 of the primary particle. Aerogels. Figure 2 shows the nitrogen adsorption isotherms of type IV in the general IUPAC classification27 obtained for the supercritical dried aerogels prepared with different SDS quantities. Table 2 shows the specific surface SBET, the specific pore volume Vp, and the mean pore size lp = 4Vp/SBET obtained from the isotherms. The BET specific surface of the aerogels was typically 1000 m2/g, and it was found not be too dependent on the SDS quantities. The pore volume and the mean pore size, however, were substantially larger in the aerogels prepared with additions of SDS compared to those in the aerogel prepared without SDS. The pore volume and the mean pore size of the aerogels were found even increasing with the quantities of SDS. The exception to this role was apparently found in sample A100 (Table 2), for which the highest pore volume was expected. The reasons for this apparent discrepancy will be discussed after analyzing the SAXS data for the aerogels. The increase of the pore volume with increasing the SDS quantity should be due to the fact that the larger the quantity of SDS micelles, the larger were the spaces filled by the surfactant during the earlier gelation process in the preparation of the wet gels. Figure 3 shows the SAXS intensity I(q) of the supercritical dried aerogels prepared with different SDS quantities. The

Table 2. Structural Properties of the Aerogels As Determined from Nitrogen Adsorption SBET (102 m2/g) A0 A1 A25 A50 A75 A100

9.74 (9) 10.2 (1) 10.3 (1) 9.78 (9) 11.9 (1) 10.2 (1)

Vp (cm3/g) 2.50 3.35 3.93 3.75 4.28 2.51

(1) (1) (1) (1) (1) (1)

lp (nm) 10.3 13.2 14.4 15.3 14.3 9.8

(1) (1) (1) (1) (1) (1)

Figure 3. SAXS patterns for the supercritical dried aerogels prepared with different additions of SDS. Curves were vertically shifted for the sake of clarity. Full lines (red lines) are fittings of eq 1, using eq 2 with B = (ξ/ξ2)D and eq 4, to experimental data (points). Straight lines with slopes −2.25 and −3.4 were drawn as a reference. 564

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Langmuir Table 3. Mass and Mass-Surface Fractal Parameters of the Aerogels ξ (nm) A0 A1 A25 A50 A75 A100

6.88 (7) 8.47 (8) 7.55 (8) 9.20 (9) 9.86 (9) 13.2 (1)

D 2.33 2.29 2.25 2.21 2.17 2.11

(3) (3) (3) (3) (3) (3)

ξ2 (nm)

Dm − Ds

0.68 0.50 0.66 0.61 0.80 0.69

−0.18 −0.09 −0.16 −0.13 −0.16 −0.11

(5) (5) (5) (5) (6) (5)

Ds 2.59 2.57 2.56 2.54 2.57 2.57

(2) (2) (2) (2) (2) (2)

Dm 2.41 2.48 2.40 2.41 2.41 2.46

(3) (3) (3) (3) (3) (3)

cluster inside of the first structure (with a cutoff at about 1 μm), so the correlation function of such a system could be written as a sum of the two corresponding correlation functions. Here, we thought the large-scale structure of the aerogels is a mass fractal, with cutoffs at ξ ∼10 nm and ξ2 ∼0.7 nm, formed by the (new) mass-surface fractal primary particles with cutoff at ξ2 ∼ 0.7 nm (the small-scale structure). Then, the cutoff at ξ2 ∼ 0.7 nm is meaning for both: (i) the characteristic size of the masssurface fractal primary particle and (ii) the high-q cutoff of the large-scale mass fractal of the aerogels (or the subunits composing the mass fractal). Using eq 4 as the factor P(q) in eq 1 seems very reasonable because the cutoff distance ξ2 just accounts for the small correlation between both the functions P(q) and S(q), although an analytical form for the correlation function associated with the product P(q)S(q) is not known. This seems to be the same motive by which the Debye−Bueche form [eq 2] has been often used for P(q) in connection with the product P(q) S(q) in eq 1.25,26 Figure 3 shows the good fitting of eq 1, with S(q) given by eq 2 with B = (ξ/ξ2)D and P(q) by eq 4, to experimental SAXS data of the aerogels. The fitting process was carried out by finding the set of parameters A, ξ, D, ξ2, and the difference Dm − Ds which best fit the experimental data in a Kratky q2I(q) versus q plot using the same nonlinear least-squares routine used for the wet gels. Only the difference Dm − Ds could be obtained from the fitting process, since both parameters Dm and Ds appear explicitly just as a unique parameter (Dm − Ds) in eq 4. Table 3 shows the values of the fitted parameters ξ, D, ξ2, and Dm − Ds for the aerogels. The characteristic size ξ of the mass-fractal domains of the aerogels increased while the mass-fractal dimension D diminished by increasing the SDS quantities. The values of the cutoff length ξ2 of the crossover to the mass-surface fractal were found all around 0.7 nm, while the difference Dm − Ds was found to be negative with the values all around −0.14. The values of the parameters ξ and D and their behavior with the SDS quantities (Table 3) were found to be in agreement with those (also obtained by SAXS) in the wet gels (Table 1). This suggests that most of the large scale mass-fractal structure of the aerogels (associated with the contribution at low- and intermediary-q) is a remnant of the same mass-fractal structure of the wet gels, except with pores instead liquid phase. To corroborate this fact, Figure 4 shows a typical aerogel SAXS curve being practically parallel to that from the precursor wet gel at low- and intermediary-q. More fundamental modifications occur in the aerogels at high scale of resolution (associated with the high-q scattering (Figure 4)) with the development of the (new) mass-surface fractal primary particle, which scatters according to eq 4. Trying to fit the data with eq 4 by fixing Dm = 3 (meaning that the (new) primary particle is homogeneous) fails because the fitting leads invariably to a parameter Ds > 3, which is physically meaningless. So, we conclude that the (new) primary particle in the aerogels is

curves in Fig. 3 show characteristics at low- and intermediary-q compatible with the same structure factor S(q) (eq 2) of the wet gels. However, the curves exhibit at about q ∼ 1/ξ2 (where ξ2 ∼ 0.7 nm) a crossover to an approximate new power-law as I(q) ∝ q−α, with the exponent α quite close to 3.4 (Fig. 3), i. e., in the range 3 < α < 4. This suggests the development of a secondary domain at a correlation distance of about 0.7 nm bounded by a surface fractal.28 We have reasons to suppose that such a secondary domain could be structured approximately as a power-law in an analogy to a (new) mass-fractal primary particle (with apparent mass-fractal dimension Dm) bounded by a surface-fractal (with surface-fractal dimension Ds). The development of the secondary mass-surface fractal domains (new primary particles) come from the evolution at a high resolution level (lower than ξ2 ∼0.7 nm) of the original structure of the wet gels, which is modified with the supercritical drying process. The scattering from a mass-surface fractal object could be interpreted as that produced by a mass-fractal domain bounded by a surface fractal.29,30 It has been shown29,30 that, for values of q sufficiently greater than the reciprocal of the characteristic cutoff size of the mass-surface fractal scaling (ξ2), the intensity is a power law on q as I(q) ∝ q−(3+Dm−Ds), where Dm and Ds are mass-fractal and surface-fractal dimensions. In the case of a homogeneous object bounded by a fractal surface, Dm is equal to 3 while Ds is in the interval 2 < Ds < 3, so the power law I(q) ∝ q−(3+Dm−Ds) becomes the classical Bale and Schmidt28 surface fractal scattering I(q) ∝ q−(6−Ds). We wondered if the SAXS curves of the aerogels could be fitted integrally by eq 1, using the structure factor S(q) of eq 2 with B = (ξ/ξ2)D to describe the mass-fractal domains, with characteristic size ξ and fractal dimension D, formed by (new) mass-surface fractal primary particles of characteristic size ξ2 with mass-fractal dimension Dm and surface-fractal dimension Ds, scattering individually according to a form factor P(q) given by30 P(q) = sin[(Dm − Ds + 2)arctan(qξ2)] /(1 + q2ξ2 2)(Dm − Ds + 2)/2 qξ2

(2) (1) (2) (1) (2) (1)

(4)

Equation 4 can be obtained by Fourier transform of a masssurface fractal correlation function γ(r) with an exponential cutoff at ξ2, or γ(r) ∝ rDm−Ds exp(−r/ξ2).30 Equation 4 has the property to be practically a constant value for qξ2 ≪ 1 and go to a power-law as P(q) ∝ q−(3+Dm−Ds) for qξ2 ≫ 1. Particularly, eq 4 goes to the Bale and Schmidt28 scattering P(q) ∝ q−(6−Ds) for qξ2 ≫ 1 and Dm = 3. It should be emphasized that the description of the masssurface fractal of the (new) primary particle using eq 4 is proposed here in a completely different sense than was done in the work by Morbidelli and co-workers.30 They have used light scattering to describe a large-scale mass-surface fractal structure (with a cutoff at about 150 μm) and a small-scale mass fractal 565

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Table 4. Structural Characteristics of the Aerogels As Inferred from SAXS and Nitrogen Adsorption r0 (nm) A0 A1 A25 A50 A75 A100

(1) (1) (1) (1) (1) (1)

1.7 2.1 1.6 1.9 1.2 1.6

(1) (1) (1) (1) (1) (1)

ρξ (g/cm3) 0.35 0.29 0.26 0.23 0.14 0.11

(3) (3) (3) (2) (1) (1)

ρN2 (g/cm3) 0.32 0.26 0.22 0.23 0.20 0.32

(2) (2) (1) (1) (1) (2)

The radius r0 of a spherical particle giving the specific surface SBET as measured by nitrogen adsorption (Table 2) could be obtained by r0 = (1/ρms)3/SBET, where ρms is the particle density. The evaluation of r0 = (1/ρms)3/SBET for the aerogels with the SBET data from Table 2 using ρms = 2.2 g/cm3 (the density often adopted for the fused silica) yields values with small dispersion around the mean value 1.3 nm, which is somewhat smaller than the mean value 1.75 nm obtained by SAXS. This suggests that the density ρms of the (new) primary particle in the aerogels should really be smaller than that of the fused silica, likely as a result of the internal structure of the (new) primary particles (as an evolution of the small-scale structure of the wet gels), for which, as it has been pointed, could well be described by a power law in an analogy to a mass fractal [m(r) ∝ rDm], even though in a limited range of the small length scale.31 The density ρms of the (new) primary particle was then evaluated as ρms = 3/r0SBET in order to match with both: the specific surface SBET as measured by nitrogen adsorption and the radius r0 as measured by SAXS. Table 4 shows the values ρms calculated for the aerogels exhibiting a fairly random dispersion around the mean value 1.7 g/cm3, rather smaller than that of the fused silica, reinforcing the hypothesis of a somewhat structured (new) primary particle in the aerogels. This result was also corroborated by the good agreement between the values of the bulk density as measured independently by SAXS and nitrogen adsorption for the aerogels, as described in the following. The increase of ξ in the aerogels with the additions of SDS (Table 3) is compatible with the increase of pore volume Vp (Table 2), since the bulk density ρξ of the mass-fractal domain should diminish (accompanying the Vp increase) with increasing the characteristic size ξ so that33

Figure 4. Direct comparison between a typical SAXS curve from an aerogel (sample A100) and that from the precursor wet gel (sample U100). The curves are practically parallel at low- and intermediary-q, but the curve of the aerogel departs from the power law behavior of the wet gel at high-q. Full lines are fittings of the models as described in the legends of Figures 1 and 3. Curves were shifted vertically for direct comparison.

internally structured, and such a structure could well be described by a power law in an analogy to a mass-fractal [m(r) ∝ rDm]. The description of power-law objects as fractal is justified, as it provides a simple model to describe ill-defined geometries, even though it applies only to a limited range of the length scale and does not imply fractality at all.31 The development of such somewhat structured (new) primary particles, with radius of about 1.8 nm (as we will see) and very rough surface, was confirmed by combining the SAXS and nitrogen adsorption data, as described in the following. The typical diameter of objects probed by the SAXS characteristic size ∼1/q is often assigned to the Bragg distance ∼2π/q.22 The meaning of the characteristic size ξ associated with the mass-fractal structure should be specified in each case. For mass-fractal clusters, the radius of gyration Rg of the clusters is given by24 Rg = [D(D + 1)/2]1/2ξ. The values of Rg of the aerogels were found increasing fairly regularly with the SDS quantity from about 20 nm (sample A0) up to about 34 nm (sample A100), as evaluated using Rg = [D(D + 1)/2]1/2ξ with the data ξ and D in Table 3, giving a mean value of about 24 nm. This mean value for the radius of gyration gives an equivalent spherical diameter of about 62 nm, which is in good agreement with the Bragg distance 2πξ typically found in the aerogels. The number of (new) primary particles per cluster in the present aerogels is simply B = (ξ/ξ2)D, so it could equally be written as32 B = (Rg/r0)D, where r0 is the true radius of the primary particle as measured by the same gauge of measurement used for Rg. Then the radius r0 of the (new) mass-surface fractal primary particles of the aerogels was evaluated as r0 = R g(ξ2/ξ)

1.8 1.4 1.8 1.6 2.1 1.8

ρms (g/cm3)

ρξ = ρms (ξ /ξ2)D − 3

(6)

bearing in mind that ξ2 is the characteristic size of the (new) mass-surface fractal primary particle with density ρms. Table 4 shows the values of ρξ calculated with ξ and ξ2 from Table 3 and ρms from Table 4. The bulk density ρN2 as measured by nitrogen adsorption was evaluated through the relationship (1/ ρN2) = (1/ρms) + Vp, with ρms from Table 4 and Vp from Table 2, and the results are shown in Table 4 in direct comparison with ρξ. The results for ρξ and ρN2 were found correspondingly in good agreement, except for the samples prepared with higher concentrations of SDS for which higher pore volumes and lower densities were expected. The agreement between ρξ and ρN2 suggests that the mesoporosity associated with the structure of the large-scale mass fractal clusters (of characteristic size ξ) corresponds to that recorded by nitrogen condensation (given by Vp). The apparent disagreement found for the samples with higher concentrations of SDS, particularly for the sample A100, could be explained on the basis of the observation that conventional nitrogen adsorption method applied to high

(5)

Table 4 shows the values of r0 obtained for the aerogels fairly randomly distributed around the mean value 1.75 nm. 566

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Langmuir

from linear fitting of eq 7 in the mentioned p/p0 domain (Figure 5), which yielded values for the surface fractal dimension Ds all being somewhat lower than 2.6. Table 3 shows the parameter Ds obtained by the fitting process from the nitrogen adsorption isotherms and the parameter Dm, as evaluated from Ds and the difference (Dm − Ds) (the last obtained from SAXS). The values Ds and Dm were found not be too dependent on the SDS quantity used in the preparation of the aerogels. The values Dm associated with the (new) mass-surface fractal primary particles were coherently found larger than the values D associated with the large scale mass-fractal structure of the aerogels (remnant of the mass-fractal structure of the wet gels), revealing some compactness (but not fully homogeneity) of the (new) primary particle structure developed in the aerogels, in agreement with the values found for ρms in Table 4. It is always possible that Dm is referring just to an ill-defined geometry in a limited range of the small-length scale and does not imply fractality at all,31 so further discussion about the influence of other factors (as very small nonfractal domains)30 on the scaling law involving this parameter will not be carried out. However, the surface fractal dimension Ds is trustworthy as it was established by independent methods.

porosity silica aerogels can seriously underestimate the pore volume in the nitrogen condensation.34,35 The adsorbate/vapor interface in a sparse silica network can adopt a surface of zero curvature while much of the larger pores remains empty, which would not occur in case of cylindrical pores.34 The extent of the phenomenon depends only on the density of the network (or on the pore/particle size ratio), not on its pore size absolute value.34 Thus, the values ρξ obtained from SAXS would be more trustworthy for the samples with high porosity. Finally, we wondered if both parameters Dm and Ds could be resolved by employing independent techniques. The surfacefractal dimension Ds of the aerogels was considered in terms of a fractal version of the Frenkel−Halsey−Hill equation,36,37 which was employed to analyze our nitrogen adsorption data. The method is based on the analysis of the multilayer adsorption to a fractal surface. After the very early stages of adsorption, in which the film/gas interface is controlled by attractive van der Waals forces between the gas and the solid (fairly equivalent to the formation of a monolayer), the further coverages occur in such a way that the interface is controlled by the liquid/gas surface tension. In this stage, the multilayer adsorption to a fractal surface occurs in such a way that36,37 ln(V /Vm) = C + (Ds − 3)ln[−ln(p/p0 )]

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(7)

CONCLUSIONS Monolithic supercritical dried silica aerogels, with typical specific surface of 1000 m2/g and density of 0.20 g/cm3, were successfully prepared from hydrolysis of TEOS with additions of SDS. The pore volume and the mean pore size increased with the increase of the SDS quantity. The original wet gels exhibited mass-fractal structure with mass-fractal dimension typically equal to 2.25 in an interval of the length scale extending from typically 10 nm to about 0.3− 0.4 nm. The interval of the length scale of the mass-fractal increased while the mass-fractal dimension diminished slightly with the increase of the SDS quantity. The aerogels presented most of the mass-fractal characteristics of the original wet gels at large length scales and exhibited at a higher resolution level a crossover (at ∼0.7 nm) to a masssurface fractal structure, with apparent mass-fractal dimension Dm ∼ 2.4 and surface-fractal dimension Ds ∼ 2.6, as inferred from SAXS and nitrogen adsorption data.

where V is the nitrogen volume adsorbed at the relative pressure p/p0, Vm is the volume of a monolayer (as determined from the BET equation), and C is a constant. Figure 5 shows eq 7 fitting well the data from the nitrogen isotherms in a large relative pressure domain, approximately

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AUTHOR INFORMATION

Corresponding Author

*Telephone +55-19 35269180. Fax +55-19 35269179. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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Figure 5. Surface fractal characteristics of the aerogels as inferred from nitrogen adsorption isotherms. Straight lines (red lines) are fittings of eq 7 to the experimental data (points) from the isotherms. Plots were shifted for different factors for the sake of clarity.

ACKNOWLEDGMENTS Research partially supported by LNLS − National Synchrotron Light Laboratory, FAPESP, and CNPq, Brazil.

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from p/p0 ∼ 0.199 to p/p0 ∼ 0.849, since eq 7 is a straight line with slope equal to (Ds − 3) in a V/Vm versus −ln(p/p0) log− log plot. It should be emphasized that p/p0 ∼ 0.199 is often assigned to the point close to the formation of a monolayer and somewhat lower than the superior limit of applicability of the two-parameter BET equation for the obtaining SBET. The values of the slope (Ds − 3) were all found around −0.4 as obtained

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