Persistence Length, Mass Fractal, and Branching in the Aggregating of

Dec 15, 2010 - Unesp - UniVersidade Estadual Paulista, IGCE, Departamento de Fısica, Cx.P. 178,. 13500-970 Rio Claro (SP), Brazil. ReceiVed: October ...
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J. Phys. Chem. C 2011, 115, 667–671

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Persistence Length, Mass Fractal, and Branching in the Aggregating of Vinyltriethoxysilane-Derived Organic/Silica Hybrids Dario A. Donatti, Carlos M. Awano, Fabio S. de Vicente, Alberto Iban˜ez Ruiz, and Dimas R. Vollet* Unesp - UniVersidade Estadual Paulista, IGCE, Departamento de Fı´sica, Cx.P. 178, 13500-970 Rio Claro (SP), Brazil ReceiVed: October 30, 2010; ReVised Manuscript ReceiVed: NoVember 25, 2010

The time evolution of the local chain persistence determined by small-angle X-ray scattering (SAXS) in the aggregation kinetics of vinyltriethoxysilane(VTES)-derived organic/silica hybrids has been analyzed by the classical Sharp and Bloomfield (SB) model, by a mass-fractal Unified Function (UF) with an arbitrary massfractal dimension D, and by a modified branching Sharp-Bloomfield (BSB) model incorporating a branching probability parameter. The SB function describes well the SAXS data, but it is restricted to stages close to the appearance of a plateau in the Kratky plots. The UF and BSB models describe well the SAXS data up to the plateau and afterward when maxima rise in the Kratky plots, the maxima increasing in intensity and shifting toward the low-q with time. The UF approach does not apply to the data at the early stages of aggregation, when D is apparently not so different from the exponent -1 of the rod-like power-law regime at high-q. The values of the persistence length lp, the contour length L, the gyration radius Rg, and the mass density per chain length M/L as determined from the different approaches were used to probe the applicability of the methods. Introduction The incorporation of organic-rich functional groups in the silica structure to produce organic/inorganic hybrids has attracted the attention of several researchers for a wide variety of applications.1-14 In a typical process, a chemically functional group is introduced by a three-functional alkoxide, while the gel backbone is formed by siloxane bonds.1,4-7,12 Vinyltriethoxysilane (VTES) has been used as a three-functional alkoxide precursor to produce a series of organic/inorganic hybrid materials with interesting optical and structural properties,7,12 coatings on film substrates with satisfactory mechanical properties and protective action,15 and highly monodisperse organic/ inorganic hybrid silica spheres via a one-step synthesis under basic conditions.3 The important feature of the organic-functionalized alkoxide polymerization is competition between the ongoing processes of hydrolysis, condensation, and phase separation, which are strongly dependent on the pH value and determine the final properties of the hybrid.3,16 In a previous work,17 we have studied the structure and the formation kinetics of VTES-derived organic/silica hybrids in strongly basic medium by means of small-angle X-ray scattering (SAXS). The time evolution of the SAXS intensity I(q), where q is the modulus of the scattering vector, was found to be compatible with formation and evolution of linear chains with local persistence. The formation of linear chains, due to steric effects, was attributed to the greater probability to form a link, which results in a linear chain with parallel alignment of the vinyl groups as compared to the minor probability to form a link that results in a branching point with other relative orientations of the vinyl groups, mainly in the early stages of the VTES condensation. The persistence length lp is a measure * Corresponding author. Phone: +55-19 35269180. Fax: +55-19 35269179. E-mail: [email protected].

of the coiling degree of the macromolecule, and it is given by the average of the sum of all projections of the chain segments on a direction given by an initial segment.18 Because rheological properties are affected by the local persistence of the chains, the measure of this persistence is naturally of scientific interest to understand the overall process of the hybrid formation and behavior. The structure of the polymeric VTES/silica hybrid system, as reported elsewhere,17 evolves apparently with time up to the formation of a perfectly Gaussian persistent chain, for which the classical Kratky/Porod model applies and the persistence length could be determined graphically.18 The scattering curve I(q) from a perfectly Gaussian coiled chain macromolecule exhibits three characteristic regions: (i) the innermost part at low-q follows approximately a Gaussian curve characterized by a radius of gyration Rg; (ii) the intermediate-q portion follows the relationship I(q) ∼ q-2 due to large subsections of the macromolecule formed by random arrangements of chain elements building an object of mass-fractal dimension D ) 2; and (iii) the outermost part at high-q follows the scattering of a needle, I(q) ∼ q-1, because the small regions of the macromolecule are portions of linear chains. In a Kratky I(q)q2 versus q plot, the Gaussian region drops toward zero at q ) 0; the intermediate-q portion I(q) ∼ q-2 becomes horizontal; and the I(q) ∼ q-1 tail end follows as ascending line, the extrapolation of which toward q ) 0 passes through the origin. The intercept q* between the extrapolation of the power-law -2 scaling (Gaussian coil) with that of the power-law -1 scaling (rod-like portion) in a Kratky plot gives the persistence length by lp ) 6/πq* in the Kratky/Porod graphical method.18 After the appearance of the plateau I(q) ∼ q-2 in the experimental data of the VTES/silica hybrid formation, the Kratky plots exhibit maxima, which increase in magnitude and shift toward the low-q region with time.17

10.1021/jp110392m  2011 American Chemical Society Published on Web 12/15/2010

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Real persistence polymers display a gradual transition between power-law -2 scaling and power-law -1 scaling. Some equations are available to describe this often gradual transition regime in polymers. The classical and most widely used is that of Sharp and Bloomfield (SB).16 Beaucage et al.19 have presented a Unified Function (UF) to describe the gradual transition between a general mass-fractal regime I(q) ∼ q-D, with arbitrary mass-fractal dimension D (valid particularly also for a Gaussian chain with D ) 2), and the persistence rod-like scaling regime q-1 of the polymer chain. Values for D lower than 2 would account for non-Gaussian chains, and values for D above 2 would account for compaction and appearance of maxima in the Kratky plots. The appearance of maxima in the Kratky plots is also characteristic of formation of randomly and nonrandomly branched polycondensates in solution.20 To account for the appearance of maxima in the Kratky plots in the VTES/silica hybrid formation kinetics, we have proposed a modified branching Sharp-Boomfield (BSB) approach,17 in which the Debye function has been replaced by a form factor, which accounts for both random and nonrandom branching of polycondensates in solution. In this work, we have a rare opportunity to probe and compare the applicability of the mentioned approaches in describing the several stages of the experimental VTES/silica hybrid formation kinetics.

Donatti et al. where I(0) is the intensity extrapolated to q ) 0, lp is the persistence length, L is the contour length (the length of the hypothetically fully extended molecule), and g(x) ) (2/x2)[exp(-x) - (1 - x)] is the Debye function,18 with x ) (Llp/3)q2 since Rg ) (Llp/3)1/2 for a Gaussian coil. Equation 1 has been found to be valid with an accuracy better than 1% for L > 20lp and lp2q2 < 2.5.16 Mass-Fractal Unified Function (UF). The Unified Function (UF) has been used to describe small-angle scattering by a sum of scattering laws from each level of structure.19 The scattering from a level of structure is composed by a Guinier regime and a structurally limited power-law regime describing the overlap of structural levels.21 For a level of structure formed by a macromolecule of radius of gyration Rg and arbitrary massfractal dimension D overlapped with a level of structure formed by a thin rod-like particle with radius of gyration rg ) 3-1/2lp, with lp being the persistence length, the scattering intensity can be written as19

I(q) ) I(0){exp(-q2Rg2/3) + [DΓ(D/2)/RgD] × exp(-q2lp2/3)[erf(qRg /61/2)]3Dq-D + (1/nk)[exp(-q2lp2 /9) + (π/2lp)[erf(qlp /181/2)]3q-1]}

(2) Experimental Section A sol of organic/silica hybrid species was prepared by acid hydrolysis (pH 2) of a mixture of vinyltriethoxisilane (VTES) (47.2 mL, Aldrich, 95%), 0.1 N HCl (16 mL, as a catalyst and water source for hydrolysis), and ethanol as a mutual solvent (55 mL, Aldrich, PA). The hydrolysis was promoted at 70 °C for 2 h under mechanical agitation. The 1 M NH4OH was dropped under magnetic stirring into a separated 10 mL of the sol up to a final base concentration equal to 0.05 M. The kinetics of the aggregation process was studied in situ by small-angle X-ray scattering (SAXS) at 25 °C up to beyond the gel point. The SAXS spectra of the sample were collected as a function of the time in minutes (min) (as it appears in the figures), immediately after addition of NH4OH. A sample of the stable sol at pH 2 was also studied by SAXS. The SAXS experiments were carried out using synchrotron radiation with a wavelength λ ) 0.1608 nm at the SAXS beamline of the LNLS synchrotron radiation facility, 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 SAXS intensity from isotropic systems 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 ) 0.077 nm-1 up to qm ) 2.347 nm-1 in intervals of ∆q ) 4.90 × 10-3 nm-1. The data were corrected by sample attenuation and parasitic scattering, and normalized with respect to the beam intensity. Theoretical Basis Sharp and Bloomfield (SB). The classical Sharp and Bloomfield function is widely used to describe the gradual transition between power-law -2 scaling and power-law -1 scaling in persistence polymers. This can be cast as18

I(q) ) I(0){g(x) + (2lp /L)[(4/15) + (7/15x) [(11/15) - (7/15x)] exp(-x)]} (1)

where

nk ) (Rg /2lp)D[(1 + 2/D)(2 + 2/D)]D/2

(3)

is the number of Kuhn units, so the contour length L ) nk2lp.19 Branching Sharp-Bloomfield (BSB). This approach has been used17 as a modified Sharp-Bloomfield function to account for the appearance of maxima in the Kratky plots17,22 which is characteristic of randomly and nonrandomly branched polycondensates and for polydisperse coils of linear chains.20 The basic idea is to replace the Debye function g(x) in eq 1 by a form factor valid for both randomly and nonrandomly branched polycondensates and, in a particular case, for polydisperse coils of linear chains, which can be written by20

g(x) ) (1 + Cx/3)/[1 + (1 + C)x/6]2

(4)

where x ) Rg2q2, and C is a dimensionless constant that is a function of the link probability for the formation of a chain point and the link probability for the formation of a branching point in a given macromolecule. Equation 4 has been used instead of the Debye function in eq 1 by keeping the constraint Rg2 ) Llp/3 in studying VTES/silica hybrids formation kinetics.17 Results and Discussion Figure 1 shows the time evolution of SAXS intensity in the growth of VTES/silica hybrids in a strongly basic solution with NH4OH concentration of 0.05 M. This set of data has been chosen for the present study as representative from a more complete set of data obtained at different concentrations of NH4OH.17 The general behavior of the scattering with time is to increase the intensity at low- and intermediate-q without changing substantially the intensity at high-q, where the intensity was found to follow a power-law regime with scattering exponent approximately equal to -1. Figure 1 (right) shows the Kratky plots for the data. The SAXS intensity evolves with

Persistence Length in VTES/Silica Hybrids

Figure 1. (Left) Time evolution of the SAXS intensity in the growth of VTES/silica hybrids in solution with NH4OH concentration of 0.05 M. (Right) Kratky plots of the data. Plots from data of ref 17.

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Figure 3. Fitting (full line) of the Unified Function (UF) to the experimental data (points) in the Kratky plots. Numbers represent time in minutes. The fitting process fails to describe the rod-like regime q-1 at high-q for times lower than about 38 min, when D is apparently lower than about 1.4. The curves are vertically shifted for the sake of clarity.

Figure 2. Fitting (full line) of the Sharp and Bloomfield (SB) function to the experimental data (points) up to the stage of the plateau appearance in the Kratky plots (118 min). Numbers represent time in minutes. The curves are vertically shifted for the sake of clarity.

time up to the appearance of a plateau I(q) ∼ q-2 in the Kratky plots, and, after that, it displays maxima at low-q in the Kratky plots, which increase in magnitude and shift toward the low-q region. The overall picture is compatible with formation of linear chains, which grow, coil, package, and, possibly, branch to form persistent chain polymeric macromolecules in solution. In the following, we analyze the applicability and limitations of the several approaches in describing the several stages of the VTES/ silica hybrid formation kinetics. Figure 2 shows the Sharp and Bloomfield (SB) model [eq 1] fitting very well to the experimental data up to a time close to the appearance of a plateau in the Kratky plots. The fitting process was carried out using a nonlinear least-squares routine (Levenberg-Marquardt algorithm) to obtain the parameters lp, L, and I(0) that minimize the square of the difference between q2I(q) with respect to the experimental ones in the Kratky plots. The fit of the SB model was found to be not so good at the instant of the plateau appearance I(q) ∼ q-2 in the Kratky plots, which occurs at about 181 min. The model does not apply anyway to stages after the appearance of the plateau, when maxima arise in the Kraty plots. Figure 3 shows the fitting of the Unified Function (UF) [eq 2] to the experimental data in the Kratky plots. The same nonlinear least-squares routine was used to obtain the parameters

Figure 4. Fitting (full line) of the modified branching Sharp-Bloomfield (BSB) to the experimental Kratky plots (points). Numbers represent times in minutes. The curves are vertically shifted for the sake of clarity.

Rg, lp, D, and I(0) that minimize the square of the difference between q2I(q) with respect to the experimental ones. The contour length L ) nk2lp was evaluated from Rg, lp, and D through eq 3. The fitting process apparently fails to describe the rod-like regime q-1 at high-q for times lower than about 38 min (as shown by an arrow in Figure 3), when D is apparently lower than about 1.4. This causes an apparent abrupt time discontinuity in the parameter lp, leading to too small and meaningless physical values for the persistence length. We then conclude that the UF approach could not describe the first stages of the process when the fractal dimension D is apparently not so different from the exponent of the rod-like regime q-1. The UF model was also found to not fit very well to the experimental data at advanced stages of the process (Figure 3). Figure 4 shows the modified branching Sharp-Bloomfield (BSB) model fitting very well to the experimental Kratky plots along all stages of the process of the VTES/silica hybrids formation kinetics.17 The same least-squares routine (Levenberg-Marquardt algorithm) was used to obtain the parameters L, lp, C, and I(0) that minimize the square of the difference between q2I(q) with respect to the experimental ones.

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Figure 6. Kratky plot of the experimental data (points) at the instant of the plateau appearance (118 min) with the fittings of the SB, UF, and SBS models and the Kratky/Porod graphical method for obtaining lp.

Figure 5. Time evolution of the structural parameters as obtained from fitting the different models (SB, UF, and BSB) to the experimental data of the VTES/silica hybrid formation kinetics. The contour length L has been evaluated in the UF model through the relationship L ) 2lp(Rg/2lp)D[(1 + 2/D)(2 + 2/D)]D/2. Zero time (t ) 0) corresponds to the very stable sol at pH 2.

The radius of gyration Rg was evaluated from L and lp through the constraint equation Rg2 ) Llp/3. Figure 5 shows the time evolution of the structural parameters as obtained from fitting the different models to the experimental Kratky plots of the VTES/silica hybrid formation kinetics. Because, as mentioned, the fitting of the UF could not describe the first stages of the process, when the fractal dimension D is apparently not so different from the exponent of the rod-like regime q-1, leading to an abrupt discontinuity in lp, with too small and likely meaningless physical values, we consider only the UF parameters obtained at times greater than 38 min. As equally mentioned, the SB model could not describe as well the data at the instant of the appearance of the plateau (118 min) in the Kratky plot and no way could describe the arising of maxima in such plots afterward. Accordingly, only SB parameters obtained up to 118 min appear in the sequence. The persistence length lp diminishes rapidly on passing from pH 2 (sol, t ) 0) to the basic step ([NH4OH] ) 0.05 M) with the increase of the length contour L (Figure 5). This means a rapid growth and coiling of the linear chains with increasing pH. The contour length L gradually increases with time in the basic step of the process, while lp diminishes rapidly at the beginning and goes to a constant value at the stage close to the appearance of the plateau in the Kratky plots (118 min). The value C ) 1 found at the beginning of the stage from t ) 0 up to t ) 118 min (Figure 5) is compatible with the formation of polydisperse coils of linear chains in solution or branched polycondensates of randomly f-functional elements.20 The properties of linear chains are obtained by setting either the branching probability to zero or the functionality of the branching units to f ) 2.20 The rapid diminution of C at about 71 min could mean an increase of the link probability to form nonrandomly branched polycondensates.20 Maxima in the Kratky plots should arise when C < 1/3. The mass fractal dimension D could be obtained just for t > 38 min (Figure 5), when D is apparently greater than 1, the modulus of the exponent of the

Figure 7. (Left Y-axis) Time evolution of the radius of gyration (Rg) evaluated through the relationship Rg ) (Llp/3)1/2 for SB and BSB models and directly fitted by the Unified Function. (Right Y-axis) Time evolution of I(0)/L as a measure proportional to the linear mass density M/L of the linear chains (t ) 0 means sol at pH 2).

rod-like power-law regime. The increase of D could mean packing of the structure. The values for lp and L determined from both SB and BSB models are in reasonable agreement among them in the description of the stage of transformation from t ) 0 up to t ≈ 118 min, but the SB model does not fit so well to the plateau. The values are also in agreement with those as determined from the UF approach, but just for t > 38 min (up to the plateau (118 min)). Figure 6 shows the Kratky plot for the data at the plateau (t ) 118 min) together with the fittings of the SB, UF, and BSB models and the Kratky/Porod graphical method for determining the persistence length lp ) 6/πq*, where q* is the intercept between the extrapolation of the power-law -2 scaling with that of the rod-like power-law -1 scaling.18 We have obtained lp ) (1.06 ( 0.03) nm from the Kratky/Porod graphical method, lp ) (0.982 ( 0.005) nm from UF, lp ) (0.926 ( 0.005) nm from BSB, and lp ) (0.807 ( 0.005) nm from SB. Among the considered models, the UF gives the value for lp that is in better agreement with that evaluated from the Kratky/Porod graphical method. For times above the appearance of the plateau, the values for lp according to the UF model were found to be practically constant, but to increase at very long times, while according to the BSB model they were found gradually diminishing with time (Figure 5). Such a disagreement was also found in the behavior of the parameters L (Figure 5) and Rg (Figure 7) for long times. Both L and Rg as determined by the UF model were found to be larger than those determined by the BSB model at advanced stages of the process. The more accentuated increase of the contour length L (Figure 5) and the radius of gyration Rg

Persistence Length in VTES/Silica Hybrids (Figure 7) in the UF model, in connection with the increase of the mass-fractal dimension D (Figure 5) and constancy of lp (Figure 5), may mean that the macromolecule keeps certain flexibility even at advanced stages of the process. The growth of the macromolecule should occur by increasing the contour length so any eventual diminution in the coiling degree would be compensated by the increase of the mass-fractal dimension D, which seems to not be enough to avoid the additional increase of the radius of gyration Rg of the macromolecule (Figure 7). On the contrary, the more discrete increase of L (Figure 5) and Rg (Figure 7) at advanced stages of the macromolecule growth in the BSB model, in connection with the diminution of the persistence length lp and increase of the branching degree (with the diminution of the parameter C) (Figure 5), means that the macromolecule loses flexibility at advanced stages of the process. These two different aspects about the flexibility of the macromolecule at advanced stages of aggregation as inferred from both the UF and the BSB models could be analyzed under another point of view. For polymeric macromolecules in solution, the independent scattering intensity extrapolated to zero, I(0), is proportional to M, where M is the mass of the macromolecule.18 Next, I(0)/L is simply proportional to M/L, where M/L is the linear mass density of the linear chains. Figure 7 shows that all models yield similar constant values for the linear mass density M/L (proportional to I(0)/L) within the range of applicability of each model along the stages of the macromolecule growth, but the values for M/L yielded by the UF and BSB models are not in agreement at very advanced stages of the process. Indeed, the values from the BSB model increased with time, while those from UF model are kept constant at very advanced stages of macromolecule growth. The constancy of M/L is compatible with the predominance of the link probability of formation of linear chains keeping some flexibility to the chains, while the increase of M/L means an increase of thickness of the chain segments due to the high degree of branching, which confers some rigidity to the macromolecule. So, the macromolecule structural properties described by the UF and the BSB models are not in agreement at very advanced stages of the macromolecule growth. These differences are even reinforced as the base concentration is increased so further stages of the macromolecule growth could be probed.17 Two main reasons could be appointed for this discrepancy: (i) the pure coiling mechanism intrinsically assumed in the UF model as lp, Rg, D, and, in connection, L changes at advanced stages of the macromolecule growth; and (ii) the constraint equation Rg ) (Llp/3)1/2 assumed in the BSB model as lp, L, C, and, in connection, Rg changes even for high branching degrees associated with the advanced stages of the macromolecule growth. It should be emphasized that the growth of the polymeric phase continues beyond the time periods probed in the present study and the structure evolves up to a completely opaque coarsened phase-separated gel, with kinetics strongly accelerated with increasing base concentration.17 Conclusions The time evolution of the local chain persistence in the aggregating of VTES/silica hybrids was studied by SAXS, and it was analyzed in terms of the classical Sharp and Bloomfield (SB) model, and of a mass-fractal Unified Function (UF) with an arbitrary mass-fractal dimension D, and of a modified branching Sharp-Bloomfield (BSB) approach, which incorporates a branching probability parameter C. The SB function describes well the SAXS data, but it is restricted to stages close to the appearance of a plateau in the

J. Phys. Chem. C, Vol. 115, No. 3, 2011 671 Kratky plots. The UF and BSB models describe well the SAXS data up to the plateau and afterward, when maxima in the Kratky plots rise and shift toward the low-q with time. The UF approach does not apply to the data at the early stages of aggregation, when D is apparently not so different from the exponent -1 of the rod-like power-law regime of the chain segments. The values of the persistence length lp, the contour length L, the gyration radius Rg, and the mass density per chain length M/L have been determined from the different approaches. At advanced stages of the macromolecule growth, both L and Rg from the UF model were found to be larger than those from the BSB model, while the linear density M/L was found to be a constant value from UF model and a value increasing with time according to the BSB model. The constancy of M/L is compatible with the predominance of the link probability of formation of linear chains, giving some flexibility to the chains, while the increase of M/L means increase of thickness of the chain segments due to the high degree of branching, conferring some rigidity to the macromolecule. Two main reasons could be appointed for this discrepancy: (i) the pure coiling mechanism intrinsically assumed in the UF model as lp, Rg, D, and, in connection, L changes at advanced stages of the macromolecule growth; and (ii) the constraint equation Rg ) (Llp/3)1/2 assumed in the BSB model as lp, L, C, and, in connection, Rg changes even for high branching degrees associated with the advanced stages of the macromolecule growth. Acknowledgment. This research was partially supported by LNLS - National Synchrotron Light Laboratory, FAPESP, and CNPq, Brazil. References and Notes (1) Zhao, D.; Huo, Q.; Feng, J.; Chmelka, B. F.; Stucky, G. D. J. Am. Chem. Soc. 1998, 120, 6024–6036. (2) Bandyopadhyay, A.; De Sarkar, M.; Bhowmick, A. K. J. Mater. Sci. 2005, 40, 5233–5241. (3) Deng, T. S.; Zhang, Q. F.; Zhang, J. Y.; Shen, X.; Zhu, K. T.; Wu, J. L. J. Colloid Interface Sci. 2009, 329, 292–299. (4) Chong, A. S. M.; Zhao, X. S. Catal. Today 2004, 93-95, 293–299. (5) Chong, A. S. M.; Zhao, X. S.; Kustedjo, A. T.; Qiao, S. Z. Microporous Mesoporous Mater. 2004, 72, 33–42. (6) Itagaki, A.; Nakanishi, K.; Hirao, K. J. Sol-Gel Sci. Technol. 2003, 26, 153–156. (7) Posset, U.; Gigant, K.; Schottner, G.; Baia, L.; Popp, J. Opt. Mater. 2004, 26, 173–179. (8) Nakane, K.; Yamashita, T.; Iwakura, K.; Suzuki, F. J. Appl. Polym. Sci. 1999, 74, 133–138. (9) Peterlik, H.; Rennhofer, H.; Torma, V.; Bauer, U.; Puchberger, M.; Hu¨sing, N.; Bernstorff, S.; Schubert, U. J. Non-Cryst. Solids 2007, 353, 1635–1644. (10) Tamaki, R.; Chujo, Y. Appl. Organomet. Chem. 1998, 12, 755–762. (11) Yano, S.; Iwata, K.; Kurita, K. Mater. Sci. Eng., C 1998, 6, 75–90. (12) Jitianu, A.; Gartner, M.; Zaharescu, M.; Cristea, D.; Manea, E. Mater. Sci. Eng., C 2003, 23, 301–306. (13) Vercaemst, C.; Friedrich, H.; de Jongh, P. E.; Neimark, A. V.; Goderis, B.; Verpoort, F.; Van Der Voort, P. J. Phys. Chem. C 2009, 113, 5556–5562. (14) Kang, K. S.; Kim, J. H. J. Phys. Chem. C 2008, 112, 618–620. (15) Eo, Y. J.; Kim, D. J.; Bae, B. S.; Song, K. C.; Lee, T. Y.; Song, S. W. J. Sol-Gel Sci. Technol. 1998, 13, 409–413. (16) Sfˇcˇ´ık, J.; McCormick, A. V. Catal. Today 1997, 35, 205–223. (17) Vollet, D. R.; Donatti, D. A.; Awano, C. M.; Chiappim, W., Jr.; Vicelli, M. R.; Iban˜ez Ruiz, A. J. Appl. Crystallogr. 2010, 43, 1005–1011. (18) Glatter, O.; Kratky, O. Small Angle X-ray Scattering; Academic Press: London, 1982. (19) Beaucage, G.; Rane, S.; Sukumaran, S.; Satkowski, M. M.; Schechtman, L. A.; Doi, Y. Macromolecules 1997, 30, 4158–4162. (20) Burchard, W. Macromolecules 1977, 10, 919–927. (21) Beaucage, G. J. Appl. Crystallogr. 1995, 28, 717–728. (22) Gommes, C. J.; Goderis, B.; Pirard, J. P.; Blacher, S. J. Non-Cryst. Solids 2007, 353, 2495–2499.

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