Dynamic Scaling Properties of TeO2-Based Gels

S. Coste,‡ A. Lecomte,*,† P. Thomas,‡ and J. C. Champarnaud-Mesjard‡

“Science des Procedes Ceramiques et de Traitements de Surface”, UMR 6638-CNRS, ENSCI,47-73 AVenue Albert Thomas, 87065 Limoges Cedex, France, and “Science des Procedes Ceramiques et

de Traitements de Surface”, UMR 6638-CNRS, Faculte des Sciences et Techniques de Limoges,123 AVenue Albert Thomas, 87060 Limoges Cedex, France

ReceiVed July 8, 2008. ReVised Manuscript ReceiVed August 28, 2008

Homogeneous, transparent, and mechanically rigid gels have been successfully synthesized in the telluriumisopropoxide-isopropanol-citric acid and water system. The sol to gel transition and the gels microstructure havebeen studied by using small angle X-ray scattering (SAXS) experiments. For any value of the two key synthesisparameters, which are the citric acid ratio and the alkoxide concentration, very small Te-rich elementary particles,about 1-1.5 nm in radius, form immediately when the water is added, leading to colloidal sols. During gelation, theseelementary particles stick progressively together to build up fractal aggregates by a pure hierarchical aggregationprocess which has been identified as a reaction-limited cluster aggregation (RLCA) mechanism. The SAXS curveanalysis, based on scaling concepts, shows that the gelling network exhibits a time and length scale invariant structurefactor characterized by self-similarity. This self-similarity is also displayed for a wide range of chemical compositionsand the gel microstructures only differ in their fractal aggregate size according to the tellurium isopropoxide concentrationas well as the citric acid ratio.

1. Introduction

Tellurium oxide-based glasses and glass ceramics are verypromising materials for nonlinear optical device technology, asthey present highly nonlinear refractive indices, low linear andnonlinear absorption coefficients, and fast nonlinear opticalresponse.1 Most research into these glasses has focused onconventional melting-quenching methods which are very diffi-cult to use to produce pure TeO2 glass,2,3 crystallized TeO2

nanoparticles,4 as well as TeO2 thin films that can be used inoptical and electronic devices. Therefore, wet chemistry andmore particularly the sol-gel technology offer the potential toavoid and overcome many of the drawbacks of this melting-quenching route. Indeed, sol-gel chemistry, based on molecularprecursors which are transformed into an oxide network byhydrolysis and condensation reactions, allows the powderlessprocessing of glasses or ceramics and exhibits many advantagesfor the synthesis of metastable phases, monodispersed powderson a nanometric scale, as well as fibers or thin coatings directlyproduced from the solution by sol-gel methods such as dipcoating, spray drying, or spin drawing.5,6 Moreover, through thechoice of precursors and polymerization conditions, thestructure-property relationships of such sol-gel materials canreadily be modified and managed.

However, the preparation of TeO2-based materials fromtellurium alkoxides has received very little investigation and issummed up to the formation of TeO2 precipitates7 and thin filmsynthesis.8,9 This limited success of TeO2 sol-gel processing ismainly attributed to difficulties encountered in controlling thehydrolysis-condensation reactions of Te(IV) alkoxides, whichare extremely rapid and cannot be managed through conventionalapproaches to steric stabilization of the Te(IV) alkoxides as wellas by using common alkoxide reaction modifiers.10,11 The useof Te(VI) compounds, like allotelluric ethyl ester, such that thelone pair electrons thought to be responsible for this high reactivityare not present, is suggested and assumed to be promising forTeO2 bulk xerogel synthesis12 but, nevertheless, has to be studiedthoroughly and optimized.

To date, various wet chemical routes12 have been successfullydeveloped to manufacture good-quality optically transparent thinfilms from these precursors, but there has been no report of thesuccessful production of bulk gels, although a study by Costeet al.13 did demonstrate that precipitation could be avoided byin situ formation of water, resulting from the esterification reactionbetween acetic acid and 2-propanol, and that bulk gels could besynthesized in a very limited composition range.

The present paper reports on the synthesis of bulk telluriumoxide based gels in the Te isopropoxide-isopropanol-citric acidand water system. Indeed, citric acid has been found to be arelevant chemical modifier to control hydrolysis-condensationreactions of tellurium (IV) isopropoxide. As the molecularstructure of the precursor and the gel network microstructure arecrucial information for engineering these materials and their

* Corresponding author. A. Lecomte, full mailing address: ENSCI, 47-73Avenue Albert Thomas, 87065 Limoges, France, Telephone: 33 (0) 5 55 4522 22, Fax: 33 (0) 5 55 79 09 98, E-mail address: andre.lecomte@unilim.fr.

‡ Faculte des Sciences et Techniques de Limoges.† ENSCI.(1) El-Mallawany, R. A. H. Tellurite Glasses Handbook: Physical Properties

and Data; CRC Press, 2001.(2) Dutreilh-Colas, M.; Thomas, P.; Champarnaud-Mesjard, J.-C.; Fargin, E.

Phys. Chem. Glasses 2003, 44(5), 349.(3) Dutreilh-Colas, M.; Charton, P.; Thomas, P.; Armand, P.; Marchet, P.;

Champarnaud-Mesjard, J.-C. J. Mater. Chem. 2002, 12, 2803.(4) Lasbrugnas, C. Materiaux doubleurs de frequence:Verres etVitroceramiques

a base d’oxyde de tellure- elaboration et caracterisation; Thesis of the Universityof Limoges, France, 2004.

(5) Livage, J.; Sanchez, C.; Babonneau, F. Molecular precursor routes toinorganic solids, chemistry of adVanced materials: an oVerView; Wiley-VCH:New York, 1998.

(6) Brinker, C. J.; Scherer, G. W. Sol-Gel Science: The Physics and Chemistryof Sol-Gel Processing; Academic Press: New York, 1990.

(7) Pierre, A.; Duboudin, F.; Tanguy, B.; Portier, J J. Non-Cryst. Solids 1992,147&148, 569.

(8) Weng, L.; Hodgson, S. N. B. J. Non-Cryst. Solids 2002, 297, 18.(9) Weng, L.; Hodgson, S. N. B. Opt. Mater. 2002, 19, 313.(10) Hodgson, S. N. B.; Weng, L. J. Non-Cryst. Solids 2000, 276, 195.(11) Coste, S. EVolution structurale et microstructurale de precurseurs d’oxyde

de tellure elabores par Voie sol-gel; Thesis of the University of Limoges, France,2003.

(12) Hodgson, S. N. B.; Weng, L. J. Mater. Sci.: Mater. Electron. 2006, 17,723.

(13) Coste, S.; Lecomte, A.; Thomas, P.; Champarnaud-Mesjard, J.-C.; Merle-Mejean, T.; Guinebretiere, R. J. Non-Cryst. Solids 2004, 345&346, 634.

12568 Langmuir 2008, 24, 12568-12574

10.1021/la802137s CCC: $40.75 2008 American Chemical SocietyPublished on Web 10/10/2008

structure-property relationships, the study mainly focuses onthe investigation of the sol to gel transformation by small angleX-ray scattering measurements (SAXS), one of the most powerfulmethods to probe density-density correlation function of thegelling network microstructure in the 1-100 nm range.6

Many investigations have clearly established that the micro-structures of a number of gels can be described by the conceptof fractals.6,14 Such fractal structures exhibit specific small anglescattering features that are briefly reviewed in section 3. In thesame section, we present the model used to accurately extractmicrostructural information from the SAXS data, such as primaryparticle and aggregate sizes and fractal dimensions. In thefollowing sections, the results are given and discussed. The timeevolution of fractal aggregate growth and the impact on the gelmicrostructure of the two essential synthesis parameters, whichare the citric acid ratio and the Te alkoxide concentration, areconsidered. The aggregation mechanism and dynamic scalingproperties have also been identified. The SAXS experimentalsetup as well as the gel synthesis are describe hereafter.

2. Experimental Details2.1. Gel Synthesis. All syntheses were performed in a glove-

box under dry air atmosphere. The tellurium isopropoxide(Te(OCH(CH3)2)4, purity >99.9%, Alfa-Aesar) was first diluted inanhydrous isopropanol (purity >99%, Prolabo) to give a 1 mol/Lsolution. Citric acid was dissolved in a mixture of this solution andanhydrous isopropanol with appropriate amounts in order to reach,at the end of the synthesis, the desired alkoxide concentration Cf andmodification ratio R ) [citric acid]/[Te alkoxide]. Then, a solutionof water diluted in isopropanol was added dropwise under vigorousmechanical stirring. Finally, the glass vessel was hermetically closedand kept at room temperature or 60°C until gelation occurred. Eachsample was identified by its precursor concentration, Cf, modificationratio, R, and hydrolysis ratio W ) [water]/[alkoxide]. In this work,W was kept equal to 4, the stoichiometric quantity to fully hydrolyzethe Te isopropoxide.

2.2. Small Angle X-ray Scattering Measurements. The smallangle X-ray scattering study was performed using an originalexperimental setup adapted to a 18 kW rotating anode X-ray generator,operating at 45 kV 270 mA with a fine 0.5 × 1 mm2 spot size. TheCu KR1 incident beam, provided by a double-channel cut germaniummonochromator, was quite parallel and directed on the sample througha system of slits with a pointlike geometry. The scattered intensityI(q) was recorded using a linear position-sensitive detector. Thesample-detector distance was 0.5 m to cover a q-range from 0.08to 4 nm-1 where q is the scattering vector, q ) 4πλ-1 sin θ, λ theCu KR1 wavelength, and 2θ the scattering angle. The sample-detectorpath was evacuated to suppress air scattering and absorption.

For time-dependent investigations, the analyzed samples wereissued from the same solution. After the synthesis, the mother solutionwas divided into several equal aliquots. Each aliquot was thenindividually aged at the desired temperature (20 or 60°C) for thechosen aging time. Then, a small amount of the aged solution wasplaced in a cell with two kapton windows, spaced about 0.5 mmapart, and the scattered intensity was collected at room temperaturewith an exposure time of 10 h. Such a long exposure time will bediscussed in the Results and Discussion section. Appropriatecorrections for background scattering from slits, kapton cell walls,residual atmosphere, and absorption effects were classically appliedto raw data.15

3. Small Angle X-ray Scattering and Fractal Feature

Many gel microstructures are made of interconnected fractalclusters resulting from the progressive aggregation of polymer

species or very small colloidal primary particles.6,14 To investigatethe fractal cluster growth of the gelling network, small angleX-ray scattering experiments have proven to be, either alone orin combination with light or neutron scattering, an extremelypowerful tool allowing the aggregates to be studied in situ withoutdisturbing their growth in any way. The SAXS intensity, I(q),produced by an isotropic set of such fractal objects, withoutcorrelation between them, is given by15

I(q) ∼ N·P(q)·S(q) (1)

where N is the number, per volume unit, of scattering monomersor primary particles that build up the aggregates, P(q) is theirform factor, and S(q) is the structure factor that accounts for thespatial correlation between the monomers or primary units insidethe aggregates. On one hand, assuming spherically shaped primaryparticles (radius a, volume V) with smooth interface and uniformdensity, their form factor P(q) is expressed by the well-knownRayleigh function16

P(q, a)) 9V2 . (sin(qa)- qa cos(qa)

(qa)3 )2(2)

On the other hand, from the definition of the mass distributionin fractal aggregates, M(r) ∼ rDf, by Fourier transform of thecorresponding density distribution and using a cutoff for thefractal correlation represented by a function such as C(r) ∼exp(-r/�), the structure factor S(q) has been established for asingle aggregate17,18

S(q, a, Df, �)) 1+ 1

(q · a)Df

Df ·Γ(Df - 1)

(1+ 1

q2 . �2)(Df-1)⁄2×

sin[(Df - 1) · tan-1(q · �)] (3)

where Df is the fractal dimensionality and Γ(x) is the gammafunction of argument x. � represents the characteristic distanceabove which the mass distribution is no longer described by thefractal law. In practice, � is assimilated to the size or correlationlength of an aggregate, and its use instead of the classical radiusof gyration Rg, deduced from Guinier’s law, comes to the samething, as these two sizes, � and Rg, are linked by a linear functiondepending on Df.18

Equation 3 better accounts low-q and large-q deviations to thefractal regime than the simplest Fisher-Burford’s approximation,19

for example. Nevertheless, S(q) as well as I(q) reduce, in arestricted q-range, to a remarkably simple behavior, I(q) ∼ q-Df,which is at the origin of the success of X-ray as well as neutronsmall angle scattering techniques in the microstructural inves-tigation of sol-gel precursors. Indeed, the fractal dimensionalityis easily and directly determined from the slope of the linearevolution when the scattering intensity is plotted versus q on abilogarithmic scale. In the following sections, this slope will benamed as an apparent fractal dimension, Dapp. This power lawdependence of I(q) is limited both at large and at small q valuesbecause of the finite size or correlation length of the aggregates,�, and because of the non-zero size of the primary particles, a,respectively. The effect of these cutoffs is the appearance of twocrossovers in the scattering curve: the first one at q ∼ 2π/�,between the Guinier and fractal regimes, and the second, at qc

∼ 2π/a, between the fractal and Porod regions.

(14) Lecomte, A.; Lenormand, P.; Dauger, A. J. Appl. Crystallogr. 2000, 33,496.

(15) Glatter, O. In Small Angle X-ray Scattering; Glatter, O., Kratky, O., Eds.;Academic Press: London, 1982; Chapter 4.

(16) Porod, G. In Small Angle X-ray Scattering; Glatter, O., Kratky, O., Eds.;Academic Press: London , 1982; Chapter 2.

(17) Freltoft, T.; Kjems, J. K.; Sinha, S. K. Phys. ReV. B 1986, 33, 269.(18) Teixeira, A. J. Appl. Crystallogr. 1988, 21, 781.(19) Fischer, M. E.; Burford, R. J. Phys. ReV. 1967, 156, 583.

Dynamic Scaling Properties of TeO2-Based Gels Langmuir, Vol. 24, No. 21, 2008 12569

At very small q, P(q) ∼ 1 and I(q) ∼ N ·S(q), leading to theevaluation of the SAXS intensity extrapolated to q ) 0 by thedevelopment of eq 3

Io ∼ I(q) 0) ∼ N·Γ(Df + 1)(�a)Df

(4)

a resulting expression similar to the one which had been previouslydirectly derived from the theoretical definition of Io.20 Thisextrapolated intensity Io is one of the most significant parametersextracted form SAXS curves, since it is equal to the square ofthe number of excess electrons in the scattering objects withrespect to surrounding matrix or solvent and irrespective of theirshape or size.16

By contrast, at large q, i.e., in the Porod domain, S(q) ∼ 1,the SAXS intensity is dominated by P(q) and is expected toexhibit an asymptotic behavior, I(q) ∼ q-4, due to the smoothingof the oscillations of the structure factor (eq 2) by experimentalsmearing and particle size polydispersity effects. As in a pointlikecollimation geometry the experimental smearing effects arenegligible, a monomodal log-normal size distribution Dn(r) hasbeen introduced to parametrize the primary particle size distribu-tion. Then, the SAXS intensity is finally expressed by15

I(q) ∼ ∫0

∞

Dn(r)·P(q, r)·S(q, r, Df, �) dr (5)

with

Dn(r)) 1

√2π

1r · L

exp(-(ln r- ln Ro)2

2L2 ) (6)

where Ro and L are the center and the width of the size distribution,respectively.

4. Results and Discussion

The alkoxide concentration Cf was varied between 0.05 and0.9 mol/L while the modification ratio R was ranging from 0 to2, with W being kept equal to 4. The evolution of the sols at roomtemperature, about 20 °C, was so slow, indeed nonexistent, thatthe gelation study was then systematically conducted at 60 °C.The final state of the samples was first assessed according tohomogeneity, transparency, and mechanical rigidity.

For 0 < R < 0.3 and any value for Cf, precipitation alwaysoccurred as soon as the water-isopropanol mixture was added.For large modification ratios, R> 1.2, gelation did not occur andprecipitation was also observed after an incubation time dependingon Cf. For 0.3 < R < 1.2, the final products were gels thatdiffered according to Cf. Near the precipitation boundaries or forthe highest concentrations, Cf > 0.7 mol/L, the gels wereinhomogeneous and became turbid. For 0.1 < Cf < 0.7 mol/L,very homogeneous and transparent gels were synthesized witha very good rigidity. When the alkoxide concentration wasdecreased to about Cf ) 0.1 mol/L, the gels turned soft. Belowthis limit, a phase separation was noticed and the samples werecomposed of a very soft gel surrounded by a liquid phase. Asgenerally observed for sol-gel chemistry,6 the gelling time, tg,for a given concentration, increases drastically when themodification ratio increases. In the same way, for a given R, tg

decreases when Cf is increased. For example, with Cf ) 0.5mol/L and W ) 4, the gelling time ranges from 0.25 h to 200 hfor R ) 0.4 and R ) 1, respectively.

From these macroscopic observations, a major finding wasthat citric acid is a relevant Te isopropoxide chemical modifier.

To better understand the basic reactions involved in the gelationprocess, a complementary study using Fourier transform infraredspectroscopy has been carried out and has shown that citric acidis strongly bonded to Te isopropoxide, slowing down thehydrolysis and condensation reactions. These results as well asthe aggregation phase diagram previously summarized have beendescribed in detail elsewhere.21

From the standpoint of gelling network nanostructure, the solto gel transition has been followed for a sol composition, Cf )0.5 mol/L, R ) 0.7, and W ) 4, located in the center of thetransparent, homogeneous, and mechanically rigid gels domain,i.e., 0.1 < Cf < 0.7 mol/L and 0.3 < R < 1.2. Then, we haveestablished the influence of the two essential parameters, thecitric acid molar ratio R and the alkoxide concentration Cf, onthe nanostructure of the gels.

4.1. The Sol to Gel Transition of a Sol Prepared with Cf

) 0.5 mol/L, R ) 0.7, and W ) 4. The viscosity of this solbecomes very high after 7 h of aging at 60 °C, and gelationoccurs after about 35 h. At 20 °C, no macroscopic evolution ofthis composition has been noticed, and this behavior has beenconfirmed by SAXS. The scattering curve of a sol, aged aboutseven months at 20 °C, presents the same features as that obtainedfrom the sol just after synthesis. This result allows us to choosea rather long acquisition time for the scattering curves, about10 h, leading to a very good signal to noise ratio. Indeed, weassume that the system does not significantly evolve during theSAXS data recording at room temperature.

Figure 1 shows the log-log plot of the scattered intensity I(q)as a function of the modulus of the scattering vector q, measuredfor the starting sol and for different aging times at 60 °C rangingfrom 5 min to 143 h, i.e., about four times the gel time tg. Fromthe beginning of the gelation process, the starting sol presentsa scattering signal that increases for low q-values, first ratherrapidly, then far more slowly, and finally saturates when theaging time reaches the gel time tg. At intermediate and highscattering q vectors, two power law regimes can be distinguished.

The first one, at intermediate q-values, extends during gelationtowards smaller and smaller scattering vectors, while the Guinierdomain simultaneously vanishes progressively. The exponent ofthis power law varies from -1.7 to -2.1, and its absolute valuewill be assigned later to an apparent fractal dimension, Dapp. Athigh q vectors, the curves overlap quite well for a wide angular

(20) Devreux, F.; Boilot, J.-P.; Chaput, F.; Lecomte, A. Phys. ReV. A 1990,41(12), 6901.

(21) Coste, S.; Lecomte, A.; Thomas, P.; Merle-Mejean, T.; Champarnaud-Mesjard, J.-C. J. Sol-Gel Sci. Technol. 2007, 41, 79.

Figure 1. Log-log plot of SAXS intensity distribution for a sol preparedwith Cf ) 0.5 mol/L, R ) 0.7, and W ) 4 and aged for different timesat 60 °C.

12570 Langmuir, Vol. 24, No. 21, 2008 Coste et al.

range; the second power law regime presents a slope very closeto -4, in agreement with the Porod law, and clearly evidencesthe colloidal nature of the sol. The lower Porod limit, at aboutqc ∼ 2 nm-1, is nearly invariant all along the sol to gel transition.Very small Te-rich particles, with size being equal here, in a firstapproximation,22 to a ) π/qc, to about 1.6 nm in radius, arereadily formed just after the water is added.

These SAXS curve features are typical of the scattering dueto mass fractal aggregates constituted of elementary particles ofsize a.17,18 During their aggregation, the increase of the scatteringintensity and the extension of the fractal correlation domain tosmaller and smaller q-values indicate that the aggregates grow.Regardless of the aging time, the monotonic decrease of thescattering intensity curves versus the scattering vector q ischaracteristic of noninteracting clusters, and the theoreticalfunction I(q) given by eq 5 fully applies.

Nonlinear least-squares fitting of eq 5 to the experimentalintensity gives excellent agreement with the experimentalscattered intensity for the whole domain of scattering vectors.Some examples of simulations corresponding to the starting soland aging times of 15 min, 4 h, and 32 h are presented in Figure2. The slight discrepancy observed at the high scattering vectorsis due either to a lack of intensity leading to an unfavorablesignal to noise ratio or to the fact that the elementary particlesare not perfectly smooth. However, with regard to their size, itseems unreasonable to qualify their surface as fractal. In thisway, all the microstructural parameters of the model, aggregatesize �, primary particle size distribution (mean primary particlesize, a, and standard deviation, σ), and fractal dimension Df wererefined and the extrapolated intensity at zero angle, I0, deduced.

The aggregate size, �, and extrapolated intensity at zero angle,I0, versus time evolve at first proportionally to power laws: � ∼tR and I0 ∼ tR′ where the kinetic exponents R and R′ are equalto 0.35 and 0.7, respectively (Figure 3). Then, near the gel time,they increase far more slowly, meaning that the system reachessaturation. During gelation, � grows from about 3 nm to 17 nm,while the elementary particle size, a, as well as the fractaldimension, Df, remain quiet constant, regardless of the aging

time, at about 1.1 and 2.1 nm, respectively (Figure 4). On thecontrary, the apparent fractal dimension Dapp, directly determinedfrom the slope of the linear behavior when the scattering intensityis plotted versus q on a bi-logarithmic scale, increases duringgelation from 1.7, for t ) 5 min, and tends to 2.1 at the gel point.This behavior discrepancy between the two fractal dimensions,Df and Dapp, is only due to a size effect and is all the weaker whenthe ratio �/a is high,18,23 which justifies their convergence forthe longest aging times. It also emphasizes that the easiest analysisof the scattered intensity by a fractal object through the remarkablysimple behavior, I(q) ∼ q-Df, in the intermediate q region, isonly relevant when both inequalities, q� . 1 and qa , 1, apply.Outside these limits, such a fractal dimension determination couldbe largely erroneous.

The fractal dimension, Df, and the elementary particle size,a, remaining constant all along the gelation process; thegrowing aggregates are so self-similar, that is to say that thegelling network microstructure remains statistically identicalto itself, whatever the moment of observation, except for ascale factor. This self-similarity can be supported by analyzing,on one hand, the behavior of the extrapolated intensity at zeroangle, I0, versus aggregate size �. The scaling law betweenIo and � is shown in Figure 5. I0 evolves according to a powerlaw of �, with an exponent of 2. This value, which could alsobe determined by calculating the R′/R kinetic exponent ratio, isvery close to the fractal dimension Df, Df ) 2.1, a well known

(22) Hasmy, A.; Vacher, R.; Jullien, R. Phys. ReV. B 1994, 50, 1305. (23) Sorensen, C. M.; Oh, C. Phys. ReV. E 1998, 58(6), 7545.

Figure 2. Log-log plot of SAXS intensity distribution for a sol preparedwith Cf ) 0.5 mol/L, R ) 0.7, and W ) 4 just after the synthesis andafter 15 min, 4 h, and 32 h of aging at 60 °C and the correspondingcalculated scattered intensity curves in a solid line. The curves are arbitraryvertically shifted for clarity.

Figure 3. Evolution of the aggregate size � (9), of the elementary particlesize a (f), and of the extrapolated intensity at zero angle I0 ((), versusthe aging time at 60 °C.

Figure 4. Evolution of the fractal dimension Df (9) and of the apparentfractal dimension Dapp ((), versus the aging time at 60°C.

Dynamic Scaling Properties of TeO2-Based Gels Langmuir, Vol. 24, No. 21, 2008 12571

result which largely occurs during the aggregation of colloidalmodel systems.24 From this evolution of Io, I0 ∼ �Df, the totalnumber, N, of elementary particles building the fractal aggregates,is deduced to remain constant (eq 4) all along the gelation, whichcorroborates the excellent superposition of all the scatteringintensity curves in the high scattering q vectors domain (Fig-ure 1).

On the other hand, all the scattering curves should collapseand make a master curve after their normalization by the individualcharacteristics of the aggregates, which are their fractal dimension,size, and mass at moment t, according to the relation25

S(q�, t) ∼ I(q�, t)) �(t)dF(q�) (7)

where d ) Df and F(q�) is a time-independent scaling function.For dense systems, the distribution of the scattered intensitygenerally displays a pronounced peak at a finite q vector. Thisexpression is usually written in the following generalized form

S(q ⁄ qm, t) ∼ I(q ⁄ qm, t)) qm(t)-dF(q ⁄ qm) (8)

where qm is the position of the scattered intensity maximum,commonly assimilated to the inverse of a correlation length andF(q/qm) is also a time-independent scaling function. For thedynamic scaling hypothesis of a decomposing system whichapplies to advanced stages,26 d is the spatial dimension, d ) 3,while for colloidal aggregation, the scaling requires d ) Df, thefractal dimension of the clusters.24,27-29

By using all the data, the best master curve is achieved witha fractal dimension adjusted to about 1.9 (Figure 6), while forthe advanced stages of gelation, the best collapse onto the samescaling curve is reached with a fractal dimension refined to about2-2.1. Obviously and logically, the 1.9 value corresponds to theaverage of the apparent fractal dimension, Dapp, and differs fromthe accurate fractal dimension of 2.1 only by the size effectpreviously discussed.

This scaling behavior during gelation states that there is onlyone length scale on which all physical quantities depend. � is the“characteristic length scale” of the system. This means that themicrostructure of the growing clusters looks the same, regardlessof the observation time, apart from a scaling factor. Thus, the

formation of the solid network of the gel results from a purelyhierarchical aggregation mechanism.

The studies, performed on very different systems such assol-gel precursors,14,20 aqueous solutions of metallic colloids,30

colloidal powders of oxides,31 emulsion of oil in water,32 orpolystyrene particles in suspension,33 attest that the aggregationprocesses exhibit a certain universal character. The theory28,29

and experiments show that these systems present two limit regimesthat are the DLCA, diffusion limited cluster aggregation, and theRLCA, reaction limited cluster aggregation. Their differentiationmainly results from the sticking probability of particles, assumedto be equal to one for the DLCA mechanism and inferior to onefor the RLCA mechanism, that is to say, that several collisionsare necessary before that two aggregates stick together. Variationsin morphology, size distribution, and fractal dimension of theaggregates appear according to the nature of the mechanism.

The DLCA mechanism is characterized by a fractal dimensionbetween 1.7 and 1.9, aggregate growth according to a power lawin time, and a relatively narrow aggregate size distribution,generally leading to the appearance of a favoured correlationlength within the samples. The particularities of the RLCA-typeprocess are a fractal dimension between 2 and 2.2, aggregategrowth that is exponential in time, and an aggregate sizedistribution normally broader than that of the DLCA mecha-nism.28,30,34-36

In point of fact, the reality is far more complex, and manyexperimental results concerning the aggregation of colloids,especially among those quoted before, demonstrate that themeasured fractal dimensions domain is much broader, since itcan extend from 1.2 to 2.8.35,36 Moreover, the aggregate growthlaw for a RLCA-type mechanism can also present a power intime growth law when the sticking probability is sufficientlyhigh.37 Similarly, the time power law exponent for the DLCA-type mechanism can also diverge from the expected theoreticalvalues.30

(24) Carpineti, M.; Giglio, M. Phys. ReV. Lett. 1992, 68(22), 3327.(25) Martin, J. E.; Wilcoxon, J. P. Phys. ReV. A 1989, 39(1), 252.(26) Furukawa, H. AdV. Phys. 1985, 34, 703.(27) Hasmy, A.; Jullien, R. J. Non Cryst. Solids 1995, 186, 342.(28) Haw, M. D.; Sievwright, M.; Poon, W. C. K.; Pusey, P. N. Physica A

1995, 217, 231.(29) Ramirez-Santiago, G.; Gonzalez, A. E. Physica A 1997, 236, 75.

(30) Olivier, B. J.; Sorensen, C. M. Phys. ReV. A 1990, 41(4), 2093.(31) Schaefer, D. W.; Martin, J. E.; Keefer, K. D. J. Phys. 1985, 46(C3), 127.(32) Bibette, J.; Mason, T. G.; Gang, H.; Weitz, D. A. Phys. ReV. Lett. 1992,

69, 981.(33) Carpineti, M.; Giglio, M. Phys. ReV. Lett. 1993, 70(24), 3828.(34) Hasmy, A.; Jullien, R. Phys. ReV. E 1996, 53(2), 1789.(35) Gonzalez, A. E.; Ramirez-Santiago, G. J. Colloid Interface Sci. 1996,

182, 254.(36) Gardner, K. H.; Theis, T. L.; Young, T. C. Colloids Surf., A 1998, 141,

237.(37) Gonzalez, A. E. Phys. Lett. A 1992, 171, 293.

Figure 5. Evolution of the extrapolated intensity at zero angle I0 versusaggregate size �. Figure 6. Scaling of the scattering curves according to the mean size

and mass of the aggregates during the gelation process.

12572 Langmuir, Vol. 24, No. 21, 2008 Coste et al.

Nevertheless, according to the determined fractal dimensionof 2.1, the absence of a favored correlation length, leading to thenon-appearance of an interference peak in the scattering curves,and the fact that the sol has to be heated at 60 °C to obtain theformation of a gel, the aggregation mechanism can reasonablybe concluded to be certainly of the RLCA type. Indeed, if themechanism depended only on elementary particle and aggregatediffusion, the gelation would occur at 20 °C as well as at 60 °C.

To sum up, the gelling network of this sol composition, i.e.,Cf ) 0.5 mol/L, R ) 0.7, and W ) 4, results from a RLCAaggregation process of Te-rich elementary particles, of 2-3 nmin diameter, readily formed from the time that the water is addedto begin the hydrolysis and condensation reactions. The gelmicrostructure is constituted of interconnected fractal aggregateswhose mean size evolves during the gelation from about 3 to 17nm, with their fractal dimension remaining constant at 2.1. Withthe gel microstructure now well-established, let us address theeffect on this gel microstructure of the two essential synthesisparameters, which are the citric acid ratio R and the alkoxideconcentration Cf.

4.2. Influence of the Citric Acid Ratio and AlkoxideConcentration on the Gel Microstructure. Sols with variablechemical compositions located in the gel domain of the phasediagram have been synthesized with the citric acid ratio R varyingfrom 0.4 to 1, the alkoxide concentration Cf from 0.1 to 0.7mol/L, and the hydrolysis ratio being again kept equal to 4. Thesols were aged at 60 °C until gelation occurred. The small angleX-ray scattering diagrams were collected, at the gel point, forall the compositions according to the previously establishedprotocol, except for the sols prepared with Cf equal to 0.1 mol/L.Indeed, since these sols were not gelled after an aging time of323 days at 60 °C, the SAXS curves were recorded at this progressstate.

Regardless of the citric acid ratio and alkoxide concentrationin the chosen composition domain, the scattering curves alwayspresent the same features, characteristic of a scattering signalgenerated by mass fractal aggregates resulting from the ag-gregation of Te-rich colloidal particles. Equation 5 fully appliesonce more, and all the experimental SAXS curves were simulated.The superposition of the experimental and calculated curves isexcellent on a broad domain of the scattering vectors and isdisplayed in Figure 7, for example, for compositions corre-sponding to a constant citric acid ratio of 0.5 and a variable

concentration. The slight divergence, observed at the highscattering vectors, is due to the same previously quoted causes,that is to say, an unfavorable signal to noise ratio or elementaryparticles whose surfaces are not well-defined. The gel networkmicrostructural parameters thus determined are summarized inTable 1.

Regardless of the citric acid ratio and alkoxide concentration,the Te-rich elementary particle diameter, constituting the fractalaggregates, is rather constant at about 2.5 nm. In the same way,the variation domains of the fractal dimension Df as well as theapparent fractal dimension Dapp are more restricted and rangebetween 2.1 and 2.2 and between 2 and 2.1, respectively. Aspreviously emphasized, the discrepancy between these two fractaldimensions decreases as the aggregates become larger. Fromthese results, the aggregation process is deduced to always bethe same, i.e., a reaction limited cluster aggregation mechanism,and the microstructure of gels is concluded to be quite independentof the citric acid ratio variation. However, the increase of the geltime, observed when the citric acid ratio increases while thealkoxide concentration is fixed, is certainly due to a slight decreasein the sticking probability of the aggregates.

Moreover, the invariance of these two microstructural pa-rameters Df and a imposes that the aggregate size at the gel pointdepends on the alkoxide concentration Cf. Indeed, the volumefraction φ, occupied by a set of fractal aggregates of size �,constituted by elementary particles of radius a, is proportionalto25,33,35,38

φ ∼ φo(�a)3-Df

(9)

where φo is the initial volume fraction of the elementary particles.Assuming that, in the gel composition domain, the chemicalcomposition of the Te-rich elementary particles remains moreor less constant, φo is then proportional to the alkoxideconcentration Cf whose variation amounts to modifying the con-centration of Te-rich elementary particles contained in the sol.

(38) Dietler, G.; Aubert, C.; Cannell, D. S.; Wiltzius, P. Phys. ReV. Lett. 1986,57(24), 3117.

Figure 7. Log-log plot of SAXS intensity distribution for the gelsprepared with Cf ) 0.1, 0.3, 0.5, and 0.7 mol/L, R ) 0.5, and W ) 4,and the corresponding calculated scattered intensity curves in solid line.

Table 1. Values of Aggregate Size �, Elementary Particle Radiusa, Fractal Dimension Df, and Apparent Fractal Dimension Dapp

for the Different Synthesized Compositions (s, MissingExperiments)

RCf ) 0.1 mol/Laged 323 days

Cf ) 0.3mol/Lgels

Cf ) 0.5mol/Lgels

Cf ) 0.7mol/Lgels

� (nm) 0.4 s s 15.5 s0.5 61 19 13 90.7 56 18 14 13.50.8 s s 15.5 s0.9 30 18.5 14.5 181 38 16 17 21

a (nm) 0.4 s s 1.2 s0.5 1.3 1.2 1.2 1.20.7 1.3 1.2 1.2 1.20.8 s s 1.2 s0.9 1.3 1.2 1.2 1.21 1.3 1.2 1.2 1.2

Df 0.4 s s 2.14 s0.5 2.10 2.15 2.18 2.220.7 2.12 2.15 2.15 2.190.8 s s 2.15 s0.9 2.12 2.15 2.15 2.081 2.10 2.16 2.12 2.12

Dapp 0.4 s s 2.00 s0.5 2.08 2.08 2.00 2.000.7 2.06 2.08 2.01 2.050.8 s s 2.05 s0.9 2.07 2.08 2.03 2.001 2.03 2.06 2.04 2.05

Dynamic Scaling Properties of TeO2-Based Gels Langmuir, Vol. 24, No. 21, 2008 12573

At the gel point, the collection of close-packed fractal aggregatesoccupies all the volume, meaning that their volume fraction isthen close to 120,25,33,35 and their size is proportional to

� ∼ aCf-1 ⁄(3-Df) (10)

The mean aggregate size �, measured at the gel point, isexpected to decrease as the alkoxide concentration increases(Table 1), a classic result for precursors issued from the sol-gelroute.6,14,25 By contrast, for a given concentration, the size � isroughly constant and independent of the citric acid ratio R inagreement with a constant fractal dimension. This assumptionis quite well confirmed for the chemical compositions locatedin the center of the transparent, homogeneous, and mechanicallyrigid gel domain, i.e., for alkoxide concentrations Cf of 0.3 and0.5 mol/L for which � is close to 18 and 15 nm, respectively.When the gel composition approaches the gel domain boundaries,the results are more dispersed. For gels having both a rather highalkoxide concentration and citric acid ratio, i.e., Cf ) 0.7 mol/Land R ) 0.9 or 1, some difficulties, mainly concerning thedissolution of citric acid, have been encountered during theirsynthesis, and their homogeneity may be not optimum. Withregard to Cf ) 0.1 mol/L, we recall that these colloidal sols arenot yet gelled and are observed for the same aging time at60 °C, i.e., 323 days. The fractal aggregate size increases ascitric acid ratio decreases, which corresponds to different degreesof progress of the aggregation process in agreement with thehypothesis of a decrease of the probability of sticking when Ris increased.

However, when Df is fixed to the determined value of 2.1, thislast scaling law (eq 10) is perfectly and somewhat less verifiedfor R ) 0.5 and R ) 0.7, respectively (Figure 8). From thisgraph, the colloidal sols with an alkoxide concentration of 0.1mol/L are deduced, on one hand, to be close to their gel pointaccordingly to their exhibited high viscosity. On the other handand more particularly for R ) 0.5, the fractal aggregates, in thegel state, are concluded to be accurately self-similar, which meansthat their microstructure remains statistically identical to itself,whatever the alkoxide concentration in the explored domain,except for a scale factor. Consequently, the correspondingscattering curves, normalized by the alkoxide concentration and

the individual characteristics of the fractal aggregates, which aretheir fractal dimension, size, and mass measured at the gel point,superpose perfectly on a master curve by using a fractal dimensionof 2, this value being very near the expected value of 2.1 (Figure9).25

This last result is particularly important and must be consideredwhen the synthesis of TeO2-based materials or thin films by thissol-gel route will be envisaged. Indeed, it will be useless toincrease the citric acid ratio far beyond the value that preventsprecipitation, its effect being negligible on the obtained gelmicrostructure. With lower citric acid ratio R, the gel time willbe shorter and the number of organic groups to eliminate duringfurther heat treatment smaller.

5. Conclusion

The gels prepared in the tellurium isopropoxide-isopro-panol-citric acid and water system are always made ofinterconnected fractal aggregates, constituted of very small Te-rich elementary particles 2-3 nm in diameter, instantaneouslyformed when the water is added. Their size and number do notevolve during the gelation process, and the fractal aggregatesresult from their hierarchical aggregation according to a reactionlimited cluster aggregation mechanism. Regardless of theobservation time, alkoxide concentration, or citric acid ratio, thefractal dimension of the growing aggregates remains constant atabout 2.1. Consequently, the microstructure of the gelling networkas well as the gel states are self-similar for a wide range of time,alkoxide concentration, or citric acid ratio, and only differs bytheir fractal aggregate size.

Citric acid has been shown to be a relevant telluriumisopropoxide chemical modifier and offers an attractive optionfor the future applications of sol-gel processed tellurite materialsand substantial practical benefits for device fabrication incomparison with conventional melting methods. This successfulhydrolysis-condensation reaction management is particularlypromising for either high-quality TeO2-based thin film fabricationor preparation of nanostructured materials such as the sol-gelsynthesis of nanometric TeO2 crystals embedded in a silica matrix.

LA802137S

Figure 8. Evolution of the aggregate size �, at the gel point, versus thealkoxide concentration Cf for R ) 0.5 and 0.7. The values in bracketscorrespond to the sols close to their gel points.

Figure 9. Scaling of the scattering curves recorded at the gel point foralkoxide concentration Cf of 0.1, 0.3, 0.5, and 0.7 mol/L according tothe size, the mass of the aggregates, and the alkoxide concentrations.

12574 Langmuir, Vol. 24, No. 21, 2008 Coste et al.