Growth versus Cyclization in the Early Stages of the Polycondensation

The early steps of the polycondensation of transition metal alkoxide have been studied from the chemical and structural points of view. Polyoxoalkoxid...
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J. Phys. Chem. B 2005, 109, 23870-23878

Growth versus Cyclization in the Early Stages of the Polycondensation of Metal Alkoxides Martin In* Laboratoire des Colloı¨des, Verres et Nanomate´ riaux, CNRS-UniVersite´ Montpellier 2, Place Euge` ne Bataillon, F-34095 Montpellier Cedex 05, France

Cle´ ment Sanchez Laboratoire de Chimie de la Matie` re Condense´ e, CNRS-UniVersite´ Pierre et Marie Curie, 4, place Jussieu, F-75252 Paris Cedex 05, France ReceiVed: May 9, 2005; In Final Form: September 30, 2005

The early steps of the polycondensation of transition metal alkoxide have been studied from the chemical and structural points of view. Polyoxoalkoxides are described like macromolecules by the composition of the repeating unit, the degree of polymerization (N), and the radius of gyration (R). The fraction p of binding sites of the coordination sphere of the metal centers occupied by terminal ligands determines N as follows: N ∝ pdf/(dA-df), where df is the fractal dimension and dA is defined by Np ∝ RdA. This approach addresses difficulties raised by both coordinative unsaturation and cyclization in the modelization of the polycondensation of metal alkoxides. The coordinative unsaturation is accounted for by a particularly small value of dA ) 1 in the very early steps, while the cyclization frequency is measured by the difference dA df. This difference is not constant along the polycondensation process, and its dependence on the extent of reaction provides clues for understanding the high apparent kinetics order of gelation often reported in the literature.

1. Introduction

SCHEME 1: Condensation Reactions

Soft chemistry routes to solid-state materials are based on inorganic polycondensation reactions.1-3 In the sol-gel processes,4 for instance, glasses or oxide ceramics are produced by pyrolysis of metalo-organic polycondensates such as polyoxoalkoxides (POAs). Such macromolecular precursors for oxides are obtained through the polycondensation of alkoxides of generic formula M(OR)z, at room temperature and in an organic solvent (M is a cation (Si, Ti, Zr, etc.) of valence z, and R is an alkyl chain). As colloidal dispersions or gels, these systems are suitable for special processing, such as molding, dip coating, and spin coating.5 Silicon alkoxide (TMOS, TEOS) polycondensation has been widely studied.5,6 It has provided model systems for experimental work on aggregation and gelation.7 The formation of POAs from transition metal alkoxides (TMAs) involves both polycondensation and coordination polymerization.1-3,8,9 The addition of water to alkoxides leads to the substitution of some alkoxo ligands (OR) by hydroxo (OH) ligands. The latter are the active functions of the condensation reactions shown in Scheme 1. (For the sake of clarity, only the ligands involved in the mentioned reactions are written.) Contrary to silicon alkoxide, TMAs react also by coordination polymerization1 because the preferred coordination number of transition metal is higher than their oxidation state. Coordination polymerization is an addition reaction which increases the coordination number of the metallic centers (Scheme 2). * Present address: LCVN UMR 5587, Universite´ Montpellier 2, Place Euge`ne Bataillon, Case Courrier 026, F-34000 Montpellier Cedex 05, France. Phone: +33 4 67 14 35 93. Fax: +33 4 67 14 46 37. E-mail: [email protected].

SCHEME 2: Addition Reactions

The POA resulting from these reactions can be described by the following formula: [MOk(OH)h-k(OR)z-h-k]N, where h is the hydrolysis ratio, k the condensation ratio, and N the nuclearity or polymerization degree.10 MOk(OH)h-k(OR)z-h-k corresponds to the repeating unit (RU) as defined by Flory.11 However, in the case of inorganic polycondensation, it has no real existence, since inorganic ligands (O, OH) lie in the core of the POA while organic remaining ligands (OR) are located at their interface with solvent. It is a virtual RU, which describes the average chemical composition of the coordination sphere of the metal. The present article reports on the early stage of the polycondensation of TMAs and attempts to address the question of the relation between the reaction extent so far described by (h, k) and the polymerization degree N. This question is relevant for the early stage of the gelation process and not necessarily in the vicinity of the gelation point. However, what happens in the early stages of the polymerization determines the size and the structure of the primary particles whose aggregation leads to gelation (in other words, the early stages of polycondensation determine the small-scale cutoff of the fractal regime). It is important for the gelation kinetics and for the processing and the properties of the final material. For instance, gelation has been shown to depend very strongly on concentration (tg ∝

10.1021/jp052410l CCC: $30.25 © 2005 American Chemical Society Published on Web 11/25/2005

Polycondensation of Metal Alkoxides SCHEME 3: Hydrolysis and Condensation of Acac Modified Zirconium Propoxide

c-10).12 It can be anticipated that this strong apparent kinetic order is related to the structure of the building units of the gel, which results from the early stage of aggregation. Previous attempts to relate N to the hydrolysis ratio (h) were based on linear association of monomeric alkoxide precursors13 or oligomeric alkoxide precursors.14,15 More recently, it was pointed out that the nuclearity of polyoxotitanates that were structurally characterized shows no correlation with the condensation ratio (k).16,17 Considering the same titanium POAs as well as zirconium POAs, we were lead to define a new chemical parameter, the poisoning ratio (p), derived from the composition of the RU and which correlates well with N. Zirconium POAs corresponding to higher extents of reaction (but low enough so that the aggregates are of finite size and so that the samples are fluid) were prepared by hydrolysis of acetylacetone (acac) modified zirconium propoxide. The relation between p and N is also established for these larger POAs and the results analyzed in the light of small-angle X-ray scattering (SAXS) experiments. 2. Experimental Section 2.1. Synthesis. Acac modified zirconium polyoxoalkoxides were prepared as follows: Zirconium n-propoxide (Johnson&Mathey, 70% in n-propanol, titrated at 73.5%) is diluted in n-propanol (Prolabo, normapur). Acac (Fluka) is then added under magnetic stirring. The modification ratio R ) acac/Zr was varied from 0.6 to 1. Under these conditions, the substitution of OR ligand by acac is stoichiometric, as shown by UV spectroscopy. Polycondensation is activated by dropwise addition of a mixture of deionized water and propanol (1:9). The nominal hydrolysis ratios H ) H2O/Zr used are 2 (precisely 1.96) and 4 (precisely 3.95). The final concentration of the colloidal solutions obtained was 5.3% in weight of equivalent zirconium oxide. Characterization of the polyoxometalates was carried out at least 3 months after synthesis, with the samples being stored at room temperature. 2.2. Chemical Characterization. The consumption of water was determined by Karl-Fischer titration using a 701 KF Titrino apparatus. To avoid any side reaction with hydroxyl groups and acac, Hydranal solvent K was used rather than methanol and the reagent used was Hydranal composite 5K (Riedel-de-Hae¨n). The amount of free acac was determined by UV spectroscopy. UV spectra were recorded in quartz UV cells 10 µm thick, to avoid any dilution of the samples. The spectra obtained were simulated by linear algebra of the spectra of acac in propanol (λmax ) 273 nm) and acac modified zirconium propoxide in propanol (λmax ) 308.5 nm). The area of each component had been preliminarily calibrated, so that the fraction of free acac could be determined from the ratio of the area of each component weighted by the proper extinction coefficient. To interpret the analysis just described in terms of the repeating unit, we considered hydrolysis and condensation reactions step by step according to Scheme 3. From the H molecules of water introduced, only h molecules of water are consumed to hydrolyze OR groups and few acac. Then, the

J. Phys. Chem. B, Vol. 109, No. 50, 2005 23871 hydroxo groups generated by hydrolysis condense with the remaining propoxo groups. Oxolation reactions (Scheme 1b) which probably occur do not need to be considered in step b of Scheme 3 because they would have generated water. We choose to consider this scheme where water and acac are involved only in the first step for simplicity. Karl-Fischer analysis yields (H-h), while UV spectroscopy gives r/R. H and R are composition variables. Weight-average molecular weights are determined from the intensity of He-Ne laser light (632.8 nm) scattered by the samples at an angle of 90°. The POAs studied are much smaller than the wavelength of the incident light and scatter light isotropically (Rayleigh limit). This technique also provides the osmotic second virial coefficient. The background subtracted intensity scattered by the POA expressed by the Rayleigh ratio (∆R) (to make it independent of the experimental setup) is related to the weight-average molecular weight (〈M〉w) by

1 Kc ) + 2〈B〉zc ∆R 〈M〉w

(1)

where K is an optical constant: K ) 2π2n2λ-4(dn/dc)2 and 〈B〉z is the z-average second virial coefficient. The refractive index increment (dn/dc) has been measured with a differential refractometer working at the D lines of sodium (λ ) 589.2 nm). It has been found to be 0.1250 ( 0.0050 mL/g in all of the samples. The Rayleigh ratio has been calibrated using toluene (R90° ) 1.406 × 10-5 cm-1 at 633 nm). In eq 1, the concentration of polyoxometalates (c) is expressed in grams per milliliter. It corresponds to the dry contents of the samples, which are not known a priori, since polycondensation reactions have released some solvent. Instead of drying the sample, which could have lead to a significant shift in the equilibria involved and hence to a modification of the composition of the RU [MOk(OH)h-k(OR)z-R-h-k(acac)R], the latter has been determined from analysis (Karl-Fischer titration and UV spectroscopy) directly carried out on the liquid samples as described above. The concentration (c) in grams per milliliter is given by c ) φFMRU/123, where F is the density (grams per milliliter), φ ) 0.053 is the equivalent metal oxide content of the sample in weight fraction, 123 is the molar weight of ZrO2, and MRU is the molar mass of the RU. The specific volume of each sample has been determined by weighing a known volume (5-10 mL). The polymerization degree (N) is deduced from the average molecular weight and the molar mass of the repeating unit: N ) 〈M〉w/MRU. 2.3. Structural Characterization. The structure of the POA was characterized by small-angle X-ray scattering (SAXS). The experiments were carried out at the synchrotron facility of the LURE (Laboratoire d’Utilisation du Rayonnement Electromagne´tique, Orsay, France) using the line D24 (λ ) 1.594 Å). Most of the samples were studied in a range of scattering vector q ) 4π/λ sin θ going from 0.04 to 0.7 Å-1. Some samples corresponding to higher extents of reaction (it means low R and high H) were studied at lower scattering vectors, down to q ) 0.01 Å-1. No calibration was done to get absolute intensities. However, the raw data were treated by the same standard procedure for normalization with regard to duration record, sample thickness, and background (propanol in Kapton cell) subtraction, allowing comparison of the measured intensity within the series. The samples have been studied by SAXS at room temperature without any dilution.

23872 J. Phys. Chem. B, Vol. 109, No. 50, 2005

In and Sanchez

TABLE 1: Characterization of the Zirconium Polyoxoalkoxides Obtained from the Hydrolysis of Acac Modified Zirconium n-Propoxidea H

R

h

1.94

0.95 0.86 0.76 0.67 0.57 0.48 0.95 0.86 0.76 0.67 0.57 0.48

1.32 1.37 1.21 1.41 1.38 1.46 1.60 1.74 1.67 1.59 1.68 1.71

3.82

RU formula [ZrOk(OPr)4-2k-r(acac)r] [ZrO1.32(OPr)0.48(acac)0.88] [ZrO1.37(OPr)0.45(acac)0.81] [ZrO1.21(OPr)0.86(acac)0.72] [ZrO1.41(OPr)0.53(acac)0.65] [ZrO1.38(OPr)0.69(acac)0.55] [ZrO1.46(OPr)0.62(acac)0.46] [ZrO1.52(OH)0.08(acac)0.88] [ZrO1.45(OH)0.29(acac)0.81] [ZrO1.61(OH)0.06(acac)0.72] [ZrO1.59(OR)0.17(acac)0.65] [ZrO1.68(OR)0.08(acac)0.56] [ZrO1.71(OR)0.10(acac)0.48]

MRU (g/mol)

p

〈M〉w (g/mol)

NLight

Rg (Å)

NSAXS

228 220 232 209 208 196 204 199 189 191 178 171

0.25 0.23 0.21 0.19 0.16 0.13 0.25 0.23 0.21 0.19 0.16 0.13

4.90 × 103

21

4.6 4.9 5.2 15.3 22

1.21 × 104

59

1.26 × 104 2.25 × 104 1.18 × 105 -

67 118 663

5.5 8.0 11.5 15.2 24.5 39

24 35 45 72 112 >180 59 70 99 168 322 978

a H, nominal hydrolysis ratio; R, modification ratio; h, real hydrolysis ratio; MRU, molar weight of the repeating unit; p ) 2r/7, poisoning ratio; 〈M〉w, weight-average molecular weight from static light scattering; NLight ) 〈M〉w/MRU, degree of polymerization from light scattering; Rg, radius of gyration from Guinier plot of the SAXS measurements; NSAXS ∝ I(0)/(cMRU) adjusted to NLight for R ) 0.95, H ) 3.82.

Figure 1. Plots of Kc/∆R against c according to eq 1: (3) R ) 0.95, H ) 1.94; (4) R ) 0.95, H ) 3.82; (O) R ) 0.76, H ) 3.82; (0) R ) 0.67, H ) 3.82; (]) R ) 0.57, H ) 3.82.

3. Results and Discussion 3.1. Chemical Composition of POAs. The results of the chemical analyses performed on acac modified zirconium propoxide hydrolyzed by two and four molecules of water are presented in Table 1. First is to notice that the water introduced is not totally consumed. 70% of the water introduced reacts for H ) 2, while only 40%, for H ) 4. The number of water molecules consumed per metal alkoxide (h) never exceeds 2. Bradley et al. pointed out the presence of remaining water,14,18,19 and we recently proposed a semiquantitative analysis of these observations.10 For the present purpose, it is important to point out that h < 2 suggests that the condensation reaction is almost stoichiometric with regard to the OH groups generated by hydrolysis (see Figure 11 of ref 10). The amount of water consumed decreases as the modification ratio (R) increases. This is related to the higher resistance of acetylacetonato complexes toward hydrolysis as compared to alkoxo complexes. UV spectroscopy shows that only a small fraction of acac complexes has been hydrolyzed (between 2 and 10%). The repeating units determined as explained in the previous section are also presented in Table 1. 3.2. Molecular Weight. Light scattering data are presented in Figure 2. These experiments provide the weight-average molecular weight (〈M〉w) and the second virial coefficient (〈B〉z) (see Table 1). For the lowest extent of reaction R ) 0.95, H ) 1.94, 〈M〉w is found to be 5000 g/mol. The largest POAs that have been characterized by light scattering were obtained for R ) 0.57, H ) 3.82 and present a molecular weight of 100 000

Figure 2. SAXS profiles of Zr-POA in propanol: (a) R ) 0.76, H ) 1.94; (b) R ) 0.76, H ) 3.82; (c) R ) 0.48, H ) 3.82. The symbols correspond to experimental data, and the lines correspond to simulations: (a) two populations of spheres with radii of 7.1 and 4.3 Å; (b) cylinder with a length of 35 Å and a diameter of 4.7 Å; (c) fractal aggregate of spheres with a radius of 5.6 Å; the overall size of the aggregate is Rg ) 36 Å, and its fractal dimension is 2.2; the scattering has been calculated using the Beaucage formula.22 For the sake of clarity, the data have been vertically shifted.

g/mol. For H ) 4, decreasing R from 1 to 0.6 increases the degree of polymerization (DP) by a factor of 10. Three typical SAXS profiles are presented in Figure 2. At low scattering vector, the scattered intensity looks independent of the scattering vector (q) in the log-log representation. This indicates that the observed length scale (1/q) is larger than the radius of gyration (Rg). At small q values, the scattered intensity actually decreases according to Guinier eq 20:

(

I(q) ≈ I(0) exp -

)

(Rgq)2 3

(2)

I(0) is the intensity scattered at q ) 0, and Rg is the radius of gyration. Comparison of I(0)/c between the samples within the series reflects the variation of the molecular weight. The degree of polymerization was determined from I(0)/(cMRU) normalized to fit light scattering data for R ) 0.95, H ) 3.82. For the other samples, the agreement is reasonably good owing to the approximation made in each method: On one hand, light scattering data have been properly extrapolated to zero concentration, but dilution might have modified the POAs. On the other hand, SAXS measurements have been carried out at only

Polycondensation of Metal Alkoxides

J. Phys. Chem. B, Vol. 109, No. 50, 2005 23873 a short range of scattering vectors (0.07 up to 0.2 Å-1). The separation of the Guinier regime and the Porod regime indicates that two characteristic lengths are needed to describe this POA sample. The q-1 dependence of the scattered intensity in this intermediate regime suggests the formation of rodlike particles, and the scattering profile compares well with the scattering profile calculated for a rod with a radius of R ) 4.8 Å and a length of L ) 35 Å and which reads

Pcyl(q) )

Figure 3. Log-log plot of the zero angle scattered intensity normalized by the concentration versus the radius of gyration. I(0)/c is proportional to the mass of the POA and is related to the radius of gyration through the fractal dimension: I(0)/c ∼ Rgdf. The lines are guides for the eyes and underline the different regimes of growth.

one concentration, but this corresponds to the native condition of POA formation. SAXS measurements confirm and complement the observation of the R and H dependence of the molecular weight by light scattering. Decreasing R from 0.95 to 0.48 leads to an increase of N by a factor of 7 for H ) 2 and by a factor of 17 for H ) 4 (see Table 1). 3.3. Size and Structure. When plotting I(0)/c versus the radius of gyration for all POAs (Figure 3), several regimes can be distinguished. In the first regime, the molecular weight varies as fast as the volume of the particle, I(0)/c ) k1Rg3, suggesting a compact isotropic structure for the POA. In the last regime corresponding to the largest POA, the molecular weight increases like the surface area of the particle, I(0)/c ) k2Rg2 and suggests tenuous structure formation. In the crossover region in between, the dependence is weak and suggests the formation of anisotropic particles. This observation reflects the modification of the structure of the POA as condensation proceeds. This interpretation is confirmed by detailed analysis of the scattering profiles of Figure 2. Each of the three typical scattering profiles corresponds to one of the regimes observed in Figure 3. The first scattering profile (Figure 2a) corresponds to a low extent of reaction (R ) 0.8, H ) 2). Rg is equal to 5.2 Å for this sample. The scattering profile observed for R ) 0.8, H ) 2 is very close to the one of a set of slightly polydispersed spheres with a mean radius of 6.7 Å. The form factor for an isolated sphere reads

Psph(q) )

[

]

3(sin(qR) - qR cos(qR)3) (qR)3

2

(3)

The dips expected at high q in the log-log representation due to zeros of eq 3 are smeared out in experimental profile 2a by the polydispersity of the sample. Simulated profile a of Figure 2 is calculated combining two types of spheres: 52% of spheres of radius 7.1 Å and 48% of spheres of radius 4.3 Å (percentages in number). Spherical structure for the particles is consistent with the first regime observed in Figure 3 where I(0)/c is proportional to Rg3. The second scattering profile corresponds to a higher extent of reaction (R ) 0.8, H ) 4) (Figure 2b). The scattered intensity at low q, I(0)/c, is about 3 times higher than that in profile a, and the radius of gyration is 12 Å. Between the Guinier regime and the Porod regime, the scattered intensity decays as q-1 over

∫0π/2

[

]

2J1(qR sin R) sin(qL cos R/2) 2 sin R dR (4) qR sin R qL cos R/2

The formation of a locally cylindrical structure is an intermediate step necessary to the formation of branched aggregates from an isotropic particle. Here, playing with both parameters R and H offers enough control to somehow quench and observe reaction products in this intermediate regime where short rods form. The third scattering profile corresponds to the highest extent of reaction (R ) 0.5, H ) 4). The intermediate regime between the Guinier and Porod regimes spans a wider range of q due to an increase of size (an estimate from a plot of ln(I) vs q2 gives Rg ) 39 Å). The intensity scattered at q ) 0 indicates a molecular weight about 30 times higher than that of the POA corresponding to the lowest extent of reaction (R ) 0.95, H ) 1.94). This has however to be considered as a minimum because of repulsive interactions that are revealed by the very weak maximum at 0.02 Å-1. However, in this range of scattering vectors, the difference between a Guinier fit and the experimental curve does not exceed 15%, indicating that the repulsive interactions are weak (max(S(q)) < 1.15). In a narrow range of scattering vectors, the intensity decays as q-2, reaching the Porod regime through a pronounced shoulder. This type of scattering profile is typical for reactionlimited cluster-cluster aggregation.21 The unified method proposed by Beaucage22 to describe the scattering profile of aggregates with structural features at various length scales has been used to fit the scattering profiles. The relation used corresponds to eq 9′ of ref 22 and reads n

I(q) )

[ ( )

Gi exp ∑ i)1

q2Rgi2

+

3

Bi exp

( )(

)]

q2Rgi+12 [erf(qRgi/x6)]3 3

q

pi

(5)

where i ) 1 refers to the lowest structural level (smallest size structure, contrary to the convention used in ref 22). At this level, the aggregate is constituted by subunits of Rg1 ) 5.6 Å. These subunits form a fractal aggregate of Rg2 ) 36 Å and fractal dimension of p2 ) 2.2. We add another level to account for the small maximum observed, but the parameters used are not of real physical significance23 and were introduced to get a better measurement of Rg2. There is a close correspondence between the three regimes observed in the plot I(0)/c versus Rg (Figure 3) and the different regions of the scattering profile of the POA obtained for R ) 0.5, H ) 4 (Figure 2c). The high q data of Figure 2c (q > 0.3 Å-1) are reminiscent of the early stages of condensation were the molecular weight of the POA is proportional to Rg3. This regime of polycondensation will be referred thereafter to as nucleation regime. The intermediate range of q of Figure 2c (0.3 < q < 0.1) reveals the locally cylindrical structure of the

23874 J. Phys. Chem. B, Vol. 109, No. 50, 2005 aggregates and is reminiscent of the crossover regime of Figure 3, where the primary particles start to associate in tenuous aggregates through a reaction-limited cluster aggregation process. Last, the features of the scattering profiles at low q (q > 0.1) reveal the regime of fractal aggregation where I(0)/c ∼ Rg2. This interesting result suggests that POAs obtained at a low extent of reaction correspond to the first stages in the formation of POAs obtained at a higher extent of reaction. The modification of TMAs by acac prior to hydrolysis would then be equivalent to “quenching in advance” the polycondensation reaction. 3.4. Interactions. Interestingly, the second virial coefficient is slightly negative for the smallest POA, while it is positive for the larger ones (see Figure 1). This means that small POAs attract each other and are not very soluble in alcohol. This is consistent with the fact that many POA of low nuclearity have been crystallized16,17 in the parent alcohol where they formed. It is however puzzling to observe that larger POAs are more soluble than smaller ones because the contrary is in general observed for organic polymers. Several possible reasons can be invoked. First, as the average molecular weight increases, the polydispersity is expected to increase as well. The structure of the POA is also expected to become less well defined, with interface of increasing roughness. This would act as an effective repulsive potential between the POAs. Second, the positive virial coefficient is observed for large POAs under conditions where more water remains. A water/alcohol mixture may be a better solvent for POAs than alcohol alone. It is worth noting that the negative second virial coefficient observed for small POAs could also result from association equilibrium24 through reversible bridging by alkoxo or hydroxo ligand (dynamical equilibrium of reversible polymerization). The negative second virial coefficient measured would in this case reflect the equilibrium constant of association. This interpretation is consistent with the well-known fact that small POAs obtained at low condensation ratios (k) are more reactive than large POAs obtained for large k.16,17 At this moment, it is difficult to conclude between van der Waals and “chemical” attraction, but this feature is confirmed by SAXS experiment. For small POAs corresponding to R ) 0.95, H ) 1.94 and R ) 0.87, H ) 1.94, the scattering profiles never completely saturate toward low q values, which is typical of a weak attraction, while, for larger POAs, the scattering profile shows a small maximum at a finite scattering vector typical of repulsive interaction. To summarize the results, we obtained polyoxoalkoxides of various degrees of polycondensation by varying the modification ratio by acac and the nominal hydrolysis ratio. For a low extent of reaction, POAs are dense isotropic objects that tend to attract each other in the corresponding alcohol. When reaction proceeds further, POAs get anisotropic and eventually branch to form typical reaction-limited aggregation clusters. Tuning of the chemical parameter was fine enough to obtain and to characterize several POAs in the crossover regime between nucleation and aggregation. Chemical analysis showed that never more than 1.7 molecules of water were consumed per metal center and that never more than 10% of acac molecules were hydrolyzed. The average composition of the coordination sphere of the metal was given as an average repeating unit (RU). In the following section, we would like to consider a possible relation between the chemical composition of the repeating unit and the degree of aggregation. 3.5. Relation between Chemical Composition and DP. When considering monodispersed polycondensates, a relation between polymerization degree and chemical composition can

In and Sanchez be derived in a very general way from Euler’s theorem (V - E + F ) 2):

N)

1-L fR 12

(6)

Euler’s formula relates the number of vertices (V), the number of edges, and the number of faces of a convex polyhedron. Equation 7 is a translation of Euler’s formula for branched polymers where N ) V is the polymerization degree, f the functionality of the monomers, R the percentage of reacted functions (NfR/2 ) E), and L ) F the number of loops (meaning intramolecular bonds). This relation is of practical interest when the number of loops is known or negligible: determination of R with a suitable analytical method yields the number-averaged degree of polymerization.11 However, in POAs, closed loops are too numerous to be neglected and taking them into account is mathematically quite involved.25-27 This is first due to the higher functionality of inorganic repeating units as compared to usual organic monomers. In silica, for example, silicon atoms have a functionality of 4, they can be described as tetrahedrons associated by vertices, and most of them belong to five- or sixmembered rings.4,6 The number of loops is higher in transition metal polycondensates because the metal centers are polyhedra tethered by edge or face sharing.28 When considering that the functionality of the metallic center is its coordination number, edge sharing and face sharing have to be considered as loops, since they correspond to multiple bonds (double bond for edge sharing and triple bond for triangular face sharing). To circumvent the difficulty raised by loop formation, we consider the functions that have not reacted (that are poisoned29) instead of the number of bonds formed. The poisoned functions are the alkoxo ligands that have reacted neither by hydrolysis/ condensation nor by coordination polymerization. In fact, any nonbridging ligand must be considered as poison. Moreover, a chelating ligand such as acac used to control the TMA reactivity certainly has to be considered more poisonous than a terminal alkoxo ligand, because it occupies more coordination sites. We propose defining the poisoning ratio as the average fraction of the metal coordination number that is occupied by nonbridging ligands.

p)

∑Cpoison ∑N Cmetal

(7)

where Cpoison is the number of donor atoms in the poisoning ligand. It is 1 for a terminal OR ligand and 2 for a bidentate ligand such as acac. Cmetal is the coordination number of the metal center. The summation is carried out on all of the N coordination centers of the POA.30 p varies from 1 to 0, with the two limiting conditions being N ) 1 for p ) 1 and N f ∞ when p f 0. Knowledge of p requires much more information on the structures of the POAs than what is usually accessible. That is why we first illustrate the use of eq 7 with zirconium and titanium alkoxides and POAs that have been structurally characterized. The structures of many alkoxides asserted from ebulliometric measurements and spectroscopic studies have been reported in the literature.8,31 Some alkoxides have been obtained as single crystals and structurally characterized.32,33 Many polyoxoalkoxides have also been obtained as single crystals and studied by X-ray diffraction.16,17 Their nuclearity varies from 2

Polycondensation of Metal Alkoxides

J. Phys. Chem. B, Vol. 109, No. 50, 2005 23875

TABLE 2: Developed Formulas of Some Titanium and Zirconium Alkoxides and Polyoxoalkoxides Structurally Characterized #

ref

developed formula

N

k

p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

8, 31 37 8, 31 16 38 39 40 41, 42 43 41 44 44 45 46 47 48 32 49 50 51 52

Ti(OPri)4 [Ti2(µ2-O)2(acac)4] [Ti3(µ2-OBun)3(OBun)9] Ti3(µ3-O)(µ3-OMe)(µ-OPri)3(OPri)6] [Ti4(µ2-OMe)2(µ2-OMe)4(OMe)10] [Ti4(µ3-O)2(µ2-OPri)2(OPri)8(acac)2] [Ti5(µ4,η2-OC2H4O)(µ3,η2-OC2H4O)3(µ2,η2-OC2H4O)(µ2-OPri)4 (OPri)9] [Ti7(µ4-O)2(µ3-O)2(µ2-OEt)8(OEt)12] [Ti8(µ2-O)2(µ3-O)4(µ2-OBz)8(OBz)12] [Ti10(µ4-O)4(µ3-O)2(µ2-O)2(µ2-OEt)10(OEt)14] [Ti11(µ3-O)10(µ2-O)3 (µ2-OPri)7(OPri)11] [Ti12(µ3-O)14(µ2-O)2 (µ2-OPri)4(OPri)12] [Ti16(µ4-O)4(µ3-O)8(µ2-O)4(µ-OEt)16(OEt)16] [Ti17(µ4-O)4(µ3-O)16(µ2-O)4(µ-OPri)4(OPri)16] [Ti18(µ5-O)(µ4-O)4(µ3-O)20(µ2-O)4(µ-OBun)14(OBun)12(acac)2] [Ti18(µ4-O)4(µ3-O)16(µ2-O)4(µ2-OH)(OBut)17] [Zr2(µ2-OPri)2(OPri)6(PrOH)2] [Zr4(µ4-O)(µ2-OPrn)6(OPrn)4(acac)4] [Zr9(µ4-O)2(µ3-O)4(µ2-OPrn)10(OPrn)10(allac)6] [Zr10(µ4-O)2(µ3-O)3(µ2-OPrn)8(OPrn)10(allac)6] [Zr13(µ4-O)8(µ2-OMe)24(OMe)12]

1 2 3 3 4 4 5 7 8 10 11 12 16 17 18 18 2 4 9 10 13

0 1 0 1/3 0 1/2 0 4/7 3/4 4/5 13/11 4/3 1 24/17 29/18 4/3 0 1/4 1/3 1/2 8/13

1 2/3 (0.67) 3/5 (0.30) 1/3 (0.33) 5/12 (0.42) 1/2 (0.5) 3/10 (0.3) 2/7 (0.29) 1/4 (0.25) 7/30 (0.23) 11/56 (0.20) 2/11 (0.18) 1/6 (0.17) 8/51 (0.16) 4/27 (0.15) 17/108 (0.16) 2/3 (0.67) 3/7 (0.43) 2/15 (0.13) 11/34 (0.32) 3/23 (0.13)

to 18 for titanium. These polyoxoalkoxides may be reaction intermediates of Ti or Zr alkoxide polycondensation.34 They could actually also constitute some dead ends in the polymerization reaction pathway or reservoir of matter.35 Whatever their role in the process going from the molecular precursor to the solid material, they are a material of choice for fundamental studies because of their tendency to crystallize in the solvent in which they have grown. The developed formulas of arbitrarily selected ones are gathered in Table 2 and the structures of two of them presented in Figure 4. The correlation between k and N is weak, as was already pointed out16,17 (Figure 5a). For instance, k ) 0 can correspond to nuclearities (i.e., degree of polymerization) from 1 to 5. This is explained by the possibility for TMAs to polymerize by an addition reaction where alkoxo ligands are shared (Scheme 2). As a matter of fact, many TMAs exist as oligomers having the generic formula [M(OR)z]N (k ) 0, N > 1).36 Also, compounds 2 and 13 in Table 2 both correspond to k ) 1, while their degrees of polymerization are very different (2 and 16, respectively). To address the relation between N and the chemical composition of the RU, the possibility for N to increase by addition reactions has to be taken into account. The definition of p addresses this point. The p dependence of N is presented in Figure 5 and is very well approximated by the following power law:

N ) p-3/2

From our definition of the poisoning ratio and from the isotropic shape of the POA, it is tempting to relate the observed exponent to the one that relates the surface area (A) and the volume (V) of spheres. Taking all of the poisoned binding sites (Np) as a measure of the surface area of the aggregate, the

(8)

Figure 4. Two examples of titanium POA isolated as single crystals and structurally characterized: (a) compound 12, Ti12O16(OPri)12; (b) compound 15, Ti18O22(OBun)26(acac)2.

Figure 5. (a) Degree of polymerization versus condensation ratio (k) for titanium (O) and zirconium (0) alkoxides and polyoxoalkoxides of Table 2. (b) Degree of polymerization versus poisoning ratio (p) for the same compounds; the curve represents N ) p-3/2.

23876 J. Phys. Chem. B, Vol. 109, No. 50, 2005

In and Sanchez

Figure 6. Degree of polymerization of Zr-POA versus poisoning ratio in log-log scale: (O) H ) 2; (() H ) 4. The dotted line represents DP ) p-3/2 (see Figure 5b) and corresponds to a growth regime where the coordination number of the metal centers increases, the continuous lines represent DP ∝ p-3, and the dashed line represents DP ∝ p-5.

poisoning ratio defined in eq 7 would be proportional to the specific surface area (p ∝ A/V). The degree of polymerization (or nuclearity) can be assumed to be proportional to the volume (hypothesis of constant density N ∝ V). This would however not lead to eq 8 but to

N ∝ p-3

(9)

Equation 9 accounts well for what is observed experimentally for the larger zirconium POA reported in Table 1 (see Figure 6). To get eq 8 would require that the surface area (A) of the particles (measured as Np) increases like the cubic square of the volume (measured as N). From a geometrical point of view, this is unacceptable but it has an important chemical meaning: In the very early stage of condensation, the tendency of the metals to reach their preferred coordination number leads them to share as many alkoxo ligands as possible. When the first hydrolysis and condensation reactions liberate some space in the coordination sphere of the metal, more alkoxo ligands can become bridging. Such a reaction mechanism was proposed to describe the formation of compound 18.49 Sharing alkoxo ligands does not increase the nuclearity (N); however, it decreases the total number of poisons. It corresponds to an intramolecular bond formation or cyclization. Equation 8 seems particularly relevant for octahedra assembled by edges. POAs consisting of polyhedra associated by vertices will lie above the curve N ) p-1.5, as compound 3 in Table 1. Face sharing will make the compounds lie below the theoretical curve, as it is for compound 4. An ideal trimer that would fit exactly on the theoretical curve would consist of three octahedra sharing edges with one µ3 and three µ2 bridges. Such a trimeric structure has not been isolated and structurally characterized but is often encountered as a building block of larger POAs or polyions.53 Zirconium atoms are 7-fold coordinated in the POAs of N > 4 and share faces. That is why zirconium POAs can lie under the idealized curve (see compounds 19 and 21). On the other hand, when POAs get larger, they get a bit anisotropic and lie above the curve (see compounds 15 and 20). However, overall, eq 8 is quiet good in describing the very early stages of the polymerization process of titanium and zirconium alkoxide of unsaturated coordination. As soon as the preferred coordination number is reached and provided that the geometry of the polycondensate remains isotropic, eq

9 should hold. The limited validity of eq 8 just discussed suggests already that the relation between N and p will require a specification of the symmetry of the POA as well as the specification of its cohesion (it means the number of multiple bonds or loops it contains). To apply our approach to the larger Zr-POA described in Table 1, p has to be determined, and this appears at a first glance as a hopeless task. However, in the case of acac modified transition metal alkoxide, reasonable approximations can be made in order to get p from the RU. The first one is to consider the few remaining alkoxo ligands as bridging ones. Second, all zirconium atoms are supposed to be 7-fold coordinated. The latter assumption is strongly supported by X-ray absorption studies.54 Both assumptions lead to p ) 2r/7. In the particular case where strong chelating ligands are used to control the reactivity of TMAs, the poisoning ratio can be obtained from UV spectroscopy. Moreover, neglecting the hydrolysis of acac, that is, taking p ) 2R/7, would not affect the determination of p by more than 5%. This means that, when using strong chelating ligands, the poisoning ratio is determined by a composition variable, namely, the modification ratio (R). Figure 6 shows the p dependence of the degree of polymerization obtained from SAXS plotted in a log-log scale. The dotted line in Figure 6 is given by eq 8 and describes the growth under coordinative unsaturation (data of Figure 5). The data for POAs obtained at H ) 2 are well described by eq 9 (N ) 0.34p-3). This result means that the POAs are isotropic and have been formed while their preferred coordination number of 7 was satisfied. The prefactor of the power law indicates essentially the intersection point with the regime of polymerization under coordinative unsaturation. It corresponds to compound 18 in Table 2 (N ) 4 and p ) 0.45). The data concerning the POAs obtained at H ) 3.94 do not lie on a power law. We can probably adjust the first points to eq 9, but with a higher prefactor (N ) 0.78p-3). For the largest POAs, the p dependence of N is clearly stronger and the dashed line running through the experimental data corresponds to p-5. This is due to the change in the structure of the polycondensates, which has been shown to shift from spherical to fractal as polymerization proceeds. When tenuous POAs grow, their interfacial area does not decrease as much as when dense isotropic aggregates are formed. The crossover from the nucleation regime to the aggregation regime occurs at p ) 0.19 for H ) 4, while it is not yet observed at p ) 0.15 for H ) 2. This difference might look small, but remember that N goes from 1 to infinity when p varies from 1 to 0. This subtle difference in crossover between regimes of polycondensation is certainly a key to understanding the strong H dependence of the gelation time12 (tgel ∝ H-7). A general relation between p and N can now be proposed. Let us call dA the exponent that relates the total number of poisoned binding sites to the characteristic size (R) of the POA:

Np ∝ RdA

(10)

The fractal dimension relates the mass of the POA to its size:55

N ∝ Rdf

(11)

From these definitions, it follows that all along the process of polycondensation the relation between N and p can be written as

N ∝ pdf/(dA-df)

(12)

Polycondensation of Metal Alkoxides The different regimes of polycondensation that we pointed out can now be described in terms of these two structural exponents. The regime of nucleation is characterized by the formation of dense isotropic aggregates and df ) 3. The regime of aggregation is characterized by df ) 2.2. Inside the regime of nucleation, two subregimes are actually distinguished. The first one is characterized by dA ) 1 and has been observed with the low nuclearity POAs (N < 20). This subregime is a peculiarity of metal alkoxides that have unsaturated coordination numbers (Ti and Zr but also Al and Ce, etc.) The second subregime of nucleation, where the coordination number does not increase anymore is characterized by dA ) 2 (in other words, the poisoned surface area corresponds to the geometrical surface area). Most of the POAs described in the present report are in this regime. However, some of them, the largest ones, are clearly in the regime of aggregation (df ) 2.2). In the first step of the aggregation regime, we determined that N ∝ p-5 (but this a quite rough estimate and does not necessarily hold at a higher extent of reaction), which means that dA ) 1.7. In the aggregation regime, dA and df are much closer to each other. The meaning of this observation is related to the frequency of cyclization during the formation of highly branched polymers. Cyclization in branched polymers is an old problem and has recently been reviewed by Suematsu,56 who suggested that dimensionality is closely related to cyclization.26 Our analysis of the relation between chemical composition and degree of polymerization leads us to the conclusion that the difference df - dA is closely related to cyclization. At the early stage of polycondensation, df - dA ) 2. This is a high value which indicates that lots of intramolecular bonds form in order for the metal centers to reach their preferred coordination number. As polycondensation proceeds, df - dA gets smaller which means that less and less loops form. First, dA increases from 1 to 2, once the coordination sphere is saturated (while df keeps constant and equal to 3). Then, in the aggregation regime, df decreases to about 2.2. This decrease of df means that the aggregates become tenuous, while the concomitant decrease of the difference df - dA means that less intramolecular bonds are established. Furthermore, we can expect that df - dA will eventually cancel (most probably by further decrease of df down to 2), since it is established that loops can be neglected at the gelation transition.56 According to eq 12, N would then vary extremely strongly with p approaching the vertical asymptote given by p ) pc. Close to the gelation point, the question of the relation between the chemical composition and the degree of polymerization becomes irrelevant, since a negligible number of new bonds make N diverging. Interestingly, the vertical asymptote is also approached by eq 6 provided that the number of loops (L) becomes negligible. Hence, when no cyclization reactions occur, the area (as described by the total number of poisoned sites) and the mass are characterized by the same exponent. The number of loops in a fractal aggregate is determined by dimension that characterized both its mass and its surface. Similar combinations of structural exponents were also used to described transport properties on fractal aggregates;57,58 however, a rigorous discussion of this last point goes much beyond the scope of this article. It is worth mentioning that POAs are certainly suitable systems for studying experimentally the relation between the interface and the mass of fractal aggregates29 because they are hybrid organic-inorganic polymers. The perimeter is organic, while the core is inorganic. 4. Conclusion The early steps of the polycondensation of transition metal alkoxides have been studied from the chemical and structural

J. Phys. Chem. B, Vol. 109, No. 50, 2005 23877 points of view. We quantitatively showed that most of the chelating ligands used to control the degree of polymerization are still complexing at the surface of the POAs when equilibrium is reached. For the range of hydrolysis ratios studied, the water consumed is always below two molecules per metal center. This suggests that all of the hydroxo groups generated by hydrolysis have condensed. With this approximation, we were able to determine the composition of the repeating unit of POA. From the structural point of view, we pointed out the existence of two regimes of polycondensation. The first one we called nucleation leads to isotropic primary particles. In the second regime, reaction-limited cluster aggregation of the primary particles proceeds and leads to tenuous aggregates. To be able to relate the chemical composition to the degree of polymerization (N), we had to introduce a new parameter, the poisoning ratio (p). It describes the fraction of the metallic coordination sphere that is occupied by terminal ligands. The degree of polymerization is shown to depend on p like N ∝ pdf/(dA-df), where df and dA characterize the POA as follows: N ∝ Rdf and Np ∝ RdA. Nucleation is characterized by df ) 3, while aggregation is characterized by df ) 2.2. In the very early stage of polycondensation, the coordination number of the metal centers increases, and this is accounted for by a particularly small value of dA ) 1. When the metallic centers reach their maximum coordination number, dA is equal to the surface fractal dimension of the POA. The difference df - dA reflects the frequency of cyclization and eventually cancels at the gelation threshold. The dependence of df - dA on the extent of reaction and the crossover thresholds between the different regimes are the clues to understanding the high apparent kinetics order of gelation often reported in the literature. Acknowledgment. Some of this work was performed at the Complex Fluid Laboratory supported by CNRS and Rhodia. The SAXS data were obtained at LURE, beam line D24, the use of which was courtesy of C. Bourgaux. B. Cabane, D. Weitz, and F. Ribot are acknowledged for discussions. References and Notes (1) Sowerby, D. B.; Audrieth, L. F. J. Chem. Educ. 1960, 37 (1), 2-10. (2) Sowerby, D. B.; Audrieth, L. F. J. Chem. Educ. 1960, 37 (2), 8691. (3) Sowerby, D. B.; Audrieth, L. F. J. Chem. Educ. 1960, 37 (3), 134137. (4) Brinker, C. J.; Scherrer, G. W. Sol-Gel Science, The Physics and Chemistry of Sol-Gel Processing; Academic Press: San Diego, CA, 1990. (5) Sol-gel Technology for Thin Films, Fibers, Preforms, Electronics and Specialty Shapes; Klein L. C., Ed.; Noyes Publication: Park Ridge, NJ, 1988. (6) Iler, R. K. The chemistry of Silica; Wiley: New York, 1979. (7) Martin, J. E.; Adolf, D. Annu. ReV. Phys. Chem. 1991, 42, 311. (8) Bradley, D. C.; Mehrotra, R. C.; Gaur, D. P. Metal Alkoxides; Academic Press: London, 1978. (9) Livage, J.; Henry, M.; Sanchez, C. Prog. Solid State Chem. 1988, 18, 259. (10) Blanchard, J.; In, M.; Schaudel, B.; Sanchez, C. Eur. J. Inorg. Chem. 1998, 1115-1127. (11) Flory, J. P. Principles of Polymer Chemistry; Cornell University Press: Ithaca, Ny, 1953. (12) In, M.; Prud’homme, R. K. Rheol. Acta 1993, 36, 556-565. (13) Boyd, T. J. Polym. Sci. 1951, VII (6), 591-602. (14) Bradley, D. C.; Gaze, R.; Wardlaw, W. J. Chem. Soc. 1955, 3977. (15) Bradley, D. C.; Gaze, R.; Wardlaw, W. J. Chem. Soc. 1957, 469. (16) Day, V. W.; Eberspacher, T. A.; Chen, Y.; Hao, J.; Klemperer, W. G. Inorg. Chim. Acta 1995, 229, 391. (17) Campana, C. F.; Chen, Y.; Day, V. W.; Klemperer, W. G.; Sparks, R. A. J. Chem. Soc., Dalton Trans. 1996, 691. (18) Bradley, D. C.; Carter, D. G. Can. J. Chem. 1961, 1434. (19) Bradley, D. C.; Carter, D. G. Can. J. Chem. 1962, 4015.

23878 J. Phys. Chem. B, Vol. 109, No. 50, 2005 (20) Neutrons, X.-Rays and Light: Scattering Methods Applied to Soft Condensed Matter, Lindner, P.; Zemb, Th. Eds; North-Holland: Amsterdam, The Netherlands, 2002. (21) Vicsek, T. Fractal Growth Phenomena; World Scientific: Singapore, 1989; Chapter 8. (22) Beaucage, G. J. Appl. Crystallogr. 1996, 29, 134-146. (23) The whole set of parameters used to simulate the profile is the following: G1 ) 2 × 104, Rg1 ) 5.6, B1 ) 0; G2 ) 3.2 × 105, Rg2 ) 36, B1 ) 190, p2 ) 2.2. (24) Tanford, C. Physical Chemistry of Macromolecules; Wiley: New York, 1961. (25) Suematsu, K.; Okamoto, T. J. Phys. Soc. Jpn. 1992, 61 (5), 15391548. (26) Suematsu, K.; Khono, M. Phys. ReV. E 2000, 62 (3), 3944-3953. (27) Suematsu, K. Phys. Chem. Chem. Phys. 2002, 4, 4161-4167. (28) Wells, A. F. Structural Inorganic Chemistry; Clarendon Press: Oxford, U.K., 1975. (29) Keefer, K. D.; Schaefer, D. W. Phys. ReV. Lett. 1986, 56, 23762379. (30) To illustrate the use of eq 7, a few examples among the compounds presented in Table 2 are now considered. Titanium isopropoxide has a monomeric structure.31 The titanium atom is 4-fold coordinated by four terminal isopropoxo ligands; the poisoning ratio in titanium isopropoxide is p ) 4/4 ) 1. Another monomeric structure that would give the same poisoning ratio is Zr(acac)4. This illustrates the generality of the definition of the poisoning ratio which depends neither on the coordination number of the metal center nor on the number of donor atoms in the ligand. One of the biggest titanium POAs which was structurally characterized is compound 13 of Table 2. All 16 titanium atoms are 6-fold coordinated; there are 16 monodentate poisoning ligands. The poisoning ratio is then (16)/(16 × 6) ) 1/6. It is worth noting that this compound has the same condensation ratio (k) as the dimeric compound 2, which corresponds to a poisoning ratio of 2/3. The correlation between p and N is expected to be much better than that between k and N. (31) Babonneau, F.; Doeuff, S.; Le´austic, A.; Sanchez, C.; Cartier, C.; Verdaguer, M. Inorg. Chem. 1988, 27, 3166. (32) Vaartstra, B. A.; Huffman, J. C.; Gradeff, P. S.; Hubert-Pfalzgraf, L. G.; Daran, J.-C.; Parraud, S.; Yunlu, K.; Caulton, K. G. Inorg. Chem. 1990, 29 (17), 3126-3131. (33) Day, V. W.; Eberspacher, T. A.; Chen, Y.; Hao, J.; Klemperer, W. G. Inorg. Chim. Acta 1995, 229, 391. (34) Gautier Luneau, I.; Mosset, A.; Galy, J.; Schmidt, H. J. Mater. Sci. 1990, 25 (8), 3739-3745.

In and Sanchez (35) Sanchez, C.; Tole´dano, P.; Ribot, F. Mater. Res. Soc. Symp. Proc. 1990, 180, 47. (36) Bradley, D. C. Nature 1958, 182, 1211. (37) Watenpaugh, K. W.; Caughlan, C. N. Inorg. Chem. 1967, 6, 963. (38) Wright, D. A.; Williams, D. A. Acta Crystallogr. 1968, B24, 1107. (39) Moran, P. D.; Rickard, C. E. F.; Bowmaker, G. A.; Cooney, R. P. Inorg. Chem. 1998, 37, 1417. (40) Pajot, N.; Papiernick, R.; Hubert-Pfalzgraf, L. G.; Vaissermann, J.; Parraud S. Chem. Commun. 1995, 1817. (41) Schmid, R.; Mosset, A.; Galy, J. J. Chem. Soc., Dalton Trans. 1999, 1991. (42) Watenpaugh, K.; Caughlan, C. N. Chem. Commun. 1967, 76. (43) Day, V. W.; Eberspacher, T. A.; Klemperer, W. G.; Park, C. W.; Rosenberg, F. S. J. Am. Chem. Soc. 1991, 113, 8190. (44) Day, V. W.; Eberspacher, T. A.; Klemperer, W. G.; Park, C. W. J. Am. Chem. Soc. 1993, 115, 8469. (45) Mosset, A.; Gally, J. C. R. Acad. Sci. Paris Serie II 1988, 307, 1747. (46) Stenou, N.; Kickelbick, G.; Boubekeur, K.; Sanchez, C. J. Chem. Soc., Dalton Trans. 1999, 3653-3655. (47) Toledano P.; In, M.; Sanchez, C. C. R. Acad. Sci. Paris Serie II 1991, 313, 1247. (48) Campana, C. F.; Chen, Y.; Day, V. W.; Klemperer, W. G.; Sparks, R. A. J. Chem. Soc., Dalton Trans. 1996, 691. (49) Toledano, P.; In, M.; Sanchez, C. C. R. Acad. Sci. Paris Serie II 1990, 311, 1161. (50) Laaziz, I.; Larbot, A.; Guizard, C.; Julbe, A.; Cot, L. Mater. Res. Soc. Symp. Proc. 1992, 271, 71. (51) Sanchez, C.; In, M.; Tole´dano, P.; Griesmar, P. Mater. Res. Soc. Symp. Proc. 1992, 271, 669. (52) Morosin, B. Acta Crystallogr., Sect. B 1977, 33, 303. (53) Sanchez, C.; Tole´dano, P.; Ribot, F. Mater. Res. Soc. Symp. Proc. 1990, 180, 47. (54) Sanchez, C.; In, M. J. Non-Cryst. Solids 1992, 147, 1-12. (55) The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers; Avnir, D., Ed.; Wiley: Chichester, U.K., 1989. (56) Suematsu, K. AdV. Polym. Sci. 2002, 156, 137-214. (57) Havlin, S. In Kinetics of Aggregation and Gelation; Family, F., Landau, D. P., Eds.; Elsevier Science Publisher: Amsterdam, 1984; pp 145-156. (58) Meakin, P.; Coniglio, A.; Stanley, H. E.; Witten, T. A. Phys. ReV. A 1986, 34 (4), 3325-3340.