29Si NMR Kinetic Study of Tetraethoxysilane and Ethyl-Substituted

Nov 15, 1995 - ethyltriethoxysilane {EtSi(OEt)3}, and diethyldiethoxysilane {Et2Si(OEt)2} at a constant initial water and catalyst concentration. A su...
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Ind. Eng. Chem. Res. 1996, 35, 117-129

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MATERIALS AND INTERFACES 29Si

NMR Kinetic Study of Tetraethoxysilane and Ethyl-Substituted Ethoxysilane Polymerization in Acidic Conditions Jorge Sanchez, Stephen E. Rankin, and Alon V. McCormick* Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Avenue SE, Minneapolis, Minnesota 55455 29Si NMR is used to quantify the polymerization kinetics of silicon alkoxide intermediates in acidic sol-gel processes well before gelation. Reaction conditions are chosen (a) to minimize interference by hydrolysis kinetics and differing degrees of hydrolysis and (b) to investigate the possibility of single-step copolymerization. At the composition chosen hydrolysis reactions rapidly reach pseudoequilibrium, so condensation rate coefficients can be easily determined. We compare condensation rate coefficients of hydrolyzed intermediates from tetraethoxysilane {Si(OEt)4}, ethyltriethoxysilane {EtSi(OEt)3}, and diethyldiethoxysilane {Et2Si(OEt)2} at a constant initial water and catalyst concentration. A surprisingly strong negative first-shell substitution effect is caused by connectivity for each system and is discussed. This substitution effect depends on the degree of condensation of both reacting groups but more so on the less connected one. We also comment on a decrease in condensation reactivity and an increase in hydrolysis reactivity and favorability caused by the nonreactive ethyl groups.

Introduction Research on sol-gel processes for silicate-based materials has focused mainly on the polymerization in alcohol/water solutions of tetrafunctional alkoxysilanes such as tetraethoxysilane (TEOS) or tetramethoxysilane (TMOS) (Assink and Kay, 1988; Bailey et al., 1990; Brinker and Scherer, 1990; Kay and Assink, 1988a,b; Peppas et al., 1988; Ro and Chung, 1989). Even for these, the most common sol-gel precursors, the detailed substitution effects for hydrolysis and condensation reactivity remain inadequately quantified. Recently, much interest has also arisen in the preparation of organically modified ceramics by copolymerization of a tetrafunctional alkoxysilane and a di- or trifunctional alkoxysilane (Sakka et al., 1986; Schmidt, 1985; Schmidt et al., 1984). Of particular interest are copolymers of silicon tetraalkoxysilanes and alkylalkoxysilanes (Schmidt, 1985). Although silicon alkylalkoxides have been known for some time (Andrianov, 1955), only recently have “hybrid” copolymer gels been successfully prepared (see, for example, Schmidt (1985)); however, these preparations have thus far required multistep approaches. In such semibatch procedures, one monomer is typically allowed to hydrolyze and condense for some time before adding the other(s), in the hope that this will promote copolymerization and homogeneity. The commercial development of these materials is rendered difficult by the absence of quantitative predictive models for the polymerization kinetics of the silicon alkoxide precursors. However, it should be possible not only to optimize the semibatch procedures mentioned above but also to develop single-step and continuous processes for the preparation of inorganic networks and inorganic/organic copolymers with adequate kinetic models. * To whom correspondence should be addressed. e-mail: [email protected]. Fax: 612-626-7246.

0888-5885/96/2635-0117$12.00/0

Condensation rate coefficients are difficult to estimate from 29Si nuclear magnetic resonance spectra because it is necessary to deal with hydrolysis transients and yet to maintain a high degree of spectral resolution to identify individual rate coefficients. In the past, simplifying assumptions have been used, but here we wish to independently measure condensation rate coefficients and therefore to test some of these assumptions. In this paper, we have begun this quantification and testing by monitoring the polymerization of tetraethoxysilane [TEOS, Si(OEt)4], ethyltriethoxysilane [EtSi(OEt)3], and diethyldiethoxysilane [(Et)2Si(OEt)2] in acidic ethanol/ water solutions. In order to perform a comparative kinetic study, we carry out the polymerization under identical conditions, i.e., same pH, same temperature, and same water, precursor, and ethanol concentrations. The solution composition used here is limited to one for which the hydrolysis kinetics have been well characterized. While this paper represents an attempt to explain why single-step copolymerization might be difficult, further compositions are being investigated (Ng and McCormick, 1995; Rankin et al., 1995) that are also of interest for multistep processing (e.g., with varying H2O/ alkoxide group). More important than the calculation of any individual rate constants, we explore the trends of substitution effects. Of particular interest is to determine how the reactivity for polymerization depends on the number of siloxane bonds or hydroxy groups carried by the silicon atoms. The implications of the relative kinetics of the early intermediates for organic-inorganic copolymerization are discussed. For these reasons we limit our current analysis to reaction times well before gelation (in fact, diethyldiethoxysilane cannot gel at all, and ethyltriethoxysilane does not gel under the conditions used here). The water composition is chosen to yield a homogeneous, low fractal dimension gel for tetrafunctional TEOS, to ensure rapid hydrolysis equilibrium, © 1996 American Chemical Society

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and to keep cyclization to three-membered rings to a minimum, since the unimolecular nature of these reactions necessarily complicates kinetic analysis. Keeping the initial water composition the same for each monomer allows us to (1) best represent the situation that might be encountered should one attempt to perform copolymerization in a simple batch reactor or a continuous stirred tank reactor (CSTR), (2) avoid ensuing complications in interpretation due to very different ionic strengths, (3) avoid serious differences in initial water and alkoxide activity, (4) avoid concerns about trivial differences in initial reaction rates due to dilution effects, (5) help evaluate why single-step copolymerizations seem so elusive, and (6) as it fortunately turns out (and will be seen herein) keep the expected number of OH groups per silicon after the establishment of hydrolysis pseudoequilibrium about the same for the different alkoxides. Ethyl substitution rather than the more common methyl is chosen to keep the order of magnitude of reaction rate constants comparable and to minimize steric effects of alkyl substitution. We will report shortly on methyl-substituted alkoxides (Rankin et al., 1995) and on the effects of varying water and acid concentrations. Experimental Methods In a 10 mm NMR tube, 2 mL of the silicon alkoxide monomer (either TEOS, ethyltriethoxysilane, or diethyldiethoxysilanesall from Hu¨ls America) was mixed with 2 mL of a 1% by weight Cr(acac)3 solution in absolute ethanol, 223 µL of deionized water, and 100 µL of a 0.10 N HCl solution (Aldrich). Cr(acac)3 was used as a paramagnetic relaxing agent, and we confirmed that at the concentration used it did not affect the kinetics of the reaction. The resulting concentrations were [EtOH] ) 7.79 M, [H2O] ) 4.15 M, [HCl] ) 2.3 × 10-3 M, and [Si alkoxide] ) 2.07 M (TEOS), 2.16 M (ethyltriethoxysilane), or 2.25 M (diethyldiethoxysilane). Polymerization was carried out at 25 °C and was monitored by 29Si NMR at times well before gelation. In this way spectral resolution remained high enough that we could unambiguously assign peaks. NMR spectra were acquired on a GE 500 spectrometer at 99.353 MHz. A 25 µs pulse width and a 10 s pulse delay were used. The pulse delay was determined to be long enough to allow complete relaxation of the nuclei. The chemical shifts were referenced to that of neat tetramethylsilane. The rate constants were calculated by iteratively solving the system of differential kinetic equations for the silicon sites to optimize the set of rate constants, obtaining the best fit to the experimental concentrations (Sanchez and McCormick, 1992b). The equations used are discussed below. Resolution of the system of differential equations was accomplished by a predictorcorrector method (Burden and Faires, 1985), and the rate constant optimization was accomplished by a Levenberg-Marquardt least-squares algorithm (Dennis and Schnabel, 1983). All calculations were performed on an HP Apollo workstation. An error analysis of the results was done to ensure the robustness of the optimized set of rate constants (Sanchez and McCormick, 1992b). Results Peak Assignments. The 29Si NMR spectra acquired during polymerization of the three precursors are shown

Figure 1. 29Si NMR spectra for the polymerization of diethyldiethoxysilane (top) and experimental (points) and fit (lines) concentration curves (bottom). The first 15 spectra were acquired every 10 min and then every 30 min, up to about 10 h of reaction. The kinetic model for the fit curves is shown in Scheme 1.

at the tops of Figures 1-3. The peak assignments for TEOS derivatives and for some of the signals for the other precursors were made according to Sugahara et al. (1992, 1994), Kelts and Armstrong (1989), and Marsmann (1981). The NMR spectra show various silicon units distinguished by their degree of hydrolysis (i.e., number of OH groups attached) and degree of condensation (i.e., number of bonds leading to other silicon atoms). In agreement with the notation commonly used (Marsmann, 1981), the silicon units are designated by their functionality: Qi j for a four-functional unit (the tetraethoxysilane system), Ti j for a three-functional unit (the triethoxysilane system), and Di j for a two-functional unit (the diethoxysilane system). The subscript (i) indicates the number of connections to other silicon units (i.e., degree of condensation), and the superscript (j) indicates the number of OH groups attached to the silicon atom (i.e., degree of hydrolysis). Similarly, Di, Ti, and Qi refer respectively to the total concentration of difunctional, trifunctional, or tetrafunctional sites of connectivity i (see eq 1 for an explicit definition). The 29Si NMR spectra for the D (diethyldiethoxysilane) system (Figure 1) are significantly different from those of the two other systems (as observed for methylsubstituted siloxane polymers (Sugahara et al., 1992; Sugahara et al., 1994)). The hydrolysis of silicon units

Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 119

Figure 2. 29Si NMR spectra for the polymerization of ethyltriethoxysilane (top) and experimental (points) and fit (lines) concentration curves (bottom). The first 10 spectra were acquired every 10 min, the next 10 spectra every 20 min, and then every 30 min, up to about 20 h of reaction. The kinetic model for the fit curves is shown in Scheme 2.

is extremely rapid and, at our water concentration, complete. No unreacted monomer is observed even in the first spectrum, and the fully hydrolyzed monomer (D02 at -5.0 ppm) is produced exclusively and nearly instantaneously. In fact, hydrolysis is so rapid and complete in our reaction conditions that the hydrolysis reaction cannot be monitored at this composition by 29Si NMR. This is consistent with previous studies (Marsmann, 1981; Schmidt, 1985). The next site to appear is the singly condensed hydrolyzed silicon atom (D11 at -13.4 ppm). Two types of doubly connected sites then appear almost simultaneously: silicon sites in trimeric cycles (D2∆0 at -9.8 ppm) and linear middle sites (D20 at -21.9 ppm). These assignments are consistent with those made by Sugahara et al. (1992) for hydrolyzed dimethyldiethoxysilane condensation products. For the T (ethyltriethoxysilane) system (Figure 2), various hydrolyzed monomers (T00 at -44.7 ppm, T01 at -42.6 ppm, T02 at -40.8 ppm, and T03 at -39.2 ppm) can be observed early in the reaction. After this, the peaks appearing in the region between -48 and -52 ppm are attributed to T1 sites with various degrees of hydrolysis (T10 at -52.1 ppm, T11 at -50.2 ppm, and T12 at -48.5 ppm), whereas the peaks appearing between -58 and -61 ppm belong to T2 sites (by analogy with trifunctional methylpolysiloxanes (Sugahara et al., 1994)). We observe two additional small peaks that

Figure 3. 29Si NMR spectra for the polymerization of tetraethoxysilane (TEOS) (top) and experimental (points) and fit (lines) concentration curves (bottom). The first 10 spectra were acquired every 10 min and then every 45 min, up to about 24 h of reaction. The kinetic model for the fit curves is shown in Scheme 3.

have not been assigned previously but which we ascribe to T2 sites by the following reasoning. By analogy with the TEOS system (Kelts and Armstrong, 1989), the peak at -54.7 ppm is attributed to T2∆1 and that at -57.0 ppm is attributed to T2∆0. It is of interest to establish that these two peaks are differently hydrolyzed forms of sites on the same-sized ring. From a study of poly(dimethylsiloxane) (PDMS) rings (Engelhardt and Jancke, 1971) it has been shown that there is a large difference in chemical shift (∼10.8 ppm) between three- and four-membered PDMS rings but that larger rings are essentially indistinguishable from linear oligomer D2 sites in a mixture. Four-membered rings are assigned ∼2.9 ppm downfield of the other D2 peaks for PDMS (Engelhardt and Jancke, 1971). For TEOS condensation products, on the other hand, a difference of 7.2 ppm separates three- and fourmembered ring sites and 1 ppm separates four-membered rings from other Q2 sites (Kelts and Armstrong, 1989). Also, in both of these studies, three-membered rings were observed downfield of the end-group (X1) peaks. Because the two peaks we see are (1) upfield of the T1 peaks, (2) separated from each other by only 2.3 ppm (consistent with a shift caused by hydrolysis), and (3) ∼3.3 ppm downfield of the other T2 sites, we propose that they are differently hydrolyzed members of the same-sized ring, most likely four-membered (the left peak being the hydrolyzed site). Note, however, that for the modeling done in the body of the paper, the only important characteristic of the peaks is that they are

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Table 1. Chemical Shift Assignments chemical shift of peak or center of group of peaks (ppm)

assignment

chemical shift of peak or center of group of peaks (ppm)

-5.0 -9.8

Diethyldiethoxysilanea D02 -13.4 D2∆0 -21.9

-39.2 -40.8 -42.6 -44.7 -48.5 -50.2

Ethyltriethoxysilaneb T03 -52.1 T02 -54.7 1 T0 -57.0 T00 -58.2 T12 -60.1 T11

-72.4 -74.2 -76.2 -78.9 -82.0 -84.3 -85.7 -86.4 -88.2

Tetraethoxysilanec Q04 -89.3 Q03 -91.1 Q02 -91.8 Q01 -93.2 Q00 -94.0 Q12 -95.7 Q2∆1 -96.4 Q11 -100 0 Q2∆ -102

jj′ kaii′

assignment

(V) D11 D20 T10 T2∆1 d T2∆0 d T21 T20

Q10 Q202 Q22 Q201 Q21 Q200 Q20 Q31 Q30

a Assignments based on Sugahara et al. (1992) and Engelhardt et al. (1971). b Assignments based on Sugahara et al. (1994). c Assignments based on Kelts and Armstrong (1989) and Marsmann (1981). d New assignment; see text for discussion.

T2 sites; their concentration is small enough that we find no need to include separate cyclization reactions. That they are differently hydrolyzed only slightly affects the results in the appendix. The other two T2 peaks observed may belong to larger cyclic or linear molecules and are described simply as T20 at -60.1 ppm and T21 at -58.2 ppm. During the reaction time shown here, the concentration of the most highly condensed site, T3 (which would appear at ∼-64 ppm (Sugahara et al., 1994)), remains negligible. The spectra for the Q (tetraethoxysilane) system (Figure 3) show the various hydrolyzed and unhydrolyzed species usually observed in acidic conditions (Kelts and Armstrong, 1989; Marsmann, 1981), including a small number of sites in three-membered rings (Q2∆j) and four-membered rings (Q20j). The chemical shifts assumed for hydrolyzed TEOS polycondensation products are summarized in Table 1, along with all other chemical shift assignments for this study. As noted in a previous paper (Sanchez and McCormick, 1992a), at this composition the monomer hydrolysis reactions rapidly reach pseudoequilibrium. Note also that in the time during which the reaction was monitored (up to 24 h), no Q4 units (which are found around -110 ppm (Kelts and Armstrong, 1989)) were detected. Kinetic Results. For the most general sol-gel reaction network, six classes of reactions must be considered: khij

Xij + H2O 98 Xij+1 + ROH krij

Xij+1 + ROH 98 Xij + H2O

0 e j e f - i - 1 (I) 0 e j e f - i - 1 (II)

kii′jj′

j-1 j′-1 + Xi′+1 + H2O Xij + Xi′j′ 98 Xi+1 0 e i(i′) e f - 1; 1 e j(j′) e f - i(i′) (III) jj′ kwrii′

j j′-1 + Xi′+1 + ROH Xij + Xi′j′ 98 Xi+1 0 e i(i′) e f - 1; 0 e j e f - i - 1; 1 e j′ e f - i′

j-1 j′-1 + Xi′+1 + H2O 98 Xij + Xi′j′ Xi+1 0 e i(i′) e f - 1; 1 e j(j′) e f - i(i′) (IV)

jj′ karii′

j j′-1 + Xi′+1 + ROH 98 Xij + Xi′j′ Xi+1 0 e i(i′) e f - 1; 0 e j e f - i - 1; 1 e j′ e f - i′

(VI) For a complete kinetic model, an extremely large number of rate constants for reactions I-VI (10 forward and 10 reverse hydrolysis, {khij} & {krij}; 55 forward and 55 reverse water-producing condensation, {kii′jj′} & jj′ {kwrii′ }; and 100 forward and 100 reverse alcohol-projj′ jj′ ducing condensation, {kaii′ } & {karii′ }) would be required to describe all first shell substitution effects (Kay and Assink, 1988b). To simplify this type of problem, Kay and Assink (Assink and Kay, 1988; Kay and Assink, 1988a,b) initially assumed equal group reactivity for both hydrolysis and condensation of TMOS (note, though, that this would appear as a statistically decreasing silicon site reactivity due to changes in the number of reactive groups on different silicon sites). This assumption means that the hydrolysis reactivity of an ethoxy group and the condensation reactivity of an OH group would not be allowed to depend on the type of silicon site to which those groups are attached. They also assumed that hydrolysis and condensation are irreversible. Pouxviel and Boilot (1987) proposed that, for systems with moderate to high water content ([H2O]/ [Si] > 4), the only condensation of significant rate involves the reaction of two OH groups (i.e., waterproducing condensation). More recently, Assink and Kay (1993) showed this to be true for TEOS at moderate water concentrations (close to those used here) at least at very early reaction times. Another contribution of that paper was to demonstrate that the initial dimerization rate is second order in the total silanol concentration for a wide range of initial water concentrations. This suggests that the condensation rate coefficient might only be a weak function of the degree of hydrolysis of the sites involved. Taking all these assumptions, a highly simplified kinetic model would require only two rate constants: one for hydrolysis and one for waterproducing condensation. However, this simple model may fail to account for some very important features of the kinetics that are necessary to understand the structure development during polymerization. As but one example, it has already been shown that hydrolysis is actually reversible (Ro and Chung, 1989; Sanchez and McCormick, 1992b). It has also been shown that the reactivity of an OH group for condensation depends on its environment (there are substitution effects) (Assink and Kay, 1988; Brunet and Cabane, 1993; Pouxviel and Boilot, 1987). The kinetic model that we use allows condensation substitution effects with degree of condensation. Since this is a bimolecular reaction, we allow the rate coefficient to depend upon the connectivity of each of the reactants. The systems studied allow us to isolate that particular effect without interference by differing degrees of hydrolysis (the average degrees of hydrolysis are nearly constant), by cyclization (the concentration of cyclic species is low), and by hydrolysis kinetics. A previous investigation showed that hydrolysis, even if slow, can approach pseudoequilibrium with composi-

Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 121 Scheme 1. Kinetic Model and Calculated Rate Coefficients for the Diethyldiethoxysilane Systema

a The rate coefficients (in L‚mol(SiOH groups)-1‚h-1) are for reactions of groups (i.e., OH groups) attached to the silicon atom. Equation 1 was used to fit the data.

Scheme 2. Kinetic Model and Calculated Rate Coefficients for the Ethyltriethoxysilane Systema

Under these assumptions, six rate constants need to be determined for TEOS polymerization; there are only three for ETEOS polymerization and for the DEDEOS system (the number is the same for these two because the amount of T3 formed over the time observed is negligible, as mentioned previously). The kinetic models used are shown in Schemes 1-3. These schemes only include water-producing condensation reactions (reaction III); hydrolysis is completely characterized by {〈j〉i}. The general set of differential equations used to model these systems is (see the appendix for derivation from a more general set of equations):

d[Xi]

f-1

) dt

∑ i′)0

{

ki-1,i′〈j〉i′[Xi′]〈j〉i-1[Xi-1] ki,i′〈j〉i′[Xi′]〈j〉i[Xi]

where

(1)

f-i

a The rate coefficients are in L‚mol(SiOH groups)-1‚h-1. Equation 1 was used to fit the data.

Scheme 3. Kinetic Model and Calculated Rate Coefficients for the Tetraethoxysilane Systema

}

[Xi] ≡

[Xij] ∑ j)0

and f-i

〈j〉i ≡

j[Xij] ∑ j)1 f-i

[Xij] ∑ j)0

a The rate coefficients are in L‚mol(SiOH groups)-1‚h-1. Equation 1 was used to fit the data.

tions such as those used here (Sanchez and McCormick, 1992a). From the spectra in Figures 1-3, we determine that this remains true for the compositions studied here. We will also show, in any event, that the average degree of hydrolysis on each silicon site in these experiments remains approximately constant over the course of the reaction. So even if we were to suppose that there is a condensation substitution effect caused by the degree of hydrolysis, it would have little effect under the conditions we have chosen (we will be able to comment more on this shortly). Thus, we can isolate condensation rate constants by directly measuring the average degree of hydrolysis for each site after a short transient. It is also possible to model this system with hydrolysis rate equations, but the appendix illustrates that the equations and results obtained are in good agreement with the approach taken here. The connection between the two models and the similarity in the reactivity trends observed is also illustrated there. Finally, note that mass action kinetics is assumed, as is commonly done for these systems (Kay and Assink, 1988a,b; Pouxviel and Boilot, 1987; Sanchez and McCormick, 1992b).

[Xij] is the concentration of Qij, Tij, or Dij units, [Xi] is the corresponding concentration of sites of connectivity i, f is the potential functionality of the monomer (e.g., f ) 4 for TEOS or 3 for triethoxysilanes), 〈j〉i is the average degree of hydrolysis of sites Xi, and ki,i′ is the rate constant for the condensation of an Xi unit with an Xi′ unit. The rate constants for condensation are taken as averaged over the degrees of hydrolysis, j and j′, of pairs of units of connectivity i and i′ such that their average degrees of hydrolysis are 〈j〉i and 〈j〉i′, respectively. We represent this averaging with a single overbar. As discussed before, this is reasonable if the average number of OH groups on a Si unit remains nearly constant over the course of the reaction studied. We will show that this is true after a short transient. In keeping with this condition, fitting is only done to data where the average degree of hydrolysis of the reactants involved actually has become almost constant. The times to reach constant average degrees of hydrolysis for each site (and the values at pseudoequilibrium) are listed in Table 2. Note that the average degrees of hydrolysis remain nearly constant not because the water has been exhausted but because hydrolysis pseudoequilibrium is reached and the water and ethanol concentrations change only very slowly after that point. The average degrees of hydrolysis will be discussed more later. Note that the rate constants are expressed with units of concentrations of the functional groups attached to

Table 2. Average Degrees of Hydrolysis at Pseudoequilibrium and Times To Reach Those Steady Values for All Sites

a

monomer

〈j〉0 (〈j〉0/〈j〉0,max)

time to 〈j〉0 (h)

〈j〉1 (〈j〉1/〈j〉1,max)

time to 〈j〉1 (h)

〈j〉2 (〈j〉2/〈j〉2,max)

time to 〈j〉2 (h)

TEOS ethyltriethoxysilane diethyldiethoxysilane

1.38 (0.35) 1.67 (0.56) 2.00 (1.0)

0.4 0.2 -OEt

On the other hand, steric hindrance increases when the ligands become more bulky, so one might expect:

bulkiness:

-OSit > -OEt ≈ -Et > -OH

The kinetic trends that we observe for condensation are difficult to rationalize since condensation involves two silicon sites. If we first consider the decrease of condensation reactivity with the degree of connectivity, the substitution of OH or OEt groups by OSi ligands should result in an increase of the positive charge on the silicon atom (Brinker and Scherer, 1990). However, it is not clear what connection exists between the partial charge on the silicon atom and the reactivity of the OH ligands for condensation. We may assume, following Iler (1979) and others, that condensation proceeds by a nucleophilic substitution on the silicon atom. If the nucleophilic attack is the rate-limiting step, an increase of the positive charge on the attacked silicon atom might result in an increase of condensation reactivity (Brinker and Scherer, 1990). It is not at all clear what trends to expect with substitution on the attacking group, although Scherer has reasoned that basicity of the attacking group would be highest for groups with the highest negative charge on the hydroxy ligand (Brinker and Scherer, 1990) and the highest rate constant that should be observed would involve sites with the maximum difference in electron-donating/electron-withdrawing character of ligands. This is not what we see; the maximum rate coefficient is for k00, while this argument implies that k02 should be highest if (as we observe) the number of OH per silicon is similar for Q0 and Q2. On the other hand, the increase of steric hindrance in more connected silicon sites might result in a decrease of condensation reactivity. Since the latter is actually observed in our results, we may speculate that steric effects dominate over inductive effects in condensation. This could explain why the rate constant for condensation involving monomers as one partner is only slightly affected by the connectivity of the other partner: since monomers are relatively mobile, they should be able to easily access sites even in fairly branched molecules. Less connected sites may also undergo inversion (Brinker and Scherer, 1990) more easily. By contrast, when both sites are highly condensed, the large steric hindrance and the lower mobility of the two partners involved decreases the condensation reactivity. It will also become important to consider cyclization effects as models are further developed. The decrease of condensation reactivity in going from the TEOS to the diethyldiethoxysilane systems is consistent, though, with a decrease of positive charge (and thus of nucleophilic character) of the silicon when ethyl groups replace ethoxy or OH groups. This effect is likely to be in competition with the reduced steric bulk of the ethyl group (compared to the silicate polymer), however. So some uncertainty remains as to which trend will be observed and under what conditions. The trend is most likely a strong function of the nature of the organic group. To explain the increase of hydrolysis reactivity and favorability in going from the Q system to the T system is more straightforward. It has been proposed that a trivalent siliconium ion might be formed as transition state (Schmidt et al., 1984). Such an intermediate state is stabilized by the negative inductive effect of the ethyl group: assuming that hydrolysis proceeds by a nucleo-

philic substitution on the silicon atom, hydrolysis would be faster. Our results are consistent with that hypothesis. It is possible that the ethyl group stabilization effect has less influence for reverse hydrolysis since the OH ligands have a strong positive inductive effect, and this may account for the small difference in reverse hydrolysis rates between the Q and T systems (see the appendix). Conclusion A comparative kinetic study of the polymerization of three different silicon alkoxide monomers has allowed us to confirm and quantify the increase of hydrolysis reactivity when more alkyl groups are attached to the silicon atom. Moreover, we have found that condensation reactivity decreases with increased ethyl substitution, thus following an opposite trend. Finally, we have also quantified the negative substitution effect of condensation, i.e., the decrease of condensation reactivity for more connected silicon sites. The effect observed here is stronger than has been reported previously and depends on the connectivity of both reactants, but not always in an additive fashion. This suggests that the additivity assumed by Brinker and Assink (1989) may work for diethyldiethoxysilane but not for ethyltriethoxysilane and tetraethoxysilane at this pH and water concentration. The resulting kinetic trends can be explained on the basis of inductive and steric effects. Not surprisingly, it was found that inductive effects dominate for reactions involving monomers and less connected sites, but steric effects are much more important when the reacting partners are more condensed. Such kinetic trends can be incorporated in polymerization models to help understand the structure development and properties of organically modified silicate copolymers. The trends shown in this paper are consistent qualitatively with previous expectations but also explain quantitatively why multistep syntheses have been necessary to prepare copolymer networks. Acknowledgment This work was supported by a grant from the Office of Naval Research, with a fellowship to J.S. from the Center for Interfacial Engineering at the University of Minnesota and with a fellowship to S.E.R. from the National Science Foundation. The authors gratefully acknowledge helpful discussions with Professor C. W. Macosko and Professor L. E. Scriven at the University of Minnesota. Nomenclature Xij ) site with functionality given by X (f ) 4 for Q, 3 for T, and 2 for D) with i siloxane bridges and j hydroxyl groups attached f ) potential functionality of a silicon site i ) number of siloxane bridges attached to a site j ) number of hydroxy groups at a site kii′jj′ ) water-producing condensation rate coefficient for reaction of Xij and Xi′j′ jj′ kaii′ ) alcohol-producing condensation rate coefficient for reaction of Xij and Xi′j′ jj′ kwrii′ ) reverse water-producing condensation rate coefficient for reaction of Xij and Xi′j′ jj′ karii′ ) Reverse alcohol-producing condensation rate coefficient for reaction of Xij and Xi′j′

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khij ) hydrolysis rate coefficient for Xij

may be written as:

krij ) reesterification rate coefficient for Xij

d[Xij] )

K ) overall hydrolysis equilibrium constant (K ) kh/kr ) ([SiOH][EtOH]/([SiOEt][H2O])) 〈j〉i ) average degree of hydrolysis for sites of connectivity i

dt kh[H2O]{(f - i - (j - 1))[Xij-1] - (f - i - j)[Xij]} + kr[EtOH]{(j + 1)[Xij+1] - j[Xij]} f-1 f-i′

Special Symbols

( ∑ ki,i′jj′[Xi′j′][Xij]) + ∑ i′)0 j′)1

θ ) averaging of quantity θ with respect to all possible degrees of hydrolysis of the sites involved θ ) averaging of quantity θ with respect to all possible degrees of hydrolysis and connectivity of the sites involved [Θ] ) concentration of species Θ

f-1 f-i′

j+1 ( ∑ ki-1,i′(j + 1)j′[Xi′j′][Xi-1 ]) ∑ i′)0 j′)1 f-1 f-1

d[EtOH] dt

(∑j[Xij]) + ∑ i)0 j)1

) -kr[EtOH]

{θR} ) vector comprised of elements θR where R is the index of the element

Appendix: An Alternative Model and Derivation of Equation 1

f-1 f-i-1

kh[H2O] d[H2O]

( ∑ (f - i - j)[Xij]) ∑ i)0 j)0

(4)

d[EtOH] )-

dt Equation 1 deals with differing time scales of hydrolysis and condensation by assuming (as is experimentally justifiable) that the average degrees of hydrolysis are approximately constant. If, as we expect, {kii′jj′} is insensitive to j and j′, this should even work when {〈j〉i} changes. In this case, only {Xi} are fit. Here we show that it is also possible to reach a reasonable fit to {Xij}, but to do that, some measure of hydrolysis and reesterification rates is required. As a first approximation, we assume that hydrolysis rate coefficients can be averaged for sites with different degrees of connectivity and hydrolysis and that condensation rate coefficients can be averaged for sites of different degrees of hydrolysis. This is formally equivalent to assuming that there are no hydrolysis substitution effects with respect to the degree of hydrolysis or connectivity and that there are no condensation substitution effects caused by hydrolysis. The first assumption (no hydrolysis substitution effects with degree of hydrolysis) actually contradicts the findings of Sanchez and McCormick (1992b). However, this is adopted only as an approximation; the ability to fit the individual species concentrations will determine how valid this approximation is. A negligible effect of connectivity on hydrolysis rate coefficients has been observed early in the reaction (Sanchez and McCormick, 1992a), but we can test this approximation over a wider range of reactions here. The neglect of a condensation substitution effect caused by hydrolysis is justified by the rapidly approached pseudoequilibrium observed in the main text. Even if the average degree of hydrolysis changes during the reaction, the effect on condensation reactivity is negligible compared to the strong effect of connectivity reported here (Ng and McCormick, 1995). Assink and Kay (1993; 1988a) also observed a negligible effect of the degree of hydrolysis on condensation rate coefficients early in the reaction. With the assumptions outlined above and as discussed in the main body of the text, the set of differential equations used to describe the evolution of the system

(3)

+ dt 1f-1

f-i f-1 f-i′

{jj′ki,i′[Xij][Xi′j]} ∑ ∑ ∑ ∑ 2i)0 j)1 i′)0 j′)1

(5)

where all variables are defined as in the body of the text. The values of i and j in all [Xij] terms must be consistent with the functionality of the monomer (i + j e f) and the definition of i and j (i, j g 0). If, in the expression above, i or j violate these restrictions, that term is by definition equal to zero. As noted earlier, the condensation rate coefficients (ki,i′) are taken as averaged over all possible j consistent with i and i′, and the hydrolysis and reesterification rate coefficients (kh and kr) are taken as averaged over all possible j and i (indicated by the double overbar). The factor of 1/2 appearing in the water equation is also worth commenting on; it arises from the indistinguishability of identical sites (by summing over all i and i′, we cancel its effect for distinguishable sites). This is similar to the accounting for the reaction of indistinguishable polymer molecules as outlined by Dotson et al. (1995) and as employed by Aris (1994) in his treatment of continuous step polymerization and as used by Kay and Assink (1988b). Before discussing the use of these sets of equations, eq 1 will be derived. Equation 3 describes the evolution of each site differentiated by degree of connectivity and degree of hydrolysis. These equations should be used to fit individual site concentrations {Xij}, and we show below that this is possible. Our concern in doing this, though, is that kh and kr are averaged over too many types of sites and therefore may not be very meaningful. Also, the number of rate coefficients being fit to the data is rather high (eight parameters for tetraethoxysilane). However, if we use the measured {〈j〉i} as data and fit only to {Xi}, we can eliminate the necessity of using kh and kr as parameters, thus reducing the number of parameters by two and increasing our confidence in the condensation rate coefficients. To do this, eq 3 can be

Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 127

summed over all values of j:

d[Xi]

d )

dt

f-i

f-1d[X j] i

∑[Xi ] ) ∑ j)0

dt j)0

j

dt

Substituting eq 3 and performing the summation gives eq 1 after some algebraic manipulation. Note that this treatment totally eliminates the need for hydrolysis and reesterification rate coefficients if {〈j〉i} can be measured directly and would do so even if they were not assumed to be independent of all substitution effects. We will show that the condensation rate coefficients found using the full set of equations (here) and the reduced set of equations (eq 1) agree well; thus, the robust model in the text provides a check of our results here. The equations laid out here were solved for the appropriate system and then fit to {Xi j} data using the techniques described in the Experimental Section. The set of equations for the D system is the same in this model as for the pseudoequilibrium model, because the degree of hydrolysis truly is constant at all times (we cannot even measure the hydrolysis transients). Therefore, the rate coefficients found are identical. For ethyltriethoxysilane and tetraethoxysilane, however, the model is somewhat different since we now model transient hydrolysis states in an approximate way. Figure 6 shows the fit to individual species concentration data for the T system, and Figure 7 shows a similar fit for the Q system. The reactions and rate coefficients found are given in Schemes 4 and 5 for the T system and the Q system, respectively (they are comparable to Schemes 2 and 3, respectively). Comparing the fits to the data here with those in the text (Figures 2 and 3), we see that both the hydrolysis pseudoequilibrium model and this model are capable of fitting the data after a certain transient period has been overcome. The fitting here does not follow the data exactly at all times, but it is close enough that the rate coefficients found are valuable. We will discuss the nature of the fit more later. For the T system, the constants that produce a good fit to this model are similar to those from the pseudoequilibrium model, as shown in Scheme 4. Comparing this to Scheme 2, the rate coefficients from this model are of the same order of magnitude as those from the pseudoequilibrium model and the trends observed are the same. The rate coefficients for all reactions are somewhat higher than those predicted by the hydrolysis pseudoequilibrium model. However, the difference between the rate coefficients found by the two models is insignificant compared to the similarity in the trend in condensation reactivity. Note also that very rapid

Figure 6. Comparison of experimentally measured individual ethyltriethoxysilane site (Ti j) concentrations (points) and the results of fitting (lines) using the model in the appendix. Shown are (a) T0, (b) T1, and (c) T2 concentrations. The kinetic model used to generate the fit curves is shown in Scheme 4.

Scheme 4. Kinetic Model with Explicit Hydrolysis and Reesterification Reactions and Calculated Rate Coefficients for the Ethyltriethoxysilane Systema

hydrolysis (kh > ki,i′) is found and there is a high equilibrium tendency for hydrolysis (characterized by K ) kh/kr ) 49.1 . 1), consistent with the findings in Table 2. For the Q system, attempts to fit directly to {Qij} were complicated by the noise in the data. Direct fitting yielded several local optima, one of which gave about the same {ki,i′} as in the body of the paper. The global minimum, however, gave a set of {ki,i′} with a similar trend as the constants in Scheme 3 but with an unrealistically low K ()kh/kr) value. An estimate of the approximate value of this equilibrium constant can be found directly from the experimental data. By plotting Kapparent ) ([SiOH][EtOH])/([SiOEt][H2O]) as a

a The rate coefficients (L‚mol(groups)-1‚h-1) are expressed in terms of the ethoxy or hydroxy groups reacting. Equations 3-5 were used to fit the data.

function of time, it was found that the value should be between 10 and 20 when pseudoequilibrium is reached.

128

Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996

Scheme 5. Kinetic Model with Explicit Hydrolysis and Reesterification Reactions and Calculated Rate Coefficients for the Tetraethoxysilane Systema

a The rate coefficients are in L‚mol(groups)-1‚h-1. Equations 3-5 were used to fit the data (see text for explanation of fitting procedure).

Therefore, we could reject sets of rate coefficients giving unrealistic equilibrium constants. To overcome the multiplicity problem, an approximate method was used to get {ki,i′}: eqs 3-5 were solved, and the results were fit to {Xi} and the total silanol concentration. This method of fitting reduced the multiplicity of solutions by making the noise in each series of data less important. Once this was done, the {ki,i′} were kept constant and only kh and kr were fit. In this way, a complete set of rate coefficients was found that fits the individual species concentrations reasonably well and is consistent with the apparent K from the experimental data. Comparison of Schemes 3 and 5 shows a consistent trend in condensation rate coefficients for the Q system. Again, the actual numbers for this model are slightly different from those of the pseudoequilibrium model, but the trend observed is the same. In addition, comparing the T and Q systems, the trends in condensation rate coefficients with degree of condensation are again the samesthe order of magnitude decreases with increasing connectivity of the least-connected site. The only difference with the findings of the hydrolysis pseudoequilibrium model is that the k01 value is lower than both the k00 and k02 values for Q. This may be a result of the fact that there is a hydrolysis substitution effect for monomers which this model fails to capture or simply of the scatter in the individual concentrations. The hydrolysis trends found are also consistent with the findings in the body of the text. The hydrolysis rate coefficient is lower for Q than for T, and the ratio of forward and reverse hydrolysis rate coefficients, K, is lower for Q (K ) 15.6), consistent with the findings of Table 2. Obviously, the instantaneous complete hydrolysis of diethyldiethoxysilane continues this trend. What we see in Figures 6 and 7 is that for X0 (monomer) sites, the fit curves agree only poorly with the individual concentrations very early in reaction but succeed later (when {〈j〉i} becomes nearly constant). This is consistent with the fact that there is a hydrolysis substitution caused by hydrolysis for TEOS monomers (Sanchez and McCormick, 1992b). However, the ability to fit the individual species concentrations improves as the sites become more connected. This implies two

Figure 7. Comparison of experimentally measured individual TEOS site (Qij) concentrations (points) and the results of fitting (lines) using the model in the appendix. Shown are (a) Q0, (b) Q1, (c) Q2, and (d) Q3 concentrations. The kinetic model used to generate the fit curves is shown in Scheme 5. See text for details about the fitting procedure.

things. First, because the simulation coincides with the Xi j concentrations for several different values of i (i g 1), the hydrolysis substitution effect caused by the degree of condensation must not be very large. If it were, we would expect the rate coefficients found by the lumped model to produce individual site concentrations consistent only with one value of i or none at all. The

Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 129

second implication of the improved coincidence of simulated and experimental concentrations with increasing i is that the hydrolysis substitution effect caused by hydrolysis is small for end groups and more connected sites and may shrink as the sites polymerize. The chemistry has proven to be such that the hydrolysis substitution effects are small enough that the predictions of the model are surprisingly consistent with the individual species concentrations for all species except monomers. We could also try allowing the hydrolysis rate coefficient to vary with j. However, this does not change the {ki,i′} trends observed and only makes a modest improvement in the fit beyond the first hour of simulation. The uncertainty in the rate coefficients found is too great to be able to interpret their values. Therefore, modeling at this level of detail for these systems is of no utility when the hydrolysis substitution effects on condensation are so small. Literature Cited Andrianov, K. A. Organic silicon compounds; State Science Publishing House for Chemical Literature: Moscow, 1955. Aris, R. “Almost discrete” Γ-distributed chemical species and reactions. Chem. Eng. Sci. 1994, 49, 581-588. Assink, R. A.; Kay, B. D. Sol-gel kinetics. III. Test of the statistical reaction model. J. Non-Cryst. Solids 1988, 107, 35-40. Assink, R. A.; Kay, B. D. The chemical kinetics of silicate solgels: functional group kinetics of tetraethoxysilane. Colloids Surf. A 1993, 74, 1-5. Bailey, J. K.; Mecartney, M. L.; Macosko, C. W. Modeling the gelation of silicon alkoxides. J. Non-Cryst. Solids 1990, 125, 208-223. Brinker, C. J.; Assink, R. A. Spinnability of silica sols. Structural and rheological criteria. J. Non-Cryst. Solids 1989, 111, 4854. Brinker, C. J.; Scherer, G. W. Sol-Gel Science; Academic Press: New York, 1990. Brunet, F.; Cabane, B. Populations of oligomers in sol-gel condensation. J. Non-Cryst. Solids 1993, 163, 211-225. Burden, R. L.; Faires, J. D. Numerical Analysis; Prindle, Weber & Schmidt: Boston, 1985. Dennis, J. E.; Schnabel, R. B. Numerical Methods for Unconstrained Optimization and Nonlinear Equations; Prentice-Hall: Englewood Cliffs, NJ, 1983. Devreux, F.; Boilot, J. P.; Chaput, F. Sol-gel condensation of rapidly hydrolyzed silicon alkoxides: A joint 29Si NMR and small-angle x-ray scattering study. Phys. Rev. A 1990, 41, 69016909. Dotson, N. A.; Galva´n, R.; Laurence, R. L.; Tirrell, M. Modeling of Polymerization Processes; VCH: New York, 1995. Doughty, D. H.; Assink, R. A.; Kay, B. D. Hydrolysis and Condensation Kinetics of Dimeric Sol-Gel Species by 29Si NMR Spectroscopy. Adv. Chem. 1990, 224, 241-250. Engelhardt, G.; Jancke, H. Concerning the 1H, 13C, and 29Si NMR chemical shifts of some linear, branched and cyclic methylsiloxanes. J. Organomet. Chem. 1971, 28, 293-300.

Huheey, J. E. The Electronegativity of Groups. J. Phys. Chem. 1965, 69, 3284-3291. Iler, R. The Chemistry of Silica; Wiley: New York, 1979. Kay, B. D.; Assink, R. A. Sol-gel kinetics. I. Functional groups kinetics. J. Non-Cryst. Solids 1988a, 99, 359-370. Kay, B. D.; Assink, R. A. Sol-gel kinetics. II. Chemical speciation modeling. J. Non-Cryst. Solids 1988b, 104, 112-122. Kelts, L. W.; Armstrong, N. J. A silicon-29 NMR study of the structural intermediates in low pH sol-gel reactions. J. Mater. Res. 1989, 4, 423-433. Marsmann, H. NMR, Basic Principles and Progress; SpringerVerlag: Berlin, 1981; p 65. Ng, L. V.; McCormick, A. V. In preparation. Ng, L. V.; Thompson, P.; Sanchez, J.; Macosko, C. W.; McCormick, A. V. Formation of Cage-like Intermediates from Non-random Cyclization during Acid-Catalyzed Sol-Gel Polymerization of Tetraethyl Orthosilicate. Macromolecules 1995, 28, 6471-6476. Peppas, N. A.; Scranton, A. B.; Tsou, A. H. Branching theory in sol-gel processing. In Better Ceramics through Chemistry III; Brinker, C. J., Clark, D. E., Ulrich, D. R., Eds.; Materials Research Society: Pittsburgh, PA, 1988; pp 43-47. Pouxviel, J. C.; Boilot, J. P. Kinetic simulations and mechanisms of the sol-gel polymerization. J. Non-Cryst. Solids 1987, 94, 374-386. Rankin, S. E.; Macosko, C. W.; McCormick, A. V. In preparation. Ro, J. C.; Chung, I. J. Sol-gel kinetics of tetraethylorthosilicate (TEOS) in acid catalyst. J. Non-Cryst. Solids 1989, 110, 2632. Sakka, S.; Tanaka, Y.; Kokubo, T. Hydrolysis and polycondensation of dimethyldiethoxysilane and methyltriethoxysilane as materials for the sol-gel process. J. Non-Cryst. Solids 1986, 82, 24-30. Sanchez, J.; McCormick, A. Kinetics and equilibrium of acidcatalyzed tetraethoxysilane hydrolysis. In Chemical Processing of Advanced Materials; Hench, L. L., West, J. K., Eds.; Wiley: New York, 1992a; pp 43-49. Sanchez, J.; McCormick, A. V. Kinetic and Thermodynamic Study of the Hydrolysis of Silicon Alkoxides in Acidic Alcohol Solutions. J. Phys. Chem. 1992b, 96, 8973-8979. Schmidt, H. New type of non-crystalline solids between inorganic and organic materials. J. Non-Cryst. Solids 1985, 73, 681-691. Schmidt, H.; Scholze, H.; Kaiser, A. Principles of hydrolysis and condensation reaction of alkoxysilanes. J. Non-Cryst. Solids 1984, 63, 1-11. Sugahara, Y.; Okada, S.; Kuroda, K.; Kato, C. 29Si-NMR study of hydrolysis and initial polycondensation processes of organoalkoxysilanes. I. Dimethyldiethoxysilane. J. Non-Cryst. Solids 1992, 139, 25-34. Sugahara, Y.; Okada, S.; Sato, S.; Kuroda, K.; Kato, C. 29Si-NMR study of hydrolysis and initial polycondensation processes of organoalkoxysilanes. II. Methyltriethoxysilane. J. Non-Cryst. Solids 1994, 167, 21-28.

Received for review April 14, 1995 Accepted August 28, 1995X IE950246Q

X Abstract published in Advance ACS Abstracts, November 15, 1995.