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29Si NMR and viscosity study of the sol-gel transition and evolution after the gel time
]OURNA L OF
Journal of Non-Crystalline Solids 147&148(1992) 686-689 North-Holland
NON-CRYSTALLIN SOLIDS E
29Si NMR and viscosity study of the sol-gel transition and evolution after the gel time L. M a l i e r a, F. D e v r e u x a, F. C h a p u t a, j . p . Boilot a a n d M . A . V . A x e l o s b a Laboratoire de Physique de la Mati~re CondensYe, Ecole Polytechnique, 91128 Palaiseau cddex, France b Laboratoire de Physico-Chimie des Macromol~cules, INRA, B.P. 527, Rue de la Gdraudi~re, 44026 Nantes cddex, France
In order to study the sol to gel transition and the early stage of aging, 29Si NMR and viscosity measurements were performed on acid-catalyzedTEOS in excess of water, where evaporation is hindered. NMR spectra evolution was followed over an interval running from 0.5tg to 4tg (tg is gelation time). The difference between static spectra and magic angle spinning (MAS) spectra, analyzed in terms of anisotropy of the chemical shift, allowed evaluation of the gel weight-fraction. Its evolution appeared to be quite slow. A kinetic model of post-gelation aggregation accounts for this time dependance. NMR measurements were not very accurate close to the transition. Rheologyexperiments over a frequency range from 10- 2 Hz to 100 Hz were performed in this region. Scalar percolation transition was observed within a very narrow interval.
1. Introduction Numerous nuclear magnetic resonance (NMR) experiments have been devoted to the study of silicon alkoxide reactions (hydrolysis as well as condensation), as a model of sol-gel phenomena [1]. Probably due to the need of specific techniques [2,3], only a very few of these focused on the late-gelation process, close to or beyond the sol-to-gel transition. The main goal of this study is to analyze N M R spectra in order to evaluate gel and sol weight fractions, the degree of condensation in the sol phase and the global one. The time dependance of these parameters will be an indication of the phenomena controlling the aggregation of the small clusters to the large one. From a chemical point of view, the gel weightfraction is a major indication of the completeness of the gelation. Moreover, it is also interesting for the physicists, for it represents the order parameter of the sol-to-gel transition.
2. Experimental The solution whose gelation is studied here was prepared in order to complete hydrolysis
before condensation takes place. This requires acidic conditions. The molar composition was TEOS : H 2 0 : ethanol = 1 : 10 : 6. Water containing gadolinium nitrate (Gd(NO3) 3) in concentration 10 .2 tool 1-1 was used; it fixed the pH at 4.3. Gadolinium paramagnetic ion was chosen because it enhances 29Si relaxation about four times as much as chromium. In order to avoid solvent evaporation which may affect the condensation, the sample containers were hermetically closed. 29Si N M R measurements were performed at 71.54 MHz, using a Brucker MSL 360 spectrometer and a Doty probe allowing magic angle spinning (MAS). During the condensation reactions, the replacement of an - O H neighbouring group by a siloxane bridge ( S i - O - S i ) leads, in 29Si NMR, to a chemical shift of - 9 . 2 _+ 0.2 ppm which permits one to distinguish between silicon atoms linked through 0, 1, 2, 3 or 4 siloxane bridges. These species are denoted, respectively, as Q0, Q1, Q2, Q3 and Q4. Thanks to the gadolinium paramagnetic ions relaxation times of every 29Si species Q" remain < 1 s all over the reaction, from the liquid state to the rigid-gel state. This permitted a repetition time of 3 s. Each spectrum, resulting from 2400 accumulations, lasted 2 h, which was short enough
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
L. Malier et al. / Sol-gel transition and evolution
$1atic ~ A ~
Static A ~
Static
_]~
':y \ MAS
.8r0 4~0
, .110 ~ -120 , ~f .gr0 .gt0 -100 , -ll0 r -12D , r -t00 Chemical Shift (ppml
Fig. 1. 29Si NMR spectra at different times. Thick lines are figuring experimental lincshapes, with or without magic angle spinning. Calculated lineshapes are presented by thin lines.
not to excessively average the kinetics (2 h represent about 1% of the gelation time, tg). M e a s u r e m e n t s were followed over an interval running from 0.5tg to 4tg. Before the gelation, liquid state N M R static spectrum allows distinction of the different species Q~, and evaluation of their relative contribution (relative concentration is denoted as q/). Beyond the time evolution of the q~, a relevant parameter to be investigated is the degree of condensation, c, defined as c = ~ i q i / 4 .
3. Results
As gelation takes place, NMR static spectra lines broaden slowly as shown in fig. 1, preventing any evaluation of the degree of condensation. This evolution occurs from the combination of two main effects. The molecular motion slows so drastically it becomes impossible to average the dipolar coupling with the protons and the anisotropic chemical shift. The rigid-lattice extension multiplies the possible neighbouring configurations of the silicon atoms, so it also spreads the isotropic chemical shift. While both these broadening causes affect static N M R measurements, MAS averages the former to zero [4]. Figure 1 compares static and MAS spectra at different
687
stages of the reaction. MAS spectra, changing only slightly in width, enable us to follow the time dependence of the different above-defined parameters. Even if dramatically broadened, static spectrum peaks are still apparent after gelation. This indicates that total gelation was not achieved at tg, some 29Si atoms still experiencing molecular motion at frequencies much larger than 150 Hz (linewidth), even far beyond tg. The sol-to-gel transition can be viewed as the occurrence of a large cluster, the gel, which ensures the rigidity of the whole system, while small aggregates remain in solution, forming a phase we call the sol. N M R measurements reveal the difference between solid-state behavior and liquid-like one. When observing the static spectra after the gelation time, one could think they roughly form a linear superposition of a final state spectrum, combined with a liquid state spectrum, whose relative contribution reduces with time. Such a decomposition, if allowable, would provide the proportions relative to the Si atoms embedded in a rigid lattice and to the ones showing a liquid-like motion. In order to proceed to this decomposition, respective references of the gel and sol signals were needed. We chose the long-time static spectrum at t = 9tg, which we checked to be steady state, as representative of a gel state (G(~0)). Doing so, we assumed the condensation rate within the gel, Cg, remains constant, which is consistent with the view of the gel as a rigid lattice. As mentioned above, the effect of magic angle spinning is to average to zero the broadening due to solid behavior. Thus, each peak 'i' provides the signal of Q/ species affected by a rapid motion (compared with 150 Hz), as is the case in the sol. For a better decomposition, we did not use simply the MAS spectrum as a single sol reference, but we separated signals, Si(~o), from the different Qi species and used them as a 'multiple sol-reference'. This allows one to vary the condensation rate in the sol, cs, independently of the global one. Contrary to what we had previously thought [5], Gaussian lineshapes rather poorly account for the long tails of these signals Si(w). As Gaussian absorption curves induced a discrepancy in the determination of the gel frac-
688
L. Malier et aL / Sol-gel transition and evolution
tion, a less restrictive hypothesis of symmetrical lineshapes was used, and MAS spectra was more faithfully exploited. Because of the slow broadening due to the isotropic part of the chemical shift, we had to perform rather frequent MAS measurements. These require a 3 kHz rotation of the sample, which certainly leads to uncontrollable perturbations of the system (at least centrifugation). We thus p r e p a r e d different containers filled with one unique freshly p r e p a r e d solution, and continuously recorded N M R spectra. Once a day, one MAS spectrum was performed, after which we changed the container. The recorded signal Y(w), as well as the references G(~o) and Si(o)), was normalized to unity. The experimental lineshapes were then decomposed by a least-square method, through minimization of the function, f , defined as f = E [Y(~o) - aG(oo) - b2S2(to)
--
b3S3((o)
~o
-- b 4 S 4 ( t o ) ] 2 ,
(1)
with regard to the parameters a and bi, whose time-dependence was then determined every 2 h. Figure 1 presents experimental and synthetic spectra at different times. Thanks to the initial normalization, the fitted p a r a m e t e r 'a' represents the massive fraction of the fixed phase, which we identified to the gel fraction, &g. The condensation rate in the sol, c s, equals Eibi/4Ebi, while the global one is
c(t) = ~bg(t)cg + (1 - Og(t))Cs(t ) .
(2)
The fig. 2(a) shows the evolution of c(t), obtained from the spectrum fit, which agrees, within experimental accuracy, with the direct MAS determination. This supports our choice of a constant gel reference. On fig. 2(b), one can see the increase of the gel fraction, q~g(t), the vicinity of the gelation point being detailed in insert.
4. D i s c u s s i o n
=0-86 l, . .,.. , 1
=
a)
0.84.
.
0.80
.
.
t = tg
"
1.00
,[
0.60
•~ 0.40
f,v,, "~
0.20 0.00
"" "
"~-...
0
200
400
600
800
t i m e (hours)
Fig. 2. (a) Degree of condensation, c(t), obtained from the spectrum fit (small dots), and MAS ones (large dots). (b) Gel weight-fraction qSg(t). The plain line presents the evolution described by eq. (4), with monodisperse conditions.
sensible discontinuity in the evolution of the condensation rate which remains linear. An important feature of the gel fraction is its rather slow evolution. At t = 2tg, about 20% of t h e silicon atoms are not connected to the gel. It shows sol-gel transition is far from meaning complete gelation, at least when solvents do not evaporate. During the interval from 0.5tg to 9tg, only a low number of new siloxane bonds are formed. The condensation rate increases from 80% to about 84%. From 1.Stg on, it almost seems to saturate, which indicates very slow kinetics. The increase of the gel fraction from zero looks linear, as described by the Flory-Stockmayer theory and contrary to the percolation predictions [1]. We tried to account for it through classical arguments. As has been shown by a preceding study [6], gelation begins after an aggregation phase which leads to highly functional units. In such a case, with the additional hypothesis of reaction rate between aggregates proportional to their mass, the Smoluchowski equations become
Onk/Ot = ½ E i+j=k
Let us first observe the results from a qualitative point of view. While the sol-to-gel transition is clearly visible on &g(t), it does not induce any
e.'* .~,;
ijnin~ - knk E J n j ,
(3)
j
where n k is the number of aggregates containing k Si atoms. If, after tg, one considers reactions between the gel and the sol as well as these
L. Malier et al. / Sol-gel transition and evolution
within the sol, the last summation term is the total mass and equals one. The solution is then given by [7] = 1 -
kn
(t = o )
k xexp[-k(t/tg)f)g(t/tg)].
(4)
Figure 2(b) presents in continuous line the evolution of the theoretical gel fraction in the case of initial monodispersity. The unique adjustable parameter is fixed at 185 h. Although giving rise to satisfactory fit, this monodisperse model is quite surprising on a physical point of view, for it assumes gelation starts at t = 0, while it is known to succeed an aggregation phase. With a more realistic small initial polydispersity and a shift on the timescale corresponding to the duration of the aggregation phase (about one third of tg, in agreement with ref. [6]), one obtains a better fit [8]. It shows that a classical description of the gelation with reaction rates (kernels) proportional to the mass of the aggregates is valid over a large range of time, if not in the very immediate vicinity of the gel point. The gelation time we determine with accuracy, as the moment at which ~bg increases from zero (about 185 h), is definitely lower than the time estimated from the macroscopic freezing of the solution (around 210 h). On the one hand, the macroscopic gel point is overestimated for it corresponds to the moment when the gel network is strong enough to support its own weight. On the other hand, our method counts as part of the gel the Si atoms which experience movement slower than 150 Hz. This should underestimate tg. In order to examine the discrepancy, we performed frequency-dependent viscosity measurements, in the immediate vicinity of the transition. The viscoelastic results, combined with the Stokes equation, indicated a radius of about 30 nm, beyond which aggregates appear static. Considering the starting aggregates of the gelation whose radius is a few nanometers, we checked the discrepancy on the determination of ~bg to be rather small because of the power law mass distribution. Moreover, being analogous to finite size effect, this estimate is strongly localized around tg. Since this might prevent a good observation of the transition, we then led complete viscosity experiment in
689
that range. As expected, intrinsic viscosity diverges and zero frequency elasticity modulus increases. Independently determined exponents s( ~ 0.73), t(~- 2), respectively associated with viscosity and elasticity, show a good agreement with scalar percolation [1,9]. These exponents can be detected over a very narrow range of _+1%tg, around the gelation point. This agrees with highly functional starting units [10]. It also justifies our inability to detect critical behavior through our N M R experiment.
5. Conclusion
Using N M R spectra, new method to determine gel mass fraction and follow condensation has been experienced. When isolated from evaporation phenomena, the condensation is slow. A kinetic model with reaction probability proportional to masses accounts for the evolution of the gel fraction. Concerning the transition, no critical behavior is sensible by NMR. Frequency-dependent viscoelasticity experiments showed a scalar percolation transition within a very restricted range.
References
[1] C.J. Brinker and G.W. Scherer, Sol-Gel Science. (Academic Press, New York, 1990). [2] W.G. Klemperer, V.V. Mainz, D.M. Millar, in: Better Ceramics Through Chemistry II, ed. C.J. Brinker, D.E. Clark and D.R. Ulrich (Materials Research Society, Pittsburgh, PA, 1986) p. 15. [3] A.J. Vega and G.W. Sherer, J. Non-Cryst. Solids 111 (1989) 153. [4] U. Haeberlen, in: High Resolution NMR in Solids. Selective Averaging (Academic Press, New York, 1976) p. 43. [5] L. Malier, F. Devreux, F. Chaput and J.P. Boilot, in: Proc. 5th Conf. on Ultrastructure Processing of Ceramics, Glasses, Composites, Ordered Polymers and Advanced Optical Materials, Feb. 1991, Orlando, FL (Wiley, New York, 1992) p. 59. [6] F. Devreux, J.P. Boilot, F. Chaput and A. Lecomte, Phys. Rev. A41 (1990) 6901. [7] R.M. Ziff, M.H. Ernst and E.M. Hendriks, J. Phys. A16 (1983) 2293. [8] L. Malier, J.P. Boilot, F. Chaput and F. Devreux, Phys. Rev. A (1992) in press. [9] H.H. Winter, Mater. Res. Soc. Bull. 16 (1991) 44. [10] P.G. de Gennes, J. Phys. (Paris) 38 (1977) L355.
Journal of Non-Crystalline Solids 147&148(1992) 686-689 North-Holland
NON-CRYSTALLIN SOLIDS E
29Si NMR and viscosity study of the sol-gel transition and evolution after the gel time L. M a l i e r a, F. D e v r e u x a, F. C h a p u t a, j . p . Boilot a a n d M . A . V . A x e l o s b a Laboratoire de Physique de la Mati~re CondensYe, Ecole Polytechnique, 91128 Palaiseau cddex, France b Laboratoire de Physico-Chimie des Macromol~cules, INRA, B.P. 527, Rue de la Gdraudi~re, 44026 Nantes cddex, France
In order to study the sol to gel transition and the early stage of aging, 29Si NMR and viscosity measurements were performed on acid-catalyzedTEOS in excess of water, where evaporation is hindered. NMR spectra evolution was followed over an interval running from 0.5tg to 4tg (tg is gelation time). The difference between static spectra and magic angle spinning (MAS) spectra, analyzed in terms of anisotropy of the chemical shift, allowed evaluation of the gel weight-fraction. Its evolution appeared to be quite slow. A kinetic model of post-gelation aggregation accounts for this time dependance. NMR measurements were not very accurate close to the transition. Rheologyexperiments over a frequency range from 10- 2 Hz to 100 Hz were performed in this region. Scalar percolation transition was observed within a very narrow interval.
1. Introduction Numerous nuclear magnetic resonance (NMR) experiments have been devoted to the study of silicon alkoxide reactions (hydrolysis as well as condensation), as a model of sol-gel phenomena [1]. Probably due to the need of specific techniques [2,3], only a very few of these focused on the late-gelation process, close to or beyond the sol-to-gel transition. The main goal of this study is to analyze N M R spectra in order to evaluate gel and sol weight fractions, the degree of condensation in the sol phase and the global one. The time dependance of these parameters will be an indication of the phenomena controlling the aggregation of the small clusters to the large one. From a chemical point of view, the gel weightfraction is a major indication of the completeness of the gelation. Moreover, it is also interesting for the physicists, for it represents the order parameter of the sol-to-gel transition.
2. Experimental The solution whose gelation is studied here was prepared in order to complete hydrolysis
before condensation takes place. This requires acidic conditions. The molar composition was TEOS : H 2 0 : ethanol = 1 : 10 : 6. Water containing gadolinium nitrate (Gd(NO3) 3) in concentration 10 .2 tool 1-1 was used; it fixed the pH at 4.3. Gadolinium paramagnetic ion was chosen because it enhances 29Si relaxation about four times as much as chromium. In order to avoid solvent evaporation which may affect the condensation, the sample containers were hermetically closed. 29Si N M R measurements were performed at 71.54 MHz, using a Brucker MSL 360 spectrometer and a Doty probe allowing magic angle spinning (MAS). During the condensation reactions, the replacement of an - O H neighbouring group by a siloxane bridge ( S i - O - S i ) leads, in 29Si NMR, to a chemical shift of - 9 . 2 _+ 0.2 ppm which permits one to distinguish between silicon atoms linked through 0, 1, 2, 3 or 4 siloxane bridges. These species are denoted, respectively, as Q0, Q1, Q2, Q3 and Q4. Thanks to the gadolinium paramagnetic ions relaxation times of every 29Si species Q" remain < 1 s all over the reaction, from the liquid state to the rigid-gel state. This permitted a repetition time of 3 s. Each spectrum, resulting from 2400 accumulations, lasted 2 h, which was short enough
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
L. Malier et al. / Sol-gel transition and evolution
$1atic ~ A ~
Static A ~
Static
_]~
':y \ MAS
.8r0 4~0
, .110 ~ -120 , ~f .gr0 .gt0 -100 , -ll0 r -12D , r -t00 Chemical Shift (ppml
Fig. 1. 29Si NMR spectra at different times. Thick lines are figuring experimental lincshapes, with or without magic angle spinning. Calculated lineshapes are presented by thin lines.
not to excessively average the kinetics (2 h represent about 1% of the gelation time, tg). M e a s u r e m e n t s were followed over an interval running from 0.5tg to 4tg. Before the gelation, liquid state N M R static spectrum allows distinction of the different species Q~, and evaluation of their relative contribution (relative concentration is denoted as q/). Beyond the time evolution of the q~, a relevant parameter to be investigated is the degree of condensation, c, defined as c = ~ i q i / 4 .
3. Results
As gelation takes place, NMR static spectra lines broaden slowly as shown in fig. 1, preventing any evaluation of the degree of condensation. This evolution occurs from the combination of two main effects. The molecular motion slows so drastically it becomes impossible to average the dipolar coupling with the protons and the anisotropic chemical shift. The rigid-lattice extension multiplies the possible neighbouring configurations of the silicon atoms, so it also spreads the isotropic chemical shift. While both these broadening causes affect static N M R measurements, MAS averages the former to zero [4]. Figure 1 compares static and MAS spectra at different
687
stages of the reaction. MAS spectra, changing only slightly in width, enable us to follow the time dependence of the different above-defined parameters. Even if dramatically broadened, static spectrum peaks are still apparent after gelation. This indicates that total gelation was not achieved at tg, some 29Si atoms still experiencing molecular motion at frequencies much larger than 150 Hz (linewidth), even far beyond tg. The sol-to-gel transition can be viewed as the occurrence of a large cluster, the gel, which ensures the rigidity of the whole system, while small aggregates remain in solution, forming a phase we call the sol. N M R measurements reveal the difference between solid-state behavior and liquid-like one. When observing the static spectra after the gelation time, one could think they roughly form a linear superposition of a final state spectrum, combined with a liquid state spectrum, whose relative contribution reduces with time. Such a decomposition, if allowable, would provide the proportions relative to the Si atoms embedded in a rigid lattice and to the ones showing a liquid-like motion. In order to proceed to this decomposition, respective references of the gel and sol signals were needed. We chose the long-time static spectrum at t = 9tg, which we checked to be steady state, as representative of a gel state (G(~0)). Doing so, we assumed the condensation rate within the gel, Cg, remains constant, which is consistent with the view of the gel as a rigid lattice. As mentioned above, the effect of magic angle spinning is to average to zero the broadening due to solid behavior. Thus, each peak 'i' provides the signal of Q/ species affected by a rapid motion (compared with 150 Hz), as is the case in the sol. For a better decomposition, we did not use simply the MAS spectrum as a single sol reference, but we separated signals, Si(~o), from the different Qi species and used them as a 'multiple sol-reference'. This allows one to vary the condensation rate in the sol, cs, independently of the global one. Contrary to what we had previously thought [5], Gaussian lineshapes rather poorly account for the long tails of these signals Si(w). As Gaussian absorption curves induced a discrepancy in the determination of the gel frac-
688
L. Malier et aL / Sol-gel transition and evolution
tion, a less restrictive hypothesis of symmetrical lineshapes was used, and MAS spectra was more faithfully exploited. Because of the slow broadening due to the isotropic part of the chemical shift, we had to perform rather frequent MAS measurements. These require a 3 kHz rotation of the sample, which certainly leads to uncontrollable perturbations of the system (at least centrifugation). We thus p r e p a r e d different containers filled with one unique freshly p r e p a r e d solution, and continuously recorded N M R spectra. Once a day, one MAS spectrum was performed, after which we changed the container. The recorded signal Y(w), as well as the references G(~o) and Si(o)), was normalized to unity. The experimental lineshapes were then decomposed by a least-square method, through minimization of the function, f , defined as f = E [Y(~o) - aG(oo) - b2S2(to)
--
b3S3((o)
~o
-- b 4 S 4 ( t o ) ] 2 ,
(1)
with regard to the parameters a and bi, whose time-dependence was then determined every 2 h. Figure 1 presents experimental and synthetic spectra at different times. Thanks to the initial normalization, the fitted p a r a m e t e r 'a' represents the massive fraction of the fixed phase, which we identified to the gel fraction, &g. The condensation rate in the sol, c s, equals Eibi/4Ebi, while the global one is
c(t) = ~bg(t)cg + (1 - Og(t))Cs(t ) .
(2)
The fig. 2(a) shows the evolution of c(t), obtained from the spectrum fit, which agrees, within experimental accuracy, with the direct MAS determination. This supports our choice of a constant gel reference. On fig. 2(b), one can see the increase of the gel fraction, q~g(t), the vicinity of the gelation point being detailed in insert.
4. D i s c u s s i o n
=0-86 l, . .,.. , 1
=
a)
0.84.
.
0.80
.
.
t = tg
"
1.00
,[
0.60
•~ 0.40
f,v,, "~
0.20 0.00
"" "
"~-...
0
200
400
600
800
t i m e (hours)
Fig. 2. (a) Degree of condensation, c(t), obtained from the spectrum fit (small dots), and MAS ones (large dots). (b) Gel weight-fraction qSg(t). The plain line presents the evolution described by eq. (4), with monodisperse conditions.
sensible discontinuity in the evolution of the condensation rate which remains linear. An important feature of the gel fraction is its rather slow evolution. At t = 2tg, about 20% of t h e silicon atoms are not connected to the gel. It shows sol-gel transition is far from meaning complete gelation, at least when solvents do not evaporate. During the interval from 0.5tg to 9tg, only a low number of new siloxane bonds are formed. The condensation rate increases from 80% to about 84%. From 1.Stg on, it almost seems to saturate, which indicates very slow kinetics. The increase of the gel fraction from zero looks linear, as described by the Flory-Stockmayer theory and contrary to the percolation predictions [1]. We tried to account for it through classical arguments. As has been shown by a preceding study [6], gelation begins after an aggregation phase which leads to highly functional units. In such a case, with the additional hypothesis of reaction rate between aggregates proportional to their mass, the Smoluchowski equations become
Onk/Ot = ½ E i+j=k
Let us first observe the results from a qualitative point of view. While the sol-to-gel transition is clearly visible on &g(t), it does not induce any
e.'* .~,;
ijnin~ - knk E J n j ,
(3)
j
where n k is the number of aggregates containing k Si atoms. If, after tg, one considers reactions between the gel and the sol as well as these
L. Malier et al. / Sol-gel transition and evolution
within the sol, the last summation term is the total mass and equals one. The solution is then given by [7] = 1 -
kn
(t = o )
k xexp[-k(t/tg)f)g(t/tg)].
(4)
Figure 2(b) presents in continuous line the evolution of the theoretical gel fraction in the case of initial monodispersity. The unique adjustable parameter is fixed at 185 h. Although giving rise to satisfactory fit, this monodisperse model is quite surprising on a physical point of view, for it assumes gelation starts at t = 0, while it is known to succeed an aggregation phase. With a more realistic small initial polydispersity and a shift on the timescale corresponding to the duration of the aggregation phase (about one third of tg, in agreement with ref. [6]), one obtains a better fit [8]. It shows that a classical description of the gelation with reaction rates (kernels) proportional to the mass of the aggregates is valid over a large range of time, if not in the very immediate vicinity of the gel point. The gelation time we determine with accuracy, as the moment at which ~bg increases from zero (about 185 h), is definitely lower than the time estimated from the macroscopic freezing of the solution (around 210 h). On the one hand, the macroscopic gel point is overestimated for it corresponds to the moment when the gel network is strong enough to support its own weight. On the other hand, our method counts as part of the gel the Si atoms which experience movement slower than 150 Hz. This should underestimate tg. In order to examine the discrepancy, we performed frequency-dependent viscosity measurements, in the immediate vicinity of the transition. The viscoelastic results, combined with the Stokes equation, indicated a radius of about 30 nm, beyond which aggregates appear static. Considering the starting aggregates of the gelation whose radius is a few nanometers, we checked the discrepancy on the determination of ~bg to be rather small because of the power law mass distribution. Moreover, being analogous to finite size effect, this estimate is strongly localized around tg. Since this might prevent a good observation of the transition, we then led complete viscosity experiment in
689
that range. As expected, intrinsic viscosity diverges and zero frequency elasticity modulus increases. Independently determined exponents s( ~ 0.73), t(~- 2), respectively associated with viscosity and elasticity, show a good agreement with scalar percolation [1,9]. These exponents can be detected over a very narrow range of _+1%tg, around the gelation point. This agrees with highly functional starting units [10]. It also justifies our inability to detect critical behavior through our N M R experiment.
5. Conclusion
Using N M R spectra, new method to determine gel mass fraction and follow condensation has been experienced. When isolated from evaporation phenomena, the condensation is slow. A kinetic model with reaction probability proportional to masses accounts for the evolution of the gel fraction. Concerning the transition, no critical behavior is sensible by NMR. Frequency-dependent viscoelasticity experiments showed a scalar percolation transition within a very restricted range.
References
[1] C.J. Brinker and G.W. Scherer, Sol-Gel Science. (Academic Press, New York, 1990). [2] W.G. Klemperer, V.V. Mainz, D.M. Millar, in: Better Ceramics Through Chemistry II, ed. C.J. Brinker, D.E. Clark and D.R. Ulrich (Materials Research Society, Pittsburgh, PA, 1986) p. 15. [3] A.J. Vega and G.W. Sherer, J. Non-Cryst. Solids 111 (1989) 153. [4] U. Haeberlen, in: High Resolution NMR in Solids. Selective Averaging (Academic Press, New York, 1976) p. 43. [5] L. Malier, F. Devreux, F. Chaput and J.P. Boilot, in: Proc. 5th Conf. on Ultrastructure Processing of Ceramics, Glasses, Composites, Ordered Polymers and Advanced Optical Materials, Feb. 1991, Orlando, FL (Wiley, New York, 1992) p. 59. [6] F. Devreux, J.P. Boilot, F. Chaput and A. Lecomte, Phys. Rev. A41 (1990) 6901. [7] R.M. Ziff, M.H. Ernst and E.M. Hendriks, J. Phys. A16 (1983) 2293. [8] L. Malier, J.P. Boilot, F. Chaput and F. Devreux, Phys. Rev. A (1992) in press. [9] H.H. Winter, Mater. Res. Soc. Bull. 16 (1991) 44. [10] P.G. de Gennes, J. Phys. (Paris) 38 (1977) L355.