- Email: [email protected]
A low frequency Raman study of fractons in partially densified silica aerogels
Journal of Non-Crystalline Solids 225 Ž1998. 248–253
A low frequency Raman study of fractons in partially densified silica aerogels Eric Anglaret a,),1, Isabelle Beurroies a,2 , Laurent Duffours a , Claire Levelut a , Marie Foret a , Pierre Delord b, Thierry Woignier a , Jean Phalippou a , Jacques Pelous b
a
a Laboratoire des Verres, UniÕersite´ Montpellier II, Montpellier, France Groupe de Dynamique des Phases Condensees, ´ UniÕersite´ Montpellier II, Montpellier, France
Abstract We report on a low-frequency Raman scattering study of base-catalyzed silica aerogels ŽBCSA. and partially densified BCSA prepared by heat treatment ŽHT., hydrostatic pressure ŽHP. and uniaxial pressure ŽUP.. Low-frequency Raman susceptibility scales with frequency, x Ž v . ; vyf , within a large range of frequencies whose extent decreases with densification. Changes in the spectra are correlated with densification-induced changes in structure and connectivity of BCSA. f is a constant independent of density for all as-prepared samples, in agreement with their mutual self-similarity. f also remains essentially constant for HT samples while it decreases for HP and UP samples. f values are discussed in terms of fractal and spectral dimensions of the samples in the light of the theoretical predictions of Alexander, Courtens and Vacher ŽACV. and recent Brillouin scattering results. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Raman study; Aerogels; Hydrostatic pressure; Densification
1. Introduction Fracton dynamics has been the subject of many investigations since the works by Alexander and Orbach w1x. Many systems have been investigated w2–5x and many relevant experiments were performed, especially with silica aerogels w6–9x. Among others, incoherent light scattering data have been reported w7,8x but their interpretation is not straight-
)
Corresponding author. Tel.: q33-4 67 14 46 38; fax: q33-4 67 14 46 37; e-mail: [email protected]. 1 Present address: Groupe de Dynamique des Phases Condensees, ´ Universite´ Montpellier II, Montpellier, France. 2 ˚ Akademi UniPresent address: Physical Chemistry Dpt., Abo versity, Turku, Finland.
forward as it requires a hypothesis on the mechanism responsible for scattering. The relation between scaling behaviour of the Raman signal and the fractal and fracton dimensions has especially been a matter of controversy w3,10x. Low frequency Raman scattering study on basecatalyzed silica aerogels ŽBCSA. is reported. Their structure has been fully and quantitatively described by comparison between SANS measurements and DLCA simulations w11–13x. Furthermore, their mutual self-similarity ŽMSS. has been evidenced and their spectral dimension has been derived from Brillouin scattering w9x. To test separately the dependence of the Raman spectra on the structure and the connectivity of the samples, we extended our investigation to partially densified BCSA, prepared by heat
0022-3093r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 2 2 - 3 0 9 3 Ž 9 8 . 0 0 3 2 3 - 8
E. Anglaret et al.r Journal of Non-Crystalline Solids 225 (1998) 248–253
treatment ŽHT., hydrostatic pressure ŽHP. and uniaxial pressure ŽUP..
2. Theoretical background Alexander, Courtens and Vacher ŽACV. w10x showed that two mechanisms could be considered to describe the interaction light-fractal media. In the dipole-induced dipole mechanism ŽDID., each particle is submitted both to the external field and to a field induced by the polarization of the other particles. In the Pockels mechanism, the local polarisability is modulated by the stress induced by atomic vibrations. In both cases, ACV predicts that the Raman response follows a scaling law x Ž v . ; vyf where f is a function of three dimensions: the fractal dimension D describing the scaling of the mass, the spectral Žor fracton. dimension d˜ describing the DOS, and an additional dimension s describing the deformation of the particles by fractons Ž s has to be larger than one and s s 1 for fully connected structures w2x.:
˜ . Ž D y 3 y s . q 1, f DID s 2 Ž drD
Ž 1.
f Pockels s 2 Ž 1 y d˜srD . ,
Ž 2.
for DID and Pockels, respectively. Several groups have investigated the validity of the ACV predictions w3–5x. In each relevant case, whatever fractal system and scattering mechanism are considered, one ob-
249
serves that extended parts of the Raman susceptibility follow a scaling law. However, the relation between the scaling exponent and the three dimensions D, d˜ and s appears not to be straightforward. Even though values of s close to 1 Žin agreement with ACV predictions. have been derived from Eq. Ž3. in some numerical investigations, especially for percolators near threshold w2–4x, they were not found to be universal w3,5x. On an experimental point of view, data were not available to test these theoretical predictions. Present results confirm that correlations do exist between the exponent in Raman and the dimensions D, d˜ and s .
3. Experimental procedures Silica gels were prepared using 4 mol of 0.05 water–ammonia solution per mol of TMOS and various amounts of ethanol. After aging, they were dried above the critical point of the solvent. The resulting aerogels were oxidized in air at 5008C for 12 h. Some as-prepared ŽAP. samples were partially densified using three different techniques. The first was sintering by heat treatment ŽHT. over various periods of time at 10508C, as detailed in Ref. w14x. Other BCSA were submitted to various hydrostatic mercury pressures ŽHP. w15x. Finally, some aerogels cut in thin slides Ž1 to 3 mm thick. were densified under various uniaxial pressures ŽUP. applied homoge-
Fig. 1. SAXS data for a AP70 BCSA densified by HT Ža. and HP Žb.. The arrows indicate approximately the frequencies of the crossover between the scaling intermediate range and the Porod regime.
250
E. Anglaret et al.r Journal of Non-Crystalline Solids 225 (1998) 248–253
different interferometers thicknesses were used. At ‘high’ frequencies, we also performed some measurements with a double monochromator Sopra 2000 spectrometer in single pass.
Fig. 2. Raman susceptibility for a series of AP BCSA. Indicated slopes are fits to the scaling part of the curves. The filled arrows indicate the maximum of the high-frequency peak. Different symbols correspond to different interferometers thicknesses Žsee text..
neously and progressively on the surface Ž S , 1 cm2 . using a set of lead weights. Weights were taken off after 15 min. Residual deformation D ere was measured 24 h later. For each technique, a series was obtained from a single as-prepared sample and various degrees of densification. Samples are labelled with their final apparent density. SAXS measurements were performed on HT and HP samples with a Bonse–Hart camera using two Ge crystals as monochromators, leading to a resolution ˚ y1 . The beam collimation was qmin , 2 = 10y3 A semi-linear and deconvolution was achieved following the Strobl ¨ procedure for isotropic media w16x. We do not present spectra for UP samples since because of their possible anisotropy, measurements would require a punctual collimation and have not been performed. However, we assume that in the direction of the pressure application, their structure is at least as modified as that of HP samples. Low-frequency Raman measurements were performed on a Sandercock spectrometer using a tandem of plane Fabry–Perot ´ interferometers in 6-pass w7,8x. To cover a large range of frequencies, five
Fig. 3. Raman susceptibility for a AP70 BCSA densified by HT Ža. and HP Žb. and a AP88 BCSA densified by UP Žc.. Indicated slopes are fits to the scaling part of the curves. The filled arrows indicate the maximum of the high-frequency peak. Different symbols correspond to different interferometers thicknesses.
E. Anglaret et al.r Journal of Non-Crystalline Solids 225 (1998) 248–253
4. Results SAXS measurements on partially densified BCSA are presented in Fig. 1 and discussed in detail elsewhere w17x. For both HT and HP samples, the extension of the intermediate scaling ‘fractal’ regime w11x decreases with densification and the low q-limit shifts to high q-values, indicating a decrease of fractal persistence length. The high q-transition to the well-known Porod regime slightly shifts to the low q-values for HT aerogels, while no shift is observed in the HP case w17x. Fig. 2 displays the Raman susceptibilities x Ž v . for a series of BCSA of various densities. x Ž v . follows a scaling law within a large range of frequencies, as already observed for aerogels w7,8x. The scaling exponent does not depend significantly on the sample density and is in agreement with a previous measurement w8x. The low-frequency limit of this range shifts to high frequencies when the density increases. The shape and position of the peak observed in the high-frequency limit is almost the same for all as-prepared samples. In Fig. 3, we report Raman susceptibilities for densified aerogels pre-
251
pared by the three techniques HT, HP and UP. The main and general feature of the spectra is the existence of an extended scaling part even for the densest samples. For each series, its extension decreases with densification as expected. The low v-limit systematically shifts to high-frequencies and relates to the phonon–fracton transition frequency. For densest samples, a depolarized Brillouin peak can be observed Žits intensity lies around some hundredths of the polarized one., which may be due to the imperfection of our polarizer but may also mirror a partial depolarisation of the acoustic modes. Concerning the exponent f of the scaling law and its high v-limit, the results clearly depend on the densification technique. For the sintered aerogels, f remains essentially constant and the high-frequency peak is found to shift to low v-values with densification. The log–log representation is not well-designed to emphasize the high-frequency maximum. The interested reader will find a detailed presentation of similar results in Ref. w18x. In contrast, the shape and position of this peak are insensitive to the HP and UP densifications but one may note that f decreases slightly with densification for the HP and UP series.
Fig. 4. Values of the scaling exponent f for AP, HT, HP and UP samples. The two lines are guides to the eyes. The right ordinate scale ˜ Žopen triangles labelled UPBS. for UP samples are derived from Brillouin assumes the validity of Eq. Ž3. for our samples. Values of drD scattering and refer to this ordinate axis Žsee text..
E. Anglaret et al.r Journal of Non-Crystalline Solids 225 (1998) 248–253
252
The effect is very pronounced for the two highest density UP samples. f values derived from fits to the data are reported for each sample in Fig. 4. We point out that the results are relatively Žand unexpectedly. similar for HP and UP samples, in agreement with the weak anisotropy of UP samples suggested by Brillouin scattering results on the same samples w19x.
5. Discussion SAXS features are consistent with the structural changes expected for each densification technique: HT induces matter displacements at all scales including the particles scale whereas the effect of HP essentially affects the long-range structure of fractal aggregates. About the fractal character of the samples w11x, one notes that the intermediate scaling range does not change so that the fractal dimension, as long as it can still be defined, will be assumed as a constant in the density range investigated. Extension of the scaling laws in Raman frequency space correlates with what is known about the structure and confirms that the scaling behaviour of the Raman susceptibility should be discussed in terms of fractons. First, it is interesting to compare the position of the high-frequency Raman peak to a fractonparticles modes crossover frequency v co 2 which should check:
vco 2 vco
s
jac
ž / aac
Dr d
,
Ž 3.
where D is the fractal dimension, measured via SANS and determined from DLCA simulations w11x, d,˜ v co and jac are the spectral dimension, the phonon–fracton crossover frequency and the ‘fracton persistence length’, respectively, and aac ' a 0 the mean particle size assuming that fractons extend down to the particle scale. Using values from Refs. w9,11–13x, i.e., D s 1.72 and a 0 s 2.3 nm, one gets vco 2 s 8 cmy1 in correct agreement with the position of the peaks in Fig. 2. The shift of the Raman peak to low frequencies for sintered samples corresponds to an increase of the mean-size of the silica particles with HT w18x, in agreement with the shift to
low q of the high q-crossover in the SAXS spectra of Fig. 1. Scaling behaviours are systematically observed in the Raman spectra. Their extension, larger than those of the SAXS scaling laws, directly relate to the dispersion relation of fractons Žsee Ref. w9x and Eq. Ž3... The same scaling exponent is measured for all MSS samples, and this exponent depends on the structure of the materials since it is significatively different for BCSA and neutrally-reacted aerogels Ž f , 0.37 w7,8x.. Moreover, it appears that f varies for pressure-densified samples and hence depends on their connectivity and mechanical properties. One may also note that whatever scattering mechanism is considered Žsee part 2., f is expected to decrease when D decreases andror d˜ andror s increases. In a schematic picture, according to the SAXS data, the fractal dimension remains essentially constant during HT and HP in this density range. No SAXS data are available for UP samples but Brillouin scattering measurements w19x as well as our Raman data suggest that their structure is close to that of HP samples. For the same samples, the exponent f of the Raman scaling law is constant for HT Žat least rather close to that of as-prepared samples., and decreases for HP and UP ŽFig. 4.. In HP and UP cases, we deduce that d˜ andror s increase during densification. As s is not expected to increase Žexcept if an important number of breakages occur as observed in the first steps of densification for acid-catalyzed aerogels w20x but this is not the case for our UP samples w19x., we conclude that d˜ increases with densification, in agreement with a strengthening of the bonds at the particle scale leading to an increase of the DOS at high frequencies. For HT samples, no significative variation of f could be detected. This may mean that d˜ and s remain constant during heat treatment which is consistent since the connectivity is strengthened at all scales during sintering. A quantitative analysis can be achieved only if one independently measures at least two dimensions. This has been done for as-prepared samples: D , 1.75 w11x and d˜, 1.1 w9x, so that one can estimate the value of s from expressions 2 and 3. A negative value is found for the DID mechanism Žy0.30. and a value close to 1 for the Pockels mechanism. For UP sam˜ has been reported ples, an increase of the ratio drD w from Brillouin measurements 19x. Assuming that the
E. Anglaret et al.r Journal of Non-Crystalline Solids 225 (1998) 248–253
Pockels mechanism is responsible for Raman scattering in BCSA Žand that s s 1., these results are compared in Fig. 4 Žstars. to the Raman results. Even if the agreement is not perfect, the Brillouin mea˜ increases with surements confirm that the ratio drD UP densification. These results for AP and UP samples suggest that the scattering mechanism could be dominated by the Pockels effect but additional work, for example through simulations, will have to be conducted to confirm it.
6. Conclusion Extended scaling laws for the Raman susceptibility are observed for silica aerogels, including densified samples. Results correlate with SAXS data and with the microscopic changes expected for each densification technique. The position of the ‘highfrequency’ peak is close to the fracton-particle mode crossover frequency. The low-frequency scaling law limit corresponds to the phonon–fracton crossover frequency. The exponent of the scaling law is constant for the as-prepared series in agreement with MSS, and decreases for pressure-densified samples in agreement with an increase of the spectral dimension. The quantitative analysis of the scaling laws is not straightforward. It implies that one knows independently the value of the exponents D, d˜ and s and that one postulates a mechanism responsible for light scattering. However, we established with samples of various structures and connectivities that correlations
253
actually exist between the scaling exponent in Raman and the three dimensions characteristic of the fractal media. Moreover, the scaling predictions for the Pockels mechanism are found to be in correct agreement with our data.
References w1x S. Alexander, R. Orbach, J. Phys. Lett. ŽParis. 43 Ž1982. L625. w2x E. Stoll, M. Kolb, E. Courtens, Phys. Rev. Lett. 68 Ž1992. 2472. w3x G. Villiani et al., Phys. Rev. B 52 Ž1995. 3346. w4x T. Terao, T. Nakayama, Phys. Rev. B 53 Ž1996. R2918. w5x A. Rahmani et al., J. Phys.: Condens. Matter 8 Ž1996. 5555. w6x A. Bernasconi et al., Phys. Rev. B 45 Ž1992. 10363. w7x Y. Tsujimi et al., Phys. Rev. Lett. 60 Ž1988. 2757. w8x R. Vacher et al., Phys. Rev. Lett 65 Ž1990. 1008. w9x E. Anglaret et al., Europhys. Lett. 28 Ž1994. 591. w10x S. Alexander, E. Courtens, R. Vacher, Physica A 195 Ž1993. 286. w11x A. Hasmy et al., Phys. Rev. B 50 Ž1994. 6006. w12x A. Hasmy et al., J. Non-Cryst. Solids 186 Ž1995. 118. w13x E. Anglaret, thesis, Universite´ Montpellier II, 1995. w14x T. Woignier, J. Phalipppou, M. Prassas, J. Mater. Sci. 25 Ž1988. 3118. w15x L. Duffours, T. Woignier, J. Phalippou, J. Non-Cryst. Solids 186 Ž1995. 321. w16x G.R. Strobl, ¨ Acta Crystallogr. A 26 Ž1970. 367. w17x T. Woignier et al., in: Proc. 7th Conf. on the Structure of Non-Crystalline Solids, Caligari, 1997, J. Non-Cryst. Solids, to be published. w18x J. Pelous et al., J. Phys. France 51 Ž1990. 433. w19x C. Levelut, E. Anglaret, J. Pelous, these Proceedings, p. 272. w20x S. Calas, C. Levelut, T. Woignier, J. Pelous, these Proceedings, p. 244.
A low frequency Raman study of fractons in partially densified silica aerogels Eric Anglaret a,),1, Isabelle Beurroies a,2 , Laurent Duffours a , Claire Levelut a , Marie Foret a , Pierre Delord b, Thierry Woignier a , Jean Phalippou a , Jacques Pelous b
a
a Laboratoire des Verres, UniÕersite´ Montpellier II, Montpellier, France Groupe de Dynamique des Phases Condensees, ´ UniÕersite´ Montpellier II, Montpellier, France
Abstract We report on a low-frequency Raman scattering study of base-catalyzed silica aerogels ŽBCSA. and partially densified BCSA prepared by heat treatment ŽHT., hydrostatic pressure ŽHP. and uniaxial pressure ŽUP.. Low-frequency Raman susceptibility scales with frequency, x Ž v . ; vyf , within a large range of frequencies whose extent decreases with densification. Changes in the spectra are correlated with densification-induced changes in structure and connectivity of BCSA. f is a constant independent of density for all as-prepared samples, in agreement with their mutual self-similarity. f also remains essentially constant for HT samples while it decreases for HP and UP samples. f values are discussed in terms of fractal and spectral dimensions of the samples in the light of the theoretical predictions of Alexander, Courtens and Vacher ŽACV. and recent Brillouin scattering results. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Raman study; Aerogels; Hydrostatic pressure; Densification
1. Introduction Fracton dynamics has been the subject of many investigations since the works by Alexander and Orbach w1x. Many systems have been investigated w2–5x and many relevant experiments were performed, especially with silica aerogels w6–9x. Among others, incoherent light scattering data have been reported w7,8x but their interpretation is not straight-
)
Corresponding author. Tel.: q33-4 67 14 46 38; fax: q33-4 67 14 46 37; e-mail: [email protected]. 1 Present address: Groupe de Dynamique des Phases Condensees, ´ Universite´ Montpellier II, Montpellier, France. 2 ˚ Akademi UniPresent address: Physical Chemistry Dpt., Abo versity, Turku, Finland.
forward as it requires a hypothesis on the mechanism responsible for scattering. The relation between scaling behaviour of the Raman signal and the fractal and fracton dimensions has especially been a matter of controversy w3,10x. Low frequency Raman scattering study on basecatalyzed silica aerogels ŽBCSA. is reported. Their structure has been fully and quantitatively described by comparison between SANS measurements and DLCA simulations w11–13x. Furthermore, their mutual self-similarity ŽMSS. has been evidenced and their spectral dimension has been derived from Brillouin scattering w9x. To test separately the dependence of the Raman spectra on the structure and the connectivity of the samples, we extended our investigation to partially densified BCSA, prepared by heat
0022-3093r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 2 2 - 3 0 9 3 Ž 9 8 . 0 0 3 2 3 - 8
E. Anglaret et al.r Journal of Non-Crystalline Solids 225 (1998) 248–253
treatment ŽHT., hydrostatic pressure ŽHP. and uniaxial pressure ŽUP..
2. Theoretical background Alexander, Courtens and Vacher ŽACV. w10x showed that two mechanisms could be considered to describe the interaction light-fractal media. In the dipole-induced dipole mechanism ŽDID., each particle is submitted both to the external field and to a field induced by the polarization of the other particles. In the Pockels mechanism, the local polarisability is modulated by the stress induced by atomic vibrations. In both cases, ACV predicts that the Raman response follows a scaling law x Ž v . ; vyf where f is a function of three dimensions: the fractal dimension D describing the scaling of the mass, the spectral Žor fracton. dimension d˜ describing the DOS, and an additional dimension s describing the deformation of the particles by fractons Ž s has to be larger than one and s s 1 for fully connected structures w2x.:
˜ . Ž D y 3 y s . q 1, f DID s 2 Ž drD
Ž 1.
f Pockels s 2 Ž 1 y d˜srD . ,
Ž 2.
for DID and Pockels, respectively. Several groups have investigated the validity of the ACV predictions w3–5x. In each relevant case, whatever fractal system and scattering mechanism are considered, one ob-
249
serves that extended parts of the Raman susceptibility follow a scaling law. However, the relation between the scaling exponent and the three dimensions D, d˜ and s appears not to be straightforward. Even though values of s close to 1 Žin agreement with ACV predictions. have been derived from Eq. Ž3. in some numerical investigations, especially for percolators near threshold w2–4x, they were not found to be universal w3,5x. On an experimental point of view, data were not available to test these theoretical predictions. Present results confirm that correlations do exist between the exponent in Raman and the dimensions D, d˜ and s .
3. Experimental procedures Silica gels were prepared using 4 mol of 0.05 water–ammonia solution per mol of TMOS and various amounts of ethanol. After aging, they were dried above the critical point of the solvent. The resulting aerogels were oxidized in air at 5008C for 12 h. Some as-prepared ŽAP. samples were partially densified using three different techniques. The first was sintering by heat treatment ŽHT. over various periods of time at 10508C, as detailed in Ref. w14x. Other BCSA were submitted to various hydrostatic mercury pressures ŽHP. w15x. Finally, some aerogels cut in thin slides Ž1 to 3 mm thick. were densified under various uniaxial pressures ŽUP. applied homoge-
Fig. 1. SAXS data for a AP70 BCSA densified by HT Ža. and HP Žb.. The arrows indicate approximately the frequencies of the crossover between the scaling intermediate range and the Porod regime.
250
E. Anglaret et al.r Journal of Non-Crystalline Solids 225 (1998) 248–253
different interferometers thicknesses were used. At ‘high’ frequencies, we also performed some measurements with a double monochromator Sopra 2000 spectrometer in single pass.
Fig. 2. Raman susceptibility for a series of AP BCSA. Indicated slopes are fits to the scaling part of the curves. The filled arrows indicate the maximum of the high-frequency peak. Different symbols correspond to different interferometers thicknesses Žsee text..
neously and progressively on the surface Ž S , 1 cm2 . using a set of lead weights. Weights were taken off after 15 min. Residual deformation D ere was measured 24 h later. For each technique, a series was obtained from a single as-prepared sample and various degrees of densification. Samples are labelled with their final apparent density. SAXS measurements were performed on HT and HP samples with a Bonse–Hart camera using two Ge crystals as monochromators, leading to a resolution ˚ y1 . The beam collimation was qmin , 2 = 10y3 A semi-linear and deconvolution was achieved following the Strobl ¨ procedure for isotropic media w16x. We do not present spectra for UP samples since because of their possible anisotropy, measurements would require a punctual collimation and have not been performed. However, we assume that in the direction of the pressure application, their structure is at least as modified as that of HP samples. Low-frequency Raman measurements were performed on a Sandercock spectrometer using a tandem of plane Fabry–Perot ´ interferometers in 6-pass w7,8x. To cover a large range of frequencies, five
Fig. 3. Raman susceptibility for a AP70 BCSA densified by HT Ža. and HP Žb. and a AP88 BCSA densified by UP Žc.. Indicated slopes are fits to the scaling part of the curves. The filled arrows indicate the maximum of the high-frequency peak. Different symbols correspond to different interferometers thicknesses.
E. Anglaret et al.r Journal of Non-Crystalline Solids 225 (1998) 248–253
4. Results SAXS measurements on partially densified BCSA are presented in Fig. 1 and discussed in detail elsewhere w17x. For both HT and HP samples, the extension of the intermediate scaling ‘fractal’ regime w11x decreases with densification and the low q-limit shifts to high q-values, indicating a decrease of fractal persistence length. The high q-transition to the well-known Porod regime slightly shifts to the low q-values for HT aerogels, while no shift is observed in the HP case w17x. Fig. 2 displays the Raman susceptibilities x Ž v . for a series of BCSA of various densities. x Ž v . follows a scaling law within a large range of frequencies, as already observed for aerogels w7,8x. The scaling exponent does not depend significantly on the sample density and is in agreement with a previous measurement w8x. The low-frequency limit of this range shifts to high frequencies when the density increases. The shape and position of the peak observed in the high-frequency limit is almost the same for all as-prepared samples. In Fig. 3, we report Raman susceptibilities for densified aerogels pre-
251
pared by the three techniques HT, HP and UP. The main and general feature of the spectra is the existence of an extended scaling part even for the densest samples. For each series, its extension decreases with densification as expected. The low v-limit systematically shifts to high-frequencies and relates to the phonon–fracton transition frequency. For densest samples, a depolarized Brillouin peak can be observed Žits intensity lies around some hundredths of the polarized one., which may be due to the imperfection of our polarizer but may also mirror a partial depolarisation of the acoustic modes. Concerning the exponent f of the scaling law and its high v-limit, the results clearly depend on the densification technique. For the sintered aerogels, f remains essentially constant and the high-frequency peak is found to shift to low v-values with densification. The log–log representation is not well-designed to emphasize the high-frequency maximum. The interested reader will find a detailed presentation of similar results in Ref. w18x. In contrast, the shape and position of this peak are insensitive to the HP and UP densifications but one may note that f decreases slightly with densification for the HP and UP series.
Fig. 4. Values of the scaling exponent f for AP, HT, HP and UP samples. The two lines are guides to the eyes. The right ordinate scale ˜ Žopen triangles labelled UPBS. for UP samples are derived from Brillouin assumes the validity of Eq. Ž3. for our samples. Values of drD scattering and refer to this ordinate axis Žsee text..
E. Anglaret et al.r Journal of Non-Crystalline Solids 225 (1998) 248–253
252
The effect is very pronounced for the two highest density UP samples. f values derived from fits to the data are reported for each sample in Fig. 4. We point out that the results are relatively Žand unexpectedly. similar for HP and UP samples, in agreement with the weak anisotropy of UP samples suggested by Brillouin scattering results on the same samples w19x.
5. Discussion SAXS features are consistent with the structural changes expected for each densification technique: HT induces matter displacements at all scales including the particles scale whereas the effect of HP essentially affects the long-range structure of fractal aggregates. About the fractal character of the samples w11x, one notes that the intermediate scaling range does not change so that the fractal dimension, as long as it can still be defined, will be assumed as a constant in the density range investigated. Extension of the scaling laws in Raman frequency space correlates with what is known about the structure and confirms that the scaling behaviour of the Raman susceptibility should be discussed in terms of fractons. First, it is interesting to compare the position of the high-frequency Raman peak to a fractonparticles modes crossover frequency v co 2 which should check:
vco 2 vco
s
jac
ž / aac
Dr d
,
Ž 3.
where D is the fractal dimension, measured via SANS and determined from DLCA simulations w11x, d,˜ v co and jac are the spectral dimension, the phonon–fracton crossover frequency and the ‘fracton persistence length’, respectively, and aac ' a 0 the mean particle size assuming that fractons extend down to the particle scale. Using values from Refs. w9,11–13x, i.e., D s 1.72 and a 0 s 2.3 nm, one gets vco 2 s 8 cmy1 in correct agreement with the position of the peaks in Fig. 2. The shift of the Raman peak to low frequencies for sintered samples corresponds to an increase of the mean-size of the silica particles with HT w18x, in agreement with the shift to
low q of the high q-crossover in the SAXS spectra of Fig. 1. Scaling behaviours are systematically observed in the Raman spectra. Their extension, larger than those of the SAXS scaling laws, directly relate to the dispersion relation of fractons Žsee Ref. w9x and Eq. Ž3... The same scaling exponent is measured for all MSS samples, and this exponent depends on the structure of the materials since it is significatively different for BCSA and neutrally-reacted aerogels Ž f , 0.37 w7,8x.. Moreover, it appears that f varies for pressure-densified samples and hence depends on their connectivity and mechanical properties. One may also note that whatever scattering mechanism is considered Žsee part 2., f is expected to decrease when D decreases andror d˜ andror s increases. In a schematic picture, according to the SAXS data, the fractal dimension remains essentially constant during HT and HP in this density range. No SAXS data are available for UP samples but Brillouin scattering measurements w19x as well as our Raman data suggest that their structure is close to that of HP samples. For the same samples, the exponent f of the Raman scaling law is constant for HT Žat least rather close to that of as-prepared samples., and decreases for HP and UP ŽFig. 4.. In HP and UP cases, we deduce that d˜ andror s increase during densification. As s is not expected to increase Žexcept if an important number of breakages occur as observed in the first steps of densification for acid-catalyzed aerogels w20x but this is not the case for our UP samples w19x., we conclude that d˜ increases with densification, in agreement with a strengthening of the bonds at the particle scale leading to an increase of the DOS at high frequencies. For HT samples, no significative variation of f could be detected. This may mean that d˜ and s remain constant during heat treatment which is consistent since the connectivity is strengthened at all scales during sintering. A quantitative analysis can be achieved only if one independently measures at least two dimensions. This has been done for as-prepared samples: D , 1.75 w11x and d˜, 1.1 w9x, so that one can estimate the value of s from expressions 2 and 3. A negative value is found for the DID mechanism Žy0.30. and a value close to 1 for the Pockels mechanism. For UP sam˜ has been reported ples, an increase of the ratio drD w from Brillouin measurements 19x. Assuming that the
E. Anglaret et al.r Journal of Non-Crystalline Solids 225 (1998) 248–253
Pockels mechanism is responsible for Raman scattering in BCSA Žand that s s 1., these results are compared in Fig. 4 Žstars. to the Raman results. Even if the agreement is not perfect, the Brillouin mea˜ increases with surements confirm that the ratio drD UP densification. These results for AP and UP samples suggest that the scattering mechanism could be dominated by the Pockels effect but additional work, for example through simulations, will have to be conducted to confirm it.
6. Conclusion Extended scaling laws for the Raman susceptibility are observed for silica aerogels, including densified samples. Results correlate with SAXS data and with the microscopic changes expected for each densification technique. The position of the ‘highfrequency’ peak is close to the fracton-particle mode crossover frequency. The low-frequency scaling law limit corresponds to the phonon–fracton crossover frequency. The exponent of the scaling law is constant for the as-prepared series in agreement with MSS, and decreases for pressure-densified samples in agreement with an increase of the spectral dimension. The quantitative analysis of the scaling laws is not straightforward. It implies that one knows independently the value of the exponents D, d˜ and s and that one postulates a mechanism responsible for light scattering. However, we established with samples of various structures and connectivities that correlations
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actually exist between the scaling exponent in Raman and the three dimensions characteristic of the fractal media. Moreover, the scaling predictions for the Pockels mechanism are found to be in correct agreement with our data.
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