Brillouin scattering on porous silica aerogels reveals the onset of the phonon-to-fracton crossover in both the wave-vector and density dependence of the spectra. The results are consistent with a fractal description, giving both the fractal dimension D and the spectral dimension d. In the hypothesis that the samples form single infinite clusters, the numerical values suggest that scalar elasticity is significant in these materials.PACS numbers: 63.50. +X, 62.20.Dc, 82.70.Gg There is much interest in the vibrational spectrum of structures crossing over from self-similarity to translational invariance as the measurement scale increases past a characteristic length £. 1_4 The phenomenon is believed relevant to thermal properties of amorphous systems at low temperatures, 5 but no spectroscopy of vibrational crossover has so far been presented. This Letter reports Brillouin-scattering results on a series of porous silica aerogels. The measurements extend up to scatteringvector values k -l/£. As explained below, the frequency co of the longitudinal acoustic mode, and its half-width T, can be scaled with k and the aerogel densities p(£). The results are interpreted in terms of the Hausdorff dimension D and the fracton dimension d, and suggest that scalar elasticity is significant in those materials. 6,7 The gels were prepared by reaction of tetramethoxysilane with water in molar ratio 1:4. The main steps are as follows: 8,9 (1) hydrolysis of the alkoxysilane, (2) polycondensation of silocon-hydroxide groups up to gelification, (3) "curing" of that alcogel, and (4) hypercritical removal of the solvent to obtain the dried aerogel. The final density is adjusted by initial dilution of the reactants with methanol. The relative rates of reactions (1) and (2), which proceed simultaneously, are very pH sensitive. 9 This considerably affects the microstructure of the products, as can be easily recognized from the strong dependence of their turbidity on pH. This fact should be considered in a comparison of our results, obtained on neutrally reacted gels, with other investigations. 10 At pH =7, hydrolysis is very slow while the polycondensation reaction rate is high. 11 Hence, (2) is limited by the sequence of binary reactions (1), in which case one expects self-similar cluster growth, at least in the early stages when cluster diffusion plays no role. All our gels were prepared with identical reaction conditions and very long curing times. During curing, all remaining free clusters are expected to attach to the infinite cluster by further growth or diffusion. This produces a shrink-age of the originally formed gel, from which a buildup of internal stresses can be inferred. Drying above the solvent critical point prevents the formation of a meniscus, and preserves the delicate microstructure of the material.With these precautions, infinite cluster theories should apply, withwhere a is the grain size below which the system is not self-similar, and d =3 the space dimension. For the six gels used here, p(£) ranged from 103 to 407 ...
