Journal of Non-Crystalline Solids 95 & 96 (I 987) I I89 - I 196 North-Holland. Amsterdam 1189

RAMAN OBSERVATION OF FRACTONS IN SILICA-AEROGELS

A. BOUKENTER*, B. CHAMPAGNON*. J. DUMAS**, E. DUVAL*, J.F. QUINSDN*** and J. SERUGHETTI**

* Physico-Chimie des Materiaux Luminescents, VA 442 CNRS ** Departement de Physique des Materiaux, UA 172 CNRS *** Laboratoire de Chimie Appliquee et Genie Chimique, UA 417 CNRS UNIVERSITE LYON I. 43 Boulevard du 11 Novembre 1918 69622 VILLEURBANNE (FRANCE)

The low frequency Raman scattering shows that acid and base catalysed silica gels dried by hypercritical evacuation have a dynamical fractal behavior between 10 cm-l and 180 cm-I. The parameter * is, in a good approximation, only a function of the spectral dimension.

1. INTRDDUCTION

Fractals objects constitute self-similar structures with dilatation

synnnetryl meanning that the geometry features are invariant to scale change.

Fractal geometry provide a quantative measurement of randomness and then

permits characterization of systems such as colloidal aggregates... and

porous materials.

From a physical point of view self-similarity and fractal dimension cannot

be applied whatever is the scale2. Molecular or atomic dimensions and

macroscopic size of the sample are limitations. We have seen in the case

of gels3 that the concept of fractal applies to a much more restricted length

scale. But the replacement of the euclidian dimension d by the fractal

dimension D is not adequate to describe the dynamic of fractals. A new

dimension is required : the spectral dimension or fracton dimension d" that

govern some properties on a fractal structure4.

In this paper, we use low frequency Raman scattering to study random

materials such as acid and base catalysed silica gels treated by hypercritical

drying.

2. LOW FREQUENCY RAMAN SCATTERING FROM FRACTONS

In a real physical system at long length scale, the system has an euclidian

behavior and is described in term of phonons. On limited length scale the

system has an fractal behavior : vibrational excitations on the fractal,

are localized and are described in term of fractons. The crossover corresponds

to the transition from phonons to fractons5.

0022-3093/87/$03.50 0 Elsev~er Science Publishers B.V. (North-Holland Physics Publishing Division)

The Raman scattered intensity3 is proportional to the dielectric

susceptibility x :

'ij a Xi (0, r) xj (w, r)

The susceptibilities are linearly related to the elastic local strains induced

by the localized vibrations :

I. 1.j

= ek ( 0, f-1 el ( w, t-1 (2)

According to Alexander et alh, the local strain is proportionnal to the

gradient of the wave function + that can be written for fractons :

@,(~,P L) % (lUa )-D/2 exp I -4 (Lll,ll)d41 (3)

with A@ (t-1 s wq @(t-J (4)

where l,,,= is the localization length that plays the same role that the

wawelength for phonons.

D is the fractal dimension and d+ a geometrical exponent that describes

the localisation in real space.

q calculated from A+ is :

ad 9 =-

D (6)

Taking account of all fractons modes the scattered intensity becomes :

'ij ( w, ~ 02q-1 g ( ~3) I n (w) + 1 I

where g ( o ) is the fracton vibrational density of states described in a

fractal space7 by :

and n ( w) is the Dose factor.

A. Roukcnrer et al. / Fracrons in silica-aem&

The reduced intensity can then be written :

with %

“C - D

I 2 d+ +DI-1

1191

(9)

(10)

v can be deduced from experimental results.

Notice that when d" = D = 3. de = 1, Y = 4 correspond to an euclidian

behavior. Fractal material is associated with a lower value of v .

3. EXPERIMENTAL RESULTS AND DISCUSSION

The experimental configuration to observe Raman scattering from a sample

is conventional3. The incident light is emitted from an argon or krypton

laser. The various lines are used in order to avoid the luminescence which

appears in some samples. A &bin Yvon U 1000 spectrograph and a photon

counting system (photomultiplier RCA 31034 and a multichannel analyser)

are used to analyse the scattered light. The power of the beam laser varied

between 100 and 300 mW. Experiments are performed at room temperature.

Two series of porous silica are prepared :

A - The first (A) is prepared by acid-catalyzed hydrolysis and condensation

of silicon tetraethoxide (TEDS) in ethanol.

To control gel formation we have used an esterification reaction between

ethanol and acetic acid in order to provide water to the molecules of

tetraethoxide8 in a very controlled way. The components are added at room

temperature in the following order ethanol 3.50 ml. catalyst, TEOS 3.36 ml

and acetic acid 3.43 ml. The solution is kept at 50°C in closed containers

until the gelation point is attained.

The gel is dried by hypercritical runs using the low critical point of

carbon dioxide. The fluid phase inside the gel is replaced by acetone in

the pore, then the sample is immersed in liquid carbon dioxide and put in

hypercritical conditions. The gel is then dried by a very slow supercri-

tical expansion. The porous structure of gels is determined by thermoporo-

metry and electron microscopy 9% ID. The porosity seems to correspond to

very tightly connected cylindrical pores. The pores size distribution is

monodispersed around 6.7 nm. The total pore volume was determined to be

1600 mrn3/g of dried matter. The specific surface is 998 mZ/g. Figure la

1192 A. Boukenter et al. / Fracmrts in silica-orrogels

is a micrograph of a sample of type A which has been hypercritically dried.

Mechanical measurements and electron microscopy seem to indicate that a

macrostructure function of polymerisation conditionsll coexists with the

mesopores in the gel. The density of sample is 0.48 g/cmS. Density

measurements are done at ambiant conditions and include a contribution from

adsorbed water.

b

-60 -40 -20 0 20 40

Raman Shift Inn-‘)

FIGURE 1 : a) Electron transmission micrograph of gels A b) Stokes and antiStokes Raman Spectra at room temperature of

gels A. Horizontal linesindicatethe baselineforeachspectrum c) log-log plot of the raman reduced intensity as function

of " = 1.79

The low frequency Raman spectrum at room temperature is shows in figure

lb. The log-log plot of the reduced intensity for Stokes and antiStokes

low frequency scattering are shown on figure lc. The value of parameter

deduced from the measurements is v = 1.79.

B - The second serie (B) is prepared by base-catalyzed hydrolisis and

condensation of silicon methoxide in alcohol. The condensation leads to

a gel dried by the same hypercritical procedure to yield a nearly trans-

parent and fragile solid. The samples used were purchased from airglass

AB (Sweden). The density is 0.09 g/cm3 for this sample.

b -60 -40 -20 0 20 40 ,

Raman Shift km-‘I

FIGURE 2 : a) Electron transmission micrograph of gels B b) Stokes and antistokes Raman Spectra at room temperature of

gels B. Horizontal linesindicatethebaseline foreachspectrum c) log-log plot of the raman reduced intensity as function

of u = 1.69

Thermoporometry measurements performed on gels (B) give the following

information : the mean pore size, the total pore volume and the specific

surface are about 12.52 nm, 2328 nnn3/g of dried matter and 690 m2/g

respectively. Figure 2a is a micrograph of gel B which has been

hypercritically dried. Figure 2b shows the Stokes and antiStokes Raman spectra

at room temperature of this aerogels. On figure 2c. the reduced intensity

1194 A. Boukenrer et al. / Froc!ons in dim-oerogel

for Stokes and antiStokes low frequency scattering is shown. The reduced

intensity versus the frequency gives a straight line with a slope v = 1.69.

The values of Y obtained for the two series of gels Y = 1.79 and v = 1.69

are strongly different from the value v = 4 expected for a non fractal

material. The fractal behavior is observed between 10 cm-l and 180 cm-l.

The slope v depends on the spectral dimension 2, the fractal dimension D

and the geometric parameter d+ .

To deduce independently the value of D and d" we can use the suggestion

of Aharony et al5 to identify d+ with the exponent c which appears in the

expression of the resistance between two points for a resistor network :

the exponent depends upon d and D :

(2 -c?j D 5 =

i

from equation (lo), v is only function of the spectral dimensionality d"

” = 3 - i

Values of d for samples (A) and (6) are respectively 1.31 and 1.21. For

sample (B) the value of 1 is close to the value 4/3 predicted bytheAlexander-

Orbach conjecture in the case of a percolating network for 6>d>2. Note that

we can use the relation d"= 3 - v only in the case of constant ramification12

and d" = 4/3. The value found for sample (A) is % = 1.21, the difference

with values found for sample (B) can be explained by a slight difference

in the ramification related with structures and textures as suggested by

electron micrographs, figures la and 2b , obtained on the two dried gels.

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