Acoustic dispersion in silica aerogels observed by brillouin scattering : Evidence for phonon-fracton crossover

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

ACOUSTIC EVIDENCE

E.

1175 - 1180

1175

DISPERSION IN SILICA AEROGELS FOR PHDNON-FRACTON CROSSOVER

COURTENS,*

J.

PELOUS,**

J.

i I.B.M. ZUrich Research ** Laboratoire de Science Universite de Montpellier

OBSERVED

PHALIPPOU,**

BY BRILLOUIN

R. VACHER,**

SCATTERING

and

:

T. WOIGNIER**

Laboratory, CH-8803 RUschlikon, Switzerland. des Materiaux Vitreux, U.A. 1119 II, F-34060 Montpellier Cedex, France.

Small angle neutron scattering and Brillouin scattering experiments have been performed in porous silica aerogels with densities ranging from 0.1 to 0.4 gem-j. The results demonstrate the fractal structure of these materials. The fractal dimension is D = 2.40 c 0.02. The high dispersion observed was interpreted in terms of phonon to fracton crossover. From the scaling exponents, we deduced the spectral dimension $ = 1.25 f 0.12. The numerical value of these exponents raises the interesting question of whichis the proper elasticity theory to be applied to these weakly connected structures.

1. INTRODUCTION Among

models

of

extremely

interesting,

culations

relating

cular,

predictions

perties

of

this

paper,

a brief

(SANS)

study

evidence

given.

The

SAMPLES

can the

are

theoretical

cal-

properties. spectrum'

the

In parti-

and

be

were

thermal

be

7.

After

0022-3093/87/$03.50 Physics

pro-

our

to

the

in

terms

fracton

a small-angle

of

on of

the

neutron

It clarifies those

aerogels

investigation

length elementary

correlation

presented.

structure of

the

5 , the

in is

of

homogeneous.

obtained aerogels

character

interpreted

is same

a crossover

of

495 .

materials presented,

and

samples the

are

elastic

regime.

CONDITIONS.

in aging

0 Elsevier Publishing

while

fractal

for

dimension

becomes

results

possible

obtained

dissolved

built,

silica

self-similar

the

material

of

fractal

are

a is

is the

porous the

phonon

gels

methoxysilane

(North-Holland

summary

AND EXPERIMENTAL

silica

to

structure which

of

for

latter from

justed

geometry'

for

physical

vibrational

materials

a new Brillouin-scattering

properties

Our

the

as real

beyond

regarding

Definitive of

about

fractal

a basis

to

a < L < E, . Here fractal

limit

controversy

results

2.

as well

range the

the

scattering the

the

which

is

In

involving

provides structures

be made

model,

in

from

length,

can

those

self-similarity

scale-invariant

of

L

units

materials,

as

fractals.

A number scales

disordered

methanol during

by

hydrolysis 6.

The

and pH of

10 days,

Science Publishers Division)

the

B.V.

polycondensation

the humid

solution gels

of was

were

tetra-

initially dried

adunder

hypercritical were

conditions

performed,

400

The

SANS experiments Leon

were

explored

in

present

the

Various

densities

initial

ranging

paper

STRUCTURE

are

fractal

pair

dilutions

frdm

100

to

this

= b2c3

Here,

0

T(x)

0

I(q)

be

noted can

presentation

of the

for

in

I(q) this

structure. 0 = 2.40 increasing

0.3

aerogel

used.

The

slices

of spectra

of

experimental

the

experi-

results

performed

on

the

the

i-l

Brillouin

description

The

SANS given

multiplied

valid

in

intensity by

g(r)

was

crossover

was

to

several

I(q)

in

authors

l",ll.

calculated

the

on

reported

same

q-region

series

of

and

of

The

appro-

of

fractal

at

large

exp(-r/S).

qa < 1 , is (qc))

dimension,

case

basis

regime

function

where

the

the

homogeneous

by a cutoff the

(Haussdorf)

that,

when

our

1. that

A fit

is

The

then

.

p

density.

The

two

larger

Eq.

1 of

relative

belief,

is

.ll

This

(l)

the

mass

using

(1)

can

that the

of 100,

curves

extension

samples used,

density,

whose

and

it

for

all

is

The

density

densities is

behavior

the

signature samples fractals

curves

different

q . This

the

a

A detailed

in

the of

give law

difficulties

12.

of

a linear

magnitude

S/a

some

literature. elsewhere

silica-,

orders

a power

explain

the

five

sample

vitreous

with

published for

lightest

fits

in

being

I(q)

fits

from

results

bulk

than is

to

the of

connnon

< 10

results

For

more S/a

f 0.03.

E,/a

intensity

Fig.

case,

to

different

diffraction

SANS

5 % of over

contrary

angle

scattered

only

the

been

be appreciably

small

shown

were

sin((O-(;)-a;;;;n

especially

= q-O

'-i.e.

for

the

fractal

that

analysing

showing

experiment,

to

Laboand

qamma-function.

must

value

the

the

X

0.0018

A detailed

eT

is

is

this

diameter

the

wavelength

exoeriments.

function for

expression,

I(q)

the

range

new measurements SANS

already

expression

appropriate

It

has

To account

scales,

the

at

spectrometer7,

AEROGELS

correlation

geometry.

PACE

previously'.

from

expression

structures

priate

given

the

OF SILICA

For

spectrometerB.

for

theoretical

in

10 mm in

been

the

By varying

vectors

sin(e/2)).

a tandem

used

on

France).

and

has

also

The

(4n/X)

on

conditions

in

aerogels. dry

performed

scattering

thichness

analysed

mental

materials

0 (q=

2 mn in

were

are

porous with

(Saclay,

angle

about

were

Brillouin

scattering

in

obtain

aerogels

kg/m3.

ratoire

3.

to

yielding

llBkg/m" is

observed

indicates of

gives

that,

a fractal a value

decreases

with

The

1 118 102

‘0.

0. t

175 l --.., 1

0.

-

s r Gi z

0.

l . .......

220

0.

?

10-S

z

_

356

:

10-4

-

i 10-6

I

-

I

I

l *.0.

l.

'.

5 l . . . 0.

0.

l.

0. 0.

0.

0.01

0.001 SCATTERING

f

5

0.

-

increases

I(q)

in

3.

We also

treatment. the

usual

smooth

0.

that

only

the

After

oxidation

dependence

There structures. wavelength scales.

is

much "Normal" X

5DD"C,

is

In contrast,

than the

is

can

the

physical

E, , since

properthe

of

The

p

the

independen-

existence

data

the

of

in

the

that

the

di-

by

par-

slope

of

that

dimension

a slope

region the

suggests

affected

the

close

a further

close

to

to heat

-4.

This

for

the

scattering

by an

assembly

I(q)

is

unaffected

by the

heat

smooths

the

of

particles

their

the

without

propagate

medium

of treatchan-

aggregation.

spectrum

X ?r C , a transition

is

CROSSOVER

vibrational to

mu-

The ana-

constituent

samples, This

vaI(0)

'*.

fractal

density

intensity

a of

OF PHONON-FRACTON the

scale

samples.

the

mension

is

property,

of

of

for its

self-simito

shows

-3.

p in

This

ii-1

than

be expected

region

'*.

SANS

is the

and

self-similarity

heaviest

oxidation

in

L

required

of

p

samples

"mutual"

a series

of

that

expressing

as a function

a surface

of

of

q = 0.1

characteristic

interest

in

region

larity,

region

shows

individually

we call

we observe

part

set

as that

which

lysis

vathe

E, to

between

tual

this

OBSERVATION

larger

same

on

expected

that

phonons

the

riations

larger

at

law)

current

the

ties

with

structure

4.,BRILLOLlIN-SCATTERING

for

confirm

in

conclude

relating

sample

remarkable

, with'

D . This

whole

value density

length

value

meaningfully

the

is

function

a function

slope

It

the

as

remaining

mass-fractal

as

fractal

slightly

(Porod The

ries

the the

correlation ,1/W)

tly

structure,

found

We therefore the

is

the

each

o . For

give of

samples.

that

q

with

a fractal

particles.

ment. ging

region

have

the

also

0.1 VECTOR

slightly

this

particles

of

fits a function

relation

FIGURE 1 Scattered intensities I(q) of a smallangle neutron scattering on silica aerogels. Samples density are included. ticles

as

same

‘0. l* l. % l **.. f f 0. l. f 0. 5 f 5 l. l. l l . . .. . f f 0. .. f l* ., 0. l. l . f f 0. f l. f 5 f . #III t 1t-01 L

284

& 9

III,

‘0..

-

m

8

5

is

as

of long

homogeneous towards

such

fractal

as their at

such

a new regime

1178

FIGURE 2 Dispersion curves on silica aerogels. Densities of the are (in kg/m-") 0

356

+

313

4

284

.

260

v

220

A

201

w(q) samples

0 189 0 WAVE

is

expected.

The

fractons.

Since

with on

X

as

for

vature

of lo-'

branch

nine the

acoustic the

increases

demonstrated,

values the

are

with

a previous

an

Excellent linear crossover.

and

here

exponent

is at

The

is

remarkable

obtained

the

scat-

that

a curas

curvature

of

excellent

Brillouin

the

We have aerogel the

curvature

agreement

SANS results

v = w/q

with

12.

The

can

be scaled

scattering

velocity

small of

porosity. different

in

acoustic

light

wavevectors The

on

are

the

by

an analysis

from

present

(qC, << l),

for

study from

quantities

interpret limit

It

the

latter

data.

In

x ,

scaling regime

to

reached

increasing

scattering

values

these

be

distance. i.e.

crossover

2 shows

can

called

vibrations

phonon-fracton

observed

be extracted

The of

phonon

already

are

acoustic

Fig.

that

interatomic density,

can

branches.

used

is the

and

of

the

densities.

Brillouin 5

study

technique.

range

decreasing

localized,

the of

this

different

of

strongly

observation with

wavevector

determination

asymptotic

become

the

of

cm-’

allows

branches

D

acoustic

direct

i,

possible

inverse

in that

the

2000

the

with

x104

21r

should

samples

times

samples', of

in

k

scattering

be

obtained

tering,

the

as

should

branches

as

vibrations Brillouin

small

aerogels

VECTOR

175

larger

exponent

X

-=V

JL

'a

I Pa I

obtained

with

q

indicates

x

is

related

(2)

.

x = 1.55 the direct

? 0.20.

onset

of to

The the

the

departure

phonon fracton

from to

the

fracton

dimension

2

, (2)

1119 independently

of

the

elasticity

model,

by

x=(D-a,

(3)

7(3-O) Under

the

extract

assumption ?!

The

significance

5.

DISCUSSION

of

sion

is

of

close

mutual

that

city

is

much closer 2,14 , while

The

occurrence

of

various

tentative

ticity,

owing

tion.

Another

dence

of

is

the

infinite

the

in

4/3

an

to

large

is

that

and

effective

be

value.

It

by

it

Several

models

of

more

complicated

return

to

this

subject

to

that

the

the

of

p

scaling

in

later

not

account

be

15

given

scalar

for

the

p

obtain

that

elasprepara-

such

a large

v,

and

pa

on more

than

one

length

exist.

depen-

7 = 0.9

of

do already

to of

elasti-

during

to

unlikely

dependence

scalings

and value

elasticity

have

shrinkage

needed

depends

The

scalar

can

really

to

x

somewhat

the

value

gels

does value

to

tensorial

a

dimen-

mechanisms

applies usual

related

for

The effective

a scalar

that

seems

solely

possible

is

the

Finally,

able

is

with

t 0.03

(D = 2.5).

reaction

former

13 .

This

percolation

can .

elsewhere

D = 2.40

the

more

one * 0.12

self-similar.

regime.

determination

However,

produced

the

stresses x

in to

The

for

One

the

measured

could

0.9.

discussed

that

density

exponent

internal

oa

high

valid

independent, 3 = 1.25

tmtuaZZy

cluster

to

is

sample

been

is

are

related

the

than

latter

has

those

presumably

explanations,

v,

twice

effect

to the

are

We obtain

SANS experiment that

aggregation

(2)

dimension

the

is

in

velocities.

and of

self-similarity

cluster-cluster 7

of

pa

the

fracton

samples,

to

and

of

this

result

series

v,

a scaling

A remarkable large

that

from

We hope

. scale. to

be

publications.

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