Elastic properties of silica aerogels

Journal of Non-Crystalhe North-HoUand. Amslrrdam

ELASTIC

Solids 95 & 96 (I 987)

1197

PROPERTIES

* Laboratoire Universite ** I.B.M.

OF SILICA

J.

T. WOIGNIER,*

I I97 - I202

PELOUS,*

AEROGELS

PHALIPPOU,*

J.

de Science de Montpellier Zirich Research

R. VACHER,*

and

E. COURTENS**

Vitreux, UA 1119, Montpellier Cedex, CH-8803 RUschlikon,

des

Materiaux II, F-34060 Laboratory,

We have measured the elastic properties in the range 0.1 to 0.4 g/cm3. The ned from three-point flexion experiments, wed the determination of the elastic frequencies. The comparison of these shows that the samples exhibit large The density dependence of the Young's to obey the same power dependence po compressibility to the shear modulus treous silica. Percolation models are tic behaviour.

France. Switzerland.

of static

silica aerogels with densities Young's modulus E was obtaiBrillouin scattering alloconstants Cl1 and Cur at hypersonic frequency-dependent determinations scale homogeneity down to a few urn . modulus and elastic constants is shown with n = 3.8 . The ratio of the is similar to that found in bulk viused for the discussion of these elaswhile

1. INTRODUCTION Aerogels smaller of

are

similar

liquid

to

of

orders

the

up

under

to here

constants

in

As shown sity

below,

study

GHz

the

particular

to paper

light-scattering at

this

2.

SAMPLE

found

predictions

materials

the of

fractal Young's of

are

are

solids of

known

from

form

self-similar

C . They

offer

modulus,

and

the

small-

of

well-characterized scale

thus

systems. of

various

p

elas-

work

materials branches,

or

models,

The present these

dispersion

density

to

the

aerogels.

with

compared

structure

moduli

compact

objects.

moduli

acoustic

is

length

percolation

reporting

elimination

those

static

all

by

prepared

a series

for

for

that

the

obtained

determination

poroand

is

a

as well also

as

presented

conference2.

PREPARATION

The gels

used

tetramethoxysilane neutral

of

elastic of

It

of

for

Their those

conditions. that

elasticity

exponents

to

are

to

correlation

range,

values

q . The scaling

companion

the

density.

compared

Aerogels

measurements the

low

hypercritical

a density-dependent

We present tic

very

magnitude

experiments1

to

possibility

of of

composition. gels

neutron-scattering

structures

the

materials

two

chemical phase

angle

in

porous

by one

0022-3093/87/$03.50

(North-Holland

were

diluted

condition. solvent

AND MEASUREMENTS

presently

above

The its

in alcogels

critical

obtained methanol. were point

by

hydrolysis

The

reaction

transformed 3,4 . Drying

0 Elsevier Science Publishers Physics Publishing Division)

B.V

and took

polycondensation place

into

aerogels

under

normal

under

of initially

by evacuation pressure

produces

of

T . Woignier

1198

a collapse to

of

the

case.

the

surface

The

bulk in

from

the

at

of

the

the

starting

bulk

and

the

Elostic

and the

monolithic

density

tration

microstructure

tension

We prepared

et al. /

properties

the

formation

liquid-gas samples

of

aerogels

increases

solution.

The

skeletal

of silica

ocrogels

of

cracks.

This

interfaces

that

are

dimensions

150

x 8.5

with volume

densities.

the

of

x 8.5

latter

being

is

by

to

edges

8ars

70 mm. The

load

was

such

conditions,

In

Here

P

and

6

is

by

loading

gave

Elastic aerogels

FIGURE versus

Modulus

1 density

for

the

deduced

from in

ture. to

Brillouin-scattering

detail

in

would

break

load.

As explained obtain

the

in

the

variation

nal

acoustic

the

results

were

fitted

the

linear

region

of

companion of

dispersion

curves.

those

paper*,

several

q

necessary

wavevector

assuming curves

were

To account a sine gives

for

the

dispersion then

for

performed

at

scattering to

not

curvature

long-wavelength

elastic can

value

modube

setup

was

room

tempera-

the of

The

bars

necessary

angles

determine

be mea-

as those

the

experimental

results Young's

aerogels

the

frequency

releast

The

light

under

high

law. the

1.

manner

Values at

The

The measurements

at

could

this

unfor

values of

The

very

the

and values

Fig.

kg/m')

experiments.

elsewhere7.

in

sured

is

deformation

average

of

li

cribed

the

determinations. given

and

d

similar

I, < 100

silica

bar,

P . Loading

modulus

-b

load,

thickness

. The experimental

are ( Kgme3)

applied

the

three

the Es

. (1)

the

of

present

Density

the are

produced

Es

)

is !

width

span

in

by

and

the

by

applied

Es = * 4eV6

e

were

spaced

center.

.5'6 Es

three-

method. on

given

1.85

modulus

a standard

flexural

supported

calculated

close Young's

measured

point

is

that

mm3 . concen-

pores

The static is

related in

tetramethoxysilane

fraction the

is

present

these

were

des-

used

longitudicurves,

of

w(q)

limit

of

in the

T. Woignier

velocity.

The

also

plotted For

the

giving

elastic in

t-1 al. / Ehric

constant

Fig.

CII

heaviest

samples,

the

of

the

high-frequency also

3.

plotted

The

value

of

Fig.

1.

in

acoustic

VL

acoustic

elastic

constant.

4Ls)Cvr/(C,,

Ed

properties

the

similar

can

be

These

1199

by

CII

mode

was

Using

- C,,)

= p.

vi

also

observed,

the

,

calculated.

, is

The

relation

(2) corresponding

values

are

and

the

oxidation

properties

blish

laws

measurements

on

As shown elastic

in

moduli

tained

over

mogeneous ratio

Let

us

1,

our

static

and

to

the

same

scaling

at

least

of

bulk

to

what

extent

exists.

for

sites

p,

is

Near . This

compressibility,

isotropic equals the

and

force t usual different

to

constants

, that case

of in from

solids that

to

the

shear

rela-

for

condition of

the

attempts

to

esta-

origins,

or

from

determinations

be

are

samples

is

ob-

are

find

that

hothe

K/G = 1.4

described is

for

moduli

modulus.

p are

expected

equally

apply

by de conductivity. elastic

to

above

Assuming Gennes In problem conductivity.

in

terms

characterized

bonds

of

should

electrical

G

decimation

value

, the

the

urn . We also

modulus

the

values

,

silica.

or

the

of

similar that

a few

could

the

conjectured

the

different

That

shear

results

elastic

electrical 12,13 of

to

vitreous

relation

as

the

as

, the

the

the

? 0.2

variation Thus,

demonstrates

be occupied,

defined p,

for

v = pa

so that

o = 3.8

of

exponent.

models,

scaling

preparation

high-frequency

down

these

In discrete p

pc)T

that

the

Ed

and

hazardous.

range

K

to

found,

by

various

2 in

Ed 0: po

were

change.

samples

are

Fig.

close

of

a factor

to

a considerable

on

samples,

compressibility

both

density

lead

threshold

K = (P-

produce

measurements

densified

as

determination that

do

scales,

see

present

significant

large

probability

cluster

5OO'C

from

models.

tion

at

the

frequency

the

a = 1.26

noteworthy

large

very

colation

to is

a very at

of

a value

and

as much be fitted

previously

obeyed.

It

without

scaling

could

CL = 3.7

similar

by

results

approximately

samples.

elastic

the

is

aerogels were investigated 8,9 . For a set of samples

variations

Those values

1

values

non-densified

MHz region

indicated

The

o = 2a t

silica

the

densities.

respectively. tion

in

results

bulk

(pH)

of

techniques

origin8,

class

amgels

DISCUSSION

ultrasonic

is

to

transverse Cbl(

Ed = (3C11the

, related

o/silica

1.

a determination

are

properries

exist.

The

which to to

of

an scale the

percola-

infinite as Young

modulus, 10 , or

scalar elasticity 11 , the exponent tensorial

T

elasticity,

belongs

to This

per-

by the

which

a universality

leads

to

a new

1200

T. Woignier

er al. / Elasric

properries

of silica

acrogds

exponent

T = f

than

t

colation

model

(the

Swiss-cheese values

much

the

of

is

the

holes

and

in

colation

,

5x1o-2

1 -1

10

measurable the

variable model. be

q,-q

-

D = 2.40

that

Young silica

Elastic aerogels

the

porosity

of

of

percolation

rogel

with

the

approximation. can

infinite

be expected

lation

to

cluster.

fractal

also

that

the

ding

infinite

sity

at

wing

be

less

noted is

scales, with

of

object

is

an

more

the

"backbone"

which

indeed

as

the

clusters,

implies

the

much p

the

the

infinite

only

an

Fractal

in

l5 ae-

the an

the

infinite

gels perco-

cluster. than

Its

2.5

implicitely

percolation

characterized

d= 3.

of be

infinite

p

in some-

D= 2.50.

aging,

smaller

with

entire

is

thus long

than of

is of

fractal,

of

connected

found

than

is

can

because

value

is

identification

models

that,

identification

finite

which these

or

l6

a mass as

is

ramified

that not

percolation

view

D = 2.0,

as well

all

Proceeding

be

is

object

of

point

One such

dimension

should

cluster

Another

the

cluster

The latter An

is

smaller

dimension

the

should

dimension

scattering1

what

an

of It

that

Haussdorf

neutron FIGURE 2 modulus versus

noted

such with

percolation

the

the

a physical-

density,

appropriate

of

per-

parameter,

gel

first

5x10-l

of

ly

elasof

requires

identification

as

the

terms

first

frac-

f = ft5/2. of

results

q)?.

volume

The analysis

i-f,

moduli

K = (q,-

q

tic

other

elastic

areexpected14, tion

per-

so-called

model),

of

Here,

larger

. In a continuum

. It means

system,

inclu-

by an

average

den-

finds

the

follo-

one

has

D = 3 .

different

possible

assumptions,

one

: 1)

If

we assume

that

the P = (P-P,? . Using f = 3.7 + 0.2 for tensorial

the

gels

values

are

infinite

t = 1.8 elasticity

percolation + 0.2

12,13,

for together

clusters scalar with

elasticity, G = 0.45tO.05

or ,15

T . Woignier

one

obtains

tively. This be

for The

of

If

the

in

3.7

t 0.2

the

The

to

is

to

meter-

which

It

should can

structures21.

0.9

f 0.5 1,

one

f 0.2

which

lo.

could

We have

expects

. Such

, respec-

3.8

systems, large

f 0.1

P

as a pc

be

needed

to

the

also be

the

does

give

be

one

dis-

a larger

a large

value

6

would

obtains'

6/a

to

with

respect

also

percolation

the is

a

model

to

which

be

too

is

a mean

interpret

is

certainly for

dia-

of

tenuous

this

value

work

aerogels.

crude

results

further

describe

than these

particle

limit

to

which

might

fracton l8

neutron

stability

incorrect

an approximation but

the

the 4/3

mechanically

the a

+ 0.2.

in

which

. In conclusion,

p,

appropriate

only

description,

on

of to

ho-

range

elasticity for

, -where

seems

with

value

Extrapolating

estimates

it

be connected

establish

2, 1000

p

of

the

? = 3.4

closer

scalar

minimum . 2o

in

give

the is

to

fraction

are 2)

used,

support the

volume

(Fig.

exponent2

some that

the values

model

30 kg/m'

is

agreement

the

velocity

noted

prepared

with

predicted

experiments

of

from

q

the

, while

indeed

In this

of forward

these

connected,

fact

porosity

independently

excellent

qualitative

the

f 0.2

densities in

more

in

models,

calculated

l9

gels these

are is

we identify

that,

7

systems. stable

to

8.2 Fig.

sufficiently 17 . elsewhere

gels

percolation

< f < 6.2

fact

0.9,

in

were

details

backbone

and

observation applies

1201

oerogels

4 f 0.5

,

our

of silica

elasticity. if

continuum

dimension

to

elasticity

more

the

properries

E = or'6

tensions

in

Finally,

les

scalar

that

for

Elastic

close

internal

aspect

tensorial 3)

of is

we assume

6 , which

allow

that

if

these

2)

value

imply

case

cussed

exponent

former

might the

the

et al. /

will

Straigthuseful

for

quantitative

a

results.

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