Behaviour of colloidal silicas during uniaxial compaction

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Behaviour of colloidal silicas during uniaxial compaction

Françoise Ehrburger, Jacques Lahaye

To cite this version:

Behaviour of colloidal silicas

_during

uniaxial

_compaction

Françoise

Ehrburger

and

_{Jacques Lahaye}

Centre de Recherches sur la

Physico-Chimie

des Surfaces Solides, C.N.R.S., 24 avenue du Président

_Kennedy,

68200 Mulhouse, France

(Reçu

le 9 décembre 1988, révisé le 31

janvier

1989,

accepté

le 2

_{février 1989)}

Résume. 2014

La

_pression

P _{transmise par des échantillons de silice colloïdale,} lors d’un compactage

uniaxial à très faible vitesse, est mesurée en fonction de la fraction

_{volumique f}

du solide. Dans le

cas des silices

_{pyrogéniques,}

deux domaines sont mis en évidence. Un

_premier

domaine, limité à

f ~

0,065 est _{caractérisé par} un _comportement de type

percolation,

P ~

_{(f -}

_fc)k.

Pour les fractions

_volumiques

_plus

élevées

_{(deuxième domaine),}

P varie comme une

_puissance

du volume

poreux

_Vp,

P ~

(Vp)-03B1

avec 03B1 ~ 3. On _{observe que le} comportement de la silice de

_{précipitation}

est très différent des

_{précédentes.}

La courbe

_{P (f) peut}

_{être lissée par deux lois de}

_puissance

différentes

_{quand f}

est inférieur ou

_supérieur

à ~ _0,13. Un mécanisme de consolidation est

proposé

et la différence de _comportement des deux _types de silice

_pourrait

être attribuée à la différence des dimensions fractales des

agrégats

(D

2 ou D >

2).

Abstract. 2014

The pressure P transmitted

by

colloidal silica

_samples

during

low

speed

uniaxial

compaction,

is measured as a function of the volume fraction

_f

of solid. In the case of

_pyrogenic

silicas,

two domains are evidenced. A first one below

_{f ~}

_0.065,

is characterized

_by

a

_percolation

type behaviour P ~

_{(f -}

_fc)k.

For

_higher

volume fractions

_{(second domain)}

P scales as the void volume

_Vp,

P ~

_{(Vp)- 03B1}

_{with 03B1 ~ 3. It appears that the behaviour of the}

_precipitated

_{silica is very} different. The

_P(f)

curve can be fitted

_by

two _{different power laws below} or above

f ~

0.13. A mechanism of consolidation is

_proposed

and the difference of the behaviour observed for the two _types of silica could be attributed to the difference of the fractal dimension of the

aggregates (D 2 or D >

2).

Classification

Physics

Abstracts 62.20 - 82.70

Introduction.

The mechanical

_properties

of submicron solid

_particle

assemblies arise a

_great

deal of interest

since such

_systems

exhibit solid-like

_properties

even at _{very low volume} fraction

_f

of solid

( f

--

0.1 ).

Such

properties

are evidenced either in

_{compacted powder}

or

_pellets

or in the

_gel

state when the

_powder

is

_dispersed

in a

_liquid.

Recent papers

[1-4]

indicate that the elastic

modulus E of a

_compacted

submicron

_powder

follows a

_percolation

law E ’"

( f - f,)’.

The

percolation

threshold

_te

obtained for silver

_powder

beams

_[1]

is 0.062 and much lower for a

pyrogenic

(Cab-0-Sil)

silica

_(te

=

0.017) [2].

Similar results _were also obtained for

Aerogel

silicas

_{[3, 4].}

In every case the

_scaling

_exponent

T was

_larger

than the universal

_conductivity

exponent t

= 2.

We have shown

_[5]

that the

_gelification

threshold

_{f g}

for

_pyrogenic

silicas

_(Aerosil)

dispersed

in a non

_{polar liquid}

was also

_ranging

between 0.024 and

_0.05,

_depending

on the

morphology

of the silica.

Moreover,

it

_appeared

that the

_{intra-aggregate}

void

_volume,

and,

therefore the

_aggregate

volume

fraction 0

in the

_gel

at the

_gelification

_threshold,

was constant

and

_independent

of

_{f g}

_[6].

However,

the

_gelification

threshold of a

_given

silica was much

higher

(about twice)

when the silica was

_dispersed

in water which is

_strongly

adsorbed on the solid surface

_[7].

From these

_experimental

results,

it was concluded that the

_gelification

of a

liquid

occurs when a connected network of silica

_aggregates

is achieved in the

_{liquid and}

we

suggested

that it could be described

_by

the

_{percolation theory similarly}

to the

_gelification

of

polymers

[8].

Due to the narrow _{range of volume fraction of silica}

_allowing

the

_gelification

of

non

_polar

_liquids

_{(f max -- 2 /c)}

and to the low value of the shear modulus of the

_gel,

the

measurement of the mechanical

_properties

of the

_gel

is delicate. Therefore we have studied

the mechanical behaviour of the

_powder

_{(silica-air system)}

as a function of the solid volume fraction and assumed _{that it should be very similar} to that of the

_gel

_(silica-non

_polar

_liquid

system),

in the same volume fraction _range.

The method used for the

_present

_study

consists in the measurement of a force

(or pressure)

transmitted

_by

the

_{sample during}

uniaxial

_compaction

in

_quasi

_equilibrium

conditions. The interest of the method is to follow

_continuously

the variation of a mechanical

_property

(mechanical

strength

to

_compaction)

as a function of the volume fraction

_f

of solid.

Expérimental

section.

SILICA SAMPLES. - Aerosil

(Degussa)

is a fume silica obtained

by pyrohydrolysis

of

SiC14-TK900

_(Degussa)

is an arc silica and the

_sample

P 170 is a

_precipitated

silica

(laboratory

sample provided by

Rhône

_Poulenc).

_Precipitated

and

_pyrogenic

silicas are all

_{hydroxylated,}

although

their surface chemical

_properties

are different

_[9].

R805

_(Degussa)

is a

chemically

dehydroxylated

Aerosil 200. The BET surface area of the different

_samples

and their

_particle

size calculated from

_SBET

are

given

in table I.

All these silicas are

_light

and

_{fluffy powders.}

The solid volume fraction

_{f o}

determined from the

_weight

of 10 cm3 of as received

_powder

without

_tapping

in a

_glass

tube

_(diameter

1.3

cm)

is very small

(Tab. II).

It increases after

_tapping

to an

_equilibrium

value

_{f o}

t

(Tab. II).

Aerosil electron

_micrographs

show the well known chain-like structure. The fractal dimension of the Aerosil

_aggregates

is close to 1.75

_[10].

On the

_contrary,

the

_precipitated

silica P 170

_aggregates

_appear to be more

_compact

and

spherical.

This observation is in

Table I. -

Table II. - Values

of

the

_different

thresholds,

exponents

and limits

_of

the _power laws :

fo

and

_fot

_are the solid volume

_{fraction of}

the loose

_powder

determined

_from

the

_{weight of}

10

cm3

_{respectively before}

and

_after

_{tapping ;}

_{f g}

is the solid volume

_{fraction corresponding}

to

the

_gelification

threshold in a non

_polar

_{liquid ; fe}

is the mechanical

_percolation

threshold obtained in the

_present

_{experiment ;}

k is the exponent

of

the

_percolation

law and

f max

the limit

_{o f}

the

_{fit ;}

a is the

_exponent

o f the

_power law P versus

_Vp

or

f /

(1 - f ) ;

m is the

exponent

of

the _power law P versus

_{f in}

the second

_step

of

_compaction

(precipitated

silica).

agreement

with the fact that the fractal dimension of

_precipitated

silicas and

_aerogels

is

_larger

than 2

_[11-13].

The arc silica TK 900 appears as a mixture of chain-like

_aggregates

and

_larger,

spherical

isolated

_particles

with scattered sizes

_[14].

COMPACTION MEASUREMENTS. - The

_{fluffy powder}

was

_poured,

as

_received,

without

tapping,

in a

_compaction

die

_(diameter

20

_mm).

The

_compaction

measurements were

performed

in an INSTRON 1195 B

_apparatus.

The force transmitted

_by

the

_{sample during}

the slow

_displacement

of the crosshead

_{(0.1 mm/min)}

was measured

_by

a _{pressure gauge} connected to a

_graphical

recorder.

In order to scan the

_{largest pressure-volume}

fraction range, the measurements were

achieved in two different

_{steps, first,}

in the low _{pressure range} and

_second,

in the

_high

pressure range. As the

samples

were

_changed

for both series of

_experiments,

_{the pressure}

range was

_overlapping

over about two decades in order to check the

_continuity

of the

corresponding

curves.

In _{the low pressure range,} a

high

_sensitivity

_gauge

(0.5 kN)

was used. The lowest limit of

detection was close to

10-3

kN

_{corresponding}

to a _pressure of about 3 kPa for the die used.

The

_plunger

was fixed to the crosshead in order to avoid the

_{precompaction}

due to its

_weight,

corresponding

to a _{pressure of 4.3 kPa. The thickness of the}

_sample

was determined from the

position

of the crosshead. _{The pressure} was recorded as a function of the thickness on a

XY

recorder,

the zero thickness

_being

obtained from the

_empty

die. The

_weight

of the

_sample

was

_adjusted

in order to fill _up about the 3/4 of the die

_(about

3.5

_cm).

A 100 _{kN gauge} was used in the

_high

_{pressure range}

_(limit

of detection about 60

kPa).

The

displacement

of the

_plunger,

_placed

now over the

_powder,

was followed

_by

a Wallace

extensometer. As _{the range of the} extensometer was 1 _cm, the

_weight

of the

_sample

was such

that the

_height

of the

_powder

did not exceed 1 cm after the

_{precompaction}

_by

the

_plunger.

Corrections were introduced in the

_{displacement-pressure}

curve in order to take into account

The void volume

_Vp

or the solid volume fraction

_f

were obtained from the thickness of the

sample

at a

_given

_{pressure and from the} true

_density

of the silica

_particles

_(p

=

2.2

_g/cm3).

When frictions between the

_plunger

and the die walls occurred

_(due

to a

_misalignment

of the die with the

_plunger

or to silica

_sticking

on the

walls),

_they

induced oscillations in the pressure curve. Such effects were _easy to detect and the measurements were

_rejected.

_Thus,

in the

_following,

_{the measured transmitted pressure could be assumed} to result from the

powder only.

Expérimental

results.

The variation of the transmitted _{pressure P} as a function of the volume fraction

f

of solid is

_represented

in

_figures

la to Id in double

_logarithmic

coordinates. The curves are

associated two

_by

two in order to _{compare the} influence of the different

_{parameters :}

BET surface area

_{(Fig. la),}

_origin

of the silica

_(Figs.

lb and

_lc)

and surface chemical

_properties

(Fig. 1 d).

Figs. 1.

-

(a,

b, c,

d)

Variation of the pressure P as a function of the solid volume fraction

_{f for}

the different

_samples.

_{Symbols (0), ( EB)}

and

_{(+), (*) correspond respectively}

to the values obtained with the low and the

_high

_pressure

_measuring

_{system). a)}

OX 50 and A 300 : same

_origin

_{(fume silicas)}

but different BET surface area.

b)

A 150 and TK 900 : same BET surface area but different

_origin

(fume

Figures

la to Id show that for a

_given

_silica,

the low pressure and the

high

_pressure curves,

overlap fairly

well. The

_{reproducibility}

of the curves in both low and

high

_{pressure range is}

examined in

_figure

2 for Aerosil 200. The main

_{discrepancies}

_{appear in the very low pressure} range. It can be

_{mainly imputed}

to the lack of

_precision

of the XY recorder but also to the

powder

itself

_(presence

of

_{small lumps,}

differences in the

_{atmospheric humidity,}

_etc.).

The

reproducibility

of the results obtained for the other

_samples

_(two

or three

measurements)

is

very similar to that of Aerosil 200.

Fig.

2. -

Reproducibility

of the

_experimental

results P

_{(f )}

for Aerosil 200

_{«0), (x)}

_{low pressure}

measuring

system and

_{(+), (*)}

_high

_pressure

_measuring

_system).

1. ANALYSIS OF THE DATA IN THE LOW PRESSURE RANGE : PERCOLATION BEHAVIOUR.

-The

_shape

of the

_{P ( f )}

curves obtained for all

_pyrogenic

silica

_samples

_suggests

a

_possible

fit

by

a

_percolation

_{equation :}

As no

_hypothesis

can be made on the value of the three

_parameters

_Po,

_fc

and

k,

the values

_leading

to the best fit were obtained

_by

the SIMPLEX

_optimization

treatment. It appears that the best values of the three

parameters

and the

_quality

of the fit

_depend

on the range of volume fraction

f.

Now,

the

percolation

behaviour is

strictly

verified when

f - fc.

In order to _{determine the upper limit}

_{f max}

for the

fit,

the SIMPLEX

_procedure

was

repeated

for

_increasing

values of

_{f max}

above

_fc.

When

_fa.

was _very

_low,

the values of the

parameters

were

_scattered,

due to the

_experimental

errors on the low P values.

_Increasing

f max

led to almost constant values of the

_parameters

and an

increasing

_quality

of the

fit,

_up to

a limit above which the

_percolation

threshold started to decrease and the

_scaling

_exponent

to

increase. The values of the

_parameters

k,

Ic

and

_fmax

collected in table II

_correspond

to the smallest mean _square deviation between the

_experimental

and the calculated P values. Under these _{conditions the least square correlation coefficient} was

_always

_higher

than 0.999.

It can therefore be concluded that all

_pyrogenic

silicas

_investigated

exhibit a

_percolation

Figs.

3. -

(a,

b, c,

d)

Variation of the pressure P as a function of

_{f - fc’ (The}

values of

fe

and of the

_{exponent k}

are

_given

in Tab.

_II).

For the

_precipitated

silica

P170,

the SIMPLEX

_procedure

led to a zero

_percolation

threshold and a

_{high scaling}

_exponent.

One obtains p

_fk

with k = 5.83

(Fig. la).

The mechanical behaviour of the

_precipitated

silica is therefore different from that of the

pyrogenic

silicas.

2. ANALYSIS OF THE DATA IN THE HIGH PRESSURE REGION. - We have shown earlier

[15]

that,

above P 0.2

_MPa,

the void volume

_Vp

of Aerosil scales as

P - 1 / «

with eï -- 3. As

vsIV p

=

fl (1

f ),

where vs

= 1 Ip

= 0.455

cm3/g

is the true

specific

volume of the silica

particles,

the pressure P scales as

[fl (1 - f )]"

or

_(v,lVp)«

_(Figs.

4a to

_4d).

The values of the

exponent

a are

_given

in table II. The fact that a > k is consistent with the increase of the

exponent

of the

_{percolation regime}

observed

_above,

_by

_increasing

the _{range of} the fit in the

domain where

_{f «}

1.

Figs.

4. -

(a,

b, c,

d)

Variation of the pressure P as a function of

_{vs/V p}

_=//(!"

_{f ).}

Discussion.

The

_experimental

results indicate

_that,

for all

_silicas,

the evolution of the transmitted pressure P or the mechanical

_strength,

as a function of the volume fraction

_{f of}

_solids,

occurs

in two different

_{steps :}

- in the first

step,

one observes a

_percolation

_type

behaviour P -

_{( f -}

_te)k

for

_pyrogenic

silicas and a _{power law p _}

tk

for the

_precipitated

_{silica ;}

- in the second

step,

P scales as the void volume of the

_pyrogenic

silicas and as the volume

fraction of solid for the

_precipitated

silica.

The cross-over

_region

between both

_steps

is located around

_f

= 0.065 for

_pyrogenic

silicas

and around

_f

= 0.13 for the

precipitated

silica.

1. FIRST STEP : LOW PRESSURE RANGE. - From table

_II,

_{it appears that the mechanical}

percolation

threshold

_{te is}

_{very close} to the volume fraction

_{f o}

t in the

tapped powder.

Moreover,

for most

_samples,

it is also close to the

_gelification

threshold

_{f g}

in a non

_polar

- the mechanical

_percolation

threshold

_te

_corresponds

to the volume fraction of solid

above which a _{pressure is} transmitted

_by

connected units

_throughout

the

_{system ;}

- these units

are the fractal blobs

_(volume

fraction

_4»

evidenced in the

_gels

_{[16, 17]}

whose

structure was shown to remain constant

_[16]

over the

_gelification

_range.

It follows that the first

_step

of the

_compaction

of

_pyrogenic

silicas

_corresponds

to the

increase of the mechanical

_{strength by increasing}

the size of the backbone or the number of bonds

_transmitting

_{the pressure. The}

_percolation

_{process and the}

_gelification

_{possibility [5]}

ends when all blobs are connected.

It is

_likely

_that,

for a

_given

random

_system,

the mechanical

_percolation

threshold is

_equal

to

the elastic

_percolation

threshold. Forsman et al.

_[2]

have studied the elastic modulus of

pyrogenic

silica beams

_(the

silica was Cab-0-Sil M5

_expected

to _{be very similar} to

Aerosil 200).

They

obtained

_te

= 0.017 ± 0.002 and T = 2.9 ± 0.2 for volume fraction of silica

_ranging

between 0.035 and 0.3. Since _{the extension of the range of}

_fitting

leads to lower

te

values and

_higher

_exponents,

it becomes difficult to draw definite conclusion about the

following points :

- the

meaning

of the low

exponent

k - 1.55 obtained for the mechanical

_strength

as

compared

with

_elasticity

exponents

ranging

between 2 in some

_{polymer gels}

[8]

and 3.8 for sintered silver

_powder

beams

_{[1] ;}

- the difference between the

percolation

threshold for Cab-0-Sil M5 and for Aerosil 200. The values of the

_percolation

threshold are

_always

much smaller than the invariant

Oc

= 0.16 introduced

by

Scher and Zallen

[18].

It confirms the fact that the

percolating

units are not the individual silica

_particles

but

_aggregates

whose

_density

is much lower. The volume

fraction of the

_dispersed

solid is obtained from the mass M of solid

_by

unit volume and from its

_density

_{p S. The volume}

fraction 0

of clusters will be

_Mlpc

where pe is the

density

of the

cluster,

generally

unknown _{but easy} to measure in

_gels

_{[6, 7].}

Furthermore when the clusters

are

_fractal,

their

_density

_{Pc scales} as

(Lla )D -3

where D is the fractal dimension and

Lla

the size of the cluster in units of

_particle

diameter a. It follows :

As the correlation between the

_gelification

threshold and the structure of the

_aggregates

will be discussed

elsewhere,

the

_present

discussion will be limited to a few comments.

Recent

_developments

in continuum

_percolation

_{[19, 20]}

involves

_also,

as in

_equation

_(2),

the effect of excluded volume on the

_percolation

threshold.

_Moreover,

the

_experimental

system

considered here deals with reversible

_aggregation

of

_primary

_irreversible

aggre-gates

[17].

Therefore the size of the

_aggregates

will be limited

_by

_{gravitational}

forces

_{[21, 22]}

and will

_depend

on the inter

_aggregate

_sticking

forces in a

_given

medium. It follows that the

structure of the

_system

can also be considered from a sedimentation

_{[23, 24]}

_point

of view.

Such effect could

_explain

the difference observed between the

_gelification

threshold

(fg =

0.052)

and the mechanical

_percolation

threshold

_{(f, = 0.027)}

in

_{dehydroxylated}

Aerosil 200

_{(R 805)}

or in Aerosil OX 50 which

_particles

are

_larger.

These

_arguments

_{may also}

_explain

the different behaviour of the

_precipitated

silica. Let us

assume

_that,

due to the fact that the

_aggregates

are more

_compact

_{(D > 2 )}

and to differences

in surface

_chemistry,

the mechanical

_strength

of the

_{tapped powder}

_{is very low. It} follows that the structure of the

_{tapped powder}

_{( f o}

₌

_{0.036 )}

could

_collapse

to about

_{fo -}

0.05 under a

very low pressure, below the

sensitivity

of the

_experimental

device. It is also

_noteworthy

that the lowest pressure measured for this

_sample

_corresponds

to a volume fraction close to that of

the

_gelification

threshold in undecane

_{(Tab. II).}

The

_{signification}

of the

_scaling

_exponent

2. SECOND STEP OF THE MECHANICAL STRENGTHENING. - Above

_{f -}

_0.065,

_{for all} pyrogen-ic

_silicas,

the mechanical

_strength

scales with the void volume as

with a - 3

_{(Tab. II).}

It follows that P scales also with the size of the voids. For low

_{dimensionality}

fractal

aggregates

(D

:

2),

the size of the voids may be assumed to

correspond

to the size of the

aggregates.

On this

_assumption,

_{equation (3)}

would describe the correlation between the mechanical

_strength

and the size of the

_aggregates

and

_suggest

mechanisms of

_breaking

_[25]

and restructuration of

_aggregates

_[26]

_up to the limit of

_primary

irreversible

_aggregates.

Further work is now _{in progress} in order to modelize this mechanism and to

_explain

the value of the

_exponent.

For the

_precipitated

silica,

the pressure scales as

_{f m}

with m = 3.75 for

f >.

0.13. It is

interesting

to note that this

_exponent

_{is very close} to the

_Young

modulus

_{scaling exponent 13}

in the law E -

_{P /3}

where

p is

the

_apparent

_density

of the

_aerogel

and

0 = 3.8

(Woignier et

al.

_[4])

or 8

= 3.7

_{(Gronauer et}

al.

_[3]).

_{It may therefore be concluded that the elastic modulus} and

the mechanical

_strength

behave

_similarly.

3. MECHANICAL STRENGTHENING MECHANISM OF THE PRECIPITATED SILICA. -

Recently

Kendall et al.

_{[27, 28]}

have

_developped

a

_theory

of

_sphere

contact to calculate the

_Young

modulus E * of an

_assembly

of

_{spherical microparticles,}

in terms of the volume

_{packing 0}

of the

_particles,

the interfacial energy r and the elastic deformation of

spheres

of size

Dp

and modulus E. The

_{resulting equation}

is :

As seen above in the silica

_systems,

the mechanical units are not the individual silica

particles

(which

volume fraction is

_fi

but the fractal blobs

_(which

volume fraction is

_{cp), cp}

and

f being

related

_{by 0 -}

_{f (Lla )3 - D(Eq.}

_(2)).

In

_equation

_(4),

the diameter

_Dp

of the

_particles

arises from the interfacial _energy term

FD p 2

and the stress and strain

_expressions.

As the

_interacting

_objects

are the silica

_particles,

the term

_Dp

_(i.e.

the size

_Lla

of the

_blobs,

has to take into account the number of

_particle

Ns

on the extemal surface of the blobs. As

_Ns-

(Lla)D-I,

then

From

₍₂₎

and

_{(5), equation (4)}

becomes :

The

_experimental

results indicate that

_during

the first

step

of

_compaction

of the

_precipitated

silica : P -

_{E* -f5.83@}

in the second

_step

P - Eu

_{f 3v5}

whereas from

_equation

₍₆₎

a

_scaling

exponent

equal

to 4 is

_expected.

An increase of the

_exponent

can

_only

be obtained :

- if the size

L/a

of the fractal blobs is not constant but correlated with

_{f ;}

- and if it is

an

_increasing

function of

f,

i.e. if the size of the mechanical units increases

and

where n and N =

(L/a )D

are

_respectively

the number of blobs and the number of

_particles

in

a blob. Let us assume in a first

_{approximation,}

that the number of blobs

_by

unit

volume,

n/VT

remains constant. It follows

_{that 0}

scales as

(Lla)’

and

_{f scales}

as

(L/a)D.

On this

assumption,

equation

(6)

becomes :

or

For fractal blobs with D = 2 _one obtains :

and for

_compact

units

_(D

=

3 )

_one obtains :

The

_similarity

between

_b2

and

_b3

and the

_exponents

obtained

_respectively

for the first and the second

_step

of

_{compaction k}

= 5.83 and k =

3.75,

is

stricking.

Moreover,

the

experimen-tal value of the

_{exponent k}

= 5.83 can be obtained

_{by taking}

D = 2.03 instead of

D = 2 in

equation

(9).

Therefore the first

_step

of

_compaction

of the

_precipitated

silica P 170 can be described

_by

the

_following

mechanism : the increase of the mechanical

_strength

to

_compaction

is achieved

by

an increase of the size of the blobs

_{by sticking leading}

to new

_larger

units

_(with

mechanical intrinsic elastic modulus

_E).

The above process, achieved at the end of the first

_step

leads to

compact blobs,

(D

=

3 ).

However,

the

scaling

exponent

calculated with D = 3 in

equation

(9)

is

_{slightly larger}

than the

_experimental

value. A further refinement is obtained

_by

_assuming

that the number of blobs

_by

unit

volume,

n/VT

is no

_longer

constant but is a function of the

solid volume

_{fraction f :}

Therefore

_{L la -}

_{f (’ -}

h }/3 and

_{equation (6)}

becomes : E * ’"

_{[4 .}

_{[(h - l }/9.}

_Intuitively,

one

expects

that

_{by increasing}

the solid volume fraction

_f,

the number of

_{aggregates n}

decreases faster when

_VT

decreases. Therefore h will be

_negative.

For h = - 1 the

_scaling

_exponent

_{of f}

is

_equal

to

_3.78,

in

_good

_agreement

with the

_experimental

value k = 3.75.

(One

may observe

that,

if relation

₍₁₀₎

is introduced for the above determination of

_b2,

one obtains

b2

= 7.83 which is too

high.)

Therefore _{it appears that} the mechanism of

_{strengthening}

of

_precipitated

_{silica is very}

different from that of the

_pyrogenic

silicas. For the

first,

strengthening

occurs

_by

an increase

of the size of the mechanical

units,

whereas for the latter as seen

_above,

it

_probably

occurs

_by

restructuration

_{(i.e. breaking}

and diminution of the size of the

_aggregates).

The

_present

result is in

_agreement

with the fact that low fractal dimension

_aggregates

_(D

:

2 )

are tenuous

whereas

_aggregates

of

_larger

fractal dimension exhibit a much

_higher

mechanical

strength

[26].

Conclusion.

The measurements of uniaxial

_compaction

allowed to follow the evolution of the mechanical

properties

of an

_assembly

of colloidal silica

_particles

as a function of the volume fraction. The

percolation

threshold of

_pyrogenic

silicas. Similarities between results obtained from

compaction

and from elastic modulus measurements

_suggests

that both mechanical

_strength

and

_Young

modulus behaves

_similarly

with the volume fraction. One

_important

conclusion of the

_present

_{paper is that the very different} mechanical behaviour between

_pyrogenic

silicas and

_precipitated

silicas could be attributed to the fact that the fractal dimension of the first is

smaller than 2 whereas for the latter it close or

_larger

than 2. Further

experiment

on different

precipitated

silicas will _{be necessary} to

_generalize

this conclusion.

References

[1]

DEPTUCK D., HARRISON J. P. and ZAWADZKI P.,

Phys.

Rev. Lett. 54

₍₁₉₈₅₎

913.

[2]

FORSMAN J., HARRISON J. P. and RUTENBERG A., Can. J.

Phys.

65

₍₁₉₈₇₎

767.

[3]

GRONAUER M., KADUR A. and FRICKE J.,

Aerogels,

Ed. J. Fricke

_{(Springer-Verlag, Berlin)}

1986,

pp.167-173.

[4]

WOIGNIER T., PHALIPPOU J., SEMPERE R. and PELOUS J., J.

_Phys.

France 49

(1988)

289.

[5]

EHRBURGER F., GUERIN V. and LAHAYE J., Colloids

_Surf.

9

₍₁₉₈₄₎

371.

[6]

EHRBURGER F., GUERIN V. and LAHAYE J., Thermochim. Acta 77

₍₁₉₈₄₎

259.

[7]

EHRBURGER F., GUERIN V. and LAHAYE J., Colloids

_Surf.

14

₍₁₉₈₅₎

31.

[8]

JOUHIER B., ALLAIN _C., GAUTHIER-MANUEL B. and GUYON E., Percolation Structures and

Processes, Eds. G. Deutcher, R. Zallen and J. Adler, An. Israel

_Phys.

Soc. 5

₍₁₉₈₃₎

167-185.

[9]

KRAEHENBUEL F., STOECKLI H. F., EHRBURGER F. and LAHAYE J., Euchem

_Symposium,

Le

Bischenberg,

September

22-24, 1988.

[10]

FORREST D. R. and WITTEN T. A., Jr., J.

_{Phys. A}

12

₍₁₉₇₉₎

L109.

[11]

KEEFER K. D. and SCHAEFER D. W.,

Phys.

Rev. Lett. 56

₍₁₉₈₆₎

2376.

[12]

SCHAEFER D. W., MRS Bulletin, XIII

(1988)

22.

[13]

Dos SANTOS D. I., AEGERTER M. A., CRAIEVICH A. F., LOURS T. and ZARZYCKI J., J. Non

Cryst.

Solids 95/96

₍₁₉₈₇₎

1143.

[14]

BARBY D., Characterization of Powder _Surfaces, Eds. G. D. Parfitt and K. S. W.

_Sing

_(Academic

Press, London) 1976, pp. 353-425.

[15]

EHRBURGER F. and LAHAYE J., C.R. Acad. _Sci., Paris 303

_{(Sdr. II)}

₍₁₉₈₆₎

1367. ,

[16]

EHRBURGER F. and LAHAYE J., Colloids

_Surf.

23

₍₁₉₈₇₎

105.

[17]

EHRBURGER F. and JULLIEN R., Characterization of Porous

_Solids,

Eds. K. K.

_{Unger et}

al.

(Elsevier, Amsterdam) 1988,

pp. 441-449.

[18]

SCHER H. and ZALLEN R., J. Chem.

_Phys.

53

₍₁₉₇₀₎

3759.

[19]

BALBERG I., Philos.

Mag.

B 56

₍₁₉₈₇₎

991.

[20]

BALBERG I.,

Phys.

Rev. B 37

₍₁₉₈₈₎

2391.

[21]

KANTOR Y. and WITTEN T. A., J.

_Phys.

Lett. France 45

₍₁₉₈₄₎

L675.

[22]

KANTOR Y.,

Scaling

Phenomena in Disordered

_Systems

_(NATO

ASI Series B, Plenum New

York),

Eds. R.

_Pynn

and A.

_{Skjeltorp (1985)}

391-396.

[23]

ANSELL G. C. and DICKINSON E., J. Chem.

_Phys.

85

₍₁₉₈₆₎

4079.

[24]

ANSELL G. C. and DICKINSON E.,

Faraday

Discuss. Chem. Soc. 83

₍₁₉₈₇₎

167.

[25]

MILLS P., J.

_Phys.

Lett. France 46

₍₁₉₈₅₎

L301.

[26]

LINDSAY H. M., LIN M. Y., WEITZ D. A., SHENG P., CHEN Z., KLEIN R. and MEAKIN _P.,

Faraday

Discuss. Chem. Soc. 83

₍₁₉₈₇₎

153.

[27]

KENDALL K., ALFORD N. McN. and BIRCHALL D. J., Nature 325

₍₁₉₈₇₎

794.

[28]

KENDALL K., ALFORD N. Mc N. and BIRCHALL J. D., Proc. R. Soc. Lond. A 412

(1987)

269.