Structure of percolating carbon black fractal aggregates dispersed in a polymer

(1)

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Structure of percolating carbon black fractal aggregates dispersed in a polymer

Laurence Salomé

To cite this version:

Laurence Salomé. Structure of percolating carbon black fractal aggregates dispersed in a polymer.

Journal de Physique II, EDP Sciences, 1993, 3 (11), pp.1647-1656. �10.1051/jp2:1993224�. �jpa-

00247931�

(2)

J. Phys. II IFance 3

(1993)

1647-1656 NOVEMBER1993, PAGE 1647

Classification Physics Abstracts

61.10 61.12 82.70

Structure of percolating carbon black fractal aggregates dispersed in

_a

polymer

Laurence Salomd

Centre de Recherche Paul Pascal, _avenue A. Schweitzer, 33600 Pessac, France

(Received

4 June 1993, accepted in final form 5 August

1993)

R4sum4. La structure d'anlas d'agr4gats primaires de Noirs de Carbone dispers4s dans

un polymbre est 4tudi4e _par diffusion de _neutrons _et _rayons X _aux petits angles. Pour des concentrations ~ croissantes _en charge, la dimension fractale diminue continilment depuis D ₌

1, 75 h ~

= 0, correspondant h la valeur _pour les agr4gats ^de d4part, h 1,5, dons le domaine 4tud14 h l'int4rieur duquel _on observe _une transition de percolation. Ce r4sultat suggbre _un m4canisme de collage _par les pointes conduisant h des objets plus t4nus _que l'agr4gation cluster-cluster

limit4e _par la diffusion.

Abstract. The structure of the clusters of primary carbon black aggregates dispersed in _a

polymer matrix is studied by small angle neutron and X-ray scattering. The fractal dimension is found _to decrease continuously for increasing concentration ~ of carbon black from D

= 1.75 at

~ = 0, corresponding to the value for the isolated primary aggregates, to 1.5 in the considered concentration range where _a percolation transition takes place. This result suggests _a "tip-to-tip"

sticking mechanism which leads to _more open ^structures than the diffusion-limited cluster-cluster aggregation model.

1 Introduction.

Besides their industrial interest in

practical applications

the carbon black filled

polymers

have

given

rise to fundamental studies

concerning

in

particular

the variation of the

conductivity

as a function of the filler _content. If it is _now established that the transition from _an insulator to _a conductor is well described

by

the

percolation

model

ill,

the

question

^of the

prediction

of the threshold

position

remains unsolved. In _a

geometrical

arguments since _we did not find _a correlation of the threshold

position

with the fractal dimension of the carbon black

primary

_aggregates _[2].

In

fact,

the materials considered here _are that of _a frozen _state of _an aggregates

suspension

_so

(3)

1648 JOURNAL DE PHYSIQUE II N°11

the

problem

is the _one of the

aggregation

of fractal clusters where the interactions

governing

the

sticking

mechanism _are to be determined.

In this paper we present ^SANS and SAXS results obtained with _two series of carbon black filled

polymers

with

particle

^volume concentration below and above the

percolation

threshold.

We found that the structure is fractal in the _same wavevector range as the

primary

_aggregates and the fractal dimension D decreases

continuously

for

increasing

carbon black volume fraction 4l from 1.75

(the

cluster-cluster model

value)

for 4l

= 0

corresponding

to the value for the isolated

primary

aggregates to 1.5 for the most concentrated

samples.

In addition to the evidence for _a relation between the

microscopic

structure and the

physical properties

of these materi-

als,

these results suggest ^that ^the

aggregates

^stick

together through

_a

"tip-to-tip"

mechanism which leads to _a more open structure than the diffusion limited cluster-cluster model. The main features of this interpretation are corroborated

by

the

preliminary

results of _a numerical

simulation.

Lastly,

_we discuss the

origin

of the rise of the scattered

intensity

observed at lower values of the wavevector for the most concentrated

samples.

2

Experimental.

2. I SAMPLE PREPARATION AND CHARACTERISTICS. Two sets of

samples

_were

prepared

with two different carbon blacks

(Raven

2000 and 7000 from the Columbian Carbon Inter-

national).

The carbon black

powder

consists of so-called

"primary"

aggregates ^of

elementary particles.

We have shown _[2]

by

_an

analysis

of TEM

images

^{of these} aggregates ^that

they

have

a fractal structure. The fractal dimension determined from the

slope

of the

log-log plot

of the measured

projected

surface _area of the aggregates as a function of their _mean size for _a hundred of them has been found close to the cluster-cluster

aggregation

model

prediction

in three di-

mensions: D

= 1.75 [3] ^as it could be

expected

for the

growth

mechanism of the carbon blacks

(see

the values for each carbon black in Tab.

I).

The

primary

_aggregates _are not

monodisperse,

the size distribution of the aggregates ^considered ^for ^the determination of D is

represented

in

figure I;

it is

noteworthy

to remark that it is

"bell-shaped"

_so _our

preceding

determination of D is not biased. The _mean size of the aggregates is close to 1000

I

_while _the _diameter _of _the

elementary particles

is about 100

I.

_Let

us notice the _poor

anisotropy

of the aggregates, the aspect ^ratio ^is

equal

to 1.3. Before _use the carbon black _was dried under _vacuum at 250 °C for 3 h in order to _remove the residual additional

products

used to

keep

the carbon black

powder

less

fluffy

and _more compact ^for the transport ^and

manipulation

convenience.

Table I. IYactal dimensions determined

by

TEM

image analysis,

volume fraction 4l at the

percolation

threshold in the

composites

and 4l range studied in the

scattering experiments

of the two carbon blacks used.

Carbon Black DIEM

4lc($lvol) 4lrange(%vol)

Raven 2000 1.67 2.8 _{1 <} 4l _< 6

Raven 7000 1.78 3.8 1 < 4l _< 8

The

polymer

matrix is _an epoxy obtained

by

the

polycondensation

of the

Diglycidyl

ether of

Bisphenol

F

(Araldite

XPY 306 from Ciba-

Geigy) by

_a diamine

(1,12-Diamino-4,9-dioxadodecan

from

Aldrich).

We have checked that the

viscosity

of the

prepolymer

is

nearly

constant

during

the

preparation

and does not exceed 50 _cp.

(4)

N°11 STRUCTURE OF PERCOLATING FRACTAL AGGREGATES 1649

is

.

Raven 2lK©

1

Raven 7lK©

#l~ IO

~

li

©

)

~

ean

ize (nm)

Fig. I. Mean size distribution of the carbon black primary aggregates.

The

polymer

filled

samples

_are

prepared

with the

following procedure during

which the tem-

perature ând ^the ^duration ôf êach step _are controlled: after the

mixing

of the

constituents,

the carbon black

powder

is

introduced,

_a

magnetic

stirrer

agitation

leads to _a

homogeneous

suspension

which is

outgassed

and at last sonicated in order to

disagglomerate

and

disperse randomly

the carbon black aggregates. ^The

curing

realized in teflon molds is initiated

just

after the last step

by placing

the

samples

at l10 °C. After five hours the

polymerization

is

completed

and the

samples

consist in solid

cylinders

which _can be cut to the desired dimensions for

scattering experiments,

^electrical

conductivity

measurements _or transmission electron

microscope imaging.

^The concentration of

particles

4l

expressed

in volume fraction _ranges from 0.I _to 7 $l

covering

the domain where the

samples

exhibit _an insulator to conductor transition, I-e-

overlapping

the

percolation

threshold 4l*. The carbon black is almost twice _as dense _as the

polymer

thus _we checked

by

direct observation of thin slices of the composites, ⁱⁿ

particular

for the dilute

samples,

that there is _no sedimentation. The

good quality

of the

dispersion

_was

also verified

by

TEM

imaging.

2.2 SCATTERING EXPERIMENTS. The SANS

experiments

_were

performed

at the spec-

trometer PAXE of the

Orph4e

reactor of the Laboratoire L40n Brillouin

(CEA-CNRS

_common

laboratory).

The neutron beam

issuing

from _a cold _neutron

guide

tube is monochromated

by

a mechanical

velocity

filter

giving

_a

wavelength

width Al

Ii

_~- 0.08; the detector consists of 40 _x 40 active cm2 cells. Two distinct

wavelengths

of the

incoming

beam _were chosen _to _scan the

largest

part ^of the

interesting

wavevector

Q

domain. In the first

configuration,

1 ₌ 20

I,

the collimation and

sample

detector distances _were 4.8 and 5 meters,

respectively.

Under these conditions the covered wavevector range is 2 x10~~ <

Q

< 2.8 _x 10~2

i~~.

_In _the _second

one, 1

= 7

I,

_both collimation and

sample

detector distances _are

equal

_to 2.5 _m and the

Q

range is I.I _x 10~~ <

Q

< 1.3 _x 10~~

i~~;

_thus

_partially overlapping

^the ^first one.

The

investigated samples

consisted of thin slices

(0.5

_mm

width)

of the filled

polymers

elaborated _as described above. The values of the measured transmission coefficients _were

sufficiently high (around 0.7)

to _ensure that _no

multiple

scattering ^could ^take

place

in the

samples.

Data _were corrected for noise and for

background by subtracting

a pure

polymer

(5)

1650 JOtmNAL DE PHYSIQUE II N°11

matrix

sample.

The scale factor between the _two covered

Q

_range is determined

simply by adjusting

the

points

in the

overlapping region.

The SAXS

experiments

_were carried _out _on the

high

resolution

X-ray

_set-up at the CRAP in Pessac. The _source is _a

Iligaku rotating

anode

(Cu:

I

= 1.54

I,18 kW).

We used _a

triple-

bounce Ge

(ill)

monochromator and _a

triple-bounce

Ge

(ill) analyzer

in the

nondispersive configuration

to obtain _a

sharp

Gaussian

in-plane

resolution function _with _rms _width _2.2

x

10~~

i~~

_The

out-of-plane

resolution function _was determined

by

_a set of

slits, yielding

_a

Gaussian

out-of-plane

resolution function of _rms width 3 _x

10~3i~~.

_Data1vere _corrected _for

the

noise,

empty cell and

polymer

matrix spectrum.

Considering

that

experimental points

are relevant when ±lie ratio of the

sample intensity

is

superior

to ten times the

intensity'

of the empty cell _we _were

limited, despite

of the resolution function

sharpness,

to momentum

transfers

superior

to 8 _x 10~~

l~~,

_which

was reached

only

for the most concentrated

samples.

3. Results and

scattering

^functions.

In table I there _are summarized for each of the carbon black used the fractal dimension D determined

by

TEM

image analysis,

the volume fraction at the

percolation

threshold in the

composites

and the 4l _range studied in the

scattering experiments.

Let _us first consider the results of the neutron

scattering experiments.

^As shown _on _a

typical log-log plot

of the scattered

intensity

I _as _a function of the wavevector

Q

the spectra

are characteristic of fractal

objects (Fig. 2).

Two domains _are

clearly distinguishable.

At low

Q, Q

<

Qo

_'~

I/ro

'~ 2 _x 10~2

i~~,

_the

_intensity

_follows _the

power law:

I(Q)

~

Q~~,

the

slope

of the line

giving

the fractal dimension D~ At

larger Q, Q

>

Qo,

the

points

still fall _on

a

straight

line but with _a

larger slope I(Q)

'~

Q~~,

this is the Porod law

expected

at scales inferior _to the

elementary particle

size when its surface is smooth and the interface

sharp.

The determination of D from the

slope

of

log-log plots

of

I(Q)

at low

Q

is rather

rough

and _can lead _to incorrect

results;

in

particular

the existence of cut-off

lengths occurring

in real fractal

objects

_may

modify

the apparent ^fractal dimension.

So,

_we

proceeded

to the fit of the

experimental

results with the

complete

theoretical

expression

of

I(Q).

When the individual

scatterers

(here

the

elementary

carbon black

particles)

_are identical the scattered

intensity

is

given by

the

product

of the form factor

F(Q)

and the structure factor

S(Q)

_:

1(Q)

~4

F(Q)S(Q) (i)

The

complete expression

^of the structure factor

including

_a cut-off distance

f

at

large

scales for the fractal

behavior, corresponding approximately

to the size of the

aggregates,

has been

shown _[4] to be

correctly given by:

S(Q)

₌ 1+

~

~~~~ j-i

sin

[(D I)tg~~(Qf)j

~2)

(Qro)

ji

~ TAPⁱ ^~ whicll reduces to

S(Q)

~-

Q~~

when

f~~

_<

Q

<

rp~

For

spherical

individual _scatterers the form factor is:

2

Fioi

r~

~ ^~° i~iii

^~"

^~~~°

^~3)

(6)

~~o

~~-i

A

lo'~

_10'~ _10'~

Q(I-11

Fig. 2. SANS eTperiments: log-log plot of the scattered intensity I _versus the wavevector Q. The solid line is the fit

(see text)

of the experimental points

IA).

However this

expression

is correct

only

in the _case of

monodisperse spheres.

One _can

easily

detect if it is the _case

by

the characteristic oscillations

developed

then

by

the spectrum

beyond Q

~-

4/ro.

As it is

obviously

not what

happens

in _our

samples,

the

elementary

carbon black

particles

present some

polydispersity la

_very small

dispersion

is

enough

to smear the

curve)

and the form factor _can be

approximated by:

FIQ)

~

[

~~

(4)

The fit of _our

experimental points

with the

equations (1, 2, 4) proceeded through

the classical least _mean _square method

with,

in addition of

D,

_To _as _a free parameter.

Concerning

the cut- off

length f,

it is

obviously larger

than 1000

I

_for _all _the

samples

since _no saturation _was discemable _on the spectra, ^and after

having

^verified ^for various concentrations that the results of the fit did not

depend

_on the choice

off,

_we fixed

definitively

this parameter to _a value

equal

to 5000

I.

_We _have _to _remark _that

although

the _mean size of the

primary

aggregates ^is ¹⁰⁰⁰

I

it is not

surprising

that _we do not detect the cut-off _on the spectra because the contribution of the

biggest

_aggregates is _more

important

and shift the curvature at smaller

Q

than it would be located with

monodisperse

_aggregates of1000

I,

_I-e- around 10~3

i~~

The values of D obtained for the best fit of the

experimental

scattered

intensity

of the _two series of

samples

_are

reported

_on the

plot

in

figure

3 _as _a function of the volume fraction 4l of carbon black. The value of _To is found

independent

of 4l and

equal

to

(45

+

2)I

for the Raven 2000 and

(40

+

2)I

for the Raven 7000.

Concerning

the last

points:

4l > 5 ~ for the Raven 2000 and 4l > 6 ~ for the Raven

7000,

we have to mention that the spectra ^seemed to exhibit _an

upward

turn at small

Q

but since it involved

only

one or two points on the I

=

f(Q)

_curve _we _were at this point unable to conclude

(7)

A

1.8 o ~

~ .

A

)

C _O

, ^A ~

fl ~

(

l.6 , O A

-

O~~

3 _A .

~ ,

i~ ~

1.5 O

IA

0 2 3 4 5 6 7 8

Volume Fraction _~l~ (ib vol)

Fig. 3. Variation of the fractal dimension determined by SANS

(A)

and SAXS

(O)

_on the compos-

ites _as _a function of tbe volume fraction of Carbon Black. The values obtained by TEM image analysis

(n)

_are reported at ~

= o. Full symbols correspond to the Raven 2000 samples and empty _ones to the Raven 7000 samples. The _arrows indicate the respective conduction thresholds.

for _a real

tendency

_or _an

experimental

artefact. This remark motivated _our

study

of the _same

samples by SAXS, raising

the

hope

^of

elucidating

this

question by working

at smaller than

with the neutron

scattering (at

least with the

equipment

_we had _a

facility access).

As

already mentioned,

_we discarded all the

experimental points

for which the empty ^cell

signal

represents more than 10 $l of the

sample signal

in a way ^to ^ensure that _we have

good

statistics and that _we _are out of the incident beam. For these

experiments

_we _expect the scattered

intensity

to be smeared

by

_a Gaussian

weighting

function _to take into _account the effect of the

imperfect

collimation

by

the slits [5].

Considering

the small value of its rms

width _we _can

neglect

the correction due to the

in-plane

resolution function. The

experimental scattering

_curves have then been fitted

by

the

following

theoretical

expression:

+oo

~j2

~~~~~~ ~«

liao

^~~~

2~(

^~

~ _~

~~~

where

I(Q)

is the

perfectly

collimated scattered

intensity given by

the equations

(1,

2,

4).

One _can _see in

figure

4 the _very

good quality

of the fit of the

experimental points.

As it is the _case for this

sample

_we did not take into account for the

fit,

when it _was

occurring,

^the

points belonging

to the curved

region

at small

Q.

The results enabled _us to confirm the rise of the

intensity

at small

Q (cf. Fig. 4) following

the _power law

behavior,

in the _same

samples

as

previously,

I.e. for the

highest

concentrations, since _we could reach for those

samples Q

_as

small _as 8 _x 10~~

i~~.

_For _the _less concentrated materials _we confirmed the fractal behavior

over

approximately

the _same range of

Q,

_a little

larger only

in _a few _cases. The overall results

concerning

the measured fractal dimension _are in quantitative agreement with those obtained

(8)

by

the SANS

experiments (cf. Fig. 3),

_we _can state that _we observe _a decrease of the measured fractal dimension _as the concentration in

particles

is increased and _a rise of the scattered

intensity

at small

Q

for the

highly

concentrated

samples.

loco

iw

~

i

cool ooi oi

Q

(I

_-1>

Fig. 4. SAXS experiments: log-log plot of the scattered intensity I _versus the wavevector Q. The solid line is the fit

(see text)

of the experimental points

(O).

4. Discussion.

Let _us first _comment

briefly

_on the values of the fractal dimension measured _at low _concentra- tions. The spectra of the

samples

with 4l less than I _To _were

impossible

to

get properly

but TEM

images

of ultrathin slices of the I $l filled

polymers

show

clearly

that _we _are in the dilute

regime

at this concentration: the

primary aggregates

are

homogeneously dispersed

and the distances between them _are much

larger

than their average size. Thus, ^the fractal dimensions found for these

samples

_are those of the

primary

aggregates. Furthermore

they

_are in quite

good

agreement with those determined

by

_our TEM

image analysis

_[2]

validating

a

posteriori

our method and

confirming

that the determination of D is not biased

by

the size distribution of the aggregates.

Lastly,

the values of D _are close to the fractal dimension

expected by

the

cluster-cluster

aggregation

model

again

this is _not

surprising

for these _vapour

phase

elaborated aggregates. ^In

conclusion,

_our

samples

consist in frozen

dispersions

of cluster-cluster grown aggregates.

In the

investigated

_range of

concentrations,

the

samples

exllibit _an insulator to conductor transition. This is known to be associated with _a

percolation

transition which

corresponds

to the formation of _a continuous

path

in _a

heterogeneous

material _or _an "infinite" cluster in

JOURNAL DE PHYSIQUE II -T 3. N'll. NOVEMBER J993 62

(9)

1654 JOURNAL DE

PHYSIQUE

II N°11

our

experiment

_[6]. Therefore _we _can _argue

that,

when the concentration is

increased,

there is

aggregation

in _our

samples;

the

problem

_we ^deal with is the _one of _a

suspension

of

aggregating

cluster-cluster _grown clusters. The main

questions

_are the nature of the

aggregation

mechanism and the

understanding

of the variation of the measured fractal dimension _as _a function of the

initial cluster concentration.

If D has _a lower value than the

original clusters,

it _means that their

aggregation

builds

more open or tenuous structures than the cluster-cluster

(Clcl)

_one. As _a _matter of fact if the

sticking

rule _was the _same _we would _not expect any

change

in the observed fractal dimension until the concentration of

original

^clusters ^reaches the

overlapping limit,

when the _sum of the embodied volumes is

equal

to the total volume

(the analog

of C* in

polymer solutions), beyond

which the

interpenetration

of the clusters will be forced.

By

the way, this would result in _a

larger

fractal dimension _as it is has been shown for various extensions of the Clcl model

considering

stronger penetration ^effects.

Consequently,

_we believe that in _our _case the clusters aggregate ^with even smaller

penetration

than in Clcl I,e.

they

stick

"tip-to-tip".

Such _a model has been

already

studied

numerically by

^Jullien _{[7] in} the _case of the cluster-cluster

aggregation

of initial

single particles.

The fractal dimension is decreased _and found

equal

to 1.42 in three dimensions. This is

clearly

_a

qualitative

^indication that _our argument is correct. We _can argue that the

progressive

decrease of D with

increasing

4l is due to the

increasing

relative proportion of "linear"

(as opposed

to

branched)

parts in the

samples

created

by

the

sticking

of the clusters.

Indeed _{if you} believe that the

aggregation

does not induce further ramifications the proportion of branched _arms will diminish. In order to

verify

this behavior _we have performed two-

dimensional numerical simulations of the

tip-to-tip

cluster-cluster

aggregation

of Clcl clusters.

For

practical

convenience

(due

to the limited number of

particles

_we could work

with)

_we

chose _to treat the

problem

in the

following

_way. The total number of

particles

was fixed

(equal

to

512)

and _we let grow clusters

following

a hierarchical model

(the

number of clusters is divided

by

two at each

step)

until

only

_one _aggregate

containing

all the particles remains.

In the first steps ^the ^clusters _grow

by

_a Clcl mechanism and in the

following

_steps

by

_a

tip-to-tip

_one. The increase of the

proportion

of "linear" parts, which _we

supposed

to result from the increase of the concentration in the

experimental

_case, _was obtained

by

_a decrease of the number of Clcl

growth

steps. ^To ^allow ^for a better comparison with _our

experimental

results _we decided _to determine the fractal dimension of the so-constructed aggregates from the square of their numerical Fourier transform

(F2)

which is

equivalent

to the scattered

intensity.

The first important

point

is that the

log-log plot

of

F~(Q)

_was

always

found

typical

to the

one of _a fractal

object. Secondly,

^the ^fractal ^dimension obtained from the _mean

slope

_over

ten different aggregates at low

Q

_was found to decrease when

going

from the Clcl to the tip-to-tip ^mechanism

(Fig. ₅₎

after

exhibiting

_an extremum

larger

than the

departure

value.

The _occurrence of _an extremum is rather

surprising

and further

work, beyond

the _scope of this

study,

would be needed in order to check if this is due to _a lack of statistics _or _an artefact. Note that the measured fractal dimensions of both the Clcl and the

tip-to-tip

aggregates are

larger

than those determined

by

Jullien _[7]

presumably

because of the smaller _size of _our

aggregates.

Nevertheless the overall behavior is in

good agreement

with _our

experimental results, validating

our

hypothesis

of _a

tip-to-tip aggregation

_process. ^The

origin

of such _a mechanism could be attributed to the triboelectrification properties ^of the Carbon

Black-polymer

_composites used in the

xerographic

_process [8]. ^In our case, the sonication of the carbon black suspensions could be

responsible

for the _appearance of interfacial

charges leading

to the

sticking

of the aggregates

by

the

tips

due to electrostatic forces.

Let _us _now discuss the

upward

turn of the scattered

intensity

at small

Q

observed in the _con- centrated

samples.

Such _an effect _was

reported

_on

aggregated gold

^colloids and attributed

by

the authors [9] ^to a

restructuring

of the aggregates

referring

to

experimen-

(10)

1.8

u o Q

1.7 Q

I

fl u o

Eu u o

i$

mZ

f i+

o

1.5

1

Number of

tip.to-tip

_steps

Fig. 5. Variation of the fractal dimension of clusters obtained by the tip-to-tip aggregation of Clcl aggregates as a function of the number of tip-to-tip growth steps. The total number of particles is

equal to 512, the growth follows

a hierarchical model and the number of clusters is divided by two at each step.

tal, by light scattering,

_on the

restructuring

_upon

ageing

of colloidal silica aggregates

leading

to _an increase of the

slope

of

log

I _vs.

log Q

for

Q

<

rp~

[10]; ^the ^other a theoretical

study

_on

the mechanical

stability

of tenuous

objects demonstrating

that low D aggregates cannot grow

over a certain size without deformation

ill].

Both of these arguments are not relevant in _our

case since the aggregates are

suspended

in _a viscous

liquid (limiting

the

gravitational

forces and thermal fluctuations

affecting

the mechanical

stability) only

_a short time before the whole gets "frozen".

Taking

into account the

rigidity

of the

primary

aggregates ^the most

probable

restructuration should be the

partial reorganization investigated by

Meakin and Jullien [12]

but this effect makes the aggregates ^become more compact on very short

length

scales and the fractal

dimension,

_on

longer length scales,

is

only slightly

increased. Therefore _we believe that the rise of

I(Q)

at small

Q

reveals correlations in the cluster

growth.

Indeed this would lead to _an Orstein-Zernicke law _at small

Q:

IIQ)

-~

sio)

rw

~(j~~~

16)

K is the inverse of the correlation

length,

and is consistent with the fact that _we detect this effect _at concentrations of the order of the

overlapping

concentration. The calculation for _a

monodisperse population

of1000

I

_clusters

gives

C*

= 6 To.

Unfortunately

_we could not check this

interpretation by

the

study

of the scattered

intensity

at lower

Q

and _we _were unable to prepare more concentrated filled

polymers

the

consistency

of the

sample

before the

curing

of the

prepolymer

_was then _no _more that of _a _viscous

liquid

but rather that of _a paste

(11)

5 Conclusion.

This work _was the first attempt to determine the structure of carbon black

dispersions

in _a

polymer.

The

problem

consisted in the

aggregation

^of initial cluster-cluster _grown clusters which had not yet been considered _to _our

knowledge.

An important

point

is that _we found

experimentally

that the structure of the

aggregated

clusters is fractal in the _range where the

primary aggregates

are self-similar.

Furthermore,

the decrease of the measured fractal dimension observed _as the concentration of the initial clusters is raised could be

interpreted by

_a

tip-to-tip sticking

mechanism which could be

physically justified

in the _case of these

suspensions.

Acknowledgements.

The author would like to thank F. Carmona for

suggesting

^this

^study,

^and

gratefuly

acknowl-

edges

R.

Jullien,

D. Roux and J. Teixeira for their

help during

this work.

References

ill

Carmona F., Physica A 157

(1989)

461-469.

[2] Salom6 L., Carmona F., Carbon 29

(1991)

599.

[3] Meakin P., Phys. Rev. Lett. 51

(1983)

ll19.

[4] Teixeira J., On Growth and Form, H-E- Stanley, N. Ostrovsky Eds.

(Dordrecht,

Nijholf,

1986).

[5] ^Guinier A., Fournet G., Small Angle Scattering of X-rays

(New

York, Wiley, 1955) p-114, equation

(lo).

[6] Balberg I., Bozowski S., Solid State Commun. 44

(1982)

551.

[7] ^Jullien R., J. Phys. A: Math. Gen. 19

(1986)

2129.

[8] ^Julien P-C-, Carbon-Black Polymer Composites E-K- Sichel Ed., (New York, Marcel Dekker,

1982) p.189-201.

[9] Dimon P., ^Sinha S-K-, Weitz D-A-, Safinya C-R-, Smith G-S-, Varady W-A-, Lindsay H-M-, Phys.

Rev. Lett. 57

(1986)

595.

[lo] Aubert C., Cornell D-S-, Phys. Rev. Lett. 56

(1986)

738.

[iii

Kantor Y., Witten T-A-, J. Phys. Lett. France 45

(1984)

L675.

[12] ^Meakin P., Jullien R., J- Phys. France 46

(1985)

1543.