Percolation in ternary composites: threshold position and critical behaviour

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Percolation in ternary composites: threshold position

and critical behaviour

C. Vinches, L. Salome, C. Coulon, F. Carmona

To cite this version:

Short Communication

Percolation

in

_ternary

_composites:

threshold

_position

and critical behaviour

C.

_Vinches,

L.

Salome,

C. Coulon and F. Carmona

Centre de Recherche Paul

Pascal,

Avenue A.

_Schweitzer,

33600

Pessac,

France

(Received

24

_July

_1990,

_accepted

6

_September

₁₉₉₀₎

Résumé. 2014 Nous

_présentons

la conductivité de

_composites

ternaires

_(fibres

de

_carbone,

_sphères

de

_carbone,

_polymère).

Une transition métal - isolant est observée au dessus d’une concentration

critique

en

_particules

conductrices. Près du seuil nous observons une

_dépendance

en loi de

_puissance

de la conductivité. Ces deux résultats sont

_analysés

dans le cadre de la théorie de la

_percolation.

En

particulier

nous discutons l’influence de la

_proportion

_{fibres-sphères}

sur le seuil de

_percolation

et sur

l’exposant critique

observé à l’aide

_{d’arguments géométriques simples.}

Abstract. 2014 The

conductivity

of _ternary

_{composites (carbon}

_fibers,

carbon

_spheres

and

_polymer)

is

presented.

A transition from an

_insultating

to a

_{conducting regime}

is observed above a critical volume fraction of

_{conducting particles.}

Near the threshold a _power law

_dependence

of the

_conductivity

is found. Both results are

_analysed

in the frame of the

_{percolation theory.}

In

_particular

the influence of the relative amount of fibers and

_spheres

on the

_percolation

threshold and on the critical _exponent is discussed and

_{quantitatively explained using simple geometrical}

_arguments.

Classification

Physics Abstracts

72.60 - 64.60A

Introduction.

Carbon -

_polymer

_composites

have been shown to be

_good

model materials to

_study

percola-tive conduction.

_{Conducting particles}

with various

_geometries

have been introduced in a

_polymer

matrix to obtain an insulator - conductor transition. In

_particular

_spherical

carbon

_particles

or

carbon fibers have been used

_{[1, 2].}

The

_study

of these

_binary

_systems

has shown that the

perco-lation threshold is

_{strongly dependent}

on the

_shape

of the

_conducting

_particles.

For the carbon

spheres

the critical volume fraction

~c ~

17%

is found to be in

_agreement

with the classical result for

_{monodisperse particles.}

It is much

_larger

than the

_{corresponding}

values

_(of

the order of one

percent)

for fibers. In this case, it is found both

_{theoretically}

and

_{experimentally}

that the critical concentration _goes like

_{0, -}

_(d/1)2

when d « 1 where d and 1 are the diameter and

_length

of the fiber

_respectively

_[2].

Close to the

_threshold,

a

_power

law

_dependence

is found with a critical

exponent

t. When the

_{conducting particles}

are

_spheres

the observed

_exponent

is t = 1.6-2.0 in

agreement

with the theoretical value found

_numerically,

i.e. 1.9

_[3].

_This

number is the universal

2506

value

_expected

close to the critical

_point,

i.e. within the critical

_region

where the fluctuations are

important.

The fact that it is observed with

_spheres

indicates a

_quite

_large

critical

_region

in this

case. On the

_contrary,

the

_exponent

found for the fibers is

_larger,

1.e. t = 3. 1Bvo alternative

explanations

have been

_proposed

for this result. On the one hand it has been

_suggested

that the

critical

_region

is

_very

small. In this case the observed

_exponent

has the mean field value t = 3

[4] .

On the other

_hand,

an anomalous critical behaviour due to the

_peculiar

stucture of the

_system

of

interconnected fibers has also been

_{anticipated [5].}

We

_present

in this

_report

new results obtained with

_{ternary systems}

_{(carbon spheres,}

carbon

fibers and

_polymer)

where the influence of the relative amount of fibers and

_spheres

is studied. In

particular

the effect on the

_percolation

threshold and on the

_conductivity

_exponent

is evidenced.

The evolution of the threshold is discussed with

_{simple geometrical}

_arguments.

In the same _way,

the width of the critical domain is estimated to

_explain

the observed critical

_exponent.

Experimental.

The

_composite

is made of an

_insultating

matrix in which

_{conducting particles}

are introduced.

Thw kinds of

_particles

were

_used,

_namely

carbon fibers and carbon

_spheres.

Carbon fibers are

high

modulus HMS from Courtaulds with a

_conductivity

of

102

0-lcm-1.

Their diameter and

length

are _{8 um and 1 mm,}

_{respectively.}

Carbon

_{spheres (type}

E from Versar

_M.C.)

have a mean

diameter of 20 _{m. The}

_polymer

matrix is an

_epoxy

resin

_(Araltide

_R,

hardener HY 905 and accelerator DY 061 from Ciba

_Geigy)

which has a

_conductivity

lower than

10-15

a-1cm-1.

To

_improve

the

_dispersion

and orientation of the

_particles

in the matrix a new _way of

prepara-tion has been

_designed.

In a first

_step

the constituents of the matrix are mixed. The

_particles

are

then introduced and

_spread

into the

_sample.

Ultrasonic

_agitation

is used to

_disperse

the

_particles

randomly

and cancel out the

_preferential

orientations which _may result from the

_previous

_mixing.

The second

_step

consists in a

_{polymerisation (}

24 hours at 65°

_C)

which freezes the

_composite

in the obtained state.

Microscopic

observations of thin slices of the

_samples

are then

_performed

to check the

effi-ciency

of the method. In most cases, no sedimentation is detected and the

_composite

_appears

to be

_homogeneous

and

_isotropic.

These

_{good quality}

_composites

are

_kept

for

_conductivity

mea-surements.

The DC

_conductivity

was measured on small

_{parallelepipedic}

_{samples using}

either an ohmeter

(when

R >

106 0)

or an electrometer

_{(when R}

>

106 Q).

Results.

Before

_studying

_ternary

_composites,

the

_{corresponding binary}

_systems

have been

_prepared

to

be used as reference materials.

_Figure

la shows the

_conductivity

of fiber

_composites

as a function of the volume fraction of fibers. As

_expected

a transition towards a

_conducting

state is observed. In

the

_present

case we find a critical concentration

_{çJc =}

_(0.95

f

_0.05)%.

The

_{corresponding}

critical

exponent

can be estimated from a

_{log-log plot}

of u as a function of

_{(0 - 0,,).}

A

_typical

_plot

is

_given

in

_figure

1b with

_0,

=

0.98%

and

gives t

= 3.1.

Changing

_sligthly

_0,

does _not affect

_noticeably

this

_exponent

which

_stays

around 3. Both results are in

_agreement

with

_previous

studies on fiber

value for

çlc

is

_{significantly higher}

than those

_{previously reported}

_[1].

However it is well known that the

_position

of the threshold is sensitive to _many

_microscopic

_parameters.

In

_particular

the

polydispersity

of the

_spheres

may

explain

this difference.

Fig.

1. -

(a) Log

of the

_{conductivity o,}

of the

_binary

fiber

_composites

as a function of the volume fraction

l/J

of

_{conducting particles. (b)}

_Log-log

_plot

of a

_against

_{(çl - oc)}

with

oc

= 0.98%. The solid line is the best

linear fit. The

_slope

_gives

the critical _exponent t = 3.1.

Fig.

2 -

_(a)

_Log

of the

_{conductivity o,}

of the

_ternary

_{fibers-spheres composites}

as a function of the total volume

fraction 0

for p = 0.5

(çlf

=

Os). (b) Log-log

plot

of o

_against

_{(0 - Oc)}

with

_Oc

= 1.6%. Ibe solid

2508

7ivo

_ternary

_composites

have been

_prepared

with _p =

0,/o

= 0.5 and 0.8

_(Ps

_{and 0}

_are the

volume fraction of

_spheres

and the total volume fraction in

_{conducting particles, respectively).}

Figure

2a

_gives

the

_conductivity

as a function

_{of 0}

for _p =

0.5,

the obtained critical concentration

is

Pc =

(1.6:i: 0.1)%.

It

_corresponds

to a fiber concentration close to the critical value determined

for

_pure

fiber

_{composite. Figure}

2b

_gives

the

_{Log-log plot}

to show the critical

_exponent

t. We find

t =

_2,

i.e. the value

_usually

obtained with

_pure

_{sphere composites}

_{[1] .}

Another series of

_samples

with _p = 0.8 has been

prepared

to estimate the

position

of the

percolation

threshold. In that

case, we obtain

_{0,,(3.9 --t- 0.5)%}

which confirms the low value of the critical concentration in the

previous sample (Fig. 3a).

We also

_report

the

_{log-log plot}

of the results to show their reasonable

agreement

with an

_exponent

_equal

to 2

_{(Fig. 3b).}

Fig.

3. -

(a) Log

of the

_{conductivity o,}

of the _ternary

_{fibers-spheres composites}

as a function of the total volume

fraction 0

for p = 0.8

(4)f

=

0.254>8).

_{(b) Log-log plot}

of a

_against

₍₀

4>c)

with

_4>c

= 3.9%. The

dashed line has a

_{slope equal}

to 2.

Discussion.

The

_striking

results obtained with these

_ternary

_composites

can be summarized as follows:

- the evolution of

0,

with

p is

strongly

non linear and this critical concentration remains close

to the value

obtained

for the fiber

_composite

in a

_large

domain of values

_{of p,}

- at the _same time the critical

exponent

t N 2 found for _p = 0.5 and 0.8 is similar to the one

reported

for

_{binary composites}

with

_spheres.

Both results are discussed in the

_following.

EVOLUTION OF THE THRESHOLD IN TERNARY SYSTEMS. - As

already

mentioned,

one

_important

parameter

which makes the difference between

_composites

with

_spheres

or fibers is the

_shape

I/d

between its

_length

and diameter. The role of this

_anisotropy

has been discussed with

geomet-rical

_arguments

_[2]

that we summarize

_briefly.

The volume of a fiber reads:

Similarly

the number of

_particles

_per

unit volume is

_(n

has the dimension of

_1-3):

where

_f

is a function of

_d/1.

The factor

4/ir

has been introduced for convenience.

The volume fraction is therefore:

This

_expression

contains the

_asymptotic

limit d « 1. In this case

_{f (dll)}

-

f (0)

_{and 0}

=

f (0)

(d/1)2

i.e. 0

is

_proportional

to

1-2.

However it can be used _{for any} value of

_d/l.

In

_particular

we can

consider that

_{spherical particles}

_correspond

to the case d = 1 and assume that both

_sphere

and

fiber

_composites

can be described

_using

the above relation with différent values of

_d/l.

This

_implies

simple

relations for

_any

_ternary

mixture

_containing

two kinds of

_particles

with

_anisotropies

_(d/1)1

and

_(d/1)2.

Both states can be obtained from a reference state of

_anisotropy

_(d/1)o

_through

a

geometrical

transformation. If the volume fractions of each

_species

are

_4Jl

and

_4J2,

_they

are related

to the

_{corresponding}

concentrations in the référence state

_4JOl

and

4J02

by:

The total concentration in

_conducting

_particles

reads:

where

Introducing

p as: we

_finally

obtain:

This

_expression

can be used to deduce the critical concentration

_{of any}

_{ternary system}

as a function

of p.

If

_{0,, ,(0)}

=

Aloo

and

çlc( 1)

=

A2PcO

are the critical concentration for the

_binary

_composites

2510

Fig.

4. - Critical volume fraction

Or,

_{as a} function

of p.

Continuous line: theoretical calculation. The

ex-perimental

critical concentration for the

_binary

_systems

_(p

= 0 and

p

= 1)

are used as

_input

_parameters

_(full

squares). Open

squares: the

experimental

critical volume fraction for _p = 0.5 and

p = 0.8.

In

_particular

we can

_apply

this formula for a

_ternary

_composite

with fibers and

_{spheres. Using}

our

experimental

determination

_çJc(0)

=

0.95%

for fibers and

çJc(1)

=

31%

for

_spheres

we obtain the

variation of the threshold

_given

in

_figure

4. The

_experimental

results obtained for p = 0.5 and

p = 0.8 are also

_{reported. They}

are in excellent

_agreement

with the theoretical

_prediction.

It is

_interesting

to mention that the theoretical evolution of

_{çlc (p)}

can be

_justified

with a _very

simple

argument.

The above formula also reads:

which

_simply

means that the two

_conducting

_species

act in

_parallel

in the clusters.

_They

are

weighted by

( 1- p)

and p

respectively.

When

çJc(0)

«

_{çJc( 1 )}

the

_connectivity

inside a cluster

is dominated

_by

the

_particles

of

_larger

_anisotropy.

z VALUE OF THE CRITICAL EXPONENT t. - We

now discuss the observed critical behaviour for _p =

0.5 and 0.8. The

_interesting

result is that the universal

_exponent

_(t

=

_1.9)

is obtained

_although

the

threshold is

_very

close to the one observed for

_pure

fiber

_binary

_systems,

i.e. when the

_connectivity

between

_conducting

_particles

inside a cluster is dominated

_by

the fibers.

As we mentioned

_previously,

two alternative

_explanations

have been

_proposed

to

_explain

the

critical behaviour of fiber

_composites.

One of them is based on the

_peculiar

stucture of the

_system

of interconnected fibers. Since we believe that

_connectivity

is still dominated

_by

the fibers in

particular

for p =

0.5,

this

_explanation

can be ruled out.

_Thus,

the width of the critical

_region

is more

_likely

at the

_origin

of the observed

_exponent.

We

_give

_arguments

which rationalize this

Quite

generally

the width of the critical

_region

is related to the size of the fluctuations. This

problem

has been discussed in détail

_by

de Gennes for the vulcanization

_process

which

_belongs

to

the same

_university

class as

_percolation

_[6].

In this case the size

à 0 *

of the critical

_region

is

given

by:

where

_eo

is the

_prefactor

for the correlation function

_ç:

a is the

microscopic length

used as reference

_(i.e.

the smallest characteristic

_length

scale in the

system).

The

_length

_eo

can be estimated with

_geometrical

_arguments.

_Following

reference

[6] :

where m

and R;

are the centers of

_gravity

of two

_neighbouring

_particles

in a

_{cluster. ( )}

indicates

an _average over all

_{possible configurations.}

This

_{simple expression}

allows the determination

_{of eo}

for either

_binary

or

_ternary

_composites.

In the case of

_binary

_systems

with

_spheres

this

_{expression gives}

eo - D

(D

is the diameter of a

sphere).

Since D is also the reference

_length

a, we

_expect

a

_large

critical

_region

as

_{experimentally}

revealed

_by

the

_exponent

t = 2 On the _other hand we

_expect

_{eo -}

1 and a - d for fibers

_(the

exact calculation of the _average mentioned above

_gives

_12/6

in the limit

_d/1=

₀₎

and therefore a

very narrow critical

_region

and a mean field

_exponent.

A similar

_explanation

is

_given

in reference

[6]

to _argue that

_{long polymer}

chains have a mean field like behaviour. Thus this

_simple

_argument

explains why t

= 3 is found for the

binary

composites

with fibers.

With similar

_arguments

we can estimate

_eo

for

_{ternary systems.}

Since the _average is taken over

all

_pairs

of

_neighbouring

_particles

we now write:

where

_{eff , ef,}

and

_e.,

are the

_averages

( ( m -

_Rj

)2)

for

_fiber-fiber,

_fiber-sphere

and

_{sphere-sphere}

couples respectively.

The p; ; are the

_{probabilities}

of

_finding

these

_couples

as nearest

_neighbours

in a cluster.

For thin fibers and small

_sphere

₍₁

» d and

_D)

one

_easily

obtains:

If the

_particles

are

spread

_randomly

the

_{probabilities}

_pij are

_{given by:}

2512

Fig.

5. - Theoretical variation of

(Vold )2

with p for the

fibers-spheres

ternary

composites . e6 2«

is propor-tional to the inverse of the width of the critical

_region.

where vs and v f are the volumes of a

_sphere

and a fiber

_{respectively.}

With these

_expressions

one

easily

obtains e2

for a

_ternary

_composite

as a function

_{of p, r =}

_vs/vf

and b =

DI 1:

Figure

5

_gives

a

_plot

of

(eo/D) 2

as a function

_{of p using}

the

_experimental

dimensions d = 8 03BCm,

D = 20 Mm, 1 = 1

mm. eo2

decreases

strongly

for small values

of p.

Independently

of details like numerical factors it is clear that

_eo

should decrease as soon as _p

increases. Since vs«vf we have nf « ns as soon as

_{p is}

not

_very

small

_(because

of the

_equality

0

=

_nv).

Then the fiber-fiber

_couples

_only

dominate for

very

small values of p. For

larger

p, the two other terms are the most

_important

and lead to a smaller value of

_03BEo.

As a

_consequence

the

critical

_region

should increase

_quickly

_{(ase 02)}

as soon as a small amount of

_spheres

is introduced in a

_binary

fiber

_composite.

With this

_respect

the result t N 2 found for both

_ternary

_composites

is consistent with our

_analysis.

Conclusion.

In

_summary,

we have discussed the behaviour of

_ternary

_composites

made of carbon fibers

and

_spheres

_spread

into an

_insulating

matrix. Our

_findings

can be summarized

_considering

the series of

_samples

with _p = 0.5

(tPs

=

çlf).

In this case we find a

_percolation

threshold close to

exponent

reported

for

_spheres.

We have shown that both

_properties

can be understood with

_simple

geometrical

arguments:

- the

position

of the threshold reflects the

_connectivity

inside the infinite cluster. For this

prop-erty

the

_particles

of

_higher

_anisotropy

are dominant. As a

_consequence,

the relation between

_çJc

and

_{p is}

_highly

non linear

_{(1/ f/Jc}

is linear with

_p);

- the

exponent t

is related to the width of the critical

_region

which in tums

_depends

on the

fluctuations

inside

the clusters. The

_pair

correlation function is

_strongly

influenced

_by

the

_spheres

in a

_large

domain of p. Thus the critical

region

can be

_{large enough}

for most of the

_ternary

composites

to observe the universal

_exponent

t = 1.9.

Although

our

_experimental

results remain

_preliminary,

there is

_clearly

a

_good

_agreement

be-tween

_theory

and

_experiment.

Further

_experimental

studies are _necessary to check the theoretical

_predictions

in more detail.

Among

other

_things,

the

_position

of the

_percolation

threshold of

_any

_ternary

_{composite (made}

with two kinds of

_particles

of different

_anisotropy)

can be tested. Moreover with

_ternary

_systems

of fibers and

_spheres

a mean field - non mean field crossover is

_{expected (presumably}

around p N

0.1)

which should be evidenced

by

a crossover between t = 3 and t N 1.9. These two values

being quite

different,

these

_ternary

_composites

are

_certainly

_good

_systems

for the

_study

of this

crossover.

References

[1]

FUG _G., CANET R. and DELHAES

R,

C.R. Hebdo. Sean. Acad. Sci. 287B

₍₁₉₇₈₎

5.

[2]

CARMONA

F.,

BARREAU

F.,

DELHAES P. and CANET

R.,

J.

_Phys.

Lett. France 41

₍₁₉₈₀₎

_L531;

BARREAU

F.,

Thèse 3e

_cycle,

Bordeaux

_(1983).

[3]

DERRIDA

B.,

STAUFFER

D.,

HERRMANN H.J. and VANNIMENUS

_{J., J.}

_Phys.

Lett. France 44

₍₁₉₈₃₎

L701.

[4]

For a review on

_scaling

and

_percolation

see for

_example:

STAUFFER

_D.,

_{Phys. Rep.}

54

₍₁₉₇₉₎

1.

[5]

CARMONA

_F.,

PRUDHON P. and BARREAU

F.,

Solid. Stat. Commun. 51

₍₁₉₈₄₎

255.

[6]

DE GENNES P.G., J.

Phys.

Lett. France 38

₍₁₉₇₇₎

L355.