Random media with temperature controlled connectivity

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Random media with temperature controlled connectivity

F. Carmona, E. Valot, L. Servant, M. Ricci

To cite this version:

F. Carmona, E. Valot, L. Servant, M. Ricci. Random media with temperature controlled connectivity.

Journal de Physique I, EDP Sciences, 1992, 2 (5), pp.503-510. �10.1051/jp1:1992111�. �jpa-00246502�

Classification Physics Abstracts

64.60A 73.40C 81.20T

Short Communication

Random media with temperature ^controlled connectivity

F-

Carmona,

E.

Valot(*),

L. Servant and M.

Ricci(**)

Centre de Recherche Paul Pascal-CNRS, Av. A. Schweitzer, 33600 Pessac, France

(Received

3 March 1992, accepted 12 March

1992)

Abstract We show that the electrical resistance of _a percolating medium made of _a polymer

filled with conductive particles _may undergo _a steep increase _{over a narrow} temperature range above _room temperature. We demonstrate that _a likely mechanism for the effect is the loss of interparticle electrical contact inducing _a weakening of the connectivity of the medium _as

temperature is increased.

There has been much interest in recent years in the non-linear electrical

properties

of random

percolative

systems

comprising

non-linear elements like fuses [1, 2], ^diodes [3], weak _supracon- ductive links [4, 5],

Analogies

with the rupture ^{of materials have}

largely

been stressed

[6-8].

Most of the

published

studies _are concerned with the theoretical

predictions

of behavior _or report computer

experiments.

^{On the other}

hand,

_very

large

and reversible increases of the resistance of

composites

made of _a

polymer

filled with conductive

particles

when heated above

room temperature have

long

been

reported

and known _as the

positive

temperature coefficient

or CTP efTect [9, 10]. ^But no

convincing interpretation

of such efTect had been

provided

_so

far

especially

when the

polymer

matrix is not

thermoplastic [11,

12]. ^{Besides it has}

long

been

reported that,

at room temperature, such disordered media

undergo

an insulator-to-conductive transition _as

predicted by percolation theory

when the conductive filler volume concentration is increased [13]. ^{In this letter} we

give

evidence that _a

likely origin

of such _a behavior is _a tem-

perature ^induced

change

of the

connectivity

of the random network of

interparticle

contacts, in other

words,

it is _an

experimental

illustration of the behavior of _a random fuse network with

temperature controlled fuse

breaking.

The materials

reported

in this letter _are made of _an epoxy whose

glass

transition temper-

ature is 50°C and conductive short carbon fibers with I _mm

length

and 8 pm diameter. The

processing

and electrical

properties

of such

composites

have been

reported previously

_[14]. It has been shown

[IS] that,

contrary to similar materials in which the

filling particles

_are carbon

blacks,

^conduction

proceeds through

direct contact among the

particles

and not

by tunneling (*)

^Now at ATOCHEM, Centre d'applications, 95 _rue Danton, 92300 Levallois, France.

(**) ^Now at DIAS Industries, BPS, 33380 Marcheprime, France.

504 JOURNAL DE PHYSIQUE I N°5

across a thin

polymer

barrier. As the volume concentration 4l of the fibers is increased from

zero, an insulator-tc-conductive transition is

observed,

the behavior of the electrical conduc-

tivity

in the

neighborhood

of the critical concentration 16*

being largely compatible

with the

power law behavior

predicted by percolation theory.

The fibers critical volume fraction and the

conductivity

exponent, ^determined from the concentration

dependence

of the _room tempera-

ture

composite conductivity

_as shown _on the

log-log plot

of

figure

I, _are 16* ₌ 0.9~ and t ₌ 3

respectively.

Both

figures

_are in

good

agreement ^with

previous findings

_on similar systems [13]. ^Let us

just

recall that the _very low value for 16* is due to the geometry of the

particles (aspect

ratio

larger

than _a

hundred)

and it is sensitive to the

physicc-chemical properties

of the

suspending

resin

(viscosity

and surface tension

essentially).

On the other hand the value for t which is

significantly larger

than the universal value t ₌ 2 is still not

fully

^understood [16].

io

P

io ~

i~,cm)

i o4

1o ⁱ

1.5 1-o o.5. o-o o.5 1-o

tog _{(tb q~#)}

Fig, i.

Log-log

plot of the concentration dependence of the resistivity of the short carbon filled composites.

Straight

line has slope t ₌ 3,o.

In

figure

2 _we

give

a

typical example

of the temperature

dependence

above _room temperature of _a

composite

resistance _on the conductive side of the transition. One observes _a

large

increase

by

^{several orders} of

magnitude

of the resistance _{over a narrow}

temperature

_range. ^{The location} of this efTect and the

apparent slope depend

_on the fiber volume concentration

(more precisely

on the relative "distance" from the

transition)

and _on the _epoxy. Outside the

abrupt change

region,

the resistance is

only weakly temperature dependent.

Below

it,

there exists _a

slight

anomily

in the

glass

transition temperature

neighbourhood.

Above this

region,

^the resistance is almost temperature

independent

and _we have also found that its value is of order I to 10 x 10~ kQ

practically independent

^{of the} fiber concentration. Note that such

large

resistance

changes

are

essentially

due to

resistivity changes,

the efTects of thermal

expansion

of the

sample being negligibly

small. However, the

resistivity change

itself is

obviously

to be attributed to the

thermal

expansion

of the

composite.

But it _cannot be

simply

^due to fiber volume concentration

changes

^induced

by

the difTerence between the

respective

coefficients of thermal

expansion (CTE)

of the

polymer

and of the carbon fibers _as

actually

observed when the temperature is decreased below _room temperature

[15]. Although

such _an efTect would

lead,

in agreement

with the

data,

to _an increase of the resistance since the

polymer expands

_more

(CTE

of order

10~~ °C~~)

than the fibers

(CTE

of order

10~~ °C~~),

it

predicts

too small

a relative

change of,

at most, a few tens of percent.

~6

R/Ra

~~5

~WW+

+*

io4

+ RJRa exp

lo ~

R/Ro calc

i o2

o i

io°

-.*'*****

+~

T(°c)

i o

50 loo lso 200

Fig.

2. Relative variations with temperature of the resistance of _a composite sample containing 4i = 2.5% carbon fibers. Full line is calculated with relations (3) ^and

(4)

^and Ts ₌

162°C,

S

=

19°C and Tc

= 187°C.

A

possible explanation

of the observed behavior is that there is _a

change

in the connectivity of the system as the temperature is

increased,

that is _{to say,-} electrical connection _among the conductive

particles

^is

changed

when the

polymer

matrix

expands.

^Such _a

conjecture

is

convincingly supported by

the thermal behavior of the electrical resistance of _a test system

consisting

in two

initially contacting

carbon filaments _or wires embedded in the _epoxy. Such _a

system

can be considered _{as an}

experimental

model of _a "contact" similar _to those present in

506 JOURNAL DE PHYSIQUE I N°5

the

composites.

One

actually

observes that it _may

reversibly

switch from about I kQ

(which

represents

essentially

^the resistance

along

the

filaments)

to about 10~ kQ when the temperature is

cyclically changed

from _room temperature to 200 °C and back. This proves that the "contact resistance" _among the wires is

negligible

at _room temperature and increases

dramatically

under

heating:

the

initially

"closed"

junction

_may

"open"

but _one must notice that in the _open state the resistance remains finite. For _a

given junction,

the

"opening"

and

"closing"

temperatures in _a

cycle

_are difTerent and

they

both _vary from _one

cycle

to the other.

Furthermore,

^when _a

set of _n identical

junctions

is

studied,

all

opening

and

closing

temperatures _are different. This

experiment

then evidences actual reversible random

openings

of the electrical contact between two wires imbedded in _an epoxy.

Figure

3 reports _a

typical histogram

of contact

openings

_as

a function of temperature,

Q(T) being

^{the cumulated}

proportion

of the _n studied

junctions (in

this

example

_n

=

20)

which _{are open} at temperature ^{T when} the temperature ^{has been}

cyclically changed

N times

(N

₌ 20 in this

example)

between _room temperature and 150°C.

Opening

_occurs

statistically

_{over a} limited temperature _range ^above room temperature in fair

similarity

with the

shape

of the

composite

resistance _versus temperature curve. Note

specially

the few contact

openings

in the

region

of the

polymer glass

transition temperature. ^{We found} that the

shape

of the

Q(T)

_curve

depends

_on the mechanical tension which is

applied

to the

junctions

^while

processed

in order _to

get "good"

electrical contact at _room temperature. ^The mechanism of contact

opening

^above room temperature,

although

not

totally clear,

is then _seen

as the result of the two

competing

actions

of,

_on the _one

hand,

the epoxy thermal expansion which tends to separate the wires

and,

_on the other

hand,

of the mechanical stress

acting

on the

junction

which _opposes to it. Besides the

applied

mechanical

tension,

this stress also

comprises

those

appearing

_on

cooling

from

polymerization

temperature

(which

is 120° C in these

materials)

and in the _course of resin

polymerization.

The latter

are known _to

reorganize locally

when the

polymer glass

transition is

crossed,

while

remaining grossly statistically

identical below and above the transition

respectively

_over successive thermal

cycles.

This insures that contact

opening

is _a random _process which _can be described

by

_a definite distribution function.

Admitting

from _{now on} that such contact

openings

_among ^conductive

particles

is _a

likely

mechanism for

connectivity change

in

composites

_{we may}

analyze

which consequences on the temperature

dependence

of the

composites resistivity

_are to be

expected.

Note that the fol-

lowing analysis

will _not consider the weak

anomaly

^around _Tg

since, although

it has the _same

origin,

it is _more difficult to model.

The

assembly

of short carbon fibers inside the

composite

is viewed _{as a} random lattice of fiber-te-fiber contacts with number

density

_p. We make the first reasonable

assumption

that there is

negligible

^Joule

heating

while

measuring

^the

^composite

resistance _so that the temperature ^{is uniform} over the

sample.

It results that contact

opening

_may involve _{any two}

initially

connected

particles

whether

they belong

to the infinite cluster _or to a finite size cluster.

We make the second

assumption

that contact

opening

statistics is describable

by

_a

probability density

function

q(T)

which has _non-zero values in _a

given

temperature _range

(Tm, TM)

above

room temperature: the

probability

for _a contact to open when the temperature T in the convenient interval is

changed by

dT will be

q(T)dT

and the

proportion

of open ^contacts

~~~~~"~~ '~~~~ ~ _"'

T

Q(T)

₌

/ _{q(r)dr (I)}

Tm

Let po be the number

density

of

interparticle

contacts at _room temperature in _a

given composite

with fiber volume concentration 16. It must be _an

increasing

function of16. The

relationship

between _po and 16 is not known. The excluded volume estimate of critical number of contacts per fiber is of _no

help

here since in the present system it has been demonstrated that contact formation

during

material

processing

is not

totally

accounted for

by geometrical

_arguments

a(T) (%)

80

40

20

O

oUnoUnco~u~~u~o-UnoUnoUnc3~c3~Comc3mCo~u~~u~~u

~u~ummm~~UnUn~o~o~~CoCoCoc3c3oo--~u~umm+~Un

Fig. 3. Typical contact opening histogram

Q(T)

obtained during 20 successive heatings from _room temperature to 160°C _{of 20} model fiber-fiber contacts,

[13, 17].

^Let _pc ^{be the} value of _po for the critical concentration 16*

(as

observed at _room

temperature).

As _a result of

heating,

the actual

density

_p will decrease _as:

P(T)

₌

Po(I Q(T)) (2)

Depending

_on both the

shape

of

q(T)

and the initial _po

value, p(T)

will

eventually equal

p~ ^at a

given

temperature T~ and the

composite

will become

insulating Q(T~)

₌ ^{~° ~~}

Pc is the

proportion

of contacts which have _to be broken in order to

bring

the random lattice of

interparticle

contacts to the

percolation

threshold. It is

expected

to increase with'the fiber volume concentration since the

probability

^for a contact to

belong

to the infinite cluster

increases with16

(or po).

^{Provided that} the _po value is such that the usual _power law

dependence (p

_pc)~ for the

conductivity

holds at _room temperature

(this

is the _case for all

samples

_{we are}

going

to consider below _{as seen} in

Fig, I),

then the temperature

dependence

of the

composite

resistivity

^is:

p cc

(p(T) pc)~~ (when

T <

Tc)

which

finally

leads to

j

~

~

_PO

~~~j

^~~

~

~ @j

^~~

~~~

Po Po Pc

Q (Tc)

where _po is _some reference material

resistivity:

it should be the _room temperature

resistivity

but,

due _to the weak

anomaly

at

Tg,

^{it will be the}

resistivity just

above it. Note that all these power laws would be _exact

if,

in the _open state, the contacts had infinite resistance which _we

508 JOURNAL DE PHYSIQUE I N°5

know not to be _true: when _a contact is

"broken",

it is

replaced by

_a

large

but finite resistance in the random lattice and for _p close

enough

to pc

(high enough temperature)

the _power law

dependence

is _no

longer

valid.

However,

^the actual

"open

state" resistance is

large enough

^for

these

relationships

to remain vali@ _up to temperatures ^which are very close to the

beginning

of the

plateau. Besides,

_as there is _{no reason} for the

relationship

between _po and F _to be critical at the _room

temperature percolation transition,

the

conductivity

exponent ^value to be used is the _one which _was deduced from the concentration

dependence

of the _room temperature

conductivity,

I,e, t ₌ 3.

The relative

change

with temperature of the

composite resistivity

_can then be calculated if

q(T)

is known. This is not the _case since

first,

it is _not

simply

related to the

experimental Q(T)

of the model

junctions (there

is _no

equivalent

in the

composit~

to the

applied

mechanical

tension)

and

second,

it

depends

_on the internal stress distribution inside the

composite

which is not known.

Assuming

for

q(T)

a function which satisfies the

requirement

of _non-zero

opening probability

within _a limited temperature range, we may on the contrary

by

_curve

fitting

to the

experimental

data obtain

q(T). Taking

for

q(T)

_a Gaussian distribution

q(T)

₌

(I /s/&)

exp

[(T ^Ts)~ /2s~j ⁽⁴⁾

we

performed

_a curve

fitting

_on the data above Tg ^with

Tc,

7~ and S _as three

adjustable

parameters. Note that Tc is

only slightly adjustable

since it _must be below and _very close to the

beginning

of the

plateau. Figure

2 shows the

typical

result obtained with the data for the

sample containing

2.5~ fibers with Tc ₌ 187 +

2°C,

Ts ₌ 162 + 4°C and S

= 19 + 1°C. These values

satisfy

the

respective requirements

of

plateau

^closeness and

negligible

contact

opening

outside _a finite temperature _range as shown

by

the

corresponding Q(T) offigure

⁴ which has been calculated with the above set of parameters. ^As

expected,

the temperature _{range over} which there is

agreement

between calculated and

experimental points

is limited _on the lower side

by

the Tg

anomaly,

and _on the _upper side

by

the finite _open state resistance effect which

actually precludes

_any resistance

divergence,

Table I reports ^{the values of the} parameters as obtained _{on a} limited series of

samples

with

varying

fiber volume concentration between I and 3.5~. Given the

experimental

and

fitting uncertainties,

_one observes the

interesting

feature of concentration

independent q(T) (constant

7~ ^and

S).

This demonstrates

that,

at least for materials with fiber concentrations in the range of this

study,

contact

opening

is _a local effect with _no correlation _among

neighbouring

contacts. In other

words,

all material and

processing

parameters

being

held constant, the

opening probability

function

depends only

_on the matrix material constants and _{not on} the

composite

material constants

(which depend on16)

this

interesting

conclusion contradicts

some earlier works. The distribution function

q(T) being independent

of

concentration,

the Tc increase is in

good

_agreement with the _necessary increase with ₁₆ ofthe

proportion

of contacts to be

opened

for

reaching

the critical _contact

density

_pc.

Noting

that the values of n and S

obtained for the

sample containing

^2.5~ fibers _{are very} close to the _means of the

respective

values for all

compositions,

_we have located the

respective

temperatures Tc

(Fig. 4).

For each

composition,

the value

Q (Tc), gives

^the

proportioi

of _{contacts to} be

opened

for

bringing

the

corresponding

medium to

percolation

threshold. It increases with the fiber concentration.

In summary, we have shown that

percolating

media made ofshort carbon fibers

dispersed

in

a

polymer

have _a

peculiar

temperature

dependence

of the resistance which _may be accounted for

by

_a

phenomenological

model of

varying connectivity.

We have shown that the latter _may be described

by

_some function

q(T)

which is concentration

independent:

it _can be used for

characterizing

different series of materials. The

relationship between,

_on the _one

hand,

the contact

opening density

function

and,

_on the other

hand,

the

polymer

thermomechanical _con- stants and

processing

_parameters has been

extensively

studied [18]: ^{the results will be}

reported

ioo

Q(T) (ib)

80

60

40

20

I'

O

loo 200

T

(oc)

Fig. 4. Cumulated contact opening function

Q(T)

in composites calculated with relations

(1)

and

(4)

and values of the parameters: Ts ₌ 162°C and S

= 19°C. Vertical lines show the location of the

respective values of Tc for the five investigated composites.

Table I. Concentration

dependence

of the parameters

7~, S,

and

Tc.

Note that the values of the first two parameters are

fairly

constant whereas Tc increases with 16.

~(i~)

158.5 19 161

2 161 19 167

2.5 162 19 187

3 164 19 199

3.5 158 20 205

510 JOURNAL DE PHYSIQUE I N°5

elsewhere. In this

article,

_we have considered the

only

case of

connectivity

_as controlled

by

temperature ^{when the material is heated.}

Temperature

increase may also be achieved

by

Joule

heating:

^there is

experimental

evidence that contact

opening

also _occurs

giving

rise to

strongly

non-linear resistance _versus

voltage

characteristics [18]. ^{Work is} now in _progress for

studying

this different _case where contact

opening

involves the

only

fibers which

belong

to the backbone of the infinite cluster.

Finally,

the _same

approach

_may be

applied

to the _case of random normal

metal/superconducting grain composites

in which

grain-tc-grain superconducting coupling

is _a function of temperature ^and

magnetic

field: the results of this

study

will be

published shortly.

Acknowledgements.

We

acknowledge

financial support of this

study by

Le Carbone Lorraine

Cy.

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