LETTER
Anomalous electrical conductivity and percolation in carbon
nanotube composites
Chunsheng Lu
Æ
Yiu-Wing Mai
Received: 18 October 2007 / Accepted: 6 March 2008 / Published online: 8 August 2008
Ó
Springer Science+Business Media, LLC 2008
Transport properties of disordered multiphase materials,
such as electrical and thermal conductivities, have been an
active research area in statistical physics for decades. In a
composite consisting of conductive Fllers dispersed in an
insulating matrix, there is a well-deFned insulator–
conductor transition when an inFnite conductive network
or path throughout the matrix is formed. This process can
be well described by percolation theory [
1
,
2
]. Recently,
carbon nanotube (CNT)-reinforced composites and sus-
pensions have attracted a great deal of attention due to their
excellent properties and many potential applications. CNTs
have a unique set of mechanical and physical properties,
including extremely high Young’s modulus, strength,
electrical, and thermal conductivities. The current experi-
ments showed that CNT-reinforced composites exhibit an
electrical percolation with addition of 0.1 vol.% or less
Fllers, at which electrical conductivity rises sharply by
several orders of magnitude [
3
–
7
]. Here, the percolation
threshold of CNTs is closely dependent on their geometric
factors (e.g., volume fraction, size, shape, and orientation)
and the interaction between them. It is a critical issue in
producing conductive composites for use in Flms, coatings,
and paints since the lower percolation threshold can reduce
the loading of expensive CNTs, leading to lighter com-
posites. In comparison with composites reinforced with
isotropic particles, however, the percolation threshold of
composites containing highly anisotropic conductor Fllers
such as CNTs is still not well understood.
The term ‘‘percolation’’ refers to the onset of a sharp
transition or an inFnite network (or cluster) at which long-
range connectivity suddenly appears [
1
]. The electrical
conductivity near the percolation threshold is anomalously
greater than that predicted by traditional theoretical models,
such as Maxwell, Hamilton-Crosser models [
8
]. Intuitively,
the electrical percolation process in CNT composites is
similar to traditional ones with the addition of isotropic
conductive particles, but with an ultra-low percolation
threshold. As shown in ±ig.
1
, near the percolation thresh-
old, the probability or fraction of a CNT,
P
, on the inFnite
cluster obeys a power law and can be described as
P
±½
/
ÿ
/
c
ð
a
Þ²
b
ð
a
Þ
;
ð
1
Þ
where
/
is the volume fraction of CNT Fllers (in vol.% or
wt.%),
/
c
is the percolation threshold, and
b
is the con-
nectivity exponent. Similarly, the electrical conductivity,
r
, in the system increases monotonically for
/
[
/
c
but
ð
/
ÿ
/
c
Þ³
1 and follows a universal power law
r
/
ÿ
/
c
ð
a
Þ²
t
ð
a
Þ
;
ð
2
Þ
where
t
is the conductivity exponent. Here, both the per-
colation threshold and connectivity (or conductivity)
exponents are a function of aspect ratios of CNTs,
a
=
L
/
D
,
with
L
and
D
being length and diameter, respectively. It is