10.4 Electrical Conductivity of Nanocomposites 243
Box 10.2 Percolating Systems
The theory of percolation is applied quite universally. In the case of nanocomposites it describes the trend of the electrical conductivity as a function of the
filler shape and content, in the case of porous materials the same theory
describes the permeability for gases as a function of the porosity, and there are
many more applications.
In the theory of percolation, the crucial question is the estimation of the
percolation threshold. Considering a composite consisting of prolate (a prolate
body is an ellipsoid where one axis is longer than the diameter) particles (as a
model, for example, for nanotubes) the diameter d and the length l of the particles rule the position of the percolation threshold. Following the theory of
Balberg [10], the percolation threshold p c , the concentration, where the first
electrical conductivity is observed, is given by
p
l
l
d
l
c = 0 7
3
3
.
.
(10.10)
The 〈〉 brackets stand for mean values of an ensemble. In any possible length
distribution,
l
l
3
3
1
≤ is valid. To simplify the considerations, fibers of equal
length l and equal diameter d are assumed; then Eq. (10.10) is simplified to:
p
d
l a
c =
∝
0 7
1
.
.
(10.11)
It is an important characteristic of percolation that the percolation threshold
increases with the mean aspect ratio a
l
d
=
. As a consequence, one sees that
the concentration of particles necessary for the onset of percolation is, for
fibers, up to a few orders of magnitude less compared to spherical particles.
Therefore, to obtain optically transparent electric conductive composites one
has to apply extremely thin long fibers, usually realized with nanotubes or
nanowires.
According to the considerations of Stauffer and Aharoni [11], above the percolation threshold p c , the electrical conductivity σ is described by an exponential
law:
σ σ
=
−
0 (
) .
p p
t
c
(10.12)
In Eq. (10.12) σ 0 is the conductivity of the conducting phase and p the volume
fraction of the conducting phase. The exponent t reflects the dimensionality of
the network; usually, it is found to be not an integer number; ranging between
1.3 and 3. This exponent is determined by plotting the experimental conductivity values linearized in a double-logarithmic graph, log(σ) versus log(p − p c ).
Box 10.2 Percolating Systems
The theory of percolation is applied quite universally. In the case of nanocomposites it describes the trend of the electrical conductivity as a function of the
filler shape and content, in the case of porous materials the same theory
describes the permeability for gases as a function of the porosity, and there are
many more applications.
In the theory of percolation, the crucial question is the estimation of the
percolation threshold. Considering a composite consisting of prolate (a prolate
body is an ellipsoid where one axis is longer than the diameter) particles (as a
model, for example, for nanotubes) the diameter d and the length l of the particles rule the position of the percolation threshold. Following the theory of
Balberg [10], the percolation threshold p c , the concentration, where the first
electrical conductivity is observed, is given by
p
l
l
d
l
c = 0 7
3
3
.
.
(10.10)
The 〈〉 brackets stand for mean values of an ensemble. In any possible length
distribution,
l
l
3
3
1
≤ is valid. To simplify the considerations, fibers of equal
length l and equal diameter d are assumed; then Eq. (10.10) is simplified to:
p
d
l a
c =
∝
0 7
1
.
.
(10.11)
It is an important characteristic of percolation that the percolation threshold
increases with the mean aspect ratio a
l
d
=
. As a consequence, one sees that
the concentration of particles necessary for the onset of percolation is, for
fibers, up to a few orders of magnitude less compared to spherical particles.
Therefore, to obtain optically transparent electric conductive composites one
has to apply extremely thin long fibers, usually realized with nanotubes or
nanowires.
According to the considerations of Stauffer and Aharoni [11], above the percolation threshold p c , the electrical conductivity σ is described by an exponential
law:
σ σ
=
−
0 (
) .
p p
t
c
(10.12)
In Eq. (10.12) σ 0 is the conductivity of the conducting phase and p the volume
fraction of the conducting phase. The exponent t reflects the dimensionality of
the network; usually, it is found to be not an integer number; ranging between
1.3 and 3. This exponent is determined by plotting the experimental conductivity values linearized in a double-logarithmic graph, log(σ) versus log(p − p c ).