10.4
Electrical Conductivity of Nanocomposites
For technical applications outside of electronics, the electrical conductivity of
nanomaterials can best be exploited as the electric-conducting phase in nanocomposites. In order to obtain electrical conductivity in a composite consisting
of conducting and nonconducting phases, the conducting particles must touch each
other to form a continuous series of conducting elements. Clearly, the probability of
forming a continuous electrical conducting system increases with increasing
concentration of the conducting phase. The concentration at which such a continuous system is formed is termed the percolation threshold.
The electrical conductivity of nanocomposites depends on percolation, the theory
of which treats the properties of two-phase mixtures consisting either of conducting
and insulating phases, or of a solid and pores. In the latter case, percolation leads to
the formation of a network of open pores. Detailed theories of percolation consider
the shape and concentration of the constituents, but in all cases the crucial question
relates to the critical concentration p c , the percolation threshold, where the minority
phase of the mixture forms a continuous network. Assuming an electric-conducting
nanocomposite – a two-phase mixture consisting of an insulating and an electrical
conductive phase – the percolation threshold describes, in simple terms, the
concentration of conductors required for the onset of electrical conductivity. At
concentrations below the percolation threshold, there is no electrical conductivity,
whereas above the threshold conductivity is observed. Above the percolation
threshold p c , the electrical conductivity s is described by [14]:
s ¼ s 0 ðp À p c Þ
a
ð10:14Þ
where s 0 is the conductivity of the conducting phase and p is the volume fraction of
the conducting phase. The exponent a reflects the dimensionality of the network;
usually, this is found not to be an integer, and experimental values of a are found to
range between 1.3 and 3. Equation (10.14) is linearized in a double logarithmic
graph, where log(s) is plotted against log(p À p c ).
As mentioned above, percolation is extremely sensitive to the shape and aspect
ratio of the second phase’s particles. Therefore, any theory describing percolation
must of necessity consider the aspect ratio of the second phase. For this, many
stochastic theories have been devised, although most are so complex that their
technical use is nearly excluded. As an example, a description of percolation for a
fiber-shaped second phase was reported by Balberg [15] and this theory led to the
following equation for the percolation threshold:
p c ¼ 0:7
L
h i
3
L
3
d
L
h i
ð10:15Þ
where L is the length of the particles and d is the particle diameter. This theory
assumes a second phase with constant diameter and a distribution of lengths. The
angular brackets k i denote mean values. It is an important characteristic that the
288j 10 Electrical Properties of Nanoparticles
Electrical Conductivity of Nanocomposites
For technical applications outside of electronics, the electrical conductivity of
nanomaterials can best be exploited as the electric-conducting phase in nanocomposites. In order to obtain electrical conductivity in a composite consisting
of conducting and nonconducting phases, the conducting particles must touch each
other to form a continuous series of conducting elements. Clearly, the probability of
forming a continuous electrical conducting system increases with increasing
concentration of the conducting phase. The concentration at which such a continuous system is formed is termed the percolation threshold.
The electrical conductivity of nanocomposites depends on percolation, the theory
of which treats the properties of two-phase mixtures consisting either of conducting
and insulating phases, or of a solid and pores. In the latter case, percolation leads to
the formation of a network of open pores. Detailed theories of percolation consider
the shape and concentration of the constituents, but in all cases the crucial question
relates to the critical concentration p c , the percolation threshold, where the minority
phase of the mixture forms a continuous network. Assuming an electric-conducting
nanocomposite – a two-phase mixture consisting of an insulating and an electrical
conductive phase – the percolation threshold describes, in simple terms, the
concentration of conductors required for the onset of electrical conductivity. At
concentrations below the percolation threshold, there is no electrical conductivity,
whereas above the threshold conductivity is observed. Above the percolation
threshold p c , the electrical conductivity s is described by [14]:
s ¼ s 0 ðp À p c Þ
a
ð10:14Þ
where s 0 is the conductivity of the conducting phase and p is the volume fraction of
the conducting phase. The exponent a reflects the dimensionality of the network;
usually, this is found not to be an integer, and experimental values of a are found to
range between 1.3 and 3. Equation (10.14) is linearized in a double logarithmic
graph, where log(s) is plotted against log(p À p c ).
As mentioned above, percolation is extremely sensitive to the shape and aspect
ratio of the second phase’s particles. Therefore, any theory describing percolation
must of necessity consider the aspect ratio of the second phase. For this, many
stochastic theories have been devised, although most are so complex that their
technical use is nearly excluded. As an example, a description of percolation for a
fiber-shaped second phase was reported by Balberg [15] and this theory led to the
following equation for the percolation threshold:
p c ¼ 0:7
L
h i
3
L
3
d
L
h i
ð10:15Þ
where L is the length of the particles and d is the particle diameter. This theory
assumes a second phase with constant diameter and a distribution of lengths. The
angular brackets k i denote mean values. It is an important characteristic that the
288j 10 Electrical Properties of Nanoparticles