Page 174 - Polymer-based Nanocomposites for Energy and Environmental Applications

148 Polymer-based Nanocomposites for Energy and Environmental Applications

5.2.6.4 Percolation theory

In a multicomponent system, there is an insulator-metal transition due to the produc-
tion of continuous networks in the polymer nanocomposite at the percolation thresh-
old (Fig. 5.4B) [187]. Generally, the percolation of fillers has been classified into two
regimes, i.e., soft and hard percolation [188]. Soft percolation is characteristic of the
interconnected form of nanoparticles, which causes a sudden drop in breakdown
strength and thus not much appreciated. The hard percolation causes local
nanoparticles packing density to a maximum, which often leads to local air voids for-
mation. Because of the accumulation of air voids, the breakdown strength does not
follow a regular trend. For the insulator and conducting fillers, dielectric constant
and conductivity are given by the power law (Eqs. 5.18–5.21) [48,69,185,189,190].
Dielectric permittivity:

ε eff ∝ε m φ φ p q for q p < q c (5.18)
c
Electrical conductivity (σ c ):

σ c ∝σ p φ φ c t for φ > φ c (5.19)

p
p
σ c ∝σ m φ φ p q for φ < φ c (5.20)

c
p

u 1 u
σ c ∝σ σ for φ φ ! 0 (5.21)

p m p c
φ p designates the volume fraction of metallic fillers; φ c is the volume fraction at per-
colation threshold; σ m and σ p denote the electric conductivities of polymer matrix and
fillers, respectively; t and q designate the critical exponents in the conducting and
insulating regions having values lying in the range of 1.6–2.0 and 0.8–1.0, respec-
tively; and u is equal to t/(t+q).
Relevance of percolation in polymer nanocomposites
Percolation results in conduction due to the generation of an interconnecting network
that causes tunneling [11]. The interparticle tunneling conductivity, σ t , is given by
Eq. (5.22) [129,191]:

l 2b
σ t ∝ exp (5.22)
d
where l is the distance between the particles of radius b and d in the typical tunneling
range. When the interparticle distance l lowers up to near the percolation threshold, σ t
begins to increase exponentially. The percolation threshold depends on the shape,
size, and orientation of the fillers [192]. For a homogeneous polymer nanocomposite
with uniformly sized spherical fillers, the percolation threshold is 0.16 (also known
as Sher-Zallen invariant), although in practical situations, it ranges from 0.013 to 0.17