Page 170 - Polymer-based Nanocomposites for Energy and Environmental Applications

144 Polymer-based Nanocomposites for Energy and Environmental Applications

2
r ψ rðÞ ¼ eε 1 X z i n i ∞ðÞe z i eψ rðÞ=kT (5.9)
i

where ψ(r) is the potential distribution function varying with distance, r, from the sur-
face of the nanoparticle, ε is the dielectric constant of the medium, k is the Boltzmann
constant, and z i and n i (∞) are the valency and concentration of ionic species i in the
bulk matrix, respectively. For a lower potential, the Debye-H€ uckel approximation can
be represented given below:

|z i eψ rðÞ=kT| < 1 (5.10)
The solution of the Eq. (5.9) is simplified to Debye-H€ uckel form as shown below:

1=2
!
2e 2 X 2
kr
ψ rðÞ ¼ ψ e ;k ¼ z n i ∞ðÞ (5.11)
o
i
εkT
i
1
where κ is the Debye-H€ uckel parameter in the units of m . The inverse of κ is called
Debye length. The solution of Eq. (5.11) for a diffused double layer is given by the
below equation:

z i eψ rðÞ z i eψ oðÞ kr
tanh ¼ tanh e (5.12)
4kT 4kT
Eq. (5.12) is also known as Gouy-Chapman equation, which represents the potential
variation in the diffused part of the double layer starting from the Stern layer [153].
Lewis argued that the properties of the polymer-filler interface would become
dominant over the bulk properties of the constituents as the size of the filler particles
decreases to the nanometer scale [35,167]. So, engineering the filler surface was a
good idea to obtain better dispersion of the dielectric filler particles. Hence, the large
surface area of the filler amplifies the unique interface properties in the nanodielectric.
Sun et al. studied this concept experimentally on epoxy-/silica-based composites and
explored the effect of the interface on the dielectric properties [13,176].

5.2.5.2 Tanaka’s model
This model explores the role of interface in defining the dielectric properties of the
nanocomposites [34]. He found that a reduction in the internal field occurred on addi-
tion of nanofiller due to the decrease in particle size [41]. As mentioned earlier, the
large particle size causes greater field distortion and larger dielectric loss as compared
with the nanocomposites. Further, the variation in the space charge distribution has
been proposed to improve the dielectric behavior in the nanocomposites [177]. There
is a significant reduction in charge accumulation on addition of a nanofiller, which
reduces the breakdown at lower fields. Tanaka et al. presented a multicore model
and discussed the interfacial structure and charge behavior (Fig. 5.5) [178].