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Polymer-based nanocomposites 147

5.2.6.2 Maxwell-Garnett equation

This model is relatively easy for modeling due to its linearity. However, this model
shows better results for lower filler loading, below the percolation threshold. It pro-
poses that the separation distances between the randomly oriented inclusions are
larger than their characteristic sizes [183,184]. This model is not restricted by the
resistivity of the filler or the polymer matrix in a polymer nanocomposite. The effec-
tive dielectric permittivity of the final mixture for spherical inclusions is given by
Eq. (5.14) [69]:

" #

3φ ε f ε m
f
ε eff ¼ ε m 1+ (5.14)
φ m ε f ε m +3ε m
However, for the randomly oriented ellipsoids, the effective dielectric permittivity for
such an isotropic mixture in the jth direction is given by Eq. (5.15) [185,186]:
f
φ X ε f ε m
3 j¼x,y,z ε m + A j ε f ε m

ε eff ¼ ε m + ε m (5.15)
f
φ X A j ε f ε m
1
3 j¼x,y,z ε m + A j ε f ε m
where A is the depolarization factor. For spherical inclusions (i.e., when A¼1/3),
Eq. (5.15) will simply revert to Eq. (5.14).

5.2.6.3 Bruggeman self-consistent effective medium
approximation
This model fits well for comparatively larger filler concentration as compared with the
Maxwell-Garnett equation, which is best for lower loadings. This model gives better
results when the fillers approach to near agglomeration [69]. The main formula of
Bruggeman’s model for spherical fillers is given by Eq. (5.16):

1 φ f + φ f ¼ 0 (5.16)
ε m ε eff ε f ε eff
ε m +2ε eff ε f +2ε eff
As can be seen from Eq. (5.16), the filler and polymer matrix are symmetrically
related. Moreover, Eq. (5.16) can be solved for the final Bruggeman’s formula as
shown in Eq. (5.17):

1 φ
ε f ε eff f ε f ε m
¼ (5.17)
ε 1=3 1=3
eff ε m
Eq. (5.17) is best for filler concentration smaller than the percolation thresholds.