a) b - École Polytechnique de Montréal

where p is the volume fraction of the conducting phase above the percolation threshold, pc is the
percolation threshold, σ is conductivity of sample at given volume fraction of p, and σc is the
conductivity of the sample at the percolation threshold. The conductivity exponent t generally
reflects the dimensionality of the system with values typically around 1.3 and 2.0 for two and
three-dimensions, respectively(Clerc, Giraud, Laugier, & Luck, 1990). Gubbels et al. found that
in a polystyrene (PS)/ polyethylene (PE)/ carbon black (CB) system, t is 2.0 when CB is
dispersed within the amorphous phase of pure PE. Its value becomes 1.3 when CB is at the
interface of a PE/PS blend with dual-phase continuity(Gubbels, Blacher, et al., 2002; Gubbels,
Jerome, et al., 2002). Al-Saleh et al.(Al-Saleh & Sundararaj, 2008) showed that selective
localization of CB at the interface of PS/PP blend is possible by introducing SBS copolymer that
can selectively localize at the interface and for which CB has the highest affinity. Addition of
5% SBS resulted in reduction of percolation threshold to less than 1%.
The classical percolation theory (Equation 2-44) applied to conductive polymer composite
systems must meet some definite conditions. First, the system should be randomly distributed,
and secondly, particles must be spherical, monodisperse and have an isotropic conductivity. If
one or all of these conditions are not fulfilled, the theoretical value will deviate far from the
realistic one. For example, Carmona et al. (Carmona, Prudhon, & Barreau, 1984) reported that in
short fiber-epoxy resin composites, the exponent t = 3.0 ± 0.6 when the length of fibers ranges
between 1.15 mm and 2.85 mm.
A simplified view of percolation transition is depicted in Figure 2-19. At low filler loadings the
conductive particles act like conductive islands in a sea of electrically insulating resin. Since
electrons moving through the composite still encounter the insulating polymer, little or no
change in conductivity is observed. Addition of further conductive material leads to the
conductive particles being more crowded and more likely to come into contact with each other.
Finally, at the percolation threshold, the first complete cluster from the bottom to the top of the
sample is generated.
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