Abstract:
Percolation phenomena are studied in a series of composites, each with a cellular
structure (small conductor particles embedded on the surfaces of large insulator
particles). The DC and AC conductivities, l/f noise and magnetic properties (in some
series) are measured in the systems consisting of Graphite, Graphite-Boron Nitride,
Carbon Black, Niobium Carbide, Nickel and Magnetite (Fe304) as the conducting
components with Talc-wax (Talc powder coated with 4% wax by volume) being the
common insulating component. Compressed discs of 26mm diameter and about 3mm
thickness (with various conductor volume fractions covering both the insulating and
conducting region) were made from the respective powders at a pressure of 380MPa
and all measurements were taken in the axial (pressure) direction.
The conductivity (σm) and dielectric constant (εm) of percolation systems obey the
equations: σm = σc( ɸ - ɸc)t for ɸ >ɸc; σm = σi( ɸc - ɸ-s and εm = εi( ɸc - ɸ-s' for ɸ < ɸc;
outside of the crossover region given by ɸc± (δdc ~=(σi/σc)1/(t+s). Here ɸc is the critical
volume fraction of the conductor (with conductivity σ = σc) and cri is the conductivity
of the insulator, t and s are the conductivity exponents in the conducting and
insulating regions respectively and S’ is the dielectric exponent. The values of s and t
are obtained by fitting the DC conductivity results to the combined Percolation or the
two exponent phenomenological equations. Both universal and non-universal values
of the sand t exponents were obtained. The dielectric exponent S’, obtained from the
low frequency AC measurements, is found to be frequency-dependent. The real part
of the dielectric constant of the systems, has been studied as a function of the volume
fraction (ɸ) of the conducting component. In systems where it is measurable beyond
the DC percolation threshold, the dielectric constant has a peak at ɸ > ɸ, which
differs from key predictions of the original Percolation Theory. This behaviour of the
dielectric constant can be qualitatively modeled by the phenomenological two
exponent equation given in Chapter two of this thesis. Even better fits to the data are
obtained when the same equation is used in conjunction with ideas from Balberg's
extensions to the Random Void model (Balberg 1998a and 1998b).
At high frequency and closer to the percolation threshold, the AC conductivity and
dielectric constant follow the power laws: σm( ɸ,שּׂ) ~ שּׂX and εm( ɸ,שּׂ) ~ שּׂ-Y
respectively. In some of the systems studied, the x and y exponents do not sum up to
unity as expected from the relation x + y = 1. Furthermore, the exponent q obtained
from שּׂ x σm( ɸ,O)q in all but the Graphite-containing systems is greater than 1, which
agrees with the inter-cluster model prediction (q = (s + t)/t). The Niobium Carbide
system is the first to give an experimental q exponent greater than the value calculated
from the measured DC s and t exponents.
l/f or flicker noise (Sv) on the conducting side (ɸ > ɸc) of some of the systems has
been measured, which gives the exponents k and w from the well-established
relationships Sv/V2 = D(ɸ - ɸc)-k and Sv/V2 = KRw. V is the DC voltage across the
sample with resistance R while D and K are constants. A change in the value of the
exponent k and w has been observed with k taking the values kl ~ 0.92 - 5.30 close to
ɸc and k2 ~ 2.55 - 3.65 further into the conducting region. Values of WI range from
0.36 -1.1 and W2 ~ 1.2 - 1.4. These values of ware generally well within the limits of
the noise exponents proposed by Balberg (1998a and 1998b) for the Random Void
model. The t exponents calculated from k2 and W2 (using t = k/w) are self-consistent
with the t values from DC conductivity measurements. Magnetic measurements in
two of the systems (Fe304 and Nickel) show unexpected behaviour of the coercive
field and remnant magnetisation plotted as a function of magnetic volume fraction.
Fitting the permeability results to the two exponent phenomenological equation gives
t values much smaller than the corresponding DC conductivity exponents.
A substantial amount of data was obtained and analysed as part of this thesis.
Experimental results, mostly in the form of exponents obtained from the various
scaling laws of Percolation Theory, are presented in tabular form throughout the
relevant chapters. The results have been tested against various models and compare
with previous studies. While there is some agreement with previous work, there are
some serious discrepancies between the present work and some aspects of the
standard or original Percolation Theory, for example the dielectric constant behaviour
with conductor volume fraction close to but above ɸc. New results have also emerged
from the present work. This includes the change in the noise exponent k with (ɸ - ɸc),
the variation of the dielectric exponent s' with frequency and some DC scaling results
from the Fe304 system. The present work has dealt with some intriguing aspects of
Percolation Theory in real continuum composites and hopefully opened avenues for
further theoretical and experimental research.