The conductivity, dielectric constant 1/f noise and magnetic properties in percolating three-dimensional cellular composites - Chiteme, Cosmas (University of the Witwatersrand, Johannesburg, 2000)
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.
Thesis (Ph.D.)--University of the Witwatersrand, Science Faculty (Physics), 2000.
Keywords: Percolation (statistical physics); Superconducting composites