A Percolation Equation for Modeling Experimental Results for Continuum Percolation Systems
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A Percolation Equation for Modeling Experimental Results for Continuum Percolation Systems
D.S. McLachlan, C. Chiteme, W.D. Heiss and Junjie Wu School of Physics and Materials Physics Institute, University of the Witwatersrand, P O Wits 2050, Johannesburg, South Africa.
ABSTRACT The standard percolation equations or power laws, for dc and ac conductivity (dielectric constant) are based on scaling ansatz, and predict the behaviour of the first and second order terms, above and below the percolation or critical volume fraction (φc), and in the crossover region. Recent experimental results on ac conductivity are presented, which show that these equations, with the exception of real σm above φc and the first order terms in the crossover region, are only valid in the limit σi/σc = 0, where for an ideal dielectric σi=ωε0εr. A single analytical equation, which has the same parameters as the standard percolation equations, and which, for ac conductivity, reduces to the standard percolation power laws in the limit σi(ωε0εr)/σc = 0 for all but one case, is presented. The exception is the expression for real σm below φc, where the standard power law is always incorrect. The equation is then shown to quantitatively fit both first and second order dc and ac experimental data over the entire frequency and composition range. This phenomenological equation is also continuous, has the scaling properties required at a second order metal-insulator and fits scaled first order dc and ac experimental data. Unfortunately, the s and t exponents that are necessary to fit the data to the above analytical equation are usually not the simple dimensionally determined universal ones and depend on a number of factors. INTRODUCTION Quantitative models for the dc and complex ac conductivity σm (dielectric constant εm) of good conductor-bad conductor composites can be used not only to understand their properties and, hopefully, design new or better electro- and other composites, but can also be used to gauge the microstructure from electrical measurements. The most often used models to interpret experimental results are effective media theories and percolation theories. These theories are reviewed in Landauer [1], Mclachlan et al [2], Bergman and Stroud [3], Nan [4], Clerc et al [5] and Mclachlan [6], where some experimental data is also presented. The standard percolation [3,4,5] equations are based on the original work of Efros and Shklovskii [7] and Bergman and Imry [8]. (Note that references [3,4,5] also discuss other approaches and models for the conductivity of composites.) It has recently become apparent that percolation theory, as given in [3,4,5] is unable to model recent ac experimental results [9,10,11,12,13,14], except in the limit where one component is a perfect conductor (σc) or the other a perfect insulator (σi =0), provided that the measurements are done at zero frequency (σi = iωε0εr =0). In this paper the ansatz upon which the standard percolation equations, or power laws, are based are given and the practical limits of their va
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