Abstract
In this paper large resistor-capacitor (RC) networks that consist of randomly distributed conductive and capacitive elements which are much larger than those previously explored are studied using an efficient algorithm. We investigate the emergent power-law scaling of the conductance and the percolation and saturation limits of the networks at the high and low frequency bounds in order to compare with a modification of the classical Effective Medium Approximation (EMA) that enables its extension to finite network sizes. It is shown that the new formula provides a simple analytical description of the network response that accurately predicts the effects of finite network size and composition and it agrees well with the new numerical calculations on large networks and is a significant improvement on earlier EMA formulae. Avenues for future improvement and explanation of the formula are highlighted. Finally, the statistical variation of network conductivity with network size is observed and explained. This work provides a deeper insight into the response of large resistor-capacitor networks to understand the AC electrical properties, size effects, composition effects and statistical variation of properties of a range of heterogeneous materials and composite systems.
Highlights
In this paper large resistor-capacitor (RC) networks that consist of randomly distributed conductive and capacitive elements which are much larger than those previously explored are studied using an efficient algorithm
In the case of binary mixtures where the components have a variable conductivity ratio, other analytical approaches are required. These include: empirically derived mixing laws, such as equation (2), which are valid in the intermediate powerlaw region [4]; averaging approaches such as the Effective Medium Approximation [23], valid away from the critical percolation probability where network size does not have an effect; approaches based on the distribution of the polezero spectra [8], which account for network size effects near the critical mixing ratio but only approximately capture the percolation and saturation limits away from criticality; as well as more speculative combinations thereof, as described
In this paper we have investigated and compared models representing the conduction flow-paths though disordered random composite media in terms of new Medium Approximation (MEMA) formula and RC networks that are much larger than previously explored
Summary
For networks of randomly sited single conductive components, classical percolation theory gives good results [22]. In the case of binary mixtures where the components have a variable conductivity ratio (here due to the capacitive components having frequency-dependent admittance), other analytical approaches are required These include: empirically derived mixing laws, such as equation (2), which are valid in the intermediate powerlaw region [4]; averaging approaches such as the Effective Medium Approximation [23], valid away from the critical percolation probability where network size (which represents granularity in a real material) does not have an effect; approaches based on the distribution of the polezero (generalised eigenvalue) spectra [8], which account for network size effects near the critical mixing ratio but only approximately capture the percolation and saturation limits away from criticality; as well as more speculative combinations thereof, as described next. This is because we have made the assumption that the networks we are considering have an infinite size
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