Effective complex permittivity and continuum percolation analysis of two-phase composite media

Abstract

Using ab initio finite-element (FE) calculations we study the dielectric properties of the continuum (off-lattice)-percolation system consisting of two-dimensional equilibrium distributions of randomly distributed circular and partially penetrable disks (or parallel, infinitely long, identical, partially penetrable circular cylinders) throughout a host matrix. Theoretical investigations of the (relative) effective complex permittivity epsiv = epsiv' - iepsiv" were conducted a hybrid modeling that combine standard Metropolis Monte Carlo (MC) algorithm and continuum-electrostatics equations which are solved by finite element calculations. We present the details of the epsiv dependence on surface fraction Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> of the disks, permittivity contrast between the two phases and arbitrary degree of impenetrability lambda (0 les lambda les 1), for wide ranges of these parameters. Careful evaluation of the critical exponents s and t governing the power-law behavior of epsiv' and epsiv" respectively, near the percolation threshold, are used to address controversial or unresolved issues, related to the underlying physics of the classical percolation model. Our results, corresponding to different values of lambda in the range 0 les lambda les 0.9 and for a wide range of phase's permittivity ratios, indicate that s and t can differ from the universal values, i.e. s = t cong 1.3 , characterizing the continuum percolation phenomena of statistically isotropic distributions of disks in a plane. As the distance to Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2c</sub> is decreased, epsiv' and epsiv" display a smooth transition from a power-law dependence, which is well fit by the standard percolation expression, to a plateau regime. We associate the plateau with finitesize effects and the short-range multipolar interactions localized in disk clusters. The radial distribution function (RDF) results are consistent with the notion that larger area fractions lead to an increase in the distance over which one disk influences another via excluded volume effect. Furthermore, we perform a quantitative test of the McLachlan (TEPPE) equation by comparing its prediction of the effective permittivity to the simulation results obtained on systems with overlapping disks (0 les lambda les 0.9). We find that the analytic equation presented by McLachlan is consistent with FE-MC simulations only for Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> < Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2c</sub> . However, the failure of the TEPPE for Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> > Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> can be attributed to a poor representation of the various degrees of disk aggregation present in the equilibrium distributions where increased aggregation results in an enhanced multipolar interaction.