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The final system to be considered is a two phase, interpenetrating composite [27,28] in which both phases are percolated in three dimensions. The phases have the following properties: σ1 = 1.0, σ2 = 0.5, k1 = 1, k2 = 10000. Phase 2 is formed by placing overlapping spheres at random locations in the digital microstructure. Periodic boundary conditions were maintained during the sphere placement process. Each sphere had a diameter of 9 pixels, and was placed in a 643 unit cell, giving volume fractions c1 = 0.66 and c2 = 0.34.
The Nyquist plot of the impedance curve of this system is shown in Fig. 7. The data points clearly show that there is more than one arc present, and most probably many arcs, with a distribution of resonant frequencies. However, using a burning algorithm on the digital image [29,30] model, both phases were found to be fully percolated and continuous. The dashed line in Fig. 7 is the Maxwell-Wagner equation [1] an effective medium theory equation based on spherical inclusions:
where x2 is the volume fraction of phase 2, the phase fraction of phase 1 is 1 − x2, and ψ1 and ψ2 are the admittances of phases 1 and 2, respectively. The Maxwell-Wagner equation has been shown, like eq. (7), to be analytically equivalent to a series combination of two RC (resistor in parallel with a capacitor) circuits [25], which will always show two arc behavior if there is a significant difference in relaxation times between the two arcs. The solid line in Fig. 7 clearly shows two well-separated arcs, which does not agree well with the numerical data points. However, the DC resistivity does matches the numerical result within a few percent.
The Maxwell-Wagner equation does not work well for the overall impedance curve in this case because it cannot take into account the overlapping of the spheres which eventually leads to the percolation of the sphere phase. Equation (8) does not have a percolation threshold. This is easily proven by letting ψ2 go to ∞, which does not produce an infinite value of ψ for any value of x2 < 1. However, letting ψ1 go to zero results in ψ being zero for any value of x2. This result means that in the Maxwell-Wagner "microstructure" implied by the equation's derivation, phase 1 is always percolated and phase 2 is never percolated.
Figure 7: Nyquist plot for interpenetrating phase composite model of overlapping spheres. Phase 1 is the matrix, phase 2 is the phase made up from the overlapping spheres. The volume fraction of phase 2 is 0.34.
The dashed line in Fig. 7 is a variation of eq. (8), often referred to as the "self-consistent" approach [32]. This theoretical value for the impedance matches the overall curve much more closely than did the Maxwell-Wagner equation. It is obtained by modifying the derivation of eq. (8) by letting the embedding medium in the derivation be the effective medium, rather than the pure matrix phase [31]. This treats phases 1 and 2 on an equal basis, thereby eliminating the distinction between "matrix" and "inclusion". The resulting prediction for the effective conductivity is:
