Next: Discussion and Conclusions Up: Main Previous: Model Verification: Exactly

# Two-Phase Interpenetrating Random Network

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:

where the parameters are the same as in eq. (8), and the square root is the complex square root. Eq. (9) is the solution to the quadratic equation that results from making eq. (8) self-consistent. Eq. (9) does display a percolation threshold. In the limit where ψ2 approaches zero, eq. (9) gives a critical value of x2 of 1/3. The actual critical value of x2 for overlapping spheres to percolate is x2 = 0.29 [33]. This is why eq. (9) matches the overall shape of the impedance curve better than did eq. (8). Although we have not checked this explicitly, it is almost certain that eq. (9) is not expressible as a finite number of RC circuits, but rather produces a continuous spectrum of resonant frequencies. The overlapping sphere microstructure, like many real random microstructures, has many differently-shaped microstructural features, and thus shows a broadened, non-circular Nyquist impedance plot.

Next: Discussion and Conclusions Up: Main Previous: Model Verification: Exactly