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In order to verify the model for random multi-phase composite microstructures, it is necessary to show that the simulation scheme described above is correct by computing the impedance of simpler microstructures for which analytical solutions exist. Four analytically solvable models will be studied numerically.
The first example is the simplest: a single phase material that has a DC conductivity σ = 0.05 S/unit (mho/unit) and a relative dielectric constant kr = 1000. Fig. 2 shows the Nyquist plot  of this system, where the negative of the imaginary part of the impedance is plotted against the real part of the impedance. The solid line is the exact impedance, and the circles are numerical results. The DC resistance of the sample (R) is shown to be 0.4Ω. Eq. (1) may be used to convert this resistance to conductivity, where A is equal to 50 x 50 = 2500 units2, and D is equal to 50 units since the image used in 503 in size. The conductivity for the system is 0.05 S/unit, as expected. The capacitance, and hence kr, is determined by R and ωo, the angular frequency at the peak of the arc in a Nyquist plot. For a homogeneous system,
where σ is the DC conductivity of the system, and kr and εo are as defined before. Since we can establish both σ and ωo, from the impedance curve, kr can be calculated directly. Approximately locating the arc maximum by interpolating between the data points displayed in Fig. 2 gives a value of kr = 1036, an error of only 3.6%. This error can be attributed to approximating the position of the arc peak, and thus the value of ωo. In this case, since we knew ωo analytically, the value of kr could have been computed even more accurately. However, the interpolation method would be that used in a simulation of a general microstructure, where no analytical results would be available. Table 1 shows the absolute magnitude of impedance, | Z |, for several different frequencies, showing the accuracy available when the frequency is precisely known. The relative errors shown in column 5 are due to computer round off only.
Fig. 2. Nyquist plot for a single phase material, whose admittance is given at all frequencies by ψ = σ + ik r ω, φ = 0.05 S/unit, kr = 1000.
The next two cases considered are two phase composites, with each phase occupying half the volume, arranged in series or parallel, as shown in Fig. 3. The DC composite conductivity for each of these cases is :
|(Series) σ = ( c1 / σ1 + c2/σ2)−1,||(5)|
|(Parallel) σ = ( c1 σ1 + c2σ2,||(6)|
Fig. 3. 2D cross sections of a series and parallel arrangement of the phases in a two-phase composite
where σ is the conductivity of the composite microstructure, and σ1, c1, σ2, and c2 are the DC conductivity and volume fraction of phases 1 and 2, respectively. The same equations apply if the admittance ψ is substituted for conductivity.
The calculated impedance curve for the series microstructure is shown in Fig. 4 along with values determined from a 503 image. Fig. 5 shows a similar plot for the parallel arrangement. In each case, the model has accurately determined the impedance curve. Numerical comparisons of | Z | at difference frequencies for both the series and parallel case are also shown in Table 1.
Comparison of computational and analytical results for the absolute value of the impedance of the single phase, series, parallel, and dilute sphere suspension examples. The applied frequency is ƒ, and |Z| = [(ReZ)2]1/2 is the absolute value of the impedance.
|single phase||0 Hz||0.40000||0.40000||0|
The last example is somewhat more complicated and consists of a dilute distribution of insulating spheres in a conducting cube. "Dilute" means that the volume fraction of the spheres is small enough so that each sphere does not interact with the other spheres. In this case, the DC conductivity of the composite is :
|σ = σ1 + 3c2(σ2 − σ1)/(2σ1 + σ2),||(7)|
Fig. 4. Nyquist plot for the series-microstructure composite material, where the admittances of the two phases is as shown in the figure.
Fig. 5. Nyquist plot for the parallel-microstructure composite material, where the admittances of the two phases is as shown in the figure.
where the terms have the same meanings as before, and the insulating sphere is phase 2. This equation may also be converted to admittance to provide an analytical solution for the impedance spectra. The electrical parameters for the example considered are σ1 = 0.1, σ2 = 0.05, k1 = 1, k2 = 1000. Both the analytical solution and model results are shown in Fig. 6, and are in good agreement. The sphere is simulated by a digital representation that has a diameter of 33 pixels, and is placed in the center of a 1003 cube, having periodic boundary conditions. The sphere has a volume fraction of 0.018. A single sphere can be simulated to give the correct result for a dilute concentration of spheres, since Eq. (7) is based on the assumption of non-interacting spheres. It is known that the admittance of a periodic array of spheres, which result from the periodic boundary conditions used, is the same as Eq. (7) up to the linear order term in the sphere volume fraction . Some error is also incurred because the insulating sphere in the model is only a digital representation of a true continuum sphere. Nevertheless, the model has accurately described the complex electrical behavior for this system, as can be seen in Table 1 and Fig. 6.
Nyquist plot for dilute suspension of spheres microstructure, where the admittances of the two phases are given in the figure. Phase 1 is the matrix, phase 2 if the spherical inclusions.
It is interesting to note that Fig. 6 clearly displays two arcs. In fact, it has been shown that the analytical solution in eq. (7) can be mapped exactly onto two RC parallel circuits arranged in series . Two-arc behavior like this is often taken to mean that a series arrangement of the phases exists, since a parallel arrangement of phases will only show one arc (see Fig. 5). However, it is physically not possible to consider the dilute sphere suspension microstructure as either a parallel or a series arrangement of phases. Rather, since the current must flow around and through the spheres, there is a degree of tortuosity present, which gives a more complicated response. This behavior will be present for any choice of the individual phase admittance parameters, although two arcs will only be visually observed when there is a significant difference between the two resonant frequencies, ω1 = σ1 / k1 and ω2 = σ2 / k2 . In Bonanos' exact mapping to two RC parallel circuits in series , the resistors and capacitors defined by eq. (7) contain terms relating to the individual phase admittances and phase fractions of both phases, so the two "phases" that appear to be in series are actually some mathematical combination of the two actual phases.
Fricke  computed the admittance of a dilute suspension of triaxial (a b c ) ellipsoids, randomly oriented, where matrix and inclusions had arbitrary admittances. He found that for this low concentration limit, the impedance could be expressed as a series combination of four RC circuits. The number four comes from the matrix phase plus one each for the three possible orientations of the ellipsoidal inclusions. These four circuits collapse to three for ellipsoids of revolution (b = c), and to two for spheres (a = b = c). Therefore inclusion geometry also contributes to determining how many arcs will appear in the impedance spectrum, and not just a material's series/parallel character, which usually cannot be defined for a complex microstructure.
One could speculate that two (or more) arc behavior will only be observed if at least one of the phases is discontinuous, as was the case in the above dilute suspensions of inclusions. The next section gives an example of how this hypothesis is not true, in general.