Computer Simulation of Impedance for 2-D Conductor-Insulator Composite
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Mat. Res. Soc. Symp. Proc. Vol. 500 © 1998 Materials Research Society
values of Pt and P2 are .iven as 10[(fI/sphere] and 10.[flsphere], respectively. Both E, and 62 are given as 8.854 X 10"' [F/sphere]. Two kinds of spheres used in the simulations correspond to two RC parallel circuits as shown in Fig. i.b. The values of spheres were chosen to reflect the case where the conductive particles were introduced into the insulating matrix of the same permittivity. The diagonal admittance matrix is newly introduced for the triangular network, and all elements of matrix are complex numbers for a.c. impedance simulation. We used Marsaglia's algorithm 5 to generate random numbers. The transfer-matrix algorithm and Marsaglia's algorithm are very simple and need less memory so that they make the a.c. impedance simulation possible on the personal computer. The detailed algorithm was given elsewhere'. In this study, all simulations are made over a mesh of 30 X 100 (N X L) spheres by an IBM compatible personal computer. . .. .L=4 -...
.
a-=~ g2 1=
-
1.f
-u)
Cd(=8.854x10' F)
C2 (=8.854x10 F)
(a)
(b)
Fig. I Closed packed hard spheres model (a) and the electrical components of spheres (b) used in simulations.
RESULTS AND DISCUSSION The impedance spectra are presented in M-plots because M-plots have better resolution than Z-plots when the difference of the time constants is due to the difference of the resistance values. The frequency in simulation ranges from 10-2 Hz to 102 MHz and reflects the frequency range of many impedance analyzers. The simulated impedance spectra with "completely-random patterns" of two spheres are given in Fig.2.a. The shape of the impedance spectra is too complex to be analyzed clearly into two components. The third semicircles are clearly seen when the filling fraction P2 is 0.2 and 0.4. When the distribution of spheres has "no-contact random patterns" as shown in Fig.2.b, the impedance spectra can be clearly analyzed by a two-component series circuit model. Series equivalent circuit model is often used to analyze the electrical components of polycrystalline materials. When a; and C2 are the conductivities of the basic constituent I and 2, respectively, and P2 is the volume fraction of the constituent 2, the series circuit model gives the following equation:
. . ...... (I) . . . =.. .. +.. where at, is the real or complex conductivity of the composite. In no-contact random patterns, black spheres are distributed randomly, but forbidden to contact each other, thus no black clusters form. The third semicircle is not observed although the filling fraction, P2 reaches as high as 0.26. When the filling fraction increases further, the distribution loses randomness since the number of sites, which may be occupied by black spheres without contacting each other, is limited. The analysis of above two patterns suggests that the appearance of the third semicircle is due to the clusters formed in the composites. When the horizontal-clusters are introduced as
shown in Fig.2.c, however, the third semicircl
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