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A Percolation Equation for Modeling Experimental Results for Continuum Percolation Systems

Published online by Cambridge University Press:  17 March 2011

D.S. McLachlan,
C. Chiteme,
W.D. Heiss and
Junjie Wu
D.S. McLachlan
Affiliation:
School of Physics and Materials Physics Institute, University of the Witwatersrand, P O Wits 2050, Johannesburg, South Africa.
C. Chiteme
Affiliation:
School of Physics and Materials Physics Institute, University of the Witwatersrand, P O Wits 2050, Johannesburg, South Africa.
W.D. Heiss
Affiliation:
School of Physics and Materials Physics Institute, University of the Witwatersrand, P O Wits 2050, Johannesburg, South Africa.
Junjie Wu
Affiliation:
School of Physics and Materials Physics Institute, University of the Witwatersrand, P O Wits 2050, Johannesburg, South Africa.
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Abstract

The standard percolation equations or power laws, for dc and ac conductivity (dielectric constant) are based on scaling ansatz, and predict the behaviour of the first and second order terms, above and below the percolation or critical volume fraction (øc), and in the crossoverregion. Recent experimental results on ac conductivity are presented, which show that these equations, with the exception of real σm above øc and the first order terms in the crossover region, are only valid in the limit σi/σc = 0, where for an ideal dielectric σi=ωε0εr.

A single analytical equation, which has the same parameters as the standard percolation equations, and which, for ac conductivity, reduces to the standard percolation power laws in the limit σi(ωε0εr)/σc = 0 for all but one case, is presented. The exception is the expression for real σm below øc, where the standard power law is always incorrect. The equation is then shown to quantitatively fit both first and second order dc and ac experimental data over the entire frequency and composition range. This phenomenological equation is also continuous, has the scaling properties required at a second order metal-insulator and fits scaled first order dc and ac experimental data. Unfortunately, the s and t exponents that are necessary to fit the data to the above analytical equation are usually not the simple dimensionally determined universal ones and depend on a number of factors.

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References

1. Landauer, R., AIP Conference Proceedings 40, New York, p2 (1978).Google Scholar
2. McLachlan, D.S., Blaszkiewics, M., and , Newnham, J. Am. Ceram. Soc. 73, 2187 (1990).Google Scholar
3. Bergman, D.J., Stroud, D., Solid State Physics 46 (Academic Press, San Diego, 1992) 147.Google Scholar
4. Nan, Ce-Wen, Prog. Mater Sci. 37, 1 (1993).Google Scholar
5. Clerc, J.P., Girand, G., Langier, J.M., and Luck, J.M., Adv. Phys. 39, 191 (1990).Google Scholar
6. McLachlan, D.S., J. Electroceramics 5, 93 (2000).Google Scholar
7. Efros, A.L. and Shklovskii, B.I., phys. stat. sol. (b) 76, 475 (1976).Google Scholar
8. Bergman, D.J. and Imry, I., Phys. Rev. Lett. 39, 1222 (1977).Google Scholar
9. Wu, J., and McLachlan, D.S., Phys. Rev. B56, 1238 (1997).Google Scholar
10. Wu, J., and McLachlan, D.S., Phys. Rev. B58, 14880 (1998).Google Scholar
11. McLachlan, D.S., Heiss, W.D., Chiteme, C. and Wu, J., Phys. Rev. B58, 13558 (1998).Google Scholar
12. Heiss, W.D., McLachlan, D.S., Chiteme, C., Phys. Rev. B62, 4196 (2000).Google Scholar
13. Chiteme, C. and Mclachlan, D.S., submitted for publication.Google Scholar
14. Chiteme, C. and Mclachlan, D.S., submitted for publication.Google Scholar
15. Mclachlan, D.S., Physica B254, 249 (1998).Google Scholar
16. McLachlan, D.S., Rosenbaum, R., Albers, A., Etyan, G., Grammatica, N., Hurwitz, G., Pickup, J. and Zaken, E., J. Phys.: Condens. Matter. 5, 4829 (1993).Google Scholar
17. Kogut, P. M. and Stanley, J. P., J. Phys. C 12, 2151 (1979).Google Scholar
18. Halperin, B. I., Feng, S. and Sen, P, Phys. Rev. Lett. 54, 239 (1985).Google Scholar
19. Feng, S., Halperin, B. I. and Sen, P, Phys. Rev B 35, 197 (1987).Google Scholar
20. Balberg, I., Phys. Rev. B 57, 13351 (1998).Google Scholar
21. Balberg, I., Phil Mag. B 56, 991 (1987).Google Scholar
22. Dijk, M. A. Van, Phys Rev. Lett. 55, 1003 (1985).Google Scholar
23. Dijk, M. A. van, Castleliejn, G., Joosten, J. G. H. and Levine, Y. K., J. Chem. Phys. 85, 626 (1986).Google Scholar
24. Jonscher, I. A. K., Universal Relaxation Law, Chelsea Dielectric Press, London, 1996. 2. A. K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectric Press, London, 1983.Google Scholar
25. McLachlan, D.S., Cai, Kefeng, Chiteme, C. and Heiss, W.D., Physica B279, 66 (2000).Google Scholar