The fundamental mechanism of the non-universality of exponent t in the classical percolation equation is discussed on the basis of the comparison of lattice percolation and continuum percolation systems in the present study. The intrinsic universality of percolation exponents is revealed by analyzing the power-law percolation relationship between two different scales (microstructure and macroscopic properties). In view that the classical percolation equation with fitting parameters lacks a clear correlation to the microstructure features, a physically based percolation model is proposed to characterize the effective electrical conductivity of particle filled composites by means of the combination of effective medium (EM) and percolation equations with universal exponents. It is shown that the present model agrees well with the experimental data. Furthermore, the effect of particle shape on the percolation threshold and the effect of electrical conductivity of particle on the effective properties of composites are discussed.
Maxwell, J.C. (1873). Treatise on Electricity and Magnetism, Clarendon Press, Oxford .
2.
Bruggeman, D. (1935). Berechnung verschiedener Physikalischer Konstanten von heterogenen Substanzen , Ann. Physik (Liepzig), 24(7): 636–679 .
3.
Stauffer, D. and Aharony, A. (1992). Introduction to Percolation Theory, Taylor and Francis, London .
4.
McLachlan, D.S. (1989). Complex Permittivity of Emulsions: An Effective Media-percolation Equation , Solid State Communications, 72(8): 831–834 .
5.
McLachlan, D.S. (1989). Measurement and Analysis of a Model Dual Conductivity Medium Using a Generalized Effective Medium Theory , Physica A, 157(1): 188–191 .
6.
McLachlan, D.S. , Blaszkiewicz, M. and Newnham, R.E. (1990). Electrical Resistivity of Composites , Journal of the American Ceramic Society, 73(8): 2187–2203 .
7.
Garboczi, E.J. , Snyder, K.A. and Douglas, J.F. (1995). Geometrical Percolation Threshold of Overlapping Ellipsoids , Physical Review E, 52(1): 819–828 .
8.
Abeles, B. , Sheng, P. , Coutts, M.D. and Arie, Y. (1975). Structural and Electrical Properties of Granular Films , Advances in Physics, 24(3): 407–461 .
9.
Hashin, Z. and Shtrikman, S. (1963). A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials , Journal of the Mechanics and Physics of Solids, 11(2): 127–140 .
10.
Deptuck, D. , Harrison, J.P. and Zawadzki, P. (1985). Measurement of Elasticity and Conductivity of a Three-dimensional Percolation System , Physical Review Letters, 54(9): 913–916 .
11.
Havlin, S. and Ben-Avraham, D. (1987). Diffusion in Disordered Media , Advances in Physics, 36(6): 695–798 .
12.
Youngs, I.J. (2002). Exploring the Universal Nature of Electrical Percolation Exponents by Genetic Algorithm Fitting with General Effective Medium Theory , Journal of Physics D, 35(23): 3127–3137 .
13.
Nan, C.W. (1993). Physics of Inhomogeneous Inorganic Materials , Progress in Materials Science, 37(1): 1–116 .
14.
Brosseau, C. (2002). Generalized Effective Medium Theory and Dielectric Relaxation in Particle Filled Polymeric Resins , Journal of Applied Physics, 91(5): 3197–3204 .
15.
Kogut, P.M. and Straley, J.P. (1979). Distribution-induced Non-universality of the Percolation Conductivity Exponents , Journal of Physics C, 12(11): 2151–2159 .
16.
Sen, P.N. , Roberts, J.N. and Halperin, B.I. (1985). Nonuniversal Critical Exponents for Transport in Percolating Systems with a Distribution of Bond Strengths , Physical Review B, 32(5): 3306–3308 .
17.
Halperin, B.I. , Feng, S. and Sen, P.N. (1985). Differences between Lattice and Continuum Percolation Transport Exponents , Physical Review Letters, 54(22): 2391–2394 .
18.
Feng, S. , Halperin, B.I. and Sen, P.N. (1987). Transport Properties of Continuum Systems near the Percolation Threshold , Physical Review B, 35(1): 197–214 .
19.
Balberg, I. (1998). Limits on the Continuum-percolation Transport Exponents , Physical Review B, 57(21): 13351–13354 .
20.
Balberg, I. (1998). Resistivity and Electrical Noise in Continuum Percolation , Trends in Statistical Physics, 2: 39–68 .
21.
Balberg, I. (1987). Tunneling and Nonuniversal Conductivity in Composite Materials , Physical Review Letters, 59(12): 1305–1308 .
22.
Azulay, D. , Eylon, M. , Eshkenazi, O. , Toker, D. , Balberg, M. , Shimoni, N. , Millo, O. and Balberg, I. (2003). Electrical-Thermal Switching in Carbon-Black–Polymer Composites as a Local Effect , Physical Review Letters, 90(23): 236601-236601 .
23.
Yin, X.H. , Yoshino, K. , Yamamoto, H. , Watanuki, T. , Isa, I. , Nakagawa, Sh. and Adachi, M. (1996). Electrical Conduction near Percolation Threshold in Polymer Composite Containing Conducting Polypyrrole-coated Insulating Polymer Fiber , Japanese Journal of Applied Physics, 35(9): 4692–4695 .
24.
Toker, D. , Azulay, D. , Shimoni, N. , Balberg, I. and Millo, O. (2003). Tunneling and Percolation in Metal-insulator Composite Materials , Physical Review B, 68(4): 041403(R)/1–041403(R)/4 .
25.
Grimaldi, C. , Maeder, T. , Ryser, P. and Strassler, S. (2003). Critical Behaviour of the Piezoresistive Response in Ruo2-Glass Composites , Journal of Physics D, 36(11): 1341–1348 .
26.
Rubin, Z. , Sunshine, S.A. , Heaney, M.B. , Bloom I. and Balberg, I. (1999). Critical Behavior of the Electrical Transport Properties in a Tunneling-percolation System , Physical Review B, 59(19): 12196–12199 .
27.
Carmona, F. , Canet, R. and Delhaes, P. (1987). Piezoresistivity of Heterogeneous Solids , Journal of Applied Physics, 61(7): 2550–2557 .
28.
Deprez, N. and McLachlan, D.S. (1988). Analysis of the Electrical Conductivity of Graphite Powders during Compaction , Journal of Physics D, 21(1): 101–107 .
29.
Heaney, M.B. (1995). Measurement and Interpretation of Nonuniversal Critical Exponents in Disordered Conductor–Insulator Composites , Physical Review B, 52(17): 12477–12480 .
30.
McQueen, D.H. , Jager, K.-M. and Peliskova, M. (2004). Multiple Threshold Percolation in Polymer/Filler Composites , Journal of Physics D, 37(15): 2160–2169 .
31.
Mendelson, K.S. and Karioris, F.G. (1980). Percolation in Two-dimensional, Macroscopically Anisotropic Systems , Journal of Physics C, 13(33): 6197–6203 .
32.
Yoon, S. and Lee, S.I. (1990). Possible Breakdown of the Universality of the Conductivity Critical Exponent in an Anisotropic Percolation System , Physica B, 167(2): 133–137 .
33.
Wu, J. and McLachlan, D.S. (1997). Percolation Exponents and Thresholds in Two nearly Ideal Anisotropic Continuum Systems , Physica A, 241(1–2): 360–366 .
34.
Yin, X.H. , Yoshino, K. , Hashizume, K. and Isa, I. (1997). Preparation and Electrical Properties of Conducting Polyaniline-Polycarbonate Composite Films , Japanese Journal of Applied Physics, 36(6): 3537–3540 .
35.
Jing, X. and Zhao, W. (2000). Effect of Particle Size on Electric Conducting Percolation Threshold in Polymer/Conducting Particle Composites , Journal of Materials Science Letters, 19(5): 377–379 .
36.
Xue, Q.Z. (2004). The Influence of Particle Shape and Size on Electric Conductivity of Metal–Polymer Composites , European Polymer Journal, 40(2): 323–327 .
37.
Torquato, S. (2002). Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer-Verlag, New York .
38.
Doyle, W.T. and Jacobs, I.S. (1992). Influence of Particle Shape on Dielectric Enhancement in Metal-insulator Composites , Journal of Applied Physics, 71(8): 3926–3936 .
39.
Agari, Y. , Ueda, A. and Nagai, S. (1991). Thermal Conductivity of a Polyethylene Filled with Disoriented Short-cut Carbon Fibers , Journal of Applied Polymer Science, 43(6): 1117–1124 .
40.
Yin, Y. and Tu, S.T. (2002). Thermal Conductivities of PTFE Composites with Random Distributed Graphite Particles , Journal of Reinforced Plastics and Composites, 21(18): 1619–1627 .
41.
Tu, S.T. , Cai, W.Z. , Yin, Y. and Ling, X. (2005). Numerical Simulation of Saturation Behavior of Physical Properties in Composites with Randomly Distributed Second-phase , Journal of Composite Materials, 39(7): 617–631 .
42.
Cai, W.Z. , Tu, S.T. and Tao, G.L. (2005). Thermal Conductivity of PTFE Composites with Three-dimensional Randomly Distributed Fillers , Journal of Thermoplastic Composite Materials, 18(3): 241–253 .
