The percolation staircase model and its manifestation in composite materials

  • I. Balberg
  • D. Azulay
  • Y. Goldstein
  • J. Jedrzejewski
  • G. Ravid
  • E. Savir
Regular Article

Abstract

We studied the tunneling percolation conductivity dependence on the site or bond occupation probability in the square lattice. The model predicts that in both, lattice and continuum systems in which there is a hierarchy of the local conductances, the dependence of the global conductivity on the site or volume occupation probability will yield a conductivity staircase. In particular we evaluate the implications of the staircase on the critical behavior of the conductivity. We then show experimental evidence for the predicted percolation-tunneling staircase in a Ag-Al2O3 granular metal system and in a carbon black-polymer composite. Following that, we propose that for carbon nanotube (CNT) polymer composites the data in the literature give ample support to a percolation-dispersion staircase behavior. The implication of the present findings on the percolation-hopping problem in composite materials is also discussed.

Keywords

Solid State and Materials 

References

  1. 1.
    D. Stauffer, A. Aharony, Introduction to Percolation Theory (Taylor & Francis, London, 1994)Google Scholar
  2. 2.
    R. Zallen, The Physics of Amorphous Solids (Wiley, New York, 1983)Google Scholar
  3. 3.
    I. Balberg, in Continuum Percolation in the Springer Encyclopedia of Complexity, edited by M. Sahimi (Springer, Berlin, 2009), Vol. 2, p. 1443Google Scholar
  4. 4.
    P.M. Kogut, J. Straley, J. Phys. C 12, 2151 (1979) ADSCrossRefGoogle Scholar
  5. 5.
    I. Balberg, Phys. Rev. Lett. 59, 1305 (1987) ADSCrossRefGoogle Scholar
  6. 6.
    H. Scher, R. Zallen, J. Chem. Phys. 53, 3759 (1970) ADSCrossRefGoogle Scholar
  7. 7.
    I. Balberg, Carbon 40, 139 (2002)CrossRefGoogle Scholar
  8. 8.
    I. Balberg, J. Phys. D 42, 064003 (2009) ADSCrossRefGoogle Scholar
  9. 9.
    S. Vionnet-Menot, C. Grimaldi, T. Maeder, S. Strassler, P. Ryser, Phys. Rev. B 71, 064201 (2005) ADSCrossRefGoogle Scholar
  10. 10.
    B. Abeles, P. Sheng, M.D. Coutts, Y. Arie, Adv. Phys. 24, 3689 (1975) CrossRefGoogle Scholar
  11. 11.
    X. Huang, C. Kim, P. Jiang, Y. Yin, Z. Lee, J. Appl. Phys. 105, 014105 (2009) ADSCrossRefGoogle Scholar
  12. 12.
    V.H. Poblete, M.P. Alvarez, V.M. Funzalida, Polym. Compos. 30, 328 (2009)CrossRefGoogle Scholar
  13. 13.
    H. Zois, L. Apekis, M. Omastova, Macromol. Symp. 170, 249 (2001) CrossRefGoogle Scholar
  14. 14.
    J. Vileakova, P. Saha, O. Quadrat, Eur. Polym. J. 38, 2343 (2002) CrossRefGoogle Scholar
  15. 15.
    W. Bauhofer, J.Z. Kovacs, Compos. Sci. Tech. 69, 1486 (2009) CrossRefGoogle Scholar
  16. 16.
    J.Z. Kovacs, B.S. Velagala, K. Schulte, W. Bauhofer, Compos. Sci. Tech. 67, 922 (2007)CrossRefGoogle Scholar
  17. 17.
    E. Tkalya, M. Ghislandi, A. Aleksev, C. Konig, J. Loos, J. Mater. Chem. 20, 3035 (2010) CrossRefGoogle Scholar
  18. 18.
    V. Panwar, B. Kang, J.-O. Park, S. Park, R.M. Mehra, Eur. Polym. J. 45, 1777 (2009) CrossRefGoogle Scholar
  19. 19.
    G. Ambrosetti, I. Balberg, C. Grimaldi, Phys. Rev. B 82, 134201 (2010) ADSCrossRefGoogle Scholar
  20. 20.
    D. Berman, B.G. Orr, H.M. Jaeger, A.M. Goldman, Phys. Rev. B 33, 4301 (1986) ADSCrossRefGoogle Scholar
  21. 21.
    S. Tyc, B.I. Halperin, Phys. Rev. B 39, 877 (1989)ADSCrossRefGoogle Scholar
  22. 22.
    I. Balberg, N. Binenbaum, Phys. Rev. A 35, 5174 (1987) ADSCrossRefGoogle Scholar
  23. 23.
    B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors (Springer, Berlin, 1984)Google Scholar
  24. 24.
    S. Galam, A. Mauger, Phys. Rev. E 53, 2177 (1996) ADSCrossRefGoogle Scholar
  25. 25.
    J.M. Ziman, Models of Disorder (Cambridge University Press, Cambridge, 1979)Google Scholar
  26. 26.
    A. Trohymachuk, I. Nezbeda, J. Jirsak, D. Henderson, J. Chem. Phys. 123, 024501 (2005) ADSCrossRefGoogle Scholar
  27. 27.
    V.S. Kumar, V. Kumaran, J. Chem. Phys. 123, 074502 (2005) ADSCrossRefGoogle Scholar
  28. 28.
    N.L. Lavik, V. Voloshin, J. Chem. Phys. 114, 9489 (2001) ADSCrossRefGoogle Scholar
  29. 29.
    O. Entin-Wohlman, Y. Gefen, Y. Shapira, J. Phys. C 16, 1161 (1983) ADSCrossRefGoogle Scholar
  30. 30.
    M. Mostefa, G. Olivier, J. Phys. C 18, 93 (1985)ADSCrossRefGoogle Scholar
  31. 31.
    Y. Zweifel, C.J.G. Plummer, H.-H. Kausch, J. Mater. Sci. 33, 1715 (1998) ADSCrossRefGoogle Scholar
  32. 32.
    B. Abeles, Appl. Solid State Sci. 6, 1 (1976)CrossRefGoogle Scholar
  33. 33.
    B. Abeles, H.L Pinch, J.I. Gittleman, Phys. Rev. Lett. 35, 247 (1975)ADSCrossRefGoogle Scholar
  34. 34.
    I. Balberg, D. Azulay, D. Toker, O. Millo, Int. J. Mod. Phys. B 18, 2091 (2004) ADSCrossRefMATHGoogle Scholar
  35. 35.
    R.W. Cohen, G.D. Cody, M.D. Coutts, B. Abeles, Phys. Rev. B 8, 3689 (1987)ADSCrossRefGoogle Scholar
  36. 36.
    M.H. Lee, I.T.H. Chang, P.J. Dobson, B. Cantor, Mat. Sci. Eng. A 179-180, 545 (1994) CrossRefGoogle Scholar
  37. 37.
    O. Mamezaki, M. Fujii, S. Hayashi, Jpn J. Appl. Phys. 40, 5389 (2001) ADSCrossRefGoogle Scholar
  38. 38.
    W. Bouwen, E. Kunnen, K. Temst, P. Thoen, M.J. Van Bael, F. Vanhoutte, H. Weidele, P. Lievens, R.E. Silverans, Thin Solid Films 354, 87 (1999)ADSCrossRefGoogle Scholar
  39. 39.
    Y. Jiang, B. Guan, X.L. Xu, Chinese Phys. Lett. 22, 730 (2005)ADSCrossRefGoogle Scholar
  40. 40.
    J. Liu, J.Z. Zhao, Z.Q. Hu, Mat. Sci. Eng. A 452-453, 103 (2007) CrossRefGoogle Scholar
  41. 41.
    D. Toker, D. Azulay, N. Shimoni, I. Balberg, O. Millo, Phys. Rev. B 68, 041403(R) (2003) ADSCrossRefGoogle Scholar
  42. 42.
    A. Hunt, R. Ewing, Percolation Theory for Flow in Porous Media (Springer, Berlin, 2009)Google Scholar
  43. 43.
    Z. Rubin, S.A. Sunshine, M.B. Heaney, I. Bloom, I. Balberg, Phys. Rev. B 59, 12196 (1999) ADSCrossRefGoogle Scholar
  44. 44.
    S. Torquato, B. Lu, J. Rubinsein, J. Phys. A 23, L162 (1990) CrossRefGoogle Scholar
  45. 45.
    S. Torquato, S.B. Lee, Physica A 167, 36 (1990)CrossRefGoogle Scholar
  46. 46.
    R.C. Picu, A. Rakshit, J. Chem. Phys. 126, 144909 (2007) ADSCrossRefGoogle Scholar
  47. 47.
    D. Brown, V. Marcdon, P. Mele, N.D. Alberola, Macromolecules 41, 1499 (2008) ADSCrossRefGoogle Scholar
  48. 48.
    J.Y. Feng, J.X. Li, C.M. Chan, J. Appl. Poly. Sci. 85, 358 (2002)CrossRefGoogle Scholar
  49. 49.
    C. Li, E.T. Thostenson, T.W. Chou, Appl. Phys. Lett. 91, 223114 (2007) ADSCrossRefGoogle Scholar
  50. 50.
    S. Samarzija-Jovanovic, V. Jovanovic, G. Markovic, M. Marinovic-Cinovic, J. Therm. Anal. Calorim. 98, 275 (2009)CrossRefGoogle Scholar
  51. 51.
    M.B. Heaney, Phys. Rev. B 52, 1 (1995)CrossRefGoogle Scholar
  52. 52.
    M.B. Heaney, Phys. Rev. B 52, 12477 (1995) ADSCrossRefGoogle Scholar
  53. 53.
    M.T. Conner, S. Roy, T.A. Ezquerra, F.J. Balta Calleja, Phys. Rev. B 57, 2286 (1998)ADSCrossRefGoogle Scholar
  54. 54.
    E. Sichel, J.I. Gittleman, P. Sheng, J. Electronic Mater. 11, 69 (1982)CrossRefGoogle Scholar
  55. 55.
    M.J. O’Connell, P. Baul, L.M. Ericson, C. Huffman, Y. Wang, E. Haroz, C. Kuper, J. Tour, K.D. Ausman, R.E. Smalley, Chem. Phys. Lett. 342, 265 (2001) CrossRefADSGoogle Scholar
  56. 56.
    G. Pécastaings, P. Delhaes, A. Derre, H. Saadaoui, F. Carmona, S. Cui, J. Nanosci. Nanotechnol. 4, 838 (2004)CrossRefGoogle Scholar
  57. 57.
    D. Untereker, S. Lyu, J. Schley, G. Martinez, L. Lohstreter, ACS Appl. Mater & Int. 1, 97 (2009)CrossRefGoogle Scholar
  58. 58.
    Y. Yu, G. Song, L. Sun, J. Appl. Phys. 108, 084319 (2010) ADSCrossRefGoogle Scholar
  59. 59.
    Y. Simsek, L. Ozyuzer, A.T. Seyhan, M. Tanoglu, Karl, Sculte, J. Mater. Sci. 42, 7689 (2007)CrossRefGoogle Scholar
  60. 60.
    D. Azulay, M. Eylon, O. Eshkenazi, D. Toker, M. Balberg, O. Millo, I. Balberg, Phys. Rev. Lett. 90, 236601 (2003) ADSCrossRefGoogle Scholar
  61. 61.
    S.I. White, R.M. Mutisu, P.M. Vora, D. Jahnke, S. Hsu, J.M. Kikakawa, J. Li, J.E. Fisher, K.I. Winey, Adv. Funct. Matter 20, 2709 (2010) CrossRefGoogle Scholar
  62. 62.
    M.-J. Jiang, Z.-M. Dang, H.-P. Xu, Appl. Phys. Lett. 90, 42914 (2007) CrossRefGoogle Scholar
  63. 63.
    P. Keblinski, F. Cleri, Phys. Rev. B 69, 184201 (2004) ADSCrossRefGoogle Scholar
  64. 64.
    I. Balberg, N. Binenbaum, C.H. Anderson, Phys. Rev. Lett. 51, 1605 (1983) ADSCrossRefGoogle Scholar
  65. 65.
    I. Balberg, B. Berkowitz, G.E. Drachsler, J. Geophys. Research: Solid Earth and Planets 96, 10015 (1991) CrossRefGoogle Scholar
  66. 66.
    N. Hu, Z. Masuda, C. Yan, G. Yamamoto, H. Fukunaga, T. Hhashida, Nanotechnology 19, 215701 (2008) ADSCrossRefGoogle Scholar
  67. 67.
    A. Maaroufi, K. Haboubi, A. El Amarti, F. Carmona, J. Mater. Sci. 39, 265 (2004)ADSCrossRefGoogle Scholar
  68. 68.
    A. Hiraiwa, T. Kobayashi, J. Appl. Phys. 70, 309 (1991)ADSCrossRefGoogle Scholar
  69. 69.
    D.C. Wright D.J. Bergman, Y. Kantor, Phys. Rev. B 33, 396 (1985)ADSCrossRefGoogle Scholar
  70. 70.
    I. Balberg, N. Wagner, D.W. Hearn, J.A. Ventura, Phys. Rev. Lett. 60, 1887 (1988) ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • I. Balberg
    • 1
  • D. Azulay
    • 1
  • Y. Goldstein
    • 1
  • J. Jedrzejewski
    • 1
  • G. Ravid
    • 1
  • E. Savir
    • 1
  1. 1.The Racah Institute of PhysicsThe Hebrew UniversityJerusalemIsrael

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