Journal of Statistical Physics

, Volume 146, Issue 1, pp 244–248 | Cite as

A Remark Concerning Percolation Thresholds in Polydisperse Systems of Finite-Diameter Rods

  • Avik P. Chatterjee


A lattice-based analysis of the percolation threshold for randomly distributed cylindrical particles is generalized to consider arbitrary joint distributions over the radii and lengths of the rods. Effects due to the finite hard core diameter of the particles are accounted for. An analogy to site percolation on a modified Bethe lattice is exploited to yield a result for the percolation threshold that is equivalent to one that has been obtained from integral equation methods in the limit of large aspect ratios for the rods.


Percolation Bethe lattice Polydisperse rods Integral equation methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ambrosetti, G., Grimaldi, C., Balberg, I., Maeder, T., Danani, A., Ryser, P.: Solution of the tunneling-percolation problem in the nanocomposite regime. Phys. Rev. B 81, 155434 (2010) CrossRefADSGoogle Scholar
  2. 2.
    Balberg, I., Anderson, C.H., Alexander, S., Wagner, N.: Excluded volume and its relation to the onset of percolation. Phys. Rev. B 30, 3933–3943 (1984) CrossRefADSGoogle Scholar
  3. 3.
    Beck-Candanedo, S., Roman, M., Gray, D.G.: Effect of reaction conditions on the properties and behavior of wood cellulose nanocrystal suspensions. Biomacromolecules 6, 1048–1054 (2005) CrossRefGoogle Scholar
  4. 4.
    Berhan, L., Sastry, A.M.: Modeling percolation in high-aspect-ratio fiber systems. I. Soft-core versus hard-core models. Phys. Rev. E 75, 041120 (2007) CrossRefADSGoogle Scholar
  5. 5.
    Boehm, R.E., Martire, D.E.: Theory of liquid-chromatographic retention and solute-transfer thermodynamics using the Bethe-Guggenheim quasi-chemical approach. J. Phys. Chem. 98, 1317–1327 (1994) CrossRefGoogle Scholar
  6. 6.
    Chatterjee, A.P.: Connectedness percolation in polydisperse rod systems: A modified Bethe lattice approach. J. Chem. Phys. 132, 224905 (2010) CrossRefADSGoogle Scholar
  7. 7.
    Dalmas, F., Cavaille, J.Y., Gauthier, C., Chazeau, L., Dendievel, R.: Viscoelastic behavior and electrical properties of flexible nanofiber filled polymer nanocomposites. Influence of processing conditions. Compos. Sci. Technol. 67, 829–839 (2007) CrossRefGoogle Scholar
  8. 8.
    Foygel, M., Morris, R.D., Anez, D., French, S., Sobolev, V.L.: Theoretical and computational studies of carbon nanotube composites and suspensions: Electrical and thermal conductivity. Phys. Rev. B 71, 104201 (2005) CrossRefADSGoogle Scholar
  9. 9.
    Hackett, A., Melnik, S., Gleeson, J.P.: Cascades on a class of clustered random networks. Phys. Rev. E 83, 056107 (2011) CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Kihara, T.: Virial coefficients and models of molecules in gases. Rev. Mod. Phys. 25, 831–843 (1953) CrossRefMATHADSGoogle Scholar
  11. 11.
    Larson, R.G., Davis, H.T.: Conducting backbone in percolating Bethe lattices. J. Phys. C 15, 2327–2331 (1982) CrossRefADSGoogle Scholar
  12. 12.
    Newman, M.E.J.: Random graphs with clustering. Phys. Rev. Lett. 103, 058701 (2009) CrossRefADSGoogle Scholar
  13. 13.
    Otten, R.H.J., van der Schoot, P.: Continuum percolation of polydisperse nanofillers. Phys. Rev. Lett. 103, 225704 (2009) CrossRefADSGoogle Scholar
  14. 14.
    Otten, R.H.J., van der Schoot, P.: Connectivity percolation of polydisperse anisotropic nanofillers. J. Chem. Phys. 134, 094902 (2011) CrossRefADSGoogle Scholar
  15. 15.
    Ounaies, Z., Park, C., Wise, K.E., Siochi, E.J., Harrison, J.S.: Electrical properties of single wall carbon nanotube reinforced polyimide composites. Compos. Sci. Technol. 63, 1637–1646 (2003) CrossRefGoogle Scholar
  16. 16.
    Philipse, A.P.: The random contact equation and its implications for (colloidal) rods in packings, suspensions, and anisotropic powders. Langmuir 12, 1127–1133 (1996) CrossRefGoogle Scholar
  17. 17.
    Potschke, P., Abdel-Goad, M., Alig, I., Dudkin, S., Lellinger, D.: Rheological and dielectrical characterization of melt mixed polycarbonate-multiwalled carbon nanotube composites. Polymer 45, 8863–8870 (2004) CrossRefGoogle Scholar
  18. 18.
    Trionfi, A., Wang, D.H., Jacobs, J.D., Tan, L.S., Vaia, R.A., Hsu, J.W.P.: Direct measurement of the percolation probability in carbon nanofiber-polyimide nanocomposites. Phys. Rev. Lett. 102, 116601 (2009) CrossRefADSGoogle Scholar
  19. 19.
    Wang, X., Chatterjee, A.P.: Connectedness percolation in athermal mixtures of flexible and rigid macromolecules: Analytic theory. J. Chem. Phys. 118, 10787–10793 (2003) CrossRefADSGoogle Scholar
  20. 20.
    Wheeler, J.C., Widom, B.: Phase equilibrium and critical behavior in a two-component Bethe-lattice gas or three-component Bethe-lattice solution. J. Chem. Phys. 52, 5334–5343 (1970) CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of ChemistrySUNY College of Environmental Science and ForestrySyracuseUSA

Personalised recommendations