Structure-correlated diffusion anisotropy in nanoporous channel networks by Monte Carlo simulations and percolation theory

  • Daria Kondrashova
  • Rustem Valiullin
  • Jörg Kärger
  • Armin Bunde
Regular Article


Nanoporous silicon consisting of tubular pores imbedded in a silicon matrix has found many technological applications and provides a useful model system for studying phase transitions under confinement. Recently, a model for mass transfer in these materials has been elaborated [Kondrashova et al., Sci. Rep. 7, 40207 (2017)], which assumes that adjacent channels can be connected by “bridges” (with probability p bridge) which allows diffusion perpendicular to the channels. Along the channels, diffusion can be slowed down by “necks” which occur with probability p neck. In this paper we use Monte-Carlo simulations to study diffusion along the channels and perpendicular to them, as a function of p bridge and p neck, and find remarkable correlations between the diffusivities in longitudinal and radial directions. For clarifying the diffusivity in radial direction, which is governed by the concentration of bridges, we applied percolation theory. We determine analytically how the critical concentration of bridges depends on the size of the system and show that it approaches zero in the thermodynamic limit. Our analysis suggests that the critical properties of the model, including the diffusivity in radial direction, are in the universality class of two-dimensional lattice percolation, which is confirmed by our numerical study.


Solid State and Materials 


  1. 1.
    G. Ertl, H. Knözinger, F. Schüth, J. Weitkamp, Handbook of Heterogeneous Catalysis, 2nd edn. (Wiley-VCH, Weinheim, 2008)Google Scholar
  2. 2.
    F. Schüth, K.S.W. Sing, J. Weitkamp, Handbook of Porous Solids (Wiley-VCH, 2002)Google Scholar
  3. 3.
    P.W. Barone, S. Baik, D.A. Heller, M.S. Strano, Nat. Mater. 4, 86 (2005)ADSCrossRefGoogle Scholar
  4. 4.
    J. Kim, W.A. Li, Y. Choi, S.A. Lewin, C.S. Verbeke, G. Dranoff, D.J. Mooney, Nat. Biotechnol. 33, 64 (2015)CrossRefGoogle Scholar
  5. 5.
    D. Cohen-Tanugi, J.C. Grossman, Nano Lett. 12, 3602 (2012)ADSCrossRefGoogle Scholar
  6. 6.
    O.K. Farha, A.O. Yazaydin, I. Eryazici, C.D. Malliakas, B.G. Hauser, M.G. Kanatzidis, S.T. Nguyen, R.Q. Snurr, J.T. Hupp, Nat. Chem. 2, 944 (2010)CrossRefGoogle Scholar
  7. 7.
    L. Canham, Handbook of Porous Silicon (Springer International Publishing, 2014)Google Scholar
  8. 8.
    D. Kovalev, M. Fujii, Adv. Mater. 17, 2531 (2005)CrossRefGoogle Scholar
  9. 9.
    K.P. Tamarov, L.A. Osminkina, S.V. Zinovyev, K.A. Maximova, J.V. Kargina, M.B. Gongalsky, Y. Ryabchikov, A. Al-Kattan, A.P. Sviridov, M. Sentis et al., Sci. Rep. 4, 7034 (2014)CrossRefGoogle Scholar
  10. 10.
    C.C. Striemer, T.R. Gaborski, J.L. McGrath, P.M. Fauchet, Nature 445, 749 (2007)ADSCrossRefGoogle Scholar
  11. 11.
    P. Barthelemy, M. Ghulinyan, Z. Gaburro, C. Toninelli, L. Pavesi, D.S. Wiersma, Nat. Photon. 1, 172 (2007)ADSCrossRefGoogle Scholar
  12. 12.
    P. Kumar, T. Hofmann, K. Knorr, P. Huber, P. Scheib, P. Lemmens, J. Appl. Phys. 103, 024303 (2008)ADSCrossRefGoogle Scholar
  13. 13.
    R. Valiullin, A. Khokhlov, Phys. Rev. E 73, 051605 (2006)ADSCrossRefGoogle Scholar
  14. 14.
    H. Jobic, D.N. Theodorou, Microporous Mesoporous Mater. 102, 21 (2007)CrossRefGoogle Scholar
  15. 15.
    M. Appel, G. Fleischer, J. Kärger, F. Fujara, S. Siegel, Europhys. Lett. 34, 483 (1996)ADSCrossRefGoogle Scholar
  16. 16.
    R. Valiullin, Diffusion NMR of Confined Systems: Fluid Transport in Porous Solids and Heterogeneous Materials (The Royal Society of Chemistry, Cambridge, UK, 2017)Google Scholar
  17. 17.
    D. Kondrashova, A. Lauerer, D. Mehlhorn, H. Jobic, A. Feldhoff, M. Thommes, D. Chakraborty, C. Gommes, J. Zecevic, P. de Jongh et al., Sci. Rep. 7, 40207 (2017)ADSCrossRefGoogle Scholar
  18. 18.
    S. Naumov, A. Khokhlov, R. Valiullin, J. Kärger, P.A. Monson, Phys. Rev. E 78, 060601 (2008)ADSCrossRefGoogle Scholar
  19. 19.
    S. Fritzsche, J. Kärger, Europhys. Lett. 63, 465 (2003)ADSCrossRefGoogle Scholar
  20. 20.
    K. Binder, Rep. Prog. Phys. 60, 487 (1997)ADSCrossRefGoogle Scholar
  21. 21.
    D. Schneider, D. Mehlhorn, P. Zeigermann, J. Kärger, R. Valiullin, Chem. Soc. Rev. 45, 3439 (2016)CrossRefGoogle Scholar
  22. 22.
    A. Bunde, S. Havlin, Fractals and Disordered Systems, 2nd edn. (Springer-Verlag, Berlin, Heidelberg, New York, 1996)Google Scholar
  23. 23.
    D. ben Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 2000)Google Scholar
  24. 24.
    Z. Sadjadi, H. Rieger, Phys. Rev. Lett. 110, 144502 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    R.L. Fleischer, P.B. Price, E.M. Symes, Science 143, 249 (1964)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Daria Kondrashova
    • 1
    • 2
  • Rustem Valiullin
    • 2
  • Jörg Kärger
    • 2
  • Armin Bunde
    • 1
  1. 1.Institut für Theoretische Physik, Justus-Liebig-Universität GiessenGiessenGermany
  2. 2.Felix Bloch Institute for Solid State Physics, University of LeipzigLeipzigGermany

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