Skip to main content
SpringerLink
Log in

Diffusion coefficients for two-dimensional narrow asymmetric channels embedded on flat and curved surfaces

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

This paper focuses on the derivation of a general position-dependent diffusion coefficient to describe the two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying cross section and non-straight midline embedded in a flat or on a curved surface. We consider the diffusion of non-interacting point-like Brownian particles under no external field. In order to project the 2D diffusion equation into an effective one-dimensional generalized Fick-Jacobs equation in both, flat and curved manifolds using the generalization of the mapping procedure introduced by Kalinay and Percus. The expression obtained is the more general position-dependent diffusion coefficient for 2D narrow channels that lies in a plane, which contains all the well-known previous results both symmetric and asymmetric channels as special cases. In a straightforward manner, previously defining the corresponding Fick-Jacobs equation on a curved surface, this result can be generalized to the case of a narrow 2D channel embedded on a no-flat smooth surface where the full position-dependent diffusion coefficient is modified according to the metric elements that accounts for the curvature of the surface. In addition, the equations for the mean first-passage time are obtained for asymmetrical channels on curved surfaces. As an example we shall solve this equation for the case of an asymmetric channel defined by straight walls embedded on a cylindrical surface having a reflecting wall at the origin and an absorbent one at distance θ L .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P.S. Burada, P. Hänggi, F. Marchesoni, G. Schmid, P. Talkner, Chem. Phys. Chem. 10, 45 (2009)

    Google Scholar 

  2. B. Hille, Ion Channels of Excitable Membranes (Sinauer, Sunderland, 2001)

  3. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell (Garland Science, New York, 2007)

  4. J. Kärger, D.M. Ruthven, Diffusion in Zeolites and Other Microporous Solids (Wiley, New York, 1992)

  5. J. Kärger, D.M. Ruthven, D.N. Theodorou, Diffusion in Nanoporous Materials, Vols. 1 & 2 (Wiley-VHC, Weinheim, 2012)

  6. I.D. Kosinska, I. Goychuk, M. Kostur, G. Schmid, P. Hänggi, Phys. Rev. E 77, 031131 (2008)

    Article  ADS  Google Scholar 

  7. A. Berezhkovskii, G. Hummer, Phys. Rev. Lett. 89, 064503 (2002)

    Article  ADS  Google Scholar 

  8. J.C.T. Eijkel, A. van den Berg, Microfluid Nanofluid 1, 249 (2005)

    Article  Google Scholar 

  9. M. Gershow, J.A. Golovchenco, Nat. Nanotechnol. 2, 775 (2007)

    Article  ADS  Google Scholar 

  10. C. Dekker, Nanotechnology 2, 209 (2007)

    Google Scholar 

  11. U.F. Keyser, B. Koeleman, S.V. Dorp, D. Krapf, R. Smeets, S. Lemay, N. Dekker, C. Dekker, Nat. Phys. 2, 473 (2006)

    Article  Google Scholar 

  12. S. Pagliara, C. Schwall, U.F. Keyser, Adv. Mater. 6, 844 (2013)

    Article  Google Scholar 

  13. A. Fick, Ann. Phys. 170, 59 (1855)

    Article  Google Scholar 

  14. M.H. Jacobs, Diffusion Processes (Springer, New York, 1967)

  15. R. Zwanzig, J. Chem. Phys. 96, 3926 (1992)

    Article  Google Scholar 

  16. D. Reguera, J.M. Rubí, Phys. Rev. E 64, 061106 (2001)

    Article  ADS  Google Scholar 

  17. P. Kalinay, J.K. Percus, J. Chem. Phys. 122, 204701 (2005)

    Article  ADS  Google Scholar 

  18. P. Kalinay, J.K. Percus, Phys. Rev. E 74, 041203 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  19. S. Martens, G. Schmid, L. Schimansky-Geier, P. Hänggi, Phys. Rev. E 83, 051135 (2011)

    Article  ADS  Google Scholar 

  20. R.M. Bradley, Phys. Rev. E 80, 061142 (2009)

    Article  ADS  Google Scholar 

  21. A.M. Berezhkovskii, A. Szabo, J. Chem. Phys. 135, 074108 (2011)

    Article  ADS  Google Scholar 

  22. E. Yariv, H. Brenner, S. Kim, SIAM J. Appl. Math. 64, 1099 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. L. Dagdug, I. Pineda, J. Chem. Phys. 137, 024107 (2012)

    Article  ADS  Google Scholar 

  24. G. Chacón-Acosta, I. Pineda, L. Dagdug, J. Chem. Phys. 139, 214115 (2013)

    Article  ADS  Google Scholar 

  25. J. Fourier, Théorie analytique de la chaleur (Paris: Firmin Didot Père et Fils, 1822)

  26. N.F. Sheppard, D.J. Mears, S.W. Straka, J. Controlled Release 42, 15 (1996)

    Article  Google Scholar 

  27. Y. Yin, Y. Lu, B. Gates, Y. Xia, J. Am. Chem. Soc. 123, 8718 (2001)

    Article  Google Scholar 

  28. M.V. Vazquez, A.M. Berezhkovskii, L. Dagdug, J. Chem. Phys. 129, 046101 (2008)

    Article  ADS  Google Scholar 

  29. L. Dagdug, M.-V. Vazquez, A.M. Berezhkovskii, S.M. Bezrukov, J. Chem. Phys. 133, 134102 (2010)

    Article  ADS  Google Scholar 

  30. P.K. Ghosh, P. Hänggi, F. Marchesoni, S. Martens, F. Nori, L. Schimansky-Geier, G. Schmid, Phys. Rev. E 85, 011101 (2012)

    Article  ADS  Google Scholar 

  31. L. Dagdug, M.-V. Vázquez, A.M. Berezhkovskii, V.Y. Zitserman, S.M. Bezrukov, J. Chem. Phys. 136, 204106 (2012)

    Article  ADS  Google Scholar 

  32. L. Bosi, P.K. Ghosh, F. Marchesoni, J. Chem. Phys. 137, 174110 (2012)

    Article  ADS  Google Scholar 

  33. I. Pineda, M.V. Vázquez, A.M. Berezhkovskii, L. Dagdug, J. Chem. Phys. 135, 224101 (2011)

    Article  ADS  Google Scholar 

  34. S. Lifson, J.L. Jackson, J. Chem. Phys. 36, 2410 (1962)

    Article  ADS  Google Scholar 

  35. R. Zwanzig, J. Stat. Phys. 30, 275 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  36. F. Li, B. Ai, Phys. Rev. E 87, 062128 (2013)

    Article  ADS  Google Scholar 

  37. J. Álvarez-Ramírez, L. Dagdug, L. Inzunza, Physica A 410, 319 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  38. D. Salgado, I. Pineda, L. Dagdug (in preparation)

  39. I. Pineda, J. Álvarez–Ramírez, L. Dagdug, J. Chem. Phys. 137, 174103 (2012)

    Article  ADS  Google Scholar 

  40. M. Bauer, A. Godec, R. Metzler, Phys. Chem. Chem. Phys. 16, 6118 (2014)

    Article  Google Scholar 

  41. T. Fujiwara, K. Ritchie, H. Murakoshi, K. Jacobson, A. Kusumi, J. Cell Biol. 157, 1071 (2002)

    Article  Google Scholar 

  42. R.G. Parton, J.F. Hancock, Trends in Cell Biology 14, 41 (2004)

    Article  Google Scholar 

  43. S. Semrau, T. Schmidt, Soft Matter 5, 3174 (2009)

    Article  ADS  Google Scholar 

  44. M. Hirtz, A. Oikonomou, T. Georgiou, H. Fuchs, A. Vijayaraghavan, Nature Comm. 4, 2591 (2013)

    Article  ADS  Google Scholar 

  45. M.J. Saxton, Biophys. J. 39, 165 (1982)

    Article  ADS  Google Scholar 

  46. A. Kusumi, Y. Sako, M. Yamamoto, Biophys. J. 65, 2021 (1993)

    Article  ADS  Google Scholar 

  47. A.D. Douglass, R.D. Vale, Cell 121, 937 (2005)

    Article  Google Scholar 

  48. C.A. Day, A.K. Kenworthy, Biochim. Biophys. Acta 1788, 245 (2009)

    Article  Google Scholar 

  49. J. Hwang, L.A. Gheber, L. Margolis, M. Edidin, Biophys. J. 74, 2184 (1998)

    Article  ADS  Google Scholar 

  50. J. Capoulade, M. Wachsmuth, L. Hufnagel, M. Knop, Nat. Biotechnol. 29, 835 (2011)

    Article  Google Scholar 

  51. H. Risken, The Fokker-Planck equation. Methods of solution and applications (Springer, Berlin, 1989)

  52. N.G. van Kampen, J. Stat. Phys. 44, 1 (1986)

    Article  ADS  MATH  Google Scholar 

  53. P. Castro-Villareal, J. Stat. Mech. P08006, 1742 (2010)

    Google Scholar 

  54. J. Balakrishnan, Phys. Rev. E 61, 4648 (2000)

    Article  ADS  Google Scholar 

  55. J. Faraudo, J. Chem. Phys. 116, 5831 (2002)

    Article  ADS  Google Scholar 

  56. A.A. García-Chung, G. Chacón-Acosta, L. Dagdug (in preparation)

  57. A.M. Berezhkovskii, M.A. Pustovoit, S.M. Bezrukov, J. Chem. Phys. 126, 134706 (2007)

    Article  ADS  Google Scholar 

  58. P. Kalinay, J. Chem. Phys. 126, 194708 (2007)

    Article  ADS  Google Scholar 

  59. M.-V. Vázquez, L. Dagdug, J. Non-Newtonian Fluid Mech. 165, 987 (2010)

    Article  MATH  Google Scholar 

  60. Y. Chávez, G. Chacón-Acosta, M.-V. Vázquez, L. Dagdug, App. Math. 5, 1218 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Physics Department, Universidad Autónoma Metropolitana Iztapalapa, San Rafael Atlixco 186, México D.F., 09340, Mexico

    I. Pineda & L. Dagdug

  2. Applied Mathematics and Systems Department, Universidad Autónoma Metropolitana Cuajimalpa, Vasco de Quiroga 4871, México D.F., 05348, Mexico

    G. Chacón-Acosta

Authors
  1. I. Pineda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Chacón-Acosta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Dagdug
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Dagdug.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pineda, I., Chacón-Acosta, G. & Dagdug, L. Diffusion coefficients for two-dimensional narrow asymmetric channels embedded on flat and curved surfaces. Eur. Phys. J. Spec. Top. 223, 3045–3062 (2014). https://doi.org/10.1140/epjst/e2014-02318-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2014-02318-4

Keywords

Search

Navigation