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Journal of Computational Electronics

, Volume 13, Issue 4, pp 801–817 | Cite as

Hierarchies of transport equations for nanopores

Equations derived from the Boltzmann equation and the modeling of confined structures
  • Clemens Heitzinger
  • Christian Ringhofer
Article

Abstract

We review transport equations and their usage for the modeling and simulation of nanopores. First, the significance of nanopores and the experimental progress in this area are summarized. Then the starting point of all classical and semiclassical considerations is the Boltzmann transport equation as the most general transport equation. The derivation of the drift-diffusion equations from the Boltzmann equation is reviewed as well as the derivation of the Navier–Stokes equations. Nanopores can also be viewed as a special case of a confined structure and hence as giving rise to a multiscale problem, and therefore we review the derivation of a transport equation from the Boltzmann equation for such confined structures. Finally, the state of the art in the simulation of nanopores is summarized.

Keywords

Boltzmann equation Model hierarchy Drift-diffusion-Poisson system Navier–Stokes equation Confined structure Nanopore 

Notes

Acknowledgments

The first author acknowledges support by the FWF (Austrian Science Fund) START project No. Y660 PDE Models for Nanotechnology. The second author acknowledges support by NSF Grant 11-07465 (KI-Net).

References

  1. 1.
    Pennisi, E.: Search for pore-fection. Science 336(6081), 534–537 (2012)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Branton, D., et al.: The potential and challenges of nanopore sequencing. Nat. Biotechnol. 26(10), 1146–1153 (2008)CrossRefGoogle Scholar
  3. 3.
    Wanunu, M.: Nanopores: a journey towards DNA sequencing. Phys. Life Rev. 9(2), 125–158 (2012)CrossRefGoogle Scholar
  4. 4.
    Hall, A.R., Scott, A., Rotem, D., Mehta, K.K., Bayley, H., Dekker, C.: Hybrid pore formation by directed insertion of \(\alpha \)-haemolysin into solid-state nanopores. Nat. Nanotechnol. 5, 874–877 (2010)CrossRefGoogle Scholar
  5. 5.
    Bell, N.A.W., Engst, C.R., Ablay, M., Divitini, G., Ducati, C., Liedl, T., Keyser, U.F.: DNA origami nanopores. Nano Lett. 12(1), 512–517 (2012)CrossRefGoogle Scholar
  6. 6.
    Burns, J.R., Stulz, E., Howorka, S.: Self-assembled DNA nanopores that span lipid bilayers. Nano Lett. 13(6), 2351–2356 (2013)CrossRefGoogle Scholar
  7. 7.
    Cercignani, C.: The Boltzmann Equation and Its Applications. Springer-Verlag, New York (1988)CrossRefzbMATHGoogle Scholar
  8. 8.
    Maxwell, J.C.: On the dynamical theory of gases. Phil. Trans. Roy. Soc. (London) 157, 49–88 (1867)CrossRefGoogle Scholar
  9. 9.
    Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer-Verlag, Berlin (1994)CrossRefzbMATHGoogle Scholar
  10. 10.
    Ben Abdallah, N.: On a hierarchy of macroscopic models for semiconductors. J. Math. Phys. 37(7), 3306–3333 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Ben Abdallah, N., Degond, P., Génieys, S.: An energy-transport model for semiconductors derived from the Boltzmann equation. J. Statist. Phys. 84(1–2), 205–231 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Heitzinger, Clemens, Ringhofer, Christian: A transport equation for confined structures derived from the Boltzmann equation. Commun. Math. Sci. 9(3), 829–857 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Ben Abdallah, N., Degond, P., Markowich, P., Schmeiser, C.: High field approximations of the spherical harmonics expansion model for semiconductors. Z. Angew. Math. Phys. 52(2), 201–230 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations. Springer-Verlag, Wien (1990)CrossRefzbMATHGoogle Scholar
  15. 15.
    Anile, A.M., Allegretto, W., Ringhofer, C.: Mathematical Problems in Semiconductor Physics. Springer-Verlag, Berlin (2003)zbMATHGoogle Scholar
  16. 16.
    Baumgartner, Stefan, Heitzinger, Clemens: Existence and local uniqueness for 3d self-consistent multiscale models for field-effect sensors. Commun. Math. Sci. 10(2), 693–716 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Markowich, P.A.: The Stationary Semiconductor Device Equations. Springer-Verlag, Wien (1986)CrossRefGoogle Scholar
  18. 18.
    Degond, P., Méhats, F., Ringhofer, C.: Quantum energy-transport and drift-diffusion models. J. Statist. Phys. 118, 625–667 (2005)CrossRefzbMATHGoogle Scholar
  19. 19.
    Baumgartner, Stefan, Heitzinger, Clemens: A one-level FETI method for the drift-diffusion-Poisson system with discontinuities at an interface. J. Comput. Phys. 243, 74–86 (June 2013)Google Scholar
  20. 20.
    Rubinstein, I.: Electro-Diffusion of Lons. SIAM, Philadelpha, PA (1990)CrossRefGoogle Scholar
  21. 21.
    Ramírez, P., Mafé, S., Aguilella, V.M., Alcaraz, A.: Synthetic nanopores with fixed charges: an electrodiffusion model for ionic transport. Phys. Rev. E (3), 68(1):011910/1-8, (July 2003)Google Scholar
  22. 22.
    van der Straaten, T.A., Tang, J.M., Ravaioli, U., Eisenberg, R.S., Aluru, N.R.: Simulating ion permeation through the ompF porin ion channel using three-dimensional drift-diffusion theory. J. Comp. Electron. 2, 29–47 (2003)CrossRefGoogle Scholar
  23. 23.
    Gardner, C.L., Nonner, W., Eisenberg, R.S.: Electrodiffusion model simulation of ionic channels: 1D simulations. J. Comput. Electron. 3, 25–31 (2004)CrossRefGoogle Scholar
  24. 24.
    Coalson, R.D., Kurnikova, M.G.: Poisson–Nernst–Planck theory approach to the calculation of current through biological ion channels. IEEE Trans. Nanobiosci. 4(1), 81–93 (2005)CrossRefGoogle Scholar
  25. 25.
    Cervera, J., Schiedt, B., Ramírez, P.: A Poisson/Nernst–Planck model for ionic transport through synthetic conical nanopores. Europhys. Lett. 71(1), 35–41 (2005)CrossRefGoogle Scholar
  26. 26.
    Liu, Y., Sauer, J., Dutton, R.W.: Effect of electro-diffusion current flow on electrostatic screening in aqueous pores. J. Appl. Phys. 103, 084701 (2008)CrossRefGoogle Scholar
  27. 27.
    Cervera, J., Ramírez, P., Manzanares, J.A., Mafé, S.: Incorporating ionic size in the transport equations for charged nanopores. Microfluid. Nanofluid. 9(1), 41–53 (2010)CrossRefGoogle Scholar
  28. 28.
    Gardner, C.L., Jones, J.R.: Electrodiffusion model simulation of the potassium channel. J. Theoret. Biol. 291, 10–13 (2011) Google Scholar
  29. 29.
    Lu, B., Hoogerheide, D.P., Zhao, Q., Yu, D.: Effective driving force applied on DNA inside a solid-state nanopore. Phys. Rev. E 86(1), 011921 (2012)CrossRefGoogle Scholar
  30. 30.
    Gummel, H.K.: A self-consistent iterative scheme for one-dimensional steady state transistor calculations. IEEE Trans. Electron Devices ED–11, 455–465 (1964)CrossRefGoogle Scholar
  31. 31.
    Scharfetter, D.L., Gummel, H.K.: Large-signal analysis of a silicon read diode oscillator. IEEE Trans. Electron Devices ED 16(1), 64–77 (1969)CrossRefGoogle Scholar
  32. 32.
    Kerkhoven, T.: A proof of convergence of Gummel’s algorithm for realistic device geometries. SIAM J. Numer. Anal. 23(6), 1121–1137 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Gartland Jr, E.C.: On the uniform convergence of the Scharfetter–Gummel discretization in one dimension. SIAM J. Numer. Anal. 30(3), 749–758 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Mao, M., Ghosal, S., Hu, G.: Hydrodynamic flow in the vicinity of a nanopore induced by an applied voltage. Nanotechnology 24(24), 245202 (2013)CrossRefGoogle Scholar
  35. 35.
    Ai, Y., Qian, S.: Electrokinetic particle translocation through a nanopore. Phys. Chem. Chem. Phys. 13(9), 4060–4071 (2011)CrossRefGoogle Scholar
  36. 36.
    van Dorp, S., Keyser, U.F., Dekker, N.H., Dekker, C., Lemay, S.G.: Origin of the electrophoretic force on DNA in solid-state nanopores. Nat. Phys. 5, 347–351 (2009)CrossRefGoogle Scholar
  37. 37.
    Jubery, T.Z., Prabhu, A.S., Kim, M.J., Dutta, P.: Modeling and simulation of nanoparticle separation through a solid-state nanopore. Electrophoresis 33(2), 325–333 (2012)CrossRefGoogle Scholar
  38. 38.
    Bearden, S., Zhang, G.: The effects of the electrical double layer on giant ionic currents through single-walled carbon nanotubes. Nanotechnology 24(12), 125204 (2013)CrossRefGoogle Scholar
  39. 39.
    Liu, Y., Huber, D.E., Tabard-Cossa, V., Dutton, R.W.: Descreening of field effect in electrically gated nanopores. Appl. Phys. Lett. 97(1–3), 143109 (2010)CrossRefGoogle Scholar
  40. 40.
    Bhatia, S., Nicholson, D.: Molecular transport in nanopores. J. Chem. Phys. 119(3), 1719–1730 (2013)CrossRefGoogle Scholar
  41. 41.
    Joseph, S., Guan, W., Reed, M.A., Krstic, P.S.: A long DNA segment in a linear nanoscale Paul trap. Nanotechnology 21(1), 015103 (2010)CrossRefGoogle Scholar
  42. 42.
    Wei, G.-W., Zheng, Q., Chen, Z., Xia, K.: Variational multiscale models for charge transport. SIAM Rev. 54(4), 699–754 (2012)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State University (ASU)PhoenixUSA
  2. 2.Institute for Analysis and Scientific ComputingVienna University of Technology (TU Vienna)WienAustria

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