Journal of Statistical Physics

, Volume 161, Issue 1, pp 227–249 | Cite as

Counter-Ions Near a Charged Wall: Exact Results for Disc and Planar Geometries

  • Ladislav ŠamajEmail author


Macromolecules, when immersed in a polar solvent like water, become charged by a fixed surface charge density which is compensated by “counter-ions” moving out of the surface. Such classical particle systems exhibit poor screening properties at any temperature and the trivial bulk regime (far away from the charged surface) with no particles, so the validity of standard Coulomb sum rules is questionable. In the present paper, we concentrate on the two-dimensional version of the model with the logarithmic interaction potential. We go from the finite disc to the semi-infinite planar geometry. The system is exactly solvable for two values of the coupling constant \(\varGamma \): in the Poisson–Boltzmann mean-field limit \(\varGamma \rightarrow 0\) and at the free-fermion point \(\varGamma =2\). We show that the finite-size expansion of the free energy does not contain universal term as is usual for Coulomb fluids. For the coupling constant being an arbitrary positive even integer, using an anticommuting representation of the partition function and many-body densities we derive a sequence of sum rules. As a result, the contact density of counter-ions at the wall is available for the disc. The amplitude function, which characterizes the asymptotic inverse-power law behavior of the two-body density along the wall, is found to be related to the particle density profile. The dielectric susceptibility tensor, calculated exactly for an arbitrary coupling and the particle number, exhibits the anticipated disc value in the thermodynamic limit, in spite of zero contribution from the bulk region. Some of the results obtained in the Poisson–Boltzmann limit are generalized to an arbitrary Euclidean dimension.


Counter-ions Logarithmic Coulomb interaction Free-fermion point Sum rules Dielectric susceptibility tensor 



The support received from Grant VEGA No. 2/0015/15 is acknowledged.


  1. 1.
    Andelman, D.: Introduction to electrostatics in soft and biological matter. In: Poon, W.C.K., Andelman, D. (eds.) Soft Condensed Matter Physics in Molecular and Cell Biology, vol. 6. Taylor & Francis, New York (2006)Google Scholar
  2. 2.
    Attard, P., Mitchell, D.J., Ninham, B.W.: Beyond Poisson-Boltzmann: images and correlations in the electric double layer. I. Counterions only. J. Chem. Phys. 88, 4987–4996 (1988)ADSCrossRefGoogle Scholar
  3. 3.
    Attard, Ph.: Electrolytes and the electric double layer. Adv. Chem. Phys. 92, 1–159 (1996)Google Scholar
  4. 4.
    Baus, M.: On the compressibility of a one-component plasma. J. Phys. A 11, 2451–2462 (1978)ADSCrossRefGoogle Scholar
  5. 5.
    Boroudjerdi, H., Kim, Y.-W., Naji, A., Netz, R.R., Schlagberger, X., Serr, A.: Statics and dynamics of strongly charged soft matter. Phys. Rep. 416, 129–199 (2005)ADSCrossRefGoogle Scholar
  6. 6.
    Cardy, J.L.: Conformal invariance and statistical mechanics. In: Brézin, E., Zinn-Justin, J. (eds.) Fields, Strings and Critical Phenomena, Les Houches, Session XLIX, 1988. North Holland, Amsterdam (1990)Google Scholar
  7. 7.
    Carnie, S.L., Chan, D.Y.C.: The Stillinger-Lovett condition for non-uniform electrolytes. Chem. Phys. Lett. 77, 437–440 (1981)ADSCrossRefGoogle Scholar
  8. 8.
    Carnie, S.L.: On sum rules and Stillinger-Lovett conditions for inhomogeneous Coulomb systems. J. Chem. Phys. 78, 2742–2745 (1983)ADSCrossRefGoogle Scholar
  9. 9.
    Choquard, Ph, Favre, P., Gruber, Ch.: On the equation of state of classical one component systems with long range forces. J. Stat. Phys. 23, 405–442 (1980)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Choquard, Ph., Piller, B., Rentsch, R.: On the dielectric susceptibility of classical Coulomb systems. J. Stat. Phys. 43, 197–205 (1986)Google Scholar
  11. 11.
    Choquard, Ph., Piller, B., Rentsch, R.: On the dielectric susceptibility of classical Coulomb systems. II. J. Stat. Phys. 46, 599–633 (1987)Google Scholar
  12. 12.
    Choquard, Ph., Piller, B., Rentsch, R., Vieillefosse, P.: Surface properties of finite classical Coulomb systems: Debye-Hückel approximation and computer simulations. J. Stat. Phys. 55, 1185–1262 (1989)Google Scholar
  13. 13.
    Forrester, P.J.: Finite-size corrections to the free energy of Coulomb systems with a periodic boundary condition. J. Stat. Phys. 63, 491–504 (1991)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Gelfand, I.M., Shilov, G.E.: Generalized functions. Academic Press, New York (1964)Google Scholar
  15. 15.
    Gulbrand, L., Jönsson, B., Wennerström, H., Linse, P.: Electrical double layer forces. A Monte Carlo study. J. Chem. Phys. 80, 2221–2228 (1984)ADSCrossRefGoogle Scholar
  16. 16.
    Hansen, J.P., Löwen, H.: Effective interactions between electric double layers. Annu. Rev. Phys. Chem. 51, 209–242 (2000)ADSCrossRefGoogle Scholar
  17. 17.
    Jancovici, B.: Exact results for the two-dimensional one-component plasma. Phys. Rev. Lett. 46, 386–388 (1981)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Jancovici, B.: Classical Coulomb systems near a plane wall. II. J. Stat. Phys. 29, 263–280 (1981)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Jancovici, B.: Classical Coulomb systems near a plane wall. I. J. Stat. Phys. 28, 43–65 (1982)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Jancovici, B.: Inhomogeneous two-dimensional plasmas. In: Henderson, D. (ed.) Inhomogeneous Fluids, pp. 201–237. Dekker, New York (1992)Google Scholar
  21. 21.
    Jancovici, B.: Classical Coulomb systems: screening and correlations revisited. J. Stat. Phys. 80, 445–459 (1995)MathSciNetADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Jancovici, B.: A sum rule for the two-dimensional two-component plasma. J. Stat. Phys. 100, 201–207 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jancovici, B.: Surface correlations for two-dimensional Coulomb fluids in a disc. J. Phys. Condens. Matter 14, 9121–9132 (2002)ADSCrossRefGoogle Scholar
  24. 24.
    Jancovici, B., Téllez, G.: Coulomb systems seen as critical systems: Ideal conductor boundaries. J. Stat. Phys. 82, 609–632 (1996)ADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Jancovici, B., Trizac, E.: Universal free energy correction for the two-dimensional one-component plasma. Physica A 284, 241–245 (2000)ADSCrossRefGoogle Scholar
  26. 26.
    Jancovici, B., Šamaj, L.: Charge correlations in a Coulomb system along a plane wall: a relation between asymptotic behavior and dipole moment. J. Stat. Phys. 105, 193–209 (2001)CrossRefzbMATHGoogle Scholar
  27. 27.
    Jancovici, B., Manificat, G., Pisani, C.: Coulomb systems seen as critical systems: finite-size effects in two dimensions. J. Stat. Phys. 76, 307–329 (1994)ADSCrossRefzbMATHGoogle Scholar
  28. 28.
    Jancovici, B., Kalinay, P., Šamaj, L.: Another derivation of a sum rule for the two-dimensional two-component plasma. Physica A 279, 260–267 (2000)ADSCrossRefGoogle Scholar
  29. 29.
    Jancovici, B., Šamaj, L.: Microscopic calculation of the dielectric susceptibility tensor for Coulomb fluids. II. J. Stat. Phys. 114, 1211–1234 (2004)ADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Kalinay, P., Markoš, P., Šamaj, L., Travěnec, I.: The sixth-moment sum rule for the pair correlations of the two-dimensional one-component plasma: exact result. J. Stat. Phys. 98, 639–666 (2000)CrossRefzbMATHGoogle Scholar
  31. 31.
    Kjellander, R., Marčelja, S.: Correlation and image charge effects in electric double-layers. Chem. Phys. Lett. 112, 49–53 (1984)ADSCrossRefGoogle Scholar
  32. 32.
    Landau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media, Chap. I, 2nd edn. Pergamon, Oxford (1963)Google Scholar
  33. 33.
    Lovett, R., Mou, C.Y., Buff, F.P.: The structure of the liquid-vapor interface. J. Chem. Phys. 65, 570–572 (1976)ADSCrossRefGoogle Scholar
  34. 34.
    Mallarino, J.P., Téllez, G., Trizac, E.: Counter-ion density profile around charged cylinders: the strong-coupling needle limit. J. Phys. Chem. B 117, 12702–12716 (2013)CrossRefGoogle Scholar
  35. 35.
    Mallarino, J.P., Téllez, G., Trizac. E.: The contact theorem for charged fluids: from planar to curved geometries. arXiv:1410.6499 (2014)
  36. 36.
    Mallarino, J.P., Téllez: Counter-ion density profile around a charged disc: from the weak to the strong association regime. arXiv:1502.05291 (2015)
  37. 37.
    Martin, Ph.A: Sum rules in charged fluids. Rev. Mod. Phys. 60, 1075–1127 (1988)Google Scholar
  38. 38.
    Messina, R.: Electrostatics in soft matter. J. Phys. Condens. Matter 21, 113102 (2009)ADSCrossRefGoogle Scholar
  39. 39.
    Netz, R.R., Orland, H.: Beyond Poisson-Boltzmann: fluctuation effects and correlation functions. Eur. Phys. J. E 1, 203–214 (2000)CrossRefGoogle Scholar
  40. 40.
    Netz, R.R.: Electrostatics of counter-ions at and between planar charged walls: from Poisson-Boltzmann to the strong-coupling theory. Eur. Phys. J. E 5, 557–574 (2001)CrossRefGoogle Scholar
  41. 41.
    Podgornik, R.: An analytic treatment of the first-order correction to the Poisson-Boltzmann interaction free energy in the case of counter-ion only Coulomb fluid. J. Phys. A 23, 275–284 (1990)ADSCrossRefzbMATHGoogle Scholar
  42. 42.
    Šamaj, L., Percus, J.K.: A functional relation among the pair correlations of the two-dimensional one-component plasma. J. Stat. Phys. 80, 811–824 (1995)ADSCrossRefzbMATHGoogle Scholar
  43. 43.
    Šamaj, L.: Microscopic calculation of the dielectric susceptibility tensor for Coulomb fluids. J. Stat. Phys. 100, 949–967 (2000)CrossRefzbMATHGoogle Scholar
  44. 44.
    Šamaj, L.: Is the two-dimensional one-component plasma exactly solvable? J. Stat. Phys. 117, 131–158 (2004)ADSCrossRefzbMATHGoogle Scholar
  45. 45.
    Šamaj, L., Jancovici, B.: Charge and current sum rules in quantum media coupled to radiation II. J. Stat. Phys. 139, 432–453 (2010)MathSciNetADSCrossRefzbMATHGoogle Scholar
  46. 46.
    Šamaj, L.: Counter-ions at single charged wall: sum rules. Eur. Phys. J. E 36, 100 (2013)CrossRefGoogle Scholar
  47. 47.
    Šamaj, L., Trizac, E.: Counterions at highly charged interfaces: From one plate to like-charge attraction. Phys. Rev. Lett. 106, 078301 (2011)ADSCrossRefGoogle Scholar
  48. 48.
    Šamaj, L., Trizac, E.: Counter-ions at charged walls: two-dimensional systems. Eur. Phys. J. E 34, 20 (2011)CrossRefGoogle Scholar
  49. 49.
    Šamaj, L., Trizac, E.: Counter-ions between or at asymmetrically charged walls: 2D free-fermion point. J. Stat. Phys. 156, 932–947 (2014)ADSCrossRefzbMATHGoogle Scholar
  50. 50.
    Stillinger, F.H., Lovett, R.: Ion-pair theory of concentrated electrolytes. I. Basic concepts. J. Chem. Phys. 48, 3858–3868 (1968)ADSCrossRefGoogle Scholar
  51. 51.
    Stillinger, F.H., Lovett, R.: General restriction on the distribution of ions in electrolytes. J. Chem. Phys. 49, 1991–1994 (1968)ADSCrossRefGoogle Scholar
  52. 52.
    Totsuji, H.: Distribution of charged particles near a charged hard wall in a uniform background. J. Chem. Phys. 75, 871–874 (1981)MathSciNetADSCrossRefGoogle Scholar
  53. 53.
    Usenko, A.S., Yakimenko, I.P.: Interaction energy of stationary charges in a bounded plasma. Sov. Tech. Phys. Lett. 5, 549–550 (1979)Google Scholar
  54. 54.
    Vieillefosse, P., Hansen, J.P.: Statistical mechanics of dense ionized matter. V. Hydrodynamic limit and transport coefficients of the classical one-component plasma. Phys. Rev. A 12, 1106–1116 (1975)ADSCrossRefGoogle Scholar
  55. 55.
    Wertheim, M.S.: Correlations in the liquid-vapor interface. J. Chem. Phys. 65, 2377–2381 (1976)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of PhysicsSlovak Academy of SciencesBratislavaSlovakia

Personalised recommendations