Structural and electrical properties of an electric double layer formed inside a cylindrical pore investigated by Monte Carlo and classical density functional theory

  • Stanisław Lamperski
  • Shiqi ZhouEmail author
Research Paper


We present the properties of an electrical double layer formed by ions inside a charged cylindrical pore studied by the grand canonical Monte Carlo simulation and classical density functional theory. The cylinder radius is 3000 pm. The wall is hard, perfectly smooth. The ions are modelled by hard spheres with a point electric charge at the centre. The hard sphere diameter is fixed at 400 pm. The monovalent ions are immersed in a continuous dielectric medium of the relative permittivity εr. The temperature is 298.15 K and the electrolyte concentration takes the following values: 1.0, 2.5 and 4.0 M. The surface charge density varies in the range from − 1.0 to + 1.0 C/m2. The ion singlet distribution results show adsorption of counter-ions and desorption of co-ions from the cylindrical electrode. At high electrode charges the second layer of counter-ions is formed, while for high electrolyte concentration the co-ion distribution curve has a small maximum at some distance from the electrode surface. In comparison to the planar electrode, the concave one attracts stronger the counter-ions and repels the co-ions. At high electrolyte concentration, the profiles of the volume charge density have a positive hump, while those of the mean electrostatic potential have a negative minimum, which indicates the overscreening effect. For low electrolyte concentrations, the differential capacitance curve has a minimum at σ = 0 surrounded by two maxima. With increasing concentration, the minimum transforms into a maximum. The differential capacitance curves run above the curves for the planar electrode at small electrode charges and below them for high negative and positive charges. The very good agreement of all the grand canonical Monte Carlo to the classical density functional theory results presented in the paper indicates the reliability of the latter approach in cylindrical pore as well as planar geometry.


Electrical double layer Cylindrical pore Grand canonical Monte Carlo simulation Classical density functional theory Differential capacitance curve 



The authors would like to thank genuinely the anonymous reviewers for the constructive comments which help in deepening the discussions. SL gratefully acknowledges the financial support from the Faculty of Chemistry, Adam Mickiewicz University in Poznań. This project is supported by the National Natural Science Foundation of China (Nos. 21373274 and 21673299).


  1. Allen MP, Tildesley DJ, Computer simulation of liquids, Oxford University Press (1987) 349Google Scholar
  2. Blum L (1975) Mean spherical model for asymmetric electrolytes I. Method of solution. Molec Phys 30:1529CrossRefGoogle Scholar
  3. Chapman DL (1913) A contribution to the theory of electrocapillarity. Philos Mag 25:475–481CrossRefGoogle Scholar
  4. Frenkel D, Smit B (1996) Understanding molecular simulation: from algorithms to applications. Academic Press, San DiegozbMATHGoogle Scholar
  5. Georgi N, Kornyshev AA, Fedorov MV (2010) The anatomy of the double layer and capacitance in ionic liquids with anisotropic ions: electrostriction versus lattice saturation. J Electroanal Chem 649:261–267CrossRefGoogle Scholar
  6. Gouy G (1910) Sur la constitution de la charge électrique à la surface d’un electrolyte. Compt Rend 149:457–468zbMATHGoogle Scholar
  7. Henderson D (1992) Fundamentals of inhomogeneous fluids. Marcel Dekker, New YorkGoogle Scholar
  8. Hribar B, Vlachy V, Bhuiyan LB, Outhwaite CW (2000) Ion Distributions in a cylindrical capillary as seen by the modified Poisson-Boltzmann theory and Monte Carlo simulations. J Phys Chem B 104:11522–11527CrossRefGoogle Scholar
  9. Jamnik B, Vlachy V (1993) Monte Carlo and Poisson-Boltzmann study of electrolyte exclusion from charged cylindrical micropores. J Am Chem Soc 115:477–481CrossRefGoogle Scholar
  10. Kierlik E, Rosinberg ML (1990) Free-energy density functional for the inhomogeneous hard-sphere fluid: application to interfacial adsorption. Phys Rev A 42:3382CrossRefGoogle Scholar
  11. Kong W, Wu J, Henderson D (2014) Density functional theory of the capacitance of single file ions in a narrow cylinder. J Col Interf Sci 449:130–135CrossRefGoogle Scholar
  12. Kornyshev AA (2013) The simplest model of charge storage in a single file metallic nanopores. Faraday Discuss 164:117–133CrossRefGoogle Scholar
  13. Lamperski S (2007) The individual and mean activity coefficients of an electrolyte from the inverse GCMC simulation. Mol Simul 33:1193CrossRefGoogle Scholar
  14. Lamperski S, Bhuiyan LB (2003) Counterion layering at high surface charge in an electric double layer. Effect of local concentration approximation. J Electroanal Chem 540:79CrossRefGoogle Scholar
  15. Lamperski S, Outhwaite CW (2008) Monte Carlo simulations of mixed electrolytes next to a plain charged surface. J Colloid Interface Sci 328:458CrossRefGoogle Scholar
  16. Lamperski S, Outhwaite CW,LB, Bhuiyan (2009a) The electric double-layer differential capacitance at and near zero surface charge for a restricted primitive model electrolyte. J Phys Chem B 113:8925CrossRefGoogle Scholar
  17. Lamperski S, Outhwaite CW, Bhuiyan LB (2009b) The electric double layer differential capacitance at and near zero surface charge for a restricted primitive model ionic solution. J Phys Chem B 113:8925–8929CrossRefGoogle Scholar
  18. Lamperski S, Sosnowska J, Bhuiyan LB, Henderson D (2014) Size asymmetric hard spheres as a convenient model for the capacitance of the electrical double layer of an ionic liquid. J Chem Phys 140:014704CrossRefGoogle Scholar
  19. Lo WY, Chan KY (1995) Non-neutrality in a charged capillary. Mol Phys 86:745CrossRefGoogle Scholar
  20. Lo WY, Chan KY, Lee M, Mok KL (1998) Molecular simulation of electrolytes in nanopores. J Electroanal Chem 450:265–272CrossRefGoogle Scholar
  21. Mashayak SY, Aluru NR (2017) Langevin-Poisson-EQT: a dipolar solvent based quasi-continuum approach for electric double layers. J Chem Phys 146:044108CrossRefGoogle Scholar
  22. Mashayak S, Aluru NR (2018) A multiscale model for charge inversion in electric double layers. J Chem Phys J Chem Phys 148:214102Google Scholar
  23. Mills P, Anderson CF, Record MT Jr (1985) Monte Carlo studies of counterion-DNA interactions. Comparison of the radial distribution of counterions with predictions of other polyelectrolyte theories. J Phys Chem 89:3984–3994CrossRefGoogle Scholar
  24. Outhwaite CW, Bhuiyan LB (1983) An improved modified Poisson–Boltzmann equation in electric-double-layer theory. J Chem Soc Faraday Trans 2:707–718, 79 CrossRefGoogle Scholar
  25. Pech D, Brunet M, Durou H, Huang P, Mochalin V, Gogotsi Y, Taberna PL, Simon P (2010) Ultrahigh-power micrometre-sized supercapacitors based on onionlike carbon. Nat Nanotechnol 5(9):651–654CrossRefGoogle Scholar
  26. Peng B (2009) Ion distributions, exclusion coefficients, and separation fractions of electrolytes in a charge cylindrical nanopore: a partially perturbative density functional theory study. J Chem Phys 131:134703CrossRefGoogle Scholar
  27. Rosenfeld Y (1989) Free-energy model for the inhomogeneous hard sphere fluid mixture and density-functional theory of freezing. Phys Rev Lett 63:980CrossRefGoogle Scholar
  28. Schmickler W, Henderson D, PCCP, On the capacitance of narrow nanotubes, 19 (2017) 20393–20400Google Scholar
  29. Stern OZ (1924) The theory of electrolytic double-layer. Electrochem 30:508–516Google Scholar
  30. Torrie GM, Valleau JP (1980) Electrical double layers. I. Monte Carlo study of a uniformly charged surface. J Chem Phys 73:5807CrossRefGoogle Scholar
  31. Vlachy V, Haymet ADJ (1989) Electrolytes in charged micropores. J Am Chem Soc 111:660–666Google Scholar
  32. Wang H, Fang J, Pilon L (2013) Scaling laws for carbon-based electric double layer capacitors. Electrochim Acta 109:316–321CrossRefGoogle Scholar
  33. Yu A (2013) Electrochemical supercapacitors for energy storage and delivery: fundamentals and applications. Taylor & Francis, Boca RatonGoogle Scholar
  34. Zhang LL, Zhao XS (2009) Carbon-based materials as supercapacitor electrodes. Chem Soc Rev 38(9):2520–2531CrossRefGoogle Scholar
  35. Zhou S (2010) Augmented Kierlik–Rosinberg fundamental measure functional and extension of fundamental measure functional to inhomogeneous non-hard sphere fluids. Commun Theor Phys 54:1023CrossRefGoogle Scholar
  36. Zhou S (2011) Enhanced KR-fundamental measure functional for inhomogeneous binary and ternary hard sphere mixtures. Commun Theor Phys 55:46CrossRefGoogle Scholar
  37. Zhou S (2015a) Electrostatic potential of mean force between two curved surfaces in the presence of counterion connectivity. Phys Rev E 92:052317CrossRefGoogle Scholar
  38. Zhou S (2015b) Three-body potential amongst similarly or differently charged cylinder colloids immersed in a simple electrolyte solution. J Stat Mech Theory Exp 2015:P11030CrossRefGoogle Scholar
  39. Zhou S (2016) Change of electrostatic potential of mean force between two curved surfaces due to different salt composition, ion valence and size under certain ionic strength. J Phys Chem Solids 89:53CrossRefGoogle Scholar
  40. Zhou S (2017a) Effective electrostatic interactions between two overall neutral surfaces with quenched charge heterogeneity over atomic length scale. J Stat Phys 169:1019MathSciNetCrossRefGoogle Scholar
  41. Zhou S (2017b) A new method suitable for calculating accurately wetting temperature over a wide range of conditions: based on the adaptation of continuation algorithm to classical DFT. J Phys Chem Solids 110:274CrossRefGoogle Scholar
  42. Zhou S (2018a) Capacitance of electrical double layer formed inside a single infinitely long cylindrical pore. J Stat Mech Theory Exp 2018:103203CrossRefGoogle Scholar
  43. Zhou S (2018b) Wetting transition of nonpolar neutral molecule system on a neutral and atomic length scale roughness substrate. J Stat Phys 170:979MathSciNetCrossRefGoogle Scholar
  44. Zhou S, Zhang M (2017) Statistical mechanics study on wetting behaviors of Ne on Mg surface. J Phys Chem Solids 103:123CrossRefGoogle Scholar
  45. Zhou S, Lamperski S, Zydorczak M (2014) Properties of a planar electric double layer under extreme conditions investigated by classical density functional theory and Monte Carlo simulations. J Chem Phys 141:064701CrossRefGoogle Scholar
  46. Zhou S, Lamperski S, Sokołowska M (2017) Classical density functional theory and Monte Carlo simulation study of electric double layer in the vicinity of a cylindrical electrode. J Stat Mech-Theory E 073207Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Physical ChemistryAdam Mickiewicz University in PoznańPoznańPoland
  2. 2.School of Physics and ElectronicsCentral South UniversityChangshaPeople’s Republic of China

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