Journal of Statistical Physics

, Volume 163, Issue 1, pp 156–174 | Cite as

A Local Approximation of Fundamental Measure Theory Incorporated into Three Dimensional Poisson–Nernst–Planck Equations to Account for Hard Sphere Repulsion Among Ions



The hard sphere repulsion among ions can be considered in the Poisson–Nernst–Planck (PNP) equations by combining the fundamental measure theory (FMT). To reduce the nonlocal computational complexity in 3D simulation of biological systems, a local approximation of FMT is derived, which forms a local hard sphere PNP (LHSPNP) model. In the derivation, the excess chemical potential from hard sphere repulsion is obtained with the FMT and has six integration components. For the integrands and weighted densities in each component, Taylor expansions are performed and the lowest order approximations are taken, which result in the final local hard sphere (LHS) excess chemical potential with four components. By plugging the LHS excess chemical potential into the ionic flux expression in the Nernst–Planck equation, the three dimensional LHSPNP is obtained. It is interestingly found that the essential part of free energy term of the previous size modified model (Borukhov et al. in Phys Rev Lett 79:435–438, 1997; Kilic et al. in Phys Rev E 75:021502, 2007; Lu and Zhou in Biophys J 100:2475–2485, 2011; Liu and Eisenberg in J Chem Phys 141:22D532, 2014) has a very similar form to one term of the LHS model, but LHSPNP has more additional terms accounting for size effects. Equation of state for one component homogeneous fluid is studied for the local hard sphere approximation of FMT and is proved to be exact for the first two virial coefficients, while the previous size modified model only presents the first virial coefficient accurately. To investigate the effects of LHS model and the competitions among different counterion species, numerical experiments are performed for the traditional PNP model, the LHSPNP model, the previous size modified PNP (SMPNP) model and the Monte Carlo simulation. It’s observed that in steady state the LHSPNP results are quite different from the PNP results, but are close to the SMPNP results under a wide range of boundary conditions. Besides, in both LHSPNP and SMPNP models the stratification of one counterion species can be observed under certain bulk concentrations.


Hard sphere repulsion Three dimensional fundamental measure theory Poisson–Nernst–Planck equations Size-modified PNP Equation of state 



The authors thank Prof. Weishi Liu, Dr. Derek Frydel and Prof. Bob Eisenberg for helpful discussions. Yu Qiao, Xuejiao Liu and Benzhuo Lu were supported by the State Key Laboratory of Scientific/Engineering Computing, the Chinese Academy of Sciences and the China NSF (91230106). Minxin Chen was supported by the China NSF (NSFC11001062) and the NSF of Jiangsu Province (BK20130278).


  1. 1.
    Abaid, N., Eisenberg, R.S., Liu, W.: Asymptotic expansions of I–V relations via a Poisson–Nernst–Planck system. SIAM. J. Appl. Dyn. Syst. 7, 1507–1526 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bazant, M.Z., Kilic, M.S., Storey, B.D., Ajdari, A.: Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions. Adv. Colloid Interface Sci. 152, 48–88 (2009)CrossRefGoogle Scholar
  3. 3.
    Boda, D., Nonner, W., Valisk, M., Henderson, D., Eisenberg, B., Gillespie, D.: Steric selectivity in Na channels arising from protein polarization and mobile side chains. Biophys. J. 93, 1960–1980 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    Boda, D., Nonner, W., Henderson, D., Eisenberg, B., Gillespie, D.: Volume exclusion in calcium selective channels. Biophys. J. 94, 3486–3496 (2008)ADSCrossRefGoogle Scholar
  5. 5.
    Boda, D., Valisk, M., Henderson, D., Eisenberg, B., Gillespie, D., Nonner, W.: Ionic selectivity in L-type calcium channels by electrostatics and hard-core repulsion. J. Gen. Physiol. 133, 497–509 (2009)CrossRefGoogle Scholar
  6. 6.
    Borukhov, I., Andelman, D., Orland, H.: Steric effects in electrolyte: a modified Poisson–Boltzmann equation. Phys. Rev. Lett. 79, 435–438 (1997)ADSCrossRefGoogle Scholar
  7. 7.
    Borukhov, I., Andelman, D., Orland, H.: Adsorption of large ions from an electrolyte solution: a modified Poisson–Boltzmann equation. Electrochim. Acta 46, 221–229 (2000)CrossRefGoogle Scholar
  8. 8.
    Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system I: interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)ADSCrossRefGoogle Scholar
  9. 9.
    Chen, D., Lear, J., Eisenberg, B.: Permeation through an open channel: Poisson–Nernst–Planck theory of a synthetic ionic channel. Biophys. J. 72, 97–116 (1997)ADSCrossRefGoogle Scholar
  10. 10.
    Chen, M., Lu, B.: Tmsmesh: a robust method for molecular surface mesh generation using a trace technique. J. Chem. Theory Comput. 7, 203–212 (2011)CrossRefGoogle Scholar
  11. 11.
    Chen, M., Tu, B., Lu, B.: Triangulated manifold meshing method preserving molecular surface topology. J. Mol. Graph. Model. 38, 411–418 (2012)CrossRefGoogle Scholar
  12. 12.
    Chu, V.B., Bai, Y., Lipfert, J., Herschlag, D., Doniach, S.: Evaluation of ion binding to DNA duplexes using a size-modified Poisson–Boltzmann theory. Biophys. J. 93, 3202–3209 (2007)ADSCrossRefGoogle Scholar
  13. 13.
    Eisenberg, B.: Crowded charges in ion channels. In: Advances in Chemical Physics, pp. 77–223, Wiley, New York (2011)Google Scholar
  14. 14.
    Evans, R.: Density functional theory for inhomogeneous fluids I: Simple fluids in equilibrium. Lectures at 3rd Warsaw School of Statistical Physics, Kazimierz Dolny 27 (2009)Google Scholar
  15. 15.
    Frink, L.J.D., Salinger, A.G.: Two- and three-dimensional nonlocal density functional theory for inhomogeneous fluids: I. Algorithms and parallelization. J. Comput. Phys. 159, 407–424 (2000)ADSCrossRefMATHGoogle Scholar
  16. 16.
    Frink, L.J.D., Salinger, A.G., Sears, M.P., Weinhold, J.D., Frischknecht, A.L.: Numerical challenges in the application of density functional theory to biology and nanotechnology. J. Phys. 14, 12167–12187 (2002)Google Scholar
  17. 17.
    Frydel, D., Levin, Y.: A close look into the excluded volume effects within a double layer. J. Chem. Phys. 137, 164703 (2012)ADSCrossRefGoogle Scholar
  18. 18.
    Gillespie, D.: A review of steric interactions of ions: why some theories succeed and others fail to account for ion size. Microfluid. Nanofluid. 18, 717–738 (2014)CrossRefGoogle Scholar
  19. 19.
    Gillespie, D., Nonner, W., Eisenberg, R.S.: Coupling Poisson–Nernst–Planck and density functional theory to calculate ion flux. J. Phys. 14, 12129–12145 (2002)Google Scholar
  20. 20.
    Gillespie, D., Nonner, W., Eisenberg, R.S.: Density functional theory of charged, hard-sphere fluids. Phys. Rev. E 68, 031503 (2003)ADSCrossRefGoogle Scholar
  21. 21.
    Hansen, J., McDonald, I.: Theory of Simple Liquids, 3rd edn. Academic Press, Cambridge (2006)MATHGoogle Scholar
  22. 22.
    Hansen-Goos, H., Mecke, K.: Fundamental measure theory for inhomogeneous fluids of nonspherical hard particles. Phys. Rev. Lett. 102, 018302 (2009)ADSCrossRefGoogle Scholar
  23. 23.
    Harris, R.C., Bredenberg, J.H., Silalahi, A.R.J., Boschitsch, A.H., Fenley, M.O.: Understanding the physical basis of the salt dependence of the electrostatic binding free energy of mutated charged ligand-nucleic acid complexes. Biophys. Chem. 156, 79–87 (2011)CrossRefGoogle Scholar
  24. 24.
    Harris, R.C., Boschitsch, A.H., Fenley, M.O.: Sensitivities to parameterization in the size-modified Poisson–Boltzmann equation. J. Chem. Phys. 140, 075102 (2014)ADSCrossRefGoogle Scholar
  25. 25.
    Horng, T.L., Lin, T.C., Liu, C., Eisenberg, B.: PNP equations with steric effects: a model of ion flow through channels. J. Phys. Chem. B 116, 11422–11441 (2012)CrossRefGoogle Scholar
  26. 26.
    Hyon, Y., Eisenberg, B., Liu, C.: A mathematical model for the hard sphere repulsion in ionic solutions. Commun. Math. Sci. 9, 459–475 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Im, W., Roux, B.: Ion permeation and selectivity of ompf porin: a theoretical study based on molecular dynamics, brownian dynamics, and continuum electrodiffusion theory. J. Mol. Biol. 322(4), 851–869 (2002)CrossRefGoogle Scholar
  28. 28.
    Ji, S., Liu, W.: Poisson–Nernst–Planck systems for ion flow with density functional theory for hard-sphere potential: I–V relations and critical potentials. Part I: analysis. J. Dyn. Differ. Equ. 24, 955–983 (2012)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Jiang, J., Cao, D., De Jiang, WuJ: Time-dependent density functional theory for ion diffusion in electrochemical systems. J. Phys. 26, 284102 (2014)Google Scholar
  30. 30.
    Jimenez-Morales, D., Liang, J., Eisenberg, B.: Ionizable side chains at catalytic active sites of enzymes. Eur. Biophys. J. 41, 449–460 (2012)CrossRefGoogle Scholar
  31. 31.
    Kamalvand, M., Keshavarzi, T.E., Mansoori, G.A.: Behavior of the confined hard-sphere fluid within nanoslits: a fundamental-measure density-functional theory study. Int. J. Nanosci. 07, 245–253 (2008)CrossRefGoogle Scholar
  32. 32.
    Kierlik, E., Rosinberg, M.L.: Free-energy density functional for the inhomogeneous hard-sphere fluid: application to interfacial adsorption. Phys. Rev. A 42, 3382–3387 (1990)ADSCrossRefGoogle Scholar
  33. 33.
    Kilic, M.S., Bazant, M.Z., Ajdari, A.: Steric effects in the dynamics of electrolytes at large applied voltages I: double-layer charging. Phys. Rev. E 75, 021502 (2007)ADSCrossRefGoogle Scholar
  34. 34.
    Knepley, M.G., Karpeev, D.A., Davidovits, S., Eisenberg, R.S., Gillespie, D.: An efficient algorithm for classical density functional theory in three dimensions: ionic solutions. J. Chem. Phys. 132, 124101 (2010)ADSCrossRefGoogle Scholar
  35. 35.
    Kurnikova, M.G., Coalson, R.D., Graf, P., Nitzan, A.: A lattice relaxation algorithm for three-dimensional Poisson–Nernst–Planck theory with application to ion transport through the gramicidin A channel. Biophys. J. 76, 642–656 (1999)CrossRefGoogle Scholar
  36. 36.
    Levesque, M., Vuilleumier, R., Borgis, D.: Scalar fundamental measure theory for hard spheres in three dimensions: application to hydrophobic solvation. J. Chem. Phys. 137, 034115 (2012)ADSCrossRefGoogle Scholar
  37. 37.
    Li, B., Liu, P., Xu, Z., Zhou, S.: Ionic size effects: generalized Boltzmann distributions, counterion stratification and modified Debye length. Nonlinearity 26, 2899–2922 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Lin, G., Liu, W., Yi, Y., Zhang, M.: Poisson–Nernst–Planck systems for ion flow with a local hard-sphere potential for ion size effects. SIAM J. Appl. Dyn. Syst. 12, 1613–1648 (2013)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Liu, J.L.: Numerical methods for the Poisson–Fermi equation in electrolytes. J. Comput. Phys. 247, 88–99 (2013)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Liu, J.L., Eisenberg, B.: Correlated ions in a calcium channel model: a Poisson–Fermi theory. J. Phys. Chem. B 117, 12051–12058 (2013)CrossRefGoogle Scholar
  41. 41.
    Liu, J.L., Eisenberg, B.: Poisson–Nernst–Planck-Fermi theory for modeling biological ion channels. J. Chem. Phys. 141, 22D532 (2014)CrossRefGoogle Scholar
  42. 42.
    Liu, W., Tu, X., Zhang, M.: Poisson–Nernst–Planck systems for ion flow with density functional theory for hard-sphere potential: I–V relations and critical potentials. Part II: numerics. J. Dyn. Differ. Equ. 24, 985–1004 (2012)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Lu, B., Zhou, Y.: Poisson–Nernst–Planck equations for simulating biomolecular diffusion-reaction processes II: size effects on ionic distributions and diffusion-reaction rates. Biophys. J. 100, 2475–2485 (2011)ADSCrossRefGoogle Scholar
  44. 44.
    Lu, B., Zhou, Y., Huber, G.A., Bond, S.D., Holst, M.J., McCammon, J.A.: Electrodiffusion: a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution. J. Chem. Phys. 127, 135102 (2007)ADSCrossRefGoogle Scholar
  45. 45.
    Lu, B., Zhou, Y., Holst, M.J., McCammon, J.A.: Recent progress in numerical methods for the Poisson–Boltzmann equation in biophysical applications. Commun. Comput. Phys. 3, 973–1009 (2008)MATHGoogle Scholar
  46. 46.
    Meng, D., Zheng, B., Lin, G., Sushko, M.L.: Numerical solution of 3D Poisson–Nernst–Planck equations coupled with classical density functional theory for modeling ion and electron transport in a confined environment. Commun. Comput. Phys. 16, 1298–1322 (2014)MathSciNetGoogle Scholar
  47. 47.
    Nauman, E., He, D.Q.: Nonlinear diffusion and phase separation. Chem. Eng. Sci. 56, 1999–2018 (2001)CrossRefGoogle Scholar
  48. 48.
    Phan, S., Kierlik, E., Rosinberg, M.L., Bildstein, B., Kahl, G.: Equivalence of two free-energy models for the inhomogeneous hard-sphere fluid. Phys. Rev. E 48, 618–620 (1993)ADSCrossRefGoogle Scholar
  49. 49.
    Pods, J., Schönke, J., Bastian, P.: Electrodiffusion models of neurons and extracellular space using the Poisson–Nernst–Planck equations-numerical simulation of the intra- and extracellular potential for an axon model. Biophys. J. 105, 242–254 (2013)ADSCrossRefGoogle Scholar
  50. 50.
    Qiao, Y., Tu, B., Lu, B.: Ionic size effects to molecular solvation energy and to ion current across a channel resulted from the nonuniform size-modified PNP equations. J. Chem. Phys. 140, 174102 (2014)ADSCrossRefGoogle Scholar
  51. 51.
    Rosenfeld, Y.: Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing. Phys. Rev. Lett. 63, 980–983 (1989)ADSCrossRefGoogle Scholar
  52. 52.
    Rosenfeld, Y.: Free-energy model for the inhomogeneous hard-sphere fluid in D dimensions: structure factors for the hard-disk (D = 2) mixtures in simple explicit form. Phys. Rev. A 42, 5978–5989 (1990)ADSCrossRefGoogle Scholar
  53. 53.
    Rosenfeld, Y.: Free energy model for inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas. J. Chem. Phys. 98, 8126–8148 (1993)ADSCrossRefGoogle Scholar
  54. 54.
    Rosenfeld, Y.: Density functional theory of molecular fluids: free-energy model for the inhomogeneous hard-body fluid. Phys. Rev. E 50, R3318–R3321 (1994)ADSCrossRefGoogle Scholar
  55. 55.
    Rosenfeld, Y.: Free energy model for the inhomogeneous hard-body fluid: application of the Gauss–Bonnet theorem. Mol. Phys. 86, 637–647 (1995)ADSCrossRefGoogle Scholar
  56. 56.
    Rosenfeld, Y., Levesque, D., Weis, J.J.: Free-energy model for the inhomogeneous hard-sphere fluid mixture: triplet and higher-order direct correlation functions in dense fluids. J. Chem. Phys. 92, 6818–6832 (1990)ADSCrossRefGoogle Scholar
  57. 57.
    Rosenfeld, Y., Schmidt, M., Löwen, H., Tarazona, P.: Dimensional crossover and the freezing transition in density functional theory. J. Phys. 8, L577–L581 (1996)Google Scholar
  58. 58.
    Rosenfeld, Y., Schmidt, M., Löwen, H., Tarazona, P.: Fundamental-measure free-energy density functional for hard spheres: dimensional crossover and freezing. Phys. Rev. E 55, 4245–4263 (1997)ADSCrossRefGoogle Scholar
  59. 59.
    Roth, R.: Fundamental measure theory for hard-sphere mixtures: a review. J. Phys. 22, 063102 (2010)Google Scholar
  60. 60.
    Roth, R., Evans, R., Lang, A., Kahl, G.: Fundamental measure theory for hard-sphere mixtures revisited: the White Bear version. J. Phys. 14, 12063–12078 (2002)Google Scholar
  61. 61.
    Santangelo, C.D.: Computing counterion densities at intermediate coupling. Phys. Rev. E 73, 041512 (2006)ADSCrossRefGoogle Scholar
  62. 62.
    Sears, M.P., Frink, L.J.D.: A new efficient method for density functional theory calculations of inhomogeneous fluids. J. Comput. Phys. 190, 184–200 (2003)ADSCrossRefMATHGoogle Scholar
  63. 63.
    Si, H.: TetGen, a delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. 41, 11:1–11:36 (2015)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Silalahi, A.R.J., Boschitsch, A.H., Harris, R.C., Fenley, M.O.: Comparing the predictions of the nonlinear Poisson–Boltzmann equation and the ion size-modified Poisson–Boltzmann equation for a low-dielectric charged spherical cavity in an aqueous salt solution. J. Chem. Theory Comput. 6, 3631–3639 (2010)CrossRefGoogle Scholar
  65. 65.
    Tarazona, P.: Density functional for hard sphere crystals: a fundamental measure approach. Phys. Rev. Lett. 84, 694–697 (2000)ADSCrossRefGoogle Scholar
  66. 66.
    Tarazona, P., Rosenfeld, Y.: From zero-dimension cavities to free-energy functionals for hard disks and hard spheres. Phys. Rev. E 55, R4873–R4876 (1997)ADSCrossRefGoogle Scholar
  67. 67.
    Tarazona, P., Rosenfeld, Y.: Free energy density functional from 0d cavities. New Approaches to Problems in Liquid State Theory, vol. 529, pp. 293–302. Springer, Netherlands (1999)CrossRefGoogle Scholar
  68. 68.
    Tu, B., Chen, M., Xie, Y., Zhang, L., Eisenberg, B., Lu, B.: A parallel finite element simulator for ion transport through three-dimensional ion channel systems. J. Comput. Chem. 34, 2065–2078 (2013)CrossRefGoogle Scholar
  69. 69.
    Xie, Y., Cheng, J., Lu, B., Zhang, L.: Parallel adaptive finite element algorithms for solving the coupled electro-diffusion equations. Mol. Based Math. Biol. 1, 90–108 (2013)CrossRefMATHGoogle Scholar
  70. 70.
    Yu, Y., Wu, J.: Structures of hard-sphere fluids from a modified fundamental-measure theory. J. Chem. Phys. 117, 10156–10164 (2002)ADSCrossRefGoogle Scholar
  71. 71.
    Zhang, L.: A parallel algorithm for adaptive local refinement of tetrahedral meshes using bisection. Numer. Math. Theory Methods Appl. 2, 65–89 (2009)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.State Key Laboratory of Scientific and Engineering Computing, National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.Department of Mathematics, Center for System BiologySoochow UniversitySuzhouChina

Personalised recommendations