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Poisson-Nernst-Planck charging dynamics of an electric double-layer capacitor: Symmetric and asymmetric binary electrolytes

Ivan Palaia1,*, Adelchi J. Asta2, Megh Dutta3, Patrick B. Warren4, Benjamin Rotenberg5, and Emmanuel Trizac6

  • *Contact author: ivan.palaia@kcl.ac.uk
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Phys. Rev. E 112, 035417 – Published 29 September, 2025

DOI: https://doi.org/10.1103/p4dg-snqf

Abstract

A parallel plate capacitor containing an electrolytic solution is the simplest model of a supercapacitor or electric double-layer capacitor. Using both analytical and numerical techniques, we solve the Poisson-Nernst-Planck equations for such a system, describing the mean-field charging dynamics of the capacitor, when a constant potential difference is abruptly applied to its plates. Working at constant total number of ions, we focus on the physical processes involved in the relaxation and, whenever possible, give its functional shape and exact time constants. We first review and study the case of a symmetric binary electrolyte, where we assume the two ionic species to have the same charges and diffusivities. We then relax these assumptions and present results for a generic strong (i.e., fully dissociated) binary electrolyte. At low electrolyte concentration, the relaxation is simple to understand, as the dynamics of positive and negative ions appear decoupled. At higher electrolyte concentration, we distinguish several regimes. In the linear regime (low voltages), relaxation is multiexponential, it starts by the buildup of the equilibrium charge profile and continues with neutral mass diffusion, and the relevant timescales feature both the average and the Nernst-Hartley diffusion coefficients. In the purely nonlinear regime (intermediate voltages), the initial relaxation is slowed down exponentially due to increased capacitance, while bulk effects become more and more evident. In the fully nonlinear regime (high voltages), the dynamics of charge and mass are completely entangled and, asymptotically, the relaxation is linear in time. We finally discuss nonideal behavior in real capacitors and provide conditions for which mean-field is expected to hold.

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Charging Dynamics of Electric Double-Layer Nanocapacitors in Mean Field

Ivan Palaia, Adelchi J. Asta, Megh Dutta, Patrick B. Warren, Benjamin Rotenberg, and Emmanuel Trizac
Phys. Rev. Lett. 135, 148002 (2025)

Article Text

References (72)

  1. P. Simon and Y. Gogotsi, Materials for electrochemical capacitors, Nat. Mater. 7, 845 (2008).
  2. S. Chen, L. Qiu, and H.-M. Cheng, Carbon-based fibers for advanced electrochemical energy storage devices, Chem. Rev. 120, 2811 (2020).
  3. P. Simon and Y. Gogotsi, Perspectives for electrochemical capacitors and related devices, Nat. Mater. 19, 1151 (2020).
  4. M. Winter and R. J. Brodd, What are batteries, fuel cells, and supercapacitors? Chem. Rev. 104, 4245 (2004).
  5. D. P. Dubal, N. R. Chodankar, D.-H. Kim, and P. Gomez-Romero, Towards flexible solid-state supercapacitors for smart and wearable electronics, Chem. Soc. Rev. 47, 2065 (2018).
  6. D. Brogioli, Extracting renewable energy from a salinity difference using a capacitor, Phys. Rev. Lett. 103, 058501 (2009).
  7. N. Boon and R. van Roij, ‘Blue energy’ from ion adsorption and electrode charging in sea and river water, Mol. Phys. 109, 1229 (2011).
  8. M. Janssen, A. Härtel, and R. van Roij, Boosting capacitive blue-energy and desalination devices with waste heat, Phys. Rev. Lett. 113, 268501 (2014).
  9. V. Vivier and M. E. Orazem, Impedance analysis of electrochemical systems, Chem. Rev. 122, 11131 (2022).
  10. G. Barbero and L. R. Evangelista, Impedance spectroscopy of a cell: The role of the ions, in Adsorption Phenomena and Anchoring Energy in Nematic Liquid Crystals (CRC Press, Boca Raton, FL, 2005).
  11. G. Jeanmairet, B. Rotenberg, and M. Salanne, Microscopic simulations of electrochemical double-layer capacitors, Chem. Rev. 122, 10860 (2022).
  12. A. A. Kornyshev, Double-layer in ionic liquids: Paradigm change? J. Phys. Chem. B 111, 5545 (2007).
  13. M. Z. Bazant, B. D. Storey, and A. A. Kornyshev, Double layer in ionic liquids: Overscreening versus crowding, Phys. Rev. Lett. 106, 046102 (2011).
  14. J. Wu, Understanding the electric double-layer structure, capacitance, and charging dynamics, Chem. Rev. 122, 10821 (2022).
  15. M. Janssen, Curvature affects electrolyte relaxation: Studies of spherical and cylindrical electrodes, Phys. Rev. E 100, 042602 (2019).
  16. A. J. Asta, I. Palaia, E. Trizac, M. Levesque, and B. Rotenberg, Lattice Boltzmann electrokinetics simulation of nanocapacitors, J. Chem. Phys. 151, 114104 (2019).
  17. C. Lian, M. Janssen, H. Liu, and R. van Roij, Blessing and curse: How a supercapacitor's large capacitance causes its slow charging, Phys. Rev. Lett. 124, 076001 (2020).
  18. K. Ma, M. Janssen, C. Lian, and R. van Roij, Dynamic density functional theory for the charging of electric double layer capacitors, J. Chem. Phys. 156, 084101 (2022).
  19. J. Yang, M. Janssen, C. Lian, and R. van Roij, Simulating the charging of cylindrical electrolyte-filled pores with the modified Poisson–Nernst–Planck equations, J. Chem. Phys. 156, 214105 (2022).
  20. L. Varela, S. Andraus, E. Trizac, and G. Téllez, Relaxation dynamics of two interacting electrical double-layers in a 1D Coulomb system, J. Phys.: Condens. Matter 33, 394001 (2021).
  21. K. Breitsprecher, C. Holm, and S. Kondrat, Charge me slowly, I am in a hurry: Optimizing charge-discharge cycles in nanoporous supercapacitors, ACS Nano 12, 9733 (2018).
  22. I. Palaia, Charged systems in, out of, and driven to equilibrium: From nanocapacitors to cement, Ph.D. thesis, Paris-Saclay University, 2019, https://cnrs.hal.science/tel-02926717/.
  23. K. Breitsprecher, M. Janssen, P. Srimuk, B. L. Mehdi, V. Presser, C. Holm, and S. Kondrat, How to speed up ion transport in nanopores, Nat. Commun. 11, 6085 (2020).
  24. M. Z. Bazant, K. Thornton, and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E 70, 021506 (2004).
  25. M. Janssen and M. Bier, Transient dynamics of electric double-layer capacitors: Exact expressions within the Debye-Falkenhagen approximation, Phys. Rev. E 97, 052616 (2018).
  26. I. Palaia, A. J. Asta, M. Dutta, P. B. Warren, B. Rotenberg, and E. Trizac, companion paper, Charging dynamics of electric double-layer nanocapacitors in mean field, Phys. Rev. Lett. 135, 148002 (2025).
  27. R. J. Hunter, Foundations of Colloid Science, 2nd ed. (Oxford University Press, Oxford, 2001).
  28. D. Andelman, Introduction to electrostatics in soft and biological matter, in Soft Condensed Matter Physics in Molecular and Cell Biology, edited by W. C. K. Poon and D. Andelman (CRC Press, Boca Raton, FL, 2006), pp. 97–122.
  29. G. Barbero and A. L. Alexe-Ionescu, Role of the diffuse layer of the ionic charge on the impedance spectroscopy of a cell of liquid, Liq. Cryst. 32, 943 (2005).
  30. G. Barbero, F. Batalioto, and A. M. Figueiredo Neto, Theory of small-signal ac response of a dielectric liquid containing two groups of ions, Appl. Phys. Lett. 92, 172908 (2008).
  31. J. R. Macdonald, Double layer capacitance and relaxation in electrolytes and solids, Trans. Farad. Soc. 66, 943 (1970).
  32. A. A. Kornyshev and M. A. Vorotyntsev, Electric current across the metal-solid electrolyte interface II. Low-amplitude alternating current, Phys. Stat. Sol. (a) 39, 573 (1977).
  33. A. Kornyshev and M. Vorotyntsev, Conductivity and space charge phenomena in solid electrolytes with one mobile charge carrier species, a review with original material, Electrochim. Acta 26, 303 (1981).
  34. L. Bocquet and E. Charlaix, Nanofluidics, from bulk to interfaces, Chem. Soc. Rev. 39, 1073 (2010).
  35. Electric double layers, in Fundamentals of Interface and Colloid Science, Vol. 2, edited by J. Lyklema (Academic Press, San Diego, CA, 1995).
  36. J. R. Macdonald, Theory of the differential capacitance of the double layer in unadsorbed electrolytes, J. Chem. Phys. 22, 1857 (1954).
  37. F. Beunis, F. Strubbe, M. Marescaux, J. Beeckman, K. Neyts, and A. R. M. Verschueren, Dynamics of charge transport in planar devices, Phys. Rev. E 78, 011502 (2008).
  38. F. Beunis, F. Strubbe, M. Marescaux, K. Neyts, and A. R. M. Verschueren, Diffuse double layer charging in nonpolar liquids, Appl. Phys. Lett. 91, 182911 (2007).
  39. D. C. Grahame, Diffuse double layer theory for electrolytes of unsymmetrical valence types, J. Chem. Phys. 21, 1054 (1953).
  40. I. Lelidis and G. Barbero, Effect of different anionic and cationic mobilities on the impedance spectroscopy measurements, Phys. Lett. A 343, 440 (2005).
  41. G. Barbero and I. Lelidis, Evidence of the ambipolar diffusion in the impedance spectroscopy of an electrolytic cell, Phys. Rev. E 76, 051501 (2007).
  42. C. Chassagne, E. Dubois, M. L. Jiménez, J. P. M. van der Ploeg, and J. van Turnhout, Compensating for electrode polarization in dielectric spectroscopy studies of colloidal suspensions: Theoretical assessment of existing methods, Front. Chem. 4, 1 (2016).
  43. A. Antonova, G. Barbero, L. R. Evangelista, and P. Tilli, Ambipolar diffusion in the low frequency impedance response of electrolytic cells, J. Stat. Mech. (2020) 043202.
  44. R. A. Robinson and R. H. Stokes, Electrolyte Solutions: Second Revised Edition (Dover, Garden City, NY, 2002), pp. 286–292.
  45. H. Zhao, Diffuse-charge dynamics of ionic liquids in electrochemical systems, Phys. Rev. E 84, 051504 (2011).
  46. D. Fertig and M. Janssen, Charging dynamics of electric double layer capacitors including beyond-mean-field electrostatic correlations, Phys. Rev. E 112, 025414 (2025).
  47. R. R. Netz and H. Orland, Beyond Poisson-Boltzmann: Fluctuation effects and correlation functions, Eur. Phys. J. E 1, 203 (2000).
  48. A. Naji, M. Kanduč, J. Forsman, and R. Podgornik, Perspective: Coulomb fluids–Weak coupling, strong coupling, in between and beyond, J. Chem. Phys. 139, 150901 (2013).
  49. E. Trizac and L. Samaj, Like-charge colloidal attraction: A simple argument, in Proceedings of the International School of Physics Enrico Fermi, Vol. 184, edited by C. Bechinger, F. Sciortino, and P. Ziherl (Società Italiana di Fisica, Bologna; IOS Press, Amsterdam, 2012), pp. 61–73.
  50. I. Palaia, A. Goyal, E. Del Gado, L. Šamaj, and E. Trizac, Like-charge attraction at the nanoscale: Ground-state correlations and water destructuring, J. Phys. Chem. B 126, 3143 (2022).
  51. A. G. Moreira and R. R. Netz, Strong-coupling theory for counter-ion distributions, Europhys. Lett. 52, 705 (2000).
  52. C. D. Santangelo, Computing counterion densities at intermediate coupling, Phys. Rev. E 73, 041512 (2006).
  53. L. Šamaj, M. Trulsson, and E. Trizac, Strong-coupling theory of counterions between symmetrically charged walls: From crystal to fluid phases, Soft Matter 14, 4040 (2018).
  54. C. Zhao, T. Yang, S. Jin, and B. Wu, Measurement of electric double layer charging dynamics on platinum electrodes in aqueous solutions of alkali sulfates and nitrates, J. Phys. Chem. C 128, 5964 (2024).
  55. V. Lobaskin and R. R. Netz, Diffusive-convective transition in the non-equilibrium charging of an electric double layer, Europhys. Lett. 116, 58001 (2016).
  56. B. Balu and A. S. Khair, Role of Stefan–Maxwell fluxes in the dynamics of concentrated electrolytes, Soft Matter 14, 8267 (2018).
  57. M. Armand, F. Endres, D. R. MacFarlane, H. Ohno, and B. Scrosati, Ionic-liquid materials for the electrochemical challenges of the future, Nat. Mater. 8, 621 (2009).
  58. B. D. Storey and M. Z. Bazant, Effects of electrostatic correlations on electrokinetic phenomena, Phys. Rev. E 86, 056303 (2012).
  59. M. Kanduč, A. Naji, J. Forsman, and R. Podgornik, Dressed counterions: Strong electrostatic coupling in the presence of salt, J. Chem. Phys. 132, 124701 (2010).
  60. H. Bonneau, V. Démery, and E. Raphaël, Temporal response of the conductivity of electrolytes, J. Stat. Mech. (2023) 073205.
  61. R. M. Adar, T. Markovich, and D. Andelman, Bjerrum pairs in ionic solutions: A Poisson-Boltzmann approach, J. Chem. Phys. 146, 194904 (2017).
  62. I. Borukhov, D. Andelman, and H. Orland, Steric effects in electrolytes: A modified Poisson-Boltzmann equation, Phys. Rev. Lett. 79, 435 (1997).
  63. D. Ben-Yaakov, D. Andelman, D. Harries, and R. Podgornik, Beyond standard Poisson-Boltzmann theory: Ion-specific interactions in aqueous solutions, J. Phys.: Condens. Matter 21, 424106 (2009).
  64. J. Jiang, D. Cao, D. E. Jiang, and J. Wu, Time-dependent density functional theory for ion diffusion in electrochemical systems, J. Phys.: Condens. Matter 26, 284102 (2014).
  65. A. Härtel, Structure of electric double layers in capacitive systems and to what extent (classical) density functional theory describes it, J. Phys.: Condens. Matter 29, 423002 (2017).
  66. A. Gupta, A. Govind Rajan, E. A. Carter, and H. A. Stone, Ionic layering and overcharging in electrical double layers in a Poisson-Boltzmann model, Phys. Rev. Lett. 125, 188004 (2020).
  67. M. Bültmann and A. Härtel, The primitive model in classical density functional theory: Beyond the standard mean-field approximation, J. Phys.: Condens. Matter 34, 235101 (2022).
  68. A. Schlaich, A. P. Dos Santos, and R. R. Netz, Simulations of nanoseparated charged surfaces reveal charge-Induced water reorientation and nonadditivity of hydration and mean-field electrostatic repulsion, Langmuir 35, 551 (2019).
  69. T. Mendez-Morales, M. Burbano, M. Haefele, B. Rotenberg, and M. Salanne, Ion-ion correlations across and between electrified graphene layers, J. Chem. Phys. 148, 193812 (2018).
  70. B. L. Werkhoven, J. C. Everts, S. Samin, and R. van Roij, Flow-induced surface charge heterogeneity in electrokinetics due to Stern-Layer conductance coupled to reaction kinetics, Phys. Rev. Lett. 120, 264502 (2018).
  71. M. Janssen and M. Bier, Transient response of an electrolyte to a thermal quench, Phys. Rev. E 99, 042136 (2019).
  72. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, New York, 2007).

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Phys. Rev. E 112, 035417– Published 29 September, 2025
  • Received 15 March 2023
  • Revised 1 June 2025
  • Accepted 20 August 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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