Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Charging Dynamics of Electric Double-Layer Nanocapacitors in Mean Field

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

  • *Contact author: ivan.palaia@kcl.ac.uk

Phys. Rev. Lett. 135, 148002 – Published 29 September, 2025

DOI: https://doi.org/10.1103/72b9-c8cq

Abstract

An electric double-layer capacitor (EDLC) stores energy by modulating the spatial distribution of ions in the electrolytic solution that it contains. We determine the mean-field timescales for planar EDLC relaxation to equilibrium after a potential difference is applied. We tackle first the fully symmetric case, where positive and negative ionic species have the same valence and diffusivity, and then the general, more complex, asymmetric case. Depending on the applied voltage and salt concentration, different regimes appear, revealing a remarkably rich phenomenology relevant for nanocapacitors.

View figure in article

Physics Subject Headings (PhySH)

See Also

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

Ivan Palaia, Adelchi J. Asta, Megh Dutta, Patrick B. Warren, Benjamin Rotenberg, and Emmanuel Trizac
Phys. Rev. E 112, 035417 (2025)

Article Text

References (45)

  1. M. Salanne, B. Rotenberg, K. Naoi, K. Kaneko, P.-L. Taberna, C. P. Grey, B. Dunn, and P. Simon, Efficient storage mechanisms for building better supercapacitors, Nat. Energy 1, 16070 (2016).
  2. P. Simon and Y. Gogotsi, Perspectives for electrochemical capacitors and related devices, Nat. Mater. 19, 1151 (2020).
  3. C. Merlet, B. Rotenberg, P. A. Madden, P. L. Taberna, P. Simon, Y. Gogotsi, and M. Salanne, On the molecular origin of supercapacitance in nanoporous carbon electrodes, Nat. Mater. 11, 306 (2012).
  4. S. A. Sherrill, P. Banerjee, G. W. Rubloff, and S. B. Lee, High to ultra-high power electrical energy storage, Phys. Chem. Chem. Phys. 13, 20714 (2011).
  5. K. V. G. Raghavendra, R. Vinoth, K. Zeb, C. V. Muralee Gopi, S. Sambasivam, M. R. Kummara, I. M. Obaidat, and H. J. Kim, An intuitive review of supercapacitors with recent progress and novel device applications, J. Energy Storage 31, 101652 (2020).
  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. M. E. Suss, S. Porada, X. Sun, P. M. Biesheuvel, J. Yoon, and V. Presser, Water desalination via capacitive deionization: What is it and what can we expect from it?, Energy Environ. Sci. 8, 2296 (2015).
  10. M. Simoncelli, N. Ganfoud, A. Sene, M. Haefele, B. Daffos, P.-L. Taberna, M. Salanne, P. Simon, and B. Rotenberg, Blue energy and desalination with nanoporous carbon electrodes: Capacitance from molecular simulations to continuous models, Phys. Rev. X 8, 021024 (2018).
  11. P. Sun and M. V. Mirkin, Electrochemistry of individual molecules in zeptoliter volumes, J. Am. Chem. Soc. 130, 8241 (2008).
  12. K. Mathwig, D. Mampallil, S. Kang, and S. G. Lemay, Electrical cross-correlation spectroscopy: Measuring picoliter-per-minute flows in nanochannels, Phys. Rev. Lett. 109, 118302 (2012).
  13. L. Rassaei, K. Mathwig, E. D. Goluch, and S. G. Lemay, Hydrodynamic voltammetry with nanogap electrodes, J. Phys. Chem. C 116, 10913 (2012).
  14. 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).
  15. 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).
  16. 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/.
  17. A. A. Kornyshev, Double-layer in ionic liquids: Paradigm change?, J. Phys. Chem. B 111, 5545 (2007).
  18. 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).
  19. Z. A. H. Goodwin, G. Feng, and A. A. Kornyshev, Mean-field theory of electrical double layer in ionic liquids with account of short-range correlations, Electrochim. Acta 225, 190 (2017).
  20. 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).
  21. A. A. Lee, S. Kondrat, D. Vella, and A. Goriely, Dynamics of ion transport in ionic liquids, Phys. Rev. Lett. 115, 106101 (2015).
  22. D. T. Limmer, C. Merlet, M. Salanne, D. Chandler, P. A. Madden, R. van Roij, and B. Rotenberg, Charge fluctuations in nanoscale capacitors, Phys. Rev. Lett. 111, 106102 (2013).
  23. S. Kondrat, P. Wu, R. Qiao, and A. A. Kornyshev, Accelerating charging dynamics in subnanometre pores, Nat. Mater. 13, 387 (2014).
  24. C. Péan, C. Merlet, B. Rotenberg, P. A. Madden, P. L. Taberna, B. Daffos, M. Salanne, and P. Simon, On the dynamics of charging in nanoporous carbon-based supercapacitors, ACS Nano 8, 1576 (2014).
  25. C. Noh and Y. Jung, Understanding the charging dynamics of an ionic liquid electric double layer capacitor: Via molecular dynamics simulations, Phys. Chem. Chem. Phys. 21, 6790 (2019).
  26. 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).
  27. L. Scalfi, M. Salanne, and B. Rotenberg, Molecular simulation of electrode-solution interfaces, Annu. Rev. Phys. Chem. 72, 189 (2021).
  28. T. Hoang Ngoc Minh, G. Stoltz, and B. Rotenberg, Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations, J. Chem. Phys. 158, 104103 (2023).
  29. G. Jeanmairet, B. Rotenberg, and M. Salanne, Microscopic simulations of electrochemical double-layer capacitors, Chem. Rev. 122, 10860 (2022).
  30. M. Z. Bazant, K. Thornton, and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E 70, 021506 (2004).
  31. 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).
  32. 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).
  33. M. Janssen and M. Bier, Transient response of an electrolyte to a thermal quench, Phys. Rev. E 99, 042136 (2019).
  34. 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).
  35. 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).
  36. A. J. Asta, I. Palaia, E. Trizac, M. Levesque, and B. Rotenberg, Lattice Boltzmann electrokinetics simulation of nanocapacitors, J. Chem. Phys. 151, 114104 (2019).
  37. B. Balu and A. S. Khair, Role of Stefan–Maxwell fluxes in the dynamics of concentrated electrolytes, Soft Matter 14, 8267 (2018).
  38. 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).
  39. R. J. Hunter, Foundations of Colloid Science, 2nd ed. (Oxford University Press, New York, 2001).
  40. I. Palaia, A. J. Asta, M. Dutta, P. B. Warren, B. Rotenberg, and E. Trizac, companion paper, Poisson-Nernst-Planck charging dynamics of an electric double-layer capacitor: Symmetric and asymmetric binary electrolytes, Phys. Rev. E 112, 035417 (2025).
  41. 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, 2006), pp. 97–122.
  42. R. A. Robinson and R. H. Stokes, Electrolyte Solutions: Second Revised Edition (Dover Publications, New York, 2002), pp. 286–292.
  43. G. Barbero and L. R. Evangelista, Adsorption Phenomena and Anchoring Energy in Nematic Liquid Crystals (CRC Press, Boca Raton, 2005), 10.1201/9781420037456.
  44. G. Barbero and I. Lelidis, Evidence of the ambipolar diffusion in the impedance spectroscopy of an electrolytic cell, Phys. Rev. E 76, 051501 (2007).
  45. 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).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation