pith. sign in

arxiv: 2606.21980 · v1 · pith:XVSN2FECnew · submitted 2026-06-20 · ⚛️ physics.chem-ph · cond-mat.mtrl-sci

The impedance of a charged flat-plate electric double-layer capacitor

Adrian L. Usler , David Fertig , Mathijs Janssen This is my paper

Pith reviewed 2026-06-26 11:10 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.mtrl-sci
keywords electric double layerimpedance spectroscopyNyquist plotWarburg elementambipolar diffusionPoisson-Nernst-Planckcapacitor
0
0 comments X

The pith

Slanted lines in Nyquist plots of biased or asymmetrically diffusing EDL capacitors originate from ambipolar salt diffusion and are described by Warburg circuit elements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper calculates the impedance of flat-plate electric double-layer capacitors using finite element simulations of modified Poisson-Nernst-Planck equations. It finds that a slanted line appears at intermediate frequencies in Nyquist plots when a bias voltage is applied or when cation and anion diffusion coefficients differ. Inspection of concentration perturbations shows this line arises from ambipolar salt diffusion in both cases. The work disproves prior claims that the line width represents EDL resistance or that its slope measures the ratio of diffusion to charging time scales. Instead it proposes two equivalent circuits using Warburg short and open elements whose prefactors relate quantitatively to the ambipolar diffusion coefficient, bulk chemical capacitance, and differential charge efficiency of the EDLs.

Core claim

The impedance spectra of biased or asymmetrically diffusing flat-plate EDL capacitors exhibit a slanted line at intermediate frequencies caused by ambipolar salt diffusion. Symmetric salt perturbations from nonzero bias are modeled by a Warburg open element while antisymmetric perturbations from unequal diffusion coefficients are modeled by a Warburg short element. The resulting circuit analysis supplies explicit relations connecting the Warburg prefactors to the ambipolar diffusion coefficient, the chemical capacitance of the bulk electrolyte, and the differential charge efficiency of the EDLs.

What carries the argument

Warburg short and open circuit elements that capture ambipolar salt diffusion in the equivalent-circuit models for the two cases of bias and unequal diffusion coefficients.

If this is right

  • The width of the slanted-line region saturates at large bias voltages.
  • The width of the slanted-line region vanishes for large ion packing fractions.
  • The overall capacitance of the closed system is limited by the finite total number of ions.
  • The Warburg prefactors are quantitatively linked to the ambipolar diffusion coefficient, bulk chemical capacitance, and EDL differential charge efficiency.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The symmetric versus antisymmetric distinction in salt perturbations may generalize to impedance analysis of other confined electrochemical cells.
  • Varying bulk ion concentration while holding bias fixed could test the predicted dependence of the Warburg prefactor on chemical capacitance.
  • The finite-ion limitation identified here suggests that closed-system corrections become essential for modeling small-volume or low-concentration EDL devices.

Load-bearing premise

The modified Poisson-Nernst-Planck continuum equations with the chosen boundary conditions and discretization faithfully represent the ion transport and electrostatics without additional microscopic effects.

What would settle it

Direct observation of the spatial and temporal form of salt concentration perturbations at the frequencies where the slanted line appears, checking whether they match the predicted symmetric or antisymmetric ambipolar diffusion patterns.

Figures

Figures reproduced from arXiv: 2606.21980 by Adrian L. Usler, David Fertig, Mathijs Janssen.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic of a flat-plate EDL capacitor subject [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (b) and (c) show that ω 0 c appears at lower an￾gular frequency (scaled to ωD) with increasing electrode separation L/λD. For ω < ω0 c , ReZ reaches a plateau [ [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Impedance spectra obtained for various electrode sep [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Characteristic quantities extracted from impedance [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Impedance spectra for varying diffusivity ratios, [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Plot of low-frequency limit of Re [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Profiles of Laplace-transformed concentration perturbations [see Eq. (6)] of cations (solid red lines) and anions (dashed [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Impedance spectra for [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Proposed equivalent circuits for the description of [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Circuit parameters [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Impedance parameters obtained by fitting circuit B [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
read the original abstract

We calculate the impedance of a flat-plate electric double-layer (EDL) capacitor by means of Finite Element Method simulations of modified Poisson--Nernst--Planck equations. In Nyquist representation, the impedance spectra show a slanted line at intermediate frequencies if the capacitor is biased by a voltage, $U_\mathrm{bias}\neq 0$, or if the cation and anion diffusion coefficients differ, $D_-\neq D_+$. By inspecting the concentration perturbations in the relevant frequency range, we confirm that the slanted line is in both cases related to ambipolar salt diffusion. On the basis of our impedance data, we disprove two previously made claims: 1) that the width $R_\mathrm{sl}$ of the slanted-line region represents an EDL resistance; and 2) that the slope $k_\mathrm{sl}$ of the slanted line is a measure of the ratio of the diffusion and charging time scales, $\tau_\mathrm{diff}/\tau_\mathrm{c}$. For the quantitative analysis of flat-electrode EDL capacitor impedance, we propose instead two equivalent circuits, for the cases $D_-\neq D_+$ and $U_\mathrm{bias}\neq 0$. These two cases give rise to antisymmetric and symmetric salt perturbations, which are best described by a Warburg short and Warburg open element, respectively. From our circuit analysis, we obtain quantitative relations that link the Warburg prefactors to the ambipolar diffusion coefficient, the chemical capacitance of the bulk electrolyte, and the differential charge efficiency of the EDLs. We thus provide a theoretical framework that explains why the width of the slanted line saturates at large biases, why it vanishes for large ion packing fractions, and how the system's overall capacitance is limited by the finite amount of ions in a closed system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript uses finite-element simulations of modified Poisson-Nernst-Planck equations to compute the impedance spectra of a flat-plate electric double-layer capacitor. It shows that a slanted line appears in Nyquist plots at intermediate frequencies when a bias voltage is applied or when cation and anion diffusivities differ, attributes this feature to ambipolar salt diffusion via concentration-perturbation analysis, disproves two prior interpretations of the slanted-line width and slope, and proposes equivalent-circuit models employing Warburg short and open elements whose prefactors are quantitatively linked to the ambipolar diffusion coefficient, bulk chemical capacitance, and differential charge efficiency of the EDLs. The framework is used to explain saturation of the slanted-line width at large bias, its disappearance at high packing fractions, and the ion-number limitation on total capacitance.

Significance. If the central results hold, the work supplies a concrete, falsifiable mapping between observable impedance features and microscopic transport parameters that is directly testable by experiment. The combination of perturbation diagnostics, circuit fitting, and explicit relations to independently defined quantities (ambipolar diffusivity, chemical capacitance) constitutes a clear advance over purely phenomenological descriptions. The explanation of bias saturation and packing-fraction dependence supplies mechanistic insight that can guide device modeling.

major comments (2)
  1. [§3] §3 (model section): the precise functional form of the 'modified' PNP equations (steric corrections, boundary conditions at the electrodes) is not reproduced in the main text; because the ambipolar-diffusion mechanism and the resulting Warburg prefactors depend on these modifications, an explicit statement of the governing equations and the chosen mesh-convergence criteria is required to allow independent reproduction.
  2. [§5.2] §5.2 (circuit analysis): the quantitative relations that map Warburg prefactors onto ambipolar diffusivity, chemical capacitance, and differential charge efficiency are obtained by fitting the same simulation data used to generate the spectra; an independent, a-priori calculation of these three quantities from the steady-state profiles (without re-fitting) would remove any residual concern about circularity.
minor comments (3)
  1. [Figure 4] Figure 4: the Nyquist plots for the two cases (bias vs. diffusivity asymmetry) use different line styles that are not explained in the caption; adding a legend or explicit labels would improve readability.
  2. Notation: the symbol for differential charge efficiency is introduced only in the abstract and the final paragraph; an explicit definition (e.g., as a derivative of surface charge with respect to bulk concentration) should appear in the methods or results section.
  3. The manuscript cites prior claims that are being disproved but does not provide the exact references or the numerical values those works reported for R_sl and k_sl; adding these citations and a short comparison table would strengthen the disproof section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The comments are constructive and we address each one below.

read point-by-point responses

  1. Referee: [§3] §3 (model section): the precise functional form of the 'modified' PNP equations (steric corrections, boundary conditions at the electrodes) is not reproduced in the main text; because the ambipolar-diffusion mechanism and the resulting Warburg prefactors depend on these modifications, an explicit statement of the governing equations and the chosen mesh-convergence criteria is required to allow independent reproduction.

    Authors: We agree that explicit reproduction of the modified Poisson-Nernst-Planck equations (including steric corrections) and the electrode boundary conditions is necessary for independent verification, given their direct role in the ambipolar-diffusion analysis. In the revised manuscript we will insert the complete set of governing equations into the main text of §3 together with the mesh-convergence criteria used for the finite-element calculations. revision: yes

  2. Referee: [§5.2] §5.2 (circuit analysis): the quantitative relations that map Warburg prefactors onto ambipolar diffusivity, chemical capacitance, and differential charge efficiency are obtained by fitting the same simulation data used to generate the spectra; an independent, a-priori calculation of these three quantities from the steady-state profiles (without re-fitting) would remove any residual concern about circularity.

    Authors: We accept the referee’s point that an independent verification would eliminate any appearance of circularity. In the revision we will add a direct, a-priori evaluation of the ambipolar diffusivity, bulk chemical capacitance and differential charge efficiency computed solely from the steady-state concentration and potential profiles; these values will then be compared with the Warburg-prefactor results to confirm consistency without re-fitting the impedance spectra. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds from FEM solutions of the modified PNP equations, direct inspection of symmetric/antisymmetric concentration perturbations to attribute the slanted Nyquist line to ambipolar diffusion, and standard Warburg-element circuit fitting whose prefactor mappings are obtained by algebraic rearrangement of independently defined quantities (ambipolar D, bulk chemical capacitance, differential charge efficiency). No step reduces a claimed prediction to a fitted parameter by construction, no load-bearing premise rests on self-citation, and the quantitative relations follow from the circuit topology rather than tautological re-labeling of simulation outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the continuum modified Poisson-Nernst-Planck description of ion transport and on the assumption that the simulated concentration perturbations directly correspond to the physical mechanism responsible for the observed impedance features.

axioms (2)
  • domain assumption Modified Poisson-Nernst-Planck equations govern the coupled ion transport and electrostatic potential inside the electrolyte domain.
    All simulation results and subsequent circuit interpretations are generated from solutions of these equations.
  • domain assumption The capacitor is a closed finite-volume system containing a fixed total number of ions.
    The abstract invokes this fact to explain saturation of the slanted-line width and the overall capacitance limit.

pith-pipeline@v0.9.1-grok · 5879 in / 1577 out tokens · 26574 ms · 2026-06-26T11:10:37.486419+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

83 extracted references · 4 canonical work pages

  1. [1]

    B. E. Conway,Electrochemical supercapacitors: scientific fundamentals and technological applications(Springer Science & Business Media, 2013)

    2013

  2. [2]

    J. R. Macdonald, Phys. Rev.92, 4 (1953)

    1953

  3. [3]

    D. R. Franceschetti and J. R. Macdonald, J. Electroanal. Chem. Interf. Electrochem.100, 583 (1979)

    1979

  4. [4]

    Barbero and I

    G. Barbero and I. Lelidis, Phys. Rev. E76, 051501 (2007)

    2007

  5. [5]

    B.-A. Mei, O. Munteshari, J. Lau, B. Dunn, and L. Pilon, J. Phys. Chem. C122, 194 (2018)

    2018

  6. [6]

    Barbero, A

    G. Barbero, A. S. Gliozzi, M. Scalerandi, and A. M. Scar- fone, J. Mol. Liq.272, 565 (2018)

    2018

  7. [7]

    J. J. L´ opez-Garc´ ıa, J. Horno, and C. Grosse, Microma- chines14, 368 (2023)

    2023

  8. [8]

    J. J. L´ opez-Garc´ ıa, J. Horno, and C. Grosse, J. Phys. Chem. C129, 12561 (2025)

    2025

  9. [9]

    M. Z. Bazant, K. Thornton, and A. Ajdari, Phys. Rev. E 70, 021506 (2004)

    2004

  10. [10]

    M. S. Kilic, M. Z. Bazant, and A. Ajdari, Phys. Rev. E 75, 021503 (2007)

    2007

  11. [11]

    Janssen and M

    M. Janssen and M. Bier, Phys. Rev. E97, 052616 (2018)

    2018

  12. [12]

    R. F. Stout and A. S. Khair, Phys. Rev. E92, 032305 (2015)

    2015

  13. [13]

    Balu and A

    B. Balu and A. S. Khair, Soft Matter14, 8267 (2018)

    2018

  14. [14]

    Babel, M

    S. Babel, M. Eikerling, and H. L¨ owen, J. Phys. Chem. C 122, 21724 (2018)

    2018

  15. [15]

    K. Ma, M. Janssen, C. Lian, and R. van Roij, J. Chem. Phys.156, 084101 (2022)

    2022

  16. [16]

    Pireddu and B

    G. Pireddu and B. Rotenberg, Phys. Rev. Lett.130, 098001 (2023)

    2023

  17. [17]

    Pireddu, C

    G. Pireddu, C. J. Fairchild, S. P. Niblett, S. J. Cox, and B. Rotenberg, Proc. Natl. Acad. Sci.121, e2318157121 (2024)

    2024

  18. [18]

    Barnaveli and R

    A. Barnaveli and R. van Roij, Soft Matter20, 704 (2024)

    2024

  19. [19]

    Palaia, A

    I. Palaia, A. J. Asta, M. Dutta, P. B. Warren, B. Roten- berg, and E. Trizac, Phys. Rev. Lett.135, 148002 (2025)

    2025

  20. [20]

    Palaia, A

    I. Palaia, A. J. Asta, M. Dutta, P. B. Warren, B. Roten- berg, and E. Trizac, Phys. Rev. E112, 035417 (2025)

    2025

  21. [21]

    Beunis, F

    F. Beunis, F. Strubbe, M. Marescaux, K. Neyts, and A. R. M. Verschueren, Appl. Phys. Lett.91, 182911 (2007)

    2007

  22. [22]

    Beunis, F

    F. Beunis, F. Strubbe, M. Marescaux, J. Beeckman, K. Neyts, and A. R. M. Verschueren, Phys. Rev. E78, 011502 (2008)

    2008

  23. [23]

    Marzantowicz, J

    M. Marzantowicz, J. R. Dygas, and F. Krok, Electrochim. Acta53, 7417 (2008)

    2008

  24. [24]

    Nakamura, H

    M. Nakamura, H. Kaminaga, O. Endo, H. Tajiri, O. Sakata, and N. Hoshi, J. Phys. Chem. C118, 22136 (2014)

    2014

  25. [25]

    R. J. Kortschot, A. P. Philipse, and B. H. Ern´ e, J. Phys. Chem. C118, 11584 (2014)

    2014

  26. [26]

    Schuster, Curr

    R. Schuster, Curr. Opin. Electrochem.1, 88 (2017)

    2017

  27. [27]

    Schalenbach, Y

    M. Schalenbach, Y. E. Durmus, H. Tempel, H. Kungl, and R.-A. Eichel, Phys. Chem. Chem. Phys.23, 21097 (2021)

    2021

  28. [28]

    C. Zhao, T. Yang, S. Jin, and B. Wu, J. Phys. Chem. C 128, 5964 (2024)

    2024

  29. [29]

    Kutbay, B

    E. Kutbay, B. Ulgut, C. Kocabas, and S. Suzer, J. Phys. Chem. Lett.16, 8778 (2025)

    2025

  30. [30]

    Barsoukov and J

    E. Barsoukov and J. R. Macdonald,Impedance spec- troscopy: theory, experiment, and applications(John Wi- ley & Sons, 2018)

    2018

  31. [31]

    M. E. Orazem and B. Tribollet,Electrochemical Impedance Spectroscopy, 2nd ed. (John Wiley & Sons, 2017)

    2017

  32. [32]

    Lasia,Electrochemical Impedance Spectroscopy and its Applications(Springer, 2014)

    A. Lasia,Electrochemical Impedance Spectroscopy and its Applications(Springer, 2014)

    2014

  33. [33]

    Levie, Adv

    D. Levie, Adv. Electrochem. and Electrochem. Eng6, 329 (1967)

    1967

  34. [34]

    C. Ho, I. D. Raistrick, and R. A. Huggins, J. Electrochem. Soc.127, 343 (1980)

    1980

  35. [35]

    E. Lust, G. Nurk, A. J¨ anes, M. Arulepp, P. Nigu, P. M¨ oller, S. Kallip, and V. Sammelselg, J. Solid State Electrochem.7, 91 (2003)

    2003

  36. [36]

    E. Lust, A. J¨ anes, and M. Arulepp, J. Electroanal. Chem. 562, 33 (2004)

    2004

  37. [37]

    E. Lust, A. J¨ anes, T. P¨ arn, and P. Nigu, J. Solid State Electrochem.8, 224 (2004)

    2004

  38. [38]

    Jurczakowski, C

    R. Jurczakowski, C. Hitz, and A. Lasia, J. Electroanal. Chem.572, 355 (2004)

    2004

  39. [39]

    Segalini, B

    J. Segalini, B. Daffos, P. L. Taberna, Y. Gogotsi, and P. Simon, Electrochimica Acta55, 7489 (2010)

    2010

  40. [40]

    Jokar, A

    E. Jokar, A. I. Zad, and S. Shahrokhian, J. Solid State Electrochem.19, 269 (2015)

    2015

  41. [41]

    Liu, Z.-Z

    H. Liu, Z.-Z. Pan, A. Aziz, R. Tang, W. Lv, and H. Nishi- hara, Adv. Funct. Mater.33, 2303730 (2023)

    2023

  42. [42]

    Zeng, B.-A

    Z. Zeng, B.-A. Mei, G. Song, M. Hamza, Z. Yan, Q. Wei, H. Feng, Z. Zuo, B. Jia, and R. Xiong, J. Energy Storage 102, 114021 (2024). 21

    2024

  43. [43]

    Hallemans, D

    N. Hallemans, D. Howey, A. Battistel, N. F. Saniee, F. Scarpioni, B. Wouters, F. La Mantia, A. Hubin, W. D. Widanage, and J. Lataire, Electrochimica Acta 466, 142939 (2023)

    2023

  44. [44]

    Li and J

    C. Li and J. Huang, Advanced Science , e75601 (2026)

    2026

  45. [45]

    Paasch, K

    G. Paasch, K. Micka, and P. Gersdorf, Electrochim. Acta 38, 2653 (1993)

    1993

  46. [46]

    Huang, Electrochim

    J. Huang, Electrochim. Acta281, 170 (2018)

    2018

  47. [47]

    Pedersen, T

    C. Pedersen, T. Aslyamov, and M. Janssen, PRX Energy 2, 043006 (2023)

    2023

  48. [48]

    Lelidis and G

    I. Lelidis and G. Barbero, Phys. Lett. A343, 440 (2005)

    2005

  49. [49]

    Gunning, D

    J. Gunning, D. Y. C. Chan, and L. R. White, J. Colloid Interface Sci.170, 522 (1995)

    1995

  50. [50]

    E. H. B. DeLacey and L. R. White, J. Chem. Soc. Lond. Faraday Trans. 278, 457 (1982)

    1982

  51. [51]

    A. D. Ratschow, A. J. Wagner, M. Janssen, and S. Hardt, Proc. Natl. Acad. Sci.122, e2504322122 (2025)

    2025

  52. [52]

    See the Supplemental Material

  53. [53]

    Borukhov, D

    I. Borukhov, D. Andelman, and H. Orland, Phys. Rev. Lett.79, 435 (1997)

    1997

  54. [54]

    A. A. Kornyshev, The Journal of Physical Chemistry B 111, 5545 (2007)

    2007

  55. [55]

    M. S. Kilic, M. Z. Bazant, and A. Ajdari, Phys. Rev. E 75, 021502 (2007)

    2007

  56. [56]

    J. Song, E. Khoo, and M. Z. Bazant, Phys. Rev. E100, 042204 (2019)

    2019

  57. [57]

    A. A. Pilla, J. Electrochem. Soc.117, 467 (1970)

    1970

  58. [58]

    S. C. Creason, J. W. Hayes, and D. E. Smith, J. Elec- troanal. Chem. Interfacial Electrochem.47, 9 (1973)

    1973

  59. [59]

    G. S. Popkirov and R. N. Schindler, Rev. Sci. Instrum. 63, 5366 (1992)

    1992

  60. [60]

    Barsoukov, S

    E. Barsoukov, S. H. Ryu, and H. Lee, J. Electroanal. Chem.536, 109 (2002)

    2002

  61. [61]

    Sihvo, N

    J. Sihvo, N. Hallemans, A. H. Tan, D. A. Howey, S. R. Duncan, and T. Roinila, IEEE Trans. Transp. Electrif. 10.1109/TTE.2026.3659482 (2026)

    work page doi:10.1109/tte.2026.3659482 2026

  62. [62]

    H. P. Van Leeuwen, D. J. Kooijman, M. Sluyters- Rehbach, and J. H. Sluyters, J. Electroanal. Chem. In- terfacial Electrochem.23, 475 (1969)

    1969

  63. [63]

    Logg, K.-A

    A. Logg, K.-A. Mardal, and G. Wells,Automated solu- tion of differential equations by the finite element method: The FEniCS book, Vol. 84 (Springer Science & Business Media, 2012)

    2012

  64. [64]

    Logg and G

    A. Logg and G. N. Wells, ACM Trans. Math. Softw.37, 10.1145/1731022.1731030 (2010)

    work page doi:10.1145/1731022.1731030 2010

  65. [65]

    M. S. Alnæs, A. Logg, K. B. Ølgaard, M. E. Rognes, and G. N. Wells, ACM Trans. Math. Softw.40, 1 (2014)

    2014

  66. [66]

    R. C. Kirby and A. Logg, ACM Trans. Math. Softw.32, 10.1145/1163641.1163644 (2006)

    work page doi:10.1145/1163641.1163644 2006

  67. [67]

    P. M. Biesheuvel and M. Z. Bazant, Phys. Rev. E81, 031502 (2010)

    2010

  68. [68]

    Jamnik and J

    J. Jamnik and J. Maier, Phys. Chem. Chem. Phys.3, 1668 (2001)

    2001

  69. [69]

    Barbero, Phys

    G. Barbero, Phys. Chem. Chem. Phys.19, 32575 (2017)

    2017

  70. [70]

    T. R. Brumlev and R. P. Buck, J. Electroanal. Chem. Interf. Electrochem.126, 73 (1981)

    1981

  71. [71]

    Jamnik and J

    J. Jamnik and J. Maier, J. Electrochem. Soc.146, 4183 (1999)

    1999

  72. [72]

    Van Soestbergen, P

    M. Van Soestbergen, P. M. Biesheuvel, and M. Z. Bazant, Phys. Rev. E81, 021503 (2010)

    2010

  73. [73]

    Mirzadeh, F

    M. Mirzadeh, F. Gibou, and T. M. Squires, Phys. Rev. Lett.113, 097701 (2014)

    2014

  74. [74]

    Tsch¨ ope, E

    A. Tsch¨ ope, E. Sommer, and R. Birringer, Solid State Ion.139, 255 (2001)

    2001

  75. [75]

    Tsch¨ ope, Solid State Ion.139, 267 (2001)

    A. Tsch¨ ope, Solid State Ion.139, 267 (2001)

    2001

  76. [76]

    M. J. Verkerk, B. J. Middelhuis, and A. J. Burggraaf, Solid State Ion.6, 159 (1982)

    1982

  77. [77]

    Vollman and R

    M. Vollman and R. Waser, J. Am. Ceram. Soc.77, 235 (1994)

    1994

  78. [78]

    X. Guo, W. Sigle, J. Fleig, and J. Maier, Solid State Ion. 154, 555 (2002)

    2002

  79. [79]

    X. Guo, W. Sigle, and J. Maier, J. Am. Ceram. Soc.86, 77 (2003)

    2003

  80. [80]

    Shirpour, R

    M. Shirpour, R. Merkle, and J. Maier, Solid State Ion. 225, 304 (2012)

    2012

Showing first 80 references.