J. Moškon and M. Gaberšček, Transmission line models for evaluation of impedance response of insertion battery electrodes and cells, J. Power Sources Adv. 7, 100047 (2021). 2666-2485 10.1016/j.powera.2021.100047
J. Nielsen and J. Hjelm, Impedance of SOFC electrodes: A review and a comprehensive case study on the impedance of LSM:YSZ cathodes, Electrochim. Acta 115, 31 (2014). 0013-4686 10.1016/j.electacta.2013.10.053
B. J. Privett, J. H. Shin, and M. H. Schoenfisch, Electrochemical sensors, Anal. Chem. 82, 4723 (2010). 0003-2700 10.1021/ac101075n
R. Kötz and M. Carlen, Principles and applications of electrochemical capacitors, Electrochim. Acta 45, 2483 (2000). 0013-4686 10.1016/S0013-4686(00)00354-6
J. Wu, Understanding the electric double-layer structure, capacitance, and charging dynamics, Chem. Rev. 122, 10821 (2022). 0009-2665 10.1021/acs.chemrev.2c00097
S. Porada, R. Zhao, A. van Der Wal, V. Presser, and P. M. Biesheuvel, Review on the science and technology of water desalination by capacitive deionization, Prog. Mater. Sci. 58, 1388 (2013). 0079-6425 10.1016/j.pmatsci.2013.03.005
A. D. Ratschow, D. Pandey, B. Liebchen, S. Bhattacharyya, and S. Hardt, Resonant nanopumps: ac gate voltages in conical nanopores induce directed electrolyte flow, Phys. Rev. Lett. 129, 264501 (2022). 0031-9007 10.1103/PhysRevLett.129.264501
M. S. Kilic, M. Z. Bazant, and A. Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson-Nernst-Planck equations, Phys. Rev. E 75, 021503 (2007). 1539-3755 10.1103/PhysRevE.75.021503
T. Aslyamov, K. Sinkov, and I. Akhatov, Relation between charging times and storage properties of nanoporous supercapacitors, Nanomaterials 12, 587 (2022). 2079-4991 10.3390/nano12040587
R. J. Tomlin, T. Roy, T. L. Kirk, M. Marinescu, and D. Gillespie, Impedance response of ionic liquids in long slit pores, J. Electrochem. Soc. 169, 120513 (2022). 0013-4651 10.1149/1945-7111/ac89b5
A. A. Lee, S. Kondrat, D. Vella, and A. Goriely, Dynamics of ion transport in ionic liquids, Phys. Rev. Lett. 115, 106101 (2015). 0031-9007 10.1103/PhysRevLett.115.106101
P. Malgaretti, M. Janssen, I. Pagonabarraga, and J. M. Rubi, Driving an electrolyte through a corrugated nanopore, J. Chem. Phys. 151, 084902 (2019). 0021-9606 10.1063/1.5110349
H. Sakaguchi and R. Baba, Charging dynamics of the electric double layer in porous media, Phys. Rev. E 76, 011501 (2007). 1539-3755 10.1103/PhysRevE.76.011501
J. Lim, J. D. Whitcomb, J. G. Boyd, and J. Varghese, Effect of electrode pore geometry modeled using Nernst-Planck-Poisson-modified Stern layer model, Comput. Mech. 43, 461 (2009). 0178-7675 10.1007/s00466-008-0322-y
M. Mirzadeh, F. Gibou, and T. M. Squires, Enhanced charging kinetics of porous electrodes: Surface conduction as a short-circuit mechanism, Phys. Rev. Lett. 113, 097701 (2014). 0031-9007 10.1103/PhysRevLett.113.097701
F. Henrique, P. J. Zuk, and A. Gupta, Charging dynamics of electrical double layers inside a cylindrical pore: Predicting the effects of arbitrary pore size, Soft Matter 18, 198 (2021). 1744-683X 10.1039/D1SM01239H
F. Henrique, P. J. Zuk, and A. Gupta, Impact of asymmetries in valences and diffusivities on the transport of a binary electrolyte in a charged cylindrical pore, Electrochim. Acta 433, 141220 (2022). 0013-4686 10.1016/j.electacta.2022.141220
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). 0021-9606 10.1063/5.0094553
S. Alizadeh and A. Mani, Multiscale model for electrokinetic transport in networks of pores, part I: Model derivation, Langmuir 33, 6205 (2017). 0743-7463 10.1021/acs.langmuir.6b03816
T. Aslyamov and M. Janssen, Analytical solution to the Poisson-Nernst-Planck equations for the charging of a long electrolyte-filled slit pore, Electrochim. Acta 424, 140555 (2022). 0013-4686 10.1016/j.electacta.2022.140555
V. S. Daniel-Bekh, Polarisation of porous electrodes. Part I: Current and potential distribution within the electrode, Zh. Fiz. Khim. SSR 22, 697 (1948).
O. S. Ksenzhek and V. V. Stender, Determination of specific surface of porous electrodes by capacitance measurement methods, Dokl. Akad. Nauk SSSR 106, 487 (1956).
R. de Levie, On porous electrodes in electrolyte solutions: I. Capacitance effects, Electrochim. Acta 8, 751 (1963). 0013-4686 10.1016/0013-4686(63)80042-0
R. de Levie, in Advances in Electrochemistry and Electrochemical Engineering, Vol. 6 (Wiley-Interscience, New York, 1967), p. 329.
M. E. Orazem and B. Tribollet, Electrochemical Impedance Spectroscopy (John Wiley & Sons, Inc., Hoboken, NJ, 2017), p. 383.
A. Lasia, Electrochemical Impedance Spectroscopy and Its Applications (Springer Science+Business Media, New York, 2014), chapter 8 and 9.
L. Gassa, J. Vilche, M. Ebert, K. Jüttner, and W. Lorenz, Electrochemical impedance spectroscopy on porous electrodes, J. Appl. Electrochem. 20, 677 (1990). 0021-891X 10.1007/BF01008882
R. Jurczakowski, C. Hitz, and A. Lasia, Impedance of porous (Equation presented) based electrodes, J. Electroanal. Chem. 572, 355 (2004). 1572-6657 10.1016/j.jelechem.2004.01.008
N. Ogihara, S. Kawauchi, C. Okuda, Y. Itou, Y. Takeuchi, and Y. Ukyo, Theoretical and experimental analysis of porous electrodes for lithium-ion batteries by electrochemical impedance spectroscopy using a symmetric cell, J. Electrochem. Soc. 159, A1034 (2012). 0013-4651 10.1149/2.057207jes
N. Ogihara, Y. Itou, T. Sasaki, and Y. Takeuchi, Impedance spectroscopy characterization of porous electrodes under different electrode thickness using a symmetric cell for high-performance lithium-ion batteries, J. Phys. Chem. C 119, 4612 (2015). 1932-7447 10.1021/jp512564f
B. E. Conway, Electrochemical Supercapacitors: Scientific Fundamentals and Technological Applications (Kluwer Academic/Plenum Publishers, New York, 1999).
J. Huang, Y. Gao, J. Luo, S. Wang, C. Li, S. Chen, and J. Zhang, Editors' choice-review-impedance response of porous electrodes: Theoretical framework, physical models and applications, J. Electrochem. Soc. 167, 166503 (2020). 0013-4651 10.1149/1945-7111/abc655
F. Posey and T. Morozumi, Theory of potentiostatic and galvanostatic charging of the double layer in porous electrodes, J. Electrochem. Soc. 113, 176 (1966). 0013-4651 10.1149/1.2423897
M. Janssen and J. Bisquert, Locating the frequency of turnover in thin-film diffusion impedance, J. Phys. Chem. C 125, 15737 (2021). 1932-7447 10.1021/acs.jpcc.1c04572
O. Barcia, E. D'Elia, I. Frateur, O. Mattos, N. Pébère, and B. Tribollet, Application of the impedance model of de Levie for the characterization of porous electrodes, Electrochim. Acta 47, 2109 (2002). 0013-4686 10.1016/S0013-4686(02)00081-6
D. Cericola and M. E. Spahr, Impedance spectroscopic studies of the porous structure of electrodes containing graphite materials with different particle size and shape, Electrochim. Acta 191, 558 (2016). 0013-4686 10.1016/j.electacta.2016.01.121
H. Keiser, K. Beccu, and M. Gutjahr, Abschätzung der porenstruktur poröser elektroden aus impedanzmessungen, Electrochim. Acta 21, 539 (1976). 0013-4686 10.1016/0013-4686(76)85147-X
K. Eloot, F. Debuyck, M. Moors, and A. Van Peteghem, Calculation of the impedance of noncylindrical pores Part I: Introduction of a matrix calculation method, J. Appl. Electrochem. 25, 326 (1995). 0021-891X 10.1007/BF00249650
H.-K. Song, Y.-H. Jung, K.-H. Lee, and L. H. Dao, Electrochemical impedance spectroscopy of porous electrodes: The effect of pore size distribution, Electrochim. Acta 44, 3513 (1999). 0013-4686 10.1016/S0013-4686(99)00121-8
G. Paasch, K. Micka, and P. Gersdorf, Theory of the electrochemical impedance of macrohomogeneous porous electrodes, Electrochim. Acta 38, 2653 (1993). 0013-4686 10.1016/0013-4686(93)85083-B
R. de Levie, The influence of surface roughness of solid electrodes on electrochemical measurements, Electrochim. Acta 10, 113 (1965). 0013-4686 10.1016/0013-4686(65)87012-8
M. Eikerling, A. Kornyshev, and E. Lust, Optimized structure of nanoporous carbon-based double-layer capacitors, J. Electrochem. Soc. 152, E24 (2005). 0013-4651 10.1149/1.1825379
M. Itagaki, Y. Hatada, I. Shitanda, and K. Watanabe, Complex impedance spectra of porous electrode with fractal structure, Electrochim. Acta 55, 6255 (2010). 0013-4686 10.1016/j.electacta.2009.10.016
C. J. Gommes and F. Chaltin, The electrical impedance of carbon xerogel hierarchical electrodes, Electrochim. Acta 433, 141203 (2022). 0013-4686 10.1016/j.electacta.2022.141203
H.-Q. Li, J.-Y. Luo, X.-F. Zhou, C.-Z. Yu, and Y.-Y. Xia, An ordered mesoporous carbon with short pore length and its electrochemical performances in supercapacitor applications, J. Electrochem. Soc. 154, A731 (2007). 0013-4651 10.1149/1.2741198
E. Lust, A. Jänes, and M. Arulepp, Influence of solvent nature on the electrochemical parameters of electrical double layer capacitors, J. Electroanal. Chem. 562, 33 (2004). 1572-6657 10.1016/j.jelechem.2003.07.034
E. Lust, A. Jänes, T. Pärn, and P. Nigu, Influence of nanoporous carbon electrode thickness on the electrochemical characteristics of a nanoporous carbon-tetraethylammonium tetrafluoroborate in acetonitrile solution interface, J. Solid State Electrochem. 8, 224 (2004). 1432-8488 10.1007/s10008-003-0396-6
M. D. Murbach, B. Gerwe, N. Dawson-Elli, and L. kun Tsui, impedance.py: A python package for electrochemical impedance analysis, J. Open Source Softw. 5, 2349 (2020). 2475-9066 10.21105/joss.02349
If an electrode is indeed a bundle of (Equation presented) parallel pores of known surface area (Equation presented) per pore, one can estimate the number of pores (Equation presented) using typical values for the specific capacitance (Equation presented).
B.-A. Mei, O. Munteshari, J. Lau, B. Dunn, and L. Pilon, Physical interpretations of nyquist plots for EDLC electrodes and devices, J. Phys. Chem. C 122, 194 (2018). 1932-7447 10.1021/acs.jpcc.7b10582
S. Babel, M. Eikerling, and H. Löwen, Impedance resonance in narrow confinement, J. Phys. Chem. C 122, 21724 (2018). 1932-7447 10.1021/acs.jpcc.8b05559
G. Pireddu and B. Rotenberg, Frequency-dependent impedance of nanocapacitors from electrode charge fluctuations as a probe of electrolyte dynamics, Phys. Rev. Lett. 130, 098001 (2023). 0031-9007 10.1103/PhysRevLett.130.098001
Reference. [16] analytically solved the PNP equations for the charging of a cylindrical pore. For thin EDLs, their solution also simplifies to TL model results.
Reference. [74] is a notable exception.
M. Janssen, Transmission line circuit and equation for an electrolyte-filled pore of finite length, Phys. Rev. Lett. 126, 136002 (2021). 0031-9007 10.1103/PhysRevLett.126.136002
J. Huang, Diffusion impedance of electroactive materials, electrolytic solutions and porous electrodes: Warburg impedance and beyond, Electrochim. Acta 281, 170 (2018). 0013-4686 10.1016/j.electacta.2018.05.136
V. F. Lvovich, Impedance Spectroscopy: Applications to Electrochemical and Dielectric Phenomena (John Wiley & Sons, Inc., Hoboken, New Jersey, 2012).
M. Itagaki, S. Suzuki, and I. Shitanda, Impedance analysis on electric double layer capacitor with transmission line model, J. Power Sources 164, 415 (2007). 0378-7753 10.1016/j.jpowsour.2006.09.077
M. Keddam, C. Rakotomavo, and H. Takenouti, Impedance of a porous electrode with an axial gradient of concentration, J. Appl. Electrochem. 14, 437 (1984). 0021-891X 10.1007/BF00610808
A. Lasia, Impedance of porous electrodes, J. Electroanal. Chem. 397, 27 (1995). 1572-6657 10.1016/0022-0728(95)04177-5
J. Bisquert, Theory of the impedance of electron diffusion and recombination in a thin layer, J. Phys. Chem. B 106, 325 (2002). 1520-6106 10.1021/jp011941g
R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. II: Mainly Electromagnetism and Matter (Basic Books, New York, 2011), Vol. 2.
G. Barbero and I. Lelidis, Analysis of Warburg's impedance and its equivalent electric circuits, Phys. Chem. Chem. Phys. 19, 24934 (2017). 1463-9076 10.1039/C7CP04032F
Ref.. [63] seems to have a minus sign error in their corresponding Eq. (45).
G. Strang and S. MacNamara, Functions of difference matrices are Toeplitz plus Hankel, SIAM Rev. 56, 525 (2014). 0036-1445 10.1137/120897572
A. A. Pilla, A transient impedance technique for the study of electrode kinetics, J. Electrochem. Soc. 117, 467 (1970). 0013-4651 10.1149/1.2407544
J.-S. Yoo and S.-M. Park, An electrochemical impedance measurement technique employing Fourier transform, Anal. Chem. 72, 2035 (2000). 0003-2700 10.1021/ac9907540
C. Montella, Voigt circuit representation model for electrochemical impedances under finite-length diffusion conditions, J. Electroanal. Chem. 879, 114785 (2020). 1572-6657 10.1016/j.jelechem.2020.114785
A. Logg, K.-A. Mardal, and G. Wells, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book (Springer-Verlag, Berlin Heidelberg, 2012), Vol. 84.
C. Geuzaine and J.-F. Remacle, Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities, Int. J. Numer. Methods Eng. 79, 1309 (2009). 0029-5981 10.1002/nme.2579
C. Pedersen,. A Finite Element solver for the axisymmetric Poisson-Nernst-Planck equation in cylindrical pores, https://github.com/christian-pedersen/FEM-Axisymmetric-Poisson-Nernst-Planck (2023).
J. Newman, Resistance for flow of current to a disk, J. Electrochem. Soc. 113, 501 (1966). 0013-4651 10.1149/1.2424003
J. E. Hall, Access resistance of a small circular pore, J. Gen. Physiol. 66, 531 (1975). 0022-1295 10.1085/jgp.66.4.531
K. Eloot, F. Debuyck, M. Moors, and A. Van Peteghem, Calculation of the impedance of noncylindrical pores Part II: Experimental verification on pores drilled into stainless steel, J. Appl. Electrochem. 25, 334 (1995). 0021-891X 10.1007/BF00249651
Even though it looks similar to Fig.. 12, note that Fig. (3) of Ref.. [16] is not to scale and corresponds to aspect ratios between (Equation presented) and 50 (private communication with F. Henrique).
S. J. Cooper, A. Bertei, D. P. Finegan, and N. P. Brandon, Simulated impedance of diffusion in porous media, Electrochim. Acta 251, 681 (2017). 0013-4686 10.1016/j.electacta.2017.07.152
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). 0031-9007 10.1103/PhysRevLett.124.076001
Y. Lin, C. Lian, M. U. Berrueta, H. Liu, and R. van Roij, Microscopic model for cyclic voltammetry of porous electrodes, Phys. Rev. Lett. 128, 206001 (2022). 0031-9007 10.1103/PhysRevLett.128.206001
L. Ji, Z. Xu, and S. Zhou, Asymptotic analysis on charging dynamics for stack-electrode model of supercapacitors, Proc. R. Soc. A: Math. Phys. Eng. Sci. 479, 20230044 (2023). 1364-5021 10.1098/rspa.2023.0044
V. Vivier and M. E. Orazem, Impedance analysis of electrochemical systems, Chem. Rev. 122, 11131 (2022). 0009-2665 10.1021/acs.chemrev.1c00876
J. Gunning, The exact impedance of the de Levie grooved electrode, J. Electroanal. Chem. 392, 1 (1995). 1572-6657 10.1016/0022-0728(95)03951-C
R. de Levie, Fractals and rough electrodes, J. Electroanal. Chem. Interfacial Electrochem. 281, 1 (1990). 0022-0728 10.1016/0022-0728(90)87025-F
T. Aslyamov, Properties of electrolyte near rough electrodes: Capacity and impedance, Curr. Opin. Electrochem. 35, 101104 (2022). 2451-9103 10.1016/j.coelec.2022.101104
T. Aslyamov, K. Sinkov, and I. Akhatov, Electrolyte structure near electrodes with molecular-size roughness, Phys. Rev. E 103, L060102 (2021). 2470-0045 10.1103/PhysRevE.103.L060102
J. Seebeck, C. Merlet, and R. H. Meißner, Elucidating curvature-capacitance relationships in carbon-based supercapacitors, Phys. Rev. Lett. 128, 086001 (2022). 0031-9007 10.1103/PhysRevLett.128.086001
P. M. Biesheuvel, Y. Fu, and M. Z. Bazant, Diffuse charge and Faradaic reactions in porous electrodes, Phys. Rev. E 83, 061507 (2011). 1539-3755 10.1103/PhysRevE.83.061507
C. K. Li, J. Zhang, and J. Huang, Impedance response of electrochemical interfaces. III. Fingerprints of couplings between interfacial electron transfer reaction and electrolyte-phase ion transport, J. Chem. Phys. 157, 184704 (2022). 0021-9606 10.1063/5.0119592
S. Kondrat, P. Wu, R. Qiao, and A. A. Kornyshev, Accelerating charging dynamics in subnanometre pores, Nat. Mater. 13, 387 (2014). 1476-1122 10.1038/nmat3916
T. L. Kirk, A. Lewis-Douglas, D. Howey, C. P. Please, and S. J. Chapman, Nonlinear electrochemical impedance spectroscopy for lithium-ion battery model parameterization, J. Electrochem. Soc. 170, 010514 (2023). 0013-4651 10.1149/1945-7111/acada7
N. Hallemans, D. Howey, A. Battistel, N. F. Saniee, F. Scarpioni, B. Wouters, F. La Mantia, A. Hubin, W. D. Widanage, and J. Lataire, Electrochemical impedance spectroscopy beyond linearity and stationarity-a critical review, ArXiv:2304.08126 (2023).
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). 1936-0851 10.1021/nn4058243
S. Bi, H. Banda, M. Chen, L. Niu, M. Chen, T. Wu, J. Wang, R. Wang, J. Feng, T. Chen, et al., Molecular understanding of charge storage and charging dynamics in supercapacitors with MOF electrodes and ionic liquid electrolytes, Nat. Mater. 19, 552 (2020). 1476-1122 10.1038/s41563-019-0598-7
G. Jeanmairet, B. Rotenberg, and M. Salanne, Microscopic simulations of electrochemical double-layer capacitors, Chem. Rev. 122, 10860 (2022). 0009-2665 10.1021/acs.chemrev.1c00925
