A Beginners’ Guide to Modelling of Electric Double Layer under Equilibrium, Nonequilibrium and AC Conditions

Abstract

In electrochemistry, perhaps also in other time-honored scientific disciplines, knowledge labelled classical usually attracts less attention from beginners, especially those pressured or tempted to quickly jam into research fronts that are labelled, not always aptly, modern. In fact, it is a normal reaction to the burden of history and the stress of today. Against this context, accessible tutorials on classical knowledge are useful, should some realize that taking a step back could be the best way forward. This is the driving force of this article themed at physicochemical modelling of the electric (electrochemical) double layer (EDL). We begin the exposition with a rudimentary introduction to key concepts of the EDL, followed by a brief introduction to its history. We then elucidate how to model the EDL under equilibrium, using firstly the orthodox Gouy-Chapman-Stern model, then the symmetric Bikerman model, and finally the asymmetric Bikerman model. Afterwards, we exemplify how to derive a set of equations governing the EDL dynamics under nonequilibrium conditions using a unifying grand-potential approach. In the end, we expound on the definition and mathematical foundation of electrochemical impedance spectroscopy (EIS), and present a detailed derivation of an EIS model for a simple EDL. We try to avoid the omission of supposedly ‘trivial’ information in the derivation of models, hoping that it can ease the access to the wonderful garden of physical electrochemistry.

Keywords

electric double layer / equilibrium / nonequilibrium / electrochemical impedance spectroscopy

References

[1] Huang J, Chen S L, Eikerling M. Grand-canonical model of electrochemical double layers from a hybrid density-potential functional[J]. J. Chem. Theory Comput., 2021,17(4):2417-2430.

[2] Trasatti S, Lust E. The potential of zero charge, in Modern aspects of electrochemistry[M]. White R E, Bockris J O’ M, Conway B E, Editors, 1999, Springer US: Boston, MA. 1-215.

[3] Schmickler W, Guidelli R. The partial charge transfer[J]. Electrochim. Acta, 2014,127:489-505.

[4] Huang J, Malek A, Zhang J B, Eikerling M. Non-monotonic surface charging behavior of platinum: A paradigm change[J]. J. Phys. Chem. C, 2016,120(25):13587-13595.

[5] Huang J, Zhou T, Zhang J B, Eikerling M. Double layer of platinum electrodes: Non-monotonic surface charging phenomena and negative double layer capacitance[J]. J. Chem. Phys., 2018,148(4):044704.

[6] Helmholtz H V. Studien über electrische grenzschichten[J]. Ann. Phys., 1879,243(7):337-382.

[7] Chapman D L, Li. A contribution to the theory of electrocapillarity[J]. Philos. Mag., 1913,25(148):475-481.

[8] Gouy G. Sur la constitution de la charge electrique a la surface d’un electrolyte.[J]. J. Phys. Theor. Appl., 1910,9(1):457-468.

[9] Stern O. Zur theorie der elektrolytischen doppelschicht[J]. Ber. Bunsenges. Phys. Chem., 1924,30(21-22):508-516.

[10] Bikerman J J. Xxxix. Structure and capacity of electrical double layer[J]. Lond. Edinb. Dubl. Phil. Mag., 1942,33(220):384-397.

[11] Grahame D C. The electrical double layer and the theory of electrocapillarity[J]. Chem. Rev., 1947,41(3):441-501.

[12] Lobenz W, Salie G. Potentialabhängigkeit und ladungsübergangskoeffizienten der tl/tl+-reaktion[J]. Z. Phys. Chem., 1961,29(5):408-412.

[13] Grahame D C. Components of charge and potential in the non-diffuse region of the electrical double layer: Potassium iodide solutions in contact with mercury at 25°¹[J]. J. Am. Chem. Soc., 1958,80(16):4201-4210.

[14] Mott N F, Watts-Tobin R J. The interface between a metal and an electrolyte[J]. Electrochim. Acta, 1961,4(2):79-107.

[15] Mott N F, Parsons R, Watts-Tobin R J. The capacity of a mercury electrode in electrolytic solution[J]. Philos. Mag., 1962,7(75):483-493.

[16] Watts-tobin R J. The interface between a metal and an electrolytic solution[J]. Philos. Mag., 1961,6(61):133-153.

[17] Schmickler W. Electronic effects in the electric double layer[J]. Chem. Rev., 1996,96(8):3177-3200.

[18] Guidelli R, Schmickler W. Recent developments in models for the interface between a metal and an aqueous solution[J]. Electrochim. Acta, 2000. 45(15):2317-2338.

[19] Trasatti S. Effect of the nature of the metal on the dielectric properties of polar liquids at the interface with electrodes. A phenomenological approach[J]. J. Electroanal. Chem. Interfacial Electrochem., 1981,123(1):121-139.

[20] Badiali J P, Rosinberg M L, Vericat F, Blum L. A microscopic model for the liquid metal-ionic solution interface[J]. J. Electro-Anal. Chem., 1983,158(2):253-267.

[21] Kornyshev A A. Metal electrons in the double layer theory[J]. Electrochim. Acta, 1989,34(12):1829-1847.

[22] Schmickler W. A jellium-dipole model for the double layer[J]. J. Electroanal. Chem., 1983,150(1):19-24.

[23] Lundqvist S, March N H. Theory of the inhomogeneous electron gas[M] //Physics of solids and liquids, Springer Science & Business Media. XIV, Plenum Press, New York, 1983: 396.

[24] Price D L, Halley J W. Molecular dynamics, density functional theory of the metal-electrolyte interface[J]. J. Chem. Phys., 1995,102(16):6603-6612.

[25] Otani M, ugino O. First-principles calculations of charged surfaces and interfaces: A plane-wave nonrepeated slab approach[J]. Phys. Rev. B, 2006,73(11):115407.

[26] Petrosyan S A, Briere J F, Roundy D, Arias T A. Joint density-functional theory for electronic structure of solvated systems[J]. Phys. Rev. B, 2007,75(20):205105.

[27] Letchworth-Weaver K, Arias T A. Joint density functional theory of the electrode-electrolyte interface: Application to fixed electrode potentials, interfacial capacitances, and potentials of zero charge[J]. Phys. Rev. B, 2012,86(7):075140.

[28] Sundararaman R, Goddard W A, Arias T A. Grand canonical electronic density-functional theory: Algorithms and applications to electrochemistry[J]. J. Chem. Phys., 2017,146(11):114104.

[29] Jinnouchi R, Anderson A B. Electronic structure calculations of liquid-solid interfaces: Combination of density functional theory and modified Poisson-Boltzmann theory[J]. Phys. Rev. B, 2008,77(24):245417.

[30] Mathew K, Sundararaman R, Letchworth-Weaver K, Arias T A, Hennig R G. Implicit solvation model for densityfunctional study of nanocrystal surfaces and reaction pathways[J]. J. Chem. Phys., 2014,140(8):084106.

[31] Mathew K, Kolluru V S C, Mula S, Steinmann S N, Hennig R G. Implicit self-consistent electrolyte model in plane-wave density-functional theory[J]. J. Chem. Phys., 2019,151(23):234101.

[32] Nishihara S, Otani M. Hybrid solvation models for bulk, interface, and membrane: Reference interaction site methods coupled with density functional theory[J]. Phys. Rev. B, 2017,96(11):115429.

[33] Nattino F, Truscott M, Marzari N, Andreussi O. Continuum models of the electrochemical diffuse layer in electronicstructure calculations[J]. J. Chem. Phys., 2018,150(4):041722.

[34] Hörmann N G, Andreussi O, Marzari N. Grand canonical simulations of electrochemical interfaces in implicit solvation models[J]. J. Chem. Phys., 2019,150(4):041730.

[35] Melander M M, Kuisma M J, Christensen T E K, Honkala K. Grand-canonical approach to density functional theory of electrocatalytic systems: Thermodynamics of solid-liquid interfaces at constant ion and electrode potentials[J]. J. Chem. Phys., 2018,150(4):041706.

[36] Le J B, Iannuzzi M, Cuesta A, Cheng J. Determining potentials of zero charge of metal electrodes versus the standard hydrogen electrode from density-functional-theory-based molecular dynamics[J]. Phys. Rev. Lett., 2017,119(1):016801.

[37] Sakong S, Groβ A. The electric double layer at metalwater interfaces revisited based on a charge polarization scheme[J]. J. Chem. Phys., 2018,149(8):084705.

[38] Sakong S, Groβ A. Water structures on a Pt(111) electrode from ab initio molecular dynamic simulations for a variety of electro-chemical conditions[J]. Phys. Chem. Chem. Phys., 2020,22(19):10431-10437.

[39] Le J B, Chen A, Li L, Xiong J F, Lan J G, Liu Y P, Iannuzzi M, Cheng J. Modeling electrified Pt(111)-Had/water interfaces from ab initio molecular dynamics[J]. JACS Au, 2021,1(5):569-577.

[40] Huang J, Li P, Chen S L. Potential of zero charge and surface charging relation of metal-solution interphases from a constant-poten-tial Jellium-Poisson-Boltzmann model[J]. Phys. Rev. B, 2020,101(12):125422.

[41] Huang J. Hybrid density-potential functional theory of electric double layers[J]. Electrochim. Acta, 2021,389:138720.

[42] Karasiev V V, Trickey S B. Chapter nine-frank discussion of the status of ground-state orbital-free DFT[M] //Advances in quantum chemistry. Sabin J R, Cabrera-Trujillo R, Editors, Academic Press, 2015: 221-245.

[43] Wang Y A, Carter E A. Orbital-free kinetic-energy density functional theory, in Theoretical methods in condensed phase chemistry[M]. Schwartz S D, Editor, Springer Netherlands: Dordrecht, 2002: 117-184.

[44] Gavini V, Bhattacharya K, Ortiz M. Quasi-continuum orbital-free density-functional theory: A route to multi-million atom non-periodic DFT calculation[J]. J. Mech. Phys. Solids, 2007,55(4):697-718.

[45] Nightingale E R. Phenomenological theory of ion solvation. Effective radii of hydrated ions[J]. J. Phys. Chem., 1959,63(9):1381-1387.

[46] 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[J]. Adv. Colloid Interface, 2009,152(1-2):48-88.

[47] Kornyshev A A. Double-layer in ionic liquids: Paradigm change?[J]. J. Phys. Chem. B, 2007,111(20):5545-5557.

[48] Zhang Y F, Huang J. Treatment of ion-size asymmetry in lattice-gas models for electrical double layer[J]. J. Phys. Chem. C, 2018,122(50):28652-28664.

[49] Zhang L L, Cai J, Chen Y X, Huang J. Modelling electrocatalytic reactions with a concerted treatment of multistep electron transfer kinetics and local reaction conditions[J]. J. Phys. Condens. Mat., 2021,33(50):504002.

[50] Zhang Z M, Gao Y, Chen S L, Huang J. Understanding dynamics of electrochemical double layers via a modified concentrated solution theory[J]. J. Electrochem. Soc., 2019. 167(1):013519.

[51] Budkov Y A. Nonlocal statistical field theory of dipolar particles in electrolyte solutions[J]. J. Phys. Condens. Mat., 2018,30(34):344001.

[52] Budkov Y A. Statistical field theory of ion-molecular solutions[J]. Phys. Chem. Chem. Phys., 2020,22(26):14756-14772.

[53] Huang J. Confinement induced dilution: Electrostatic screening length anomaly in concentrated electrolytes in confined space[J]. J. Phys. Chem. C, 2018,122(6):3428-3433.

[54] Gao Y, Huang J, Liu Y W, Yan J W, Mao B W, Chen S L. Ion-vacancy coupled charge transfer model for ion transport in con-centrated solutions[J]. Sci. China. Chem., 2019,62(4):515-520.

[55] Bard A J, Faulkner L R. Electrochemical methods: Fundamentals and applications[M]. 2nd ed. Russ. J. Electrochem., New York: Wiley, 2002,38:1364-1365.

[56] Bockris J O’ m, Devanathan M A V, Müller K. On the structure of charged interfaces[J]. Roy. Soc. A, 1963,274(541):55-79.

[57] Huang J, Li Z, Liaw B Y, Zhang J B. Graphical analysis of electrochemical impedance spectroscopy data in bode and Nyquist representations[J]. J. Power Sources, 2016,309:82-98.

[58] Li C K, Huang J. Impedance response of electrochemical interfaces: Part i. Exact analytical expressions for ideally polarizable electrodes[J]. J. Electrochem. Soc., 2021,167(16):166517.

[59] Klotz D, Schönleber M, Schmidt J, Ivers-Tiffée E. New approach for the calculation of impedance spectra out of time domain data[J]. Electrochim. Acta, 2011,56(24):8763-8769.

[60] Nussbaumer H J. The fast fourier transform, in fast fourier transform and convolution algorithms[M]. Nussbaumer H J, Editor, Springer Berlin Heidelberg: Berlin, Heidelberg, 1981: 80-111.