M

Mean-Field Theoryof the Electrical Double Layerin Ionic Liquids

Zachary A. H. Goodwin1, J. Pedro de Souza2,Martin Z. Bazant2,3 and Alexei A. Kornyshev1,41Department of Chemistry, Imperial College ofLondon, Molecular Science Research Hub,London, UK2Department of Chemical Engineering,Massachusetts Institute of Technology,Cambridge, MA, USA3Department of Mathematics, MassachusettsInstitute of Technology, Cambridge, MA, USA4Institute of Molecular Science and Engineering,Imperial College of London, London, UK

Introduction

Ionic liquids (ILs) are superconcentrated electro-lytes of great interest in energy storage devices.The lack of electrolysable solvents, since ILs aresolely composed of cations and anions, means thevoltages that ILs can withstand (�3 V) areroughly twice that of conventional aqueous elec-trolytes [1]. As the energy stored increases withthe voltage, ILs are promising candidates as elec-trolytes for electrical double layersupercapacitors.

In such a capacitor, electrodes are in contactwith an electrolyte. The ions in the electrolyteredistribute such that they reside in energetically

favorable electrostatic environments, forming alayer rich in countercharges that screens the elec-trostatic fields arising from the charged electrodes.This structure is commonly referred to as theelectrical double layer (EDL). The work doneseparating charges in the EDL is stored as energyin EDL capacitors.

The intensive study of EDL capacitors with ILshas burgeoned theories for the EDL. The theoryone develops for the EDL sensitively depends onthe geometry of the interface: In highly porouselectrodes with nanoconfined, overlapping EDLs,the structure differs from isolated planar elec-trodes. Here we focus on the equilibrium EDLstructure in ILs at planar interfaces (note we donot review specific surface interactions with theions but only focus on the space-charge of the IL).Importantly, the equilibrium theory serves as astarting point for describing ionic transport out-of-equilibrium [2, 3]. The case of nano-confinement exhibits very different physics,which depends both on the charge screening ofthe electrode material as well as the electrolyte.For a review of possible and observed effects innanoconfinement, see Ref. [1].

In pure ILs, the concentration of the electrolyteis approximately determined by the size of thecomposing ions. As the cations are typicallylarge organic molecules and the anions are oftenbulky inorganic molecules, the concentration ofions is in the range of 1–10M [1, 4]. Since there isno solvent present in pure ILs, there are strongelectrostatic and steric interactions between the

© Springer Nature Singapore Pte Ltd. 2021S. Zhang (ed.), Encyclopedia of Ionic Liquids,https://doi.org/10.1007/978-981-10-6739-6_62-1

ions. Furthermore, the structures and chemistriesof the ions can be vastly different, which meansthat despite two ILs having a similar concentra-tion, their properties can be quite distinct. There-fore, one requires a general theory of the EDL inILs which not only includes strong electrostaticand steric interactions, but also retains at leastsome elements of chemical specificity.

These three requirements clearly manifestthemselves in the EDL of ILs, where there aremainly two regimes of behavior. At small voltages,there is the overscreening regime [1, 5, 6]. This iswhere the charge density exhibits decaying oscil-lations from the interface, with a period of the orderof an ion diameter, that propagates many ion diam-eters into the liquid [1, 5, 6]. The overscreeningstructure is a direct consequence of electrostaticinteractions between densely packed ions [7]. Atlarger voltages, the overscreening structure givesway to a crowding regime, where layers of coun-terions line up before the overscreening structureappears at distances further from the electrode,where the majority of the electrode charge isalready screened [1, 5–7]. The crowding regimeis dominated by the steric interactions [7–9].Depending on the chemistry of the ions, the elec-trochemical signatures of overscreening and over-crowding are quite sensitive [1]. For example,when the cation has a long alkyl chain, the over-screening regime is further complicated by theorientation of the cations: The charged part of thecation prefers to be closer to the interface, causinga peak in the capacitance at moderate voltagesfrom the compression (“electrostriction”) of theEDL [10].

These features (electrostatic, steric, and spe-cific interactions) can be captured by simplemean-field theories of the EDL with varyinglevels of complexity. We first outline the generalformalism used for such theories of the EDL. Weshall summarize theories of the EDL of neat ILswithin the local density approximation, and thenmove onto more accurate theories. We also brieflyreview some advancements of solvent impuritiesand mixtures of ILs. Finally, we summarize theunderscreening paradox, and other directions ofresearch.

General Formalism

Here theories based on classical density functionaltheory (cDFT) for the EDL in ILs near planarinterfaces are reviwed. The free energy, F , ofILs in proximity to electrified interfaces can beseparated into two components

F ¼ F el þ F chem, ð1Þ

(i) the electrostatic contribution, F el, whichdescribes the energetics of the electrostatic fieldsand the electrostatic interactions between the ions,and also (ii) the chemical contribution, F chem,which accounts for energy and entropic contribu-tions from excluded volume effects and specificinteractions. These contributions can be furthersubdivided into ideal contributions – thosewhich apply for dilute solutions – and excesscontributions – which are required for concen-trated solutions.

In a mean-field theory, assuming that the effec-tive dielectric response (in the way that it countsfor degrees of freedom not explicitly accountedfor in the theory) of the IL is a constant, the idealelectrostatic contribution, F id

el , to the free energyis given by

F idel f,r½ � ¼

ðdr � e

2∇f rð Þ½ �2 þ r rð Þf rð Þ

n o,

ð2Þ

where ∇ is the differential operator and r and fare the mean-field charge density and electrostaticpotential of the IL, respectively. Within this mean-field approximation, the electrostatic potential is

f rð Þ ¼ Ðdr0 r r0ð Þ

4pejr�r0j. Here the dielectric constant εof the IL includes the contributions to systempolarizability not explicitly accounted for in therest of the theory. There can be internal contribu-tions from the electronic polarizability of ions anddipolar contributions, if present. In addition, therecan be contributions from the effective dipolarmoments of ionic pairs and clusters. These allcontribute to the, so-called, high frequency dielec-tric constant that is measured to be 10–20 [4]. Inthe EDL, this value can actually change due to

2 Mean-Field Theory of the Electrical Double Layer in Ionic Liquids

effects that we will discuss later. For now, let usconsider ε to be constant.

Taking the functional derivative of the freeenergy with respect to the electrostatic potential,δF /δf ¼ 0 yields

e∇2f rð Þ ¼ �r rð Þ: ð3Þ

This is the Poisson equation, with furtherknowledge of the charge density being required.The charge density is r(r) ¼ q[c+(r) � c�(r)],where q is the charge of an ion (note that whenone is considering systems more complex thanneat ILs, the expression for the charge densitymust be generalized), and c+(r) and c�(r) are theconcentrations of cations and anions, respectively.There can be additional electrostatic contributionsto the free energy, F ex

el , which either includeelectrostatic correlations or dielectric responses,that modify the Poisson equation. These excesselectrostatic contributions shall be discussed later.

The ideal contribution to the chemical freeenergy, accounts for the entropy of ions that canmove without any restrictions. Here kB isBoltzmann’s constant, T is temperature, and Λ isthe thermal de Broglie wavelength. This idealterm is the basis of the classical Gouy-Chapmantheory of the EDL of dilute electrolytes [1, 2].

F idchem½cþ, c�� ¼kBT

Xi¼�

ðdrfciðrÞ½lnfL3ciðrÞg � 1�g, ð4Þ

The (electro)chemical potential of cations(anions), m+ (m�), is determined from minimizingthe free energywith respect to the concentration ofcations (anions). At equilibrium, the chemicalpotential of the cations (anions) in the bulk mustequal the electrochemical potential of the cations(anions) in the EDL. Therefore, the equations forthe concentration of ions are given by

c� ¼ c exp b �qfþ Dmex�� �� �

, ð5Þ

where β ¼ 1/kBT, c is the bulk concentration ofsalt, and Dmex� is the change in excess chemicalpotential (between the excess chemical potential

in the bulk and the excess electrochemical poten-tial in the EDL), which can be both chemical andelectrostatic in origin. Inserting these equationsfor concentration into the charge density yields ageneral Poisson-Boltzmann (PB) equation

e∇2f ¼ �qc exp b �qfþ Dmexþ� �� ��

� exp b þqfþ Dmex�� �� �Þ: ð6Þ

It is by specifying Dmex� and solving such mod-ified PB equations that we can describe the EDLof ILs. By linearizing this PB equation in theelectrostatic potential for the case Dmexþ ¼ Dmex� ,we obtain the Helmholtz equation

∇2f ¼ k2f, ð7Þ

where k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2q2c=kBTe

pis the inverse Debye

length, which is a key length scale that describesthe ability of a dilute electrolyte to screen charge.

There are two physical observables of the EDLthat we shall discuss: ionic density profiles anddifferential capacitance. The signatures of ionicdensity profiles are accessible through surfaceforce measurements, but also molecular simula-tions can provide a great wealth of information onthe interfacial arrangement of ions [1]. The differ-ential capacitance is readily obtained from imped-ance experiments or from molecular simulations[1]. The differential capacitance describes the dif-ferential charge accumulated at the interface perchange in electrode potential, as defined by

C ¼ dsdf0

, ð8Þ

where s ¼ �n � ε ∇ f|0 is the surface chargedensity, with n denoting the vector normal tothe planar interface, j0 signifying evaluation atthe interface, and f0 is the voltage drop acrossthe entire EDL.

Mean-Field Theory of the Electrical Double Layer in Ionic Liquids 3

Neat Ionic Liquids

The “local density approximation” shall bereviewed first, where excluded volume effects,specific interactions, and electrostatics are onlydescribed with local, mean-field variables, afterwhich the developments beyond the simplest the-ories of the EDL in ILs shall be discussed.

Local Density ApproximationTo have qualitative agreement between theory andexperiments/simulations for the differentialcapacitance, one must include excluded volumeeffects [1]. Before ILs, there were pioneering the-ories for excluded volume effects of ions in theEDL [2]. These approaches came ahead of theirtime, having not attracted the attention theydeserved by those who worked with diluted elec-trolytic solutions. These approaches were inde-pendently “rediscovered” [1, 11], triggered bythe study of the importance of these effects in ILs.

One of the first and simplest theories forexcluded volume effects in ILs was an ideallattice-gas model [8]. The multiplicity of ionsdistributed on the lattice, in the thermodynamiclimit, gives an excess term which logarithmicallydiverges as the concentration of ions approachesthe maximum concentration

mex� ¼ �kBT ln 1� vcþ � vc�f g, ð9Þ

where v is the volume of the ions/lattice sites,which is equal to the inverse of the maximumconcentration of ions (assumed to be the samefor cations and anions). For ILs, this was firstproposed by Kornyshev [8], leading to a paradigmshift in the field, although an equivalent formula-tion had been introduced by Bikerman [12] andsince rediscovered by several others [2, 11, 13] inthe context of moderately concentrated electro-lytes. As such, this term is often referred to asthe Bikerman excess chemical potential, whichtakes the form mex ¼ �kBT ln 1� 2vcf g in thebulk. The g ¼ 2vc term is referred to as thecompacity of the IL [8], as it is a measure ofhow much the electrolyte can be compressed. Indilute electrolytes, 2vc � 0, such that this excessterm is negligible (unless one is at large voltages

[11, 13]). For ILs however, 2vc � 1, as the size ofions/lattice sites determines the bulk concentra-tion, meaning that excluded volume effects areextremely important.

Including this excess term results in the follow-ing modified PB equation

∇2f ¼ 2qce

sinh bqfð Þ1� 2vcþ 2vc cosh bqfð Þ : ð10Þ

The excluded volume effects of the latticemodel are naturally present in the Fermi-like func-tions for the concentration of ions. At very largepositive (negative) voltages, the concentration ofanions (cations) reaches the maximum concentra-tion, 1/v, and the concentration of cations (anions)tends toward 0. When the concentration reachesthe maximum value, the system is said to be in thecrowding regime. An advantage of the approachof Kornyshev [8] is that a closed-form analyticalsolution of the differential capacitance can beobtained

C ¼ C0cosh u0=2ð Þ

1þ 4vc sinh 2 u0=2ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4vc sinh 2 u0=2ð Þln 1þ 4vc sinh 2 u0=2ð Þ� �

s, ð11Þ

where C0 ¼ εk is the Debye capacitance andu0 ¼ βqf0 is the voltage drop across the EDL inunits of thermal volts. It is interesting to note thatthis formula was also obtained by Kilic, Bazant,and Ajdari at the same time [11], but in the contextof moderately concentrated electrolytes. In fact, asreviewed in Refs. [1, 2], this equation was firstderived by Freise [14] based on Bikerman’smodel [12].

The capacitance-voltage response predicted byEq. (11) is qualitatively different depending on thecompacity [8]. When the compacity tends to zero,the U-shape Gouy-Chapman differential capaci-tance curve [C ¼ C0 cosh (u0/2)] of dilute electro-lytes is recovered [8]. For values of compacity thatare not vanishingly small and satisfy 2vc < 1=3,the differential capacitance curve has a character-istic “camel”-shape [8]. That being, the differential

4 Mean-Field Theory of the Electrical Double Layer in Ionic Liquids

capacitance initially increases with applied volt-age, as the electrolyte can compress, but then theelectrolyte reaches the crowding regime and thedifferential capacitance decreases with furtherapplied voltage. This can be seen in Fig. 1 forγ ¼ 0.1. While, for values of 2vcP1=3, the differ-ential capacitance monotonically decreases in a“bell”-shape [8]. At very large voltages, the differ-ential capacitance follows C � C0=

ffiffiffiffiffiffiffiffiffiffiffiffi4vcu0

p, which

is a law that describes how the effective length of theEDL grows with voltage in the crowding regime[8], which is model independent up to a numericalfactor (for a discussion, see Ref. [1]).

In ILs, “bell”- or “camel”-shaped differentialcapacitance curves are often found [1], which wasa major success of the theory [8]. However, thevolume used for an ion had to typically be largerthan the intrinsic volume of an ion, as theBikerman term underestimates steric repulsion[2]. As such, the compacity has been used as aneffective fitting parameter. Moreover, the value ofthe Debye capacitance was predicted to be of theorder of 100 mFcm�2 (at room temperature withthe effective dielectric constant of ILs in the rangeof 10–20ε0 [4]) and to sharply vary over a voltagerange of 0.1 V. Experiments and simulations mea-sured values of the Debye capacitance which were

of the order of 10 mFcm�2 and varying on thescale of 1 V [1].

To correct the differential capacitance, a Sternlayer was typically added to suppress the capaci-tance at zero charge [1]. A Stern layer arisesbecause the ions can only approach within anion radius of the interface, meaning that there isa linear potential drop until the charges arereached. The capacitance of the Stern layer,given by Cs ¼ ε/R where R is the radius of anion, and EDL occur in series, which results in thesmallest contribution dominating. At small volt-ages, the Stern layer capacitance is of the order of10 mFcm�2, but at large voltages the capacitanceof the crowding regime can be much lower.

Later, Goodwin, Feng, and Kornyshev [15]introduced regular-solution terms to the freeenergy (also investigated earlier by di Caprioet al. [16]). These virial-coefficient termsrepresented additional short-range interactionsbeyond the Bikerman excess chemical potential,such as additional steric interactions or favorableinteractions between cations and anions [2]. Gen-erally, introducing regular-solution terms resultsin a transcendental equation which can only benumerically solved. Using a perturbativeapproach, Goodwin et al. [15] showed that a con-tribution to the excess chemical potential could beapproximated as

Dmex� ¼ Av c� � cð Þ þ Bv c� � cð Þ� � a� 1ð Þqf0, ð12Þ

where a ¼ 1þ vcb A� Bð Þ½ ��1 with A andB representing the short-range interactionsbetween ions of the same sign and between cat-ions and anions, respectively [15]. When A ispositive and B is negative, the value of α is lessthan 1, which makes it harder to form the EDL[15]. It was assumed that Eq. (12) worked for allapplied voltages, and the resulting equation fordifferential capacitance recovered the same formof Eq. (11), but where the Debye capacitance wasmultiplied by a factor of

ffiffiffia

p, and the voltage

dependence was multiplied by a factor of α[15]. This causes a suppression of the Debyecapacitance and its variation with applied voltageto be significantly smoothed [17]. Moreover, at

Mean-Field Theory of the Electrical Double Layerin Ionic Liquids, Fig. 1 Differential capacitance [calcu-lated from Eq. (11) following Ref. [8] in units of Debyecapacitance as a function of applied voltage in units ofthermal volts. Differential capacitance curves are plottedfor three values of the compacity, demonstrating the qual-itatively different capacitance-voltage responses. Theshape of the differential capacitance curve goes from a“camel” to a “bell” shape for γ > 1/3

Mean-Field Theory of the Electrical Double Layer in Ionic Liquids 5

large voltages, the universal law (C � C0=ffiffiffiffiffiffiffiffiffiffiffiffi4vcu0

p)

is recovered. These features can all be seen in Fig. 2,where the capacitance-voltage response is shownfor several values of α. This approach allowed dif-ferential capacitance curves to be fitted at theexpense of an additional parameter in the theory, α.

The regular solution type modification of themean-field theory proposed in Ref. [15] intro-duces a new cumulative parameter to the theory.Although it has a clear physical origin, its value isnot known, so that, strictly speaking, it comes outas a fitting parameter of the theory. Note that,earlier, in order to improve the correspondencewith Monte Carlo simulations, Fawcett and Ryan[18] introduced a number of fitting parametersintended to correct for finite size and correlationeffects beyond the Bikerman excess term. In theirapproach, the values of those parameters weredetermined by comparison of the theory withrestricted primitive model Monte Carlo simula-tions of electrolytes at charged interfaces up to2 M in concentration.

The differential capacitance curves of ILs aretypically quite asymmetric, however, whileEq. (11) is an even function of the applied voltage.One reason why there are asymmetric differentialcapacitance curves is because the sizes of ions in

ILs can often be quite different. For example, aprototypical IL is [Emim]Cl. The Emim+ cation isa large, organic molecule, while the anion is aninorganic Cl� ion that is much smaller. To intro-duce this, Kornyshev phenomenologically param-eterized the value of compacity to depend onvoltage, such that the lattice volume tends to thatof cations (anions) in the limit of large negative(positive) voltages [8].

Similar results have been obtained by Han,Huang, and Yan [19] in a more rigorous asymmet-ric version of the lattice-gas model, while onlyintroducing a single additional parameter whichis the ratio of the volume of (the lattice sites of)anions to cations x ¼ v�/v+. An advantage of theapproach of Han et al. [19] is that analytical solu-tions for the differential capacitance can beobtained, curves of which can be seen in Fig. 3.At large negative voltages, the capacitance isapproximately C � C0=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi4vþcu0

p, but at large

positive voltages it reduces to C � C0=ffiffiffiffiffiffiffiffiffiffiffiffiffiffi4v�cu0

p,

which is a reflection of the EDL being larger forbulkier ions. The asymmetric ion approaches withdifferent lattice-gas types have been investigatedby Gongadze and Iglič [20], Maggs andPodgornik [21], and M. Popović and A. Šiber[22], and also Yin et al. [23] incorporated short-range interactions with asymmetric steric interac-tions into one theory.

The Bikerman style excess chemical potentialis quite a crude approximation, however. TheCarnahan-Starling equation of state is signifi-cantly more accurate than the Bikerman one, andgeneralizations to multicomponent, asymmetricfluids are possible [2]. With this equation of statein the local density approximation, however, onerequires a numerical solution for the differentialcapacitance, and the interfacial layering and over-screening structure are not described. Therefore,one needs to go beyond local variables to achievemore accurate descriptions of the EDL.

Beyond the Local Density ApproximationIn the previous section, advancements in theunderstanding of the capacitance-voltageresponse of ILs was reviewed. In those works,the charge density profiles always monotonicallydecayed. The strong charge-charge correlations in

Mean-Field Theory of the Electrical Double Layerin Ionic Liquids, Fig. 2 Differential capacitance [calcu-lated using a modified version of Eq. (11) in Ref. [15] inunits of Debye capacitance as a function of potential dropin units of thermal volts. Several values of the short-rangeinteraction parameter α are displayed for a fixed compacityof γ ¼ 1. As α decreases, a suppression and smoothing ofthe capacitance-voltage response is observed

6 Mean-Field Theory of the Electrical Double Layer in Ionic Liquids

ILs cannot be accurately described by those mean-field theories, however, as they excessively smearout the detailed structure of the IL. Furthermore,the treatment of steric interactions with local den-sities cannot describe the layered structure ofcrowded ions at interfaces. Both electrostatic andsteric interactions are inherently nonlocal interac-tions. Therefore, to obtain more accurate descrip-tions of the system, covering, in particular,overscreening, one needs to develop a theorythat includes effects beyond a local description,either in terms of higher-order derivatives or inte-grals of the electrostatic potential and/or ionicdensities.

To go beyond the limitations of mean-fieldelectrostatics, Bazant, Storey, and Kornyshev(BSK) [7] introduced an additional term in theelectrostatic energy

F exel f½ � ¼ �

ðdr

el2c2

∇2f rð Þ� �2, ð13Þ

where lc is the phenomenological correlationlength, of the order of an ion diameter in ILs[7]. The term lowers the free energy when thereis curvature in the electrostatic potential, andtherefore, it favors oscillations in the electrostaticpotential. Taking the functional derivative of the

free energy with respect to the electrostatic poten-tial yields the modified PB equation

e 1� l2c∇2

� �∇2f rð Þ ¼ �r rð Þ, ð14Þ

which has fourth-order derivatives too. Thismeans additional boundary conditions arerequired to obtain a solution. In the bulk, thepotential and its derivative should go to zero,and at the interface one boundary condition isfixed by the voltage of the electrode. Whilenumerous other boundary conditions have beenapplied to close the system [7], a final boundarycondition arises naturally from the mechanicalequilibrium constraint at a charged interface[24]. The boundary condition implicitly accountsfor the short-ranged part of the electrostatic inter-actions within a correlation length.

In the charge density of Eq. (14), the chemicalpart of the free energy from the previous sectionscan be utilized. At small voltages, one can linear-ize the charge density, which practically becomesindependent of the equation of state which is usedto account for excluded volume effects. Solvingthe resulting modified Helmholtz equation for aplanar surface yields for the electrostatic potential,in the linear response approximation

Mean-Field Theory of the Electrical Double Layerin Ionic Liquids, Fig. 3 Differential capacitance [calcu-lated from a generalized form of Eq. (11) in Ref. [19]] inunits of Debye capacitance as a function of voltage dropacross the EDL in units of thermal voltage. We display the

curve for a value of compacity as γ ¼ 0.1 (a) and γ ¼ 0.5(b) for a range of values of the ion asymmetry ratio, x. Thecapacitance profile becomes asymmetric when x 6¼ 1, sincethe smaller ions experience less crowding and more easilypopulate the EDL

Mean-Field Theory of the Electrical Double Layer in Ionic Liquids 7

f ¼ f0 exp �k1xð Þ cos k2xð Þ þ A sin k2xð Þ½ �,ð15Þ

where k1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2dc þ 1

p=2dc , k2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2dc � 1

p=2dc ,

and A ¼ (1 � δck1)/(δck2) with δc ¼ lc/lD > 0.5.Clearly, the electrostatic interactions accountedfor by the square of the Laplacian of the potentialresults in overscreening at small voltages, whichgives way to crowding at higher voltages [7], asseen in Fig. 4. This extension of the mean-fieldtheory was a major advancement toward under-standing the EDL of ILs, as it included all of thequalitative features observed in molecular simu-lations and experiments, while retaining the sim-plicity of the PB equation with onephenomenological parameter, lc.

However, as seen in Fig. 4, the overscreeningstructure from the BSK theory is actually quiteweak. There is perhaps only one oscillation in theelectrostatic potential (equivalently the chargedensity) before it is exponentially suppressed

[7]. In surface force measurements and molecularsimulations, it is typical to observe many oscilla-tions in the ionic density near an interface. More-over, one needs to introduce either short-rangeinteractions or a Stern layer to obtain reasonablevalues of the differential capacitance [7].

To achieve more pronounced charge densityoscillations, Gavish and Yochelis [25] developeda phase-field type model of ILs. This model relieson the Cahn-Hilliard regular solution terms in thefree energy functional, which are postulated todescribe nonelectrostatic like-charge attractionthat leads to cation-anion phase separation. Simi-larly, Limmer developed a Landau-Ginzburg the-ory of the EDL which relies on the assumptionthat cations and anions phase separate [26]. Thesesorts of models predict snake-like patterns of ionicphase separation in the bulk, reminiscent ofspinodal decomposition, but a microscopic theoryof the underlying like-charge attraction has yet tobe developed.

In the outlined approaches above, thecrowding regime is described by a continuouslyextending region where the concentration is thatof the macroscopic maximal concentration, asseen in Fig. 4 for the BSK theory. In atomic-force measurements and atomic simulations, how-ever, discrete layers of countercharge clearly formin the crowding regime. In these layers, there is acoherence in where the center of the ions resides,which causes a much larger concentration of ions(than the macroscopic maximum allowed) in asmall region separated by regions where ions donot reside. Upon integrating the ion profiles, themaximal macroscopic concentration is recovered.

To describe the layered structure of thecrowding regime, a very accurate description ofthe steric interactions is required. Commonly, thereference system of hard sphere ions with animplicit solvent, known as the primitive model,is used to describe these structures mathematicallywithin the framework of classical Density Func-tional Theory (cDFT). In cDFT, the excess freeenergy, and thus the excess chemical potential, isrepresented in a nonlocal manner by variousapproximations. While no exact equation of stateis known for inhomogeneous hard-sphere fluids,the most accurate functional is based on

Mean-Field Theory of the Electrical Double Layerin Ionic Liquids, Fig. 4 BSK theory [7] predictions [cal-culated from combining Eqs. (10) and (14) as shown inRef. [7]] for the concentration profiles of ions as a functionof distance from the electrode in units of ion diameter, d.Here, the compacity is γ¼ 0.5 and the correlation length isklc¼ 10. This corresponds to an ion diameter of d¼ 1 nm,a permittivity of ε¼ 5ε0, a temperature of T¼ 450 K, and aconcentration of c ¼ 0:5 M, implying lc ¼ 1.33d [7]. Theelectrode potential in units of thermal volts is u0 ¼ 100. Atsuch large voltages, the counterion density saturates to itsmaximal value, and the dense “layers” of counterions arefollowed by an overscreening layer of opposite charge

8 Mean-Field Theory of the Electrical Double Layer in Ionic Liquids

Fundamental Measure Theory, which representsthe excess free energy density in terms of variousweighted densities of the ions [26, 27]. The Fun-damental Measure Theory approach can accu-rately describe the layering for a hard-spherefluid at an interface [27, 28].

Along with describing the excess free energydue to crowding, cDFT approaches also providecorrections for the finite size of the ions in terms ofelectrostatics, as well as the correlations betweenions. One common approach is to include a con-tribution to the chemical potential from convolu-tions of ionic densities with direct correlationfunctions [28, 29], which is itself supplied as aninput from the Mean Spherical Approximation.This approach, and other related approaches, cangive overscreening and charge ordering at aninterface that is characteristic of concentratedelectrolytes and ILs [29, 30].

The nonlinear integro-diferential equationsthat result from the cDFT theories are far lesscomputationally expensive than Monte Carlo orMolecular Dynamics simulations. Even so, theydo not have the same analytical simplicity andphysical transparency of the simple modified PBapproaches described above that they seek toreplace. For example, due to the complexity ofthe functionals, it can be significantly more diffi-cult to implement the functional numerically thanthe simple PB theories [28]. Furthermore, thefunctionals do not output simple formulas forimportant quantities such as decay lengths froman interface or differential capacitance, in contrastto the modified PB theories. While the predictivepower of cDFT functionals for the microscopicstructure of the EDL is strong, the application ofless accurate local density approximationapproaches as described in previous sections canbe more practical.

Recently, de Souza et al. [9] took inspirationfrom integral theories of the EDL and developed asimplified version for ILs. In the model, the elec-trostatic part of the free energy is expressed interms of the weighted ionic densities, homoge-nized over the size of a single ion. The approachis similar to preceding theories including intramo-lecular ionic charge distributions and also to thecharged-shell functionals derived from the Mean

Spherical Approximation [53–59]. When appliedto the highly concentrated limit of ILs, the theoryreproduces the layered structure at interfaces, asseen in Fig. 5. At high concentrations, the effec-tive screening length for the decay of oscillationsscales inversely with the Debye length, as quali-tatively observed in experiments and simulations.The resulting theory is successful in quantitativelyreproducing the charge density distributions anddifferential capacitance observed in molecularsimulations as well as experimentally observedEDL structures involving crowding of large cat-ions [30]. Interestingly, Adar et al. [32] also mod-ified the electrostatic term and obtained a similardecay length.

Practically all of these theories neglect internaldegrees of freedom of the ILs. Many ILs ions areactually quite large organic molecules, sometimeswith lengthy alkyl chains [1, 10]. To include theseeffects, approaches have either coarse grained[29] the ions into a few representative spheres ordeveloped approaches from polymer physics andapplied them to ILs [33, 34]. These theories for theEDL of ILs can account for internalrearrangement of ions in applied field, which isoften observed in molecular simulations.

Mean-Field Theory of the Electrical Double Layerin Ionic Liquids, Fig. 5 Ionic concentration profile aspredicted by the weighted-density functional in Ref. [9]as a function of distance from the electrode in units of iondiameter. Here, the parameters are T ¼ 300 K, ε ¼ 2ε0,c ¼ 5M, d ¼ 0.5 nm, and s ¼ 1.2 C/m2. Due to the use ofweighted densities, the theory predicts sharp overcrowdingpeaks and a long tail of decaying oscillations in chargedensity further from the interface

Mean-Field Theory of the Electrical Double Layer in Ionic Liquids 9

Solvents and Mixtures

The EDL of ILs in their pure form is perhaps asimplified picture. Some ILs are hydroscopic,meaning that they sorb water from the environ-ment, but actually ILs in their pure form are pos-sibly not the most interesting for applications[1]. Introducing organic solvents in ILs increasesthe ionic conductivity of these electrolytes dra-matically [1]. It is, therefore, practically evenmore interesting to have a picture of the EDL inILs with added solvent. Alternatively, several ILscan be mixed to tune the exact physiochemicalproperties that are desired, or more conventionalinorganic salts can be dissolved in ILs [1], andEDLs in such cases would also be worthy ofdescription.

As has been typical in theories of dilute elec-trolytes, the solvent has often been modeled asdielectric continuum in which the ions reside[1, 2]. This simply means one alters the salt con-centration and dielectric constant in the abovetheories. Such an approach can often capture thequalitative changes to the differential capacitancewhich occurs upon dilution of ILs [1, 2]. However,this approach misses the competition whichoccurs between the ions and solvent in the EDLor in an adsorbed state at the electrode, whichmanifests itself in differential capacitance mea-surements, for example [17]. Therefore, onerequires theories of the EDL which are generaliz-able to more than two components, with theresponses to electrostatic fields possibly beingdifferent for each component [20].

The competition between solvent and ions inILs, or even different types of ions in mixtures ofILs, is a balance between the steric, specific, andelectrostatic interactions. For excluded volumeeffects, generalizations of the above equations ofstate exist to any number of components withdifferent sizes [35]. One quickly loses analyticaltractability when the number of components ismore than 3–4, which means it is favorable toopt for more accurate equations of state, such asmulticomponent generalizations of the CS,Eq. (2).

Specific interactions can also become quiteimportant when solvent or water is present

[36]. The cations of ILs are often hydrophobic,but the anions are usually not (an example ofwhich is [Bmim][PF6]), which leads to largeasymmetry in the response of water in the EDLdepending on the polarity of the interface[37]. Interestingly in Ref. [38], it was shown thathydrophilic ILs may have less water at the elec-trodes, the signatures of which were experimen-tally confirmed through cyclic voltammetry.Alternatively, if lithium salts are dissolved, thewater will coordinate to the lithium cations, dras-tically changing the response [35].

It is also essential to include how the speciesrespond to electrostatic potentials. For mixtures ofILs, the charge of the ions is typically the domi-nant effect, but for solvent the responses can bequite different. If the solvent has a large dipolemoment, such as water, one can include aLangevin term [36] to describe the response ofthe dipole to external fields (note, this approachhas limitations, and one often needs to use effec-tive dipole moments to recover bulk properties ofthe solvents). In the absence of significant dipolemoments, the polarizability of the solvent couldbe accounted for [39].

As an example of these competitions, Budkovet al. [36] investigated a theory of a small amountof water in an IL. The dipolar response of thewater and its affinity for anions caused an enrich-ment of water in the EDL at moderate positivevoltages, but a depletion at moderate negativevoltages took place owing to unfavorable specificinteractions with cations [37]. This caused a peakto occur in the differential capacitance at moderatepositive voltages, which is often seen in experi-ments [17]. At large (negative and positive)applied voltages, there is always enrichment ofwater in the EDL as it is favorable for water toreside in regions of large electric fields.

Underscreening and Beyond

As previously mentioned, surface force measure-ments can reveal the layered structure of ILs atinterfaces [40], which the more involved EDLtheories can capture [28]. These theories predictthat the charge density has decaying oscillations

10 Mean-Field Theory of the Electrical Double Layer in Ionic Liquids

until the bulk is reached. However, past the lay-ered structure, these measurements actually findmonotonically decaying interactions that extendfar beyond what one expects. This was firstreported by Gebbie et al. [41], where the effectwas interpreted as ILs behaving as a dilute elec-trolyte with 99.997% of the ions being bound upin ion pairs. This observation was further con-firmed by a number of groups, with Smith et al.[40] demonstrating that moderately concentratedelectrolytes (1 M) exhibit this monotonic tail,referred to as underscreening [42], that wouldextend further into the electrolyte with increasingconcentration.

This motivated the investigation of ion pairingeffects in the EDL of ILs. If ILs are actually diluteelectrolytes, with the concentration of free chargecarriers of the order of 0.003%, then one canreinterpret the compacity parameter as the con-centration of “free” ions (not bound up into ionpairs or aggregates) over the maximal concentra-tion of “free” ions [1, 15]. Upon doing so with0.003% of “free” ions, the values of Debye capac-itance are significantly smaller than what is foundin experiments. Moreover, for a “dilute electro-lyte,” there should be a strong U-shaped differen-tial capacitance curve consistent with Gouy-Chapman, but a bell shape is often found in exper-iments of ILs. This was further investigated byMaet al. [43] in a cDFT theory, where it was foundthat the Debye capacitance when 99.997% of theions were paired was not consistent with experi-mental observations.

The extent of ion pairing is still an open ques-tion in ILs [44–47]. Aside from mean-field theo-ries [43, 44], the balance between ions in a“clustered state” and in “free” state has beenexplored using molecular dynamics simulations,such as Refs. [46, 48]. The former study showedthat for typical ILs, the percentage of free ions(which is equivalent to the revised meaning ofcompacity) is of the order of 10–20%, dependingon temperature – the larger percentage for highertemperatures following the Arrhenius-like lawwith very small free energy difference betweenthe states of the order of 1 kBT.

Importantly, one could expect that the extent ofion pairing and clustering will “effectively

change” in the EDL, due to changed balancebetween the two states (paired vs. free). Indeed,with ions clustered, and majorly into neutral clus-ters, they contribute less or no charge into theEDL, increasing the effective (high-frequency)dielectric constant. The effect expected here isthe electric field-induced destruction of clusters –“dipoles” – in the EDL in favor of free ions –“monopoles” (the physics of this effect is analo-gous to the Frumkin effect of electric field-induced desorption of organic molecules fromelectrodes in favor of electrosiorption of ions ofelectrolyte [49]). In Ref. [50], a simple mean-fieldtheory was constructed to describe this effect. Themain new prediction was, in spite of the low valueof compacity in the bulk of IL, it effectivelyincreases in the EDL and gives rise to the “bell”shape of differential capacitance rather than the“camel” shape that a simpler theory would sug-gest for low values of compacity.

In such concentrated electrolytes as ILs, aggre-gates larger than ion pairs can exist, motivatingtheories of ionic aggregation and cluster-ing [51]. Recently, McEldrew et al. [52] appliedpolymer theories to superconcentrated electro-lytes, where aggregates of arbitrarily large sizecan be systematically accounted for in a generalframework. It was predicted there that an infinite,percolating network of ions can form, which isreferred to as the gel phase. The existence andproperties of the gel phase in ILs, and its relationto the fragile glass phase, are open questions.

Beyond underscreening, there are still manyother interesting avenues of research. For exam-ple, the dynamics of charging the EDL in super-capacitors (electrokinetic phenomena atinterfaces) and surface chemistry of any defectsare all areas of great importance in improving theperformance of EDL supercapacitors which haveILs as electrolytes.

Cross-References

▶All-Atom Models of Ionic Liquids▶Capacitance with Different Electrode Topology▶ Interfacial Microstructure of Ionic Liquids▶ Ion Pairing in Ionic Liquids

Mean-Field Theory of the Electrical Double Layer in Ionic Liquids 11

▶Mixture of Ionic Liquids/Water▶Monte Carlo Simulation of Ionic Liquids▶Nanopore-confined Ionic Liquids▶ Property Prediction of Ionic Liquids by Molec-ular Dynamics

▶ Structure Heterogeneity in Ionic Liquids

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