Underscreening, Overscreening and Double-Layer Capacitance

Zachary A. H. Goodwina,∗, Alexei A. Kornysheva,∗ aImperial College London, Department of Chemistry, Imperial College Rd, London, SW7 2AZ

Abstract There have recently been reports of extraordinarily long screening lengths in concentrated electrolytes and which vary non-monotonically with ion concentration. There were several attempts to explain these puzzling observations. In this communication we further explore possibilities to rationalise those results. Our analysis does not yet close the problem, but it highlights con- ﬂicting consequences of the models designed to reproduce those observations. Keywords: concentrated electrolytes, room temperature ionic liquids, Debye length, mean-ﬁeld, underscreening, overscreening

1. Introduction The ﬁrst report of exceptionally long Debye length (11 ± 2 nm) in room temperature ionic liquids (RTILs), signiﬁcantly longer than expected under the assumption of complete ion dissociation (∼ 0.1 A˚), was by Gebbie et al. [1]. Initially, that observation was interpreted as incomplete dissociation of the studied RTIL, owing to ion pair formation. To obtain the screening length extracted from those measurements, the extent of ion pairing was massive, with only 0.003% of ions dissociated [1]. An estimate, based on dissociation energy of an ion pair in vacuum rescaled to relative permittivity of the RTIL, was found to agree with experimental data [1, 2]. However, since the massive ion pairing conclusion did not seem congruent with several other

∗Corresponding author Email addresses: [email protected] (Zachary A. H. Goodwin), [email protected] (Alexei A. Kornyshev)

Preprint submitted to Electrochemistry Communications July 10, 2017 properties of RTILs, it was met with skepticism [3–5]. The main concerns that this communication highlights 1 about the claim that RTILs are ‘dilute electrolytes’ [1, 2] had the following roots: (i) The qualitative shapes of measured [6–8] and simulated [9–13] double-layer capacitance curves as a function of electrode potential and values of linear-response capacitance did not match that conclusion [5, 14–19]. (ii) This conjecture disagreed with theoretical prediction of the ion pairing extent [20]. Lately, Perkin et al. [21] reported similar values of the screening length in pure RTILs, and, furthermore, when considering mixtures with solvents found a non-monotonic dependence of the screening length on RTIL concentration [21]. A critical review of this puzzle was recently presented in Ref. 22. Our communication further explores options that could rationalise these observations. We rest our arguments on a diﬀerent approach, based on a revisited/amended ‘lattice’ model, and a tutorial style analysis on the im- plications of pronounced charge density oscillations. The analysis reveals conditions under which such perplexing behaviour could emerge, and how realistic those could be.

2. ‘Lattice’ model Consider a ‘lattice’ model of 1:1 electrolyte in which each of the N sites are occupied with either participating ions, spectating ions or diluent

N = N+ + N− + Ns + Nd. (1)

Here, Nd, is the number of solvent molecules; participating ions comprise of N+ cations and N− anions; spectating ions, Ns, belong to ion pairs or neutral aggregates [23], and are assumed to bear no charge. Just as electrons in semiconductors thermally populate the conduction band from valence band, spectating ions can be thermally activated to participating ions. Hence, each ion can spectate or participate, but at any given time there is some average population of ‘free’, screening ions. Conservation of the total number of ions, N±, reads

1Another concern includes conductivity measurements, the discussion for which shall be presented elsewhere.

2 N± = N+ + N− + Ns, (2) ¯ with electroneutrality in the bulk, N+ = N− = N/2. Voids [14] are neglected here [24]. The free energy can be approximated by

F = eΦ(N+ − N−) + Up − TSp (N+ + N−) + Us − TSs (N± − N+ − N−) zA N 2 zA N 2 N N N (N − N ) N (N − N ) + + + + − − + zB + − + zB + ± + zB − ± 2 N 2 N N + N − N ( ) N! − kBT ln . (3) (N+)!(N−)!(N± − N+ − N−)!(N − N±)!

The first term describes interactions of participating ions with the mean-field electrostatic potential, Φ; where e is elementary charge. Terms two and three are intrinsic free energy of participating, p, and spectating, s, ions.2 Short- range correlations, of a regular solution type [16, 26], between participating ions are described with terms four through six; where z, A+, A− and B are coordination number, and correlation constants between cations, between anions, and between cations and anions, respectively [14, 17].3 Short-range correlations between cations and solvent, and anions and solvent are described by terms seven and eight; where B+ and B− are the corresponding constants. For simplicity, short-range correlations involving spectating ions have been neglected. Finally, the last term describes lattice entropy [5, 14, 17]. The meaning of γ = N/N¯ ,4 which previously appeared as compacity [5, 14, 17, 18], is different here. It now represents the participating ion fraction. As spectating ions have been introduced in place of voids, spectating ions appear in an analogous manner; hence, γ still measures the extent to which

2According to Debye-H¨uckel theory [25] the self-energy of an ion depends on the ionic atmosphere surrounding the ion, and therefore, on the concentration of ions [19, 20]. The presented formalism does not consider any dependence of intrinsic free energy on ion concentration; intrinsic free energy is a constant at a given temperature. 3See Refs. 27–30 for density functional theories, and 14, 26, 31–33 for numerical account of non-electrostatic interactions, which treat correlations in a more sophisticated manner. 4 Also expressed as γ =c/c ¯ max, wherec ¯ and cmax are bulk and maximum concentration of participating ions, respectively.

3 participating ions can concentrate in the double-layer. The participating ion fraction is now a variational parameter that depends, in part, on the ‘band gap’ [Up − Us − T (Sp − Ss)]; this is in contrast to compacity, which was a constant that could be determined from simulations and experiments [9–11, 14–16]. Applying Stirling’s formula Eq. (3), converting to a dimensionless free energy (f = F/kBTN), and minimising f with respect to γ, obtains an equation on participating ion fraction in the bulk ( ) ∂f γ/2 = χ1γ/2 + χ0 + ln = 0; (4) ∂γ γ± − γ where a + a χ = + − + b, (5) 1 2 1 χ = (u − T s ) − (u − T s ) + (b + b )(1 − γ ). (6) 0 p p s s 2 + − ±

Here γ± = N±/N is ion fraction in the bulk; short-range correlation constants and intrinsic free energy terms have been rescaled to thermal energy, kBT , denoted by lower case letters. It is expected that short-range correlation constants a+ and a− are positive and b is negative [17]. Furthermore, repulsive constants have a larger magnitude than attractive, owing to the interplay between steric and electrostatic interaction [17]; therefore χ1 > 0. If χ1 is small (χ1 ≈ 0), Eq. (4) simpliﬁes to a Fermi function γ γ = ± . (7) 1 1 + exp χ 2 0

If χ0 = 0 then γ = 2γ±/3, which is consistent with the estimate for pure RTIL [20]. For large positive χ0, Eq. (7) simpliﬁes to a Boltzmann distribution [1, 2]. Example dependences prescribed by Eq. (7) are displayed in Fig. 1. Within the used model, solvent [15, 16] and spectating ions act as latent voids. Furthermore, in the double-layer equations short-range correlations between solvent and participating ions will be neglected. Then, for the symmetric case, a+ = a− = a, equations from Ref. 17 can be utilised. The rescaled inverse Debye length is

4 Figure 1: Qualitatively diﬀerent dependences of the participating ion fraction on RTIL fraction can be obtained for diﬀerent χ0. Participating ion fraction as a function of RTIL fraction; χ0: -100 (dotted black - complete dissociation) and 0 (solid blue - entropically determined [20]); b+ + b− and (up − T sp) − (us − T ss) are -20 and 10 (dotted-dashed red - massive ion pairing), and -2 and 2 (dashed green), respectively.

√ κ˜ = ακ, (8) and diﬀerential capacitance

v u 2 ˜ cosh αu0/2 2γ sinh αu0/2 C = C0 · u ; (9) 1 + 2γ sinh2 αu /2 t n 2 o 0 ln 1 + 2γ sinh αu0/2 where 1 α = γ . (10) 1 + (a − b) 2 ˜ Here C0 is rescaled Debye capacitance, given by 0κ˜, in SI units, where 0 is the permittivity of free space; and u0 is dimensionless potential drop across the double-layer, seen as eΦ0/kBT [14, 17].

5 Typically polar solvents, such as propylene carbonate [20], have a larger relative permittivity than RTILs [5, 21, 34, 35]. To describe the Debye length at diﬀerent ion fractions, relative permittivity must change as a function of ion fraction [21, 36]. A linear interpolation, that has the required limiting behaviour, shall be utilised as a rough parameterization

(il) (d) (il) (γ±) = + ( − )(1 − γ±), (11) where (d) and (il) are relative permittivities of solvent and RTIL, respectively. Under the condition of χ1 ≈ 0 for the symmetric case, the rescaled screening length, λ = 1/κ˜, takes on v u " # u(γ±) λ = λmaxt 2 + exp{χ0} + γ±(a − b) , (12) 2γ± which can deviate from monotonic decrease with increasing ion concentration, owing to non-monotonic dependence of the participating ion fraction on ion 2 fraction. Here, λmax, given by 0kBT/e cmax, is a Debye length at maximum ion concentration. In the limit of χ0 → −∞, Eq. (12) recovers the rescaled Debye length with full ion dissociation [17]. Whereas, for large positive χ0, the screening length is signiﬁcantly larger than the native one.

3. Are RTILs ‘Dilute Electrolytes’? If spectating ions are favoured for a concentrated system, from a positive band gap, and participating ions for a dilute system, due to negative b++b−, a non-monotonic Debye length can be obtained, as seen in Fig. 2. To reproduce extraordinarily long screening lengths [1, 2, 21], the participating ion fraction is in the range of 10−4 - 10−5 for neat RTIL. Assumption of such small extent of participating ions, however, contra- dicts the values and shapes of typically reported diﬀerential capacitance for RTILs [5–18]. To illustrate this point, example curves are displayed in Fig. 3, at diﬀerent RTIL fractions, for the cases in Fig. 2. Evidently, for moderate ion pairing [20], a transition from camel to bell shape occurs as RTIL fraction increases, which is consistent with the expected transition in shape from a dilute to concentrated electrolyte [5, 14–17].

6 Figure 2: Dependence of Debye length on RTIL fraction displaying monotonic or non-monotonic behaviour. Debye length as a function of RTIL fraction calculated (il) (d) with Eq. (12). Here T = 300 K, cmax = 6 M, = 12, = 64 (corresponding to propylene carbonate [21]), and a − b = 50; b+ + b− and (up − T sp) − (us − T ss) were -20 and 10 (dotted-dashed red curve), and 0 and 0 (solid blue curve), respectively.

For the case of massive ion pairing there is no such transition. The diﬀerential capacitance curve retains camel shape for all considered values of RTIL fraction.5 Furthermore, for pure RTIL, linear-response capacitance is almost zero, while typical values are in the range of 5-25 µFcm−2 [5–18]. Similar contradiction would have been obtained in the theory of Ref. 41, had capacitance been calculated. Recently, Adar et el. [42] performed a Poisson-Boltzmann analysis that also conditioned the non-monotonic Debye length dependence. They did not, however, explored the consequences of such eﬀect on double-layer capacitance.

5The assumption that solvent acts as latent voids [15, 16] and relative permittivity is a constant do not weaken the conclusions drawn; linear-response capacitance and qualitative shapes of capacitance curves will not be substantially affected by this. The only contribution to differential capacitance discussed here is from the diffuse double-layer. This is a reasonable approximation for comparison with differential capacitance curves near metal- lic electrodes [5], but not graphite or low-dimensional carbon electrodes, which require inclusion of electrodes space charge layer or quantum capacitance [37–40].

7 Figure 3: No qualitative changes in differential capacitance curves occur for the case of massive ion pairing (b) at different RTIL fractions, which is in contrast to moderate ion pairing (a). Differential capacitance as a function of electrode potential at the indicated RTIL fractions for the same values as those in Fig. 2.

The consequences of ion pairing in differential capacitance curves was also theoretically investigated by Ma et al. [43]. It was found that with increase of ion pairing, the differential capacitance curve transitioned from a bell to camel shape, which is in agreement with the presented results. For the case of more than 99.9% ions paired, linear-response capacitance was only displayed when the double-layer within a artificially defined distance was imposed. Implementing a cutoff significantly changed differential capacitance curves; thus, such procedure does not seem to be justified, due to a still substantial contribution to screening of electrode charge outside the cutoff. In order to reconcile underscreening [19] with existing capacitance data, Lee and Perkin suggested that the potential drop resides within a monolayer

8 of ions [44], and thereby, the distribution of ions beyond that layer should be unimportant for capacitance. Actually, it goes back to the conjecture of Baldelli [45], that the double-layer in RTILs is one layer thick [44]. Lee and Perkin attributed such effect to image forces, which creates an interfacial melted monolayer, with the remaining RTIL in a quasi-dielectric state. Such situation has not been seen so far in simulations, performed at constant electrode potential (accounts for image forces) [12, 46, 47]. In simpler constant charge simulations, with Lennard-Jones charged spheres, such situation was encountered only at specific large values of electrode charge, far from linear-response [48]. The short range structure of RTILs near smooth surfaces exhibits pronounced layering of ions with alternating sign [21, 22, 49–54]. This double- layer structure can not be captured with the utilised mean-field theory [5, 14– 17], nor can it be adequately described by the BSK theory [18].

4. Effect of Oscillations on Capacitance Here we investigate the consequence of charge density oscillations, in the presence of a ‘ultra diffuse’ tail. Consider a profile of induced charge density forming in response to charging the electrode. This profile is assumed to have a short range decaying oscillatory part and a long, exponentially decaying tail, approximated as

ρ = A1 exp{−κ1x} sin(qx) + A2 exp{−κ2x}. (13)

Here, A1 and A2 are constants which, in part, control the dominance of each term; κ1 and q are short range inverse decay length and wavenumber of oscillations, q ≈ 2π/a, where a is ion pair diameter; κ2 is the long range inverse screening length (κ2 << κ1). These parameters depend on the system considered and the charge of the electrode, all related by the charge conservation law Z ∞ qA1 A2 −σ = dxρ = 2 2 + . (14) 0 κ1 + q κ2

Assuming κ1, κ2 and q are independent constants, there is a linear rela- tionship between A1 and A2, and σ. Introducing renormalised constants, ˜ ˜ A1 = −σκ1A1 and A2 = −σκ2A2, gives

9 " # ρ ˜ κ1κ2q − = A1 κ1 exp{−κ1x} sin(qx) − 2 2 exp{−κ2x} + κ2 exp{−κ2x}. σ κ1 + q (15) ˜ By varying A1, as seen in Fig. 4, the double-layer structure can be tuned from an exponential decay to decaying oscillation. Overscreening occurs when the ﬁrst layer of countercharge, σ1, has a larger magnitude than the surface charge of the electrode [9, 18]. The former can be determined by R a/2 integrating charge density over an ion diameter, σ1 = 0 ρdx, to obtain a criterion for overscreening

2 2 ˜ κ1 + q 1 A1 > . (16) κ1q 1 + exp{−(κ1 − κ2)a/2} For the limiting case of decaying oscillations, with no long tail, overscreening is always observed: σ1 = −σ[exp{−κ1a/2} + 1].

Figure 4: Double-layer structure varied through A˜1. Charge density scaled to minus surface charge density as a function of displacement from the electrode for the 9 −1 9 −1 indicated A˜1; while κ1, κ2, a are 1x10 m , 0.1x10 m , and 1.4 nm, respectively.

To determine linear response capacitance, the distance to the center of mass of countercharge [5], l, shall be calculated

10 C = 0 ; (17) l where 1 Z ∞ l = − xρdx. (18) σ 0 For the trial function, Eq. (15), we obtain

2 2 2 ! ˜ 2κ1κ2q − κ1q[κ1 + q ] 1 l = A1 2 2 2 + . (19) κ2[κ1 + q ] κ2 Oscillations in charge density can thus reduce l, and therefore, reconcile capacitance with long screening lengths. For the examples considered in Fig. ˜ ˜ ˜ 4, l is 9.8 nm (A1 = 0.1), 7.9 nm (A1 = 1), and 1.6 nm (A1 = 4). Thus, if there are pronounced oscillations, a long decaying tail of induced charge density may not necessarily lead to small values of the double-layer capacitance.

5. Coulomb Critically An alternative interpretation of the long decay lengths observed in the interaction between charged mica surfaces mediated by concentrated electrolytes [22], may be related with the phenomenon of Coulomb criticality [55]. It has been found both experimentally and theoretically (see Ref. 56 and references cited within), that there is a phase diagram on the ‘electrolyte concentration - temperature’ plane, which separates disordered (liquid) and ordered (quasi-crystalline) states of ionic assemblies, with an Ising type transition between those states. The position of the critical line depends on the system, but close to the line there will be the two phases coexist. If such situation realises for the considered RTILs and concentrated solutions [1, 2, 21], interaction between the surfaces might be mediated by domains of the ordered phase. A theory needs to be developed to show how this would aﬀect the interaction.

6. Conclusion Using simple mean-ﬁeld analysis, we found a non-monotonic Debye length dependence on electrolyte concentration, as reported by Perkin’s group, can

11 be justified under assumption of massive ion pairing, as hypothesised initially by Israelashvili’s group. Within the set of parameters that provided such massive ion pairing, we calculated the electrical double-layer capacitance. The results show that the shape of capacitance-voltage dependence resulting from such calculation is qualitatively different from what has been experimentally observed. However, this controversy might not arise in the theory that would take into account strong overscreening. There are still other facts that speak against remarkably small amounts of ‘free’ ions in RTILs and highly concentrated electrolyte solutions, and we may still need to look for another, yet unknown explanation of that non-monotonic dependence. Equally, more independent experiments are needed to confirm the existence of underscreening, as well as the degree of ion clustering.

7. Acknowledgements Discussions with Prof. F. Bresme, Prof. S. Kondrat, and O. Robotham are greatly appreciated. The work was finalised during our guest-stay at the Institute of Theoretical Physics II: Soft Matter - Heinrich-Heine-Universität Düsseldorf.AAK acknowledges the support of the Alexander von Humboldt foundation through post-award visits program for Humboldt Awardees; ZG acknowledges Junior Research Fellowship of the Thomas Young Centre (UK) and the support from the host institute; we are thankful to our host Prof. H. Löwen for hospitality.

8. References [1] M. A. Gebbie, M. Valtiner, X. Banquy, E. T. Fox, W. A. Henderson, J. N. Israelachvili, Ionic liquids behave as dilute electrolyte solutions, PNAS 110 (2013) 9674–9679.

[2] M. A. Gebbie, H. A. Dobbs, M. Valtiner, J. N. Israelachvili, Long-range electrostatic screening in ionic liquids, PNAS 112 (2015) 7432–7437.

[3] S. Perkin, M. Salanne, P. Madden, R. Lynden-Bell, Is a Stern and diﬀuse layer model appropriate to ionic liquids at surfaces?, PNAS 110 (2013) E4121.

12 [4] M. A. Gebbie, M. Valtiner, X. Banquy, W. A. Henderson, J. N. Is- raelachvili, Reply to Perkin et al.: Experimental observations demon- strate that ionic liquids form both bound (Stern) and diﬀuse electric double layers, PNAS 110 (2013) E4122.

[5] M. V. Fedorov, A. A. Kornyshev, Ionic liquids at electriﬁed interfaces, Chem. Rev. 114 (2014) 2978–3036.

[6] M. T. Alam, J. Masud, M. M. Islam, T. Okajima, T. Ohsaka, Diﬀer- ential capacitance at Au(111) in 1-alkyl-3-methylimidazolium tetraﬂu- oroborate based room-temperature ionic liquids, J. Chem. Phys. C 115 (2011) 19797–19804.

[7] M. T. Alam, M. M. Islam, T. Okajima, T. Ohsaka, Measurements of diﬀerential capacitance at mercury/room-temperature ionic liquids interfaces, J. Phys. Chem. C 111 (2007) 18326–18333.

[8] M. M. Islam, M. T. Alam, T. Ohsaka, Electrical double-layer structure in ionic liquids: A corroboration of the theoretical model by experimental results, J. Phys. Chem. C 112 (2008) 16568–16574.

[9] M. V. Fedorov, A. A. Kornyshev, Towards understanding the structure and capacitance of electrical double layer in ionic liquids, Electrochim. Acta 53 (2008) 6835–6840.

[10] M. V. Fedorov, A. A. Kornyshev, Ionic liquid near a charged wall: Struc- ture and capacitance of electrical double layer, J. Phys. Chem. B 112 (2008) 11868–11872.

[11] N. Georgi, A. A. Kornyshev, M. V. Fedorov, The anatomy of the double layer and capacitance in ionic liquids with anisotropic ions: Electrostric- tion vs. lattice saturation, J. Electroanal. Chem. 649 (2010) 261–267.

[12] C. Merlet, B. Rotenberg, P. A. Madden, M. Salanne, Computer simulations of ionic liquids at electrochemical interfaces, Phys. Chem. Chem. Phys. 15 (2013) 15781–15792.

[13] J. Vatamanu, O. Borodin, D. Bedrov, G. D. Smith, Molecular dynamics simulation study of the interfacial structure and diﬀerential capacitance of alkylimidazolium bis(triﬂuoromethanesulfonyl)imide [Cnmim][TFSI]

13 ionic liquids at graphite electrodes, J. Phys. Chem. C 116 (2012) 7940– 7951.

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

[15] M. S. Kilic, M. Z. Bazant, A. Ajdari, Steric eﬀects in the dynamics of electrolytes at large applied voltages. I. Double-layer charging, Phys. Rev. E 75 (2007) 021502.

[16] M. Z. Bazant, M. S. Kilic, B. D. Storey, A. Ajdari, Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions, Adv. Colloid Interface Sci. 152 (2009) 48–88.

[17] Z. A. H. Goodwin, G. Feng, A. A. Kornyshev, Mean-ﬁeld theory of electrical double layer in ionic liquids with account of short-range correlations, Electrochim. Acta 225 (2017) 190–196.

[18] M. Z. Bazant, B. D. Storey, A. A. Kornyshev, Double layer in ionic liquids: Overscreening versus crowding, Phys. Rev. Lett. 106 (2011) 046102.

[19] A. A. Lee, C. S. Perez-Martinez, A. M. Smith, S. Perkin, Under- screening in concentrated electrolytes, Faraday Discuss. (2017) DOI: 10.1039/C6FD00250A.

[20] A. A. Lee, D. Vella, S. Perkin, A. Goriely, Are room-temperature ionic liquids dilute electrolytes?, J. Phys. Chem. Lett. 6 (2015) 159–163.

[21] A. M. Smith, A. A. Lee, S. Perkin, The electrostatic screening length in concentrated electrolytes increases with concentration, J. Phys. Chem. Lett. 7 (2016) 2157–2163.

[22] M. A. Gebbie, A. M. Smith, H. A. Dobbs, A. A. Lee, G. G. Warr, X. Banquy, M. Valtiner, M. W. Rutland, J. N. Israelachvili, S. Perkin, R. Atkin, Long range electrostatic forces in ionic liquids, Chem. Com- mun. 53 (2017) 1214–1224.

[23] H. K. Stassen, R. Ludwig, A. Wulf, J. Dupont, Imidazolium salt ion pairs in solution, Chem. Eur. J. 21 (2015) 8324 – 8335.

14 [24] N. Gavish, A. Yochelis, Theory of phase separation and polarization for pure ionic liquids, J. Phys. Chem. Lett. 7 (2016) 1121–1126.

[25] E. H¨uckel, P. Debye, The theory of electrolytes. I. lowering of freezing point and related phenomena, Phys. Z 24 (1923) 185–206.

[26] M. Z. Bazant, Theory of chemical kinetics and charge transfer based on nonequilibrium thermodynamics, Acc. Chem. Res. 46 (2013) 1144–1160.

[27] J. Forsman, C. E. Woodward, M. Trulsson, A classical density functional theory of ionic liquids, J. Chem. Phys. B 115 (2011) 4606–4612.

[28] J. Wu, T. Jiang, D. en Jiang, Z. Jin, D. Henderson, A classical density functional theory for interfacial layering of ionic liquids, Soft Mater 7 (2011) 11222–11231.

[29] A. Härtel,S. Samin, R. van Roij, Dense ionic fluids confined in planar capacitors: in- and out-of-plane structure from classical density functional theory, J. Phys.: Condens. Matter 28 (2016) 244007.

[30] J. Forsman, A simple correlation-corrected Poisson-Boltzmann theory, J. Chem. Phys. C 108 (2004) 9236–9245.

[31] H. Zhao, Eﬀects of electrostatic correlations on electrokinetic phenomena, Phys. Rev. E 86 (2012) 051502.

[32] D. di Caprio, Z. Borkowska, J. Stafiej, Specific ionic interactions within a simple extension of the Gouy-Chapman theory including hard sphere effects, J. Electroanal. Chem. 572 (2004) 51–59.

[33] Y. Lauw, M. D. Horne, T. Rodopoulos, , F. A. M. Leermakers, Room- temperature ionic liquids: Excluded volume and ion polarizability eﬀects in the electrical double-layer structure and capacitance, Phys. Rev. Lett. 103 (2009) 117801.

[34] H. Weing¨artner, Understanding ionic liquids at the molecular level: Facts, problems, and controversies, Angew. Chem. Int. Ed. 47 (2008) 654.

[35] H. Weing¨artner,The static dielectric permittivity of ionic liquids, J. Mol. Liq. 192 (2014) 185–190.

15 [36] A. Stoppa, J. Hunger, G. Hefter, , R. Buchner, Structure and dynamics of 1-N-Alkyl-3-N-Methylimidazolium Tetraﬂuoroborate + Acetonitrile mixtures, J. Phys. Chem. B 116 (2012) 75097521.

[37] T. Fang, A. Konar, H. Xing, D. Jena, Carrier statistics and quantum capacitance of graphene sheets and ribbons, Appl. Phys. Lett. 91 (2007) 092109.

[38] H. Xu, Z. Zhang, L.-M. Peng, Measurements and microscopic model of quantum capacitance in graphene, Appl. Phys. Lett. 98 (2011) 133122.

[39] A. J. Pak, E. Paek, G. S. Hwang, Relative contributions of quantum and double layer capacitance to the supercapacitor performance of carbon nanotubes in an ionic liquid, Phys. Chem. Chem. Phys. 15 (2013) 19741– 19747.

[40] A. A. Kornyshev, N. B. Luque, W. Schmickler, Diﬀerential capacitance of ionic liquid interface with graphite: the story of two double layers, J. Solid State Electrochem. 18 (2014) 1345–1349.

[41] J. Zwanikken, R. van Roij, Inﬂation of the screening length induced by Bjerrum pairs, J. Phys.: Condens. Matter 21 (2009) 424102.

[42] R. M. Adar, T. Markovich, D. Andelman, Bjerrum pairs in ionic solutions: A Poisson-Boltzmann approach, J. Chem. Phys. 146 (2017) 194904.

[43] K. Ma, J. Forsman, C. E. Woodward, Inﬂuence of ion pairing in ionic liquids on electrical double layer structures and surface force using classical density functional approach, J. Chem. Phys. 142 (2015) 174704.

[44] A. A. Lee, S. Perkin, Ion-image interactions and phase transition at electrolyte-metal interfaces, J. Phys. Chem. Lett. 7 (2016) 2753–2757.

[45] S. Baldelli, Surface structure at the ionic liquid-electriﬁed metal interface, Acc. Chem. Res. 41 (2008) 421–431.

[46] K. Breitsprecher, K. Szuttor, C. Holm, Electrode models for ionic liquid- based capacitors, J. Phys. Chem. C 119 (2015) 22445–22451.

16 [47] C. Merlet, C. P´ean,B. Rotenberg, P. A. Madden, P. Simon, M. Salanne, Simulating supercapacitors: Can we model electrodes as constant charge surfaces?, J. Phys. Chem. Lett. 4 (2013) 264–268.

[48] V. Ivaniˇstˇsev,K. Kirchner, T. Kirchner, M. V. Fedorov, Restructuring of the electrical double layer in ionic liquids upon charging, J. Phys.: Condens. Matter 27 (2015) 102101.

[49] A. M. Smith, A. A. Lee, S. Perkin, Switching the structural force in ionic liquid-solvent mixtures by varying composition, Phys. Rev. Lett. 118 (2017) 096002.

[50] R. Hayes, G. G. Warr, R. Atkin, Structure and nanostructure in ionic liquids, Chem. Rev. 115 (2015) 6357–6426.

[51] R. Atkin, S. Z. E. Abedin, R. Hayes, L. H. S. Gasparotto, N. Borisenko, F. Endres, AFM and STM studies on the surface interaction of [BMP]TFSA and [EMim]TFSA ionic liquids with Au(111), J. Chem. Phys. C 113 (2009) 13266–13272.

[52] F. Endres, N. Borisenko, S. Z. E. Abedin, R. Hayes, R. Atkin, The interface ionic liquid(s)/electrode(s): In situ STM and AFM measurements, Faraday Discuss. 154 (2012) 221–233.

[53] N. Hjalmarsson, R. Atkinb, M. W. Rutland, Switchable long-range double layer force observed in a protic ionic liquid, Chem. Commun. 53 (2017) 647–650.

[54] R. Hayes, G. G. Warr, R. Atkin, At the interface: solvation and design- ing ionic liquids, Phys. Chem. Chem. Phys. 12 (2010) 1709–1723.

[55] M. E. Fisher, The story of coulombic critiality, J. Stat. Phys. 75 (1994) 1–36.

[56] V. Kobelev, A. B. Kolomeisky, M. E. Fisher, Lattice models of ionic systems, J. Chem. Phys. 116 (2002) 7589–7598.