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Underscreening, Overscreening and Double-Layer Capacitance

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Underscreening, Overscreening and Double-Layer Capacitance 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 concen- tration. 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 concen- tration [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 an- ions, and between cations and anions, respectively [14, 17].3 Short-range cor- relations 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 pos- itive and b is negative [17]. Furthermore, repulsive constants have a larger magnitude than attractive, owing to the interplay between steric and elec- trostatic 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 sym- metric 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. p 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, respec- tively. Under the condition of χ1 ≈ 0 for the symmetric case, the rescaled screen- ing length, λ = 1=κ~, takes on v u " # u(γ±) λ = λmaxt 2 + expfχ0g + 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.
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