The Role of Ion–Ion Correlations for the Differential Capacitance of

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C: Energy Conversion and Storage; Energy and Charge Transport

The Role of Ion-Ion Correlations for the Differential Capacitance of Ionic Liquids Rachel Downing, Guilherme Volpe Bossa, and Sylvio May J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b09756 • Publication Date (Web): 27 Nov 2018 Downloaded from http://pubs.acs.org on December 3, 2018

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The Journal of Physical Chemistry

The Role of Ion-Ion Correlations for the Differential Capacitance of Ionic Liquids Rachel Downing,† Guilherme Volpe Bossa,‡ and Sylvio May∗,† Department of Physics, North Dakota State University, Fargo, ND 58108-6050, USA, and Department of Physics, São Paulo State University (UNESP), Institute of Biosciences, Humanities and Exact Sciences, São José do Rio Preto, SP,15054-000, Brazil E-mail: [email protected]

∗

To whom correspondence should be addressed Department of Physics, North Dakota State University, Fargo, ND 58108-6050, USA ‡ Department of Physics, São Paulo State University (UNESP), Institute of Biosciences, Humanities and Exact Sciences, São José do Rio Preto, SP,15054-000, Brazil †

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract We employ the quasi-chemical approximation to incorporate ion-ion correlations into a lattice gas model for the electric double layer formed by a compact, sizesymmetric ionic liquid. The resulting differential capacitance, which we calculate for planar and weakly curved electrodes up to first order in curvature, transitions from a bell-shape to a camel-shape profile for increasing correlation strength. No such transition is present if the quasi-chemical approximation is replaced by a random mixing approximation. The bell-to-camel shape transition is, up to first order, independent of curvature. If Coulomb interactions dominate on molecular length scales, the differential capacitance has a tendency to adopt a camel-shape profile. Hence, our model offers a physical interpretation for the observed camel shape (or even U-shape) of the differential capacitance in many ionic liquids. Correlations also cause “underscreening” where the characteristic decay length of the electric double layer grows with the correlation strength and can become much larger than the size of a single ion.

Introduction Ionic liquids are molten salts that remain in their fluid phase at ambient temperatures. Ions that form ionic liquids are typically bulky, with internal degrees of freedom, and low levels of structural symmetry – some possess hydrocarbon side chains that render the molecule amphiphilic. Much of the immense interest in ionic liquids results from their energy-related applications, especially energy conversion and storage. 1–5 Among the fundamental properties of ionic liquids is the ability to form an energy-storing electric double layer (EDL) when facing a charged electrode. The double layer can be viewed as a capacitor formed by the charges at the electrode and the counterions that are recruited from the ionic liquid. This capacitor is associated with a differential capacitance Cdif f = dσ/dΦ0 that characterizes the relation between the surface charge density σ of an electrode and the surface potential Φ0 . It is convenient to introduce a scaled (dimensionless) differential 2


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capacitance C¯dif f through Cdif f = 0 C¯dif f /l, where is the dielectric constant of the ionic p liquid, 0 the permittivity of vacuum, and l = ν/(4πlB ) a length that reflects the ion volume ν and the Bjerrum length lB = e2 /(4π0 kB T ) at which the electrostatic interaction between two elementary charges e equals the thermal energy unit kB T (Boltzmann’s constant times absolute temperature). One of the most commonly used approaches to describe the EDL, which can be applied to the entire concentration range from dilute to highly concentrated electrolytes, employs a non-interacting lattice gas. 6–9 When each lattice site is occupied by an ion (an anion or a cation, all of the same volume ν), the non-interacting lattice gas model describes a solventfree, compact ionic liquid with size-symmetric ions that predicts for the scaled differential capacitance 8 |tanh Ψ0 | , C¯dif f = √ 2 ln cosh Ψ0

(1)

where Ψ0 = eΦ0 /(kB T ) is the dimensionless surface potential of the electrode. Eq. 1 applies to a strictly planar electrode. The dependence of C¯dif f on curvature has been studied, so far, on the level of the classical Poisson-Boltzmann model 10 and using density functional theory, 11,12 yet not for ionic liquids. The shape of C¯dif f (Ψ0 ) according to Eq. 1 is bell-like; see the curve marked ω = 0 below in each of the two diagrams of Fig. 1. The large diversity of electrode properties and chemical structures of ionic liquids impedes a straightforward classification of experimental observations: there are examples for bell-shape profiles, 13–15 but there are also many examples for camel-shape profiles of the differential capacitance. 13,15–23 Camel-shape profiles possess a region around which C¯dif f (Ψ0 ) exhibits a local minimum. For some ionic liquids this region dominates the entire profile; such profiles are sometimes referred to as U-shaped. 13,15,17,22 Camel-shape profiles for ionic liquids have also been reported to emerge from computer simulations. 24–26 The non-interacting lattice gas model predicts a bell-to-camel shape transition as function of dilution with a solvent or, in an alternative interpretation, as function of ion compacity. 8 Quantitatively, the presence of solvent or voids can be cast into a compacity parameter γ (with 0 < γ ≤ 1), 3



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implying 8 C¯dif f =

γ |sinh Ψ0 | p . [1 + γ(cosh Ψ0 − 1)] 2 ln [1 + γ(cosh Ψ0 − 1)]

(2)

The absence of solvent or void lattice sites (γ = 1) renders Eqs. 1 and 2 identical. Eq. 2 predicts the camel-to-bell shape transition at γ = 1/3, where one third of all lattice sites are occupied by ions. Clearly then, based on the non-interacting lattice gas model, bell-like profiles of C¯dif f are expected for solvent-free ionic liquids. Whenever the non-interacting lattice gas model according to Eq. 2 has been used in the past to fit experimental data of camel-shape profiles for the differential capacitance, 20,22,23 the predicted compacity was lower than γ = 1/3, implying that more than 2/3 of the cells in the underlying lattice are empty instead of being filled with ions. As was pointed out by Trulsson et al, 26 compacity values of γ ≤ 1/3 are generally too low for a liquid. This raises the question if the compacity γ is a meaningful quantity to rationalize the camel shape of the differential capacitance for ionic liquids. It is widely recognized 27 that the description of ionic liquids using the non-interacting lattice gas model is problematic. The only length scale in the model is l which, based on the typical values ν = 1 nm3 and lB = 5 nm, amounts to about 0.1 nm, much smaller than the size of an individual ion. In contrast, ample experimental evidence suggests the existence of a characteristic decay length ξ much larger than the size of an ion and further growing with lB . 27–30 This so-called “underscreening” has received significant attention, prompting multiple modeling approaches with the goal to account for short-range ion-ion interactions and correlation effects. 31–39 Among those are also approaches that supplement the lattice gas (or models based on more general underlying equations of state 40 ) with mean-field expressions 41–44 of short-ranged interactions between the ions. Here, ion-ion interactions are accounted for on the level of the Bragg-Williams free energy 45 with its underlying random mixing approximation (RMA). Depending on the level of sophistication, these models are able to predict damped oscillating potentials 34,40,43,44 with a characteristic decay length ξ much larger than the ion size. However, oscillating potentials only appear if the short-range 4


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interactions favor like-charged ions over ions of different charge. This is opposite to what is expected if electrostatic interactions dominate on short length scales. A conceptually different approach to address the origin of “underscreening” emerges from the consideration of complexes between oppositely charged ions, so-called Bjerrum pairs. If, hypothetically, all ions of the ionic liquid engaged in the formation of such pairs, only a few remaining counterions would be able to form a diffuse layer – with a correspondingly large screening length. Bjerrum pair formation is a consequence of short-range ion-ion correlations, which are most pronounced in solvent-free ionic liquids. Hence, ionic liquids were postulated to exhibit properties of dilute electrolyte solutions, 46 and subsequent theoretical studies have characterized the nonmonotonic behavior of the relevant screening length as function of electrolyte concentration: 47–50 large for dilute electrolytes and large again for concentrated electrolytes, including ionic liquids. In the present work, we study the influence of ion-ion correlations on the differential capacitance of planar and weakly curved electrodes. To this end, we apply the quasi-chemical approximation (QCA) 45,51 to the lattice gas model of a compact ionic liquid. QCA considers only correlations between pairs of ions, which is reminiscent of the above-mentioned Bjerrum pair formation. We derive an expression for C¯dif f , including its first-order dependence on the electrode curvature. For increasing correlation strength, the characteristic decay length ξ of the diffuse double layer grows and, as a consequence, C¯dif f undergoes a bell-to-camel shape transition. The first-order curvature correction of C¯dif f exhibits the same transition. Rather than being a consequence of adding solvent or decreased ion compacity, the bell-to-camel shape transition of C¯dif f exclusively reflects short-ranged ion-ion interactions. In addition, the bell-to-camel shape transition is absent if short-range ion-ion interactions are described on the basis of RMA instead of QCA. Hence, we argue that differences in the shape of C¯dif f for solvent-free ionic liquids can be regarded as emerging from different ion-ion correlation strengths as determined by the Bjerrum length, ion size, and additional short-ranged nonelectrostatic ion-ion interactions. Our finding that the ion-ion correlations strength can cause

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the bell-to-camel shape transition of the differential capacitance in ionic liquids constitutes the main result of the present work.

Theory We first apply QCA to a uniform, solvent-free, compact ionic liquid that consists of N monovalent ions contained in a volume V . No electrode is present, implying that we model bulk properties. To describe the ionic liquid we employ a lattice model, where each lattice site can be occupied by either an anion (index “1”) or a cation (index “2”). The fixed fractions φ1 and φ2 of lattice sites occupied by anions and cations, respectively, obey the relation φ1 +φ2 = 1. QCA incorporates nearest-neighbor interactions by assuming that pairs of nearest-neighbors are statistically uncorrelated. 45,51 The total number of pairs supported by the lattice is N z/2, where z is the lattice coordination number. For example, z = 2 for a one-dimensional lattice, and z = 6 for a three-dimensional cubic lattice. If we denote the fraction of anion-anion pairs by φ11 , anion-cation pairs by φ12 , cation-anion pairs by φ21 , P P and cation-cation pairs by φ22 , then 2i=1 2j=1 φij = φ11 + φ12 + φ21 + φ22 = 1, and the free energy per lattice site f = F/N can be expressed as

f = (1 − z)

2 X i=1

2

2

z XX φi ln φi + (φij ln φij + ωij φij ) . 2 i=1 j=1

(3)

Note that in Eq. 3 and everywhere below we express energies in units of the thermal energy unit kB T . The ωij ’s in Eq. 3 define the anion-anion, anion-cation, cation-anion, and cation-cation pair-interactions. In the general case, only ω12 = ω21 is required by symmetry. However, targeting electrostatic correlations suggests to use

ω11 = ω22 = −ω12 = −ω21 = ω.

6


(4)

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This introduces the parameter ω ≥ 0, which characterizes the magnitude of the effective short-range Coulomb attraction between an anion and a cation. In the following, we view ω as an unspecified parameter, yet with the expected scaling ω ∼ lB /ν 1/3 . The interaction P P term 2i=1 2j=1 ωij φij in the free energy can then be expressed as −2ω(φ12 + φ21 ) plus an irrelevant constant. It is important to point out that the emergence of a single interaction strength 1 ω11 + ω22 − ω12 ω= 2 2

(5)

is also correct in the general case of distinct ω11 , ω12 , and ω22 . The results of this work therefore apply to any combinations of short-range interactions between ions, not only those for which ω11 = ω22 = −ω12 . The φi ’s and the φij ’s are not independent from each other: the two conservation relations φ1 = φ11 + φ12 and φ2 = φ21 + φ22 as well as the symmetry requirement φ12 = φ21 leave us with one single degree of freedom that we define as φ¯ = φ12 + φ21 . Hence, we can express

φ12 =

φ¯ φ¯ φ¯ φ¯ , φ21 = , φ11 = φ1 − , φ22 = φ2 − 2 2 2 2

(6)

¯ Minimization of f in Eq. 3 with respect to φ¯ then yields the directly as function of φ. condition 4ω

e

=

φ1 −

1¯ 1¯ φ φ 2 2 1¯ φ φ2 2

−

=

1¯ φ 2

φ12 φ21 , φ11 φ22

(7)

¯ Eq. 7 can be rewritten as which constitutes a quadratic equation for φ.

K = 4e4ω =

φ¯2 (φ12 + φ21 )2 = , φ11 φ22 φ11 φ22

(8)

which embodies the mass action law corresponding to the chemical “reaction” of an anionanion and cation-cation pair forming two anion-cation pairs, AA + CC 2AC, with equilibrium constant K. We finally note that the special case ω = 0 implies φ¯ = 2φ1 φ2 and thus P φ11 = φ21 , φ22 = φ22 , and φ12 = φ21 = φ1 φ2 . Inserting this into f yields f = 2i=1 φi ln φi , 7



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which demonstrates the consistence of the free energy expression in Eq. 3 with the familiar limit of a non-interacting lattice gas. Next, we use f in Eq. 3 as underlying free energy contribution in modeling the (nonuniform) EDL of an ionic liquid. To this end, we consider a planar electrode of large lateral area A and fixed surface charge density σ next to an extended ionic liquid. Let the x-axis of a Cartesian coordinate originate at the surface and point normal into the electrolyte. It is common to employ the local density approximation according to which the total free energy of the system F can be expressed as an integral over a local density. 52 The density varies along the x-direction, implying that the (dimensionless) electrostatic potential Ψ = Ψ(x), and the fractions φi = φi (x), φij = φij (x) are local, x-dependent quantities. With this we write for the total free energy per unit area of the ionic liquid 1 F = A ν

Z∞ 0

(

2

X l2 0 [Ψ (x)]2 + (1 − z) φi (x) ln φi (x) dx 2 i=1

"

2 2 z XX + φij (x) ln φij (x) − 2ω [φ12 (x) + φ21 (x)] 2 i=1 j=1 )

− µ1 φ1 (x) − µ2 φ2 (x) .

#

(9)

The first term on the right-hand side of Eq. 9 characterizes the energy stored in the electric field; the prime denotes the derivative with respect to the argument, Ψ0 (x) = dΨ/dx. The p length l = ν/(4πlB ) also appears in Poisson’s equation l2 Ψ00 (x) = φ1 (x) − φ2 (x) that the potential Ψ(x) must fulfill. The remaining terms in Eq. 9 characterize the mixing entropy and nearest-neighbor interactions of the lattice model according to QCA as discussed above. Finally, the fixed chemical potentials µ1 and µ2 maintain thermal equilibrium with the bulk, x → ∞, where φ1 (x → ∞) = φ2 (x → ∞) = 1/2 ensure electroneutrality. The free energy F contains two functional degrees of freedom. One of them characterizes the degree of anion-cation association at every location x. In analogy to Eq. 6 we use again

8


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¯ the quantity φ¯ = φ(x), defined through ¯ ¯ φ(x) φ(x) , φ12 (x) = , 2 2 ¯ ¯ φ(x) φ(x) φ22 (x) = φ2 (x) − , φ21 (x) = . 2 2 φ11 (x) = φ1 (x) −

(10)

The other degree of freedom accounts for the ability of the ionic liquid to adjust its local mole fractions φ1 (x) and φ2 (x) subject to φ1 (x) + φ2 (x) = 1. It is thus convenient to write

φ1 (x) =

1 + η(x) , 2

φ2 (x) =

1 − η(x) 2

(11)

with the function η = η(x) characterizing the second degree of freedom. ¯ η)/A subject to Poisson’s equation l2 Ψ00 (x) = η(x) we To functionally minimize F (φ, insert the relations of the φij (x) from Eq. 10 and of the φi (x) from Eq. 11 into Eq. 9. The variation of the free energy in Eq. 9 then yields δF A

Z∞ ( " δσ 1 µ1 µ2 = Ψ0 + dx δη − Ψ − − + e ν 2 2 0 1 1+η 1 1−η ln − ln + (1 − z) 2 2 2 2 # z 1 1 + η − φ¯ 1 1 − η − φ¯ + ln − ln 2 2 2 2 2 1 1 + η − φ¯ 1 1 − η − φ¯ z + δ φ¯ − ln − ln 2 2 2 2 2 ¯ ¯ 1 φ 1 φ + ln + ln − 2ω . 2 2 2 2

(12)

Fixed surface charge density σ implies the term Ψ0 δσ/e is zero and, therefore, vanishing variation δF/A = 0 demands the two terms in square brackets must equal zero. Taking the first one in the bulk, at x → ∞ where η = 0 and Ψ = 0, implies µ1 = µ2 . With that the two

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vanishing terms yield the two algebraic relations φ¯2 ¯ 2 − η2 , (1 − φ) 1 + η z 1 + η − φ¯ + ln 2Ψ = (1 − z) ln 1 − η 2 1 − η − φ¯

e4ω =

(13)

¯ that specify the functions η = η(Ψ) and φ¯ = φ(Ψ) for any given z and ω explicitly in terms of the potential Ψ. The function η = η(Ψ) enters into Poisson’s equation l2 Ψ00 (x) = η[Ψ(x)], which is subject to the two boundary conditions Ψ0 (x = 0) = −s/l and Ψ(x → ∞) = 0, where we have defined the dimensionless surface charge density s = νσ/(le) = 4πlB lσ/e. R Ψ(x) Applying the first integration of Poisson’s equation, l2 [Ψ0 (x)]2 /2 = 0 dΨ η(Ψ) to the position x = 0 and using the boundary condition Ψ0 (x = 0) = −s/l yields the explicit R 1/2 Ψ0 relation s = 2 0 dΨ η(Ψ) between the dimensionless surface charge density s and the dimensionless surface potential Ψ0 = Ψ(x = 0). The scaled differential capacitance ds |η(Ψ0 )| C¯dif f = =s dΨ0 Ψ R0 2 dΨη(Ψ)

(14)

0

then follows immediately. Eqs. 13 and 14 fully define the (scaled) differential capacitance for any given z and ω. Of course, for ω = 0 we recover the result for the non-interacting lattice gas φ¯ = 1/(2 cosh Ψ) and η(Ψ) = tanh Ψ, leading to the result in Eq. 1.

Results and Discussion The choice z = 2 for the lattice coordination number is of special interest because the underlying statistical behavior of the ions is then identical to that of spins according to the one-dimensional Ising model with an external field. The role of the external field is played by the electrostatic potential, which is determined by Poisson’s equation. We obtain from

10


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Eq. 13 the explicit solution η(Ψ) = p

sinh Ψ e4ω + sinh2 Ψ

,

(15)

which upon insertion into Eq. 14 yields for the (scaled) differential capacitance |sinh Ψ0 | . √ 2 4ω cosh Ψ0 + e +sinh Ψ0 2 e4ω + sinh2 Ψ0 ln 1+e2ω

C¯dif f = s

(16)

In the upper diagram of Fig. 1 we display plots of C¯dif f (Ψ0 ) according to Eq. 16 for different values of the interaction parameter ω. For ω = 0 we obtain the familiar bell-shape profile of

1

ω=0

z=2

0.8 C¯dif f 0.6 0.4 ω=1

0.2 1

ω=0

z=6

0.8 C¯dif f 0.6 0.4 0.2 −6

ω=1 −4

−2

0

2

4

6

Ψ0

Figure 1: Scaled differential capacitance C¯dif f as function of the dimensionless surface potential Ψ0 for lattice coordination numbers of z = 2 (upper diagram) and z = 6 (lower diagram) and different values of ω, varying from ω = 0 to ω = 1 in steps of 0.1 (black curves). The red curve in each diagram marks the bell-to-camel shape transition, adopted at ω = ωc with ωc = (1/4) ln 3 = 0.275 for z = 2 and ωc = 0.182 for z = 6. the non-interacting lattice gas model as specified in Eq. 1. Growing ω leads to a transition 11



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from bell-shape to camel-shape curves. The transition occurs at the specific interaction strength ω = ωc where the condition 0 = (d2 C¯dif f /dΨ20 )Ψ0 =0 is satisfied. Using the result in Eq. 16 we find ωc = (1/4) ln 3 ≈ 0.275. The red curve in the upper diagram of Fig. 1 shows C¯dif f (Ψ0 ) for z = 2 and ω = ωc = 0.275. Three-dimensional liquids have coordination numbers z > 2. In this case we cannot extract a closed form of the relation η(Ψ) from Eqs. 13. However, numerical determination of η(Ψ) and the ensuing C¯dif f (Ψ0 ) suggests qualitatively the same behavior as in Eq. 16. To support this claim we show in the lower diagram of Fig. 1 the relation C¯dif f (Ψ0 ) for z = 6 and different values for ω. Here too, as for any other finite value z > 0, we observe a transition of bell-shape to camel-shape curves as ω is increased. The value ωc at which the transition occurs fulfills the relation

z=

4 2−

3e2ωc

+ e6ωc

.

(17)

Fig. 2 shows ωc (z) according to Eq. 17. For z = 2 we recover ωc = (1/4) ln 3. For z = 6 the

0.3

0.25

ωc

0.2

0.15

0.1

2

4

6

8

10

12

14

16

z

Figure 2: The interaction strength ωc at which the transition of bell-shape to camel-shape behavior occurs as function of the lattice coordination number z according to Eq. 17. Note the limit ωc → 0 for z → ∞.

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prediction ωc = 0.182 corresponds to the red curve in the lower diagram of Fig. 1. Note that we have calculated Eq. 17 by expanding the function η(Ψ) = b1 Ψ + b3 Ψ3 in the vicinity of √ √ Ψ = 0 up to third order. The resulting differential capacitance C¯dif f = b1 + 3b3 Ψ20 /(4 b1 ) has a vanishing second derivative at Ψ0 = 0 for b3 = 0. Solving that equation yields z(ωc ) as specified in Eq. 17. The values of ωc predicted by Eq. 17 and plotted in Fig. 2 are substantially smaller than the bare Coulomb interaction lB /ν 1/3 between two neighboring ions. For example, ν = 1 nm3 and lB = 5 nm implies lB /ν 1/3 = 5, much larger than ωc ≈ 0.2. Although we do not know what the most realistic choice of ω is, the bare Coulomb interaction lB /ν 1/3 should provide the correct scale and serve as an upper limit. Hence, we expect shortrange electrostatic interactions are able to cause a camel-shape (or U-shape) profile of the differential capacitance. Short-range ion-ion interactions have been accounted for previously on the level of the random mixing approximation (RMA). 40,42–44 Because RMA neglects correlations whereas QCA accounts for them (albeit on the most simple level), a comparison of both models is expected to reveal the role of ion-ion correlations. Within the framework of modeling a compact ionic liquid, RMA gives rise to an underlying free energy (per lattice site) of fRM A = φ1 ln φ1 + φ2 ln φ2 + χφ1 φ2 , where χ = z[ω12 − (ω11 + ω22 )/2] is a nonideality parameter that is related to ω in Eq. 5 via χ = −2zω. As was demonstrated previously, 44 when replacing f in Eq. 3 by fRM A , the (scaled) differential capacitance becomes C¯dif f = q

1 zω −

ln(1−c2 ) c2

(18)

where the constant c is the solution of the transcendental equation Ψ0 = arctanh(c)+zωc. In Fig. 3 we compare both approaches, random mixing approximation (RMA, dashed lines) and quasi-chemical approximation (QCA, solid lines), for z = 6 and different choices of ω. When ω is sufficiently small, the two approaches make the same predictions. For RMA, increasing ω merely widens the profile of C¯dif f without changing its shape. This was also pointed out

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0.8 z=6 ω = 0.2

0.6

ω = 0.5 C¯dif f 0.4

0.2

ω=1

ω=2 0

−10

−5

0

5

10

Ψ0

Figure 3: Scaled differential capacitance C¯dif f according to RMA (dashed lines) and QCA (solid lines). Different colors correspond to different values of ω: ω = 0.2 (black), ω = 0.5 (red), ω = 1 (blue), and ω = 2 (green). The predictions for RMA are based on Eq. 18, those for QCA on Eqs. 13 and 14. All results are calculated for z = 6. by Goodwin et al 42 based on a perturbation approach for a not necessarily compact ionic liquid. In contrast, QCA predicts a bell-to-camel shape transition at ω = ωc . For large ω the differences between the two approaches become significant, with qualitatively different behaviors and, perhaps unexpectedly, the largest influence of correlations for small electrode potentials. For those, correlations decrease Cdif f whereas for higher electrode potentials, correlations tend to increase Cdif f . We emphasize that our predictions are based on a comparison of QCA with RMA. The account of three-body and higher-order correlations will further refine them. When expanding η(Ψ) = b1 Ψ only up to linear order, we obtain the linear equation l2 Ψ00 (x) = b1 Ψ(x) with b1 = 1/[1+(z/2)(e2ω −1)]. The characteristic length of the exponential solution Ψ(x) = Ψ0 e−x/ξ is l ξ= √ =l b1

r 1+

14

z 2ω (e − 1), 2


(19)

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where we recall l2 = ν/(4πlB ). Any choice ω > 0 will increase ξ. If we use for ω the bare Coulomb interaction lB /ν 1/3 between two neighboring ions, Eq. 19 reads s ξ=

ν 4πlB

z 2lB 1+ exp −1 . 2 ν 1/3

(20)

For example, ν = 1 nm3 , lB = 5 nm, and z = 6 yields ξ ≈ 30 nm, which is consistent with the experimental observation of decay lengths that can be many times larger than the molecular size of individual ions. 27 Also, our account of nearest neighbor interactions leads to an increase of ξ with increasing Bjerrum length lB . Eq. 20 predicts this increase to be exponential whereas experimental observations and a scaling model suggest a linear dependence. 53 Next, we investigate how ion-ion correlations affect the differential capacitance C¯dif f of a weakly curved electrode. To this end, we express the first-order dependence of C¯dif f on the two principal curvatures c1 and c2 of a weakly curved electrode in terms of a single dimensionless parameter τ through (0) C¯dif f (c1 , c2 ) = C¯dif f [1 + τ l(c1 + c2 )] ,

(21)

(0) where C¯dif f = C¯dif f (c1 = c2 = 0) is the scaled differential capacitance of a planar electrode.

Note that the curvatures c1 and c2 are measured at the electrode surface; positive c1 and c2 imply outward bending such that the space available to the mobile ions of the EDL increases. Negative c1 and c2 decrease that space, and saddle curvatures c1 = −c2 do not influence the differential capacitance up to first order in curvature. The differential capacitance increases if the curvature c1 + c2 and τ carry the same sign. It decreases for opposite signs. Bossa et al 54 have recently shown how the function η(Ψ) in Poisson’s equation l2 Ψ00 (x) = η[Ψ(x)] can be used to calculate the relation between the scaled surface charge density s and surface potential Ψ0 up to second order in curvature. This relation defines the derivative ds/dΨ0 ,

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including its curvature dependence, leading to  τ=

d  1  dΨ0 η(Ψ0 )

ZΨ0



ZΨ

dΨ 2 0

1/2  ¯ η(Ψ) ¯   dΨ .

(22)

0

For z = 2 the function η(Ψ) is specified in Eq. 15; this allows us to calculate τ directly from "p sinh2 Ψ0 + e4ω × sinh Ψ0 !1/2 # p ZΨ0 cosh Ψ + sinh2 Ψ + e4ω × dΨ 2 ln . 1 + e2ω

d τ = dΨ0

(23)

0

In the upper diagram of Fig. 4 we show τ for z = 2 according to Eq. 23 for different choices of ω. For larger z (that is, z > 2) we use a numerical representation of η(Ψ), extracted from Eq. 13, to compute τ as function of Ψ0 ; the result for z = 6 is displayed in the lower diagram of Fig. 4. We observe that the extent of short-range ion-ion interactions, as expressed by ω, drastically affects τ . For ω = 0 the increase of C¯dif f for outward bending (that is, positive c1 + c2 ) always grows with |Ψ0 |. Above ω = ωc , with ωc being specified in Eq. 17 and plotted in Fig. 2, the increase of C¯dif f for outward bending passes through a minimum as function of |Ψ0 |. This non-monotonic behavior becomes more pronounced with growing ω. For sufficiently large ω, outward bending even decreases C¯dif f within two regions of the surface potential that are centered symmetrically around Ψ0 = 0. Most importantly, (0)

vanishing of the two second derivatives (d2 C¯dif f /dΨ20 )Ψ0 =0 and (d2 τ /dΨ20 )Ψ0 =0 occurs at the same value, ω = ωc . This shows that, up to first order, the curvature of an electrode does not affect the camel-to-bell shape transition that is induced by short-range ion-ion interactions. Although our theoretical model is only valid up to first order in curvature, we may attempt a qualitative comparison with simulation results. Increases of C¯dif f for spherical geometry have been reported in molecular dynamics simulations of room-temperature ionic liquids 55,56 and based on classical density functional theory. 11 Specifically, Feng et al 55 as

16


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3 z=2 2

ω=1

τ 1 ω=0

0 3 z=6

ω=1

2 τ 1

0 −6

ω=0 −4

−2

0

2

4

6

Ψ0

Figure 4: Curvature expansion parameter τ (defined in Eq. 21) as function of the surface potential Ψ0 for z = 2 (upper diagram) and z = 6 (lower diagram). Different black curves correspond to different values of ω, varying from ω = 0 to ω = 1 in steps of 0.1. The red curves in each diagram mark the (inverted) bell-to-camel shape transitions, adopted (as in Fig. 1) at ω = ωc with ωc = (1/4) ln 3 = 0.275 for z = 2 and ωc = 0.182 for z = 6. well as Ma et al 11 compared planar and spherical electrodes with different radii of curvature, observing increased values and flattened shapes of C¯dif f (Ψ0 ) when the curvature increases. Both observations are qualitatively consistent with the predictions in Fig. 4. First, τ in Fig. 4 is positive, implying that C¯dif f increases for spherical goemetry with c1 = c2 > 0. Second, for small ω, where C¯dif f is bell-shaped for a flat surface, τ is small for small |Ψ0 | and large for large |Ψ0 |, implying a flattening of C¯dif f (Ψ0 ). Conversely, for large ω, where C¯dif f is camel-shaped for a flat surface, τ is large for small |Ψ0 | and small for larger |Ψ0 |, again leading to a flattened shape of the curve C¯dif f (Ψ0 ). We finally note that the increase of C¯dif f for spherical and decrease for inverse spherical geometry predicted by the positive

17



Page 18 of 26

sign of τ in Fig. 4 is also consistent with model calculations of exohedral versus endohedral supercapacitors by Huang et al. 57 Our model has introduced ω as a parameter to characterize short-range Coulomb interactions between neighboring ion pairs. The nature of this interaction suggests a positive sign for ω, but it is difficult to specify the exact magnitude because mean-field electrostatic interactions are already accounted for by the first term in the integral of Eq. 9 so that the choice lB /ν 1/3 is likely to overestimate ω. A pragmatic approach to estimate ω can be based on comparing our model’s prediction for Cdif f with that of the non-interacting lattice gas in Eq. 2. Clearly, a number of experimental results for Cdif f have been fitted in the past to the compacity parameter γ, yielding values in the range of 0.1 ≤ γ ≤ 0.3 for camel-shaped profiles of Cdif f . 20,22,23 Using the model for Cdif f according to Eq. 16 (and similarly for other

1

0.8

C¯dif f 0.6

0.4

0.2 −8

−6

−4

−2

0

2

4

6

8

Ψ0

Figure 5: Scaled differential capacitance C¯dif f according to Eq. 2 (curves colored blue) and Eq. 16 (curves colored red) for. The upper pair of curves was calculated for γ = 0.8 (blue) and ω = 0.12 (red), the middle pair for γ = 0.2 (blue) and ω = 0.89 (red), and the lower pair for γ = 0.1 (blue) and ω = 1.27 (red). choices of z) would then yield fitting parameters in the range of 0.9 ≤ ω ≤ 1.3. This is suggested by matching the models for Cdif f in Fig. 5, where the predictions from the noninteracting lattice gas model with compacity γ (see Eq. 2) are shown in blue color and those 18


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from QCA with a best-matching value of ω (see Eq. 16) in red color. Hence, we expect values on the order ω = 1.

Conclusions We have included short-range ion-ion correlations into the lattice model of a compact ionic liquid that is in contact with a planar or weakly curved electrode. Growing correlation strengths induce a transition from bell-shape to camel-shape profiles of the differential capacitance. Both types of profiles are commonly observed and have been rationalized in terms of a compacity parameter that is related to the packing density of the ions. Our present work proposes an alternative explanation that may act individually or in conjunction with the compacity of the ionic liquid. Ion-ion correlations increase the width of the EDL at small surface potentials and thus decrease the differential capacitance. For large magnitudes of the surface potential, stronger condensation of the EDL increases the differential capacitance. We point out that our model accounts for ion-ion correlations on the basis of QCA but not beyond that. However, it is possible to extend the number of mutually correlated ions that enter the QCA scheme from two to a larger number. 58 Also, the addition of void- or solvent-containing lattice sites would include the compacity of the ionic liquid and extend our present model to the entire concentration range of an electrolyte. Finally, the addition of compositional gradient terms into the free energy, Eq. 9, would lead to damped oscillating potentials that are needed to capture spatially oscillating net charge densities known as “overscreening” effects. 27,31,50

Acknowledgments G.V. Bossa acknowledges a post-doctoral fellowship from Sao Paulo Research Foundation (FAPESP, Grant No. 2017/21772-2).

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Table of Content Entry

Differential Capacitance with correlations

without correlations Electrode Potential

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The Role of Ion–Ion Correlations for the Differential Capacitance of

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