Differential Capacitance of Electrolytes at Weakly Curved Electrodes

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C: Energy Conversion and Storage; Energy and Charge Transport

Differential Capacitance of Electrolytes at Weakly Curved Electrodes Guilherme Volpe Bossa, Rachel Downing, Jacob Abrams, Bjorn K Berntson, and Sylvio May J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b10933 • Publication Date (Web): 18 Dec 2018 Downloaded from http://pubs.acs.org on December 23, 2018

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The Journal of Physical Chemistry

Differential Capacitance of Electrolytes at Weakly Curved Electrodes Guilherme Volpe Bossa,† Rachel Downing,‡ Jacob Abrams,‡ Bjorn K. Berntson,¶ and Sylvio May∗,‡ †Department of Physics, São Paulo State University (UNESP), Institute of Biosciences, Humanities and Exact Sciences, São José do Rio Preto, SP,15054-000, Brazil ‡Department of Physics, North Dakota State University, Fargo, ND 58108-6050, USA ¶Department of Mathematics, North Dakota State University, Fargo, ND 58108-6050, USA E-mail: [email protected]

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Abstract The differential capacitance characterizes the ability of an electric double layer to store energy and is thus of fundamental importance for applications that employ electrostatic double-layer capacitors to store or rapidly convert energy. It is well known that the differential capacitance depends not only on the chemical structure of the electrode and electrolyte but also on the electrode’s geometry. However, attempts to describe how the curvature of an electrode affects the differential capacitance are sporadic and have focused on a few specific geometries. In the present work we carry out a systematic expansion of the differential capacitance up to second order in electrode curvature. The expansion, which applies to a large class of underlying theoretical models for the electric double layer, leads to three parameters that we calculate generally and then exemplify for a number of commonly used mean-field models: the classical Poisson-Boltzmann model, the lattice gas model, and a model that employs the Carnahan-Starling equation of state. The curvature of an electrode affects the differential capacitance in a rather complex manner depending on the electrode charge and concentration of ions in the bulk of the electrolyte. In most cases, spherical curvature tends to increase the differential capacitance whereas saddle curvature leaves it largely unaffected or decreases it slightly.

Introduction The electric double layer (EDL) is able to store energy. Yet, a large-scale technological utilization 1,2 of this ability has only started recently with the development and optimization of supercapacitors. These capacitors typically contain carbon-based nanostructured electrodes of various geometries in direct contact with electrolyte solutions. Among the reported geometries are nanorods, nanotubes, fulleres and nano-onions. 3–5 Nanostructuring aims primarily toward increasing the specific area of an electrode. Yet, there are also secondary, curvatureand confinement-related 6 influences on the ability of the EDL to store energy. These sec-

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ondary effects are manifested, in part, through the dependence of the differential capacitance Cdif f on the curvature of the electrode. The role of curvature for the structure of an EDL and its influence on the differential capacitance addresses a fundamental question that has been investigated in the past using molecular simulations and theoretical modeling. Monte Carlo and Molecular Dynamics simulations for flat and cylindrical geometries predict oscillating ion densities as function of the distance to the electrode due to strong electrostatic correlations at large ion concentrations. 7,8 Highly curved and confined geometries then lead to an interference of EDLs and thus to oscillations of the differential capacitance as function of the confinement volume. 9 Even more, when confined in nanopores of conductive materials, ion-ion interactions are strongly diminished by image forces, which has been predicted 10,11 to allow for a “superionic state” where like-charged ions may line up thus breaking the usual Coulomb ordering. Indeed, this has recently been confirmed experimentally. 12 Simulation work also suggests that Cdif f tends to increase when the radius of a cylindrical 13 or spherical 14 nanostructure decreases or when the separation between two flat electrodes becomes smaller. 10,11,15 Theoretical models of EDLs at curved geometries operate mostly on the mean-field level. When they employ the linearized Poisson-Boltzmann model, 16,17 analytic expressions for Cdif f can be extracted for any radii of curvature. For the nonlinear Poisson-Boltzmann model, which still assumes point-like ions, numerical solutions of the electrostatic potential for curved and confined geometries have been used to predict Cdif f . 18,19 In a more detailed approach, Wang et al 20 investigated a spherical geometry using a lattice gas model with an additional Stern layer and a field-dependent dielectric permittivity. Density functional theory operates beyond the mean-field level by including ion–ion correlations. Reindl et al 21 have used that approach to calculate Cdif f for spherically curved electrodes and compared the results with predictions from the classical Poisson-Boltzmann model, finding reasonable agreement when electrode curvature, ion concentration and ion size mismatch are not too large. Clearly, virtually all previous approaches have investigated a small set of specific

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geometries as function of curvature, often ranging from small (or vanishing) to large values. 22 No approach exists yet that predicts Cdif f for small curvatures but is otherwise valid for all geometries and a diverse range of electrolyte models. In the present work we study how the differential capacitance Cdif f of an EDL depends on the curvature of the electrode. Two conditions must be fulfilled for our model to be valid: first, the curvature must be sufficiently small and, second, the theoretical description of the EDL must fit into a class of models that add electrostatic interactions on the mean-field level to an underlying electrolyte equation of state. Commonly used theoretical models for the EDL employ the ideal gas, the lattice gas, or the Carnahan-Starling equation of state; they all fit into our general framework and are used to exemplify our calculations. Apart from the two imposed conditions, our calculation of the curvature dependence of Cdif f is general. That is, it applies to an electrode of any geometry and any fixed surface charge density (or potential). Actual calculations of curvature-dependent differential capacitances require only one single input function, which is related to the osmotic pressure predicted by the underlying equation of state. The final result are three constants (denoted below τ , b, and ¯b) that fully describe the curvature dependence of Cdif f up to second order. We calculate these constants for the general case and for our three selected examples. For one specific example, the classical Poisson-Boltzmann model (which employs the ideal gas as underlying equation of state), we find closed-form analytic expressions for τ , b, and ¯b. Our concluding discussion extracts general trends from the surprisingly complex dependence of Cdif f on curvature.

Theory We consider a size- and charge-symmetric 1:1 electrolyte near an extended, weakly curved electrode that carries a charge density σ and has a corresponding surface potential Φ0 . (Although the assumption of size- and charge- symmetry is convenient, our model can easily be generalized to size- and charge- asymmetric electrolytes.) Anions and cations are present

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in the bulk of the electrolyte with fixed concentration n0 = φ0 /ν each, where ν denotes the effective excluded volume occupied by each ion and 2φ0 the total volume fraction of the anions and cations. Note that we define ν such that φ0 = 1/2 is the maximal volume fraction, corresponding to a densely packed solvent-free ionic liquid. We assume the electrolyte has a uniform dielectric constant that does not depend on the local ion concentration or electric field. The dependence Φ0 (σ) determines the differential capacitance Cdif f = dσ/dΦ0 that we aim to calculate in this work. In order to formulate our model efficiently, it is convenient to cast Φ0 , σ, and ν into three new quantities: the dimensionless surface charge density s, the dimensionless surface potential Ψ0 , and a screening length l, defined through r s=σ

ν , 0 kB T

eΦ0 Ψ0 = , kB T

√ l=

ν0 kB T . e

(1)

Here, e denotes the elementary charge, 0 the permittivity of free space, kB Boltzmann’s constant, and T the absolute temperature. The differential capacitance can then be expressed as Cdif f =

0 ds . l dΨ0

(2)

We note two other lengths that often appear in models of the EDL, the Bjerrum length √ lB = e2 /(4π0 kB T ) = ν/(4πl2 ) and the Debye length lD = (8πlB n0 )−1/2 = l/ 2φ0 . We also note that the dimensionless surface potential Ψ0 is the value that the dimensionless potential Ψ = Ψ(r) = eΦ(r)/(kB T ) adopts on the surface of the electrode. Here, r denotes a position within the electrolyte, including the electrode surface, and Φ(r) is the electrostatic potential, measured in Volts. A majority of continuum models (especially mean-field models) for the EDL that have been proposed in the past give rise to a self-consistency differential equation for the dimen-

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sionless electrostatic potential Ψ = Ψ(r) of the form

l2 ∇2 Ψ = f (Ψ)f 0 (Ψ),

(3)

which involves a function f (Ψ) and its first derivative f 0 (Ψ) = df /dΨ. The symbol ∇2 on the left-hand side of Eq. 3 denotes the Laplacian. The function f (Ψ), which is required to vanish for vanishing argument, f (Ψ = 0) = 0, can be calculated from the right-hand side of Eq. 3 through v u Ψ u Z u ¯ (Ψ)f ¯ 0 (Ψ). ¯ f (Ψ) = t2 dΨf

(4)

0

For example, the self-consistency relation for the classical Poisson-Boltzmann theory is 2 ∇2 Ψ = sinh Ψ, implying the corresponding function f (Ψ) = 2(l/lD ) sinh(Ψ/2). Some lD

continuum models go beyond the local density approximation, 23 incorporate ion-ion correlations into a dielectric response function, 24 or employ a function f that contains spatial derivatives of Ψ; 25,26 these approaches do not fit into the framework of Eq. 4. Those that do fit differ in their underlying equation of state of the electrolyte. 27–30 For a size-symmetric electrolyte (where both anions and cations are of identical size and shape) it is sufficient to consider a one-component gas consisting of N particles, each of volume ν, enclosed in a fixed volume V at temperature T , where φ = νN/V is the volume fraction occupied by the particles and P the pressure they exert. The equation of state can then be cast into the form PV g(φ) =1− + g 0 (φ), N kB T φ

(5)

with a function g(φ) (and its first derivative g 0 (φ)) that describes the deviation from ideal behavior. Eq. 5 gives rise to an excess osmotic pressure 31,32 kB T ∆P (Ψ) = ν

ZΨ

¯ tanh Ψ ¯ × h−1 2φ0 eg0 (2φ0 ) cosh Ψ ¯ dΨ

0

6


(6)

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due to the presence of an applied external field Ψ. The function h−1 in Eq. 6 is the inverse 0

of the function h(φ) = φeg (φ) such that h−1 (h(φ)) = φ. The function f (Ψ) in Eq. 4 then follows from the excess osmotic pressure through 30 r f (Ψ) =

2ν ∆P (Ψ). kB T

(7)

Eqs. 6 and 7 link the function f (Ψ) to the underlying equation of state. For example, the classical Poisson-Boltzmann model is based on the ideal gas as underlying equation of state, implying g(φ) = 0 and thus h(φ) = φ. With that we obtain from Eq. 6 an excess osmotic pressure ∆P = 2n0 kB T (cosh Ψ − 1), which indeed implies f (Ψ) = 2(l/lD ) sinh(Ψ/2) as specified above. We point out that Maggs and Podgornik 30 have stated a general relationship between the excess osmotic pressure ∆P (Ψ) and the equation of state in Eq. 5; this can be used to apply our current model also to size-asymmetric electrolytes. Eq. 3 together with two appropriate boundary conditions fully specify the relationship Ψ0 (s) – and thus the differential capacitance Cdif f according to Eq. 2 – for any fixed electrode geometry. The two boundary conditions are, first, vanishing potential in the bulk of the electrolyte and, second, ∇Ψ·n = −s/l at the electrode surface, where ∇ denotes the gradient and where the unit vector n points normal to the electrode surface into the electrolyte. In the present work we focus on a weakly curved electrode, where the two principal curvatures c1 and c2 of the surface that represents the electrode are small as compared to 1/l (recall that l is the only length scale in Eq. 3). Positive c1 and c2 imply outward bending of the electrode such that the space available to the ions within the EDL increases; see the illustration in Fig. 1. We can expand the differential capacitance up to quadratic order in curvature,

Cdif f =

(0) Cdif f

b 2 2 2 ¯ 1 + τ l(c1 + c2 ) + l (c1 + c2 ) + b l c1 c2 , 2

(8)

(0)

where Cdif f is the differential capacitance for a flat electrode. The three dimensionless constants τ , b, and ¯b describe the dependence of Cdif f on curvature: τ the first-order de7



EDL

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EDL

electrode has positive curvature, c > 0

electrode has negative curvature, c < 0

Figure 1: Schematic illustration of an EDL near a curved electrode. The curvature c is measured at the electrode surface and may be identified with either one of the two principal curvatures c1 or c2 ; it is positive in the left diagram and negative in the right diagram. pendence and b as well as ¯b the second-order dependence. The second-order term has two distinct and independent contributions, one related to the mean curvature (c1 + c2 )/2 and one to the Gaussian curvature c1 c2 . We emphasize that the curvature expansion in Eq. 8 is general, subject only to the electrolyte being an isotropic fluid. Our goal in what follows is (0)

to calculate Cdif f , τ , b, and ¯b. Consider first planar geometry, c1 = c2 = 0. Here, Eq. 3 reads l2 Ψ00 (x) = f (Ψ(x))f 0 (Ψ(x)) where Ψ(x) depends only on the distance x to the electrode. Combining a first integration lΨ0 (x) = −f (Ψ(x)) with the boundary condition Ψ0 (0) = −s/l yields f (Ψ(0)) = s. Identifying Ψ(0) with the surface potential Ψ0 yields for the differential capacitance at planar geometry (0)

Cdif f =

0 0 f (Ψ0 ). l

(9)

For example, from f (Ψ) = 2(l/lD ) sinh(Ψ/2) for the classical Poisson-Boltzmann model we (0)

(0)

find immediately Cdif f = (0 /lD ) cosh(Ψ0 /2). Note that in Eq. 9, Cdif f is expressed as a (0)

function of the surface potential Ψ0 . We can also express Cdif f as a function of the (scaled) surface charge density s. To this end we invert the relationship f (Ψ(0)) = s by writing Ψ(0) = f −1 (s) where f −1 denotes the inverse function of f such that f (f −1 (s)) = s. We

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then find for the differential capacitance at planar geometry (0)

Cdif f =

0 l

1 df −1 (s)

.

(10)

ds

For example, for the classical Poisson-Boltzmann model, inverting s = f (Ψ0 ) = 2(l/lD ) sinh(Ψ0 /2) yields Ψ0 = 2arsinh(slD /2l) and thus (0)

Cdif f =

0 q, lD

(11)

where we have defined s q=

1+

s lD 2 l

2

r =

σ 2 1 + 2πlB lD . e

(12)

We will make use of the quantity q with its definition in Eq. 12 below when we discuss the classical Poisson-Boltzmann model for curved geometries. Whether it is more useful to (0)

express Cdif f as function of Ψ0 (as in Eq. 9) or as function of s (as in Eq. 10) depends on the physical conditions. Metal electrodes operate at fixed surface potential Ψ0 but a physical interpretation tends to be simpler for given s. In this work we will present our results for (0)

(0)

given s, but we point out that both representations, Cdif f as function of Ψ0 and Cdif f as function of s, have the same information content and can be converted into each other. In order to calculate τ , b, and ¯b, it is sufficient to focus on the two special cases of cylindrical (c1 = c and c2 = 0) and spherical (c1 = c2 = c) geometry. We expand the surface potential Ψ0 = ψ0 + lc ψ1 + (lc)2 ψ2 up to second order in lc. As was recently shown by Bossa et al, 31 the coefficients of this curvature-expansion can be expressed directly in terms of the

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scaled surface charge density s,

ψ0 ψ2

I = f (s), ψ1 = −n , f f 0 ψ0 =f −1 (s) 2 n2 1 d I = 0 2 f f dψ0 f f 0 ψ0 =f −1 (s)   Zψ0 1 I(Ψ)  + n(1 − n)  0 dΨ , ff f (Ψ) −1

0

(13)

ψ0 =f −1 (s)

where f = f (ψ0 ), f 0 = f 0 (ψ0 ), and I = I(ψ0 ) =

R ψ0 0

dΨf (Ψ). Eqs. 13 represent two sets

of coefficients, one for cylindrical geometry, where n = 1, and one for spherical geometry, where n = 2. Eqs. 13 enable us to calculate the curvature dependence of the differential capacitance through lCdif f 0

= +

1 dΨ0 ds

1 dψ0 ds

=

1 dψ0 ds

− lc

+

1 lc dψ ds

dψ1 ds dψ0 2 ds

(14)

2 + (lc)2 dψ " ds 2

+ (lc)2

dψ1 ds dψ0 3 ds

−

dψ2 ds dψ0 2 ds

# ,

where the expression in the second line represents an expansion of lCdif f /(0 ) up to second order for either cylindrical (c1 = c, c2 = 0, and n = 1) or spherical (c1 = c2 = c, and n = 2) geometry. To find τ , b, and ¯b, we compare Eq. 14 with Eq. 8, which reads for (0)

cylindrical geometry Cdif f = Cdif f [1 + τ lc + b(lc)2 /2] and for spherical geometry Cdif f = (0) Cdif f [1 + 2τ lc + (2b + ¯b)(lc)2 ]. A comparison of the expansion coefficients for each of the two

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geometries yields our final result

d I τ = , dψ0 f f 0 ψ0 =f −1 (s) ( 2 2 ) d I 1 d I d b = 2 , − 0 0 dψ0 f f dψ0 f f dψ0 f f 0 −1 ψ0 =f (s)    ψ 0 Z  d I(Ψ)  ¯b = 2  1 . dΨ  dψ0 f f 0 f (Ψ)  0

(15)

ψ0 =f −1 (s)

Eqs. 10 and 15 fully specify the differential capacitance according to Eq. 8 up to second order in curvature for any underlying equation of state, given it yields a self-consistency relation of the form specified in Eq. 3. Eqs. 15 constitute the principal results of the present work. In the following, we apply our general formalism to three specific examples, the classical Poisson-Boltzmann model, the non-interacting lattice gas model of the EDL, and a model of the EDL that is based on the Carnahan-Starling equation of state.

Results Classical Poisson-Boltzmann theory We start our analysis with the classical Poisson-Boltzmann theory, which treats the mobile ions on the level of an ideal gas. With f (Ψ) = 2(l/lD ) sinh(Ψ/2), f 0 (Ψ) = (l/lD ) cosh(Ψ/2), f −1 (s) = 2 arsinh(slD /2l), and I(Ψ) = 8(l/lD ) sinh2 (Ψ/4) we obtain from Eqs. 15 lD 1 1 τ = − , l q2 1 + q 2 lD 2(1 − q) 1 b = − 2 + 4 , l q3 q (1 + q)2 2 2 l 1 1 − 2q 2 D ¯b = 2 + ln , l2 q(1 + q) q 2 (1 − q 2 ) 1 + q

11


(16)


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where we recall (lD /l)2 = 1/(2φ0 ) and the definition of q in Eq. 12. The results in Eqs. 16 fully characterize the curvature-dependence of the differential capacitance for the classical Poisson-Boltzmann model, given that |c1 | 1/lD and |c2 | 1/lD . Recall that the result (0)

for a flat electrode, Cdif f , is stated in Eq. 11. Numerical values of τ , b, and ¯b require the specification of two parameters, the total bulk volume fraction 2φ0 of the anions and cations and the scaled surface charge density s of the electrode. In Fig. 2 we plot τ , b, and ¯b (see the dashed lines) according to Eqs. 16 for different choices of φ0 : 0.5 (red), 0.05 (blue), 0.01 (green), and 0.005 (black). The limit of small surface charge densities, |s| 1, is known as the Debye-H¨ uckel regime. In this 2 ∇2 Ψ = Ψ. The function limit, the Poisson-Boltzmann equation can be linearized, yielding lD √ f (Ψ) = (l/lD ) Ψ = 2φ0 Ψ is then linear. When used in Eqs. 15, this gives rise to

τ=√

1 , 8φ0

b=−

1 , 8φ0

¯b = 1 . 4φ0

(17)

Of course, Eqs. 17 can also be obtained directly from Eqs. 16 in the limit q → 1. The predictions of Eqs. 17 are marked by color-matching open diamonds on the left axis of Fig. 2. In the opposite limit, |s| 1, the classical Poisson-Boltzmann model predicts that τ , b, and ¯b all vanish; see the dashed lines in Fig. 2 in the limit of large |s|. Note however that the classical Poisson-Boltzmann model, which assumes point-like ions and thus ignores ion excluded volume effects, cannot be valid for large surface charge densities.

Non-interacting lattice gas model In order to predict Cdif f for large surface charge densities of the electrode, the steric size of the ions must be accounted for. One of the most commonly used methods to do so is based on a lattice gas model that has been developed by multiple groups; 33–37 for a summary about its history see Kornyshev 38 or Bazant et al. 39 The model employs a simple lattice statistics

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6 A 4 τ

2 0 −2 B 20

b

0

−20

60 C 40 ¯b

20 0 −20 0.01

0.1

1

10

|s|

Figure 2: The constants τ (diagram A), b (diagram B), and ¯b (diagram C) as function of the scaled surface charge density |s|. Dashed lines correspond to the classical PoissonBoltzmann results (Eqs. 16) and solid lines to the lattice gas model (based on Eqs. 18 and 20). Different colors represent φ0 = 0.5 (red), 0.05 (blue), 0.01 (green), and 0.005 (black). The color-matching open diamonds (“”) on the left axes mark the limit of small |s|. The black open circles (“◦”) describe the limit of large |s|, specified in Eqs. 21. of non-interacting particles that gives rise to the equation of state 40 PV 1 = − ln(1 − φ). N kB T φ 13


(18)


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Although the lattice gas model underestimates the pressure for spherical particles, it nev(0)

ertheless has become a valuable tool because it makes an analytic prediction for Cdif f (see below in Eq. 25) and thus can easily be used to fit experimental results 41,42 or serve as starting point for extensions such as the account of different ion sizes or the presence of ionion correlations 24 and interactions. 43 The lattice gas model gives rise to a self-consistency relation for the dimensionless electrostatic potential that is often referred to as modified Poisson-Boltzmann equation 28 or Poisson-Fermi equation 38

l 2 ∇2 Ψ =

2φ0 sinh Ψ , 1 + 2φ0 (cosh Ψ − 1)

(19)

which, according to Eq. 4, corresponds to

f (Ψ) =

p 2 ln[1 + 2φ0 (cosh Ψ − 1)],

and the inverse function f −1 (s) = arcosh[1 + (es

2 /2

(20)

− 1)/(2φ0 )]. Of course, f (Ψ) can also be 0

computed directly from g(φ) = φ + (1 − φ) ln(1 − φ) and h(φ) = φeg (φ) = φ/(1 − φ) using Eqs. 6 and 7. The predictions of the lattice gas model for τ , b, and ¯b are shown by the solid lines in Fig. 2 for the same set of φ0 -values as are displayed by the dashed lines for the classical PoissonBoltzmann model. As expected, the lattice gas model and the classical Poisson-Boltzmann model agree in their predictions for small |s|, and the agreement extends substantially beyond the Debye-H¨ uckel regime, |s| 1. For large |s|, the lattice gas model approaches a universal behavior that is independent of φ0 . We can characterize this behavior by inserting into p Eqs. 15 the function f (Ψ) = 2|Ψ|, valid according to Eq. 20 in the limit |Ψ| 1. This gives rise to τ = s,

2 b = − s2 , 3

¯b = 2 s2 , 3

(21)

which reflects the formation of densely packed layers of counterions near a highly charged

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electrode. The diffuse part of the EDL becomes irrelevant in this case. We have marked the limiting behavior according to Eqs. 21 in Fig. 2 by a sequence of black open circles. The behavior of τ , b, and ¯b, exhibited in Fig. 2 is quite intricate. According to the classical Poisson-Boltzmann model (see Eqs. 16), τ switches from positive to negative values √ √ at |s| = 2 φ0 [2(1 + 5)]1/4 . When the steric size of the ions is accounted for on the basis of the lattice gas model, τ eventually switches back to positive sign, implying that it is negative for an intermediate range of |s|. In fact, for concentrated electrolytes with φ0 & 0.02 the coefficient τ remains positive for all |s|. We can make an attempt to rationalize the behavior of τ , b, and ¯b by bending a parallel-plate capacitor. To this end, we impose curvatures c1 and c2 on one of the two plates, thereby keeping the other plate as a parallel surface at fixed distance lD . The first plate, that of curvatures c1 and c2 and fixed lateral area A, represents the curved electrode. The other plate represents the diffuse ion cloud; it has curvatures 2 c1 /(1 + c1 lD ) and c2 /(1 + c2 lD ) and a lateral area A[1 + (c1 + c2 )lD + c1 c2 lD ]. Keeping the

distance lD fixed (that is, independent of curvature) is an ad hoc assumption, motivated (0)

by the Debye-H¨ uckel prediction Cdif f = 0 /lD (which coincides with the capacitance of a parallel-plate capacitor with a plate-to-plate distance lD ). A simple calculation yields for the capacitance for the weakly curved parallel-plate capacitor

Cdif f

2 2 lD lD lD 0 2 1 + (c1 + c2 ) − (c1 + c2 ) + c1 c2 . = lD 2 12 3

(22)

√ √ With lD /l = 1/ 2φ0 this entails τ = 1/ 8φ0 , b = −1/(12φ0 ), and ¯b = 1/(6φ0 ). The value for τ coincides with that in Eqs. 17, but the values for b and ¯b do not. Hence, a parallelplate capacitor with a fixed plate-to-plate distance lD serves as an adequate model for the Debye-H¨ uckel regime of an EDL only up to first order in curvature, but not beyond that. Nevertheless, the simple parallel-plate capacitor offers an explanation for the positive sign of τ : imposing positive curvature (c1 + c2 > 0) on the plate that represents the electrode increases the lateral area of the plate that represents the diffuse ion cloud by a factor of

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1+(c1 +c2 )lD up to linear order in curvature. The larger lateral area implies a smaller surface charge density and thus a smaller potential difference across the two plates. Decreasing the potential difference for fixed surface charge density at the plate that represents the electrode leads to a larger capacitance as compared to the flat capacitor. Hence, τ must be positive in the Debye-H¨ uckel regime, where |s| 1 is small. An analogous argument can also be used for |s| 1 when the excluded volume of the ions leads to the formation of a capacitor with one charged plate and a uniform volume charge density representing the counterions. In fact, analyzing this capacitor leads directly to our result in Eqs. 21. We finally note that the result for ¯b in diagram C of Fig. 2, which is the leading curvature contribution for saddle geometry c1 = −c2 , is similarly complex as that for τ : positive for small and large |s|, but negative for an intermediate region if φ0 is sufficiently small. The dominating positive sign of ¯b can be rationalized by the decrease of the area a(d) = a0 (1 − c2 d2 ) of a parallel surface located at distance d away from an electrode-representing surface of area a0 that has a saddle curvature c = c1 = −c2 . The smaller area causes an increase in the distance between the two surfaces of an EDL-representing parallel-plate capacitor upon imposing a small saddle curvature to one of its plates. This entails a decrease in Cdif f , given that ¯b is positive. (Below, in the lower diagram of Fig. 4 we indeed observe a tendency of Cdif f to slightly decrease for saddle curvature.)

Model based on Carnahan-Starling equation of state As pointed out above, the pressure predicted by a lattice gas according to Eq. 18 is smaller than that observed in simulations of spherical particles. A more accurate equation of state for an ensemble of spherical particles is the Carnahan-Starling equation of state 44,45 1 + φ + φ2 − φ3 PV = , N kB T (1 − φ)3

16


(23)

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0

which implies g(φ) = φ2 (4−3φ)/(1−φ)2 . Here, the inverse h−1 of the function h(φ) = φeg (φ) cannot be expressed in a closed form. Nevertheless, f (Ψ) can be calculated numerically using Eqs. 6 and 7, which enables us to compute τ , b, and ¯b for any given s and φ0 . Fig. 3 compares τ , b, and ¯b as determined by the underlying Carnahan-Starling equation of state (Eq. 23, solid lines) and the lattice gas equation of state (Eq. 18, dashed lines). Curves of different color correspond to different choices of φ0 , with the same color code as in Fig. 2. Note that for the lattice gas model (see Eqs. 19 and 20) the effective volume of an ion equals the volume of a single lattice cell, whereas for the Carnahan-Starling equation of state the ion volume refers to the volume of a spherical particle. To make the comparison between these two approaches meaningful, we have therefore adjusted the lattice gas model such that each mobile ion occupies a volume fraction α = π/6 of a cubic lattice site. 31 Hence, we replace Eq. 20 by f (Ψ) =

p

2α ln[1 + 2φ0 (cosh Ψ − 1)/α].

(24)

The dashed lines in Fig. 3 are based on using Eq. 24 with α = π/6, and the solid lines on a numerical representation of f (Ψ) as calculated using Eq. 23. Both models lead to the same predictions in the limits of |s| 1 and |s| 1. Indeed the details of the underlying equation of state do not matter in the limits of infinite dilution or close packing. For intermediate |s| the model based on the Carnahan-Starling equation of state predicts larger τ , smaller b, and larger ¯b as compared to the lattice gas model. The variations of τ , b, and ¯b are generally smaller for the Carnahan-Starling equation of state.

Direct comparison of different models Because of the complexity that we observe for τ , b, and ¯b in Figs. 2 and 3, we find it beneficial to compare Cdif f directly for differently curved electrode geometries. In Fig. 4 we display predictions of lCdif f /(0 ) for a spherical electrode with lc1 = lc2 = 1/10 (red curves), for a cylindrical electrode with lc1 = 1/10 and lc2 = 0 (green), for a planar electrode

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6 A 4 τ

2 0 −2 B 20

b

0

−20

60 C 40 ¯b

20 0 −20 0.01

0.1

1

10

|s|

Figure 3: The constants τ (diagram A), b (diagram B), and ¯b (diagram C) as function of the scaled surface charge density |s|. Dashed lines correspond to the lattice gas equation of state (Eq. 18, which entails Eq. 24) and solid lines to the Carnahan-Starling equation of state (Eq. 23). Different colors represent φ0 = 0.5 (red), 0.05 (blue), 0.01 (green), and 0.005 (black). The color-matching open diamonds (“”) on the left axes mark the limit of small |s|. The black open circles (“◦”) describe the limit of large |s|, specified in Eqs. 21. with c1 = c2 = 0 (black curves), and for a saddle-shaped electrode with lc1 = −lc2 = 1/10 (lightblue curves). Upper and lower diagrams of Fig. 4 represent predictions according to the

18


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1.2 1

lattice gas

0.8 lCdif f 0.6 0 0.4 0.2 0 1.2 1

CarnahanStarling

0.8 lCdif f 0.6 0 0.4 0.2 0

−2

−1

0 s

1

2

Figure 4: Scaled differential capacitance lCdif f /(0 ) as function of the scaled surface charge density s. Different colors correspond to different electrode geometries: spherical with lc1 = lc2 = 1/10 (red); cylindrical with lc1 = 1/10 and lc2 = 0 (green); planar with c1 = c2 = 0 (black); and saddle-like with lc1 = −lc2 = 1/10 (lightblue). The three sets of curves for each color correspond to φ0 = 0.5 (solid lines), φ0 = 0.05 (dashed lines), and φ0 = 0.005 (dotted lines). The upper diagram displays results predicted by the lattice gas model (see Eq. 18); the lower diagram displays predictions according to the Carnahan-Starling equation of state (see Eq. 23). lattice gas equations of state (see Eq. 18) and according to the Carnahan-Starling equation of state (see Eq. 23), respectively. The three sets of four curves in each diagram correspond to φ0 = 0.5 (solid lines), φ0 = 0.05 (dashed lines), and φ0 = 0.005 (dotted lines). For a planar electrode we observe in Fig. 4 the familiar camel- and bell-shaped curves, the former for small and the latter for large φ0 , with transition points between them at φ0 = 1/6 × α = 1/6 × π/6 = 0.0873 according to the lattice gas model and φ0 = 0.0256

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according to the Carnahan-Starling equation of state. 32 Hence, the black broken lines in the two diagrams of Fig. 4, both calculated for φ0 = 0.05, have (upper diagram) and have not (lower diagram) undergone the transition from a bell shape to a camel shape. As already mentioned above, the lattice gas model makes an analytic prediction for the differential capacitance, 38 r Cdif f

0 α = l |s|

e

s2 2α

s2 φ0 2α −1 e −1+4α s2

.

(25)

e 2α

The black curves in the upper diagram of Fig. 4 are all described by Eq. 25 with α = π/6. Typical experimental results 42 are on the order of Cdif f = 0.1 F/m2 . This agrees well with the order of magnitude predicted by Eq. 25. For example, lCdif f /(0 ) = 1 is equivalent to Cdif f = 1.0 F/m2 for = 20, ν = 1 nm3 , and T = 300 K. The same set of parameters leads to l = 0.17 nm. Hence, the radius of curvature that we have selected for all curved geometries in Fig. 4 is 1/|c| = 10l = 1.7 nm. This is not much larger than the dimension ν 1/3 ≈ 1 nm of the individual ions. Hence, the curvature-induced changes of Cdif f that we display in Fig. 4 refer to a large curvature. In other words, within the framework of our present model, even high curvatures give rise to only moderate changes of the differential capacitance. This suggests that recently observed large effects 46 of electrode porosity on Cdif f are most likely not pure curvature effects. Instead, they may result from the discreteness of the ions 10,21,47 and from spatial variations of the local dielectric constant 20,48 (which are both ignored in our continuum model), or from the appearance of image charges 10,11 and confinement effects. 9,14,49 Confinement effects apply equally to curved and flat electrodes, and they are related to the mutual overlap and interference of the EDLs originating in two apposed electrodes that are separated by a small distance. We finally summarize the different influences that spherical (red curves in Fig. 4) and saddle curvature (lightblue curves in Fig. 4) have on Cdif f . Generally, spherical curvature (and similarly for cylindrical curvature) tends to increase Cdif f whereas saddle curvature usually leads to a small decrease. There is one exception when using the lattice gas model

20


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for the EDL: for sufficiently small φ0 and an intermediate range of s (see the dotted lines in the upper diagram of Fig. 4 at around |s| = 1) we observe a decrease of Cdif f for spherical curvature and an increase for saddle-curvature.

Conclusions The main result of the present work is the second-order curvature expansion of the differential capacitance, with the expansion coefficients specified in Eqs. 15. The expansion is general in the sense that it can be applied to any underlying model of the EDL that complies with the self-consistency relation in Eq. 3. Given that is the case, the differential capacitance of a single electrode of any shape (yet, with a weakly curved surface everywhere) can be computed. We have applied our formalism to three commonly used models of the EDL, the classical Poisson-Boltzmann model, the non-interacting lattice gas model, and a model of the EDL that employs the Carnahan-Starling equation of state. In the present work we have fixed the surface charge density of the electrode, but it is also possible to consider charge regulation models. This could proceed along the lines of a recently proposed 50 general model for charge regulation at a flat electrode. Other applications such as the extension of the lattice gas model to asymmetric ion sizes 29 or the incorporation of short-range ion-ion interactions through the Bragg-Williams model 43 (or both 51 ) are straightforward. When short-range ion-ion interactions or ion-ion correlations 24 are accounted for, a new characteristic length scale that is much larger than that of mean-field models appears. This effect, which is often referred to as “underscreening”, 52 is not only in agreement with recent experimental results, 53 it will also lead to a stronger curvature dependence of Cdif f . Hence, it will be interesting to generalize the present curvature expansion of Cdif f to self-consistency relations that go beyond Eq. 3 by accounting for ion-ion correlations. An example is the fourth-order self-consistency equation proposed by Bazant et al 24 and followed-up by oth-

21



ers. 25,26,54

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

electrolyte

weakly curved electrode

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Differential Capacitance of Electrolytes at Weakly Curved Electrodes

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