Home
Search
Collections
Journals
About
Contact us
My IOPscience
A mean-field theory on the differential capacitance of asymmetric ionic liquid electrolytes
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 284103 (http://iopscience.iop.org/0953-8984/26/28/284103) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 140.117.111.1 This content was downloaded on 24/08/2014 at 15:18
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 284103 (10pp)
doi:10.1088/0953-8984/26/28/284103
A mean-field theory on the differential capacitance of asymmetric ionic liquid electrolytes Yining Han, Shanghui Huang and Tianying Yan Tianjin Key Laboratory of Metal- and Molecule-Based Material Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education), Collaborative Innovation Center of Chemical Science and Engineering (Tianjin), Institute of New Energy Material Chemistry, College of Chemistry, Nankai University, Tianjin 300071, People’s Republic of China E-mail: [email protected] Received 24 October 2013, revised 12 January 2014 Accepted for publication 20 January 2014 Published 12 June 2014
Abstract
The size of ions significantly influences the electric double layer structure of room temperature ionic liquid (IL) electrolytes and their differential capacitance (Cd ). In this study, we extended the mean-field theory (MFT) developed independently by Kornyshev (2007 J. Phys. Chem. B 111 5545–57) and Kilic, Bazant, and Ajdari (2007 Phys. Rev. E 75 021502) (the KKBA MFT) to take into account the asymmetric 1:1 IL electrolytes by introducing an additional parameter ξ for the anion/cation volume ratio, besides the ionic compressibility γ in the KKBA MFT. The MFT of asymmetric ions becomes KKBA MFT upon ξ = 1, and further reduces to Gouy–Chapman theory in the γ → 0 limit. The result of the extended MFT demonstrates that the asymmetric ILs give rise to an asymmetric Cd , with the higher peak in Cd occurring at positive polarization for the smaller anionic size. At high potential, Cd decays asymptotically toward KKBA MFT characterized by γ for the negative polarization, and characterized by ξ γ for the positive polarization, with inverse-square-root behavior. At low potential, around the potential of zero charge, the asymmetric ions cause a higher Cd , which exceeds that of Gouy–Chapman theory. Keywords: electric double layer, mean-field theory, asymmetric ions, differential capacitance, ionic liquids (Some figures may appear in colour only in the online journal)
Composed solely of ions, the electric double layer structure of a room temperature ionic liquid (IL) electrolyte has attracted considerable attention [1–4]. An invaluable experimental measurement that probes the EDL is the differential capacitance (Cd ), which measures the electrode electric potential (ϕ0 )-dependent capacitance as the dependence of EDL on ϕ manifests itself in the Cd curve. Though the Cd s of IL electrolytes are often complicated by the specific adsorption [5, 6], the ionic geometry of different ILs [7– 9], the electrode materials [10–12], the morphology and the reconstruction of electrode [13–15], etc, the general trend can often be qualitatively described within the mean-field theory (MFT) framework [1]. More importantly, MFT provides a 0953-8984/14/284103+10$33.00
clear picture for understanding the EDL structure and the electrolyte organization at the electrode interface in response to the applied ϕ0 . An interesting phenomenon of IL electrolytes is that Cd is often found to be bell-shaped or camel-shaped experimentally [6, 9–13, 15–19], computationally [5, 7, 8, 14, 20–31], by the classical density functional theory (DFT) [32–34] and theoretically [1, 35–37]. The camel-shaped Cd may be understood in the framework of the traditional Gouy–Chapman–Stern (GCS) theory [38], in which Cd increases from the potential of zero charge (PZC) until a threshold is imposed by the Helmholtz layer. On the other hand, the bell-shaped Cd is hardly explained by the GCS theory. The reason why the 1
c 2014 IOP Publishing Ltd
Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
GCS theory fails on the bell-shaped Cd is because it assumes dimensionless ions, so that ions can accumulate on the EDL with infinite local concentration in order to completely screen the electric field from the electrode. In reality, this is not the case, because ions have a finite size, and thus the local concentration is limited. The above point naturally leads to the development of a lattice–gas model for the strong electrolyte system [39, 40], which, upon applying on the IL electrolytes, leads to the celebrated Kornyshev MFT [1], which successfully explains the bell-shaped to camel-shaped Cd transition of IL electrolytes. The same Cd expression was also derived independently for a concentrated electrolyte near the charged surface by Kilic, Bazant and Ajdari [35]. Therefore, in the rest of this paper we call the above theory Kornyshev–Kilic– Bazant–Ajdari mean-field theory, or KKBA MFT [1, 35]; this takes into account the steric effects imposed by symmetric ions of equal sizes in Cd . In Kornyshev’s derivation [1], an additional parameter, compressibility (γ ), was introduced in order to account for the maximum local concentration imposed by the finite ionic size. The parameter of the same physical picture, ν, characterizing the nondiluteness, was used in Kilic, Bazant and Ajdari’s derivation [35]. The resulting Cd may be bell-shaped or camel-shaped, depending on γ > 1/3 or not [1]. As γ → 0, KKBA MFT [1, 35] reduces to the traditional Gouy–Chapman theory [38] for the dimensionless ions. It is of interest to note that by imposing γ = 1, Gouy–Chapman theory can also give a bell-shaped Cd [36], in agreement with the upper limiting case of KKBA MFT [1, 35]. In order to take into account the asymmetric ionic sizes of cations and anions, an semi-empirical potential-dependent compressibility, γ (u 0 ), is adopted [1], in which u 0 denotes the scaled dimensionless ϕ0 . γ (u 0 ) depends on two different cationic and anionic compressibilities, i.e. γ+ and γ− , by interpolating smoothly from a cation-rich layer to an anion-rich layer. Such treatment reproduces the asymmetric Cd caused by different ionic sizes reasonably well [1, 20]. In this study, we extend KKBA MFT [1, 35] by taking into account asymmetric ions with an additional parameter, ξ , for the volume ratio between anions and cations. By assuming hard spherical ions in the framework of MFT with the lattice–gas model, it is equivalent to adopting ξ and γ , or γ+ and γ− , but the former choice makes the derivation clearer. In spite of the additional complexity imposed by an additional parameter ξ , compared with KKBA MFT [1, 35] of symmetric ions, we show that an analytical expression for Cd can be derived for a 1:1 IL electrolyte. Comparisons with KKBA MFT [1, 35] and Gouy–Chapman theory [38] are also made. The derivation starts with the Helmholtz free energy of the system, i.e. [1] F = eϕ(N+ − N− ) − kT ln .
Figure 1. A two-dimensional representation of the lattice–gas
model, in which a blue circle represents a cation and a red circle represents an anion. The space is divided into lattices by black solid lines to accommodate cations, and then subdivided by blue dotted lines to accommodate anions, in order to take into account different cationic and anionic sizes. For this particular two-dimensional representation, the total number of available lattices N = 9, the number of cations N+ = 3 and the anion/cation volume ratio ξ = 1/4. Assuming this represents the bulk of the electrolyte, where the electric potential ϕ = 0, the compressibility is γ = 2N+ /N = 2/3. It should be noted that the model is actually three dimensional and ξ = 1/8 for spherical ions. Furthermore, the ionic partition is different at different layer in EDL (see equations (9) and (10) in the text).
of possible distributions of the ions. In the framework of MFT, the correlations between the individual ions are neglected, and each ion experiences a mean-field, i.e. the electric potential ϕ. In equation (1), may be estimated via two steps, i.e. = + − , as depicted in figure 1. Denoting the total number of available lattices that can be occupied by the cations by N , + = N !/N+ !(N − N+ )!. By taking into account asymmetric ions, a new parameter ξ is introduced in order to characterize the volume ratio between anion and cation, i.e. ξ=
3 V− r− = 3. V+ r+
(2)
The second expression in equation (2) is obtained by considering the anions and cations as hard spheres of radius r− and r+ , respectively, and ξ < 1 for most ILs. Thus, the number of lattices left for the anions is (N − N+ )/ξ , in which [] denotes the integer part of (N − N+ )/ξ and − = [(N − N+ )/ξ ]!/N− ! {[(N − N+ )/ξ ] − N− }!. Then
(1)
In the above equation, N+ and N− denote the number of cations and anions, respectively, ϕ is the mean-field electrostatic potential, and e denotes the magnitude of the elementary electron charge. The second term of equation (1) represents the entropic contribution to the free energy, in which k is the Boltzmann constant, T is the temperature and is the number
=
N! [(N − N+ )/ξ ]! . N+ !(N − N+ )! N− !{[(N − N+ )/ξ ] − N− }!
(3)
It is notable that equation (3) reduces to = N !/N+ !N− !(N − N+ − N− )! for the case of ξ = 1, in which cations and anions are the same in size [1]. Using the Stirling approximation, i.e. 2
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
ln N ! ≈ N ln N − N for N 1, we have
Using equations (2) and (12), the different compressibilities of cation and anion can be recast as
ln = N ln N − N+ ln N+ − (N − N+ ) ln(N − N+ ) N − N+ N − N+ − N− ln N− + ln ξ ξ N − N+ N − N+ − − N− ln − N− . ξ ξ
γ+ = γ (4)
∂F ∂ N+
= eϕ − kT ln(N − N+ ) − ln N+ 1 N − N+ 1 N − N+ + ln (5) − N− − ln ξ ξ ξ ξ ∂F µ− = ∂ N− N − N+ = − eϕ − kT ln − N− − ln N− . (6) ξ
h i1 ξ exp(u)+η ξ −1 exp(−u) − exp(u) ξ +η 2ec0 = i 1 . (15) h γ ξ exp(−u) + (ξ + η) ξ exp(u)+η ξ +η Despite the complex form of equation (15), it is notable that by taking ξ to be 1, the expression becomes [1, 35] ρ(u) = −2ec0
N− 2 c− = = c0 N0 γ
exp(u)
h
ξ exp(u)+η ξ +η
exp(−u) + (ξ + η)
h
(9)
(10)
ξ
ρ(u) ∼ 2 2 = = ec0 γ γ+ ρ(u) ∼ 2 2 lim =− . =− u→∞ ec0 ξγ γ− u→−∞
(11)
and compressibility γ [1] and porosity η are defined as 2N0 N 2 η = − 1 − ξ. γ
γ=
(16)
(17)
lim
where the dimensionless potential u is eϕ u= kT
.
Due to the complex expression of equation (15), a clear physical picture is lost. Thus, it deserves some close inspection. Figure 2 shows the dimensionless ρ(u)/ec0 (black), calculated at different levels of theory, for two quite extreme cases of γ = 1, ξ = 0.5 in figure 2(a) and γ = 0.1, ξ = 0.1 in figure 2(c), respectively. Figures 2(b) and (d) show the same plots as in figures 2(a) and (c) for low u around zero. Also shown are the ρ(u) of KKBA MFT [1, 35] (equation (16)) for γ = 1 (red) and γ = 0.5 (blue) in figures 2(a) and (b), as well as for γ = 0.1 (red) and γ = 0.01 (blue) in figures 2(c) and (d). ρ(u) of Gouy–Chapman theory (equation (17)) is shown by a green line for comparison. Due to the asymmetry of the ions, the main difference is that equation (15) is no longer an odd function like equations (16) and (17). It can be seen clearly that equation (15) goes asymptotically toward equation (16) for γ+ = γ at the negative branch, and γ− = ξ γ at the positive branch. The high potential limit of equation (15) gives
ξ
i1
1 + 2γ sinh2 (u/2)
ρ(u) = −2ec0 sinh(u).
i 1 −1
ξ exp(u)+η ξ +η
sinh(u)
Thus, as ξ = 1, equation (15) is consistent with KKBA MFT [1, 35], i.e. equation (16), which further reduces to Gouy–Chapman theory for the γ = 0 case, i.e.
Rewriting in terms of ionic concentration, and denoting the average ionic concentration in the bulk of the IL electrolyte as c0 , the potential-dependent ionic concentration, c+ and c− , may be written as exp(−u) h i1 ξ exp(−u) + (ξ + η) ξ exp(u)+η ξ +η
(14)
ρ(u) = e(c+ − c− )
For an equilibrium system, the thermodynamic relationship is that the chemical potential of each component is equal everywhere. Thus, we can equalize the chemical potential of each kind of ion at a given potential to that in the bulk of the IL electrolyte where ϕ = 0 and N+ = N− = N0 , i.e. N+ N − N+ − ln eϕ − kT ln N − N0 N0 1 N − N+ − ξ N− 1 N − N+ + ln − ln =0 (7) ξ N − N0 − ξ N0 ξ N − N0 N− N − N+ − ξ N− − ln = 0. (8) − eϕ − kT ln N − N0 − ξ N0 N0
c+ N+ 2 = = c0 N0 γ
γ− = ξ γ
because N in equation (12) is the total number of available lattices that can be occupied by the cations. In that sense, γ− = ξ γ is the compressibility of the anions for the hard spherical ions. Therefore, η is proportional to the number of empty sites. With the ionic concentration c0 already known, the potential-dependent charge density distribution may be written as
Thus, the chemical potential may be written as µ+ =
and
(18)
Such lattice saturation or ‘crowding’ from steric effects [1, 35, 37, 39] as in figures 2(a) and (c) distinguishes equations (15) and (16) from Gouy–Chapman theory in equation (17) which grows infinitely with u. For small γ , both equations (15) and (16) are close to equation (17) at low u, as shown in figure 2(d).
(12) (13) 3
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
Figure 2. The dimensionless ρ(u)/ec0 , calculated at different levels of theory: (a) and (b) equation (15) with γ = 1, ξ = 0.5 (black), equation (16) with γ = 1 (red) and γ = 0.5 (blue), equation (17) (green); (c) and (d) equation (15) with γ = 0.1, ξ = 0.1 (black), equation (16) with γ = 0.1 (red) and γ = 0.01 (blue), equation (17) (green).
It is of interest to note that at low u, ρ(u) of equation (15) exceeds that of equation (16) and even equation (17). Such behavior can be clearly seen in figure 2(b), and the same trend is retained in figure 2(d). Thus, the asymmetric ions not only cause the asymmetric ρ(u), but also alter it at low u. Approximated to the first order, the charge density at low u is 1−ξ ρ(u) ∼ u. (19) lim =− 2+ u→0 ec0 ξ +η
in which L D is the Debye length given by s εε0 kT LD = 2e2 c0
and using equations (11), (15) and (22), equation (20) may be written as h i1 ξ exp(u)+η ξ −1 2 exp(−u) − exp(u) ξ +η d du 2 =− i1 . h du dX γ ξ exp(u)+η ξ exp(−u) + (ξ + η) ξ +η
Note that ξ + η = 2/γ − 1 by equation (13), and this effect is amplified as ξ → 0 and γ → 1, as comparing figures 2(b) and (d) shows. Fixing γ , the charge density at low u is enhanced for small ξ . As ξ → 1, equation (19) becomes −2u, i.e. the first order approximation of equations (17) and (18). The result shown in equation (19) is interesting, because the charge density of the asymmetric ions of finite size may well exceed that of dimensionless ions. Such a feature does not exist for ions of equal size—as can be seen, the absolute charge density of equation (16) can never exceed that of equation (17). We may further work out the potential by solving the one-dimensional Poisson’s equation d2 ϕ 1 d = dx 2 2 dϕ
dϕ dx
2
ρ(ϕ) =− εε0
(23) Integrating equation (23) and using the fact that u = 0 and du/dX = 0 in the bulk where X → ∞, we have s du 2 = −sgn(u) dX γ v " u 1 # u ξ exp(u) + η ξ γ t × ln exp(−u) + (ξ + η) + ln (24) ξ +η 2
in which sgn denotes the sign function. Note that the term ln(γ /2) in the square root of equation (24) is fixed by the boundary condition of du/dX = 0 as u = 0. With ξ = 1, the above equation becomes KKBA MFT, i.e. [1, 35] s r h u i du 2 = −sgn(u) . (25) ln 1 + 2γ sinh2 dX γ 2
(20)
in which ε0 and ε denote the permittivity of free space and the dielectric constant, respectively, and x denotes the distance to the electrode surface where x = 0. By recasting x to a dimensionless distance with [1] X=
x LD
(22)
The above expression further reduces to the Gouy–Chapman electric field in the γ → 0 limit. Using the identity limu→0 ln
(21) 4
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
Figure 3. Electric potential profile, u(X ), at different electrode potentials u 0 , estimated by equation (27) with different γ and ξ (black),
equation (28) with different γ (red), and equation (29) (green): (a) u 0 = −0.2; (b) u 0 = 0.2; (c) u 0 = −2; (d) u 0 = 2; (e) u 0 = −20; (f) u 0 = 20.
[1 + 2γ sinh2 (u/2)] ∼ = 2γ sinh2 (u/2), equation (25) becomes u du = −2 sinh . (26) dX 2 From equations (24)–(26), the potential distributions of the imposed electrode potential u 0 for the MFT in this study, KKBA MFT [1, 35], and Gouy–Chapman theory, respectively, are given by r γ X (u) = sgn(u 0 ) 2 Z u0 du × s 1 u ξ ln exp(−u) + (ξ + η) ξ exp(u)+η + ln γ2 ξ +η
Equations (27) and (28) can be easily solved numerically, while an analytical expression is presented in equation (29) for Gouy–Chapman theory. Instead of inversing equations (27)– (29) in order to get the potential drop across the EDL, u(X ), it can be achieved by rotating X (u) by 90◦ , because X (u) is a single valued function [1]. Figure 3 shows the comparisons of u(X ) at different electrode potentials u 0 (equation (27)) for a variety of γ and ξ , with the negative u 0 on the left column and positive u 0 on the right column, due to the asymmetric ions. For comparison, the u(X ) of KKBA MFT [1, 35] (equation (28)), of different γ , as well as u(X ) of Gouy–Chapman theory (equation (29)) are also displayed. The u(X ) of equation (28) are displayed with γ+ = γ for the negative u 0 and γ− = ξ γ for the positive u 0 , respectively. In all the figures, the u(X ) of equation (28) decay more slowly than those of equation (29), though the former may be infinitely close to the latter at low u 0 . It can be seen from figures 3(a) and (b) that u(X ) of this work decays faster than that of KKBA MFT [1, 35] and Gouy–Chapman theory at such low u 0 = ±0.2, while the
(27) r Z u 0 γ du q X (u) = sgn(u 0 ) 2 u ln 1 + 2γ sinh2 tanh(u/4) X (u) = sgn(u 0 ) ln . tanh(u 0 /4)
(28) u 2
(29) 5
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
latter two gives almost identical results. Indeed, at such low u, the charge density ρ(u) from equation (15) well exceeds that from equations (16) and (17), by comparing ρ(u)/ec0 in equation (19) with the low u limit of −2u from equations (16) and (17). Thus the screening is better for equation (19), and causes a faster potential drop. Furthermore, equation (19) also indicates better screening with lower ξ by fixing γ , which is demonstrated by comparing the u(X ) of ξ = 0.5 and ξ = 0.1 in both figures 3(a) and (b). Figures 3(c) and (d) display the u(X ) at an intermediate electrode potential u 0 = ±2. It can be seen that the u(X ) of equation (27) decay faster than both equations (29) and (28) for γ− = ξ γ with the same ξ and γ at u 0 = 2, but such a trend reverses at u 0 = −2. Such asymmetric behavior may be understood in terms of the smaller anion, which screens the positive electrode potential more effectively than does the cation on the negative electrode. For a high electrode potential of u 0 = ±20, as shown in figures 3(e) and (f), equation (29) decays remarkably faster, as expected, due to the unlimited accumulation of charges in the EDL. The u(X ) of equations (27) and (28) are almost indistinguishable at u 0 = −20 in figure 3(e), which may be understood in the high potential limit of equation (18). On the other hand, equation (27) decays faster at γ = 1, ξ = 0.5, as compared to equation (28) with γ = 0.5 at u 0 = 20 in figure 3(f). Indeed, for positive polarization, the repulsion of the bulky cation by the electrode leaves more space for packing the smaller anion, resulting in more effective screening. According to Gauss’ law, the electrode charge density σ is determined by electric displacement, i.e. du dϕ = − 2ec0 L D = σ. (30) D = − εε0 dx x=0 dX X =0
Using equations (31)–(33), Cd for the MFT in this study, KKBA MFT [1, 35] and Gouy–Chapman theory, respectively, may be written as ) ( 1 ξ exp(u 0 ) + η ξ −1 Cd (u 0 ) − exp(−u 0 ) = sgn(u 0 ) exp(u 0 ) C0 ξ +η " 1 # p ξ exp(u 0 ) + η ξ × 2γ exp(−u 0 ) + (ξ + η) ξ +η −1 v " u 1 # u ξ ξ exp(u 0 ) + η γ + ln × tln exp(−u 0 ) + (ξ + η) ξ +η 2 (35)
cosh(u 0 /2) Cd (u 0 ) = C0 1 + 2γ sinh2 (u 0 /2) s 2γ sinh2 (u 0 /2) × (36) ln 1 + 2γ sinh2 (u 0 /2) u Cd (u 0 ) 0 = cosh . (37) C0 2 The above equations are the expressions of Cd at different levels of theory, in which C0 is the Debye capacitance, given by εε0 . (38) C0 = LD It can be easily verified that equation (35) becomes equation (36) when ξ = 1, and further reduces to equation (37) as γ → 0. The scaled dimensionless capacitance Cd /C0 of various γ and ξ , based on equation (35), the Cd /C0 of various γ based on equation (36), as well as that based on equation (37) are shown in the left column of figure 4. Since equation (36) gives symmetric Cd for equal size ions, Kornyshev proposed an semi-empirical treatment of potential-dependent compressibility, γ (u 0 ), by interpolating from γ− to γ+ via [1]
Using equations (24)–(26), the electrode surface charge density σ , for the MFT in this study, KKBA MFT [1, 35], and Gouy–Chapman theory, respectively, at electrode potential u 0 can be solved by s σ (u 0 ) 8 = sgn(u 0 ) ec0 L D γ v " u ξ1 # u γ ξ exp(u ) + η 0 + ln (31) ×tln exp(−u 0 ) + (ξ + η) ξ +η 2 s r h u i σ (u 0 ) 8 0 = sgn(u 0 ) ln 1 + 2γ sinh2 ec0 L D γ 2 u σ (u 0 ) 0 = 4 sinh . ec0 L D 2
γ (u 0 ) = γ− +
dσ (ϕ0 ) . dϕ0
(39)
in which γ+ = γ and γ− = ξ γ as in equation (14). Thus, by incorporating equation (39) into equation (36), the effect of asymmetric IL is taken into account semi-empirically, and the resulting Cd /C0 are plotted in figure 4 for comparison. The electrode surface charge density by the above semi-empirical treatment can be integrated numerically, i.e. Z u0 σ (u 0 ) cosh(u/2) =2 du ec0 L D 1 + 2γ (u) sinh2 (u/2) 0 s 2γ (u) sinh2 (u/2) × (40) ln 1 + 2γ (u) sinh2 (u/2)
(32) (33)
in which L D is the Debye length in equation (22). It is the σ − u 0 curve that is determined in the computer simulation. Also, it can be seen that the PZC is located at u 0 = 0 for all levels of the theory in equations (31)–(33). Experimentally, σ (u 0 ) is difficult to obtain due to uncertainty in the PZC. Instead, the differential capacitance, Cd , is measured as a function of the electrode potential ϕ0 = kT u 0 /e, i.e. Cd (ϕ0 ) =
γ+ − γ− 1 + exp(u 0 )
in which γ (u) is given by equation (39). The dimensionless surface charge densities, σ (u 0 )/ec0 L D , of various γ and ξ , at different levels of theory are shown in the right column of figure 4. A general trend in figures 4(a), (c) and (d) is that the asymmetric ionic size is reflected in asymmetric Cd curves for equation (35), as well as for equation (36) with semi-empirical
(34) 6
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
Figure 4. C d /C 0 and σ/ec0 L D at different levels of theory. (a) C d /C 0 by equation (35) with γ = 1, ξ = 0.5 (black), equation (36) with γ = 1 (yellow) and γ = 0.5 (blue), equation (36) with semi-empirical γ (u 0 ) in equation (39) for γ+ = γ = 1 and γ− = ξ γ = 0.5 (red), equation (37) (green). (b) The corresponding σ/ec0 L D by equation (31) (black dashed) and equation (40) (red dashed), Cd /C0 by equation (35) (black) and equation (36) with γ (u 0 ) in equation (39) (red) with the same parameters in (a) are also shown to highlight the asymmetric Cd profiles; (c) and (d) show the comparisons of Cd /C0 and σ/ec0 L D of γ = 0.5, ξ = 0.1 with KKBA MFT and Gouy–Chapman theory; (e) and (f) show the comparison of Cd /C0 and σ/ec0 L D of γ = 0.1, ξ = 0.1 with KKBA MFT and
Gouy–Chapman theory. The line color and corresponding equations in (c)–(f) follow (a) and (b).
∼ γ− at high positive polarization, compared with the while γ = results of KKBA MFT [1, 35] for the corresponding γ s. The above feature is also captured by Kornyshev’s semi-empirical treatment with the potential-dependent γ (u 0 ) in equation (39), which interpolates γ (u 0 ) from the cation-rich layer to the anion-rich layer. At low electrode potential u 0 , the Cd of Gouy–Chapman theory in equation (37) imposes the upper limit of KKBA MFT [1, 35] in equation (36) with either constant γ or with semi-empirical potential-dependent γ (u 0 ) in equation (39). On the other hand, this is not the case for equation (35). With decreasing γ , both equations (35) and (36) approach asymptotically to equation (37) from the upper and lower sides. At the low u 0 limit around PZC, equation (35) may be approximated, up to second order, as
γ (u 0 ) in equation (39). The above property is correlated with the parameter ξ , with higher asymmetry the more ξ deviates from unity, while both equations (36) and (37) give symmetric Cd for symmetric ions. The result of the asymmetric Cd curve is in good agreement with DFT [33, 34] and simulations [20]. Besides, the Cd curves by equations (35) and (36) with semi-empirical γ (u 0 ) agree well in the region of high polarization. For a high electrode potential wing, it is obvious that equation (35) becomes 1 1 √ =√ , for high positive u 0 Cd ∼ 2γ− u 0 2ξ γ u 0 (41) = 1 1 C0 √ , for high negative u 0 . =√ 2γ+ |u 0 | −2γ u 0 Such inverse-square-root behavior resembles Kornyshev’s derivation [1]. It is reasonable to see that at high electrode potential, Cd is exclusively determined by the single γ of the counter ion, so that γ ∼ = γ+ at high negative polarization,
Cd = 30 + 31 u 0 + 32 u 20 + O(u 30 ) C0 7
(42)
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
In this study, we extended the mean-field theory (MFT) developed independently by Kornyshev [1] and Kilic, Bazant and Ajdari [35] (KKBA MFT) to take into account asymmetric 1:1 IL electrolytes, by introducing an additional parameter ξ for the anion/cation volume ratio. The MFT of asymmetric ions becomes KKBA MFT [1, 35] upon ξ = 1, and further reduces to Gouy–Chapman theory in the γ → 0 limit. The result of the extended MFT in this work demonstrates that the asymmetric ILs result in asymmetric Cd , with the higher peak in Cd occurring at positive polarization for the smaller anionic size. At high potential, Cd grows asymptotically toward KKBA MFT characterized by γ for negative polarization, and characterized by ξ γ for positive polarization. At low potential around the potential of zero charge, the asymmetric ions cause a higher Cd , which exceeds that of Gouy–Chapman theory. The current study also shows reasonable agreement between the MFT in this study and the semi-empirical treatment of asymmetric ions by interpolating a potential-dependent compressibility γ from γ− to γ+ for anions and cations [1], especially for smaller γ of camel-shaped Cd , while the deviation becomes apparent for larger γ of bell-shaped Cd . Finally, we should note that the result for MFT at low electrode potential should be viewed with caution. It is found in many computer simulations [5, 20–25, 28–31, 41–45] and experiments [46–51] that charge density oscillates around zero near a charged electrode of low potential characterized by distinctive ionic layers, i.e. the overscreening effect. However, the current MFT does not capture the overscreening effect, but manifests it in an average manner, as inherent from KKBA MFT [1, 35]. On the other hand, the lattice saturation or ‘crowding’ of Cd at high electrode potential, characterized by the inverse-square-root behavior [1] due to the steric effects imposed by the lattice–gas model [39, 40], is indeed captured and is in good agreement with KKBA MFT [1, 35]. A recent theory [37] successfully reproduces both overscreening at low potential and ‘crowding’ at high potential by taking into account the local ionic correlation, and thus is beyond the MFT framework.
in which s
1 1−ξ 2 ξ +η 1 1 (1 − ξ ) 31 = η+ 330 γ (ξ + η)2 1 P(γ , ξ ) 32 = − 4 9630 γ (ξ + η)3 (η + 2/γ ) P(γ , ξ ) = 384 − 1920γ + 2880γ 2 − 1920γ 3 30 =
1+
(43)
+ 600γ 4 − 72γ 5 + (768γ − 2304γ 2 + 2208γ 3 − 840γ 4 + 108γ 5 )(1 − ξ ) + (672γ 2 − 1104γ 3 + 502γ 4 − 54γ 5 )(1 − ξ )2 + (240γ 3 − 172γ 4 + 9γ 5 )(1 − ξ )3 + 28γ 4 (1 − ξ )4 . It is of interest to note that as u 0 = 0, 30 ≥ 1, so that Cd /C0 may exceed 1, the value that is predicted by KKBA MFT [1, 35], the semi-empirical γ (u 0 ) by equation (39) [1] and Gouy–Chapman theory at the PZC. For small u 0 around the PZC, the situation is quite complicated. However, we note that as ξ = 1, equation (42) reduces to Kornyshev’s derivation, i.e. [1] 1 − 3γ 2 Cd =1+ u0. C0 8
(44)
Thus, the transition from bell-shaped to camel-shaped Cd /C0 at γ = 1/3 is apparent, as denoted by Kornyshev for the ξ = 1 case. Also, it can be seen that equation (44) never exceeds the low u 0 limit of Gouy–Chapman theory of Cd /C0 = 1 + u 20 /8. On the other hand, for ξ < 1 as the MFT in this study, the situation is more complicated. Note that ξ + η = 2/γ − 1 by equation (13), Cd /C0 is amplified as ξ → 0 and√γ → 1, and in that limiting case, the maximum Cd /C0 is 3/2 at the PZC. Fixing γ , the Cd /C0 at low u is enhanced for small ξ , consistent with enhanced charge density as discussed in equation (19). As can be seen, especially for high γ cases in figures 4(a) and (c), Cd of equation (35) well exceeds that of equation (37) around the PZC. The higher Cd at low u 0 is manifest in the σ (u 0 ) curve; this can be seen in the slightly higher charge density in the positive branch of equation (31) than that of equations (32) and (40), as shown clearly in figures 4(b) and (d), though the corresponding Cd at high u 0 of equation (35) is slightly lower than that of equation (36) as well as that with semi-empirical γ (u 0 ) in equation (39) in figures 4(a) and (c). In all cases, σ (u 0 ), estimated via equation (40), with γ (u 0 ) in equation (39), is lower than that by equations (31) and (32), probably due to the over-mixing of higher γ+ in the positive polarization. With very low γ , Cd and σ (u 0 ) curves are almost indistinguishable for equation (35) and Kornyshev’s semi-empirical treatment, i.e. equation (36), with γ (u 0 ) in equation (39), as shown in figure 4(e). The same trend is also reflected in σ (u 0 ), as shown in figure 4(f) for equations (31) and (40). Such better agreement at smaller γ of Cd and σ (u 0 ) may be attributed to the fact that smaller γ gives camel-shaped Cd for both cation and anion, which gives a smaller deviation near the PZC, and the deviation becomes apparent for the larger γ of bell-shaped Cd .
Acknowledgments
This work is supported by NSFC (21373118, 21073097), Natural Science Foundation of Tianjin (12JCYBJC13900), NCET-10-0512 and the MOE Innovation Team (IRT13022) of China. We are grateful to an anonymous referee for naming the term η porosity. References [1] Kornyshev A A 2007 Double-layer in ionic liquids: paradigm change? J. Phys. Chem. B 111 5545–57 [2] Su Y Z, Fu Y C, Wei Y M, Yan J W and Mao B W 2010 The electrode/ionic liquid interface: electric double layer and metal electrodeposition Chem. Phys. Chem. 11 2764–78 and references cited therein [3] Merlet C, Rotenberg B, Madden P A and Salanne M 2013 Computer simulations of ionic liquids at electrochemical interfaces Phys. Chem. Chem. Phys. 15 15781–92 8
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
[4] Jiang D-e and Wu J 2013 Microscopic insights into the electrochemical behavior of nonaqueous electrolytes in electric double-layer capacitors J. Phys. Chem. Lett. 4 1260–7 [5] Si X, Li S, Wang Y, Ye S and Yan T 2012 Effects of specific adsorption on the differential capacitance of imidazolium-based ionic liquid electrolytes Chem. Phys. Chem. 13 1671–6 [6] Zhang Q, Han Y, Wang Y, Ye S and Yan T 2014 Comparing the differential capacitance of two ionic liquid electrolytes: effects of specific adsorption Electrochem. Commun. 38 44–6 [7] Fedorov M V, Georgi N and Kornyshev A A 2010 Double layer in ionic liquids: the nature of the camel shape of capacitance Electrochem. Commun. 12 296–9 [8] Georgi N, Kornyshev A A and Fedorov M V 2010 The anatomy of the double layer and capacitance in ionic liquids with anisotropic ions: electrostriction vs. lattice saturation J. Electroanal. Chem. 649 261–7 [9] Su Y, Yan J, Li M, Zhang M and Mao B 2013 Electric double layer of Au(100)/imidazolium-based ionic liquids interface: effect of cation size J. Phys. Chem. C 117 205–12 [10] Alam M T, Mominul Islam M, Okajima T and Ohsaka T 2007 Measurements of differential capacitance in room temperature ionic liquid at mercury, glassy carbon and gold electrode interfaces Electrochem. Commun. 9 2370–4 [11] Alam M T, Islam M M, Okajima T and Ohsaka T 2008 Capacitance measurements in a series of room-temperature ionic liquids at glassy carbon and gold electrode interfaces J. Phys. Chem. C 112 16600–8 [12] Lockett V, Horne M, Sedev R, Rodopoulos T and Ralston J 2010 Differential capacitance of the double layer at the electrode/ionic liquids interface Phys. Chem. Chem. Phys. 12 12499–512 [13] Su Y-Z, Fu Y-C, Yan J-W, Chen Z-B and Mao B-W 2009 Double layer of Au(100)/ionic liquid interface and its stability in imidazolium-based ionic liquids Angew. Chem. Int. Edn 48 5148–51 [14] Vatamanu J, Cao L, Borodin O, Bedrov D and Smith G D 2011 On the influence of surface topography on the electric double layer structure and differential capacitance of graphite/ionic liquid interfaces J. Phys. Chem. Lett. 2 2267–72 [15] Lin L G, Wang Y, Yan J W, Yuan Y Z, Xiang J and Mao B W 2003 An in situ STM study on the long-range surface restructuring of Au(111) in a non-chloroaluminumated ionic liquid Electrochem. Commun. 5 995–9 [16] Roling B, Druschler M and Huber B 2012 Slow and fast capacitive process taking place at the ionic liquid/electrode interface Faraday Discuss. 154 303–11 [17] Druschler M, Borisenko N, Wallauer J, Winter C, Huber B, Endres F and Roling B 2012 New insights into the interface between a single-crystalline metal electrode and an extremely pure ionic liquid: slow interfacial processes and the influence of temperature on interfacial dynamics Phys. Chem. Chem. Phys. 14 5090–9 [18] Baldelli S 2008 Surface structure at the ionic liquid–electrified metal interface Acc. Chem. Res. 41 421–31 [19] Islam M M, Alam M T and Ohsaka T 2008 Electrical double-layer structure in ionic liquids: a corroboration of the theoretical model by experimental results J. Phys. Chem. C 112 16568–74
[20] Fedorov M V and Kornyshev A A 2008 Ionic liquid near a charged wall: structure and capacitance of electrical double layer J. Phys. Chem. B 112 11868–72 [21] Feng G, Li S, Atchison J S, Presser V and Cummings P T 2013 Molecular insights into carbon nanotube supercapacitors: capacitance independent of voltage and temperature J. Phys. Chem. C 117 9178–86 [22] Paek E, Pak A J and Hwang G S 2013 A computational study of the interfacial structure and capacitance of graphene in BMIM PF6 ionic liquid J. Electrochem. Soc. 160 A1–A10 [23] Vatamanu J, Borodin O, Bedrov D and Smith G D 2012 Molecular dynamics simulation study of the interfacial structure and differential capacitance of alkylimidazolium bis(trifluoromethanesulfonyl)imide [Cn mim][TFSI] ionic liquids at graphite electrodes J. Phys. Chem. C 116 7940–51 [24] Vatamanu J, Borodin O and Smith G D 2011 Molecular simulations of the electric double layer structure, differential capacitance, and charging kinetics for N-methyl-N-propylpyrrolidinium Bis(fluorosulfonyl)imide at graphite electrodes J. Phys. Chem. B 115 3073–84 [25] Feng G, Jiang D-e and Cummings P T 2012 Curvature effect on the capacitance of electric double layers at ionic liquid/onion-like carbon interfaces J. Chem. Theory Comput. 8 1058–63 [26] Vatamanu J, Borodin O and Smith G D 2010 Molecular insights into the potential and temperature dependences of the differential capacitance of a room-temperature ionic liquid at graphite electrodes J. Am. Chem. Soc. 132 14825–33 [27] Hu Z, Vatamanu J, Borodin O and Bedrov D 2013 A molecular dynamics simulation study of the electric double layer and capacitance of [BMIM][PF6 ] and [BMIM][BF4 ] room temperature ionic liquids near charged surfaces Phys. Chem. Chem. Phys. 15 14234–47 [28] Fedorov M V and Kornyshev A A 2008 Towards understanding the structure and capacitance of electrical double layer in ionic liquids Electrochim. Acta 53 6835–40 [29] Feng G, Zhang J S and Qiao R 2009 Microstructure and capacitance of the electrical double layers at the interface of ionic liquids and planar electrodes J. Phys. Chem. C 113 4549–59 [30] Feng G, Huang J, Sumpter B G, Meunier V and Qiao R 2011 A ‘counter-charge layer in generalized solvents’ framework for electrical double layers in neat and hybrid ionic liquid electrolytes Phys. Chem. Chem. Phys. 13 14723–34 [31] Merlet C, Salanne M, Rotenberg B and Madden P A 2013 Influence of solvation on the structural and capacitive properties of electrical double layer capacitors Electrochim. Acta 101 262–71 [32] Wu J, Jiang T, Jiang D-e, Jin Z and Henderson D 2011 A classical density functional theory for interfacial layering of ionic liquids Soft Matter 7 11222–31 [33] Lauw Y, Horne M D, Rodopoulos T, Nelson A and Leermakers F A M 2010 Electrical double-layer capacitance in room temperature ionic liquids: ion-size and specific adsorption effects J. Phys. Chem. B 114 11149–54 [34] Jiang D-e, Meng D and Wu J 2011 Density functional theory for differential capacitance of planar electric double layers in ionic liquids Chem. Phys. Lett. 504 153–8 9
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
[35] Kilic M S, Bazant M Z and Ajdari A 2007 Steric effects in the dynamics of electrolytes at large applied voltages. I. Double-layer charging Phys. Rev. E 75 021502 [36] Oldham K B 2008 A Gouy–Chapman–Stern model of the double layer at a (metal)/(ionic liquid) interface J. Electroanal. Chem. 613 131–8 [37] Bazant M Z, Storey B D and Kornyshev A A 2011 Double layer in ionic liquids: overscreening versus crowding Phys. Rev. Lett. 106 046102 Bazant M Z, Storey B D and Kornyshev A A 2012 Phys. Rev. Lett. 109 149903 (erratum) [38] Bard A J and Faulkner L R 2000 Electrochemical Methods: Fundamentals and Applications 2nd edn (New York: Wiley) [39] Borukhov I, Andelman D and Orland H 1997 Steric effects in electrolytes: a modified Poisson–Boltzmann equation Phys. Rev. Lett. 79 435–8 [40] Borukhov I, Andelman D and Orland H 2000 Adsorption of large ions from an electrolyte solution: a modified Poisson–Boltzmann equation Electrochim. Acta 46 221–9 [41] Reed S K, Lanning O J and Madden P A 2007 Electrochemical interface between an ionic liquid and a model metallic electrode J. Chem. Phys. 126 084704 [42] Pinilla C, Del P´opolo M G, Kohanoff J and Lynden-Bell R M 2007 Polarization relaxation in an ionic liquid confined between electrified walls J. Phys. Chem. B 111 4877–84 [43] Fedorov M V and Lynden-Bell R M 2012 Probing the neutral graphene-ionic liquid interface: insights from molecular dynamics simulations Phys. Chem. Chem. Phys. 14 2552–6
[44] Lynden-Bell R M, Frolov A I and Fedorov M V 2012 Electrode screening by ionic liquids Phys. Chem. Chem. Phys. 14 2693–701 [45] Wang S, Li S, Cao Z and Yan T 2010 Molecular dynamic simulations of ionic liquids at graphite surface J. Phys. Chem. C 114 990–5 [46] Mezger M et al 2008 Molecular layering of fluorinated ionic liquids at a charged sapphire (0001) surface Science 322 424–8 [47] Hayes R, Warr G G and Atkin R 2010 At the interface: solvation and designing ionic liquids Phys. Chem. Chem. Phys. 12 1709–23 [48] Perkin S, Crowhurst L, Niedermeyer H, Welton T, Smith A M and Gosvami N N 2011 Self-assembly in the electrical double layer of ionic liquids Chem. Commun. 47 6572–4 [49] Zhang X, Zhong Y-X, Yan J-W, Su Y-Z, Zhang M and Mao B-W 2012 Probing double layer structures of Au (111)-BMIPF6 ionic liquid interfaces from potential-dependent AFM force curves Chem. Commun. 48 582–4 [50] Li H, Endres F and Atkin R 2013 Effect of alkyl chain length and anion species on the interfacial nanostructure of ionic liquids at the Au(111)-ionic liquid interface as a function of potential Phys. Chem. Chem. Phys. 15 14624–33 [51] Zhou H et al 2012 Nanoscale perturbations of room temperature ionic liquid structure at charged and uncharged interfaces ACS Nano 6 9818–27
10
Search
Collections
Journals
About
Contact us
My IOPscience
A mean-field theory on the differential capacitance of asymmetric ionic liquid electrolytes
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 284103 (http://iopscience.iop.org/0953-8984/26/28/284103) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 140.117.111.1 This content was downloaded on 24/08/2014 at 15:18
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 284103 (10pp)
doi:10.1088/0953-8984/26/28/284103
A mean-field theory on the differential capacitance of asymmetric ionic liquid electrolytes Yining Han, Shanghui Huang and Tianying Yan Tianjin Key Laboratory of Metal- and Molecule-Based Material Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education), Collaborative Innovation Center of Chemical Science and Engineering (Tianjin), Institute of New Energy Material Chemistry, College of Chemistry, Nankai University, Tianjin 300071, People’s Republic of China E-mail: [email protected] Received 24 October 2013, revised 12 January 2014 Accepted for publication 20 January 2014 Published 12 June 2014
Abstract
The size of ions significantly influences the electric double layer structure of room temperature ionic liquid (IL) electrolytes and their differential capacitance (Cd ). In this study, we extended the mean-field theory (MFT) developed independently by Kornyshev (2007 J. Phys. Chem. B 111 5545–57) and Kilic, Bazant, and Ajdari (2007 Phys. Rev. E 75 021502) (the KKBA MFT) to take into account the asymmetric 1:1 IL electrolytes by introducing an additional parameter ξ for the anion/cation volume ratio, besides the ionic compressibility γ in the KKBA MFT. The MFT of asymmetric ions becomes KKBA MFT upon ξ = 1, and further reduces to Gouy–Chapman theory in the γ → 0 limit. The result of the extended MFT demonstrates that the asymmetric ILs give rise to an asymmetric Cd , with the higher peak in Cd occurring at positive polarization for the smaller anionic size. At high potential, Cd decays asymptotically toward KKBA MFT characterized by γ for the negative polarization, and characterized by ξ γ for the positive polarization, with inverse-square-root behavior. At low potential, around the potential of zero charge, the asymmetric ions cause a higher Cd , which exceeds that of Gouy–Chapman theory. Keywords: electric double layer, mean-field theory, asymmetric ions, differential capacitance, ionic liquids (Some figures may appear in colour only in the online journal)
Composed solely of ions, the electric double layer structure of a room temperature ionic liquid (IL) electrolyte has attracted considerable attention [1–4]. An invaluable experimental measurement that probes the EDL is the differential capacitance (Cd ), which measures the electrode electric potential (ϕ0 )-dependent capacitance as the dependence of EDL on ϕ manifests itself in the Cd curve. Though the Cd s of IL electrolytes are often complicated by the specific adsorption [5, 6], the ionic geometry of different ILs [7– 9], the electrode materials [10–12], the morphology and the reconstruction of electrode [13–15], etc, the general trend can often be qualitatively described within the mean-field theory (MFT) framework [1]. More importantly, MFT provides a 0953-8984/14/284103+10$33.00
clear picture for understanding the EDL structure and the electrolyte organization at the electrode interface in response to the applied ϕ0 . An interesting phenomenon of IL electrolytes is that Cd is often found to be bell-shaped or camel-shaped experimentally [6, 9–13, 15–19], computationally [5, 7, 8, 14, 20–31], by the classical density functional theory (DFT) [32–34] and theoretically [1, 35–37]. The camel-shaped Cd may be understood in the framework of the traditional Gouy–Chapman–Stern (GCS) theory [38], in which Cd increases from the potential of zero charge (PZC) until a threshold is imposed by the Helmholtz layer. On the other hand, the bell-shaped Cd is hardly explained by the GCS theory. The reason why the 1
c 2014 IOP Publishing Ltd
Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
GCS theory fails on the bell-shaped Cd is because it assumes dimensionless ions, so that ions can accumulate on the EDL with infinite local concentration in order to completely screen the electric field from the electrode. In reality, this is not the case, because ions have a finite size, and thus the local concentration is limited. The above point naturally leads to the development of a lattice–gas model for the strong electrolyte system [39, 40], which, upon applying on the IL electrolytes, leads to the celebrated Kornyshev MFT [1], which successfully explains the bell-shaped to camel-shaped Cd transition of IL electrolytes. The same Cd expression was also derived independently for a concentrated electrolyte near the charged surface by Kilic, Bazant and Ajdari [35]. Therefore, in the rest of this paper we call the above theory Kornyshev–Kilic– Bazant–Ajdari mean-field theory, or KKBA MFT [1, 35]; this takes into account the steric effects imposed by symmetric ions of equal sizes in Cd . In Kornyshev’s derivation [1], an additional parameter, compressibility (γ ), was introduced in order to account for the maximum local concentration imposed by the finite ionic size. The parameter of the same physical picture, ν, characterizing the nondiluteness, was used in Kilic, Bazant and Ajdari’s derivation [35]. The resulting Cd may be bell-shaped or camel-shaped, depending on γ > 1/3 or not [1]. As γ → 0, KKBA MFT [1, 35] reduces to the traditional Gouy–Chapman theory [38] for the dimensionless ions. It is of interest to note that by imposing γ = 1, Gouy–Chapman theory can also give a bell-shaped Cd [36], in agreement with the upper limiting case of KKBA MFT [1, 35]. In order to take into account the asymmetric ionic sizes of cations and anions, an semi-empirical potential-dependent compressibility, γ (u 0 ), is adopted [1], in which u 0 denotes the scaled dimensionless ϕ0 . γ (u 0 ) depends on two different cationic and anionic compressibilities, i.e. γ+ and γ− , by interpolating smoothly from a cation-rich layer to an anion-rich layer. Such treatment reproduces the asymmetric Cd caused by different ionic sizes reasonably well [1, 20]. In this study, we extend KKBA MFT [1, 35] by taking into account asymmetric ions with an additional parameter, ξ , for the volume ratio between anions and cations. By assuming hard spherical ions in the framework of MFT with the lattice–gas model, it is equivalent to adopting ξ and γ , or γ+ and γ− , but the former choice makes the derivation clearer. In spite of the additional complexity imposed by an additional parameter ξ , compared with KKBA MFT [1, 35] of symmetric ions, we show that an analytical expression for Cd can be derived for a 1:1 IL electrolyte. Comparisons with KKBA MFT [1, 35] and Gouy–Chapman theory [38] are also made. The derivation starts with the Helmholtz free energy of the system, i.e. [1] F = eϕ(N+ − N− ) − kT ln .
Figure 1. A two-dimensional representation of the lattice–gas
model, in which a blue circle represents a cation and a red circle represents an anion. The space is divided into lattices by black solid lines to accommodate cations, and then subdivided by blue dotted lines to accommodate anions, in order to take into account different cationic and anionic sizes. For this particular two-dimensional representation, the total number of available lattices N = 9, the number of cations N+ = 3 and the anion/cation volume ratio ξ = 1/4. Assuming this represents the bulk of the electrolyte, where the electric potential ϕ = 0, the compressibility is γ = 2N+ /N = 2/3. It should be noted that the model is actually three dimensional and ξ = 1/8 for spherical ions. Furthermore, the ionic partition is different at different layer in EDL (see equations (9) and (10) in the text).
of possible distributions of the ions. In the framework of MFT, the correlations between the individual ions are neglected, and each ion experiences a mean-field, i.e. the electric potential ϕ. In equation (1), may be estimated via two steps, i.e. = + − , as depicted in figure 1. Denoting the total number of available lattices that can be occupied by the cations by N , + = N !/N+ !(N − N+ )!. By taking into account asymmetric ions, a new parameter ξ is introduced in order to characterize the volume ratio between anion and cation, i.e. ξ=
3 V− r− = 3. V+ r+
(2)
The second expression in equation (2) is obtained by considering the anions and cations as hard spheres of radius r− and r+ , respectively, and ξ < 1 for most ILs. Thus, the number of lattices left for the anions is (N − N+ )/ξ , in which [] denotes the integer part of (N − N+ )/ξ and − = [(N − N+ )/ξ ]!/N− ! {[(N − N+ )/ξ ] − N− }!. Then
(1)
In the above equation, N+ and N− denote the number of cations and anions, respectively, ϕ is the mean-field electrostatic potential, and e denotes the magnitude of the elementary electron charge. The second term of equation (1) represents the entropic contribution to the free energy, in which k is the Boltzmann constant, T is the temperature and is the number
=
N! [(N − N+ )/ξ ]! . N+ !(N − N+ )! N− !{[(N − N+ )/ξ ] − N− }!
(3)
It is notable that equation (3) reduces to = N !/N+ !N− !(N − N+ − N− )! for the case of ξ = 1, in which cations and anions are the same in size [1]. Using the Stirling approximation, i.e. 2
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
ln N ! ≈ N ln N − N for N 1, we have
Using equations (2) and (12), the different compressibilities of cation and anion can be recast as
ln = N ln N − N+ ln N+ − (N − N+ ) ln(N − N+ ) N − N+ N − N+ − N− ln N− + ln ξ ξ N − N+ N − N+ − − N− ln − N− . ξ ξ
γ+ = γ (4)
∂F ∂ N+
= eϕ − kT ln(N − N+ ) − ln N+ 1 N − N+ 1 N − N+ + ln (5) − N− − ln ξ ξ ξ ξ ∂F µ− = ∂ N− N − N+ = − eϕ − kT ln − N− − ln N− . (6) ξ
h i1 ξ exp(u)+η ξ −1 exp(−u) − exp(u) ξ +η 2ec0 = i 1 . (15) h γ ξ exp(−u) + (ξ + η) ξ exp(u)+η ξ +η Despite the complex form of equation (15), it is notable that by taking ξ to be 1, the expression becomes [1, 35] ρ(u) = −2ec0
N− 2 c− = = c0 N0 γ
exp(u)
h
ξ exp(u)+η ξ +η
exp(−u) + (ξ + η)
h
(9)
(10)
ξ
ρ(u) ∼ 2 2 = = ec0 γ γ+ ρ(u) ∼ 2 2 lim =− . =− u→∞ ec0 ξγ γ− u→−∞
(11)
and compressibility γ [1] and porosity η are defined as 2N0 N 2 η = − 1 − ξ. γ
γ=
(16)
(17)
lim
where the dimensionless potential u is eϕ u= kT
.
Due to the complex expression of equation (15), a clear physical picture is lost. Thus, it deserves some close inspection. Figure 2 shows the dimensionless ρ(u)/ec0 (black), calculated at different levels of theory, for two quite extreme cases of γ = 1, ξ = 0.5 in figure 2(a) and γ = 0.1, ξ = 0.1 in figure 2(c), respectively. Figures 2(b) and (d) show the same plots as in figures 2(a) and (c) for low u around zero. Also shown are the ρ(u) of KKBA MFT [1, 35] (equation (16)) for γ = 1 (red) and γ = 0.5 (blue) in figures 2(a) and (b), as well as for γ = 0.1 (red) and γ = 0.01 (blue) in figures 2(c) and (d). ρ(u) of Gouy–Chapman theory (equation (17)) is shown by a green line for comparison. Due to the asymmetry of the ions, the main difference is that equation (15) is no longer an odd function like equations (16) and (17). It can be seen clearly that equation (15) goes asymptotically toward equation (16) for γ+ = γ at the negative branch, and γ− = ξ γ at the positive branch. The high potential limit of equation (15) gives
ξ
i1
1 + 2γ sinh2 (u/2)
ρ(u) = −2ec0 sinh(u).
i 1 −1
ξ exp(u)+η ξ +η
sinh(u)
Thus, as ξ = 1, equation (15) is consistent with KKBA MFT [1, 35], i.e. equation (16), which further reduces to Gouy–Chapman theory for the γ = 0 case, i.e.
Rewriting in terms of ionic concentration, and denoting the average ionic concentration in the bulk of the IL electrolyte as c0 , the potential-dependent ionic concentration, c+ and c− , may be written as exp(−u) h i1 ξ exp(−u) + (ξ + η) ξ exp(u)+η ξ +η
(14)
ρ(u) = e(c+ − c− )
For an equilibrium system, the thermodynamic relationship is that the chemical potential of each component is equal everywhere. Thus, we can equalize the chemical potential of each kind of ion at a given potential to that in the bulk of the IL electrolyte where ϕ = 0 and N+ = N− = N0 , i.e. N+ N − N+ − ln eϕ − kT ln N − N0 N0 1 N − N+ − ξ N− 1 N − N+ + ln − ln =0 (7) ξ N − N0 − ξ N0 ξ N − N0 N− N − N+ − ξ N− − ln = 0. (8) − eϕ − kT ln N − N0 − ξ N0 N0
c+ N+ 2 = = c0 N0 γ
γ− = ξ γ
because N in equation (12) is the total number of available lattices that can be occupied by the cations. In that sense, γ− = ξ γ is the compressibility of the anions for the hard spherical ions. Therefore, η is proportional to the number of empty sites. With the ionic concentration c0 already known, the potential-dependent charge density distribution may be written as
Thus, the chemical potential may be written as µ+ =
and
(18)
Such lattice saturation or ‘crowding’ from steric effects [1, 35, 37, 39] as in figures 2(a) and (c) distinguishes equations (15) and (16) from Gouy–Chapman theory in equation (17) which grows infinitely with u. For small γ , both equations (15) and (16) are close to equation (17) at low u, as shown in figure 2(d).
(12) (13) 3
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
Figure 2. The dimensionless ρ(u)/ec0 , calculated at different levels of theory: (a) and (b) equation (15) with γ = 1, ξ = 0.5 (black), equation (16) with γ = 1 (red) and γ = 0.5 (blue), equation (17) (green); (c) and (d) equation (15) with γ = 0.1, ξ = 0.1 (black), equation (16) with γ = 0.1 (red) and γ = 0.01 (blue), equation (17) (green).
It is of interest to note that at low u, ρ(u) of equation (15) exceeds that of equation (16) and even equation (17). Such behavior can be clearly seen in figure 2(b), and the same trend is retained in figure 2(d). Thus, the asymmetric ions not only cause the asymmetric ρ(u), but also alter it at low u. Approximated to the first order, the charge density at low u is 1−ξ ρ(u) ∼ u. (19) lim =− 2+ u→0 ec0 ξ +η
in which L D is the Debye length given by s εε0 kT LD = 2e2 c0
and using equations (11), (15) and (22), equation (20) may be written as h i1 ξ exp(u)+η ξ −1 2 exp(−u) − exp(u) ξ +η d du 2 =− i1 . h du dX γ ξ exp(u)+η ξ exp(−u) + (ξ + η) ξ +η
Note that ξ + η = 2/γ − 1 by equation (13), and this effect is amplified as ξ → 0 and γ → 1, as comparing figures 2(b) and (d) shows. Fixing γ , the charge density at low u is enhanced for small ξ . As ξ → 1, equation (19) becomes −2u, i.e. the first order approximation of equations (17) and (18). The result shown in equation (19) is interesting, because the charge density of the asymmetric ions of finite size may well exceed that of dimensionless ions. Such a feature does not exist for ions of equal size—as can be seen, the absolute charge density of equation (16) can never exceed that of equation (17). We may further work out the potential by solving the one-dimensional Poisson’s equation d2 ϕ 1 d = dx 2 2 dϕ
dϕ dx
2
ρ(ϕ) =− εε0
(23) Integrating equation (23) and using the fact that u = 0 and du/dX = 0 in the bulk where X → ∞, we have s du 2 = −sgn(u) dX γ v " u 1 # u ξ exp(u) + η ξ γ t × ln exp(−u) + (ξ + η) + ln (24) ξ +η 2
in which sgn denotes the sign function. Note that the term ln(γ /2) in the square root of equation (24) is fixed by the boundary condition of du/dX = 0 as u = 0. With ξ = 1, the above equation becomes KKBA MFT, i.e. [1, 35] s r h u i du 2 = −sgn(u) . (25) ln 1 + 2γ sinh2 dX γ 2
(20)
in which ε0 and ε denote the permittivity of free space and the dielectric constant, respectively, and x denotes the distance to the electrode surface where x = 0. By recasting x to a dimensionless distance with [1] X=
x LD
(22)
The above expression further reduces to the Gouy–Chapman electric field in the γ → 0 limit. Using the identity limu→0 ln
(21) 4
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
Figure 3. Electric potential profile, u(X ), at different electrode potentials u 0 , estimated by equation (27) with different γ and ξ (black),
equation (28) with different γ (red), and equation (29) (green): (a) u 0 = −0.2; (b) u 0 = 0.2; (c) u 0 = −2; (d) u 0 = 2; (e) u 0 = −20; (f) u 0 = 20.
[1 + 2γ sinh2 (u/2)] ∼ = 2γ sinh2 (u/2), equation (25) becomes u du = −2 sinh . (26) dX 2 From equations (24)–(26), the potential distributions of the imposed electrode potential u 0 for the MFT in this study, KKBA MFT [1, 35], and Gouy–Chapman theory, respectively, are given by r γ X (u) = sgn(u 0 ) 2 Z u0 du × s 1 u ξ ln exp(−u) + (ξ + η) ξ exp(u)+η + ln γ2 ξ +η
Equations (27) and (28) can be easily solved numerically, while an analytical expression is presented in equation (29) for Gouy–Chapman theory. Instead of inversing equations (27)– (29) in order to get the potential drop across the EDL, u(X ), it can be achieved by rotating X (u) by 90◦ , because X (u) is a single valued function [1]. Figure 3 shows the comparisons of u(X ) at different electrode potentials u 0 (equation (27)) for a variety of γ and ξ , with the negative u 0 on the left column and positive u 0 on the right column, due to the asymmetric ions. For comparison, the u(X ) of KKBA MFT [1, 35] (equation (28)), of different γ , as well as u(X ) of Gouy–Chapman theory (equation (29)) are also displayed. The u(X ) of equation (28) are displayed with γ+ = γ for the negative u 0 and γ− = ξ γ for the positive u 0 , respectively. In all the figures, the u(X ) of equation (28) decay more slowly than those of equation (29), though the former may be infinitely close to the latter at low u 0 . It can be seen from figures 3(a) and (b) that u(X ) of this work decays faster than that of KKBA MFT [1, 35] and Gouy–Chapman theory at such low u 0 = ±0.2, while the
(27) r Z u 0 γ du q X (u) = sgn(u 0 ) 2 u ln 1 + 2γ sinh2 tanh(u/4) X (u) = sgn(u 0 ) ln . tanh(u 0 /4)
(28) u 2
(29) 5
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
latter two gives almost identical results. Indeed, at such low u, the charge density ρ(u) from equation (15) well exceeds that from equations (16) and (17), by comparing ρ(u)/ec0 in equation (19) with the low u limit of −2u from equations (16) and (17). Thus the screening is better for equation (19), and causes a faster potential drop. Furthermore, equation (19) also indicates better screening with lower ξ by fixing γ , which is demonstrated by comparing the u(X ) of ξ = 0.5 and ξ = 0.1 in both figures 3(a) and (b). Figures 3(c) and (d) display the u(X ) at an intermediate electrode potential u 0 = ±2. It can be seen that the u(X ) of equation (27) decay faster than both equations (29) and (28) for γ− = ξ γ with the same ξ and γ at u 0 = 2, but such a trend reverses at u 0 = −2. Such asymmetric behavior may be understood in terms of the smaller anion, which screens the positive electrode potential more effectively than does the cation on the negative electrode. For a high electrode potential of u 0 = ±20, as shown in figures 3(e) and (f), equation (29) decays remarkably faster, as expected, due to the unlimited accumulation of charges in the EDL. The u(X ) of equations (27) and (28) are almost indistinguishable at u 0 = −20 in figure 3(e), which may be understood in the high potential limit of equation (18). On the other hand, equation (27) decays faster at γ = 1, ξ = 0.5, as compared to equation (28) with γ = 0.5 at u 0 = 20 in figure 3(f). Indeed, for positive polarization, the repulsion of the bulky cation by the electrode leaves more space for packing the smaller anion, resulting in more effective screening. According to Gauss’ law, the electrode charge density σ is determined by electric displacement, i.e. du dϕ = − 2ec0 L D = σ. (30) D = − εε0 dx x=0 dX X =0
Using equations (31)–(33), Cd for the MFT in this study, KKBA MFT [1, 35] and Gouy–Chapman theory, respectively, may be written as ) ( 1 ξ exp(u 0 ) + η ξ −1 Cd (u 0 ) − exp(−u 0 ) = sgn(u 0 ) exp(u 0 ) C0 ξ +η " 1 # p ξ exp(u 0 ) + η ξ × 2γ exp(−u 0 ) + (ξ + η) ξ +η −1 v " u 1 # u ξ ξ exp(u 0 ) + η γ + ln × tln exp(−u 0 ) + (ξ + η) ξ +η 2 (35)
cosh(u 0 /2) Cd (u 0 ) = C0 1 + 2γ sinh2 (u 0 /2) s 2γ sinh2 (u 0 /2) × (36) ln 1 + 2γ sinh2 (u 0 /2) u Cd (u 0 ) 0 = cosh . (37) C0 2 The above equations are the expressions of Cd at different levels of theory, in which C0 is the Debye capacitance, given by εε0 . (38) C0 = LD It can be easily verified that equation (35) becomes equation (36) when ξ = 1, and further reduces to equation (37) as γ → 0. The scaled dimensionless capacitance Cd /C0 of various γ and ξ , based on equation (35), the Cd /C0 of various γ based on equation (36), as well as that based on equation (37) are shown in the left column of figure 4. Since equation (36) gives symmetric Cd for equal size ions, Kornyshev proposed an semi-empirical treatment of potential-dependent compressibility, γ (u 0 ), by interpolating from γ− to γ+ via [1]
Using equations (24)–(26), the electrode surface charge density σ , for the MFT in this study, KKBA MFT [1, 35], and Gouy–Chapman theory, respectively, at electrode potential u 0 can be solved by s σ (u 0 ) 8 = sgn(u 0 ) ec0 L D γ v " u ξ1 # u γ ξ exp(u ) + η 0 + ln (31) ×tln exp(−u 0 ) + (ξ + η) ξ +η 2 s r h u i σ (u 0 ) 8 0 = sgn(u 0 ) ln 1 + 2γ sinh2 ec0 L D γ 2 u σ (u 0 ) 0 = 4 sinh . ec0 L D 2
γ (u 0 ) = γ− +
dσ (ϕ0 ) . dϕ0
(39)
in which γ+ = γ and γ− = ξ γ as in equation (14). Thus, by incorporating equation (39) into equation (36), the effect of asymmetric IL is taken into account semi-empirically, and the resulting Cd /C0 are plotted in figure 4 for comparison. The electrode surface charge density by the above semi-empirical treatment can be integrated numerically, i.e. Z u0 σ (u 0 ) cosh(u/2) =2 du ec0 L D 1 + 2γ (u) sinh2 (u/2) 0 s 2γ (u) sinh2 (u/2) × (40) ln 1 + 2γ (u) sinh2 (u/2)
(32) (33)
in which L D is the Debye length in equation (22). It is the σ − u 0 curve that is determined in the computer simulation. Also, it can be seen that the PZC is located at u 0 = 0 for all levels of the theory in equations (31)–(33). Experimentally, σ (u 0 ) is difficult to obtain due to uncertainty in the PZC. Instead, the differential capacitance, Cd , is measured as a function of the electrode potential ϕ0 = kT u 0 /e, i.e. Cd (ϕ0 ) =
γ+ − γ− 1 + exp(u 0 )
in which γ (u) is given by equation (39). The dimensionless surface charge densities, σ (u 0 )/ec0 L D , of various γ and ξ , at different levels of theory are shown in the right column of figure 4. A general trend in figures 4(a), (c) and (d) is that the asymmetric ionic size is reflected in asymmetric Cd curves for equation (35), as well as for equation (36) with semi-empirical
(34) 6
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
Figure 4. C d /C 0 and σ/ec0 L D at different levels of theory. (a) C d /C 0 by equation (35) with γ = 1, ξ = 0.5 (black), equation (36) with γ = 1 (yellow) and γ = 0.5 (blue), equation (36) with semi-empirical γ (u 0 ) in equation (39) for γ+ = γ = 1 and γ− = ξ γ = 0.5 (red), equation (37) (green). (b) The corresponding σ/ec0 L D by equation (31) (black dashed) and equation (40) (red dashed), Cd /C0 by equation (35) (black) and equation (36) with γ (u 0 ) in equation (39) (red) with the same parameters in (a) are also shown to highlight the asymmetric Cd profiles; (c) and (d) show the comparisons of Cd /C0 and σ/ec0 L D of γ = 0.5, ξ = 0.1 with KKBA MFT and Gouy–Chapman theory; (e) and (f) show the comparison of Cd /C0 and σ/ec0 L D of γ = 0.1, ξ = 0.1 with KKBA MFT and
Gouy–Chapman theory. The line color and corresponding equations in (c)–(f) follow (a) and (b).
∼ γ− at high positive polarization, compared with the while γ = results of KKBA MFT [1, 35] for the corresponding γ s. The above feature is also captured by Kornyshev’s semi-empirical treatment with the potential-dependent γ (u 0 ) in equation (39), which interpolates γ (u 0 ) from the cation-rich layer to the anion-rich layer. At low electrode potential u 0 , the Cd of Gouy–Chapman theory in equation (37) imposes the upper limit of KKBA MFT [1, 35] in equation (36) with either constant γ or with semi-empirical potential-dependent γ (u 0 ) in equation (39). On the other hand, this is not the case for equation (35). With decreasing γ , both equations (35) and (36) approach asymptotically to equation (37) from the upper and lower sides. At the low u 0 limit around PZC, equation (35) may be approximated, up to second order, as
γ (u 0 ) in equation (39). The above property is correlated with the parameter ξ , with higher asymmetry the more ξ deviates from unity, while both equations (36) and (37) give symmetric Cd for symmetric ions. The result of the asymmetric Cd curve is in good agreement with DFT [33, 34] and simulations [20]. Besides, the Cd curves by equations (35) and (36) with semi-empirical γ (u 0 ) agree well in the region of high polarization. For a high electrode potential wing, it is obvious that equation (35) becomes 1 1 √ =√ , for high positive u 0 Cd ∼ 2γ− u 0 2ξ γ u 0 (41) = 1 1 C0 √ , for high negative u 0 . =√ 2γ+ |u 0 | −2γ u 0 Such inverse-square-root behavior resembles Kornyshev’s derivation [1]. It is reasonable to see that at high electrode potential, Cd is exclusively determined by the single γ of the counter ion, so that γ ∼ = γ+ at high negative polarization,
Cd = 30 + 31 u 0 + 32 u 20 + O(u 30 ) C0 7
(42)
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
In this study, we extended the mean-field theory (MFT) developed independently by Kornyshev [1] and Kilic, Bazant and Ajdari [35] (KKBA MFT) to take into account asymmetric 1:1 IL electrolytes, by introducing an additional parameter ξ for the anion/cation volume ratio. The MFT of asymmetric ions becomes KKBA MFT [1, 35] upon ξ = 1, and further reduces to Gouy–Chapman theory in the γ → 0 limit. The result of the extended MFT in this work demonstrates that the asymmetric ILs result in asymmetric Cd , with the higher peak in Cd occurring at positive polarization for the smaller anionic size. At high potential, Cd grows asymptotically toward KKBA MFT characterized by γ for negative polarization, and characterized by ξ γ for positive polarization. At low potential around the potential of zero charge, the asymmetric ions cause a higher Cd , which exceeds that of Gouy–Chapman theory. The current study also shows reasonable agreement between the MFT in this study and the semi-empirical treatment of asymmetric ions by interpolating a potential-dependent compressibility γ from γ− to γ+ for anions and cations [1], especially for smaller γ of camel-shaped Cd , while the deviation becomes apparent for larger γ of bell-shaped Cd . Finally, we should note that the result for MFT at low electrode potential should be viewed with caution. It is found in many computer simulations [5, 20–25, 28–31, 41–45] and experiments [46–51] that charge density oscillates around zero near a charged electrode of low potential characterized by distinctive ionic layers, i.e. the overscreening effect. However, the current MFT does not capture the overscreening effect, but manifests it in an average manner, as inherent from KKBA MFT [1, 35]. On the other hand, the lattice saturation or ‘crowding’ of Cd at high electrode potential, characterized by the inverse-square-root behavior [1] due to the steric effects imposed by the lattice–gas model [39, 40], is indeed captured and is in good agreement with KKBA MFT [1, 35]. A recent theory [37] successfully reproduces both overscreening at low potential and ‘crowding’ at high potential by taking into account the local ionic correlation, and thus is beyond the MFT framework.
in which s
1 1−ξ 2 ξ +η 1 1 (1 − ξ ) 31 = η+ 330 γ (ξ + η)2 1 P(γ , ξ ) 32 = − 4 9630 γ (ξ + η)3 (η + 2/γ ) P(γ , ξ ) = 384 − 1920γ + 2880γ 2 − 1920γ 3 30 =
1+
(43)
+ 600γ 4 − 72γ 5 + (768γ − 2304γ 2 + 2208γ 3 − 840γ 4 + 108γ 5 )(1 − ξ ) + (672γ 2 − 1104γ 3 + 502γ 4 − 54γ 5 )(1 − ξ )2 + (240γ 3 − 172γ 4 + 9γ 5 )(1 − ξ )3 + 28γ 4 (1 − ξ )4 . It is of interest to note that as u 0 = 0, 30 ≥ 1, so that Cd /C0 may exceed 1, the value that is predicted by KKBA MFT [1, 35], the semi-empirical γ (u 0 ) by equation (39) [1] and Gouy–Chapman theory at the PZC. For small u 0 around the PZC, the situation is quite complicated. However, we note that as ξ = 1, equation (42) reduces to Kornyshev’s derivation, i.e. [1] 1 − 3γ 2 Cd =1+ u0. C0 8
(44)
Thus, the transition from bell-shaped to camel-shaped Cd /C0 at γ = 1/3 is apparent, as denoted by Kornyshev for the ξ = 1 case. Also, it can be seen that equation (44) never exceeds the low u 0 limit of Gouy–Chapman theory of Cd /C0 = 1 + u 20 /8. On the other hand, for ξ < 1 as the MFT in this study, the situation is more complicated. Note that ξ + η = 2/γ − 1 by equation (13), Cd /C0 is amplified as ξ → 0 and√γ → 1, and in that limiting case, the maximum Cd /C0 is 3/2 at the PZC. Fixing γ , the Cd /C0 at low u is enhanced for small ξ , consistent with enhanced charge density as discussed in equation (19). As can be seen, especially for high γ cases in figures 4(a) and (c), Cd of equation (35) well exceeds that of equation (37) around the PZC. The higher Cd at low u 0 is manifest in the σ (u 0 ) curve; this can be seen in the slightly higher charge density in the positive branch of equation (31) than that of equations (32) and (40), as shown clearly in figures 4(b) and (d), though the corresponding Cd at high u 0 of equation (35) is slightly lower than that of equation (36) as well as that with semi-empirical γ (u 0 ) in equation (39) in figures 4(a) and (c). In all cases, σ (u 0 ), estimated via equation (40), with γ (u 0 ) in equation (39), is lower than that by equations (31) and (32), probably due to the over-mixing of higher γ+ in the positive polarization. With very low γ , Cd and σ (u 0 ) curves are almost indistinguishable for equation (35) and Kornyshev’s semi-empirical treatment, i.e. equation (36), with γ (u 0 ) in equation (39), as shown in figure 4(e). The same trend is also reflected in σ (u 0 ), as shown in figure 4(f) for equations (31) and (40). Such better agreement at smaller γ of Cd and σ (u 0 ) may be attributed to the fact that smaller γ gives camel-shaped Cd for both cation and anion, which gives a smaller deviation near the PZC, and the deviation becomes apparent for the larger γ of bell-shaped Cd .
Acknowledgments
This work is supported by NSFC (21373118, 21073097), Natural Science Foundation of Tianjin (12JCYBJC13900), NCET-10-0512 and the MOE Innovation Team (IRT13022) of China. We are grateful to an anonymous referee for naming the term η porosity. References [1] Kornyshev A A 2007 Double-layer in ionic liquids: paradigm change? J. Phys. Chem. B 111 5545–57 [2] Su Y Z, Fu Y C, Wei Y M, Yan J W and Mao B W 2010 The electrode/ionic liquid interface: electric double layer and metal electrodeposition Chem. Phys. Chem. 11 2764–78 and references cited therein [3] Merlet C, Rotenberg B, Madden P A and Salanne M 2013 Computer simulations of ionic liquids at electrochemical interfaces Phys. Chem. Chem. Phys. 15 15781–92 8
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
[4] Jiang D-e and Wu J 2013 Microscopic insights into the electrochemical behavior of nonaqueous electrolytes in electric double-layer capacitors J. Phys. Chem. Lett. 4 1260–7 [5] Si X, Li S, Wang Y, Ye S and Yan T 2012 Effects of specific adsorption on the differential capacitance of imidazolium-based ionic liquid electrolytes Chem. Phys. Chem. 13 1671–6 [6] Zhang Q, Han Y, Wang Y, Ye S and Yan T 2014 Comparing the differential capacitance of two ionic liquid electrolytes: effects of specific adsorption Electrochem. Commun. 38 44–6 [7] Fedorov M V, Georgi N and Kornyshev A A 2010 Double layer in ionic liquids: the nature of the camel shape of capacitance Electrochem. Commun. 12 296–9 [8] Georgi N, Kornyshev A A and Fedorov M V 2010 The anatomy of the double layer and capacitance in ionic liquids with anisotropic ions: electrostriction vs. lattice saturation J. Electroanal. Chem. 649 261–7 [9] Su Y, Yan J, Li M, Zhang M and Mao B 2013 Electric double layer of Au(100)/imidazolium-based ionic liquids interface: effect of cation size J. Phys. Chem. C 117 205–12 [10] Alam M T, Mominul Islam M, Okajima T and Ohsaka T 2007 Measurements of differential capacitance in room temperature ionic liquid at mercury, glassy carbon and gold electrode interfaces Electrochem. Commun. 9 2370–4 [11] Alam M T, Islam M M, Okajima T and Ohsaka T 2008 Capacitance measurements in a series of room-temperature ionic liquids at glassy carbon and gold electrode interfaces J. Phys. Chem. C 112 16600–8 [12] Lockett V, Horne M, Sedev R, Rodopoulos T and Ralston J 2010 Differential capacitance of the double layer at the electrode/ionic liquids interface Phys. Chem. Chem. Phys. 12 12499–512 [13] Su Y-Z, Fu Y-C, Yan J-W, Chen Z-B and Mao B-W 2009 Double layer of Au(100)/ionic liquid interface and its stability in imidazolium-based ionic liquids Angew. Chem. Int. Edn 48 5148–51 [14] Vatamanu J, Cao L, Borodin O, Bedrov D and Smith G D 2011 On the influence of surface topography on the electric double layer structure and differential capacitance of graphite/ionic liquid interfaces J. Phys. Chem. Lett. 2 2267–72 [15] Lin L G, Wang Y, Yan J W, Yuan Y Z, Xiang J and Mao B W 2003 An in situ STM study on the long-range surface restructuring of Au(111) in a non-chloroaluminumated ionic liquid Electrochem. Commun. 5 995–9 [16] Roling B, Druschler M and Huber B 2012 Slow and fast capacitive process taking place at the ionic liquid/electrode interface Faraday Discuss. 154 303–11 [17] Druschler M, Borisenko N, Wallauer J, Winter C, Huber B, Endres F and Roling B 2012 New insights into the interface between a single-crystalline metal electrode and an extremely pure ionic liquid: slow interfacial processes and the influence of temperature on interfacial dynamics Phys. Chem. Chem. Phys. 14 5090–9 [18] Baldelli S 2008 Surface structure at the ionic liquid–electrified metal interface Acc. Chem. Res. 41 421–31 [19] Islam M M, Alam M T and Ohsaka T 2008 Electrical double-layer structure in ionic liquids: a corroboration of the theoretical model by experimental results J. Phys. Chem. C 112 16568–74
[20] Fedorov M V and Kornyshev A A 2008 Ionic liquid near a charged wall: structure and capacitance of electrical double layer J. Phys. Chem. B 112 11868–72 [21] Feng G, Li S, Atchison J S, Presser V and Cummings P T 2013 Molecular insights into carbon nanotube supercapacitors: capacitance independent of voltage and temperature J. Phys. Chem. C 117 9178–86 [22] Paek E, Pak A J and Hwang G S 2013 A computational study of the interfacial structure and capacitance of graphene in BMIM PF6 ionic liquid J. Electrochem. Soc. 160 A1–A10 [23] Vatamanu J, Borodin O, Bedrov D and Smith G D 2012 Molecular dynamics simulation study of the interfacial structure and differential capacitance of alkylimidazolium bis(trifluoromethanesulfonyl)imide [Cn mim][TFSI] ionic liquids at graphite electrodes J. Phys. Chem. C 116 7940–51 [24] Vatamanu J, Borodin O and Smith G D 2011 Molecular simulations of the electric double layer structure, differential capacitance, and charging kinetics for N-methyl-N-propylpyrrolidinium Bis(fluorosulfonyl)imide at graphite electrodes J. Phys. Chem. B 115 3073–84 [25] Feng G, Jiang D-e and Cummings P T 2012 Curvature effect on the capacitance of electric double layers at ionic liquid/onion-like carbon interfaces J. Chem. Theory Comput. 8 1058–63 [26] Vatamanu J, Borodin O and Smith G D 2010 Molecular insights into the potential and temperature dependences of the differential capacitance of a room-temperature ionic liquid at graphite electrodes J. Am. Chem. Soc. 132 14825–33 [27] Hu Z, Vatamanu J, Borodin O and Bedrov D 2013 A molecular dynamics simulation study of the electric double layer and capacitance of [BMIM][PF6 ] and [BMIM][BF4 ] room temperature ionic liquids near charged surfaces Phys. Chem. Chem. Phys. 15 14234–47 [28] Fedorov M V and Kornyshev A A 2008 Towards understanding the structure and capacitance of electrical double layer in ionic liquids Electrochim. Acta 53 6835–40 [29] Feng G, Zhang J S and Qiao R 2009 Microstructure and capacitance of the electrical double layers at the interface of ionic liquids and planar electrodes J. Phys. Chem. C 113 4549–59 [30] Feng G, Huang J, Sumpter B G, Meunier V and Qiao R 2011 A ‘counter-charge layer in generalized solvents’ framework for electrical double layers in neat and hybrid ionic liquid electrolytes Phys. Chem. Chem. Phys. 13 14723–34 [31] Merlet C, Salanne M, Rotenberg B and Madden P A 2013 Influence of solvation on the structural and capacitive properties of electrical double layer capacitors Electrochim. Acta 101 262–71 [32] Wu J, Jiang T, Jiang D-e, Jin Z and Henderson D 2011 A classical density functional theory for interfacial layering of ionic liquids Soft Matter 7 11222–31 [33] Lauw Y, Horne M D, Rodopoulos T, Nelson A and Leermakers F A M 2010 Electrical double-layer capacitance in room temperature ionic liquids: ion-size and specific adsorption effects J. Phys. Chem. B 114 11149–54 [34] Jiang D-e, Meng D and Wu J 2011 Density functional theory for differential capacitance of planar electric double layers in ionic liquids Chem. Phys. Lett. 504 153–8 9
J. Phys.: Condens. Matter 26 (2014) 284103
Y Han et al
[35] Kilic M S, Bazant M Z and Ajdari A 2007 Steric effects in the dynamics of electrolytes at large applied voltages. I. Double-layer charging Phys. Rev. E 75 021502 [36] Oldham K B 2008 A Gouy–Chapman–Stern model of the double layer at a (metal)/(ionic liquid) interface J. Electroanal. Chem. 613 131–8 [37] Bazant M Z, Storey B D and Kornyshev A A 2011 Double layer in ionic liquids: overscreening versus crowding Phys. Rev. Lett. 106 046102 Bazant M Z, Storey B D and Kornyshev A A 2012 Phys. Rev. Lett. 109 149903 (erratum) [38] Bard A J and Faulkner L R 2000 Electrochemical Methods: Fundamentals and Applications 2nd edn (New York: Wiley) [39] Borukhov I, Andelman D and Orland H 1997 Steric effects in electrolytes: a modified Poisson–Boltzmann equation Phys. Rev. Lett. 79 435–8 [40] Borukhov I, Andelman D and Orland H 2000 Adsorption of large ions from an electrolyte solution: a modified Poisson–Boltzmann equation Electrochim. Acta 46 221–9 [41] Reed S K, Lanning O J and Madden P A 2007 Electrochemical interface between an ionic liquid and a model metallic electrode J. Chem. Phys. 126 084704 [42] Pinilla C, Del P´opolo M G, Kohanoff J and Lynden-Bell R M 2007 Polarization relaxation in an ionic liquid confined between electrified walls J. Phys. Chem. B 111 4877–84 [43] Fedorov M V and Lynden-Bell R M 2012 Probing the neutral graphene-ionic liquid interface: insights from molecular dynamics simulations Phys. Chem. Chem. Phys. 14 2552–6
[44] Lynden-Bell R M, Frolov A I and Fedorov M V 2012 Electrode screening by ionic liquids Phys. Chem. Chem. Phys. 14 2693–701 [45] Wang S, Li S, Cao Z and Yan T 2010 Molecular dynamic simulations of ionic liquids at graphite surface J. Phys. Chem. C 114 990–5 [46] Mezger M et al 2008 Molecular layering of fluorinated ionic liquids at a charged sapphire (0001) surface Science 322 424–8 [47] Hayes R, Warr G G and Atkin R 2010 At the interface: solvation and designing ionic liquids Phys. Chem. Chem. Phys. 12 1709–23 [48] Perkin S, Crowhurst L, Niedermeyer H, Welton T, Smith A M and Gosvami N N 2011 Self-assembly in the electrical double layer of ionic liquids Chem. Commun. 47 6572–4 [49] Zhang X, Zhong Y-X, Yan J-W, Su Y-Z, Zhang M and Mao B-W 2012 Probing double layer structures of Au (111)-BMIPF6 ionic liquid interfaces from potential-dependent AFM force curves Chem. Commun. 48 582–4 [50] Li H, Endres F and Atkin R 2013 Effect of alkyl chain length and anion species on the interfacial nanostructure of ionic liquids at the Au(111)-ionic liquid interface as a function of potential Phys. Chem. Chem. Phys. 15 14624–33 [51] Zhou H et al 2012 Nanoscale perturbations of room temperature ionic liquid structure at charged and uncharged interfaces ACS Nano 6 9818–27
10
