Dipolar Self-Consistent Field Theory for Ionic Liquids: Effects of

Article pubs.acs.org/JPCC

Dipolar Self-Consistent Field Theory for Ionic Liquids: Effects of Dielectric Inhomogeneity in Ionic Liquids between Charged Plates Issei Nakamura* State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin 130022, China ABSTRACT: We studied ionic liquids confined between charged plates to develop a dipolar self-consistent field theory (DSCFT) for incompressible states and a hybrid of DSCFT combined with Monte Carlo simulation for compressible states. Our theory, which has no adjustable parameters, accounts for the differences between the dipole moments and the molecular volumes of the cation and anion, the hard-core nature of the ions, and the electrical double layer formed through the strong association of the ions with the electrodes. We illustrate that a spatial change in the distributions of cations and anions with different dipole moments causes a significant spatial change in the dielectric function and hence gives rise to spatial asymmetry in the electrostatic field between the charged plates. Notably, this effect can be comparable to that caused by an asymmetry in the molecular volumes of the cation and anion. Moreover, the hard-core nature of ionic liquids causes oscillations in the density profile near the charged plates. We also demonstrate that a contrast in the dielectric values of the cations and anions in ionic liquids causes a substantial decrease in capacitance as the applied voltage is increased. The magnitude of this variation can be noticeably altered by changing the dipole moments of the cation and anion.

1. INTRODUCTION Ionic liquids are molten salts that consist entirely of cations and anions, which are typically several angstroms in radius. Their high solubilities at low temperatures, their extremely low vapor pressures, and their high electrical conductivities are promising properties that confer mechanical and electrical robustness on recently developed electrochemical devices such as fuel cells,1−3 supercapacitors,4−6 and electroactive actuators.7−9 Although ionic liquids exhibit certain electrochemical properties analogous to those of atomic ions dissolved in solvents, their dielectric response and steric interactions are the keys to controlling their electrochemical characteristics.10,11 The confinement of ionic liquids to nanometer-scale regions is of particular interest. The length scale for such confinement typically varies from 1 nm to a few nanometers. For example, Nafion membranes for electroactive actuators7 and fuel cells12 or carbon-based electric double-layer capacitors4 can form nanopores of 1−4 nm. In experiments with ionic polymer actuators,13 direct observation of ion distributions near electrodes has revealed their oscillatory behavior. According to theories concerning uncharged hard spheres, such oscillations in ion concentrations should be caused predominantly by the hard-core nature of the particles.14 Indeed, density functional theory15 and molecular dynamics simulations16 have demonstrated that models based on charged hard spheres also give rise to similar oscillations in ion concentrations. In other words, a theory must be found to account for the formation of a condensed layer of counterions near the electrodes, when the ion concentrations or electrostatic fields are relatively high.17 Numerous detailed studies have emerged on this subject in recent decades. Thus, it has been determined that the asymmetry in the sizes of the cations and anions, the dielectric © 2015 American Chemical Society

values of the ionic liquids, and the applied voltage (or surface charge density) on the electrodes are certainly the key factors that can be adjusted to improve the electrochemical and thermal stabilities of such systems. Despite the large number of studies, however, the effects of the orientational reorganization of the dipoles have remained largely unnoticed. Indeed, there is a large class of ionic liquids that exhibit large or moderate polarity caused by the different dielectric responses of the cation and anion.18 Although the dipole moments of many cations and anions, and the resultant dielectric constants, have already been determined in a large number of experiments,18−21 the dielectric response of ionic mixtures under external fields is not likely to be simply captured by a single parameter in the same way that the bulk dielectric constant εr appears in the Coulomb interaction, e2/(4πϵ0εrr). First, the orientational reorganization of dipoles should occur in the vicinity of charges;22,23 this process is often referred to as “dielectric saturation.” Second, the theories of Onsager24 and Kirkwood25 indicate that the dielectric value should depend on the number density of the constituent species, but both the ion distribution and the orientational order of the dipoles are spatially inhomogeneous in nanometer-scale regions bounded by electrodes. Thus, it is important to note that the resultant dielectric value should also depend on the positions and concentrations of the cations and anions. Indeed, recent coarsegrained theories have suggested that these dependencies should be primarily responsible for the substantial variations in the dielectric values of ion-containing pure liquids,26,27 polymer blends and block copolymer mixtures.28,29 Received: November 25, 2014 Revised: February 19, 2015 Published: March 5, 2015 7086

DOI: 10.1021/jp511770r J. Phys. Chem. C 2015, 119, 7086−7094

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The Journal of Physical Chemistry C In this paper, we develop a new dipolar self-consistent field theory for ionic liquids between charged plates. Our theory accounts for the permanent dipoles, the molecular volumes, and the hard-core nature of both the cations and the anions. Accordingly, our theory provides dielectric inhomogeneity without assuming a linear volume-fraction-weighted average for the local dielectric value.30 In Section 2.1, we first consider incompressible ionic liquids. It should be noted that our fieldtheoretical framework leads to a modified Poisson−Boltzmann equation that systematically accounts for the size asymmetry between the cations and anions, and for the spatial variation in the ion concentrations. We also demonstrate that our theory accounts for charge condensation near strongly charged electrodes by illustrating the relatively constant behavior of the ion concentrations in such cases. We thus suggest that our theory can also be reasonably extended to other geometries or species. Our conceptually novel finding in this study is that the effects of the orientational reorganization of dipoles in response to external electrostatic fields on the dielectric value can be of the same magnitude as the effects caused by the differences in the molecular volumes of the cation and anion. Thus, the dipole moments and molecular volumes are equally important to consider in the selection of ionic liquids for electrochemical devices. In Section 2.2, we consider compressible ionic liquids. To capture the hard-core nature of ionic liquids, we construct a hybrid self-consistent field theory that combines the dipolar self-consistent field theory with Monte Carlo simulations. We then demonstrate the oscillatory behaviors of the ion concentration and the dielectric function. The dielectric function exhibits spatial asymmetry caused by the difference in the dipole moments of the cations and anions. Notably, the Onsager and Kirkwood equations for the bulk dielectric constant cannot capture the spatial variation of the dielectric function.

volumes of the cations and anions are v+ and v−, respectively. The position of the ith ion s (s = + or −) is denoted by r ⃗ si, and its permanent dipole moment is denoted by p ⃗ si. We can then write the grand canonical partition function as follows: Z=

∑ ∏ N+ , N − s =±

⎡ 1 exp⎢ − ⎣ 2

λsNs Ns!

Ns

∫ ∏ d rsi⃗ dpsi⃗ δ[ ∑ vsnŝ ( r ⃗) − 1] i=1

s =±

⎤

∫ d3rd3r′ ρ ̂( r ⃗)v( r ⃗ − r ⃗ ′)ρ ̂( r ⃗ ′)⎥⎦

(1)

where v(r ⃗ − r ⃗ ′) =e2/ (4π ϵ0|r ⃗ −r ⃗ ′|) is the Coulomb potential in a vacuum, λs is the fugacity of ion s, and vs is the molecular volume of ion s. We use energy units of kT. T is set to room temperature. In eq 1, the hard-core repulsion between the particles is modeled by the δ function, which enforces the incompressibility of the liquid mixture outside the ions, and n̂s(r ⃗ ) = ∑Ni=1s δ(r ⃗ − r ⃗ si) is the number density of ion s. The total charge density ρ̂(r ⃗ ) is given by23,31,32 Ns

∑ ∑ qsδ( r ⃗ − rsi⃗ ) + ∑ ρŝ ( r ⃗) + ρf ( r ⃗)

ρ (̂ r ⃗) =

s =+, − i = 1

s =±

Ns

ρs ̂ ( r ⃗) = −∑ psi⃗ ·∇δ( r ⃗ − rsi⃗ ) i=1

(2)

where ρ̂s(r ⃗ ) is the charge density resulting from the dipoles and ρf(r ⃗ ) is a fixed charge distribution. The dipole moment of the ith ion s constitutes an intrinsic (permanent) contribution p ⃗ si. Using standard field-theoretical techniques from polymer physics33 and following the procedure described in ref 23, we can derive the free-energy functional from the partition function given in eq 1 in terms of field variables (see the Appendix):

2. THEORY 2.1. Dipolar Self-Consistent Field Theory (DSCFT) for Incompressible Ionic Liquids. We consider an ionic liquid consisting of N0 cations and N0 anions that is confined between two oppositely charged plates separated by a distance D (Figure 1). Note that because the numbers of cations and anions are the same, the system is charge neutral overall. The cations and anions have charges of q+ and q−, respectively. The molecular

F=

1 8πl0 −

∫

d r ⃗ ψ ( r ⃗)∇2 ψ ( r ⃗) −

∑∫

d r ⃗ ns( r ⃗)ωs( r ⃗)

s =±

∑ 4πλs ∫

d r ⃗ exp[−qsψ ( r ⃗) − ωs( r ⃗)]

s =±

×

sinh(ps |∇ψ ( r ⃗)|) ps |∇ψ ( r ⃗)|

+

∫

d r ⃗ ρf ( r ⃗)ψ ( r ⃗)

(3)

In this equation, l0= e2/(4πϵ0) is the vacuum Bjerrum length; ns and ωs are the density field and its conjugate field, respectively, for the ion s; and ψ is the electrostatic potential scaled by the elementary charge. Here, we have replaced iψ(r ⃗ ) and iωs(r ⃗ ) with ψ(r ⃗ ) and ωs(r ⃗ ) because the original fields ψ(r ⃗ ) and ωs(r ⃗ ) are purely imaginary at the saddle point of the free-energy functional F.23 Extremizing the free-energy functional F with respect to the field variable ψ (r ⃗ ) yields ∇·[εr ( r ⃗)∇ψ ( r ⃗)] = − 4πl0ρf ( r ⃗) − (4π )2 l0 ∑ λsqs s =±

exp[−qsψ ( r ⃗) − ωs( r ⃗)] ×

Figure 1. Schematic illustration of dipolar ionic liquids confined between charged plates. The colors of the circles denote cations (yellow) and anions (green). σ and −σ are the surface charge densities on the plates.

sinh(ps |∇ψ ( r ⃗)|) ps |∇ψ ( r ⃗)|

(4)

Note that eq 4 is the Poisson−Boltzmann equation for ionic liquids, where εr(r ⃗ ) represents the local dielectric function, 7087


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The Journal of Physical Chemistry C εr ( r ⃗) = 1 + (4π )2 l0 ∑ λs exp[−qsψ ( r ⃗) − ωs( r ⃗)] s =±

×

ps2 F(ps |∇ψ ( r ⃗)|)

(5)

Fs is related to the Langevin function, 3 (x) = 1/tanh x − 1/x, as follows: F = 3 (x) sinh x/x2. Similarly, the extremization of eq 3 with respect to the other field variables leads to the following set of equations: ⎛v ⎞ ω−( r )⃗ = ⎜ − ⎟ω+( r ⃗) ⎝ v+ ⎠ ns( r ⃗) = 4πλs exp[− qsψ ( r ⃗) − ωs( r ⃗)] ×

sinh(ps |∇ψ ( r )⃗ |) ps |∇ψ ( r ⃗)|

Figure 2. Volume fractions of the cation and anion, and the dielectric function between the charged plates. p+ = p− = 2 D. D = 1 nm. σ = 1.91 × 10−3 C/m2 for weakly charged surfaces and 1.83 × 10−2 C/m2 for strongly charged surfaces. v+ = v− = 163 Å3. (a) The volume fractions of the ionic liquids. The curves correspond to ϕ+ (black) and ϕ− (red) for weakly charged surfaces and to ϕ+ (purple) and ϕ− (blue) for strongly charged surfaces. (b) The dielectric values of the ionic liquids. The curves correspond to weakly charged surfaces (black) and strongly charged surfaces (blue). The inset illustrates the electrostatic fields, E(z)p+/kT.

(6)

where we have the incompressibility condition 1=

∑ vsns( r ⃗) (7)

s =±

Thus, eq 4 can be cast in a simpler form, ∇·[εr ( r ⃗)∇ψ ( r ⃗)] = −4πl0ρf ( r ⃗) − 4πl0 ∑ qsns( r ⃗) s =±

⎡ 3(p |∇ψ ( r ⃗)|) ⎤ s ⎥ εr ( r ⃗) = 1 + 4πl0 ∑ ps2 ns( r ⃗)⎢ ⎢ |∇ p ⎣ s ψ ( r ⃗)| ⎥⎦ s =±

respectively. Here, we choose the distance between the charged plates to be D = 1 nm because this length scale is the typical order of magnitude for nanostructured electrochemical materials.4,7,12 Typically, in such nanochannels, ions permeate through small regions bounded by charged surfaces. Note that the volume fractions for the strongly charged surfaces become relatively constant near the surfaces, whereas no such double-layer structure appears in the case of the weakly charged surfaces (Figure 2a). This result suggests one advantage of the self-consistent field-theoretical approach for ionic liquids over the conventional Poisson−Boltzmann (PB) equation for electrolytes, which cannot describe strong affinity (or complexation) between charges. This difference in performance can be primarily attributed to the fact that our theory accounts for finite ion volumes, whereas charged species are typically treated as volumeless particles in the conventional PB equation.36 Another notable result with respect to the dielectric properties is that the degree of dipole reorientation is substantially affected by the strength of the surface charges. Note that the dielectric constant in Figure 2b is spatially uniform because the dipole moments of the cation and anion are the same. Despite the spatial variation in the ion concentrations, the electrostatic field is also constant between the charged plates.37 Although this constant feature remains intact, the dielectric constant significantly decreases as the surface charge density increases (Figure 3). This phenomenon is primarily caused by the electrostatic response of the dipoles in the ionic liquids under the electrostatic field from the charged plates. Our present result is also consistent with Booth’s theoretical prediction of variations in the bulk dielectric constant.38 Notably, the classical theories of Onsager24 and Kirkwood25 do not account for the dependence of the dielectric constant on external electrostatic fields. Therefore, their theories appear to be impractical for applications to a system that consists of ionic liquids confined in nanochannels. We now consider the dielectric inhomogeneity that is caused primarily by the difference between the dipole moments of the cation and anion. In Figure 4a and b, we demonstrate that the

(8)

Equations 6−8 constitute a modified Poisson−Boltzmann equation for dipolar ionic liquids. Note that ω−(r ⃗ ) is replaced by ω+(r ⃗ ) using eq 6. Thus, we solve the two independent equations, eqs 7 and 8, self-consistently with respect to the two independent variables, ψ(r ⃗ ) and ω+(r ⃗ ). The condition for the self-consistency is set to |ψnew(r ⃗ ) − ψold(r ⃗ )| and |ωnew + (r ⃗ ) − −8 in the computation of the self-consistent ωold + (r ⃗ )| < 10 equations. From eq 8, we can identify the electric field and the electric displacement as E ⃗ (r ⃗ ) ≡ −∇ψ(r ⃗ ) and D⃗ (r ⃗ ) ≡ εr(r ⃗ )E ⃗ (r ⃗ ), respectively. We note that the dielectric function εr(r ⃗ ) takes the same form as that in ref 34. For weak fields with ps|∇ ψ(r ⃗ )| ∼ 0, εr(r ⃗ ) is given by εr(r ⃗ ) ∼ 1 + 4π l0∑s=±p2s ns(r ⃗ )/ 3, which is a formula derived from the Langevin dipole model in the linear response regime. For strong fields with |∇ ψ(r ⃗ )| → ∞, however, we obtain εr(r ⃗ ) = 1. These two limiting cases indicate that the effects of saturated dipoles (i.e., the dielectric saturation) are self-contained in our theory without any further assumptions. We now employ Gauss’s law in the region between the oppositely charged plates. Equation 8 then leads to εr (z)E(z) = σ +

∫0

z

dξ

∑ qsns(ξ) s =±

(9)

where z and σ are the distance from the left-hand plate and the surface charge density of the left-hand plate, respectively. Note that because of the overall charge neutrality, the integral becomes 0 when z = D. Therefore, the boundary conditions are given by εr(0)E(0) = εr(D)E(D) = σ.35 It is also important to note that the dielectric function εr(z) depends on the external electrostatic field through E(z) in eq 9. In Figure 2, by setting p+ = p−, we first illustrate the results obtained regarding the effects of weak and strong surface charges with no dielectric inhomogeneity. ϕs(r ⃗ ) ≡ vsns(r ⃗ ) denotes the volume fractions of the cation and anion for s = ± , 7088


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The Journal of Physical Chemistry C

Figure 5. Volume fractions of the cation and anion, and the dielectric function between the charged plates. p+ = 2 D and p− = 4 D. D = 1 nm. σ = 3.83 × 10−3 C/m2. (a) The volume fractions of the ionic liquids. The curves correspond to ϕ+ (black) and ϕ− (red) for (v+, v−) = (163 Å3, 263 Å3) and to ϕ+ (purple) and ϕ− (blue) for (263 Å3, 163 Å3). (b) The dielectric values of the ionic liquids.

Figure 3. Changes in the dielectric constant εr with respect to the surface charge density σ. D = 1 nm. The curves correspond to p+ = p− = 5 D (black), p+ = p− = 3 D (red), and p+ = p− = 2 D (blue).

Figure 4. Volume fractions of the cation and anion, and the dielectric function between the charged plates. p+ = 2 D and p− = 4 D. D = 1 nm. σ = 2.60 × 10−3 C/m2 for weakly charged surfaces and 1.48 × 10−2 C/m2 for strongly charged surfaces. v+ = v− = 163 Å3. (a) The volume fractions of the ionic liquids. The curves correspond to ϕ+ (black) and ϕ− (red) for weakly charged surfaces and to ϕ+ (purple) and ϕ− (blue) for strongly charged surfaces. (b) The dielectric values of the ionic liquids. The curves correspond to weakly charged surfaces (black) and strongly charged surfaces (blue).

Figure 6. Scale-free structure of the saturated layer in the case of a large separation D = 4 nm. p+ = 2 D and p− = 4 D. v+ = v− = 163 Å3. σ = 5.99 × 10−3 C/m2. (a) The volume fractions of the ionic liquids. The curves correspond to ϕ+ (black) and ϕ− (red). (b) The dielectric values of the ionic liquids. The inset illustrates the electrostatic field, E(z)p+/kT. The red dashed lines indicate the values corresponding to the case in which there is no difference in the dipole moments, i.e., p+ = p− = 2 D.

resultant spatial variation in the volume fractions of the cation and anion cause spatial inhomogeneity in the dielectric function. Thus, we suggest that in addition to the modification of the difference in the volume fractions of the cation and anion, the selection of the dipolar properties of the ionic liquids also provides an alternative strategy for controlling the spatially inhomogeneous electrostatic fields between charged plates. Indeed, we also illustrate in Figure 5 that the effects of molecular-volume asymmetry on the volume fractions of the cation and anion are comparable to those of asymmetry in the molecular dipole moments; thus, molecular volumes and dipoles can be equally important in the design of electrochemical devices that consist of ionic liquids bounded by charged surfaces, such as electroactive actuators7−9 and fuel cells.1−3 Similarly, the effects of the spreading of charge in ion structures39,40 and ionic charges40 should also be of further interest. Despite the large separation, D = 4 nm, between the charged plates, the interface between the cation-rich and anion-rich regions and the step in the dielectric function lie at approximately the center of the system at z = D/2 (Figure 6a

and b. Note that Figures 2−4 for D = 1 nm also exhibit the same behavior at z = D/2. Thus, the effects of dielectric inhomogeneity are fairly scale-free at the nanometer scale. Although the counterions are strongly associated with the charged plates, as demonstrated by the saturated layers of ion concentrations, there is no region in which the cation and anion are uniformly mixed. This effect is primarily responsible for the large spatial asymmetries in the dielectric function and the electrostatic field. Finally, we demonstrate that the dielectric contrast causes substantial variations in the capacitance of the ionic liquids in response to an applied voltage. Figure 7 illustrates the capacitance per unit area C = σ/Δψ, where Δψ = ψ(0) − ψ(D) is the applied voltage. When p+ ≠ p−, an increase in Δψ causes a decrease in C. The slope of the capacitance approaches zero as the applied voltage is decreased to Δψ < 10−1 [V] (see the inset in Figure 7). Incidentally, this characteristic behavior in close proximity to zero voltage is analogous to the results of molecular dynamics simulation41 and Landau−Ginzburg-type 7089


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spheres in nanoslabs become oscillatory because the local hardsphere packing plays a key role.14 The hard-core nature of such particles remains significant under external electrostatic fields, as demonstrated by the fact that the density profile is still oscillatory.15,16 Therefore, to investigate the coupled effects of the dielectric inhomogeneity and the hard-core nature of molecules, we developed a new hybrid method that combines the dipolar self-consistent field theory (DSCFT) presented above with Monte Carlo simulations. Hence, our theory serves as a new strategy that enables us to overcome the deficiencies in the modeling of short-range repulsive forces within the framework of the self-consistent mean-field theory. We first consider the two-point correlation function gs,s′(r ⃗ , r ⃗ ′) for uncharged, compressible hard spheres with no electrostatic interactions. The grand canonical partition function then takes the same form as that of eq 1, except without the δ function, which represents incompressibility, and yet this partition function still leads to the total free-energy functional F given in eq 3 (see the Appendix). Through this fieldtheoretical procedure, however, we enforce the following constraint on the partition function:

Figure 7. Capacitance of ionic liquids per unit area as a function of the applied voltage Δψ. The distance between the charged plates is D = 1 nm. The solid curves correspond to (p+ [D], p− [D], v+ [Å3], v− [Å3]) = (2, 4, 163, 163) (black), (2, 4, 163, 263) (red), (2, 4, 263, 163) (blue), (2, 2, 163, 163) (purple), and (2, 2, 163, 263) (green). The black dashed line is plotted for (2, 4, 163, 163) using the hypothetical dielectric constant at zero voltage εr(Δψ = 0) in eq 8 for comparison. The inset shows an enlarged section of the solid black line.

continuum theory17 for the cation and anion having the same dielectric property. However, for p+ = p−, C remains nearly constant as Δψ varies. We also obtain nearly constant capacitances when we solve eqs 6−8 with p+ ≠ p− by fixing the dielectric function εr(r ⃗ ) to a hypothetical dielectric value at zero voltage. Thus, the change in the capacitance is primarily caused by the change in the dielectric function. Because our current theory does not account for uncharged alkyl chains in the molecular structures, the resultant capacitance does not exhibit the characteristic “camel shape”.42,43 The magnitude of the variation in the case of dipole-moment asymmetry is of the same order as that of the variations in the capacitance C derived through molecular dynamics simulations5 and using a recently developed Landau−Ginzburg-type continuum theory of solvent-free ionic liquids;17 however, we note that our results arise from mean-field effects that are primarily caused by the dielectric contrast in the ionic liquids, whereas the cited studies account for ion−ion correlations through a spatially uniform dielectric constant. Thus, the difference between the dipole moments of the cation and anion can also be of critical importance in achieving the desired capacitances. Importantly, the dielectric contrast and the difference in molecular volumes between the cation and anion have equal effects on the magnitude of the capacitance C. We thus suggest that Figure 7, which illustrates the value of C for several combinations of the molecular parameters, may facilitate the selection of ionic liquids for efficient energy harvesting or optimized electrostatic energy in supercapacitors. 2.2. Hybrid DSCFT and Monte Carlo Simulation Method for Compressible Ionic Liquids. In the previous subsection, we demonstrated the significant effects of the dielectric inhomogeneity caused by differences in the dipole moments of incompressible cations and anions. However, if the liquids are compressible, then the self-consistent field theory (SCFT) for incompressible states is a gross approximation at the atomic scale, primarily because of the lack of short-range repulsive forces.33 Although a density-functional, mean-field form of the molecular interactions has often been invoked to capture compressibility, the hard-core nature of the molecules is missed in the standard formulation of the SCFT.44,45 Indeed, previous studies based on molecular dynamics simulations have demonstrated that the density profiles of uncharged hard

∑ ∫

Hconst =

d r ⃗ d r ⃗ ′μs , s ′( r ⃗ , r ⃗ ′)[⟨nŝ ( r ⃗)nŝ ′( r ⃗ ′)⟩0

s , s ′=±

− gs , s ′( r ⃗ , r ⃗ ′)]

(10)

Here, we have introduced the Lagrange multiplier μs,s′(r ⃗ , r ⃗ ′). The subscript 0 indicates that there is no electrostatic interaction (i.e., no charge or dipole). The expectation value denoted by the brackets is defined as follows: ⟨nl̂ ( r )⃗ nm̂ ( r ⃗ ′)⟩0 N

≡

∑N

+,N −

∏s =±

λs ,0s Ns !

∫ ∏iN=s 1 d rsi⃗ nl̂ ( r )⃗ nm̂ ( r ⃗ ′) exp[−H0( rsi⃗ )] N

∑N

+,N −

∏s =±

λs ,0s Ns !

∫ ∏iN=s 1 d rsi⃗ exp[−H0( rsi⃗ )] (11)

where H0 and λs,0 are the Hamiltonian and the fugacity, respectively, for uncharged, compressible hard spheres. We then extremize F with respect to the field variables, ns, ωs, and ψ. This leads to ωs( r ⃗) =

∫ dx ⃗ [2μs ,s ( r ⃗ , x ⃗)⟨ns(x ⃗)⟩0 + μs ,s′( r ⃗ , x ⃗)⟨ns′(x ⃗)⟩0 ]

ns( r ⃗) = 4πλs exp[−qsψ ( r ⃗) − ωs( r ⃗)] ×

sinh(ps |∇ψ ( r ⃗)|) ps |∇ψ ( r ⃗)| (12)

For the electrostatic term, we again obtain the modified Poisson−Boltzmann equation given by eq 4 and thus obtain eq 9 for oppositely charged plates. Note that without electrostatic interactions (i.e., qs = 0 and ps = 0), both the cation and anion simply become uncharged, nonpolar hard spheres. Equation 12 can then be reduced to ns ,0( r ⃗) = 4πλs ,0 exp[−ωs( r ⃗)]

(13)

Using eq 13, eq 12 can be transformed into 7090


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The Journal of Physical Chemistry C ⎛ λ ⎞ sinh(ps |∇ψ ( r ⃗)|) ns( r ⃗) = ⎜⎜ s ⎟⎟ns ,0( r ⃗) exp[−qsψ ( r ⃗)] × ps |∇ψ ( r ⃗)| ⎝ λs ,0 ⎠ = λs̃ ns ,0( r ⃗) exp[−qsψ ( r ⃗)] ×

for both the self-consistent equations and the Monte Carlo simulation.35 In Figure 8, we demonstrate our hybrid SCFT-MC method for ionic liquids consisting of relatively small dipolar cations

sinh(ps |∇ψ ( r ⃗)|) ps |∇ψ ( r ⃗)| (14)

where, for simplicity and without loss of generality, we have introduced the scaled fugacity λs̃ ≡ λs/ λs,0. Notably, our theoretical framework is generically applicable for any choice of the reference two-point correlation function in eq 10. Note that the number of ions s is given by Ns = ∫ d r ⃗ ns(r ⃗ ). Using Ns, we perform a Monte Carlo simulation of hard spheres interacting with each other purely through the hard-core potential as follows: ⎧ (| r ⃗ − r ⃗ ′| > d) ⎪ 0 v(| r ⃗ − r ⃗ ′|) = ⎨ ⎪ ′ ⎩ ∞ (| r ⃗ − r ⃗ | ≤ d )

(15)

where d is the ionic diameter of both the cation and the anion. From the statistical average calculated in this simulation, we can determine the number density ns, 0(r ⃗ ) for uncharged, nonpolar hard spheres. Therefore, to clarify that Ns is the input value to the Monte Carlo simulation, we indicate the variable Ns in the notation of the hard-sphere number density as follows: ns,0(r ⃗ ; N+, N−). Thus, we write the coupled self-consistent equations in the following forms: Ns

=

∫

d r ⃗ ns( r ⃗) Figure 8. Volume fractions of the cation and anion, and the dielectric function between the charged plates. p+ = 2 D and p−= 4 D. The ionic diameter is d = 6 Å, and the bulk packing fraction is η = 0.4. The curves in panels a and c correspond to the volume fractions of the cation (black) and anion (red). The red dashed lines in panels b and d represent the bulk dielectric constant εr calculated using Onsager’s formula. The insets in panels b and d illustrate the electrostatic fields, E(z)p+/kT. (a) The volume fractions for a surface charge density σ = 2.73 × 10−4 C/m2. (b) The dielectric value of the ionic liquid. (c) The volume fractions for a surface charge density σ = 1.09 × 10−3 C/m2. (d) The dielectric value of the ionic liquid.

ns( r ⃗) = λs̃ ns ,0( r ⃗ ; N+ , N −) exp{−qsψ ( r ⃗)} sinh(ps ∇ψ ( r ⃗) ) ps ∇ψ ( r ⃗)

(16)

It should be noted that Ns becomes a stochastic number through the iterative solution of eq 16 using an MC simulation of uncharged, nonpolar hard spheres. Here, we use the subscript k to denote the variable determined in the kth iteration while solving eq 16. Thus, Ns(k) =

∫

d r ⃗ λs̃ ns(,0k)( r ⃗ ; N+(k − 1), N −(k − 1)) × exp[− qsψ (k)( r ⃗)]

and relatively large dipolar anions. A marked difference between the conventional SCFT based on the density functional, mean-field form of molecular interactions and our hybrid SCFT-MC method is that the density profile exhibits oscillatory behavior. We note that this result is also consistent with the behavior of uncharged, nonpolar hard spheres between hard walls.14 Thus, our self-consistent field-theoretical approach simultaneously accounts for the hard-core nature and the intrinsic dipole moments of the particles. Moreover, simultaneous increases in the densities of both the cation and the anion in the vicinity of the weakly charged plates mark a difference between the diffuse layers of our current theory and the results of the conventional Poisson−Boltzmann theory (Figure 8a). The present result thus indicates the formation of ion pairs near the weakly charged plates. Incidentally, this ion pairing is analogous to that featured in the radial distribution functions of ionic liquids near the uncharged surface of a C60 fullerene.46 These ion pairs tend to dissociate as the surface charge density σ increases (Figure 8c).

sinh(ps |∇ψ (k)( r )⃗ |) ps |∇ψ (k)( r )⃗ | ns(,0k)( r ⃗ ; N (k − 1)) = M[N+(k − 1), N −(k − 1)] (17)

where M symbolically indicates the Monte Carlo process. To address the issue of statistics in the Monte Carlo simulation, we consider the following analysis: Ns(k) =

∫ dr ⃗ λs̃

n(k)s ,0( r ⃗ ; N+(k − 1) , N −(k − 1)) ×

exp{−qsψ (k)( r ⃗)}

sinh(ps |∇ψ (k)( r ⃗)|) ps |∇ψ (k)( r ⃗)|

(18)

where n(k)s ,0 is the average n(k)s, 0 obtained from the (k − 1) iterations of the Monte Carlo process as indicated in eq 17. We can then achieve rapid convergence in the iterative processes 7091


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an electrochemical actuator is known to cause asymmetrical volume expansion that gives rise to a mechanical force that can bend the soft membrane-like structure of the actuator.13 Although a large number of recent studies have focused on the effects of size asymmetry on the cation and anion distributions, the effects of the dielectric contrast has been largely neglected in both theoretical and experimental studies. However, our results illustrate that the effects of the dielectric inhomogeneity and the volume asymmetry on the spatial distributions of the cations and anions are of approximately equal magnitude and are hence equally important. For example, a contrast in the dielectric values of the cations and anions causes a substantial decrease in capacitance as the applied voltage is increased (Figure 7). Therefore, we suggest that the orientational reorganization of dipoles in ionic liquids is also a key factor in the strategic control of electrochemical properties in nanostructured materials. The present results yet require quantitative validation based on experimental studies and molecular dynamics simulations. Although the current study does not account for electronic polarization, incorporating induced dipoles into DSCFT can be achieved using the deformation energy for electronic polarization within the harmonic approximation.23,28 The effects of the electronic polarizability of solvents then cause the nonmonotonic variation in the dielectric function under strong electrostatic fields.23 In this context, further in-depth studies of the electronic polarization of ionic liquids between the charged plates would also be intriguing. We also note that the capacitance resulted from our current DSCFT for incompressible ionic liquids does not exhibit the characteristic “camel shape”.10 To capture this property, further development of DSCFT that accounts for uncharged alkyl chains in the molecular structures and electrostriction would be interesting.42,43 We demonstrated that the electrostatic interactions in the vicinity of each electrode should be of different magnitude because of the dielectric contrast between the cations and anions (Figures 4, 5, 6, and 8). Thus, we suggest that the importance of the dielectric inhomogeneity can be manifested through the application of our DSCFT to the kinetics of ionic liquids between charged plates.48 Along the same lines, we anticipate further in-depth studies of the dynamics of ionic liquids in terms of the dielectric inhomogeneity as the importance of the spreading of charge in ion structures and ionic charges was demonstrated.39,40 Finally, we note that the Marcus theory for electron transfer in ionic liquids is expected to be modified primarily because of the spatial dispersion of dielectric permittivity.48 In this context, the effects of the dielectric inhomogeneity also seem to be manifested by looking at electrode current at a given electrode. This work was supported by the National Natural Science Foundation of China (21474112). We are grateful to the Computing Center of Jilin Province for essential support.

The compressible nature of ionic liquids leads to a notable feature of the dielectric function near the charged plates. Note that Onsager’s formula for the bulk dielectric constant of a liquid mixture is given by εr = ∑s=+,−nsp2s /(9π kT) = (εr − n2)(2εr + n2)/[εr(n2 + 2)2],24 where the refractive index n is 1 for nonpolarizable species and the bulk number density ns is given by the bulk packing fraction η = πd3(n+ + n−)/6. By solving this equation, we obtain εr = 6 for Figure 8. Although these magnitudes are close to those of our results for the dielectric value εr(z), compressible ionic liquids exhibit notable spatial variations in εr(z), as do incompressible ionic liquids. Furthermore, our theory indicates that the oscillatory behavior of the dielectric values is strongly correlated with that of the density profiles. Thus, the short-range repulsive potential plays an important role in determining the dielectric properties of compressible ionic liquids confined in nanochannels. Note that the dielectric inhomogeneity is caused primarily by the difference between the dipole moments of the cation and anion. Accordingly, the electrostatic field also exhibits spatial variation and hence exhibits spatial asymmetry between the plates, as shown in the insets in Figure 8. From this result, we now anticipate that the electrochemical properties of ionic liquids in nanochannels can be substantially modified by the orientational reorganization of molecular dipoles and the strong correlation of this phenomenon with the hard-core nature of ionic liquids. The correlation between the image charge and the dielectric saturation is also expected to be significant because a Gaussian-fluctuation variational approach to the image charges in nanochannels has demonstrated a marked contrast to the predictions of the conventional Poisson−Boltzmann theory.47 Among others, fullerenes solvated by ionic liquids are of immediate importance with respect to the recent interest in ionic liquid−carbon nanosystems. Our theory should be suitable for addressing the highly structured solvation layers around a C60 fullerene, which feature oscillations in the radial distribution functions.46

3. SUMMARY AND CONCLUSION In summary, we developed a DSCFT23,28,29 for incompressible ionic liquids and a hybrid theory combining DSCFT with Monte Carlo simulations for compressible ionic liquids. In our theory, we described the cation and anion as charged molecules with permanent dipoles and finite molecular volumes. To incorporate the short-range excluded-volume interaction between ions, we used the hard-core repulsive potential. We note that our theory draws upon no adjustable parameters. Because of the hard cores of the ions, compressible ionic liquids confined between charged plates exhibit oscillations in their density profiles (Figure 8) in contrast to previous coarsegrained lattice-gas models.10,11 Using DSCFT for incompressible states, we illustrated the electrical double layer formed through the strong affinity of the ionic liquids for the charged plates (Figures 2, 4, 5, 6, and 8). Specifically, our theory accounts for the difference between the dipole moments of the cation and anion. Thus, a spatial variation in the distribution of an ionic liquid causes a spatial variation in the dielectric value (Figures 4, 5, 6, and 8). Therefore, the dielectric constants derived from the classical theories of Onsager24 and Kirkwood25 can be regarded as gross approximations of the true situation in nanochannels. Notably, the spatial inhomogeneity of the cations and anions is of considerable importance in certain recently developed electrochemical devices. Indeed, the asymmetry between the distributions of cations and anions in

■

APPENDIX

Derivation of the Free-Energy Functional for Ionic Liquids

We first introduce the coarse-grained number density field ns(r ⃗ ) for the ion s using the identity 7092


Article

The Journal of Physical Chemistry C ns

∫ +nsδ[nŝ ( r ⃗) − ns( r ⃗)] = ∫ +ns +ωs × exp{i ∫ dr ω ⃗ s( r ⃗)[nŝ ( r ⃗) − ns( r ⃗)]}

1=

∫

d r ⃗ nŝ ( r ⃗)ωs( r ⃗) =

i=1

=

⎡

∫ +ρ exp⎣⎢− 12 ∫ +i

∫ +ρ+n++n−+ψ +ω++ω− ∑ ∏ N ! Ns

⎤

⃗ ⃗ ′ ρ( r ⃗)v( r ⃗ − r ⃗ ′)ρ( r ⃗ ′)⎥⎦ ∫ drdr

exp{i ∑

∫ dr ⃗ ωs( r ⃗)[ns( r ⃗) − nŝ ( r ⃗)]} ×

Z=

∫ dr ⃗ ψ ( r ⃗)[ρ( r ⃗) − ρ (̂ r ⃗)]}

F=

⃗ ⃗ ′ρ( r ⃗)v( r ⃗ − r ⃗ ′)ρ( r ⃗ ′) ∫ drdr

∫

s =±

ns

ns

+

∫ ∏ ∏ drsi⃗ dpsi⃗ i = 1 s =±

■

s =± i = 1

∫

∑ 4πλs ∫

dr ⃗

s =±

d r ⃗ ρf ( r ⃗)ψ ( r ⃗) −

sinh(ps |∇ψ ( r ⃗)|) ps |∇ψ ( r ⃗)|

∑∫

d r ⃗ ns( r ⃗)ωs( r ⃗) (24)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

s =±

Notes

The authors declare no competing financial interest.

■

⃗ ⃗ ′ρ( r ⃗)v( r ⃗ − r ⃗ ′)ρ( r ⃗ ′) ∫ drdr

∫ dr ρ⃗ ( r ⃗)ψ ( r ⃗) − i ∫ dr ⃗ ρf ( r ⃗)ψ ( r ⃗) + i ∑ ∫ dr ⃗ ns( r ⃗)ωs( r ⃗) + ∑ λs ∫ drdp ⃗ s⃗ s =±

exp[−iqsψ ( r ⃗) − iωs( r ⃗) − ips ⃗ ·∇ψ ( r ⃗)]}

REFERENCES

(1) Lee, S. Y.; Ogawa, A.; Kanno, M.; Nakamoto, H.; Yasuda, T.; Watanabe, M. Nonhumidified Intermediate Temperature Fuel Cells Using Protic Ionic Liquids. J. Am. Chem. Soc. 2010, 132, 9764−9773. (2) Sood, R.; Iojoiu, C.; Espuche, E.; Gouanve, F.; Gebel, G.; MendilJakani, H.; Lyonnard, S.; Jestin, J. Proton Conducting Ionic Liquid Doped Nation Membranes: Nano-Structuration, Transport Properties and Water Sorption. J. Phys. Chem. C 2012, 116, 24413−24423. (3) Diaz, M.; Ortiz, A.; Ortiz, I. Progress in the Use of Ionic Liquids as Electrolyte Membranes in Fuel Cells. J. Membr. Sci. 2014, 469, 379− 396. (4) Largeot, C.; Portet, C.; Chmiola, J.; Taberna, P. L.; Gogotsi, Y.; Simon, P. Relation between the Ion Size and Pore Size for an Electric Double-Layer Capacitor. J. Am. Chem. Soc. 2008, 130, 2730−2731.

+i

s =±

d r ⃗ ψ ( r ⃗)∇2 ψ ( r ⃗) −

Here, we have replaced iψ(r ⃗ ) and iωs(r ⃗ ) with ψ(r ⃗ ) and ωs(r ⃗ ) because the original fields ψ(r ⃗ ) and ωs(r ⃗ ) are purely imaginary at the saddle point.

ns

∫ +ρ+n++n−+ψ +ω++ω− δ[ ∑ vsns( r ⃗) − 1] × ⎧ ⎪ 1 exp⎨− ⎪ ⎩ 2

∫

s =±

− ipsi⃗ ·∇ψ ( rsi⃗ )} =

1 8πl0

exp[−qsψ ( r ⃗) − ωs( r ⃗)] ×

exp{− i ∑ ∑ qsψ ( rsi⃗ ) − i ∑ ∑ ωs[ rsi⃗ (t )] s =± i = 1

∫ +n++n−+ψ +ω++ω− δ[ ∑ vsns( r )⃗ − 1] × exp(−F)

where the free-energy functional of the system F is

∫ dr ρ⃗ ( r ⃗)ψ ( r ⃗) − i ∫ dr ⃗ ρf ( r ⃗)ψ ( r ⃗)

+ i∑

⎤ d r ⃗ d r ⃗ ′iψ ( r ⃗)v−1( r ⃗ − r ⃗ ′)iψ ( r ⃗ ′)⎥ ⎦

(23)

s =±

⎤ λ ns dr ⃗ ns( r ⃗)ωs( r ⃗)⎥∑ s ⎥⎦ ns ns!

∫

s =±

∫ +ρ+n++n−+ψ +ω++ω− δ[ ∑ vsns( r ⃗) − 1] ×

+i

⎤ d r ρ⃗ ( r ⃗)ψ ( r ⃗)⎥ ⎦

(22)

⎡ 1 exp⎢ − ⎣ 2

⎡ 1 exp⎢ − ⎢⎣ 2

d r ⃗ d r ⃗ ′ρ( r ⃗)v( r ⃗ − r ⃗ ′)ρ( r ⃗ ′)

where v−1(r ⃗ − r ⃗ ′) is the inverse of the Coulomb operator v−1(r ⃗ − r ⃗ ′) = −(4π l0)−1∇2δ(r ⃗ − r ⃗ ′), and 5v is the thermodynamically inconsequential normalization term resulting from the Gaussian functional integral. Thus, eq 20 can be written in the form

s =±

s =±

=

∫

⎡1 = 5 −v 1 exp⎢ ⎣2

s

∫ ∏ drsi⃗ dpsi⃗ δ[ ∑ vsns( r ⃗) − 1] ×

exp{i

(21)

where the summation over the particle number uses the N x identity ∑∞ N=0(x /N!) = e . Performing the Gaussian integral over the charge density field ρ(r ⃗ ) (the Hubbard-Stratonovich transformation) transforms the Coulomb interaction term as follows:

λsNs

i=1

∑ ωs( rsi⃗ ) i=1

where the right-hand side of the equation results from the Fourier representation of the δ function. ωs(r ⃗ ) is the Fourier conjugate field to ns(r ⃗ ). A similar procedure is performed for the total charge density ρ̂(r ⃗ ) using the charge density field ρ(r ⃗ ), which introduces the conjugate field ψ(r ⃗ ). The partition function Z in eq 1 can then be cast in the form of a functional integral as follows:

N+ , N − s =±

∑ δ( r ⃗ − rsi⃗ )ωs( r ⃗)

ns

(19)

Z=

∫

dr ⃗

(20)

Using the identity operator given in eq 19, we have replaced the instantaneous particle density nŝ (r ⃗ ) with the coarse-grained (average) density ns(r ⃗ ). Moreover, we have performed a calculation of the following type: 7093


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Dipolar Self-Consistent Field Theory for Ionic Liquids: Effects of

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