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Effects of the Dielectric Response of Single-Component Liquids and Liquid Mixtures on Electrochemical Properties between Charged Plates Hongbo Chen and 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 study the dielectric properties of single-component liquids and liquid mixtures between charged plates and develop a modified Poisson−Boltzmann (PB) equation using Booth’s theory. Our theory accounts for the permanent dipole moment and polarizability of solvents and the hydrogen bonds between solvents. We demonstrate that these key parameters are crucial for capturing the reorientation of solvent dipoles under external electrostatic fields. Our results show that variations in the capacitance of ionic solutions can be nonmonotonic, whereas the conventional PB theory predicts monotonic behavior. Specifically, the effect of hydrogen bonding produces highly nonmonotonic behavior that depends on the surface charge density and applied voltage. The resultant capacitance can be enhanced by hydrogen bonding, but a significant decline in the capacitance can also occur as the applied voltage is increased. Furthermore, our novel finding is that field-induced changes in the dielectric value may cause the unconventional order−order phase transition of two immiscible liquids. Thus, the dielectric interface may preferentially become parallel to the charged plates, whereas the linear-dielectric theory predicts that the parallel interface is always energetically unfavored. We then demonstrate that a favored orientation generally exhibits a larger capacitance. in both a single-component liquid13 and a liquid mixture.14,15 From a theoretical perspective, the dielectric value must vary spatially in response to the spatial variation in electrostatic fields.16−20 Furthermore, the dielectric contrast (dielectric inhomogeneity) between different species produces qualitatively new results that cannot be captured by the lineardielectric theory.14,15,21−23 Accordingly, the linear-dielectric expression is uncertain when describing liquids under relatively strong electrostatic fields. For example, the capacitance for the plate area S and distance D between plates is commonly given by C = ε0εS/D; however, this formula is not likely to express a universal feature of capacitors. Indeed, ionic liquids confined in nanoporous electrodes exhibit nonmonotonic variations in the capacitance with respect to the applied voltage.3,24 The dipoles of polar liquids reorientate under external electrostatic fields.13,16,25 Accordingly, the reorganized dipoles become saturated, decreasing the dielectric value. Despite its critical importance, the dielectric saturation still poses a significant challenge in theory and simulations26 because the permanent and induced dipole moments of solvents and hydrogen bonds between solvents must be accounted for simultaneously. In this study, we first derive a modified Poisson−Boltzmann (PB) equation for single-component liquids between charged

1. INTRODUCTION Ion-containing solvents confined in molecular-sized regions have attracted considerable interest in recent decades. A vast amount of literature has emerged on this subject, including recent applications for electrochemical devices such as fuel cells1 and supercapacitors.2 Supercapacitors have potential applications in portable electronics because they exhibit rapid charge−discharge properties and a longer cycle lifetime. However, achieving high energy densities remains a substantial challenge to overcome.3,4 Experimental studies5 and molecular dynamics simulations6 have revealed fundamental properties of the pore size of nanoporous electrodes, and salt doping in electrochemical devices has been a continuous central issue. By contrast, the effect of the reorientation of dipoles under external electrostatic fields on the energy efficiency has recently emerged as an interesting topic.7−12 Of particular importance is the dielectric response of solvents through electrostatic interactions. A simple way to illustrate the electrostatic interaction between charged particles is to refer to the Coulomb potential q1q2/(4πεε0r), where ε, ε0, and r are the dielectric constant, the vacuum permittivity, and the distance between charge q1 and charge q2, respectively. For given charges at a fixed distance r, the strength of the electrostatic interaction depends on the dielectric constant ε, which is a parameter that accounts for the dielectric response of the medium. The literature commonly assumes a constant value for ε by invoking the linear-dielectric theory; however, this assumption fails to explain some key features of ion solvation © XXXX American Chemical Society

Received: July 11, 2015 Revised: September 13, 2015

A

DOI: 10.1021/acs.jpcc.5b06675 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C Table 1. Values of the Molecular Parameters at T = 300 K solvent with hydrogen bonding without hydrogen bonding

α

β

N0 [m−3]

n

28/[3(73) ]

1/2

(73) /6

1.33

3.34 × 10

4/3

1/2

1.33

3.34 × 1028

1/2

28

μv [D] 2.1 1.37 3.2 2.1

ε̅ 80 37 80 37

response of solvents beyond the linear-dielectric theory, we invoke Booth’s theory for the dielectric value ε(r)⃗ .27 This theory simultaneously accounts for the permanent and induced dipole moments of solvents and the hydrogen bonds between solvents. Specifically, the dielectric value tends to decrease as the external electrostatic field is increased. A recent study based on the lattice Monte Carlo simulation showed that the theory of Booth is consistent with the observed variation in the bulk dielectric constant of ion-dissolved aqueous solutions.20 The analytical formulation is thus given by the following:

plates (section 2.1). Our theory accounts for the permanent dipole moment and polarizability of the solvents and the hydrogen bonds between solvents. The molecular parameters used in this work are listed in Table 1. To rationalize the strong dielectric response of solvents under electrostatic fields, we draw upon Booth’s theory.27 Although a simple hybrid of the PB equation and the field-dependent dielectric function was developed in previous studies,7,8,21,28,29 our theoretical formulation begins with the free energy functional. We then derive a modified PB equation by extremizing the free energy functional. This free energy minimization produces a new nonlinear term that was largely missed in previous theories and computational simulations.7,8,21,28,29 In the current study, we consider practical electrostatic fields that do not cause electrical breakdown for typical liquids. We believe this value to be smaller than 109 V/m. In section 2.1, we demonstrate that when considering the hydrogen bonds between solvents, the new term obtained from the free-energy minimization may yield a substantial nonmonotonic behavior of the capacitance of ion-containing liquids with hydrogen bonding. We then examine the role of the hydrogen bonds between solvents in the capacitance in section 2.2. We demonstrate that the effect of hydrogen bonding causes a hump in the capacitance when strengths are typical of the practical electrostatic fields. Accordingly, the capacitance may be enhanced through the hydrogen bonding in a certain range of the surface charge density and applied potential. In section 2.3, we consider immiscible liquid mixtures between charged plates. We show that the field-dependent dielectric response of liquids causes order−order transitions related to the orientation of the dielectric interface, which are not predicted by the lineardielectric theory. We also illustrate that the capacitance changes because of the morphological transitions.

ε( E ) = n 2 +

απN0(n2 + 2)μv ⎡ β(n2 + 2)μv E ⎤ ⎥ L⎢ ⎢⎣ ⎥⎦ 4πε0E kBT

(1)

where n is the optical refractive index, N0 is the number of molecules per unit volume, ε0 is the vacuum dielectric constant, μv is the dipole moment of a water molecule, E is the strength of the electrostatic field, kB is the Boltzmann constant, and T is the temperature. L(x) is the Langevin function, L(x) = coth(x) − 1/x. α and β are numerical factors related to the molecular interactions involving the hydrogen bonds between solvents. The selection of these parameters indicates the effect of hydrogen bonding on the solvent. Booth showed that the parameters are simply given by a geometrical analysis: (α = 28/ [3(73)1/2], β = (73)1/2/6) and (α = 4/3 and β = 1/2) for molecules with and without hydrogen bonding, respectively.27 For example, μv = 2.1 D and μv = 3.2 D yield the bulk dielectric constant of water ε̅ = 80 with and without hydrogen bonding, respectively. Similarly, the bulk dielectric constant of ethylene glycol ε̅ = 37 can be reproduced using μv = 1.37 D and μv = 2.1 D with and without hydrogen bonding. The molecular parameters used in this work are listed in Table 1. Note that eq 1 can be approximated as ε(E ∼ 0) = n2 + [(αβπN0(n2 + 2)2μv2)/(12πε0kBT)] + O(E) when the electrostatic field E is weak. Thus, the effect of hydrogen bonding on the dielectric response can be described primarily by the single parameter αβ, which is an analogue of the Kirkwood g-factor in the theory of Kirkwood.30 Under strong electrostatic fields, however, this expression becomes a gross approximation.27 In previous studies, eq 1 was simply combined with the PB equation.7,8,21,28,29 In our study, however, we show that a simple hybrid of the two equations is a drastic simplification of the true mean-field equation derived from the free energy of the system. Thus, critically important effects of the dielectric response under external electrostatic fields are missed. To illustrate this fact, we start with the free energy functional:

2. THEORY 2.1. Single-Component Liquids. We consider singlecomponent liquids and monovalent ions confined between two oppositely charged plates separated by distance D (Figure 1). The two plates are located at z = ±D/2 and have surface charge densities of ± σ. The system is charge-neutral because the numbers of ions are identical. To rationalize the dielectric

F=

∫

⎤ ⎡ ε ε( r ⃗)(∇ψ ( r ⃗))2 d r ⃗⎢ − 0 + ρ ( r ⃗ ) ψ ( r ⃗ )⎥ 2 ⎦ ⎣

(2)

where ρ(r)⃗ = en+(r)⃗ − en−(r)⃗ . We ascribe the charge densities of cations and anions to the Boltzmann distribution, n± = n0 exp[ ∓eψ(r)⃗ /kBT].31,32 n0 is the bulk concentration of the ions, and ψ(r)⃗ is the electrostatic potential. Interested readers are referred to a more complete derivation of the free energy

Figure 1. Schematic illustration of ion-containing liquids confined between charged plates. B


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The Journal of Physical Chemistry C and charge densities via a field-theoretical treatment in Appendix A. Extremizing the free energy functional with respect to the field potential ψ yields ∇·[ε( r ⃗)∇ψ ( r ⃗)] −

∫ d r ⃗ ∂∂ψε((rr⃗)⃗)

ρ( r ⃗) [∇ψ ( r ⃗)]2 =− ε0 2 (3)

A simple hybrid of the PB and Booth equations7,8,21,28,29 ignores the second term on the left-hand side of eq 3. In this study, however, we demonstrate that significant nonlinear effects are caused by the new additional term. Using the dielectric function in eq 1 to calculate the second term, we obtain the following: ⎡ − L(β′E)∇ψ ( r ⃗) ∂ε( r ⃗) G(E)∇ψ ( r ⃗) ⎤ = 2α′⎢ + ⎥ ·∇δ( r ⃗ − x )⃗ ⎣ ⎦ ∂ψ (x ⃗) E3 E2 (4)

We can then cast eq 3 into ⎧⎡ ⎫ ⎛ L(β′E) ⎞⎤ ρ( r ⃗) ∇·⎨⎢ε( r ⃗) − α′⎜ − G ( E )⎟ ⎥ E ⃗ ( r ⃗ ) ⎬ = ⎝ ⎠ ε0 E ⎣ ⎦ ⎩ ⎭

Figure 2. Electrostatic field E between the charged plates for different values of the surface charge density σ′. The bulk ion concentration is n0 = 0.01 M. (a) σ′ = 1.5 V/Å and (b) σ′ = 2.0 V/Å. The colors of the lines correspond to the conventional PB theory (red dashed line), the conventional PB equation simply combined with the Booth equation (blue dot-dashed line), and the modified PB equation (black solid line). The inset shows the corresponding cation distributions.

(5)

where α′ = (απN0(n + 2)μv)/(8πε0), β′ = (β(n2 + 2)μv)/kBT, and G(E) = ∂L(β′E)/∂E = −(β′/sinh2(β′E)) + 1/(β′E2). The electrostatic field is given by E⃗ (r)⃗ = −∇ψ (r)⃗ . Interested readers are referred to details of the derivation of eq 5 in Appendix B. Note that L(β′E)/E − G(E) is (2β′3E2)/45 + O(E)3 in the high temperature or weak electrostatic field limit with μvE/ (kBT) ≪ 1. Up to O(E), therefore, eq 5 reduces to ∇· [ε(r)⃗ E⃗ (r)⃗ ] = ρ(r)⃗ /ε0. Thus, a simple hybrid of the PB and Booth equations is the limiting case of our modified PB equation. We solve eq 5 for the electrostatic potential ψ(r)⃗ with respect to the surface charge density σ. We then calculate the capacitance per unit area as follows: σ σ C= = Δψ ψ (D/2) − ψ ( −D/2) (6) 2

the capacitor as an energy storage device. Figure 3 shows that notably nonmonotonic variations in the capacitance are caused by the effect of the nonlinear dielectric response of hydrogenbonded solvents. The capacitance derived from the conventional PB equation simply increases as the surface charge density σ′ increases (Figure 3a). This increase is purely the result of the addition of ions because otherwise the capacitance would have been constant within the conventional equation C = ε0ε/d. ̅ By contrast, a simple hybrid of the PB equation and dielectric function ε(r)⃗ qualitatively alters this behavior (Figure 3b), and the capacitance tends to exhibit weak nonmonotonicity as the surface charge density σ′ increases. As illustrated in Figure 2b, a simple hybrid of the two equations can be a drastic simplification of the mean-field equation derived from the free-energy minimization. Accordingly, we show the results of our modified PB equation in Figure 3c. Remarkably, our modified PB equation predicts highly nonmonotonic variations in the capacitance, even with relatively low surface charge densities σ′ or low ion concentrations (n0 < 0.1 M). This behavior is caused primarily by the decrease in the dielectric value subject to the increase in the electrostatic field. Thus, our modified PB theory suggests that the field-dependent dielectric response of solvents may significantly affect the capacitance both qualitatively and quantitatively. Our modified PB theory indicates the substantial hump in the capacitance. In the case of ionic liquids, this hump in the capacitance has attracted considerable attention in the form of theoretical,33,34 experimental,3 and computer-simulation studies.3,24,35−37 Indeed, simple mean-field theories33,34 suggest that this behavior could be caused by the compressible nature of ionic liquids. Moreover, a study based on Monte Carlo simulations indicated that when ions consist of charged heads and neutral counterparts, the latter component behaves like “voids” in compressible fluids and produces the hump in the

For the model parameters, we consider the water to be at T = 300 K. The optical refractive index is n = 1.33, and N0 = 3.34 × 1028 m−3, as calculated from the density of pure water. The dipole moment of water is set to the value for the liquid phase, μv = 2.1 D.27 The boundary conditions are given by ε(−D/ 2) E(−D/2) = ε(D/2) E(D/2) = σ′ according to Gauss’ law.31 For descriptive conciseness, we used σ′ ≡ σ/ε0. We choose the distance between the charged plates to be D = 1 nm. We first illustrate the results of the electrostatic field using α and β for hydrogen bonding of water (Figure 2). We compare the results of the conventional PB equation, the conventional PB equation combined with the dielectric function, and the modified PB equation (i.e., our new equation). The differences observed in these comparisons reflect the effect caused by the second term in eq 3. Importantly, the differences between the theories become more pronounced as the surface charge density σ′ is increased. Our results indicate the difference between the conventional PB equation with the bulk dielectric constant ε̅ and the modified PB equation with the fielddependent dielectric function. Furthermore, we find that a simple hybrid of the PB equation and dielectric function ε(r)⃗ overestimates the ion distribution (see the inset in Figure 2b). We now demonstrate that the differences caused by the fielddependent dielectric function substantially alter the efficiency of C


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The Journal of Physical Chemistry C

listed for water in the previous section. We also choose the distance between the charged plates to be D = 1 nm. Figures 4 and 5 show the comparison between the capacitances of the ion-containing solvents with and without

Figure 3. Capacitance C per unit area for different ion concentrations as a function of the surface charge density σ′. The colors of the lines correspond to 0.001 M (purple), 0.01 M (blue), 0.08 M (green), 0.2 M (red), and 0.5 M (black). (a) The conventional PB theory predicts monotonic behavior of the capacitance. (b) A simple hybrid of the PB equation and the local dielectric function predicts weak nonmonotonicity in the capacitance as σ′ increases. (c) The modified PB equation with the local dielectric function predicts highly nonmonotonic behavior of the capacitance.

capacitance with respect to the applied voltage.35 The effects of electrostriction and the dense packing of ions were also shown to contribute to the hump in the capacitance.36 In our case, however, the field-dependent dielectric response of solvents is the primary cause of the hump. Therefore, our result is strikingly different from previously reported findings. In this context, we anticipate that the current prediction yet requires further in-depth studies based on computational simulations for specific solvents. 2.2. Effects of Hydrogen Bonding in Single-Component Liquids. Of further interest is the effect of hydrogen bonding on the dielectric response of solvents. To illustrate this effect, we compare two solvents with and without hydrogen bonding with the same bulk dielectric constant ε̅ under no electrostatic field. For ε̅ = 80, the dipole moments are set to μv = 2.1 D27 and μv = 3.2 D for solvents with and without hydrogen bonding, respectively. For a relatively low-dielectric solvent with ε̅ = 37, we use μv = 1.37 D and μv = 2.1 D for solvents with and without hydrogen bonding, respectively. Other parameters in the Booth equation are identical to those

Figure 4. Capacitances of solvents with (red circles) and without (black squares) hydrogen bonding for different ion concentrations. The bulk dielectric constant is ε̅ = 80. The ion concentrations are (a) n0 = 0.001 M, (b) 0.01 M, (c) 0.2 M, and (d) 0.5 M.

hydrogen bonding. The effects of the hydrogen bonds between solvents cause a highly nonmonotonic behavior of the capacitance (Figures 4 and 5). Of particular importance is the effect of salt ions. When the surface charge σ′ (or applied voltage) is relatively small, the effect of hydrogen bonding tends to enhance the capacitance of the salt-doped conductors (Figures 4c,d and 5c,d). However, at large surface charge densities, the significant decline in the capacitance may be caused by hydrogen bonding. Therefore, we suggest that the hydrogen bonds between solvents play a crucial role in the D


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Figure 6. Schematic illustration of two immiscible liquids confined between two charged plates. The distance between the plates is L, and the potential difference is Δψ. The bulk dielectric constants of liquid 1 and liquid 2 are ε1 and ε2, respectively. (a) Parallel state: The dielectric interface is parallel to the charged plates and located at z = L/2. (b) Perpendicular state: The dielectric interface is perpendicular to the charged plates.

predicts a difference between the free energy densities of the two states, f 0para − f 0perp = [ε0(ε1 − ε2)2(Δψ/L)2]/[4(ε1 + ε2)] ≥ 0.46 Thus, the dielectric interface is always perpendicular to the charged plates when the surface interaction near the plates is insignificant. In this section, however, we demonstrate that the effect of the strong field-dependence of the dielectric response can cause the order−order transition between the parallel and perpendicular states, altering the capacitance. The key to our theory is that we calculate the fielddependent values of ε1 and ε2 using the Booth equation in eq 1. The boundary condition of the dielectric interface in the parallel state is given by ε1(E1)E1 = ε2(E2)E2. For the electrostatic potential, we write Δψ = (E1 + E2)L/2. We thus obtain E1 = {2ε2(E2)/[ε1(E1) + ε2(E2)]}(Δψ/L) and E2 = ε1(E1)E1/ε2(E2). The surface area of the plate is denoted by S. The free energy for the parallel states is then given by the following: LSε0 ⎡ 1 ⎤ 1 ε1(E1)E12 + ε2(E2)E2 2 ⎥ ⎦ 2 ⎢⎣ 2 2 2 LSε0ε1(E1) ε2(E2) ⎛ Δψ ⎞ ⎜ ⎟ =− [ε1(E1) + ε2(E2)] ⎝ L ⎠

Fpara = −

(7)

For the dielectric interface perpendicular to the charged plates, the electrostatic field E0 in the two liquids is simply given by E0 = Δψ/L. Because the surface of each liquid is S/2, we write the free energy for the perpendicular states as follows: Figure 5. Capacitances of solvents with (red circles) and without (black squares) hydrogen bonding for different ion concentrations. The bulk dielectric constant is ε̅ = 37. The ion concentrations are (a) n0 = 0.001 M, (b) 0.01 M, (c) 0.2 M, and (d) 0.5 M.

Fperp = −

⎛ Δψ ⎞2 LSε0 ⎟ [ε1(E0) + ε2(E0)]⎜ ⎝ L ⎠ 4

(8)

We then obtain the phase diagram by calculating (Fperp − Fpara)/(LS). Incidentally, this difference without the fielddependence of the dielectric values reduces to f 0para − f 0perp = [ε0(ε1 − ε2)2(Δψ/L)2]/[4(ε1 + ε2)] ≥ 0; however, the fielddependence of the dielectric values alters this inequality as discussed in the subsequent paragraphs. In our numerical calculations, the distance between two charged plates is L = 2 nm, and the difference in the electrostatic potential is Δψ = 1 V. To illustrate the important features, we consider a liquid mixture with and without hydrogen bonding. We chose values for α and β in the Booth equation accordingly. Other parameters are identical to those for water at the room temperature: (T = 300 K) and (n1, n2, N0) = (1.33, 1.33, 3.34 × 1028 m−3). Table 1 refers to fixed values for the molecular parameters. In Figure 7, we show the behavior of the free energy. The x-axis denotes the bulk dielectric constant ε2 of the solvent (liquid 2) without hydrogen

energy storage efficiency, but the applicable range of the surface charge density or applied voltage must also be considered. 2.3. Liquid Mixtures. We next consider a liquid mixture consisting of two immiscible solvents. In-depth knowledge of this system is of particular importance for understanding the morphological transitions of polymer mixtures under strong electrostatic fields.38−45 The volume fractions of the solvents are set to be equal. In general, the two solvents should have different dielectric responses under electrostatic fields. The orientation of the dielectric interface of the two liquids can be perpendicular (Figure 6a) or parallel (Figure 6b) to the electrostatic field arising from the charged plates.46−49 Therefore, the capacitance is strongly correlated with the morphology of liquid mixtures between the charged plates. However, a simple mean-field theory within the linear-dielectric level E


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energetically favorable when the model parameters are in a certain range. A remarkable reentrance into the perpendicular state through the parallel state is observed. Our current results clearly contrast the prediction of the linear-dielectric theory. We thus suggest that further experiments and computer simulations are required to validate these predictions. Changing the electrostatic field Δψ/L is likely to be relatively experimentally easy. For the dielectric constants, we propose that changing the temperature would be a good strategy. In accordance with the orientation transitions, the capacitance is also altered. Note that the parallel and perpendicular states can be treated as two capacitors connected in series and parallel, respectively. The capacitance of the two states per unit area is then given by the following (see Appendix C):46

Figure 7. Free energy per volume for the parallel (black line) and perpendicular (red line) states. The difference in the electrostatic potential is fixed at Δψ/L = 0.5 V/nm. The x-axis denotes the dielectric constant of liquid 2 without hydrogen bonding under no external electrostatic field. The dipole moment of liquid 1 with hydrogen bonding is fixed at μv = 2.1 D, and hence, ε1 = 80.

bonding under no external electrostatic field. We note that the variations in the slope of the free energy are discontinuous as the morphology changes, indicating a first-order transition. We also calculated the free energies for n2 = 1.5 and 2.0 by fixing the dielectric constants of liquid 1 and liquid 2. However, the changes in the values of ε2 for the order−order transition are within 7%. Thus, the effect of the optical refractive index on the phase diagram is relatively small. In Figure 8a, the dipole moment of liquid 2 and the difference in the electrostatic potential change when the dipole moment of liquid 1 is fixed at the value for water, μ1 = 2.1 D. Similarly, in Figure 8b, the dielectric constants of the two liquids ε1 and ε2 change when the difference in the electrostatic potential is fixed at Δψ/L = 0.3 V/nm. It should be noted that the dielectric interface parallel to the charged plates can be

Cpara =

2ε0ε1(E1)ε2(E2) [ε1(E1) + ε2(E2)]L

Cperp =

ε0[ε1(E0) + ε2(E0)] 2L

(9)

When we consider the linear-dielectric theory, that is, no fielddependence of the dielectric values, eq 9 leads to the following: ΔC linear = Cperp − Cpara =

ε0(Δε)2 ≥0 2(ε1 + ε2)L

(10)

where Δε = ε1 − ε2 is the dielectric contrast between the two liquids. Thus, the capacitance for the perpendicular state is always larger. However, the capacitance should be calculated according to the orientation of the dielectric interface. To illustrate this, we transform the free energies in eqs 7 and 8 for the parallel and perpendicular states into the following: Fpara LS Fperp LS

=− =−

Cpara(Δψ )2 2L Cperp(Δψ )2 2L

(11)

It should be noted that given the difference in the electrostatic potential Δψ is fixed, the energetically favorable state has a larger capacitance. In other words, the phase diagrams in Figure 8 can be interpreted as a diagram of the capacitance. For example, when the perpendicular state is favored, the capacitance of the perpendicular state is larger than that of the parallel phase. For visual guidance, we plotted the capacitance for μ2 = 2.4 D with L = 2 nm in Figure 9 that corresponds to the horizontal line in Figure 8a. This result suggests that the observed capacitance measures the morphological transition; the sharp variation (or “kink”) in the capacitance appears to indicate the order−order transition between the parallel and perpendicular states. Furthermore, our theory predicts that the morphological change forced by introducing the surface interaction between the plate and liquid results in a decline in the capacitance. Thus, a trade-off between the morphology and energy efficiency is likely to occur. Finally, we refer to the following two cases: (a) two liquids with hydrogen bonding and (b) no liquid with hydrogen bonding. In both cases, however, we found no noteworthy results for the phase behavior. Thus, the difference in the dielectric responses of two liquids enhanced by hydrogen

Figure 8. Phase diagrams of an immiscible liquid mixture consisting of liquid 1 with hydrogen bonding and liquid 2 without hydrogen bonding. (a) The phase diagram in the Δψ/L−μ2 plane. The dipole moment of liquid 1 is fixed at μ1 = 2.1 D. Along the dashed line, we plot the capacitance in Figure 9. (b) The phase diagram in the ε1 (liquid 1)−ε2 (liquid 2) plane. The difference in the electrostatic potential is fixed at Δψ/L = 0.3 V/nm. F


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plates can be sensitive to the ionic polarization. Our current results thus suggest the necessity of studying the correlation between hydrogen bonding and ionic polarization. In section 2.3, we demonstrated that the field-dependence of the dielectric response of two immiscible liquids (Figure 6) causes a new type of order−order phase transition. We have not considered the surface interaction between the liquid and electrode. Linear-dielectric theories without field-dependent dielectric constants predict that the dielectric interface always becomes perpendicular to the charged plates when the surface interaction is insignificant.46 However, our theory suggests that the field-dependence of the dielectric value causes the firstorder phase transition between the parallel and perpendicular states (Figures 7 and 8). Notably, we observed the reentrance into the perpendicular state from the parallel state in the phase diagrams (Figure 8). In Figure 8, we illustrated that the phase transitions can be induced by changing the permanent dipole moment, the applied voltage, and the bulk dielectric constants of the liquids. Here, we also reported that the hydrogen bonds between solvents play a crucial role in these phase transitions, significantly affecting the dielectric value. Using eq 11 for the relationship between the free energy and capacitance, we then showed that given the insignificant surface interaction, the capacitance of the energetically favored state becomes larger than that of the other state. In this context, we anticipate that the coexistence of the two different morphologies will cause a decline in the capacitance. Thus, the trade-off between the morphology and energy efficiency is likely to be an issue. We thus suggest that the observed capacitance measures the morphological transitions of two immiscible liquids between the charged plates. The capacitance should exhibit sharp variations, possibly kinks, in its slope through the morphological transitions. Finally, we state that our current theory does not account for the condensation of ions near charged plates and, hence, the Stern layer. However, a previous study based on the PB equation combined with Booth’s theory showed notable effects of the field-dependent dielectric response in the Stern layer.7 Thus, our present results should motivate further investigations into the effect of the field-dependent dielectric value on the capacitance and morphological transition. For example, the surface interaction, which is critically important for the morphological transitions,46,47,49 will likely be substantially modified in the Stern layer. In this context, further in-depth studies of the correlation between the electrostatic interaction and hydrogen bonding would be intriguing.

Figure 9. Capacitance of an immiscible liquid mixture consisting of liquid 1 with hydrogen bonding and liquid 2 without hydrogen bonding. The colors of the lines correspond to the parallel (black) and perpendicular (red) states. The dipole moment of liquid 2 is fixed at μ2 = 2.4 D. This figure corresponds to changes in the potential difference Δψ/L along the horizontal dashed line in Figure 8a.

bonding is essential to feature the results presented in this subsection.

3. SUMMARY AND CONCLUSION In summary, we studied the effects of the reorientation of solvent dipoles on the electrochemical properties of singlecomponent liquids and liquid mixtures between charged plates. Our theoretical formulation starts with the free energy functional for the liquids, which accounts for the permanent and induced dipole moments of solvents and the hydrogen bonds between solvents. To capture the saturation of solvent dipoles under electrostatic fields, we invoke Booth’s theory at the level of the free energy functional (or the partition function in Appendix A). By extremizing the free energy functional, we obtain a mean-field equation for the systems beyond the lineardielectric level. Our theoretical framework differs from the empirical application of a simple hybrid of the PB equation and the field-dependent dielectric function. Indeed, the nonlinear term arising from the derivative of the free energy functional produces strikingly new features. (a) A simple hybrid of the PB equation and field-dependent dielectric function, such as Booth’s dielectric function, misses significant effects of the nonlinear dielectric response of solvents (section 2.1). Thus, the electrostatic field and ion distributions differ substantially from those obtained using the more fundamental, free energyminimization principle (Figure 2). Accordingly, our theory showed notably nonmonotonic behavior of the capacitance of salt-doped single-component liquids (Figure 3). Of technological importance is the fact that the capacitance exhibits significant declines as the surface charge density (or applied voltage) is increased. (b) Our theory predicts a notable hump in the capacitance of a single-component liquid with hydrogen bonding when the ion concentrations are relatively large. Although recent mean-field theories and computer simulations for ionic liquids suggest similar humps primarily caused by the compressible nature of the species,33−37 our hump arises from the nonlinear dielectric response of hydrogen-bonded solvents. (c) The capacitance of salt-doped, hydrogen-bonded solvents tends to be larger at relatively small surface charge densities than that of solvents without hydrogen bonding (Figures 4 and 5). However, the relationship is reversed at high surface charge densities. For electrochemical applications, the applicable ranges of the surface charge density and applied voltage must also be considered. Of further interest would be the effect of the polarization of ions on the dielectric value, which is ignored in our current theory. Indeed, in ref 32, the ion distribution near charged

■

APPENDIX A: DERIVATION OF THE FREE-ENERGY FUNCTIONAL FOR AN ION-CONTAINING LIQUID We start with the partition function of ions in a liquid as follows:

∑

Z=

n+ , n−

λ+n+ λ−n− n+! n−!

n0

∫ ∏ d ri⃗(+) d ri⃗(−) exp(−Hc) (12)

i=1

λ+ and λ− are the fugacities of the monovalent cation and anion, respectively. The positions of the ith cation and anion are n0 (s) denoted by r(+) and r(−) i⃗ i⃗ . ĉs(r)⃗ = ∑i = 1 δ(r ⃗ − ri⃗ ) is the number density of ion s. The total charge density ρ̂(r)⃗ is given by ρ̂(r)⃗ = ĉ+(r)⃗ − ĉ−(r)⃗ . The Hamiltonians are then given by Hc = G

1 2

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

(13)


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The Journal of Physical Chemistry C where v(r ⃗ − r′⃗ ) = l0/(ε|r ⃗ − r′⃗ |) is the Coulomb potential. l0 = e2/(4πϵ0kBT) is the vacuum Bjerrum length. We now introduce the coarse-grained charge density field ρ(r)⃗ using the identity

Here, we have replaced iϕ(r)⃗ by ϕ(r)⃗ because the original fields ϕ(r)⃗ is purely imaginary at the saddle point. Extremizing the free energy functional F with respect to ϕ(r)⃗ yields ∇·[ε( r ⃗) ∇ϕ( r ⃗)] −

∫ +ρδ[ρ( r ⃗) − ρ (̂ r ⃗)] = ∫ +ρ +ϕ exp{i ∫ dr ϕ ⃗ ( r ⃗)[ρ( r ⃗) − ρ (̂ r ⃗)]}

1=

■

APPENDIX B: DERIVATION OF THE MODIFIED POISSON−BOLTZMANN EQUATION We provide details of the calculation for eqs 4 and 5. Note that E⃗ (r)⃗ = −∇ψ(r)⃗ and |∇ψ(r)⃗ | = {[∇ψ(r)⃗ ]2}1/2. Using eq 1, we can obtain the following:

⎧ ⎪ ⎪

⎩

⎫ ⎪

∫ d r ⃗ ρ( r ⃗) ϕ( r ⃗)⎬ ∑ ⎪

⎭ n+ , n−

⎡ −∇ψ ( r ⃗) ⎤ απN0(n2 + 2)μv ∂ε( r ⃗) = L(β′E)⎢ ⎥ ·∇δ( r ⃗ − x ⃗) ⎣ E3 ⎦ ∂ψ (x ⃗) 4πε0

λ+n+λ−n− n+! n−!

+

n0

×

∏ d ri⃗(+) d ri⃗(−) exp{−i[ϕ( ri⃗(+)) − ϕ( ri⃗(−))]}

∫ +ρ+ϕ exp[− 12 ∫ d r ⃗ d r ′⃗ ρ( r ⃗) v( r ⃗ − r ′⃗ ) ρ( r ′⃗ )

+i

∫ d r ⃗ ρ( r ⃗) ϕ( r ⃗) + ∑ λsVQ s]

Here, we have performed the following calculation, ∂ ⎡ 1 ⎤ ⎢ ⎥ ∂ψ (x ⃗) ⎣ |∇ψ ( r ⃗)| ⎦ ∂ {[∇ψ ( r ⃗)]2 }−1/2 = ∂ψ (x ⃗) ∇ψ ( r ⃗ ) =− ·∇δ( r ⃗ − x ⃗) E3 ∂L(β′E) ∂L(β′E) ∂|∇ψ ( r ⃗)| = · ∂ψ (x ⃗) ∂E ∂ψ (x ⃗) ∇ψ ( r ⃗ ) = G (E ) ·∇δ( r ⃗ − x ⃗) E

where where Q± = (1/V)∫ dr ⃗ exp[∓iϕ(r)⃗ ]. Here, we have performed the following calculation, n0

∫ d r ⃗ ρ ̂( r ⃗) ϕ(̂ r ⃗) = ∑ [ϕ(̂ ri⃗(+)) − ϕ(̂ ri⃗(−))]

(16)

i=1

and the summation over the particle number using the identity, ∑N∞= 0(xN/N!) = ex. We then perform the Gaussian integral over the charge density field ρ(r)⃗ , referred to as the Hubbard-Stratonovich transformation, as follows: ⎡

⎤

∫ d r ⃗ ρ( r ⃗) ϕ( r ⃗)⎥⎦

⎡1 = 5 −v 1 exp⎢ ⎣2

−

⎤

∫ d r ⃗ d r ′⃗ iϕ( r ⃗) v−1( r ⃗ − r ′⃗ )iϕ( r ′⃗ )⎥⎦

where v−1 (r ⃗ − r′⃗ ) is the inverse of the Coulomb operator v(r)⃗ = −∇· [ε0ε(r)⃗ ∇ v(r)⃗ ] = δ(r)⃗ , and 5v is the field-independent normalization term resulting from the Gaussian functional integral. We then cast eq 15 into

∫

+

∫ d r ⃗ ∂∂ψε((rr⃗)⃗)

[∇ψ ( r ⃗)]2 2

⎡

∫ d r ⃗2α′⎢⎣ −L(β′EE)3∇ψ ( r ⃗)

G(E)∇ψ ( r ⃗) ⎤ [∇ψ ( r ⃗)]2 ⎥ ·∇δ( r ⃗ − x ⃗) 2 E2 ⎦

⎡ ⎛ −L(β′E) ⎞ = ∇·[ε( r ⃗)∇ψ ( r ⃗)] + ∇·⎢α′⎜ + G (E )⎟ ∇ ⎠ E ⎣ ⎝

⎛ F ⎞ +ϕ exp⎜ − ⎟ ⎝ kBT ⎠

1 F =− 8πl0 kBT

ρ( r ⃗) = ∇·[ε( r ⃗)∇ψ ( r ⃗)] − ε0 = ∇·[ε( r ⃗)∇ψ ( r ⃗)] −

(17)

Z=

(21)

where α′ = (απN0(n2 + 2)μv)/(8πε0), β′ = (β(n2 + 2)μv)/kBT, and G(E) = ∂L(β′E)/∂E = −β′/sinh2(β′E) + 1/(β′E2). Using eq 20, we cast eq 3 into

∫ +ρ exp⎣⎢− 12 ∫ d r ⃗ d r ′⃗ ρ( r ⃗) v( r ⃗ − r ′⃗ ) ρ( r ′⃗ ) +i

4πε0E

⎡ ∇ψ ( r ⃗ ) ⎤ G (E )⎢ ⎥ ·∇δ( r ⃗ − x ⃗) ⎣ E ⎦

(20)

(15)

s =±

απN0(n2 + 2)μv

⎡ −L(β′E)∇ψ ( r ⃗) G (E )∇ ψ ( r ⃗ ) ⎤ = 2α′⎢ + ⎥ ·∇δ( r ⃗ − x ⃗) 3 ⎣ ⎦ E E2

i=1

=

(19)

When ascribing the fugacities of the cation and anion to λ+ = λ− = n0, and replacing ϕ with eψ/kbT, we obtain eq 3.

∫ +ρ+ϕexp⎨− 12 ∫ d r ⃗ d r ′⃗ ρ( r ⃗) v( r ⃗ − r ′⃗ ) ρ( r ′⃗ )

+i

∂ε( r ⃗) [∇ϕ( r ⃗)]2 2 ∂ϕ( r ⃗)

= −4πl0{λ+ exp[−ϕ( r ⃗)] − λ−exp[ϕ( r ⃗)]} (14)

where the right-hand side of the equation results from the Fourier representation of the δ function with ϕ(r)⃗ being the Fourier conjugate field to ρ(r)⃗ . Using the identity operator in eq 14, the partition function Z in eq 12 can be cast into a functional integral as follows: Z=

∫ dr ⃗

⎧⎡ ⎫ ⎤ ⎛ L(β′E) ⎞⎤ × ψ ( r ⃗)⎥ = ∇·⎨⎢ε( r ⃗) − α′⎜ − G ( E )⎟ ⎥ ∇ ψ ( r ⃗ ) ⎬ ⎝ E ⎠⎦ ⎦ ⎭ ⎩⎣

∫ d r ⃗ ε( r ⃗)[∇ϕ( r ⃗)] − {λ+ ∫ d r ⃗ exp[−ϕ( r ⃗)] + λ− ∫ d r ⃗ exp[ϕ( r )]} ⃗ 2

(22)

This equation is equivalent to eq 5. Here, we have used the integration by parts for the second term in eq 3:

(18) H


Article


∫

∂ε( r ⃗) [∇ψ ( r ⃗)]2 2 ∂ψ ( r ⃗) ⎡ −L(β′E)∇ψ ( r ⃗) G ( E )∇ ψ ( r ⃗ ) ⎤ = α′ d r ⃗ ⎢ + ⎥ 3 ⎦ ⎣ E E2

Electric Double Layer Capacitors. J. Electrochem. Soc. 2015, 162, A5158−A5178. (5) Chmiola, J.; Yushin, G.; Gogotsi, Y.; Portet, C.; Simon, P.; Taberna, P. L. Anomalous Increase in Carbon Capacitance at Pore Sizes Less Than 1 Nanometer. Science 2006, 313, 1760−1763. (6) Shim, Y.; Kim, H. J. Nanoporous Carbon Supercapacitors in an Ionic Liquid: A Computer Simulation Study. ACS Nano 2010, 4, 2345−2355. (7) Wang, H. N.; Pilon, L. Accurate Simulations of Electric Double Layer Capacitance of Ultramicroelectrodes. J. Phys. Chem. C 2011, 115, 16711−16719. (8) Sharma, P.; Mišković, Z. L. Capacitance of Graphene in Aqueous Electrolytes: The Effects of Dielectric Saturation of Water and Finite Fize of Ions. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 125415. (9) Bonthuis, D. J.; Gekle, S.; Netz, R. R. Profile of the Static Permittivity Tensor of Water at Interfaces: Consequences for Capacitance, Hydration Interaction and Ion Adsorption. Langmuir 2012, 28, 7679−7694. (10) Gongadze, E.; Iglič, A. Decrease of Permittivity of an Electrolyte Solution Near a Charged Surface due to Saturation and Excluded Volume Effects. Bioelectrochemistry 2012, 87, 199−203. (11) Nakayama, Y.; Andelman, D. Differential Capacitance of the Electric Double Layer: The Interplay between Ion Finite Size and Dielectric Decrement. J. Chem. Phys. 2015, 142, 044706. (12) Sin, J.-S.; Im, S.-J.; Kim, K.-I. Asymmetric Electrostatic Properties of an Electric Double Layer: A Generalized PoissonBoltzmann Approach Taking into Account Non-Uniform Size Effects and Water Polarization. Electrochim. Acta 2015, 153, 531−539. (13) Kornyshev, A. A.; Sutmann, G. Nonlocal Dielectric Saturation in Liquid Water. Phys. Rev. Lett. 1997, 79, 3435. (14) Nakamura, I.; Shi, A.-C.; Wang, Z.-G. Ion Solvation in Liquid Mixtures: Effects of Solvent Reorganization. Phys. Rev. Lett. 2012, 109, 257802. (15) Nakamura, I. Dipolar Self-Consistent Field Theory for Ionic Liquids: Effects of Dielectric Inhomogeneity in Ionic Liquids between Charged Plates. J. Phys. Chem. C 2015, 119, 7086−7094. (16) Debye, P. Polar Molecules; Dover Publications, Inc.: Mineola, NY, 1929. (17) Gong, H.; Freed, K. F. Langevin-Debye Model for Nonlinear Electrostatic Screening of Solvated Ions. Phys. Rev. Lett. 2009, 102, 057603. (18) Gong, H. P.; Hocky, G.; Freed, K. F. Influence of Nonlinear Electrostatics on Transfer Energies between Liquid Phases: Charge Burial is Far Less Expensive Than Born Model. Proc. Natl. Acad. Sci. U. S. A. 2008, 105, 11146−11151. (19) Jha, A. K.; Freed, K. F. Solvation Effect on Conformations of 1,2: Dimethoxyethane: Charge-Dependent Nonlinear Response in Implicit Solvent Models. J. Chem. Phys. 2008, 128, 034501. (20) Duan, X.; Nakamura, I. A New Lattice Monte Carlo Simulation for Dielectric Saturation in Ion-Containing Liquids. Soft Matter 2015, 11, 3566−3571. (21) Aguilella-Arzo, M.; Andrio, A.; Aguilella, V. M.; Alcaraz, A. Dielectric Saturation of Water in a Membrane Protein Channel. Phys. Chem. Chem. Phys. 2009, 11, 358−365. (22) Nakamura, I. Ion Solvation in Polymer Blends and Block Copolymer Melts: Effects of Chain Length and Connectivity on the Reorganization of Dipoles. J. Phys. Chem. B 2014, 118, 5787−5796. (23) Nakamura, I. Synergistic Effects of Ion Pairs on the Dielectric Properties of Diblock Copolymer Melts. Soft Matter 2014, 10, 9596− 9600. (24) Kondrat, S.; Georgi, N.; Fedorov, M. V.; Kornyshev, A. A. A Superionic State in Nano-Porous Double-Layer Capacitors: Insights from Monte Carlo Simulations. Phys. Chem. Chem. Phys. 2011, 13, 11359−11366. (25) Sutmann, G. Structure Formation and Dynamics of Water in Strong External Electric Fields. J. Electroanal. Chem. 1998, 450, 289− 302.

dr ⃗

∫

× [∇ψ ( r ⃗)]2 ·∇δ( r ⃗ − x ⃗) ⎡ ⎛ −L(β′E) ⎤ ⎞ = −∇·⎢α′⎜ + G ( E ) ⎟∇ ψ ( r ⃗ ) ⎥ ⎠ E ⎣ ⎝ ⎦

(23)

■

APPENDIX C: CAPACITANCE OF LIQUID MIXTURES BETWEEN CHARGED PLATES In this section, we demonstrate the derivation of eq 9. For the parallel state, the boundary condition of the dielectric interface is given by ε0ε1(E1)E1 = ε0ε2(E2)E2 = σ and the electrostatic potential is written as Δψ = (E1+E2)L/2. We thus obtain E1 = 2ε2(E2)/(ε1(E1) + ε2(E2))(Δψ/L) and E2 = ε1(E1)E1/ε2(E2). For the parallel state, the capacitance per unit area can be calculated as follows: Cpara =

2ε ε (E )E 2ε0ε1(E1)ε2(E2) σ = 01 1 1 = Δψ (E1 + E2)L [ε1(E1) + ε2(E2)]L

(24)

Similarly, the capacitance per unit area for the perpendicular state is given by σ ⎤ 1 ⎡ σ1 + 2 ⎥ ⎢ 2 ⎣ Δψ Δψ ⎦ ⎡ ε ε (E )E ⎤ 1 ε ε (E )E = ⎢ 0 1 0 0 + 0 2 0 0⎥ E 0L 2 ⎣ E 0L ⎦

Cperp =

=

ε0[ε1(E0) + ε2(E0)] 2L

(25)

where we have used σ1 = ε0ε1(E0)E0 and σ2 = ε0 ε2(E0)E0. Note that the numerical factor 1/2 arises from the fact that the surface area S is divided into two equivalent surface areas according to two liquids.

■

AUTHOR INFORMATION

Corresponding Author

*Phone: +86-431-85262696. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (21474112). We are grateful to the Computing Center of Jilin Province for their essential support.

■

REFERENCES

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