# Double-Layer Capacitances Caused by Ion–Solvent Interaction in the Form of Langmuir-Typed Concentration Dependence

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−3}. This increase was classically explained in terms of the Gouy–Chapman (GC) equation combined with the Stern model. Unfortunately, measured DLCs were neither satisfied with the Stern model nor the GC theory. Our model suggests that salts destroy hydrogen bonds at the electrode–solution interface to orient water dipoles toward the external electric field. A degree of the orientation depends on the interaction energy between the salt ion and a water dipole. The statistical mechanic calculation allowed us to derive an equation for the DLC as a function of salt concentration and the interaction energy. The equation took the Langmuir-type in the relation with the concentration. The interaction energy was obtained for eight kinds of salts. The energy showed a linear relation with the interaction energy of ion–solvent for viscosity, called the B-coefficient.

## 1. Introduction

## 2. Materials and Methods

_{3}PO

_{4}+ HNO

_{3}+ CH

_{3}COOH, vol. 2:1:1) to remove oxides on the surface [48]. It was immersed into the aqueous test solution by a given length, ca. 8 mm, which was determined accurately with an optical microscope. Since the wire was not sealed with any insulating material, it can avoid floating capacitive current at a gap between the electrode and the insulator. The counter electrode was a platinum wire with the area 100 times larger than that of the working electrode, and the reference electrode was Ag–AgCl in saturated KCl solution. The test solution was deaerated for 20 min before the voltage application. Aqueous solutions of LiCl, NaCl, KCl, HCl, CsNO

_{3}, KNO

_{3}, NaNO

_{3}, and NaOH were prepared with reagents at the analytical grade and the deionized water prepared by CPW-100 (Advantec, Tokyo).

## 3. Theory

^{+}and Na

^{+}in the vicinity of an electrode for a common anion. Li

^{+}is more strongly hydrated than Na

^{+}to exhibit a hydration radius of Li

^{+}larger than of Na

^{+}. It makes the dipoles oriented more radially than Na

^{+}. As a result, the diploes located between the Li

^{+}and the electrode may relax more strongly in the external field than those for Na

^{+}, yielding higher DLCs in Li

^{+}solution. A thermodynamic measure of hydration is the interaction energy between ions and water molecules. We incorporate this energy to the ionic effect on DLCs.

_{B}T with the Boltzmann constant k

_{B}at the temperature T. Since N ions are selected from M dipoles randomly, there are M!/N!(M − N)! combinations so that the probability is given by P(N) = {M!/N!(M − N)!} exp[β(μ + u)N]. Since each probability takes part in the orientation, the observed probability is a sum for N ranging from 0 to M:

_{N}

_{ = 0}

^{M}P(N) = (1 + exp[β(μ + u)])

^{M}

_{B}T ln(c/c°). Eliminate μ from Equation (1) yields

^{M}

_{N}

_{ = 0}

^{M}NP(N)/Ξ = k

_{B}T (∂Ξ/∂u)/Ξ

## 4. Results and Discussion

_{1}, vs. the imaginary one, Z

_{2}, at the Pt wire electrode in 0.5 M of LiCl, HCl, and KCl solutions fell on each line with slope ranging from 5 to 8, as shown in Figure 2. The deviation of the slopes from the infinite (a vertical line) stands for the frequency dispersion of the DLC. Letting the application of ac voltage be V = V

_{ac}e

^{iωt}for the angular velocity ω and the imaginary unit i, the observed ac current density responding to the voltage is expressed as the time derivative of the charge, CV, i.e., [15]

^{2}(dC/df) + iωC}

_{1}+ iY

_{2}= −2πf

^{2}(dC/df) + iωC

_{1}= (Z

_{1}− R

_{s})/{(Z

_{1}− R

_{s})

^{2}+ Z

_{2}

^{2}}

_{2}= −Z

_{2}/{(Z

_{1}− R

_{s})

^{2}+ Z

_{2}

^{2}}

_{s}is the solution resistance determined by extrapolating the Nyquist plot to Z

_{2}= 0. Equations (6) and (7) state that Y

_{2}/Y

_{1}= −Z

_{2}/(Z

_{1}− R

_{s}), of which values are constant from the slopes in Figure 2. We set the constant slope to be 1/λ. Replacing Y

_{1}and Y

_{2}by −2πf

^{2}(dC/df) and ωC, respectively, by use of Equations (6) and (7), we obtain the differential equation, −2πf

^{2}(dC/df)/ωC = λ or −f(dC/df)/C = λ. A solution is

_{1}f

^{−λ}

_{1}is the capacitance value at f = 1 Hz. Elimination of C in Equation (6) by use of Equation (8) yields the expression for the admittance

_{1}f

^{−λ + 1}

_{2}= log(2πC

_{1}) + (1 − λ)log f

_{2}against log f showed lines with correlation coefficients close to 0.9999 for 1 Hz < f < 5000 Hz. Values of λ obtained from the slopes (1 − λ) ranged from 0.09 to 0.11. Therefore, Equation (9) holds commonly for all the salts and the concentration ranges used here. Non-zero values of λ suggest the CPE behavior. Plots of log Y

_{1}vs. log f showed the behavior similar to those of log Y

_{2}. They included noises more than log Y

_{2}because of Y

_{1}≈ Y

_{2}/10.

_{1}from the intercepts of the lines of the plots in Equation (10) at several concentration of salts, c, and plotted C

_{1}against c in Figure 3 for some salts. Values of C

_{1}at low concentrations increased linearly with an increase in c, and approached each constant value. The ratio of the increasing amount to C

_{1}was at most 1.5, depending on the kind of salts. A line extrapolated to c = 0 means an imaginary value of C

_{1}, which might be determined for c = 0 or in pure water. Therefore, the extrapolated values should be independent of the kinds of salts. Unfortunately, (C

_{1})

_{c}

_{ = 0}varied actually with a set of experimental runs. Iterative runs showed that the variation of (C

_{1})

_{c}

_{ = 0}was not caused by salts but was by history of the Pt electrode. A long-term use of the electrode increased (C

_{1})

_{c}

_{ = 0}probably because of an increase in the surface roughness by the iteratively chemical treatment with the mixed acid. In order to avoid the aging effect, we normalized C

_{1}at any value of c with (C

_{1})

_{c}

_{ = 0}at a given salt in the following discussion of the salt effects.

^{1/2}, which inspired us to make C

_{1}vs. c

^{1/2}plots in the inset of Figure 3. The linear relation is found for c < 0.2 M although the GC theory is not only valid for c < 0.05 mM [12,52]. Consequently, our data are not satisfied with the GC theory. The c

^{1/2}dependence is caused by the Poisson–Boltzmann distribution, which is based on the electrostatic interaction of ion–field and ion–ion. The invalidity of the c

^{1/2}dependence implies that the ionic effect on the DLCs does not result from the electrostatic interaction.

_{Hm}, with the ionic capacitance, C

_{ion}, by the GC theory, the observed DLC values should have a linear relation of 1/C

_{1}with c

^{−1/2}(in Figure 4) through 1/C

_{1}= 1/C

_{Hm}+ 1/C

_{ion}, = 1/C

_{Hm}+ kc

^{−1/2}, where C

_{ion}is proportional to c

^{1/2}by the GC theory. The variations in Figure 4 have no linearity, and, hence, the Stern model is not suitable for the concentration variations of the DLCs. The concentration dependence in the previous report [53] has suggested a parallel combination of C

_{Hm}and C

_{ion}. This model is based on the following concept: Dipoles of solvent molecules are oriented both with and without aid of ionic interaction by the external electric field. We let the DLCs generated by the former and the latter dipoles be (C

_{Hm})

_{c}

_{ ≠ 0}and (C

_{Hm})

_{c}

_{ = 0}, respectively. Since the equivalent circuit of the two DLCs is a parallel combination, the observed DLC is given by the simple sum of the two

_{1}= (C

_{1})

_{c}

_{ = 0}+ (C

_{1})

_{c}

_{ ≠ 0}

_{1})

_{c}

_{ = 0}corresponds to the averaged value of 30 μF cm

^{−2}in Figure 3. In contrast, (C

_{1})

_{c}

_{ ≠ 0}corresponds to the increasing parts in Figure 3.

_{1})

_{c}

_{ ≠ 0}. In order to apply Equation (3) to our experimental results, we normalized (C

_{1})

_{c}

_{ ≠ 0}with the difference between (C

_{1})

_{c}

_{ = high}and (C

_{1})

_{c}

_{ = 0}to denote

_{1})

_{c}

_{ ≠ 0}− (C

_{1})

_{c}

_{ = 0}]/[(C

_{1})

_{c}

_{ = high}− (C

_{1})

_{c}

_{ = 0}]

_{1}/(C

_{1})

_{c}

_{ = 0}≈ 1 + Hc

_{1})

_{c}

_{ = high}/(C

_{1})

_{c}

_{ = 0}− 1}/K

_{0}= 1 + Bc, where η and η

_{0}are viscosities, respectively, for any concentration and c = 0. Figure 3 has demonstrated the linear relation of C

_{1}with c at low concentrations. Therefore, the term H can be regarded as equivalence to the B-coefficient. It is plotted in Figure 6 against the B-coefficients for eight kinds of salts. The enhancement of H with an increase in B supports that the ionic variations of the DLCs are caused by the interaction between ions and solvents. A difference of H from the B coefficients lies in the fact that signs of H were always positive while B can take negative values. In other words, ionic effects always enhance the DLCs whereas they increase or decrease viscosity depending on species.

_{B}T, being a part of −log K, on the B coefficients. A linear relation indicates that the B coefficient should represent the ionic interaction energy with solvents rather than intensity variables such as viscosity or DLCs. The linear relation seems to be in abnormal behavior in that the ionic interaction energy has the linear relation with concentrations rather than logarithmic ones. This is due to the complicatedly dynamic mechanisms of Jones-Dole B-coefficients rather than a simple difference in free energy of interactions.

_{hyd}G [44]. The dimensionless energies are plotted in Figure 8, exhibiting a positive correlation except for HCl. The correlation indicates that the ionic enhancement of the DLCs should be brought about by hydration of ions, which corresponds to the destruction of hydrogen bonds by ions to facilitate orientation by the field. The slope of the linearity is 0.03. This extremely small value suggests only 3% contribution of the hydration to the DLCs, not only because of the one-directional orientation by the external electric field from multi-directional orientations by ions but also because of difference in the number of oriented dipoles.

## 5. Conclusions

- (i)
- values of DLCs increase with ionic concentrations and reach saturated values at most 1.5 times as large as the ion-free value;
- (ii)
- the values are not at all proportional to c
^{1/2}by the GC theory; - (iii)
- the Stern model is not valid in the context of the concentration dependence;
- (iv)
- ionic effects are caused by ion–solvent interaction rather than ion-ion interaction;
- (v)
- variation of C
_{1}with c depends on ionic properties.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Variations of <N>/M with c/c° calculated from Equation (3) for (μ° + u)/k

_{B}T = (a) 0, (b) 1, (c) 2, and (d) 4.

**Figure 3.**Variations of the capacitances at f = 1 Hz with concentrations, c, of (a) NaOH, (b) LiCl, (c) KCl, and (d) KNO

_{3}, where values of C

_{1}is normalized to the averaged value, 30 μF cm

^{−2}. The inset is plots of C

_{1}against c

^{1/2}of the type of the GC theory.

**Figure 5.**Dependence of the inverse of the normalized DLCs, y, on the inverse of concentrations of the salts specified in Figure 3.

**Figure 7.**Dependence of u/k

_{B}T obtained in Figure 5 on the viscosity B-coefficients, where no value of u/k

_{B}T for CsNO

_{3}was determined explicitly.

**Figure 8.**Variation of the interaction energy of the DLCs with the Gibbs free energy of the hydration.

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**MDPI and ACS Style**

Aoki, K.J.; He, R.; Chen, J.
Double-Layer Capacitances Caused by Ion–Solvent Interaction in the Form of Langmuir-Typed Concentration Dependence. *Electrochem* **2021**, *2*, 631-642.
https://doi.org/10.3390/electrochem2040039

**AMA Style**

Aoki KJ, He R, Chen J.
Double-Layer Capacitances Caused by Ion–Solvent Interaction in the Form of Langmuir-Typed Concentration Dependence. *Electrochem*. 2021; 2(4):631-642.
https://doi.org/10.3390/electrochem2040039

**Chicago/Turabian Style**

Aoki, Koichi Jeremiah, Ridong He, and Jingyuan Chen.
2021. "Double-Layer Capacitances Caused by Ion–Solvent Interaction in the Form of Langmuir-Typed Concentration Dependence" *Electrochem* 2, no. 4: 631-642.
https://doi.org/10.3390/electrochem2040039