Solvent Variables Controlling Electric Double Layer Capacitance at

Article pubs.acs.org/JPCC

Solvent Variables Controlling Electric Double Layer Capacitance at the Metal−Solution Interface Yongdan Hou, Koichi Jeremiah Aoki,* Jingyuan Chen, and Toyohiko Nishiumi Department of Applied Physics, University of Fukui, 3-9-1, Fukui 910-0017, Japan ABSTRACT: This paper aims at finding the experimentally controllable variables of solvent for the electric double layer capacitance obtained at two parallel platinum wire electrodes in the polarized potential domain. The equivalent circuit used is the frequency-dependent double layer impedance in series with solution resistance. The evaluated capacitance shows no systematic relation with the dielectric constants, viscosity, boiling temperatures, or dipole moments of the solvents but is proportional to the inverse of the lengths of field-oriented molecules. The proportionality indicates common saturated dielectric constants, 6, of 13 solvents. The variables controlling the capacitance are the saturated dielectric constants and the lengths of solvent molecules along the dipole.

1. INTRODUCTION Capacitances of an electric double layer (EDL) vary empirically with kinds and concentration of ions,1,2 solvents, applied dcpotential and ac-frequency, materials and structure of electrodes,3 separation of electrodes, and conductivity of solutions and electrodes. Several examples can be listed here. Morita4 observed enhancement of the capacitance with an increase in electrolytic conductivity by mixing organic solvents. Jänes and co-workers5−7 examined the intensity of the capacitance varying with organic solvents and ascribed the variations to the solution resistance and relaxation time constant of the EDL. Kim8 found out that the small solvent molecules can penetrate into the porous carbon electrode to enhance the capacitance, associated with a shift of potential of zero charge. The capacitances were larger with a decrease in size of the cations4,8−11 and anions.2,4,12−21 It sometimes varies with dc-potential when nonionic substances are adsorbed on, for example, carbon electrodes.8,22,23 Big variations have often been found at the acfrequency, called the frequency dependence of capacitance or frequency dispersion. The capacitance decreases generally with an increase in frequency. The frequency dispersion has been investigated as a constant phase element (CPE).24−40 Effects of these variables on the capacitances depend strongly on unpredictable conditions especially when adsorption occurs. The unpredictable conditions are caused not only by specific adsorption but also by insufficient shield of the electrode with an insulator, geometrically uncontrolled solution resistance, limited resolution of instruments, and leakage of possibly disturbing ions from a reference electrode. Some of them have been technically solved when the two parallel wire electrodes, without insulating walls, were used.41,42 As a result, the capacitances were independent of the concentrations and kinds of halide ions and almost independent of dc-potentials.43 The invariance to the dc-potential has also been observed in aqueous solutions with halide ions on a well-defined platinum electrode,13 in 30 wt % H2SO4 aqueous solutions on glassy carbon electrodes, in 0.1 M HCl at the Ir(100) electrode,44 and © 2014 American Chemical Society

in 1-methyl-3-hexylimidazolium chloride at the glassy carbon electrode.11 The independence of the EDL capacitance from the dc-potential at solid metal electrodes differs from that at mercury electrodes because mercury shows surface tension much larger than platinum does.45,46 The frequency dependence, which was explained phenomenologically in terms of CPE, has been attributed to the logarithmic variation of the capacitance with frequency.41 Our concept of the frequency dispersion of the EDL is to introduce a frequency-varying capacitance, C(ω), where ω is the applied ac-frequency. When ac voltage, V = V0 exp(iωt), for the amplitude, V0, is applied to an EDL, the responding current should equal the time derivative of the double layer charge, q, i.e. I=

d V (t ) d(C(ω)V (t )) ∂C(ω) dω = C(ω) + V (t ) dt dt ∂ω dt (1)

The first term on the right-hand side is iωCV, belonging to the out of phase. In contrast, the second term includes no imaginary number and hence belongs to the in phase, i.e., a resistive component. Since the current in eq 1 is a sum of the real and the imaginary currents, the equivalent circuit can be represented as a parallel combination of C (defined as Cp) and the resistance (1/(∂C/∂t) = Rp). The parallel circuit corresponds to the conventional EDL capacitance, Cd, which is frequency dependent and in series with the solution resistance, Rs, as illustrated in Figure 1. This equivalent circuit allows us to evaluate the frequency-dependent capacitance. The EDL capacitance is mainly brought about by solvent rather than the ionic concentration and the type of ions in the absence of specific adsorption.43 Unfortunately, it is not clear at present which properties of solvent vary systematically the Received: February 21, 2014 Revised: April 23, 2014 Published: April 23, 2014 10153

dx.doi.org/10.1021/jp5018289 | J. Phys. Chem. C 2014, 118, 10153−10158

The Journal of Physical Chemistry C

Article

Figure 2 shows the Nyquist plots obtained in DMSO, PC, and DCM solutions including 0.1 mol dm−3 THAClO4 at

Figure 1. Equivalent circuit of the series combination of the double layer impedance, Cd, and the solution resistance, Rs. The double layer impedance is composed of a parallel combination of the capacitance, Cp, and the resistance, Rp, both being frequency dependent.

Figure 2. Nyquist plots obtained in (circles) DMSO, (squares) PC, and (triangles) DCM. High frequencies correspond to large values of Z1.

capacitance. In this paper, our focus is on seeking the key properties of solvents which influence the capacitance. Impedance is obtained at zero dc-potential at room temperature.

frequencies ranging from 10 Hz to 50 kHz. They fell on each line, suggesting only the participation of the double layer impedance rather than that of faradaic impedance in the Nyquist plots. The other ten solvents in Table 1 showed linear

2. EXPERIMENTAL SECTION The two-electrode system was used for the ac-impedance measurement. Working and counter electrodes were platinum wires, 10 mm in immersion depth and 0.1 mm in diameter, in parallel with each other. The electrodes were cleaned by distilled water and acetone orderly and dried in air. The manufacture of the platinum wire electrode was descried in our previous paper.41 The potentiostat was Compactstat (Ivium, Netherlands), equipped with a lock-in amplifier. The impedance measurements were carried out at zero dc-voltage with amplitude of 10 mV at room temperature. The electrode surface was conditioned by cyclic voltammetry in some polarized potential domains before the impedance measurements. We confirmed that the prepotential application by cyclic voltammetry did not alter the impedance data, as was in accord with our previous paper.43 The solvents used were selected to cover a wide range of dielectric constants, dipole moments, molar volume, and viscosity. Ion-exchange distilled water was used for all aqueous solutions. Each organic solvent was dried by molecular sieves for 20 min right before supporting electrolyte was dissolved. Tetra-n-hexylammonium perchlorate (THAClO4) was dried in a volumetric flask under vacuum at 40 °C for 10 h and stored in an atmospheric desiccator. Sodium tetrafluoroborate was treated at 80 °C by the above method. Potassium chloride was used as commercially obtained. NaBF4 was dissolved in glycerol at 80 °C and cooled to room temperature before use. It was dissolved in ethylene glycol and formamide at room temperature. THAClO4 was dissolved in the other organic solvents at room temperature as well. The dissolution processes were done in a vacuum environment, except for glycerol. Delay of the potentiostat was checked in our previous work.41 No abnormality was observed so far as |Z2|/Z1 > 0.04, where Z1 and Z2 are the real and the imaginary components of the ac impedance, respectively. For frequencies larger than |Z2|/ Z1 < 0.04, |Z2| values were overestimated. Most experimental conditions of the double layer measurements were in the domain of |Z2|/Z1 > 0.04.

Table 1. Values of the EDL Capacitance, dmv and dor, for 13 Solvents no. 1 2 3 4 5 6 7 8 9 10 11 12 13

solvents

(Cp)1Hz/μF cm−2

dor/nm

dmv/nm

(εr)sat

11 12 10 15

0.400 0.640 0.536 0.263

0.444 0.534 0.557 0.475

5.0 8.8 5.8 4.5

14

0.369

0.491

5.7

11 20 24 17 16

0.316 0.415 0.304 0.543 0.544

0.460 0.453 0.405 0.495 0.461

4.0 9.3 8.1 10.6 10.2

9

0.614

0.521

6.4

11

0.431

0.513

5.4

35

0.158

0.312

6.3

acetonitrile aniline benzyl alcohol dichloromethane (DCM) dimethyl sulfoxide (DMSO) ethanol ethylene glycol formamide glycerol n-methylformamide (NMF) propylene carbonate (PC) tetrahydrofuran (THF) water

Nyquist plots similar to those in Figure 2. The extrapolation of the line to Z2 = 0 for infinite frequency yields the solution resistance, Rs. If the EDL impedance were to be expressed by a simple series combination of an ideal resistance and an ideal capacitance, the Nyquist plot should be a vertical line. The experimental values of the slopes ranging from 4.5 to 9.0 in Figure 2 indicate that the resistance should vary with frequency. A meaning of the slope will be discussed later. The equivalent circuit in Figure 1 of two electrodes leads to the expression of the in phase and the out of phase components, as follows 2 Z1 + iZ 2 = R s + 1/R p + iωC Z1 − R s =

3. RESULT AND DISCUSSION 3.1. Frequency-Dependence of Capacitance.

2R p

, 1 + (ωCR p)2

Z2 =

−2ωCR p2 1 + (ωCR p)2

(2)

The explicit forms of Cp and Rp are given by 10154


The Journal of Physical Chemistry C Cp =

−2Z 2 1 , ω (Z1 − R s)2 + Z 22

Rp =

Article

(Z1 − R s)2 + Z 22 2(Z1 − R s)

The ratio looks to vary logarithmically with the frequency. If the more precise approximation of Cp ≈ (Cp)1Hz − k ln f + k1(ln f)2 is used, the ratio is given by

(3)

Values of Cp and Rp were determined from Z1 and Z2 at each frequency by use of eq 3. They were ca. half values of Cd (= −2/ωZ2). Figure 3 shows variations of Cp with logarithm of

−Z 2/(Z1 − R s) =

(Cp)1Hz − k ln f k − k1 ln f

Since values of (Cp)1Hz/k were close empirically to those of k/ k1, the logarithmic term is canceled. Consequently, the Nyquist plot is represented by −Z 2/Z1 ≈ −Z 2/(Z1 − R s) = k /k1

frequency (f = ω/2π) in DMSO, PC, and DCM solutions. Values of Cp show linear relations with the logarithmic frequency, regardless of kinds of solvents, i.e. (4)

where k is a constant and (Cp)1Hz is the capacitance at f = 1 Hz by the extrapolation in Figure 3. The linear variation has been demonstrated to hold for aqueous solutions including several kinds of salts and their concentrations.43 Values of (Cp)1Hz and k depended on solvents. Figure 4 shows logarithmic variation of Rp against the logarithm of frequency for the three solvents. The plots for all

Figure 5. Variation of Cp with dc-potential and cyclic voltammogram (scan rate 0.02 V s−1) in 0.1 M THAClO4 + propylene carbonate at the two platinum electrodes.

Figure 4. Logarithmic plots of Rp evaluated from eq 3 in (circles) DMSO, (squares) PC, and (triangles) DCM solutions including 0.1 mol dm−3 THAClO4 against log( f).

solution, where the cyclic voltammogram is also shown. No variation of (Cp)1Hz was found in the polarized potential domain. Consequently a potential of zero charge cannot be determined. This is also true in the aqueous solution, previously reported.43 Our observation seems inconsistent with a wellknown minimum in a capacitance vs potential curve at a mercury electrode.1 The minimum is a property of the diffusion double layer capacitance, CD, predicted from the Gouy− Chapman theory. Since mercury has much higher surface tension than platinum, the Helmholtz (inner) layer capacitance, CH, at mercury is larger than that at platinum. As a result, 1/CH < 1/CD at mercury electrodes,1 whereas 1/CH > 1/CD at platinum electrodes. Then the observed capacitance, obeying 1/Cp = 1/CH + 1/CD, at the platinum electrode is controlled by CH, which has no minimum.

the solvents used showed linear relation with the slope −1. Therefore, Rp should be inversely proportional to the frequency. Inserting eq 4 into the definition of Rp (1/(∂C/∂t) = Rp) and carrying out the differentiation, we obtain R p = 1/2πfk

(7)

Thus, values of Z2/Z1 are constant. This is the demonstration of providing a line of the Nyquist plots (in Figure 2) for the double layer impedance. In summary, the linearity of the Nyquist plots indicates that the EDL impedance can be expressed by the equivalent circuit in Figure 1, which is based on the frequency dependence of the EDL capacitance. The linearity was found for all the solvents. The EDL resistance is inversely proportional to the frequency. It does not exist physically but is observed inevitably through eq 1 when the impedance is measured as a response of time variation. The capacitance decreases linearly with the logarithmic frequency, obeying eq 4. We selected a characteristic variable of the capacitance at 1 Hz, denoted by (Cp)1Hz. Values at any other frequency can be used for the comparison of Cp of different solvents so far as they have a linear relation with log f. A frequency of 1 Hz may be convenient for comparison with the capacitance by cyclic voltammetry in the future. 3.2. Key Property of Solvents to EDL Capacitance. Variation of Cp at 1 Hz with voltages between two platinum electrodes is shown in Figure 5 for propylene carbonate

Figure 3. Plots of Cp evaluated from eq 3 in (circles) DMSO, (squares) PC, and (triangles) DCM solutions including 0.1 mol dm−3 THAClO4 against log( f).

Cp = (Cp)1Hz − k ln f

(6)

(5)

This is the verification of the inverse proportionality, which was derived by inserting eq 4 into the second term of eq 1 and can carry out the differentiation. Taking the ratio of eq 2 yields −Z2/(Z1 − Rs) = ωRpCp. When eqs 4 and 5 are inserted into the above equation, we have −Z 2/(Z1 − R s) = (Cp)1Hz /k − ln f 10155



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Table 1 shows a list of values of (Cp)1Hz obtained in 13 solvents. Figure 6 shows the plot of (Cp)1Hz vs dielectric constants of the solvents, where the numbers in the marks correspond to those in Table 1.

Figure 8. Variations of the capacitance with (a) dipole moments, (b) boiling temperatures, and (c) viscosity.

electrode area. If the electrode is microscopically rough, the capacitance values must be smaller than those in Table 1 by the roughness factor. The plots in Figures 6−8 are, however, not independent essentially of the roughness because the y-axis is inversely proportional to the roughness factor. The highest correlation in Figures 6−8 is the linearity of the capacitance with 1/dmv. The use of dmv for d implies that solvent molecules in the EDL take the thermodynamic average of the diameters by thermal fluctuation or rotation. This is an unrealistic assumption because the molecules should be arranged in higher order by the strong electric field or solvent−electrode interaction than in the bulk. Molecules with strong hydrogen bonds such as of water (13) and formamide (8) might be compressed in the EDL to take smaller values of d than dmv. Then true values of d in the EDL might make the plot in Figure 7 pass through the origin. Since molecules in the EDL are oriented by the external field, values of d controlling the capacitance may be the molecular length oriented in the direction of the dipole rather than the diameter estimated from molar volume. A technique of estimating the controlling d is to use a model of molecules oriented in the direction of the dipole. The 3D structures of molecules were simulated and optimized by means of Chemoffice software which was based on the AM1 method.48 Lengths of oriented solvent molecules, dor, were estimated from bond lengths and bond angles in the optimized structure and corrected by atomic sizes. We take DMSO for example, as shown in Figure 9(a). The 2D molecular structure was first plotted in ChemDraw of Chemoffice and then pasted in Chem3D. Optimization of the molecular structure yielded the bond length and band angles. The dipole vector was obtained from the Models 360 database (Chemical Education Digital Library (online)49). The length of a molecule along the dipole was an orthogonal projection of the 3D image on the vector. Figure 9(b) shows some illustrations of the oriented molecular structures, together with dielectric moments which are perpendicular to the electrode surface. The values of dor, listed in Table 1, are mostly smaller than dmv for molecules. Figure 10 shows the variation of the capacitances with the longest lengths of the oriented molecules, dor. The error bar of each data point corresponds to any possible tilt of solvent dipole by forces caused other than the electric field. The proportionality is found, i.e.

Figure 6. Variation of the capacitance with dielectric constant of the solvents.

According to the Helmholtz layer of Stern’s model, dielectric constant is a crucial variable of the capacitor at a unit area through C = εoεr/d,47 where d is the separation of the parallel plate model of the inner layer, which may be close to a diameter of solvent. Although the capacitance of the parallel plate model should be proportional to the dielectric constant on the assumption of a common value of d, the measured capacitances have no relation with the dielectric constants.7,8 Dielectric saturation may occur in the double layer, as has been pointed out.7,8 If saturated dielectric constants take a common value to the 13 solvents, (Cp)1Hz should be inversely proportional to d. Values of d can be estimated from the molar volume to be designated as dmv = (M/ρNA)1/3. Here M is the molecular weight, ρ the density of solvent, and NA the Avogadro constant. Figure 7 shows dependence of (Cp)1Hz on 1/dmv. Although

Figure 7. Variation of the capacitance with the inverse of molecular diameter obtained from the molar volume.

linearity is found, the plot is not inversely proportional to dmv (see the negative value of the intercept). Consequently the distance is not a crucial variable for the capacitance. The other possible properties of the solvents are (a) dipole moment, μ, (b) boiling point, Tbp, and (c) viscosity, η. The dipole moment may contribute to molecular orientation in the EDL, while the boiling point can represent interaction among oriented molecules in the EDL. Viscosity may take part in slow relaxation of the frequency dependence of the capacitance. Figure 8 show variations of the capacitance with these variables. All the points were scattered. Consequently, the EDL capacitance cannot be determined by these variables. The values of (Cp)1Hz in Table 1 and Figures 6−8 were the determined capacitance values divided by the geometrical

(Cp)1Hz = εo(εr)sat /dor

(8)

where (εr)sat is the saturated dielectric constant. The proportionality in Figure 9 means that (εr)sat is common to the 13 solvents. Since values of dor were evaluated from the orthogonal projection of the molecules on the dipole vectors, 10156



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Both bulk and interfacial properties of the metal−solution system were examined to seek for the variables controlling the EDL capacitance in the absence of specific adsorption. 1. The capacitance cannot be represented by simple relation with the dielectric constant, viscosity, boiling temperature, or dipole moment of solvents. It increases linearly with an increase in reciprocal diameter of solvent molecules evaluated from the molar volumes, showing a negative intercept. 2. The capacitance is inversely proportional to the length of a solvent molecule along the dipole moment. 3. The solvents take an approximately common value of the saturated dielectric constant.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone +81 776 27 8665. Fax: +81 776 27 8750. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was financially supported by Grants-in-Aid for Scientific Research (Grant 22550072) from the Ministry of Education in Japan.

■

Figure 9. Models of solvent molecules: (a) an example of estimating dor of DMSO molecule from ChemOffice and (b) solvent molecules with orientation at the interface and in bulk.

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Solvent Variables Controlling Electric Double Layer Capacitance at

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