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Molecular interaction model for frequency-dependence of double layer capacitors
Electrochimica Acta 188 (2016) 545–550
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Electrochimica Acta journal homepage: www.elsevier.com/locate/electacta
Molecular interaction model for frequency-dependence of double layer capacitors Koichi Jeremiah Aoki1,* Electrochemistry Museum, 3-1304 Takagichuou, Fukui 910-0804, Japan
A R T I C L E I N F O
A B S T R A C T
Article history: Received 29 September 2015 Accepted 5 December 2015 Available online 10 December 2015
Electric double layer capacitances at electrodes exhibit frequency-dependence, decreasing with the frequency in accordance with the power law even at frequency as low as 1 Hz. Since the capacitance is mainly caused by orientation of dipoles of solvent molecules in the Helmholtz layer, the frequency dependence may be associated with the rate of the orientation. The frequency of the orientation on molecular scale is of the order of GHz, whereas the observed frequency domain is less than 10 kHz. The difference implies that the frequency dependence should not be controlled by simple flips of the dipoles but be caused by a slow process induced by the orientation. This belongs to a cooperative phenomenon. We consider here as the inducing causes the solvent-solvent interaction, the image force of the dipoles to the electrode and the orientation by the external field. The energy balance indicates that most dipoles are confined regularly on the electrode surface as a 2D phase. The 2D-Monte Carlo simulation similar to that for the Ising model is performed in the light of the competition of the interaction and the orientation. The simulation demonstrates aggregation of the oriented dipoles which are hindered by the interaction. The aggregation and hence the capacitance grow gradually with the time of voltage application. The increasing rate obeys the experimentally observed power law. ã 2015 Elsevier Ltd. All rights reserved.
Keywords: Frequency-dependence of electrical double layer capacitance Monte Carlo simulation by Ising model Solvent–solvent interaction Formation of two-dimensional phase
1. Introduction Capacitances of electric double layers are known to vary with applied ac-frequency in the polarized potential domain [1–3]. The frequency dependence can be empirically diminished with a decrease in surface roughness [4–7] as well as with a use of single crystal surfaces [8–10]. It has been considered to be caused by fractal dimensions of electrodes[11,12], adsorption [13–16], lateral charge spreading in the double layer [17], harmonic components [18], and heterogeneities of electrodes on the atomic scale [19]. It is also associated with dielectric loss [1], cracks on electrodes and/or insulators [2] and cell geometry [20]. Quantitative work on the frequency dependence has not yet been reported, to our knowledge, and the subject is still under debate [21,22]. The frequency dependence has often been analyzed by use of equivalent circuits composed of ideal resistances and ideal capacitances. It is difficult not only to represent the frequency dependence as combinations of ideal electric elements but also to interpret the physical meaning of the equivalent circuits. A strategy
* Corresponding author. Tel.: +81 90 8095 1906; fax: +81 776 27 8750. E-mail addresses: [email protected], [email protected] (K.J. Aoki). ISE member.
1
http://dx.doi.org/10.1016/j.electacta.2015.12.049 0013-4686/ ã 2015 Elsevier Ltd. All rights reserved.
of the analysis without complicated equivalent circuits is to introduce an intrinsically non-ideal capacitance, which has a function of the capacitance as the frequency. When the capacitance, C, varies with the frequency or the time, t, the current caused by the capacitive charge, q = CV, is given by I = dq/dt = CdV/dt + VdC/ dt for a time-varying voltage V. The second term is nothing but the functional form to be determined. When the ac-voltage, V ¼V0ei2pft, is inserted into the above equation, the current can be rewritten as [23] I ¼ 2pf V ½iC f ðdC=df Þ
ð1Þ
The first term in the bracket is an imaginary number, whereas the second one is a real one. The simple sum of an imaginary admittance and a real admittance suggests a parallel combination of an ideal capacitance and an ideal resistance in the equivalent circuit. We evaluated the frequency dependence of the parallel component of C (denoted by Cp) at Pt [23,24–26] and highly oriented pyrolytic graphite (HOPG) electrodes in KCl aqueous solutions in the polarized potential domain by use of Eq. (1) [27]. Fig. 1 shows an example of the plot of the log Cp against log f at the HOPG electrode [27]. The linearity indicates Cp = (Cp)1Hz fl, where l is a positive constant close to 0.1. By inserting this empirical relation into Eq. (1), the term f (dC/df) is reduced to lC.
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Fig. 1. Plots of log(Cp) at the HOPG electrode against the logarithm of the acfrequency obtained in 0.5 M KCl solution [27]. The inset is the variation of Cp with the time which is equal to 1/2pf .
Consequently Eq. (1) is rewritten as I ¼ 2pf VC p ði þ lÞ
ð2Þ
where C in Eq. (1) has been replaced by Cp. The sum i + l demonstrates clearly the validity of the parallel equivalent circuit for the double layer impedance. We have proved that the double layer resistance, 1/(2pfCpl), in Eq. (2) is almost inversely proportional to f [23,24,25,26]. The form of Eq. (2) agrees with the constant phase element (CPE) [11,28–30], which has the parameter a satisfied with tan(ap/2) = 1/l [27]. The power law of the frequency dependence has been seen in commercially available electrolytic capacitors, as found in technical notes supplied from each industry [31]. A question is why the frequency dependence appears as a result of surface roughness or morphology of electrodes. The capacitance is mainly controlled by localization of charge of solvent dipoles due to external field rather than ions, as has been demonstrated with the dependence of the capacitance on ionic concentrations [24,26] and kinds of solvents [25]. If the orientation of the dipoles were to depend only on the electric field, it should take a period of the nano-second order for the orientation. The actual relaxation time as long as 1 s in Fig. 1 cannot be realized by motion of a single molecule, but should be attributed to any macroscopic behavior including a huge number of collisions of dipoles. Development of a microscopic, molecular motion to a macroscopic behavior belongs to cooperative phenomena such as due to ferromagnetism by spinspin interaction and phase formation of alloys and liquid crystals through molecular interactions [32]. It has been embodied with the Ising models and other extended models [33]. Analysis of these models indicates that next neighboring interaction among particles can generate macroscopic phase in large domains after a huge number of collisions [34–36]. It is predicted that the twodimensional arrangement of dipoles activated by the electric field would exhibit the power law of the time as in Fig. 1 owing to the dipole-dipole and dipole-electrode interactions. This report deals with modelization of the capacitance in the Helmholtz layer, and then with carrying out computer simulation of the two-dimensional arrangement of dipoles in the context of the power law and the phase formation. 2. Molecular Model in Helmholtz Layer We discuss semi-quantitatively the delay of the response of the capacitance in aqueous solution. Dipoles of water are not oriented on an electrode until they are free from the confinement by the dipole–dipole interaction and the dipole–electrode interaction. We describe here the energetic relation of the orientation with these interactions.
Water molecules may be adsorbed on the electrode owing to the electrostatic image force between the partial charge of water molecules and the electrode. This force is always attractive for any charge to an electric conductor. When a point charge, q, is located from an planar electrode by the distance, l, in vacuum, the force between the charge and the electrode is given by q2/(16peol 2), according to the image force calculation [37], where eo is the permittivity of vacuum. Since we deal with water molecules, q is the partial negative charge 0.33e at the oxygen atom [38,39], where e is the elementary charge. It is assumed that the oxygen atom of the adsorbed water molecule is separated from the electrode surface by a half the averaged distance of closest neighboring water molecules, a, i.e., l = a/2. When a is regarded as the cubic-root of the molar volume of the water molecule, [M/ (NAd)]1/3, (M : molar mass, NA: Avogadro constant, d: density) the electrostatic energy of the adsorbed oxygen atom is given by UOad = (0.33e)2/(16peol) = 4.0 1020 J. The hydrogen atom in water has 0.165e, and hence the adsorption energy of two H atoms is given by U2H-ad = 2(0.165e)2/(16peol) = 2.0 1020 J. Then the total adsorption energy by the image force per water molecule is U ad ¼ U Oad þ U 2Had ¼ 6:0 1020 J
ð3Þ
Since the absolute value of Uad is ca. 15 times larger than the thermally fluctuation energy kBT ( = 0.41 1020 J) at 25 C, most molecules are adsorbed on the electrode, where kB is the Boltzmann constant. Quantitatively, the ratio of the number of the adsorbed molecules to that of the free molecules in equilibrium is 2.3 106 ( = exp(Uad/kBT)): 1. Ions of supporting electrolytes would be adsorbed more strongly than water because the inoic valence numbers are integers rather than fractional number of H-atom (0.16) and O-atom (0.33). However, we have to take into account the following three effects. (i) Since the hydration energy of ions is ca. 300 time larger than kBT, the ions should necessarily be hydrated strongly enough against the solvent–solvent interaction and the force by the applied electric field. The hydration varies the electric permittivity around the ion by er-times, where er is the dielectric constant, which suppresses the contribution of the adsorption. (ii) If the ion is surrounded with one layer of water molecules, the adsorbed ion may be separated from the electrode by ca. 1.5a . The longer separation decreases obviously the adsorption force. (iii) Ionic concentrations increase the entropic contribution of the adsorption free energy by kBT ln(55.5/c), where c is the molar concentration of univalent salt. When these effects are combined, the predicted adsorbed energy of ions for c = 0.1 mol dm3 and er = 78 is given by Usalt-ad = e2/(16peoer(1.5a)) + kBT ln(55.5/0.1) = 2.4 1020 J. The positive value means that ions are not adsorbed electrostatically on the electrode. As a result, ions do not participate directly in the capacitance in the Helmholtz layer. Water molecules are attracted each other with hydrogen bonds to form the tetragonal structure like sp3. This structure is geometrically not consistent with the planar arrangement of the triangular plane of the adsorbed H–O–H parallel to the electrode. The hydrogen bonding energy of water, 20 2 kJ mol1 [40] or U hb ¼ ð3:3 0:3Þ 1020 J per molecule
ð4Þ
is of the same order in magnitude as Uad. Therefore, the tetragonal structuration by the hydrogen bond coexists with the parallel arrangement like sp2 by the adsorption, which generates a dangling hydrogen bond. Since the dangling bonds decrease the force of both the adsorption and the hydrogen bond, the actual energy may be higher than the values in Eqs. (4) and (3). Fig. 2 illustrates a possible structure composed mainly of the adsorbed water molecules with the sp2 type mixed with the sp3 type. Most of
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capacitance is caused by fractional amount of the oriented dipoles. A value of b can be estimated from the experimental results, as follows. The density of the double layer capacitance at platinum electrode for 1 Hz in aqueous solution was ca 30 mF cm2 [23,24]. Then the electrostatic energy of the capacitance is CdV2/ 2 = 1.5 109 J cm2 for the applied voltage V = 0.01 V. Since the number density of water molecules adsorbed on the electrode is (NAd/WM)2/3 = 1.0 1015 molecules cm2, a dipole bringing about the capacitance has the energy 1.5 1024 J. This is only 1.5 104 times of |UH-ad|, and hence this fraction of the adsorbed dipoles can participate in the capacitance, i.e., p = 1.5 104. Applying this value of p to Eq. (6), i.e., 1.5 104 = 1/[exp((0.5 1020 J bUhb)/ kBT) + 1], we can estimate b = 1. This value implies that the number of the field-oriented dipoles are so small that the hydrogen bonds are still alive for the orientation. The estimation of the energy of water molecules near the electrode leads us to the following conclusions: (i) Water molecules are adsorbed on the electrode owing to the image force of the localized charge on H and O atoms. The adsorption energy is 15 times larger than kBT . (ii) Ions of supporting electrolyte are not so strongly adsorbed as water molecules are because of strong hydration of ions. (iii) The hydrogen bonding energy competes with the adsorption energy. The competition brings about a molecular image of the mixture of sp3 and sp2 types of the adsorbed water molecules. (iv) A very small fraction of the dipoles are oriented with the field, the other dipoles being confined on the electrode by the adsorption and the hydrogen bond. The available capacitance is much smaller than the latent capacitance.
3. Ising-type Model for Frequency Dependence
Fig. 2. Illustration of water molecules adsorbed on a flat electrode, in which (A) two sp3 structures are mixed with sp2 ones, and (B) a top view and side views when electric field is zero (middle) and applied (the bottom). The arrows are vector of a water dipole moment.
dipoles with the sp2 type are predominant, but the honeycomb form cannot be completed only with the sp2 type. Let the free energies of the adsorbed and hydrogen-bonded Hatom be UHad and UHhb, respectively. Then the sum U1 = UHad + UHhb is close to the energy of the adsorbed and hydrated H-atom without the orientation by the electric field. When the field by applying a negative voltage orients the dipole along the field, the orientation desorbs partially the adsorbed H-atom, keeping the counter part (the O-atom) adsorbed on the electrode, as is illustrated in Fig. 2B. Since the adsorption distance varies from l to a, the adsorbed energy increases by 0.5 1020 J. Then the hydrogen bond may contribute less by fraction b (0 < b < 1). As a result, the partially desorbed state has the energy U 2 ¼ UHad þ ð1 bÞU Hhd þ 0:5 1020 J
ð5Þ
When the energy varies from U1 to U2, the probability of the orientation which is proportional to the capacitance is given by p¼
expðU 2 =kB TÞ 1 ¼ expðU 1 =kB TÞ þ expðU 2 =kB TÞ expðU 1 U 2 Þ=kB T þ 1
ð6Þ
The difference in the energy, U2 U 1 ( = 0.5 1020 J bU hb) yields at least p < 0.2, where 0.2 corresponds to b = 0. Thus the
A question arises on whether the small fractions of the dipoles are oriented randomly or collectively. The orientation is initiated at dislocated points of the adsorbed, hydrogen-bonded structure to yield further dislocation. As a result, the oriented dipoles may be aggregated to form a phase against the structured phase. Formation of phase by molecular interactions belong to cooperative phenomena of statistical mechanics in the context of how microscopic, molecular interactions can generate a macroscopically extended phase [32]. The theory has been developed for the phase formation [41] by use of the Ising models, the Heisenberg models, Bethe approximations, and Bragg– Williams approximations. Most of the theoretical work have been directed to stability of phases for magnetic spins and binary alloys. They have been extended to electrochemical fields of redox interactions [42,43] and the double layer capacitances [44,45]. Monte-Carlo simulation is a powerful technique in the light of less restriction to complicated interaction. Dynamics of phase formation is also a target of research in that it takes long periods to complete the phase [46–50]. It may explain the frequency dependence of the double layer capacitances. We carry out here the Monte-Carlo simulation of the adsorbed dipoles which interact with next neighboring dipoles. We represent the sum of the adsorption energy of the H- and O-atoms, the hydrogen bonding energy and the dipole-dipole interaction energy without any field as Unf, (nf: non-field), which may be close to U1. On the other hands, the energy including the oriented dipole is represented as Uf, which may be close to U2. We count the number of oriented dipoles after applying a given field as a function of time, where the time is equivalent to the test number of trying to orient the dipoles. We use the 2D-square lattice model for the dipoles, so that a dipole interacts with four neighboring dipoles.
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Our Monte-Carlo simulation program is composed of (i) preparation of the input data, (ii) establishment of the equilibrium distribution of directions of the dipoles without applied voltage, (iii) application of the field corresponding to UE = Uf Unf to randomly selected dipoles, (iv) judgement of whether the dipole can acquire UE by means of the Metropolis method, (v) count of the number of the oriented dipoles, and (vi) plot of the variation of the dimensionless capacitance against the time. The dipoles are allowed to turn in three directions (upward, flat, downward) in 1D, and hence to turn six directions in 3D. The rotational interaction energies between neighboring two dipoles with dipole moment m are–m2/4pe0a3,m2/4pe0a3 (=1.1 1020 J) and 0, respectively, for the antiparallel, the parallel, and perpendicular arrangements. The energy by the field acting on the dipoles is expressed by the product of the dipole moment and the electric field. If the electric field, E, is uniform over the effective domain of the field for the 0.01 V application, the energy is given by U E ¼ mE ffi mðV=aÞ ¼ 0:2 1020 J
The first term in Eq. (9) represents the static capacitance, whereas the second one is the dynamic one because it varies with N. Values of N/N0 can be obtained by counting the number of oriented dipoles in the simulation for given values of N0, m, V and a under given experimental conditions. Fig. 3 shows time-variations of N/N0 for some values of Uf/kBT . Here N/N0 is proportional to the dynamic capacitance through Eq. (9), and the time, nt, is the trial number of flipping dipoles divided by N02. Since the capacitance was initially set to be in equilibrium at zero field, the interaction between dipoles, corresponding to m2/4pe0a3, makes the dipole to arrange in antiparallel. As a result, the dynamic capacitance becomes zero. It increases gradually with the time. When Unf > 0 or Unf >> Uf, the dipoles are not confined on the electrode but are fluctuated thermally to take the antiparallel form. Most of them respond to the electric field immediately after the potential step. Consequently the curve of N/N0 vs. nt rises up suddenly, as is shown in Fig. 3(e).
ð7Þ
We carried out the simulation for the lattice number, N0 = 300 300. The software was coded with Visual Basic and Visual C. In order to relate the capacitance with the orientation of the dipoles, we define two kinds of the polarization on the microscopic and the macroscopic time scale, Pmi and Pma, respectively. The former is caused by the electronic polarization and the ionic polarization, whereas the latter is by the orientation polarization caused by some interactions. Our concern is directed to the latter. Since Pmi is proportional to the electric field, it can be written as Pmi = xe0E by use of the electric susceptibility x [51]. The surface charge density involving the dielectric polarization, s p, is the subtraction of the dielectric polarization (Pmi + Pma) from that charge density without polarization, s np, i.e., s p = s np–(Pmi + Pma). Applying the Gauss’ law to s p near the conductor (electrode), we obtain e0E = s np xe0E Pma, or e0E (1 + x) = s np Pma. The voltage between the charge and the electrode with distance a is assumed to be V = aE for the uniform field. Then the capacitance is expressed by Cd ¼
s np V
¼
e0 a
ð1 þ xÞ þ
Pma V
ð8Þ
Pma is a sum of oriented dipoles, i.e. Pma = Npma, where N is the number of oriented dipoles in the available number of N0, and pma is the polarization per oriented dipole, i.e., the density of the dipole moment which equals m/a3. Then Cd is rewritten as Cd ¼
e0 a
ð1 þ xÞ þ
N N 0 m e0 N N0 m2 ¼ ð1 þ xÞ N0 Va3 N 0 a4 U E a
ð9Þ
N/N0
1
(e) (d)
0.5
(c) (b) (a)
0 0
100
nt
200
Fig. 3. Simulated time-variations of the dimensionless capacitances for UE/kBT = (a) 2.4, (b) 2.6, (c) 2.7, (d) 2.8 and (e) 4.0 at Unf/kBT = 0.69.
Fig. 4. Distributions of rising dipoles simulated at UE/kBT = 1.0 and Unf/kBT = 4.1 for N0 = 100 when nt = (A) 84 and (B) 210.
K.J. Aoki / Electrochimica Acta 188 (2016) 545–550
0
-1
log( N / N0 )
-2
0.02
-3 0
1
- UE / kBT
log(N / No)
(N / N0)(-kBT/UE)
0.03
549
2
0.01
0 0
1
2
- UE / kBT
-5
-10
3
4
log(nt No2)
5
6
Fig. 7. Logarithmic plots of the dimensionless dynamic capacitance against the time for UE/kBT = (squares) 2.4, (circles) 2.7 and (triangle) 2.9 at Unf/kBT = 0.69.
Under the conventional conditions of |UE| < |Unf|, the orientation by the field is hindered by the interactions to exhibit the delay. Fig. 4(A) and (B) show distributions of the rising dipoles (black dots) in the lying dipole (white) at different periods after the potential step. The aggregation of black dots is developed with the time. It must be caused by the attractive force among the adsorbed and hydrogen-bonded molecules (white) rather than the rising dipoles (black). When Unf/kBT < 1, the distributions of black dots were almost uniform, because the orientation randomly occurs by thermal fluctuation. The term 1/V in the middle equality of Eq. (9) indicates that Cd seems to decrease with an increase in the voltage. We obtained the simulated variations of Cd with V or UE. Fig. 5 shows the plot of the dynamic capacitance, (N/N0)(kBT/UE), against UE/kBT at three periods after the potential step. The capacitance is constant for |UE/kBT | < 1 and then increases with |UE/kBT| for|UE/kBT| > 1. The experimental values of the capacitance do not vary with applied ac-voltages for |V| < 0.2 V [24], which is equivalent to the inequality |UE/kBT| = mV/kBT < 1 for m = 1.85 Debye. Therefore, the simulated voltage dependence is consistent with the experimental one. The increase in (N/N0)(kBT/UE) for |UE/kBT | > 1 can be explained as follows. When U2 and U1 in Eq. (6) are replaced by Unf and Uf, respectively, and use the relation UE = Uf Unf, we obtain N N0 exp(UE/kBT) approximately for large values of |UE/kBT|. The variation of log (N) with UE/kBT is shown in the inset of Fig. 5, demonstrating the linearity. Curve fitting of (N/N0)(kBT/UE) by use of the exponential dependence yielded the variation of the dashed curve in Fig. 5. It is predicted that the phases are generated by a balance between UE and Unf. We obtained the simulated curves of N/N0 for various combinations of UE and Unf, monitoring both the dimensionless capacitance vs. time and the time variation of the
patterns like in Fig. 4. When values of |UE| were increased from zero at a given value of Unf, the patterns varied from the assemble of antiparallel dipoles (white domains in Fig. 4), via islands of the rising dipoles (black in Fig. 4) to the cover with the whole rising dipoles, denoted as P, P–R and R domains in Fig. 6, respectively. The capacitance vs. time curves in the P domain rose instantaneously after the voltage application, like curve (e) in Fig. 3. Those in the P–R domain increased gradually to reach a constant for a long time, like curves (b)–(d) in Fig. 3. The constant values increased with an increase in |UE|. Those of the R domain reached unity soon after the voltage application. The pattern formation is restricted to the P–R domain, in which the capacitance vs. time curves exhibit the delay. Therefore the frequency dependence should be associated with the pattern formation caused by the interactions. A feature of the experimentally obtained frequency dependence is the power law given by Cp = (Cp)1Hz fl. Fig. 7 shows logarithmic plots of the simulated capacitance against the time after the application of the voltage. A linear relation is found in a given time domain, indicating that the capacitance can be approximated to the power of the time. Since the time is equivalent to 1/(2pf), the power law in Fig. 7 can be regarded as the power law of the frequency. Unfortunately we have to wait for further theoretical work on the reason why the power law appears.
- Unf / kBT
Fig. 5. Variations of the dimensionless capacitance with UE/k BT for nt = (circles) 10, (squares) 20, and (triangles) 30 at Unf/kBT = 0.69.
10
P
P-R
5
0 0
R
1
- UE / kBT
2
Fig. 6. Phase diagram composed of three phases with (P) anti-parallel dipoles, (R) rising dipoles and (P–R) mixture of the two.
4. Conclusion If dipoles of solvents were to be oriented immediately after application of the electric field, the capacitance should respond to the field on a time scale of GHz. The frequency dependence observed even at 1 Hz can be elucidated in terms of the hindrance of the orientation against the confinement of the adsorbed solvent molecules. The confinement is caused by the solvent–solvent interaction (hydrogen bonds for water) and electrostatic adsorption of solvents owing to the image force. Since the structure formation belongs to phase transition, it takes macroscopic time of 1 Hz to reach the stable structure. The phase formation is ascribed to the fact that the absolute values of hydrogen-bonding energy and the image force energy are by one-order in magnitude larger than the thermal fluctuation energy. Therefore the orientation of dipoles is controlled by the phase formation rate rather than the flip rate of dipoles. The simulation similar to the Ising model exhibits aggregation of oriented dipoles rather than random distribution. The aggregation grows gradually with the time of voltage application, and hence the capacitance increases gradually. The increasing rate is obeyed empirically by the experimentally observed power law.
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[29] G.J. Brug, A.L.G. Van Den Eeden, M. Sluyters-Rehbach, J.H. Sluyters, J. Electroanal. Chem. 176 (1984) 275. [30] P. Zoltowski, J. Electroanal. Chem. 443 (1998) 149. [31] For example, Nippon Chemi-con, Cat. No. E1001L, http://www.mouser.com/ pdfDocs/UCC_ElectrolyticCapacitorTechnicalNotes.pdf. [32] R. Kubo, Statistical Mechanics–An Advanced Course with Problems and Solutions, North-Holland Amsterdam, Seventh Ed. 1988, Chapter 5, pp. 302– 318. [33] M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics, Oxford University Press, 1999. [34] M. Nicodemi, A. Coniglio, H.J. Herrmann, Phys. Rev. E 55 (1997) 3962. [35] B. Kim, S.J. Lee, Prog. Theo. Phys. Sup. 126 (1997) 349. [36] M. Suzuki, R. Kubo, J. Phys. Soc. Jpn. 24 (1968) 51. [37] D.J. Griffiths, Introduction to Electrodynamics, 2nd ed., Prentice-Hall, Inc, 1989, pp. 121–123. [38] K. Wendler, J. Thar, S. Zahn, B. Kirchner, J. Phys. Chem. A 114 (2010) 9529. [39] M. Chaplin, Water’s hydrogen Bond Strength, http://arxiv.org/ftp/arxiv/ papers/0706/0706.1355pdf. [40] S.J. Suresh, V.M. Naik, J. Chem. Phys. 113 (2000) 9727. [41] M. Plischke, B. Bergersen, Equilibrium Statistical Physics, 3rd ed., World Scientific Publishing Co., Singapore, 2005, pp. 63–108 Chp. 3. [42] K. Aoki, J. Chen, J. Electroanal. Chem. 380 (1995) 35. [43] K. Aoki, T. Kakiuchi, J. Electroanal. Chem. 452 (1998) 187. [44] J.R. MacDonald, A.P. Lehnen, J. Phys. Chem. Solids 43 (1982) 39. [45] S.L. Marshall, B.E. Conway, J. Electroanal. Chem. 337 (1992) 45. [46] J.C. Angles d’Auriac, R. Maynard, R. Rammal, J. Statistical Phys. 28 (1982) 307. [47] S. Miyashita, H. Takano, Prog. Theo. Phys. 73 (1985) 1122. [48] T. Tome, M.J. de Oliveira, Phys. Rev. A 41 (1990) 4251. [49] P.L. Krapivsky, S. Redner, E. Ben-Naim, A Kinetic View of Statistical Physics, Cambridge University Press, 2010 Chapter 8. [50] R. Weinkamer, P. Fratzl, H.S. Gupta, O. Penrose, J.L. Lebowitz, Phase Trans. 77 (2004) 433. [51] D.J. Griffiths, Introduction to Electrodynamics, 2nd ed., Prentice-Hall, Inc., 1989, pp. 158–164.
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Electrochimica Acta journal homepage: www.elsevier.com/locate/electacta
Molecular interaction model for frequency-dependence of double layer capacitors Koichi Jeremiah Aoki1,* Electrochemistry Museum, 3-1304 Takagichuou, Fukui 910-0804, Japan
A R T I C L E I N F O
A B S T R A C T
Article history: Received 29 September 2015 Accepted 5 December 2015 Available online 10 December 2015
Electric double layer capacitances at electrodes exhibit frequency-dependence, decreasing with the frequency in accordance with the power law even at frequency as low as 1 Hz. Since the capacitance is mainly caused by orientation of dipoles of solvent molecules in the Helmholtz layer, the frequency dependence may be associated with the rate of the orientation. The frequency of the orientation on molecular scale is of the order of GHz, whereas the observed frequency domain is less than 10 kHz. The difference implies that the frequency dependence should not be controlled by simple flips of the dipoles but be caused by a slow process induced by the orientation. This belongs to a cooperative phenomenon. We consider here as the inducing causes the solvent-solvent interaction, the image force of the dipoles to the electrode and the orientation by the external field. The energy balance indicates that most dipoles are confined regularly on the electrode surface as a 2D phase. The 2D-Monte Carlo simulation similar to that for the Ising model is performed in the light of the competition of the interaction and the orientation. The simulation demonstrates aggregation of the oriented dipoles which are hindered by the interaction. The aggregation and hence the capacitance grow gradually with the time of voltage application. The increasing rate obeys the experimentally observed power law. ã 2015 Elsevier Ltd. All rights reserved.
Keywords: Frequency-dependence of electrical double layer capacitance Monte Carlo simulation by Ising model Solvent–solvent interaction Formation of two-dimensional phase
1. Introduction Capacitances of electric double layers are known to vary with applied ac-frequency in the polarized potential domain [1–3]. The frequency dependence can be empirically diminished with a decrease in surface roughness [4–7] as well as with a use of single crystal surfaces [8–10]. It has been considered to be caused by fractal dimensions of electrodes[11,12], adsorption [13–16], lateral charge spreading in the double layer [17], harmonic components [18], and heterogeneities of electrodes on the atomic scale [19]. It is also associated with dielectric loss [1], cracks on electrodes and/or insulators [2] and cell geometry [20]. Quantitative work on the frequency dependence has not yet been reported, to our knowledge, and the subject is still under debate [21,22]. The frequency dependence has often been analyzed by use of equivalent circuits composed of ideal resistances and ideal capacitances. It is difficult not only to represent the frequency dependence as combinations of ideal electric elements but also to interpret the physical meaning of the equivalent circuits. A strategy
* Corresponding author. Tel.: +81 90 8095 1906; fax: +81 776 27 8750. E-mail addresses: [email protected], [email protected] (K.J. Aoki). ISE member.
1
http://dx.doi.org/10.1016/j.electacta.2015.12.049 0013-4686/ ã 2015 Elsevier Ltd. All rights reserved.
of the analysis without complicated equivalent circuits is to introduce an intrinsically non-ideal capacitance, which has a function of the capacitance as the frequency. When the capacitance, C, varies with the frequency or the time, t, the current caused by the capacitive charge, q = CV, is given by I = dq/dt = CdV/dt + VdC/ dt for a time-varying voltage V. The second term is nothing but the functional form to be determined. When the ac-voltage, V ¼V0ei2pft, is inserted into the above equation, the current can be rewritten as [23] I ¼ 2pf V ½iC f ðdC=df Þ
ð1Þ
The first term in the bracket is an imaginary number, whereas the second one is a real one. The simple sum of an imaginary admittance and a real admittance suggests a parallel combination of an ideal capacitance and an ideal resistance in the equivalent circuit. We evaluated the frequency dependence of the parallel component of C (denoted by Cp) at Pt [23,24–26] and highly oriented pyrolytic graphite (HOPG) electrodes in KCl aqueous solutions in the polarized potential domain by use of Eq. (1) [27]. Fig. 1 shows an example of the plot of the log Cp against log f at the HOPG electrode [27]. The linearity indicates Cp = (Cp)1Hz fl, where l is a positive constant close to 0.1. By inserting this empirical relation into Eq. (1), the term f (dC/df) is reduced to lC.
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Fig. 1. Plots of log(Cp) at the HOPG electrode against the logarithm of the acfrequency obtained in 0.5 M KCl solution [27]. The inset is the variation of Cp with the time which is equal to 1/2pf .
Consequently Eq. (1) is rewritten as I ¼ 2pf VC p ði þ lÞ
ð2Þ
where C in Eq. (1) has been replaced by Cp. The sum i + l demonstrates clearly the validity of the parallel equivalent circuit for the double layer impedance. We have proved that the double layer resistance, 1/(2pfCpl), in Eq. (2) is almost inversely proportional to f [23,24,25,26]. The form of Eq. (2) agrees with the constant phase element (CPE) [11,28–30], which has the parameter a satisfied with tan(ap/2) = 1/l [27]. The power law of the frequency dependence has been seen in commercially available electrolytic capacitors, as found in technical notes supplied from each industry [31]. A question is why the frequency dependence appears as a result of surface roughness or morphology of electrodes. The capacitance is mainly controlled by localization of charge of solvent dipoles due to external field rather than ions, as has been demonstrated with the dependence of the capacitance on ionic concentrations [24,26] and kinds of solvents [25]. If the orientation of the dipoles were to depend only on the electric field, it should take a period of the nano-second order for the orientation. The actual relaxation time as long as 1 s in Fig. 1 cannot be realized by motion of a single molecule, but should be attributed to any macroscopic behavior including a huge number of collisions of dipoles. Development of a microscopic, molecular motion to a macroscopic behavior belongs to cooperative phenomena such as due to ferromagnetism by spinspin interaction and phase formation of alloys and liquid crystals through molecular interactions [32]. It has been embodied with the Ising models and other extended models [33]. Analysis of these models indicates that next neighboring interaction among particles can generate macroscopic phase in large domains after a huge number of collisions [34–36]. It is predicted that the twodimensional arrangement of dipoles activated by the electric field would exhibit the power law of the time as in Fig. 1 owing to the dipole-dipole and dipole-electrode interactions. This report deals with modelization of the capacitance in the Helmholtz layer, and then with carrying out computer simulation of the two-dimensional arrangement of dipoles in the context of the power law and the phase formation. 2. Molecular Model in Helmholtz Layer We discuss semi-quantitatively the delay of the response of the capacitance in aqueous solution. Dipoles of water are not oriented on an electrode until they are free from the confinement by the dipole–dipole interaction and the dipole–electrode interaction. We describe here the energetic relation of the orientation with these interactions.
Water molecules may be adsorbed on the electrode owing to the electrostatic image force between the partial charge of water molecules and the electrode. This force is always attractive for any charge to an electric conductor. When a point charge, q, is located from an planar electrode by the distance, l, in vacuum, the force between the charge and the electrode is given by q2/(16peol 2), according to the image force calculation [37], where eo is the permittivity of vacuum. Since we deal with water molecules, q is the partial negative charge 0.33e at the oxygen atom [38,39], where e is the elementary charge. It is assumed that the oxygen atom of the adsorbed water molecule is separated from the electrode surface by a half the averaged distance of closest neighboring water molecules, a, i.e., l = a/2. When a is regarded as the cubic-root of the molar volume of the water molecule, [M/ (NAd)]1/3, (M : molar mass, NA: Avogadro constant, d: density) the electrostatic energy of the adsorbed oxygen atom is given by UOad = (0.33e)2/(16peol) = 4.0 1020 J. The hydrogen atom in water has 0.165e, and hence the adsorption energy of two H atoms is given by U2H-ad = 2(0.165e)2/(16peol) = 2.0 1020 J. Then the total adsorption energy by the image force per water molecule is U ad ¼ U Oad þ U 2Had ¼ 6:0 1020 J
ð3Þ
Since the absolute value of Uad is ca. 15 times larger than the thermally fluctuation energy kBT ( = 0.41 1020 J) at 25 C, most molecules are adsorbed on the electrode, where kB is the Boltzmann constant. Quantitatively, the ratio of the number of the adsorbed molecules to that of the free molecules in equilibrium is 2.3 106 ( = exp(Uad/kBT)): 1. Ions of supporting electrolytes would be adsorbed more strongly than water because the inoic valence numbers are integers rather than fractional number of H-atom (0.16) and O-atom (0.33). However, we have to take into account the following three effects. (i) Since the hydration energy of ions is ca. 300 time larger than kBT, the ions should necessarily be hydrated strongly enough against the solvent–solvent interaction and the force by the applied electric field. The hydration varies the electric permittivity around the ion by er-times, where er is the dielectric constant, which suppresses the contribution of the adsorption. (ii) If the ion is surrounded with one layer of water molecules, the adsorbed ion may be separated from the electrode by ca. 1.5a . The longer separation decreases obviously the adsorption force. (iii) Ionic concentrations increase the entropic contribution of the adsorption free energy by kBT ln(55.5/c), where c is the molar concentration of univalent salt. When these effects are combined, the predicted adsorbed energy of ions for c = 0.1 mol dm3 and er = 78 is given by Usalt-ad = e2/(16peoer(1.5a)) + kBT ln(55.5/0.1) = 2.4 1020 J. The positive value means that ions are not adsorbed electrostatically on the electrode. As a result, ions do not participate directly in the capacitance in the Helmholtz layer. Water molecules are attracted each other with hydrogen bonds to form the tetragonal structure like sp3. This structure is geometrically not consistent with the planar arrangement of the triangular plane of the adsorbed H–O–H parallel to the electrode. The hydrogen bonding energy of water, 20 2 kJ mol1 [40] or U hb ¼ ð3:3 0:3Þ 1020 J per molecule
ð4Þ
is of the same order in magnitude as Uad. Therefore, the tetragonal structuration by the hydrogen bond coexists with the parallel arrangement like sp2 by the adsorption, which generates a dangling hydrogen bond. Since the dangling bonds decrease the force of both the adsorption and the hydrogen bond, the actual energy may be higher than the values in Eqs. (4) and (3). Fig. 2 illustrates a possible structure composed mainly of the adsorbed water molecules with the sp2 type mixed with the sp3 type. Most of
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capacitance is caused by fractional amount of the oriented dipoles. A value of b can be estimated from the experimental results, as follows. The density of the double layer capacitance at platinum electrode for 1 Hz in aqueous solution was ca 30 mF cm2 [23,24]. Then the electrostatic energy of the capacitance is CdV2/ 2 = 1.5 109 J cm2 for the applied voltage V = 0.01 V. Since the number density of water molecules adsorbed on the electrode is (NAd/WM)2/3 = 1.0 1015 molecules cm2, a dipole bringing about the capacitance has the energy 1.5 1024 J. This is only 1.5 104 times of |UH-ad|, and hence this fraction of the adsorbed dipoles can participate in the capacitance, i.e., p = 1.5 104. Applying this value of p to Eq. (6), i.e., 1.5 104 = 1/[exp((0.5 1020 J bUhb)/ kBT) + 1], we can estimate b = 1. This value implies that the number of the field-oriented dipoles are so small that the hydrogen bonds are still alive for the orientation. The estimation of the energy of water molecules near the electrode leads us to the following conclusions: (i) Water molecules are adsorbed on the electrode owing to the image force of the localized charge on H and O atoms. The adsorption energy is 15 times larger than kBT . (ii) Ions of supporting electrolyte are not so strongly adsorbed as water molecules are because of strong hydration of ions. (iii) The hydrogen bonding energy competes with the adsorption energy. The competition brings about a molecular image of the mixture of sp3 and sp2 types of the adsorbed water molecules. (iv) A very small fraction of the dipoles are oriented with the field, the other dipoles being confined on the electrode by the adsorption and the hydrogen bond. The available capacitance is much smaller than the latent capacitance.
3. Ising-type Model for Frequency Dependence
Fig. 2. Illustration of water molecules adsorbed on a flat electrode, in which (A) two sp3 structures are mixed with sp2 ones, and (B) a top view and side views when electric field is zero (middle) and applied (the bottom). The arrows are vector of a water dipole moment.
dipoles with the sp2 type are predominant, but the honeycomb form cannot be completed only with the sp2 type. Let the free energies of the adsorbed and hydrogen-bonded Hatom be UHad and UHhb, respectively. Then the sum U1 = UHad + UHhb is close to the energy of the adsorbed and hydrated H-atom without the orientation by the electric field. When the field by applying a negative voltage orients the dipole along the field, the orientation desorbs partially the adsorbed H-atom, keeping the counter part (the O-atom) adsorbed on the electrode, as is illustrated in Fig. 2B. Since the adsorption distance varies from l to a, the adsorbed energy increases by 0.5 1020 J. Then the hydrogen bond may contribute less by fraction b (0 < b < 1). As a result, the partially desorbed state has the energy U 2 ¼ UHad þ ð1 bÞU Hhd þ 0:5 1020 J
ð5Þ
When the energy varies from U1 to U2, the probability of the orientation which is proportional to the capacitance is given by p¼
expðU 2 =kB TÞ 1 ¼ expðU 1 =kB TÞ þ expðU 2 =kB TÞ expðU 1 U 2 Þ=kB T þ 1
ð6Þ
The difference in the energy, U2 U 1 ( = 0.5 1020 J bU hb) yields at least p < 0.2, where 0.2 corresponds to b = 0. Thus the
A question arises on whether the small fractions of the dipoles are oriented randomly or collectively. The orientation is initiated at dislocated points of the adsorbed, hydrogen-bonded structure to yield further dislocation. As a result, the oriented dipoles may be aggregated to form a phase against the structured phase. Formation of phase by molecular interactions belong to cooperative phenomena of statistical mechanics in the context of how microscopic, molecular interactions can generate a macroscopically extended phase [32]. The theory has been developed for the phase formation [41] by use of the Ising models, the Heisenberg models, Bethe approximations, and Bragg– Williams approximations. Most of the theoretical work have been directed to stability of phases for magnetic spins and binary alloys. They have been extended to electrochemical fields of redox interactions [42,43] and the double layer capacitances [44,45]. Monte-Carlo simulation is a powerful technique in the light of less restriction to complicated interaction. Dynamics of phase formation is also a target of research in that it takes long periods to complete the phase [46–50]. It may explain the frequency dependence of the double layer capacitances. We carry out here the Monte-Carlo simulation of the adsorbed dipoles which interact with next neighboring dipoles. We represent the sum of the adsorption energy of the H- and O-atoms, the hydrogen bonding energy and the dipole-dipole interaction energy without any field as Unf, (nf: non-field), which may be close to U1. On the other hands, the energy including the oriented dipole is represented as Uf, which may be close to U2. We count the number of oriented dipoles after applying a given field as a function of time, where the time is equivalent to the test number of trying to orient the dipoles. We use the 2D-square lattice model for the dipoles, so that a dipole interacts with four neighboring dipoles.
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Our Monte-Carlo simulation program is composed of (i) preparation of the input data, (ii) establishment of the equilibrium distribution of directions of the dipoles without applied voltage, (iii) application of the field corresponding to UE = Uf Unf to randomly selected dipoles, (iv) judgement of whether the dipole can acquire UE by means of the Metropolis method, (v) count of the number of the oriented dipoles, and (vi) plot of the variation of the dimensionless capacitance against the time. The dipoles are allowed to turn in three directions (upward, flat, downward) in 1D, and hence to turn six directions in 3D. The rotational interaction energies between neighboring two dipoles with dipole moment m are–m2/4pe0a3,m2/4pe0a3 (=1.1 1020 J) and 0, respectively, for the antiparallel, the parallel, and perpendicular arrangements. The energy by the field acting on the dipoles is expressed by the product of the dipole moment and the electric field. If the electric field, E, is uniform over the effective domain of the field for the 0.01 V application, the energy is given by U E ¼ mE ffi mðV=aÞ ¼ 0:2 1020 J
The first term in Eq. (9) represents the static capacitance, whereas the second one is the dynamic one because it varies with N. Values of N/N0 can be obtained by counting the number of oriented dipoles in the simulation for given values of N0, m, V and a under given experimental conditions. Fig. 3 shows time-variations of N/N0 for some values of Uf/kBT . Here N/N0 is proportional to the dynamic capacitance through Eq. (9), and the time, nt, is the trial number of flipping dipoles divided by N02. Since the capacitance was initially set to be in equilibrium at zero field, the interaction between dipoles, corresponding to m2/4pe0a3, makes the dipole to arrange in antiparallel. As a result, the dynamic capacitance becomes zero. It increases gradually with the time. When Unf > 0 or Unf >> Uf, the dipoles are not confined on the electrode but are fluctuated thermally to take the antiparallel form. Most of them respond to the electric field immediately after the potential step. Consequently the curve of N/N0 vs. nt rises up suddenly, as is shown in Fig. 3(e).
ð7Þ
We carried out the simulation for the lattice number, N0 = 300 300. The software was coded with Visual Basic and Visual C. In order to relate the capacitance with the orientation of the dipoles, we define two kinds of the polarization on the microscopic and the macroscopic time scale, Pmi and Pma, respectively. The former is caused by the electronic polarization and the ionic polarization, whereas the latter is by the orientation polarization caused by some interactions. Our concern is directed to the latter. Since Pmi is proportional to the electric field, it can be written as Pmi = xe0E by use of the electric susceptibility x [51]. The surface charge density involving the dielectric polarization, s p, is the subtraction of the dielectric polarization (Pmi + Pma) from that charge density without polarization, s np, i.e., s p = s np–(Pmi + Pma). Applying the Gauss’ law to s p near the conductor (electrode), we obtain e0E = s np xe0E Pma, or e0E (1 + x) = s np Pma. The voltage between the charge and the electrode with distance a is assumed to be V = aE for the uniform field. Then the capacitance is expressed by Cd ¼
s np V
¼
e0 a
ð1 þ xÞ þ
Pma V
ð8Þ
Pma is a sum of oriented dipoles, i.e. Pma = Npma, where N is the number of oriented dipoles in the available number of N0, and pma is the polarization per oriented dipole, i.e., the density of the dipole moment which equals m/a3. Then Cd is rewritten as Cd ¼
e0 a
ð1 þ xÞ þ
N N 0 m e0 N N0 m2 ¼ ð1 þ xÞ N0 Va3 N 0 a4 U E a
ð9Þ
N/N0
1
(e) (d)
0.5
(c) (b) (a)
0 0
100
nt
200
Fig. 3. Simulated time-variations of the dimensionless capacitances for UE/kBT = (a) 2.4, (b) 2.6, (c) 2.7, (d) 2.8 and (e) 4.0 at Unf/kBT = 0.69.
Fig. 4. Distributions of rising dipoles simulated at UE/kBT = 1.0 and Unf/kBT = 4.1 for N0 = 100 when nt = (A) 84 and (B) 210.
K.J. Aoki / Electrochimica Acta 188 (2016) 545–550
0
-1
log( N / N0 )
-2
0.02
-3 0
1
- UE / kBT
log(N / No)
(N / N0)(-kBT/UE)
0.03
549
2
0.01
0 0
1
2
- UE / kBT
-5
-10
3
4
log(nt No2)
5
6
Fig. 7. Logarithmic plots of the dimensionless dynamic capacitance against the time for UE/kBT = (squares) 2.4, (circles) 2.7 and (triangle) 2.9 at Unf/kBT = 0.69.
Under the conventional conditions of |UE| < |Unf|, the orientation by the field is hindered by the interactions to exhibit the delay. Fig. 4(A) and (B) show distributions of the rising dipoles (black dots) in the lying dipole (white) at different periods after the potential step. The aggregation of black dots is developed with the time. It must be caused by the attractive force among the adsorbed and hydrogen-bonded molecules (white) rather than the rising dipoles (black). When Unf/kBT < 1, the distributions of black dots were almost uniform, because the orientation randomly occurs by thermal fluctuation. The term 1/V in the middle equality of Eq. (9) indicates that Cd seems to decrease with an increase in the voltage. We obtained the simulated variations of Cd with V or UE. Fig. 5 shows the plot of the dynamic capacitance, (N/N0)(kBT/UE), against UE/kBT at three periods after the potential step. The capacitance is constant for |UE/kBT | < 1 and then increases with |UE/kBT| for|UE/kBT| > 1. The experimental values of the capacitance do not vary with applied ac-voltages for |V| < 0.2 V [24], which is equivalent to the inequality |UE/kBT| = mV/kBT < 1 for m = 1.85 Debye. Therefore, the simulated voltage dependence is consistent with the experimental one. The increase in (N/N0)(kBT/UE) for |UE/kBT | > 1 can be explained as follows. When U2 and U1 in Eq. (6) are replaced by Unf and Uf, respectively, and use the relation UE = Uf Unf, we obtain N N0 exp(UE/kBT) approximately for large values of |UE/kBT|. The variation of log (N) with UE/kBT is shown in the inset of Fig. 5, demonstrating the linearity. Curve fitting of (N/N0)(kBT/UE) by use of the exponential dependence yielded the variation of the dashed curve in Fig. 5. It is predicted that the phases are generated by a balance between UE and Unf. We obtained the simulated curves of N/N0 for various combinations of UE and Unf, monitoring both the dimensionless capacitance vs. time and the time variation of the
patterns like in Fig. 4. When values of |UE| were increased from zero at a given value of Unf, the patterns varied from the assemble of antiparallel dipoles (white domains in Fig. 4), via islands of the rising dipoles (black in Fig. 4) to the cover with the whole rising dipoles, denoted as P, P–R and R domains in Fig. 6, respectively. The capacitance vs. time curves in the P domain rose instantaneously after the voltage application, like curve (e) in Fig. 3. Those in the P–R domain increased gradually to reach a constant for a long time, like curves (b)–(d) in Fig. 3. The constant values increased with an increase in |UE|. Those of the R domain reached unity soon after the voltage application. The pattern formation is restricted to the P–R domain, in which the capacitance vs. time curves exhibit the delay. Therefore the frequency dependence should be associated with the pattern formation caused by the interactions. A feature of the experimentally obtained frequency dependence is the power law given by Cp = (Cp)1Hz fl. Fig. 7 shows logarithmic plots of the simulated capacitance against the time after the application of the voltage. A linear relation is found in a given time domain, indicating that the capacitance can be approximated to the power of the time. Since the time is equivalent to 1/(2pf), the power law in Fig. 7 can be regarded as the power law of the frequency. Unfortunately we have to wait for further theoretical work on the reason why the power law appears.
- Unf / kBT
Fig. 5. Variations of the dimensionless capacitance with UE/k BT for nt = (circles) 10, (squares) 20, and (triangles) 30 at Unf/kBT = 0.69.
10
P
P-R
5
0 0
R
1
- UE / kBT
2
Fig. 6. Phase diagram composed of three phases with (P) anti-parallel dipoles, (R) rising dipoles and (P–R) mixture of the two.
4. Conclusion If dipoles of solvents were to be oriented immediately after application of the electric field, the capacitance should respond to the field on a time scale of GHz. The frequency dependence observed even at 1 Hz can be elucidated in terms of the hindrance of the orientation against the confinement of the adsorbed solvent molecules. The confinement is caused by the solvent–solvent interaction (hydrogen bonds for water) and electrostatic adsorption of solvents owing to the image force. Since the structure formation belongs to phase transition, it takes macroscopic time of 1 Hz to reach the stable structure. The phase formation is ascribed to the fact that the absolute values of hydrogen-bonding energy and the image force energy are by one-order in magnitude larger than the thermal fluctuation energy. Therefore the orientation of dipoles is controlled by the phase formation rate rather than the flip rate of dipoles. The simulation similar to the Ising model exhibits aggregation of oriented dipoles rather than random distribution. The aggregation grows gradually with the time of voltage application, and hence the capacitance increases gradually. The increasing rate is obeyed empirically by the experimentally observed power law.
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