Decrease in the double layer capacitance by faradaic current

Abstract

This study describes the reverse of the well-known double layer effects on charge transfer kinetics in the relationship between a cause and an effect. The reversible redox reaction of a ferrocenyl derivative decreased the capacitive values of the double layer impedance up to negative values, corresponding to an inductive component. This observation was disclosed by the subtraction of the real admittance from the imaginary admittance, which can extract the net double layer capacitance from the Warburg impedance. The inductance-like behavior is caused due to two reasons: (i) the double layer capacitance in the polarized potential domain is determined by the low concentrations of the field-oriented solvent dipoles that are considered as the conventionally employed redox concentrations and (ii) the double layer capacitance is caused by the orientation of the dipoles in the same direction as that of the electric field, whereas the redox reaction generates charge in the direction opposite to the field. The faradaic effect was demonstrated via the ac-impedance data obtained for the ferrocenyl compound in a KCl solution in the unpolarized potential domain between 1 Hz and 3 kHz frequency. The negative admittance was proportional to the frequency. The theory of negative capacitance was presented by combining the mirror-image surface charge with the Nernst equation.

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1. Introduction

Double layer (DL) effects on charge transfer kinetics are a classical subject, which was first reported in 1933 by Frumkin.¹ These effects deal with the variations of the charge transfer rate with the concentration and/or type of supporting electrolyte as well as with the specifically adsorbed electrolyte.^2,3 In particular, the effect of the supporting electrolyte is called the Frumkin's DL effect.^1,4 It includes the correction of the potential at the closest approach of the redox species, which may be varied with the concentration of the supporting electrolyte; well-known examples are cadmium ions,⁵ zinc ions,⁶ and some aromatic compounds.⁷ However, there are some ambiguities in the participation of the specifically adsorbed species,^8–12 in charge interactions,^13,14 in modification by finite ionic size,^15–18 in the geometrical effects^19–23 of the Poisson–Nernst–Planck equation, in simplification of the plane of the closest approach,²⁴ in complications by ions^25–28 and by the orientation of the solvent molecules,²⁹ in charge distribution within a redox species,³⁰ and in coupling of the charged redox species and supporting electrolyte.^31,32 The Frumkin's correction has recently been reviewed from a theoretical point of view.³³

The fundamental concept of estimating the DL effects is that the faradaic reactions should occur in the DL structure firmly established with solvents and/or salts independent of the reactions.^34–37 This is based on the intuition that the concentration of the redox species in the voltammetry is much smaller than the concentration of the solvent molecules and/or supporting electrolyte. However, our experimental results on the frequency-dependence of the DL impedance indicate that (i) the DL capacitance is independent of the concentration of the salt,^38,39 the type of the salt,⁴⁰ and the dc-potential;^40,41 (ii) it is caused by the field-driven orientation of the solvent molecules on the electrode;⁴² and (iii) the interaction energy of the dipole–dipole of solvent molecules or hydrogen bonding energy is larger by one order in magnitude than the externally driven orientation energy such that a small molar concentration of the solvent molecules participate in the DL impedance.^43,44 The behavior (i) with regard to the frequency dispersion is in accordance with the results reported in the literature.^45–49 The frequency-dependence has been attributed to the microscopic surface roughness,^50,51 non-uniform current distribution on the electrode surfaces,^52–55 fractal surface geometry,^56–59 and molecular interaction on the electrode surfaces.⁴³ It is necessary to take into account the relationship between the DL impedance and the faradaic impedance without intuition. A technical problem is how to separate the two impedances without ambiguity, especially as for the frequency-dependence of the DL capacitance and electrode kinetics.

A simple redox reaction is generally controlled by both the charge transfer step and diffusional step. The former is possibly complicated with the DL impedance because of involvement of a surface process, whereas the latter may be clearly distinguished from the DL effect because the process is carried out in the solution phase. Thus, we first examined the latter, i.e. the effects of the Warburg impedance on the DL impedance. If both impedances are independent of each other, they can be represented as a parallel equivalent circuit, as depicted in Fig. 1(A). A real part of the Warburg impedance for infinite diffusion has the same value as that of the imaginary part, according to the solution of the diffusion equation,^60,61 although there are some complications in the nanoscale diffusion domains.⁶² As a result, the values obtained for the observed real admittance subtracted from those of the imaginary admittance should be those of the DL impedance. This was our fundamental technique to examine the relationship between the Warburg impedance and the DL impedance.

Fig. 1 Equivalent circuits of (A) the conventional combination of the Warburg impedance (real part W_Re and imaginary part W_im), the double layer capacitance C_d and the solution resistance R_S, and (B) the suggested combination including the inductance-like capacitance caused by the faradaic reaction and the parallel DL impedance with the real R_p and imaginary C_p components.

To obtain the relationship, it is necessary to both experimentally and theoretically specify the diffusion-controlled conditions. Herein, we used (ferrocenylmethyl) trimethylammonium (FcTMA) as a redox species with negligible effects of the heterogeneous rates. Thus, the problems such as the (a) accuracy of the impedance data, (b) floating capacitance, (c) participation of harmonic components, (d) participation of the electrode kinetics of FcTMA, and (e) effects of the frequency-dependence of the DL capacitance on the Warburg impedance need to be solved in detail. We have discussed (a) and (b) in the Experimental section, (c) in the Appendix, and (d) and (e) in the Results and discussion section.

2. Experimental

FcTMA purchased from Tokyo Chemical Industry was in the iodide form. Iodide ion was substituted into PF₆⁻ by adding NH₄PF₆ to FcTMA⁺-I⁻ and recrystallizing the product. All the chemicals were of analytical grade. Water was distilled and ion-exchanged prior to use.

Since a disk-exposed working electrode often yields large background currents, which depend on the mechanical polish, chemical treatments and pre-electrolysis, the instability was empirically ascribed to the crevices at the boundary between the electrode and the insulator. To avoid the instability, we used a platinum wire electrode, 0.5 mm in diameter, without shielding just by inserting it into a solution up to a given length (ca. 7.5 mm).^38–40 The length was determined each time when the wire was set in the solution. The surface was polished with buff including alumina powder, subjected to ultrasonication in a solution of mixed acid composed of H₃PO₄ + HNO₃ + CH₃COOH (Vol. 2 : 1 : 1), and then rinsed with distilled water. The length was controlled using an optical Z-stage and was evaluated using an optical microscope. The reference electrode and the counter electrodes were Ag|AgCl in a saturated KCl solution and a platinum coil, respectively. The potentiostat used for the ac-impedance measurements was a Compactstat (Ivium, the Netherlands). The ac-impedance data slightly varied with the selected current ranges as well as the frequencies of the filter and were confirmed to be consistent within the experimental error.

The concentration of KCl was 1 M (=mol dm⁻³). The solution resistance was determined from the intercept of the real impedance at zero imaginary resistance in the Nyquist plot. The solution resistance at the electrode, 7.6 mm long, was (3.2 ± 0.2) Ω for a series of ac-experimental runs carried out by changing the dc-voltage.

The delay of the potentiostat was examined using a series combination of a carbon resistance (1, 10 kΩ) and a film capacitor (0.1 μF) over the frequency range from 1 Hz to 10 kHz. No abnormality was observed so far as |Z₂|/Z₁ > 0.04. For these high frequencies that yield |Z₂|/Z₁ < 0.04, the |Z₂| values were overestimated by a few percentages. Most of the experimental conditions of the DL impedance were in the domain of |Z₂|/Z₁ > 0.04. When the frequency was over 5 kHz, the real part of the impedance was observed to be larger than the calculated values, which was attributed to the delay of the potentiostat decreasing the current. We analyzed the impedance data in the frequency domain from 1 Hz to 3 kHz. This domain will be discussed to present the faradaic effect.

3. Results and discussion

3.1 The presence of negative capacitance in the DL

The ac-impedance was obtained for several concentrations of FcTMA at various dc-potentials in the ac frequency f ranging from 1 Hz to 3 kHz at the Pt wire electrode. Fig. 2 shows the Nyquist plots (the imaginary impedance, Z₂, vs. the real one, Z₁) for three dc-potentials. The plots (a) at which the FcTMA was electroinactive were almost overlapped with those without FcTMA in a KCl solution. They overlapped the lines for which the values of the slope were over 5. Finite values of the slopes, as shown in Fig. 2, instead of a vertical line suggests the CPE behavior^63–66 or the power law of the frequency-dependence.^38–42 When the dc potential was in the electroactive domain (b) of FcTMA, the values of −Z₂ were smaller than those in the inactive potential domain (a), due to the contribution of the faradaic current. The plot at the formal potential (c) exhibited a line for which the slope value was one. The equi-values of Z₁ and −Z₂ suggest the Warburg impedance. No hemi-circle due to the kinetic control was observed even at a frequency as high as 5 kHz. Consequently, the impedance for (c) should be controlled by the diffusion current of FcTMA. The diffusion-controlled step was confirmed using cyclic voltammetry by the proportionality of the peak currents to the square-roots of the scan rates less than 0.5 V s⁻¹ in 0.4–3.0 mM FcTMA solutions. The interaction between the DL impedance and the Warburg impedance may be revealed in the variation of their admittance with the frequency rather than the Nyquist plots because the two impedances can be regarded as a parallel combination in the equivalent circuit (Fig. 1(A)).

Fig. 2 The Nyquist plots obtained in 3 mM FcTMA + 1 M KCl solution at E_dc = (a) 0.1, (b) 0.3, and (c) 0.4 V vs. Ag|AgCl. (×) denote the values at f = 30 Hz.

We considered the equivalent circuit at which C_d exhibited frequency-dependence when the ac voltage V_ac = V₀ e^iωt was applied to the electrode, where V₀ is the amplitude of the ac-voltage, ω is the angular velocity of the ac-voltage, and i is the imaginary unit. The current density through the frequency-dependent DL impedance can be represented in terms of the time-derivative of the charge density σ_D (=C_dV_ac) stored in the DL capacitance as³⁸

j_DL = dσ_D/dt = C_ddV_ac/dt + V_acdC_d/dt (1)

The first term becomes iωC_dV_ac, whereas the second term can be rewritten as V_ac(dC_d/dω)(dω/dt′), where ω = 1/t′. When the empirical expression for the frequency-dependence f^−λ at the frequency f (=ω/2π) and a positive constant of λ (ref. 38, 39 and 41) are inserted into eqn (1), the DL admittance is given by

Y_DL = (λ + i)ωC_p (2)

where C_d is mentioned as C_p to stress the parallel combination of the real admittance (λωC_p) and the imaginary admittance (ωC_p) in the equivalent circuit.

On the other hand, the faradaic ac-current density is given by⁶¹

with

ζ_dc = (E − E⁰)F/RT

where c* is the sum of the bulk concentration of the oxidized species and the reduced species and D is the diffusion coefficient common to both species. Eqn (3) is valid on the assumption of neglecting harmonics. It is necessary to examine the assumption for eqn (3) before analyzing the experimental data. The derivation of eqn (3) using the Laplace transformation has been described in the Appendix. The derivation shows that eqn (3) is valid within 8 mV of the E_dc error at V₀ = 10 mV even if the current includes harmonic components. If the DL impedance is considered to be independent of the Warburg impedance, the observed current is a simple sum of the currents through the DL and through the Warburg impedance, as illustrated in Fig. 1(A). Then, the real admittance Y₁ and the imaginary admittance Y₂ are given, respectively, by

Y₁ = λωC_p + Y_W (4)

Y₂ = ωC_p + Y_W (5)

where

We determined the numerical values of Y₁ and Y₂ using the following equation:

Y₁ = (Z₁ − R_S)/[(Z₁ − R_S)² + Z₂²], Y₂ = Z₂/[(Z₁ − R_S)² + Z₂²]

where the solution resistance R_S was determined from the intercept of Z₁ in the Nyquist plot by extrapolation to −Z₂ → 0. According to eqn (4) and (5), the subtraction Y₂ − Y₁ cancels the Warburg impedance. When the empirical frequency-dependence

C_p = C_{p,1 Hz}f^−λ (7)

is inserted into the subtraction, we obtain

Y₂ − Y₁ = (1 − λ)ωC_p = 2π(1 − λ)C_{p,1 Hz}f^1−λ (8)

Fig. 3 shows the variations of Y₂ − Y₁ at f = 100 Hz with the dc-surface concentration c* cosh⁻²(ζ_dc/2). The values of Y₂ − Y₁ decreased with an increase in the dc-surface concentration. Those at concentrations larger than 2 mM were decreased to negative values. The variation, as shown in Fig. 3, is the intuitive proof of the dependence of the DL capacitance on the faradaic reaction.

Fig. 3 The dependence of (Y₂ − Y₁)/A at f = 100 Hz on the surface concentration causing ac currents in the solution including (filled circles) 3 mM, (filled triangles) 2.4 mM, (triangles) 2 mM, (squares) 1.6 mM, and (circles) 0.4 mM FcTMA, where E⁰ = 0.401 V vs. Ag|AgCl and A is the area of the electrode.

Fig. 3 predicts that the values of Y₂ − Y₁ depend on the frequency. Fig. 4 shows the logarithmic variations with the frequency at two dc-potentials. The plots at the formal potential of FcTMA (0.4 V, (b)) show a slope of 0.5 for the Warburg impedance, in accordance with eqn (6). In contrast, the plots for the partial participation (a) of the faradaic reaction show a change of 0.5 in the slopes at a low frequency and 1 at a high frequency. The former is controlled by the Warburg impedance, whereas the latter is controlled by the DL impedance. The plots of log Y₂ against log f are approximately linear for all the dc-potentials.

Fig. 4 The variations of (a1, b1) log(Y₁) and (a2, b2) log(Y₂) with log f at E_dc = (a) 0.30 V and (b) 0.40 V vs. Ag|AgCl for c* = 3 mM. The slopes of the solid line and the dashed line are 0.5 and 1.0, respectively.

Fig. 5 shows variations of |Y₂ − Y₁| with f on a logarithmic scale for c* = 3 mM. The values of Y₂ − Y₁ for 0.38 V < E_dc < 0.42 V were negative over the frequency domain. This was not due to errors in the subtraction because 0.06 < |Y₂ − Y₁|/Y₂ < 0.2. The negative values indicate that (1 − λ)C_p < 0 (from eqn (8)). If 1 − λ < 0 and C_p > 0, the admittance given by ωC_p = 2πC_{p,1 Hz}f^1−λ might work as an inductance since an increase in the frequency decreases the admittance. If 1 − λ > 0 and C_p < 0, the capacitance is an inductance with a negative sign. The negative values have been discussed in the next section. In the dc-potential domain for Y₂ − Y₁ > 0, we evaluated λ and C_{p,1 Hz} from the slope and the intercept, as shown in Fig. 5.

Fig. 5 The variations of log|Y₂ − Y₁| with log f at E_dc = (a) 0.30 V and (b) 0.40 V vs. Ag|AgCl in a solution at c* = 3 mM. The slopes of the lines are (a) 0.82 and (b) 1.0.

The variations of λ and C_{p,1 Hz} with the dc-potential are shown in Fig. 6, together with the cyclic voltammograms. The values of λ are almost independent of the dc-potentials in the potential domains E_dc < 0.36 and E_dc > 0.44 V, at which the FcTMA is electrochemically inactive. The independence is in accordance with eqn (8). The values of λ cannot be determined in the electroactive potential domain because Y₂ − Y₁ < 0. This is a clear demonstration of the presence of the interaction between the DL impedance and the Warburg impedance. A similar variation was found in the dependence of C_{p,1 Hz} on the dc-potential. The values of C_{p,1 Hz} suddenly decreased at the boundary between the active and inactive potential domains and become negative in the electroactive potential domain. A negative capacitance is not a remarkable phenomenon. Despite being contentious, this phenomenon has been observed in the ferroelectric superlattices of the films.^67–69

Fig. 6 The potential dependence of λ (triangles) and C_{p,1 Hz} (circles) from the depolarized potential domain at c* = 3 mM. The upper graph is the cyclic voltammogram of FcTMA at the scan rate 50 mV s⁻¹.

The observed interaction between two types of the impedance may be due to (i) the low density of dipole moments causing the DL capacitance and (ii) the generation of surface charge via faradaic reactions.

(i) The orientation of water dipoles by the external electric field, which is mainly responsible for the DL capacitance, is largely suppressed by hydrogen bonding as well as the image force of the dipole on the electrode.⁴³ Since the field-orientation energy is one order in magnitude smaller than the hydrogen bonding energy and the imaging force energy, the concentration of the oriented molecules is only 10⁻⁴ times that of the bulk molecules.^43,44 This corresponds to milli-molar concentration, which is close to the concentration of the redox species in conventional voltammetry. Therefore, the DL capacitance is sensitive to other disturbances such as faradaic currents.

(ii) We temporarily considered oxidation for the faradaic reaction, which generates a charge more positive than that generated by reduction on the electrode. The dipole of the solvent induced by the field is oriented such that it may relax the external field, as illustrated in Fig. 7. In contrast, the charge generated by oxidation enhances the field. This behavior is opposite to the orientation of the dipole and hence the capacitance decreases.

Fig. 7 Schematic of the generation of a positive charge by oxidation, which enhances the field against the orientation of the solvent dipoles.

3.2 The theory of negative capacitance

First, we define capacitance when the charge density σ_E is mounted on one of two parallel electrodes with the distance L. The electric field in vacuum is given by φ_v = σ_E/ε₀ using the Gauss law, which provides the voltage V_v = Lφ_v. Then, the capacitance in vacuum is represented by C_v = σ_E/V_v = ε₀/L. When a solvent with a dipole is inserted into the electrodes, the dipoles are oriented by the field to yield the polarization vector P_s, as illustrated in Fig. 7. The polarization uniformly decreases the field by P_s/ε₀ to yield the voltage V_s = L(σ_E − P_s)/ε₀. Then, the capacitance is given by the following equation:

C_s = σ_E/V_s = ε₀/L(1 − P_s/σ_E) = ε₀ε_d/L (9)

where ε_d is the saturated dielectric constant given by 1/(1 − P_s/σ_E). The charge density has almost been demonstrated to be independent of the dc-voltage in the polarized potential domain⁴⁰ because the measured DL capacitance is controlled by the field-oriented dipoles in the Helmholtz layer rather than ion distribution.^38,39,42

When the elementary charge e is generated near the electrode surface by the electrode reaction, the field profile on the electrode is deformed from lines towards the solution radiating curves around the cation (Fig. 7). The profile can be evaluated using the mirror image technique,⁷⁰ for which a negative charge −e may be located in the electrode symmetrically with respect to the electrode surface. Let the average distance between two neighboring redox species responding to the ac voltage be l. Then, the charge density on the electrode becomes σ_E − e/l². Since the deformation of the voltage profile is included in the abovementioned charge density, the voltage evaluating the capacitance is V_s. Then, the capacitance is expressed by the following equation:

If c* is concentration of the molecules, the number of the molecules in a unit volume is l⁻³. When the numbers of the oxidized and reduced species are denoted by l_O⁻³ and l_R⁻³, respectively, the sum is given by l_O⁻³ + l_R⁻³ = 2c*N_A, where N_A is the Avogadro constant. The Nernst equation for the ac-voltage is given by l_O⁻³/l_R⁻³ = exp(ζ_dc + ζ_ac). Eliminating l_R⁻³ and expressing l_O⁻³ as the Taylor expansion on the assumption of |ζ_ac| ≪ |ζ_dc|, we obtain the average ac-voltage 2^−1/2|ζ_ac|.

Since l is the distance of redox species responding only to the ac-voltage, it should correspond to the second term in eqn (11), which is equal to l_R⁻³. Inserting the second term into eqn (10) yields the following expression:

At high concentrations and at electroactive potentials (ζ_dc ≈ 0), the second term in the parenthesis is predominant, and hence, the capacitance takes negative values. The negative sign in C_rx means a function of inductance. Zero of the charge density at the potential of zero charge in eqn (12) may make C_rx be minus infinity. We did not find any large negative values for the capacitance in our dc-potential domain. Moreover, we do not have data on the potential of zero charge under our experimental conditions at present.

Eqn (2) shows that the DL admittance has been observed as a sum of the real and the imaginary parts of the current due to the frequency-dependence. Since the diffusion of the redox species is independent of the DL properties, the Warburg impedance can be set in a parallel combination with the DL. Fig. 1(B) shows the predicted equivalent circuit, for which admittance is a simple sum of the five admittances, given by the following equation:

Y/A = (λ + i)ωC_p + iωC_rx + (1 + i)Y_W (13)

The difference between the real component and the imaginary component is

−(Y₂ − Y₁) = 2πfX_A (14)

whereon the assumption of high concentrations and ζ_dc ≈ 0. Eqn (14) indicates that Y₂ − Y₁ should be proportional to f if X is enough large such that the frequency-dependence of C_p (=C_{p,1 Hz}f^−λ) is negligible. The plot in Fig. 5(b) demonstrates the proportionality of Y₂ −Y₁ with f. The average value of the power of f at different dc-potentials is 1.04 ± 0.06 for 0.38 ≤ E_dc ≤ 0.42 V vs. Ag|AgCl at c* = 3 mM. The unity value of the power means that C_rx is independent of the frequency. When X is not enough large that the frequency-dependence appears as for c* < 2 mM, the low and high frequencies may make −(Y₂ − Y₁) negative and positive, respectively. Then, the plot log|Y₂ − Y₁| against log f should display divergent behavior at the frequency for which Y₂ = Y₁, as shown in Fig. 8 where the sign of Y₂ − Y₁ changes at log f = 1.8. This type of variation is an evident demonstration for the negative capacitance that can be obtained using a single experimental run of varying frequency.

Fig. 8 The variation of log |Y₂ − Y₁| with log f at E_dc = 0.40 V in a solution at c* = 2 mM. The slope of the dashed line is 1.

The intercept of the line, as shown in Fig. 5(b), should be log(2πXA), according to eqn (14). The values of X were plotted against cosh^−4/3(ζ_dc/2) in Fig. 9, falling on one line. The linearity indicates the justification of the present concept of the contribution of the redox charge to the DL capacitance. The slope of the line is 22 μF, which corresponds to (c*N_A|ζ_ac|)^2/3eε₀ε_r/2Lσ_E. For the values of ε_r = 5 (ref. 42) and L = 0.1 nm without surface roughness, we obtained σ_E = 3 × 10⁻⁴ C m⁻², equivalent to the area (22 nm²) occupied by one charge on the electrode. Consequently, the redox charge has a large enough contribution to the DL impedance. If the adsorbed redox charge causes a charge transfer reaction keeping the adsorbed state, only the imaginary admittance should increase. At this point, our result was different from that of the adsorbed charge reported by Unwing's group.⁷¹

Fig. 9 The variation of X with cosh^−4/3(ζ_dc/2) in a solution of 3 mM FcTMA + 1 M KCl.

A question arises that why the negative capacitance represented by Y₂ < Y₁ was only observed in the high frequency domain, as shown in Fig. 8. This behavior results in the formation of the redox charge density e/l², as shown in eqn (10). Since the formation rate increases in proportion to f^1/2, a higher frequency provides a larger decrease in the capacitance, which mainly contributes to Y₂. As a result, Y₂ < Y₁ is remarkable only in the high frequency region. This observation may also be applied to currents after a short time via chronoamperometry and cyclic voltammetry. The currents may be smaller than the sum of the theoretically evaluated faradaic current and the capacitive current. This will be explored in our future work. The steady-state currents should have no contribution to the DL impedance and hence agree with the theoretically diffusion-controlled values.

4. Conclusions

The DL capacitance can be separated from the Warburg impedance by taking the difference between the real admittance and the imaginary impedance. The extracted DL capacitance decreases with the faradaic current of FcTMA. The decrease is caused by the generation of surface charge via diffusion-controlled redox reactions, which cancels the DL capacitive charge. This result is opposite to Frumkin's effect in the relationship between an effect and a cause. The parameter for the frequency-dependent power law λ was hardly influenced by the redox reactions. When the redox concentration is over 2 mM, the DL impedance was observed to be negative. The negative values suggest an inductance, but its frequency-dependence is opposite to that of an ideal coil.

Appendix

The equation for the ac-impedance controlled by the diffusion of redox species for R ↔ O + e⁻ has been derived herein without taking an infinite limit of the heterogeneous rate constant. The diffusion equations for species i = R and O with a common diffusion coefficient are

∂c_i/∂t = D∂²c_i/∂x² (A1)

The boundary conditions at the electrode surface (x = 0) are given by the Nernst equation

and the balance of the fluxes

D(∂c_R/∂x)_x=0 + D(∂c_O/∂x)_x=0 = 0 (A3)

where and are the bulk concentrations and ζ_ac(t) = V₀ e^iωtF/RT. The oxidation current density is defined as follows:

j = FD(∂c_R/∂x)_x=0 (A4)

The initial conditions are the same as the boundary conditions in the bulk.

The upper bar of c_i below denotes the Laplace transform of c_i. Carrying out the Laplace transformations of eqn (A1)–(A3) and solving the ordinary differential equations satisfied with the boundary conditions, we obtain an expression for the concentrations at the electrode surface:

Their inverse Laplace transforms are given by the following expressions:

Inserting eqn (A5) into (A2) yields,

where ζ_dc is the dimensionless dc potential given by . The integral equation with Abel's type can be solved intowhere β = exp(ζ_dc).

Applying Leibniz's theorem for the differentiation of an integral⁷² to eqn (A7), the following equation was obtained:

Using the following differentiations for X = exp(ζ_ac(t − u)):

Then, eqn (A8) can be rewritten as follows:

When we use the conventional approximation ζ_ac < 1, we get e^z ≈ 1 + z + z²/2 for z(t) = ζ_ac(t) = (FV₀/RT)e^iωt to obtain,

The integral of e^−au(a/πu)^1/2 from 0 to t becomes the error function, erf((at)^1/2). Since the asymptotic form of the error function is erf((iωt)^1/2) ∼1 − π^−1/2(iωt)^−1/2 exp[(iωt)^1/2], it tends to unity for a long application of the ac-voltage. Then, eqn (A9) is reduced to

When multiplying eqn (A10) by and applying the following relationships: , (1 − β)/(1 + β) = −tanh(ζ_dc/2), and (1 + β)²/3β = (4/3)cosh²(ζ_dc/2), we obtain

where

The ac-current density can be expressed as a simple sum of the fundamental frequency and its harmonics. Since |G₂| and |G₃| are independent of ω, |j| is proportional to ω^1/2 even if the current includes harmonics. Fig. 10 shows variations of the maximum of the dimensionless admittance densities

with the dc-potential for the ac-amplitude of 10 mM. The variation of Y₃ is close to Y₁ for the observed current. Harmonics, especially the second harmonics (G₂ζ₀), cause a potential shift of up to 8 mV. They have no effect on the peak current.

Fig. 10 The variations of Y_n with E_dc, where G_nζ₀ⁿ⁻¹ is each component of Y_n, calculated using eqn (A12) and (A13) for V₀ = 10 mM.

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