# Similarity of Heterogeneous Kinetics to Delay of Double-Layer Capacitance Using Chronoamperometry

^{1}

^{2}

^{*}

## Abstract

**:**

^{−0.9}, which is similar to the decay of double-layer capacitive currents. The surface concentration decays with t

^{−0.4}-dependence.

## 1. Introduction

^{−1}but with much ambiguity, while steady-state voltammetry at ultra-micro electrodes yields rates constants that are too fast [32]. The inconsistency of the observed rates has been explained in terms of solvent reorientation dynamics or solvent friction effects [33]. For example, rates of 1,4-phenylenediamine vary linearly with the inverse of dielectric relaxation time [34], as supported by theories [35,36]. Rates of redox species have been associated with solvent dynamics [29,37], hydrodynamic radius [38], and viscosity [39]. Since other effects such as neutralization are significantly slower than the motion of electrons through electric fields, exponential dependence is not likely to have a true rate, but overall rates may be controlled via a time-dependent pre-exponential factor. It is necessary to suggest a new form of heterogeneous kinetics that includes time variables rather than exponential dependence.

## 2. Theory

^{+}by applying an oxidation potential. The oxidation is associated with charge neutrality via a counterion (Cl

^{−}) to form a redox dipole (indicated as two parallel arrows). Water dipoles neighboring Fc

^{+}− Cl

^{−}are oriented in the direction opposite to the orientation of Fc

^{+}− Cl

^{−}(in imaginary cells (a) and (a′)) in order to decrease the dipole–dipole interaction energy. Those next to (a) should be directed toward the electrode in (b), which can participate in the formation of DLC. The dipolar structure in (a′) is similar to that in (a). If Fc

^{+}in (a′) is not oxidized, the neutral form of Fc has no effect on the orientation of the neighboring water diploes in (b′), which may be thermally fluctuated, reaching the sum of the dipoles and the zero dipole moment in (b′). The interaction energy in [A] is the same as that in [B] for dipole–dipole interactions, resulting in the same chemical potential in [A] as in [B]. Consequently, the concentration on the electrode is different from the activity controlled by the Nernst equation.

_{dl}= λVC

_{1s}t

^{λ}

^{−1}

_{1s}is the DLC per area at t = 1 s, and λ is the number close to 0.1, according to previous results [40]. It is not in the form of exp(−t/R

_{s}C) for a solution resistance, R

_{s}and the DLC, C, because it largely deviates from an ideal capacitance due to frequency dispersion. Since no point of zero charge has been found on a platinum electrode in various aqueous solutions [9,41], V is the step voltage independent of the formal potential of a redox species. Although the decay speed (t

^{−0.9}) is larger than the Cottrell’s decay value (t

^{−0.5}), the DLC current still survives even at 1 s. This long relaxation caused by the power laws may be involved in the heterogeneous kinetics via dipole interactions.

_{s}, and the observed current density, j

_{ob}[42] as follows:

_{s}= c* − F

^{−1}(πD)

^{−1/2}∫

_{0}

^{t}j

_{ob}(u)(t − u)

^{−1/2}du

_{N}, from Cottrell’s current density, j

_{C}:

_{ob}(t) = j

_{C}− j

_{N}

_{C}= c*F(D/πt)

^{1/2}

_{N}= μVC

_{rx}t

^{μ}

^{−1}

_{rx}is the capacitance associated with a redox reaction. The functional form of j

_{N}is assumed to be similar to that of DLC (Equation (1)) because of similarity of the current source by the dipole–dipole interaction. Inserting Equation (3) into Equation (2) and carrying out integrations via the Beta function formula cited with Equation (6.2.1) in [43], we have:

_{s}(t) = (μVC

_{rx}/FD

^{1/2}){Γ(μ)/Γ(μ + 1/2)}t

^{μ}

^{−1/2}

^{μ}

^{−1/2}dependence. Although sufficiently oxidized electrical potential is applied to the electrode, the concentration of the reduced species may not instantaneously reduce to zero. The observed currents lower than the Cottrell’s current mean that the concentration values on the electrode are detectable.

_{C}− I

_{N}+ I

_{dl}. Each contribution is examined here in order to find which current contributes the most to c

_{s}. Chronoamperometric curves were calculated using Equations (1), (4) and (5) for our experimental values of c*, D [21], λ [40] and μ, where the gamma functions were evaluated with use of the approximate Equation (6.1.35) in [43]. They are shown in Figure 3A, and the Cottrell plots (I vs. t

^{−1/2}) are shown in Figure 3B. The lower deviation of the current, I

_{C}− I

_{N}, from the Cottrell equation is noticeable at a time shorter than 0.1 s (t

^{−1/2}> 3 in Figure 3B) and is a kinetic effect. The time-dependent current must include the DLC current (c). The observed current, I

_{C}− I

_{N}+ I

_{dl}, is predicted to be curved (d), which is close to the Cottrell equation without the capacitive current. When c* is less than 0.1 mM, values of I

_{C}− I

_{N}are close to those of I

_{dl}. Thus, pulse voltametric currents for low redox concentrations always suffer from capacitive currents [44,45,46].

_{BV}= c*FD

^{1/2}Λ exp(Λ

^{2}t) erfc(Λt

^{1/2})

^{−1/2}{exp[β(E

_{dc}− E°)F/RT] + exp[−α(E

_{dc}− E°)F/RT]}

_{dc}− E°. The Cottrell plots for Butler–Volmer kinetics exhibit a nonlinear shape for E

_{dc}− E° < 0.10 V, while those for negative capacitance fall on each line with almost zero intercepts. These two kinds of kinetics can be distinguished from the logarithmic plots more clearly than from the Cottrell plots at the two points: (i) the plotted lines for negative capacitance are shown, and (ii) the slopes of the lines are independent of E

_{dc}.

## 3. Experimental Section

^{−3}mol dm

^{−3}) (ferrocenylmethyl)trimethyl ammonium (FcTMA) and 0.5 M KCl were prepared in distilled and then ion-exchanged water using an ultrapure water system, CPW-100 (Advance, Tokyo, Japan). All the measurements were recorded at a room temperature.

## 4. Results and Discussion

_{dc}= 0.489 V). The linear sweep voltammogram is shown in the inset to specify the potential domain. The difference in the currents of the sweep voltammogram and the chronoamperogram is caused by the large difference in the time scale. The solution resistance was evaluated using Nyquist plots via AC impedance and was found to be 100 Ω. If 4 mV of IR-drop is added to the applied voltage, the effectively maximum current, 40 μA, corresponds to the chronoamperometric time 0.006 s according to the Cottrell equation for D = 0.7 × 10

^{−5}cm

^{2}s

^{−1}. On the other hand, the longest and most efficient time deviation from the Cottrell equation is the incidence time for cylindrical diffusion. The current at a cylindrical electrode of radius a is approximately expressed as a(πDt)

^{−1/2}+ 0.422 [49]. If 3% errors are allowed to be involved in the current, the longest time, t

_{L}, satisfying a(πDt

_{L})

^{−1/2}> 0.422 × 0.03 is t

_{L}< 0.2 s. Consequently, the time domain for the analysis is 0.006 s < t < 0.2 s.

^{−1/2}. The Cottrell plot theoretically calculated for c* = 1 mM and D = 0.7 × 10

^{−5}cm s

^{−1}is shown in Figure 5(d). The DLC-included current (a) is ironically close to the Cottrell current (d), indicating that the contribution of kinetics should be numerically similar to the DLC current. The difference between (d) and (c) seems to lie in the contribution of the negative capacitance. This resembles the deviation of the plots of the peak current against the square root of potential scan rates in cyclic voltammetry [21].

_{N}= I

_{C}+ I

_{dl}− I

_{ob}, has t

^{μ−1}dependence. In order to examine the power law in I

_{N}, we logarithmically plotted I

_{N}against t for two values of E, as shown in Figure 6. All the points at each E fell on each line, of which slopes were common and determined to be μ = 0.1. Values of VC

_{rx}were obtained from the intercept, as shown in Figure 6, and were plotted against E, as shown in Figure 7. They fell on a line, the slope of which represents C

_{rx}, according to Equation (5). The value was C

_{rx}= 67 μF cm

^{−2}, which is close to 64 μF cm

^{−2}[18] and was obtained via AC impedance and 60 μF cm

^{−2}[21] using fast cyclic voltammetry.

_{s}using Equation (6) for the determined values of μ and C

_{rx}at c* = 1 mM. Figure 8 shows the calculated time variation in c

_{s}and a dotted line averaged over t < 0.2. The value of c

_{s}might be almost zero without any interaction. The average concentration is 0.1 mM, suggesting that it is ineffective in oxidation. Although the potential is in the limiting current domain, the concentration of the reduced species on the electrode is not zero, as illustrated in Figure 1D. Oxidation energy is consumed by decreasing the dipole–dipole interaction on the electrode until the interaction becomes stable. This interaction is also a source of negative capacitance, and hence the non-zero surface concentration is equivalent to negative capacitance. The times at which the concentration at the surface becomes 5% and 2% of the bulk concentration are 0.5 s and 5 s, respectively. This slow relaxation is close to the frequency dispersion of the DLC.

^{2}) erfc(x) = (2/√π)/{x + (x

^{2}+ 4/π)

^{1/2}}

^{2}= 1 + 4/πΛ

^{2}t

^{1/2}/j

_{BV}

^{2}against 1/t, as shown in Figure 9, at E = 0.393 V. Some data ranging from 10 s

^{−1}< t

^{−1}< 60 s

^{−1}fell on a line, of which the intercept was forced to pass through 1, according to Equation (10). The slope provided Λ and k° = 3 × 10

^{−4}cm s

^{−1}via Equation (8) for α = β = 0.5. However, the plot in Figure 9 does not justify a linear variation in that the slopes decrease over longer periods of time. Values of k° thus decreased with an increase in E and became negative for E < 0.42 V. Since they were not uniquely determined, they are physiochemically insignificant.

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Illustrations of the orientation of solvent dipoles responding to positive voltage at an electrode when (

**A**) the interaction among the dipoles is neglected, (

**B**) it is stronger than the field-orientation energy, (

**C**) it is balanced with the latter energy and thermally fluctuated, and (

**D**) a redox species is oxidized (+) with a help of a counterion (−).

**Figure 2.**Arrangements of redox dipoles (black arrows) and solvent dipoles (blue). Since dipole interactions are cancelled (

**a**,

**a**′) with the solvent dipoles, the dipole interaction in (

**A**) is the same as (

**B**). The activity of (

**A**) is the same as (

**B**). Dipoles of (

**b**,

**b**′) work as DLC.

**Figure 3.**Chronoamperometric curves (

**A**) and their variations with t

^{−1/2}(

**B**) for (a) I

_{C}, (b) I

_{C}− I

_{N}, (c) I

_{dl}, (d) I

_{C}− I

_{N}+ I

_{dl}, calculated from Equations (1), (4) and (5) at c* = 1 mM, D = 0.7×10

^{−5}cm

^{2}s

^{−1}, μ = 0.1, λ = 0.1, V = 0.2 V, C

_{1s}= 30 μF cm

^{−2}, C

_{rx}= 70 μF cm

^{−2}and the electrode area 1.77 mm

^{2}.

**Figure 4.**Variations of (

**A**) I

_{BV}(bold solid curves) and I

_{C}− I

_{N}(dashed curves) with t

^{−1/2}for E

_{dc}− E° = (a) 0.15, (b) 0.05 and (c) 0.0 V, and (

**B**) their logarithmic plots when k° = 0.005 cm s

^{−1}, α = β = 0.5.

**Figure 5.**Cottrell plots of the observed chronoamperograms in the solution of 0.5 M KCl (a) with and (b) without 1 mM FcTMA for the potential stepped from 0.26 V to 0.489 V and for (d) the calculated potential of D = 0.7 × 10

^{−5}cm

^{2}s

^{−1}. Plots for the subtracted current against t

^{−}

^{1/2}are shown in (c). The inset shows the current–voltage curve in the solution of 1 mM FcTMA and 0.5 M KCl at a scan rate of 0.015 Vs

^{−1}. The arrow means that plots (b) follow the right ordinate.

**Figure 6.**Logarithmic plots of the negatively capacitive current against time for the current–time curves observed at the stepped potential from 0.26 V to E

_{dc}= (a) 0.440 and (b) 0.489 V.

**Figure 8.**Time variation of the concentration on the electrode surface, calculated using Equation (6) for μ = 0.1, C

_{rx}= 67 μF cm

^{−2,}and D = 0.7 × 10

^{−5}cm

^{2}s

^{−1}at E

_{dc}= 0.489 V vs. Ag|AgCl. The dashed line is the average concentration for 0 < t < 0.2 s.

**Figure 9.**Plots of y of Equation (9) against 1/t to determine Λ at E = 0.393 V when BV kinetics is applied.

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**MDPI and ACS Style**

Liu, Y.; Aoki, K.J.; Chen, J.
Similarity of Heterogeneous Kinetics to Delay of Double-Layer Capacitance Using Chronoamperometry. *Electrochem* **2023**, *4*, 301-312.
https://doi.org/10.3390/electrochem4020021

**AMA Style**

Liu Y, Aoki KJ, Chen J.
Similarity of Heterogeneous Kinetics to Delay of Double-Layer Capacitance Using Chronoamperometry. *Electrochem*. 2023; 4(2):301-312.
https://doi.org/10.3390/electrochem4020021

**Chicago/Turabian Style**

Liu, Yuanyuan, Koichi Jeremiah Aoki, and Jingyuan Chen.
2023. "Similarity of Heterogeneous Kinetics to Delay of Double-Layer Capacitance Using Chronoamperometry" *Electrochem* 4, no. 2: 301-312.
https://doi.org/10.3390/electrochem4020021