Capacitive Currents Flowing in the Direction Opposite to Redox Currents

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Capacitive Currents Flowing in the Direction Opposite to Redox Currents Koichi Jeremiah Aoki, Jingyuan Chen, and Peng Tang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b03335 • Publication Date (Web): 25 Jun 2018 Downloaded from http://pubs.acs.org on July 7, 2018

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The Journal of Physical Chemistry

Capacitive Currents Flowing in the Direction Opposite to Redox Currents Koichi Jeremiah Aoki1, Jingyuan Chen2∗, Peng Tang2, 1

Electrochemistry Museum, Fukui, 910-0804 Japan

2

Department of Applied Physics, University of Fukui, 3-9-1 Bunkyo, Fukui, 910-0017

Japan

Abstract A simple electrode reaction provides negative values of the double layer (DL) capacitance because a dipole of electrochemically generated charge coupled with the imaged charge on the electrode is oriented in the direction opposite to that of solvent. Some properties such as dependence of the capacitive values on the dc-potentials, the frequency dependence and localization of the redox charge are herein discussed in order to comprehend the negative capacitance. The redox species used are a ferrocenyl derivative, hexaammineruthenium and hexachloroiridium, of which heterogeneous reaction rates are so fast that the faradaic currents should be controlled by diffusion or the Warburg impedance. The observed ac-current can be represented by a simple sum of the diffusion-controlled current and the DL charging current by solvent and that by the redox species. Subtraction of the observed imaginary part of the admittance from the real one eliminates the diffusion-contribution to extract the charging admittance. This admittance takes negative values at dc-potentials near the standard redox potential. It is formulated by the dipole of the redox charge paired with the mirror-imaged charge. The negative capacitance has frequency dispersion similar to the conventional DL

∗

Corresponding author, e-mail [email protected] (J. C.), phone +81 9082625425, fax +81 776

592070 1

ACS Paragon Plus Environment

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capacitance. A pseudo-capacitor has enhanced cooperatively the charge from the sum of individual DL charge and the redox charge, but this advantage is open to question.

1. Introduction

Nyquist plots by ac-impedance measurements show a semicircle when the circuit is regarded as a parallel combination of an ideal capacitance and an ideal resistance. The electrochemical meaning of the circuit is a combination of a DL capacitance and a charge transfer resistance by redox reactions. In contrast, Nyquist plots without the parallel resistance show a vertical line theoretically, although they exhibit experimentally a line with a slope of more than 5 owing to the frequency dispersion of the DL impedance. Visually clear distinction between a line and a semicircle confirms the presence of charge transfer resistances.1 The charge transfer resistances, Rct, observed as a semicircle can be brought about not only by electron transfer rates represented by the Butler-Volmer equation but also by suppressed transfer rates of redox charge through films, grain boundaries of electrodes, and follow-up chemical complications, as has been addressed as disadvantages of ac-impedance techniques.2 Thus, it is dangerous to assign Rct as a heterogeneous rate constant. The unambiguous assignment would require quantitative analysis of the frequency-dependence such as in the Warburg impedance of which both the real and the imaginary parts are proportional to f -1/2,3 where f is ac-frequency. Nyquist plots do not show explicit variation with frequency. A technique of examining the frequency dependence theoretically is the plot of cotangent of the phase angle of the admittance against f1/2.

4,5

Unfortunately, it is unsuccessful practically for evaluating Rct because of

difficulty in subtracting capacitive contributions.

2


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The equivalent circuit for a Randles cell, 6 which is composed of a parallel combination of a capacitance and a charge transfer resistance including the Warburg impedance, is based on the assumption that faradaic currents are independent of the capacitive ones. Delahay claimed it to be valid only for very fast reactions.7 The independence has been considered for multi-component redox species8 on the concept that observed current is a simple sum of faradaic currents and capacitive ones. 9 However, the independence is not always justified, because the capacitance is brought about with a few millimolar of field-oriented solvent dipoles,10 which are competitive with a capacitance generated by redox charge. An electrochemically generated positive charge, for example as illustrated in Figure 1, forms a mirror-imaged negative charge in the electrode in order to make the electric lines enter into the electrode surface perpendicularly.11 Since the electrostatic interaction of the two attracting charges is higher than the thermal energy,12 it can work as such a dipole as forming the DL impedance. The redox dipole by the external field is oriented in the direction opposite to that of the solvent. As a result, it decreases the DL capacitance, leading it possibly to a negative value at the cost of the solvent-driven capacitance.10 The well-known interaction of the faradaic current with the capacitance is Frumkin's effect, 13 at which the observed charge transfer resistances depend on concentration of salts as well as specifically adsorbed electrolyte,13,14,15,16 as reviewed theoretically from the points of ionic distributions.17 Since we are concerned with voltammetric conditions of high concentrations of supporting electrolyte, the capacitance caused by the redox reactions belongs to a category different from Frumkin's effect. In other words, the present interaction in Figure 1 is opposite to Frumkin's effect in a relationship between a cause and an effect. In order to find the effect of redox reactions on the DL impedance, it is necessary to extract the DL component from observed ac-currents by removing the faradaic

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currents. Since the real component of the Warburg admittance is the same as the imaginary one, the subtraction of the real admittance from the imaginary one leaves the DL component behind.10 However, this technique includes errors resulting from the subtraction between two large faradaic currents. It is important to confirm the validity of the subtraction. It is also of interest to examine whether the DL capacitance dispersion, called the constant phase element,18,19,20,21,22 is retained or not by the redox reactions. This research is directed to elucidating the four points; i) errors included in the DL components subtracted by the Warburg admittance, ii) a technique of determining the negative capacitance, iii) frequency-dependence and dc-potential-dependence of the negative capacitance and iv) presentation of a molecular model for the negative capacitance. The first three will be obtained from the ac-impedance of ferrocenylmethyl tetramethylammmonium (FcTMA), hexaammineruthenium and hexachloroirridium in aqueous solution as very fast charge transfer species causing diffusion-controlled currents. The forth subject leads to interpret the negative values of the capacitances by the redox reactions. redox reaction sol vent

0

E1 E2

sol vent

sol vent

2d

Electrode Figure 1. Orientation of a solvent molecule by the external field E1, and generation of the redox positive charge which forms the field E2 by the mirror imaged charge.

2. Experiments

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FcTMA originally with an iodide type was substituted into BF4- by addition of NH4BF4 and recrystallization. All the chemicals were of analytical grade. Water used was distilled and ion-exchanged. Concentration of a stock solution was determined from the slope of the proportional line of the peak currents of cyclic voltammograms to the square-roots of the scan rates. A platinum wire electrode 0.5 mm in diameter was used for the working electrode without shielding just by inserting it into a solution by a given length (ca. 8 mm). It does not yield floating capacitive currents, unlike a disk-exposed electrode at which floating current appears from crevices of the boundary between the electrode and the insulator. 23 , 24 Its surface was polished with buff including alumina powder, was ultrasonicated in the solution of the mixture, H3PO4 + HNO3 + CH3COOH (Vol 2:1:1), and then was rinsed with distilled water. The length inserted into the solution was controlled with an optical Z-stage, and was evaluated with an optical microscope. The reference electrode and the counter electrode were Ag|AgCl in saturated KCl solution and a platinum coil, respectively. The potentiostat for ac-impedance measurements was Compactstat (Ivium, Netherland). The ac-impedance data were obtained by applying ac-voltage 10 mV in amplitude which was super-imposed on a given dc-potential. The frequency ranged from 1 Hz to 10 kHz.

3. Results and Discussion 3.1 Confirmation of the negative capacitances

Figure 2 shows the Nyquist plots, i.e. the imaginary impedance Z2 vs. the real one

Z1, of 1 M (= mol dm-3) KCl aqueous solutions including 2.4 mM [Ru(NH3)]3+ at (a) electroinactive and (b) active dc-potentials. All the plots in a wide frequency domain showed approximately lines, regardless of dc-potentials. The extrapolation of each line

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toward Z2 = 0, which provides the solution resistance, Rs being close to 10 Ω, was independent of the dc-potentials, Edc. Plot (a) (at the electroinactive potential) is almost the same as the plot without [Ru(NH3)]3+. Values of the slopes range from 5 to 9, the behavior of which can be assigned to the constant phase element18,19,20,21,22 or the power law of the frequency.23,24,25,26,27 Plot (b) shows a slope -(Z1-RS)/Z2 = 0.95, slightly smaller than unity, except for the domain of -Z2 < 15 Ω or f > 400 Hz (see (B)). The slope close to unity suggests the diffusion-controlled current through the Warburg impedance. In contrast, the plot for -Z2 < 15 Ω in (b) deviated from the line. We simulated conventionally the deviated part in terms of a semicircle. The diameter of the semicircle, assigned frequently to the charge transfer resistance, yield Rct = 32 Ω, which corresponds to the exchange current density, 2.0 mA cm-2, for the electrode area 0.16 cm2 or the standard rate constant, 0.08 cm s-1. The value of the rate constant is much smaller than the values (> 10 cm s-1) estimated from the steady-state voltammetry at nanoelectrodes.28 It is probably not by a charge transfer reaction, as will be discussed later. FcTMA and [IrCl6]2- showed the deviation similar to in Figure 2(B,b) but were not so clear to be simulated with a semicircle. Since the DL impedance is arranged in parallel to the Warburg impedance in the equivalent circuit, admittance can be used more conveniently for algebraic operations such as addition and subtraction in order to extract the DL capacitance than the impedance. We define here the Rs-subtracted admittance as, Y = Y1 + iY2 = 1 / (Z − RS ) = (Z1 − RS − iZ 2 ) / Z − RS

2

(1)

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2000

(A)

60

(B)

1500

(a)

1000 500 0 0

(b)

500

- Z2 / Ω

(a)

- Z2 / Ω

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40 (b)

20 0

RS

-20 0

Rct +RS

20

40

60

Z1 / Ω

Z1 / Ω

Figure 2. Nyquist plots for 2.4 mM Ru(NH3)2+ solution including 1 M KCl at Edc = (a) 0.1 V vs. Ag|AgCl (electroinactive domain) and (b) -0.14 V (active one). Figure (B) is a magnification of (A) for high frequency. where Y1 and Y2 are the real and the imaginary part of the admittance. A real value of the Warburg admittance is the same as the imaginary one, whereas the DL impedance is mainly included in Y2. Therefore, we predict Y2 > Y1 under conventional conditions. However, line (b) in Figure 2 suggests Z1 - Rs > -Z2, as shown in the dashed line of Z1 Rs = -Z2. The inequality corresponds to Y1 > Y2 through Eq. (1). Both values of Y2 and Y1 at high concentrations of the redox species are so large that the sign of the difference, Y2 - Y1, may be quite sensitive to errors involved in Y2 and Y1. In order to examine the sign of Y2-Y1 without errors, values of the normalized differences, (Y2-Y1)/|Y| for FcTMA, are plotted against log f in Figure 3 for several dc-potentials, Edc. If Y2 and Y1 can be evaluated with 3% precisions, Y2 - Y1 < 0 is guaranteed for 10 Hz < f < 10,000 Hz and 0.28 V ≤ Edc ≤ 0.42 V. Fig. 4 shows variations of (Y2-Y1)/|Y| at 100 Hz with Edc for 2.4 mM FcTMA, [Ru(NH3)6]2+/3+ and [IrCl6]2-/3- solutions. The negative values of Y2-Y1 appeared at the middle of the anodic and the cathodic peak potentials of their cyclic

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voltammograms. This fact indicates that the negative values should be responsible for the redox reaction rather than errors.

0.2

(Y2 - Y1) / |Y |

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(d)

0.1 (c) 0

-0.1 0

(b) (a) 1

2

3

log(f / Hz) Figure 3. Frequency-dependence of normalized difference, Y2-Y1, for FcTMA at Edc = (a) 0.40, (b) 0.38, (c) 0.36 and (d) 0.34 V vs. Ag|AgCl.

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0.2 (a)

(b)

(c)

(Y2 - Y1) / |Y |

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0

-0.2 0

0.5

Edc / V vs. Ag|AgCl Figure 4. Variation of (Y2-Y1)/|Y| for (a) FcTMA, (b) [Ru(NH3)6]2+/3+ and (c) [IrCl6]2-/3- at f = 100 Hz with dc-potential. Arrows denote middle points of the anodic and the cathodic peak potentials of cyclic voltammograms. 3.2 Model of redox capacitance

We will suggest a source of generating the negative values theoretically at first, and then analyze the experimental data. When the faradaic current is controlled by diffusion of the redox species in solution, the observed ac-current is composed of the three; diffusion-controlled current caused by a charge transfer reaction, charging current of the DL by orientation of solvent dipoles,23,24,25,26 and the charging current caused by the formation of the redox dipole associated with the mirror-imaged charge (in Figure 1).10 The redox dipole takes the direction opposite to the external electric field and the solvent dipole. The two charging current densities are caused by the time-derivative of the charge displacement, Dc, which is composed of the electric polarization by solvent dipoles, Psl, the electric polarization by the redox charge, -Prx, and the intrinsic

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displacement, εoEe, for the external electric field, Ee. The negative sign in -Prx is due to the direction of the polarization opposite to Psl. Then we have the expression for the charging current density jc = dDc/dt = d(εoEe + Psl – Prx)/dt

(2)

We let the positive charge at the one end of the dipole for Prx be me, and the length of the dipole be 2d (see in Figure 1), where m is an average value of the reduced charge and the oxidized one, and e is the elementary charge. Then we can express the polarization Prx as the sum of the N dipole moments, Nme(2d), divided by the volume Sd including N dipoles on the surface, i.e. Prx = Nme(2d)/Sd = 2Nme/S. This is equal to twice the charge density of the redox species, Nme/S (= σrx), i.e Prx = 2σrx

(3)

σrx can be represented by the two ways, one being the product of the capacitance of the redox species by the ac-voltage, CrxEac, and the other being the surface charge which may proportional to the volume concentration at the surface without adsorption. We assume for the first way that Crx obeys a power law similar to Csl, i.e. σrx = CrxEac = Crx,1f -λ'Eac, where λ' is a constant and Crx,1 is Crx at f = 1 Hz. The redox reaction in the second way is assumed to be a one-electron charge transfer reaction obeyed by the Nernst equation without adsorption or chemical complications. Then, the surface charge density of reduced species with the charge number ν and the density of the oxidized species are proportional to the surface concentrations per volume, σr = kνFcr and σo = k(ν+1)Fco, respectively, where k is proportional constant with the dimension of length for converting the volume concentrations (cr and co) to the surface concentration. The sum of them yields σrx = σr+σo = kF[ν(cr+co)+nco]. When values of the diffusion coefficients of the reduced and the oxidized species are common, the sum of the volume concentrations, cr + co, at the electrode is equal to the bulk one, c*. The Nernst equation at the electrode potential, E, is given by exp[(E-Eo)F/RT] = co/cr = c*/cr-1. Eliminating cr from the Nernst equation yields 10


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σrx = kFc*{ν + (1+e-ζ)-1}

(4)

where ζ = F(Edc + Eac - Eo)/RT. The charging current density for Prx, defined by jrx = dPrx/dt = 2dσrx/dt, can be expressed by the two forms: (jrx)1 = ω(λ'+i) CrxEac

(5)

(jrx)2 = 2kFc*(FEac/4RT)iω sech2(ζ/2)

(6)

where the operation, dσrx/dt, has been made through the relation, d/dt' = (d/df)(df/dt'), after replacing t by t'. The equality of Eq. (5) to Eq. (6) for λ'

Capacitive Currents Flowing in the Direction Opposite to Redox Currents

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