Charge Transfer Rates Controlled by Frequency Dispersion of Double-Layer Capacitances

Abstract

Reported rate constants of charge transfer reactions (CTs) have ranged widely, depending on techniques and timescales. This fact can be attributed to the time-dependent double-layer capacitance (DLC), caused by solvent interactions such as hydrogen bonds. The time variation of the DLC necessarily affects the heterogeneous electrode kinetics. The delay by the solvation, being frequency dispersion, is incorporated into the CT kinetics in this report on the basis of the conventional reaction rate equations. It is different from the absolute rate theory. This report insists on a half value of the transfer coefficient owing to the segregation of the electrostatic energy from the chemical one. The rate equation here is akin to the Butler–Volmer one, except for the power law of the time caused by the delay of the DLC. The dipoles orient successively other dipoles in a group associated with the delay, which resembles that in the DLC. The delay suppresses the observed currents in the form of a negative capacitance. The above behavior was examined with a ferrocenyl derivative by ac impedance methods. The delay from diffusion control was attributed to the negative capacitance rather than the CT, even if the conventional DLC effect was corrected.

1. Introduction

Observed rates of a simple electrode reaction are composed of steps of (1) charge transfer, (2) mass transfer, and (3) solvation of the redox species, according to a conventional concept for charge transfer. They are accompanied by (4) charge–charge interactions as a latent step. Successively occurring steps could be partially distinguished so that the slowest step would be observed. However, actual steps are coupled. The frequent coupling is a combination of (1) and (2), which has been elucidated with the diffusion-corrected Butler–Volmer equation. Unfortunately, rates reported even for ferrocene have ranged from 0.02 to 220 cm s⁻¹ [1], varying with techniques, conditions, and research groups. They may be complicated with rates of (3) and (4), which are relevant to a slow relaxation of double-layer capacitances.

Solvents in electrochemical processes have played a role in keeping high concentrations of redox species such as carbon powders [2,3], polymers [4,5,6], and battery materials [7,8,9,10]. The high solubility is attributed not only to simple physical properties of viscosity, dielectric constants, boiling and melting points, surface tensions, and solubility of ions but also complications by hydrogen bonding acidity, bipolarity, and a Hildebrand solubility parameter [11,12]. Static effects of solvents on redox reactions have been revealed to standard potentials, exemplified with the potential dependence of ferrocenyl derivatives [13,14] and C₆₀ [15,16,17]. In contrast, a large number of dynamical solvent effects have been reported in practical work on mixed solvents. However, their physicochemical reasons have not been discussed well. Even without redox species, the solvent effects have been exhibited in the double-layer capacitances (DLCs) through the wetting abilities of solvents [18], participation in field-oriented solvent dipoles [19], combinations of ionic liquids [20], and dependence on solvent conductivity and viscosity [21].

Electrode kinetics vary with the viscosity of solvents [22], their polarization [23,24], solvent-driven reorganization energy at the metal coordination [25,26], and diffusion coefficients through weak coordination [27,28]. A property common to these kinetic effects is a long period for the reorganization of solvent dipoles surrounding a redox molecule [23], as has been expressed in an easy-to-digest term, ‘stickiness’ [26]. The delay by solvent dipoles has also been found in the DLC as the power law of the frequency [29,30,31]. The delay has been explained with the constant phase elements [32,33,34,35,36], resistive capacitive contributions of field-driven ionic separation [36], and non-uniform current distribution by surface roughness [37]. Similar behavior has been reported in other fields as the comparison of diffusion distances [38] with the movable distance of a molecule and that of reaction rates [38] with the collision frequency of molecules.

A simple charge transfer (CT) of an electrode reaction normally undergoes the following four steps:

(a)

Change in the charge of a redox molecule by a high electric field at the electrode–solution interface;

(b)

Formation of a dipole by a counterion for charge neutralization;

(c)

Rearrangement in terms of solvent dipoles for stabilization, like in the DLC;

(d)

Current is produced by the flux of the redox concentration at the electrode.

Processes (a) and (b) are caused by the motion of one particle, whereas (c) and (d) are caused by mass behavior. Therefore, the observed rate is controlled with steps (c) and (d). The rearrangement (c) of the DLC is obeyed by the power law of the time, t^λ, for a positive value of λ close to zero [19,29,30,31]. An electrode potential can control the activity of the redox species through the Nernst equation, and then the activity is converted into concentrations [39] through (c) as a result of interaction among solvents.

The delay associated with the conversion of the activity to the concentration will be incorporated into the one-electron CT kinetics in this report in the context of solvent interaction. The kinetic equation will be derived in the form of the conventional chemical reaction rates rather than the absolute rate theory. It will be applied to the ac impedance technique to obtain a practical form useful for applications and will be verified experimentally by the use of a ferrocenyl derivative.

2. Theory

2.1. Kinetic Equation with Dipole Interaction

Our electrode reaction involves the one-electron transfer, R ↔ O + e, at an electrode without chemical complications. The ‘e’ is not a molecule because of the instability in solution. The expression ‘O + e’ represents the presence of O in the state of the field, causing a reduction rather than a molecular collision of O with the electron in solution. When we express a state of the hole as H and that of the electron as E, the reaction can be rewritten in the form of a bimolecular reaction as

R + H → O (k_f) and O + E → R (k_b)

(1)

where k_f and k_b are the forward and backward rate constants, respectively. These states are controlled not only by electrode voltage but also by the environments of H and E, such as solvation and/or charge neutralization by salts. They are different from those in a vacuum, as can be imagined by the dependence of redox potentials on kinds of solvents and salts. The surface concentration of R, Γ_R, is generated from a supply of volume concentration of O, c_Os, multiplied by the intensity of the reductive state of H, and is consumed by the amount of that of R, c_Rs, multiplied by the intensity of the oxidative state of E, where the subscript ‘s’ in c_s means the concentration at the electrode surface. Since the intensive quantities of H and E are equivalent to the number of the active reaction sites per area, they can be represented as surface concentrations, Γ_H, and Γ_E for H and E, respectively. Then, the rate equation becomes

dΓ_R/dt = k_b c_Os Γ_E − k_f c_Rs Γ_H

(2)

where a unit of k_k and k_b is, for example, mol⁻¹ m³ s⁻¹.

When voltage V is applied to the electrode, a fraction of V, βV for 0 < β < 1, contributes to the oxidation by the electrostatic energy −βVe. In contrast, the remaining V, (1 − β)V, is due to the reduction by (1 − β)Ve. The densities of the reaction sites, H and E, seem to be proportional to their Boltzmann distributions in order to express the energy in terms of concentrations. The term consisting of the Boltzmann factor represents activity rather than the concentration, which includes interaction with solvents or salts for electric neutralization. Interaction of the redox charge with solvents, especially water, causes a macroscopic delay as long as 1 s, given by the power law of the time, (t/t₀)^λ in the DLC [19,30,31], where t₀ is a standard time and λ is a constant close to zero. The time delay is required to convert the activity to the concentration [40]. Then, Γ_H and Γ_E can be reformed as

Γ_H = (t/t₀)^λ A_H exp(βVe/k_BT)

(3)

Γ_E = (t/t₀)^λ A_E exp[−(1 − β)Ve/k_BT]

(4)

where k_B is the Boltzmann constant and A_H and A_E are constants, including the chemical activation energy.

The logarithm of Equation (3) has the form of lnΓ_H(t/t₀)^−λ = lnA_H + βVe/k_BT. The first term on the right-hand side represents the chemical activation energy, whereas the second one represents the electrostatic energy. The whole energy, lnΓ_H(t/t₀)^−λ, is given by a simple sum of the two energies. The simple sum has been classically used for the definition of electrochemical potentials. Complete separation into the two energies implies that the electrostatic energy should provide either the oxidation state or the reduction one with equal probability without chemical properties. As a result, we have β = 1 − β, or β = 1/2. Inserting Equations (3) and (4) into Equation (2) and using the current density, j = −FdΓ_R/dt, we have

j/F = (t/t₀)^λ[k_f A_H c_Rs e^VF/2RT − k_b A_E c_Os e^−VF/2RT]

(5)

The rate has been assumed to be a first-order reaction with respect to the reactants, whereas the dipole interaction may occur with a high order of dipoles. The latter can be represented by (t/t₀)^λ. The equilibrium occurs for V = V_e, c_Rs = c_R* (the bulk concentration) and c_Os = c_O* at j = 0, satisfying

k_f A_H c_R* exp(V_eF/2RT) = k_b A_E c_O* exp(−V_eF/2RT)

(6)

We can rewrite Equation (5) as

j = (t/t₀)^λ FΛ(e^η/2 c_Rs/c_R* − e^−η/2 c_Os/c_R*)

(7)

where η = F(V − V_e)/RT and

Λ = (k_bk_fA_HA_E c_O*c_R*)^1/2

(8)

Here, a typical unit of Λ is mol m⁻² s⁻¹. Equation (7) is a general CT kinetic equation independent of any electrochemical technique. Equation (7) has the same form as the Butler–Volmer (BV) equation if substitutions of A_HA_E = 1 and F(k_bk_fc_O*c_R*)^1/2 = j₀ (the exchange current density) are made for the transfer coefficient 0.5 and λ = 0.

The prediction of β = 1/2 infers an equal contribution of the electrostatic energy to both the oxidation and the reduction. In order to demonstrate this equality, we define the current densities of the oxidation and the reduction for a long time (t ≈ t₀) in uniform concentrations as j_O = FΛ e^βη and j_R = FΛ e^(1−β)η, respectively, in Equations (2) and (4). The derivatives with the potential, being admittance, become

dj_O/dη = FΛβe^βη

(9)

dj_R/dη = FΛ(1 − β)e^(1−β)η

(10)

The equality, dj_O/dη = dj_R/dη, only holds when β = 1/2.

Equation (7) is applied here to a voltage-controlled technique under the linear diffusion of O and R with a common value of the diffusion coefficient D. Inserting the relations of concentrations at the surface with the current density

c_Os = c_O* + {1/F(πD)^1/2}∫₀^t j(t − u)u^−1/2du

(11)

c_Rs = c_R* − {1/F(πD)^1/2}∫₀^t j(t − u)u^−1/2du

(12)

into Equation (6), we have

j(t/t₀)^−λ/FΛ = 2sinh(η/2) − [e^η/2/c_R* + e^−η/2/c_O*]∫₀^tj(t − u)u^−1/2du/F(πD)^1/2

(13)

If FΛ is regarded as the exchange current density at λ = 0, Equations (9) and (10) have the same form as the concentration-included BV for β = 1/2.

2.2. AC Impedance

We derive expressions for ac currents responding to the sinusoidal voltage with the overpotential V − V_e at the frequency f or the angular velocity ω = 2πf. The integral in Equation (9) for the ac current density, j = |j|e^iωt, becomes

∫₀^tj(u)(t − u)^−1/2du ≈ |j|e^iωt∫₀^∞e^−iωuu^−1/2du = |j|e^iωt(π/2ω)^1/2(1 − i)

(14)

By use of approximations of sinh(η/2) ≈ η/2 and e^±η/2 ≈ 1 for a small value of η = F(V − V_e)/RT as well as the relation

1/c_R* + 1/c_O* = (4/c*)cosh²(ζ_dc/2)

(15)

Equation (13) is simplified to

η = j{4(1 − i)cosh²(ζ_dc/2)/c*F(2Dω)^1/2 + (ω/ω₀)^λ/FΛ}

(16)

where t and t₀ are replaced by 1/ω and 1/ω₀, respectively, ζ_dc = F(E_dc − E^o)/RT} for the dc voltage E_dc, and c* = c_O* + c_R*. The kinetic real impedance Z₁ and the imaginary one Z₂, defined by Z = Z₁ + iZ₂ = (RT/F)η/j are

Z₁ = (RT/F²Λ)(ω/ω₀)^λ + 1/σω^1/2

(17)

Z₂ = − 1/σω^1/2

(18)

where

σ = c*F²D^1/2 sech²(ζ_dc/2)/2^3/2RT

(19)

Z₂ expresses the impedance caused by diffusion. On the other hand, the CT term can be extracted from the following sum:

Z₁ + Z₂ = (f/f₀)^λRT/F²Λ

(20)

If plots of log(Z₁ + Z₂) against log f fall on a line, the slope provides λ. The intercept at f = f₀, for example, f₀ = 1 Hz, provides Λ through log(RT/F²Λ). The specific case for λ = 0 expresses the conventional ac impedance analysis, as shown in Figure 1. The difference between (a) from the BV (c) in Figure 1 is remarkable in the high-frequency domain, where the kinetic effect is discerned from the diffusion behavior (b).

2.3. Contribution of Negative Capacitance

Electrode kinetics is necessarily disturbed by the current of the DLC because the capacitive current is largely involved in observed currents in such a high-frequency domain that the kinetics may be remarkable. This disturbance can be predicted from the conventional equivalent circuit in Figure 2, except for NC, where Wb is the Warburg impedance and R_s is the solution resistance. The other obstacle to the kinetics is the negative capacitance [31] (NC), which is brought about with relaxation of the CT associated with rearrangements of solvent dipoles. The NC current flows in the direction opposite to the DLC current because of the inverse orientation of solvent dipoles. Since the NC has a form similar to the DLC, except for the sign, the admittance of the NC can be expressed as

Y_NC = (λ + I )ωC_NC = (λ + i)C_NC,1Hz ω^1−λω₀^λ

(21)

where C_NC,1Hz is a negative value of the NC at 1 Hz. The NC current is proportional to any faradaic current, including the diffusion-controlled currents, irrespective of the extent of kinetics. The observed current decreases by the amount of the NC current because it can be regarded as a parallel combination with the NC in the equivalent circuit, as shown in Figure 2. Then, the corrected admittance for the NC is Y″ = Y′ + Y_NC, where Y′ is the admittance corrected for the solution resistance and the DLC.

3. Experimental Results and Discussion

The ac impedance was obtained with the application of ac voltage with a 10 mV amplitude and frequency ranging from 0.5 Hz to 5 kHz for the working electrode of a platinum wire 0.5 mm in diameter inserted into a test solution 5 mm long, as written previously [40]. The test solution was 1 mM (ferrocenylmethyl)trimethylammonium (FcTMA) + 0.5 M KCl to be deaerated before the electrochemical measurements. The reference electrode was Ag|AgCl in 0.5 M KCl.

3.1. Voltammetric Features

Nyquist plots at all the applied dc potentials took lines, as shown in the left inset of Figure 3, with (R_s, slope) = (12 Ω, 3.8) and (10 Ω, 0.97) as well as E_dc = 0.20 V and 0.36 V, respectively. Their slopes at E_dc near E⁰ were united, whereas those at E_dc far from E⁰ were more than 6. The former means the diffusion-controlled step [41], while the latter means the DLC control associated with the frequency dispersion [19,28,35]. No kinetic evidence, such as the appearance of a semicircle, was found at any dc potential. According to the BV equation combined with diffusion, Z₁ and −Z₂ should have a linear relationship with f^−1/2, as shown in Figure 3 (a) and (b), respectively. The intercept of the line in −Z₂ vs. f^−1/2 was zero, implying complete diffusion control. On the other hand, that for Z₁ vs. f^−1/2 was 4.2 Ω, which seems to represent the CT resistance, R_CT = RT/F²Ak⁰c* [41], where k⁰ is the standard rate constant. The calculated value of k⁰ would be 0.7 cm s⁻¹. Since the values of the impedance included the solution resistance 10.5 Ω from the Nyquist plot, the net R_CT must take a negative value [40]. A possible reason for the negative value is a contribution of the charging current, which is more remarkable at higher frequencies. Currents of the DLC have to be subtracted from the observed currents.

The observed impedance of the simplest redox reactions is controlled not only by the CT rates mixed with diffusion but also by (I) the ohmic drop mainly in the solution, by (II) the delay caused by the DLC, and by (III) the NC associated with the faradaic currents [31]. The above data were also obtained at gold and carbon electrodes but included less reproducibility than at the platinum wire.

3.2. Subtraction of the DLC

The ohmic resistance, R_s, was subtracted in the Nyquist plots from Z₁ by the use of the intercept on the Z₁-axis at extrapolated infinite frequency so that the corrected impedance was Z₁ − R_s + iZ₂. The observed DLC without FcTMA was almost independent of E_dc, Ref. [19] is opposed to the V-shape predicted by the Gouy–Chapman theory. This property allows us to apply the frequency-dependent DLC at the polarized potential to those at every depolarized potential. Since the DLC component is connected in parallel to the faradaic one at the equivalent circuit in Figure 2, admittance Y rather than impedance should be additive. Z₁ − R_s + iZ₂ was altered to Y (=Y₁ + iY₂), from which the two components of the DLC admittance, Y_1,DLC and Y_2,DLC, were subtracted to yield Y′. Y′ was transformed into the impedance Z₁′ + iZ₂′. Variations of Z₁′ and −Z₂′ with f^−1/2 are shown in Figure 3 (c), (d). The proportionality as well as the overlap of Z₁′ and −Z₂′ implies the diffusion-controlled step without including any kinetics. Equations (17) and (18) indicate that Z₁′ + Z₂′ should be logarithmically linear to the frequency. Figure 4 shows the plots at three values of E_dc, which deviated largely from a line predicted from Equations (17) and (18). Since the kinetic effect is exhibited more clearly at a higher frequency, we drew a line by force on the plots for log f > 2 in Figure 4 (b). The slope λ was less than 0.05, being as small as for an ideal capacitance (λ = 0). The intercepts varied largely with the applied potentials, inconsistent with Equations (17) adn (18). These unsatisfied results would be caused by the NC associated with faradaic reactions.

3.3. Subtraction of Negative Capacitive Currents

If the ac current is controlled by diffusion, values of C_NC can be obtained through the logarithmic plots of Y₁ − Y₂ (=Y_NC) against f by the use of Equation (21). This analysis is based on the concept that the inequality, Y₁ > Y₂, is caused by the NC. Equation (21) can be rewritten as

(Y₁ − Y₂)_NC = −(1 − λ)C_NC,1Hz ω^1−λω₀^λ

(22)

where C_NC,1Hz has a negative sign. On the other hand, the difference between Y₁ and Y₂ for the CT is given from Equation (21) by

(Y₁ − Y₂)_CT = σω ^1/2χ/{(1 + χ)² + 1}

(23)

where

χ = σω^1/2(ω/ω₀)^λRT/F²Λ

(24)

Since the value of χ for c* = 1 mM, D = 10⁻⁵ cm² s⁻¹ and f = 1 kHz, a monomolecular layer adsorption for Λ (=10⁻⁹ mol cm⁻² s⁻¹) is close to 100, and Y₁ − Y₂ is simplified to

(Y₁ − Y₂)_CT ≈ σω^1/2/χ = (ω/ω₀)^−λF²Λ/RT

(25)

Subtracting Equations (22) and (23) from the R- and DLC-corrected admittance, Y′, yields

Y₁″ − Y₂″ = (λ − 1)C_NC,1Hz ω^1−λω₀^λ + (ω/ω₀)^−λF²Λ/RT

(26)

Figure 5 shows logarithmic plots of the observed values of Y₁″ − Y₂″ against f for three values of E_dc. The linearity over the entire domain of log f indicates that either of the two terms on the right-hand side of Equation (26) should be valid. Positive values of the slopes (λ = 0.17) in Figure 5 suggest that (Y₁ − Y₂)_NC >> (Y₁ − Y_k2)_CT. Therefore, the CT contribution for FcTMA is completely overtaken by the NC one. In other words, the plots in Figure 5 are represented as

log(Y₁″ − Y₂″) = log[(λ − 1)C_NC,1Hz] + (1 − λ)log f

(27)

In order to confirm the significance of the NC term, we plotted the values of the intercept, log[(λ − 1)C_NC,1Hz], against E_dc in Figure 6. A bell shape was found, which was able to be approximated as log[sech²(ζ_dc/2)]. Since the potential dependence of sech²(ζ_dc/2) has been found in the diffusion-controlled ac currents [31], C_NC,1Hz should be caused by the faradaic current of FcTMA. From the peak value of the curve at E_dc = 0.355 V vs. Ag|AgCl, we evaluated C_NC,1Hz = −45 μF cm⁻².

Of interest is a quantitative explanation of (Y₁ − Y₂)_NC >> (Y₁ − Y_k2)_CT. Inserting Equations (22) and (23) into this inequality, we have

FΛ << (RT/F)ωC_NC,1Hz (λ − 1)(ω₀/ω)^2λ

(28)

Equation (28) indicates that the CT rate becomes much smaller than the NC rate with the dependence of ω^1−2λ as the frequency is increased. Therefore, it is in vain to attempt to increase the frequency conventionally to determine high rates of the CT. Both Λ and C_NC,1Hz are proportional to concentrations of the redox species [31]. As a result, it is difficult for us to distinguish the CT from the NC.

Figure 5. Logarithmic variations of corrected values of Y₁″ − Y₂″ with f at E_dc = (a) 0.30, (b) 0.36, and (c) 0.40 V vs. Ag|AgCl.

Figure 5. Logarithmic variations of corrected values of Y₁″ − Y₂″ with f at E_dc = (a) 0.30, (b) 0.36, and (c) 0.40 V vs. Ag|AgCl.

Figure 6. Variation of the intercept of lines in Figure 5 with E_dc. The dashed curve was calculated by curve fitting from log[sech²(ζ_dc/2)].

Figure 6. Variation of the intercept of lines in Figure 5 with E_dc. The dashed curve was calculated by curve fitting from log[sech²(ζ_dc/2)].

4. Conclusions

The plots of Z₁ and Z₂ of raw impedance data against f^−1/2 lead to erroneous rate constants, although they have been conventionally used for evaluating rates. Even if impedance data are corrected for effects of solution resistance and the DLC, they show unpredicted frequency dependence of f^−λ for the real impedance. The dependence arises from the delay in the conversion from the voltage-based activity to the concentration. Furthermore, the NC decreases the observed current. These two contributions should be considered for the CT kinetics.

The transfer coefficient in the BV is 1/2 if the electrostatic energy of the activation does not include any chemical properties. This conclusion is essential for the role of electrostatic energy, which should be segregated completely from chemical energy. Various values reported in Tafel slopes different from 0.5 suggest rate-determining steps irrelevant to the CT process.

Time-dependent measurements are necessarily complicated with capacitive currents because they include not only the DLC but also the NC. The current by the NC suppresses the observed diffusion current as if the kinetic current were to be included. It is difficult to distinguish the CT from the NC. Since the delay for FcTMA from diffusion control was attributed to the NC, no CT data could be obtained.