Distinction between Effects of IR-Drop and Negative Capacitance of Fast Cyclic Voltammograms

1. Introduction

Linear sweep cyclic voltammetry has been conventionally made at scan rates v ranging 0.01 < v < 0.2 V s^–1. The slowest scan has been used on the assumption of the steady state for hydrodynamic voltammetry [1] and microelectrode voltammetry [2]. It often provides a well-defined voltammogram but takes a long period of a few minutes. An advantage of the slow scan rates is to distinguish faradaic, diffusion-controlled currents from background currents by a double layer capacitance (DLC). For example, the diffusion current at concentration 0.5 mM (M = mol dm^–3) redox species is roughly I_d = 3v^1/2 μA at a 1 mm² planar electrode for v in the unit of V s^–1. In contrast, a DLC of a platinum electrode in aqueous solutions, being C = 30 μF cm^–2, provides the capacitive current I_c = 0.3v μA. Since the ratio is I_c/I_d = 0.1 v^1/2, the diffusion currents at v = 0.2 V s^–1 can be distinguished from the background with 5 % errors. A disadvantage is a time-consumed measurement, during which redox species might be altered through chemical complications. If faradaic reactions occur under an adsorbed state, the advantage is not perceptive because the background current has the same v-dependence as I_c. Slow responses fail to detect kinetic properties.

Study on homogeneous and heterogeneous kinetics requires such a wide range of v that reaction rates can be covered. A range of the kinetics is often much wider than the conventional scan rates (0.01 – 0.2 V s^–1), and hence fast scan voltammetry is desirable for kinetic work principally [3,4,5,6,7,8,9,10] and practically [11,12,13,14] for applications to neurotransmitter detection. Unfortunately, the increase in scan rates is associated with at least five problems to be considered. (A) IR-drop, in which applied voltage is lost by solution resistance, makes voltammograms deformed and shifted in a scanned direction [15,16]. (B) Even if difference in v-dependence of I_d and of I_c enables us to evaluate I_d accurately [17], the adsorption-controlled current, I_a, prohibits the extraction of I_a from the observed current because of the same v-dependence as I_c. (C) A delay of a potentiostat underestimates true currents at short time [7,8,17]. (D) Heterogeneous kinetics represented by the Butler-Volmer equation makes the peak potential shifted and the peak current deviated lower than the predicted diffusion-controlled peak current [18] with an increase in v. (E) Fast scans more than 0.5 V s^–1 generate the negatively capacitive current, which is caused by formation of dipoles of a reaction product with counterions at an electrode [19,20,21,22,23].

A level of these problems depends on magnitude of v. We suggest three classes; v < 0.2 V s^–1, v < 5 V s^–1 and v < 500 V s^–1. Advantages and possible risks for each class are listed in Table 1.

The usage of fast scan rates prevents clear distinction of faradaic reactions from backgrounds. The difficulty in the distinction in class III has been overcome with efforts of trial and error, as exemplified with the rapid detection of neurotransmitters,[11–13]. The developed techniques are specific to given redox species under limited conditions. In contrast, those in class II have been carried out on the theoretical base of the difference in the v-dependence when faradaic reactions proceed in diffusion-control. If problems (A)-(E) are intertwined complicatedly, corrections are not enough sufficient for finding expected behavior of the faradaic waves. For class I, the intertwinement is largely solved except for adsorption in the v-dependence.

We examine here subjects (A)-(E) carefully. Since no stereotyped redox species has been found for the Butler-Volmer kinetics yet [24], (D) may be excluded from the present work on ferrocenyl derivatives [25,26,27]. If we pay attention only to diffusion-controlled currents under category (II), subjects (B) and (C) have been automatically resolved. Both (A) and (E), being necessarily involved in fast scan voltammetry, exhibit similar behavior in spite of different concepts. We examine in this work how voltammetric peak potentials and currents belonging to class (II) are varied with controlling parameters for (A) and (E) by use of fast scan voltammetry of a ferrocenyl derivative. Unfortunately, effects of IR-drop (A) have not been discussed systematically yet. Thus, we present here the experimental results and the theory.

2. Materials and Methods

2.1. Chemicals

All the chemicals were of analytical grade. Solutions were prepared in doubly distilled water prepared by CPW-100 (Advance, Tokyo, Japan). Aqueous solutions with different concentrations of KCl (0.005, 0.05, 0.1, 0.5 and 5 M) were prepared at the aim of varying solution resistance. The electrochemical charge transfer species was 1 mM (ferrocenylmethyl)trimethyl ammonium (FcTMA).

2.2. Voltammetry

Voltammetry was made in a three-electrode cell equipped with a working platinum wire electrode 0.03 mm, 0.5 mm in diameter or a 1.6 mm disk electrode. The tip of the wire was inserted into the solution by a given length (ca. 1 mm) which was measured with a microscope. The wire was not sealed with any insulator to expose an active area in order to avoid floating capacitance at crevices of electrode|insulator. The electrode was pretreated by being socked in the mixed acid (0.26 M acetic + 0.33 M nitric acid + 0.73 M phosphoric acid) for 1 min to clean the surface. A counter electrode was a platinum coil. A reference electrode was a homemade Ag|AgCl (x M KCl). A columnar porous glass was fixed to a tip of taper glass tube with a heat shrink tube, into which a Ag|AgCl wire was inserted together with x M KCl solution. In order to detect any leakage of chloride ions during the experiment, we measured the resistance values of the solution before and after the experimental test, which were almost the same.

A delay of the potentiostat, Compactstat by Ivium (Netherlands), was examined by using a carbon resistance 1 kΩ for a dummy cell by applying the voltage with 1 V. The maximum difference in the currents at the forward and the reverse scan was 3% at 8 V s^–1. Our voltammetry for FcTMA and various concentrations of KCl was made at scan rate less than 7 V s^–1. Ac-impedance was made for the 10 mV amplitude and frequency domain from 1 Hz to 10 kHz. All measurements were made at temperature of 25 ± 1 ℃.

3. Results

Figure 1A shows voltammograms at the 0.5 mm platinum wire electrode inserted in 1 mm length in the solution of 1.0 mM FcTMA + 5 mM KCl for several scan rates. FcTMA or related ferrocenyl derivatives have been recognized as such fast charge transfer reacting species that voltammetric peak potentials, E_p, may be independent of voltammetric scan rates [19,24,28,29,30]. However, the peak potential in Figure 1A shifted in the forward direction at the fast scan rates. The potential shift has often been attributed to the IR-drop. The resistance values were estimated from the inverse slope of the line connecting the peaks (solid lines) [18,28]. The slope for the anodic waves was the same as that for the cathodic one. This equality partially supports the effect of IR-drop on the potential shift. The inverse slope was close to the resistance (2.7 kΩ) obtained through Nyquist plots by ac-impedance. The potential at which the lines are extrapolated to zero current can be regarded as the peak potential without IR-drop. The difference between the extrapolated anodic potential and the cathodic one was 60 mV, indicating the diffusion- controlled behavior [18]. The voltammograms in Figure 1B at the disk electrode 1.6 mm in diameter were drawn out to exhibit the peak-connecting line with the inverse slope of 5.5 kΩ, which was larger than the resistance value of 4.2 kΩ by ac-impedance. Except for this point, the behavior was almost the same as in (A).

There may be a reason for the larger values of the slopes of voltammetric peaks than the R_s-values by ac-impedance. Figure 2 shows logarithmic plots of two kinds of the resistance against concentrations of KCl. R_s-values (a, b) by ac-impedance were linear with the logarithmic form of [KCl] with the slope –0.91, independent of the electrode geometry. The approximately inverse proportion of R_s to [KCl] agrees with the concept of ionic mobility by electric field [31]. The deviation of the slope (–0.91) from the ideal one (–1.0) can be attributed to the frequency dispersion of the double layer capacitance, of which values increase slightly with salt concentrations [23,32,33]. It can be understood in terms of an upward shift of Nyquist plots with an increase in salt concentrations [33]. It is noteworthy to pay attention to the invalidity of simple RC-relaxation of the double layer capacitances [8,9]. In contrast, the resistances evaluated from voltammetric peaks (c, d) were kept almost constant for the low salt concentrations. They are provided through the oxidation current of FcTMA rather than the capacitive currents of the electric double layer. The potential-shifts only caused by faradaic reactions can be attributed to effects of (i) electric migration [34,35,36], (ii) the heterogeneous kinetics for the charge transfer reaction [18], (iii) following chemical reactions [37,38], and (iv) the negative capacitance associated with the charge transfer reaction [19,20,21,22,23]. We attempt to estimate these effects briefly here.

(i) The flux of a uni-valent cationic redox species controlled both with diffusion and electric migration is expressed by the Nernst-Planck equation, j/F = –Ddc/dx – (DF/RT)c(dφ/dx) [39], where c is its concentration, D is the diffusion coefficient, and φ is the potential in solution at a distance from the electrode x. It can be approximated to be j/F = –Dc*/δ – (DF/RT)c*(Δφ/δ), on the basis of a concept of diffusion layer [40]. Here, its thickness at the peak current, I_p, is given by δ = Dc*FA/I_p, where A is the electrode area. The ratio of the current component of the migration to that of diffusion is (F/RT)Δφ. When Δφ is replaced by R_sI_p, the ratio becomes less than 0.5 % under our experimental conditions. Therefore, migration has no contribution to the potential shift in our case.

(ii) The electrode kinetics has often caused potential shifts. A quantitatively accessible kinetics is represented by the Butler-Volmer equation. The theoretical peak currents and potentials for different scan rates can be obtained from the analytical equation at a given transfer coefficient α and a heterogeneous rate constant k⁰ [18]. Figure 3c–e shows the variation of I_p with E_p for several scan rates at the α = 0.5 and some values of k⁰ by use of our software of the kinetics so that I_p vs. E_p were close to the experimental ones (a). However, the experimental plots (a) is different from theoretical variations for any different values of k⁰. The inconsistency was also applied to other values of R_s (b). Since no potential shift was found in 0.1 M KCl solution (b), it is not reasonable to explain the shift in terms of the heterogeneous kinetics. However, the explanation only by the kinetics has often been reported [3,[3–,4,5,6,7,8,9,10].

(iii) A following chemical reaction causes a potential shift, as can be understood from the Nernst equation for reaction rates faster than voltammetric rates. Ferrocenyl compounds, however, are not satisfied with the condition of the rates, and hence, item (iii) is unsuitable of explaining the present potential shifts.

(iv) The negative capacitance is brought about through the following steps in Figure 4: a charge transferred redox species (Fc) is coupled with a counterion (Cl^–) for electric neutrality by responding to the externally applied field E_ap to yield an electric dipole(Fc⁺-Cl^–); the dipole with the dipole moment p is oriented in the direction for enhancing the external field by -pc/ε₀ to yield the effective field E_ef, which generates a capacitance with a sign opposite to a double-layer capacitance [17,41]. The negative capacitance has been obtained for ferrocenyl derivative [19,20], ruthenium complex, iridium complex [20] and hemin [41] by ac-impedance. Since the voltammetric current from the negative capacitance is proportional to the scan rate, it depresses the diffusion tail to cause the potential shift [17]. The negative capacitance varies with electrode areas and scan rates, as the IR-drop does. The similarity in the properties stimulates us to distinguish their properties theoretically.

4. Theory of effects of IR-drop

The electrode reaction here is a simple Nernstian charge-transfer reaction with the mass transfer controlled by x-directional diffusion. The diffusion coefficient of the reduced species, D, is assumed to have a common value to the oxidized species. Letting the solution resistance between a working electrode and a reference one be R_s, the scan rate be v and the initial potential be E_in, we can express the linearly scanned potential participating efficiently in the oxidation for the current I as E(t) = E_in + vt – I(t)R_s at the time t. The common value of D implies that the sum of concentrations of both species is equal to the bulk concentration, c^*, in any time and any location. Thus, the Nernst equation for the reduced species tends to

c_x=0 = c^*/{1 + exp((E_in + vt – I(t)R_s)F/RT)}

(1)

On the other hand, the surface concentration is given by a solution of the diffusion equation [18] as a function of the current density j

c_x=0 = c^* – F^–1(πt) ^–1/2∫₀^t j(t – u)u^–1/2du

Inserting this equation into Equation (1) and extracting j by the technique previously described [42] yields

j/c^*FD^1/2 = d∫₀^t (πu) ^–1/2{1 + exp[– (E(t–u)F/RT]} ^–1du / dt

(2)

Application of Leibniz's theorem [43] makes the combination of the integral and the differentiation be reduced to

I/Ac^*FD^1/2 = (1/4)∫₀^t {π(t – u)} ^–1/2{sech²(E(u)F/2RT)}{v – R_sdI(u)/du)}(F/RT) du

where A is the area of the electrode.

We replace the time by the potential through y = (E_in + vu)F/RT, and define the dimensionless current as

f(t) = I(t)/I_p0

(3)

where I_p0 is the peak current without solution resistance, given by I_p0 =0.446Ac^*F(FDv /RT)^1/2. The current has the dimensionless parameter for the resistance

ρ = I_p0R_sF/RT

(4)

and can be rewritten as

f(t) = π^–1/2∫_ζin^ζap (ζ_ap – y)^–1/2g(y – 2.24ρf(y)){1 – ρdf(y)/dy} dy

(5)

where g(x) = (1/4)sech²(x/2), and ζ_in and ζ_ap are the dimensionless initial voltage E_inF/RT and the dimensionless applied voltage (E_in+vt)F/RT, respectively. Function f was evaluated by numerical computation by discretization of ζ_ap –ζ_in = nh for a small value of h, as was described in the supporting information.

Figure 5 shows dimensionless voltammograms for several values of resistance parameter ρ (Equation (4)). With an increase in the values of ρ, voltammograms shift in the positive direction accompanied with a decrease in the peak currents. An exemplified value of ρ = 4 corresponds to I_p0 = 0.1 mA, R_s = 1 kΩ.

Peaks were calculated for various values of ρ. The E_p increased from 0.029 V convexly with ρ, as shown in Figure 6a. The intuitively predicted increase in E_p is I_p0R_s or (26 mV)ρ,

(E_p)_IR,prdct = I_p0R_s + 0.026 V

(6)

as shown in the line of Figure 6b. The tangent line at ρ = 0 has the slope, 33 mV (Figure 6c). The slight shift from 26 mV is ascribed to the smaller decrease in the current for E > E_p than that for E < E_p (see Figure 5) owing to the diffusion tail. Consequently, the E_p-values calculated from Equation (5) for ρ < 3 are larger than the line (Equation (6)). Since the variable ρ is proportional to v^1/2 through I_p0, the effect of IR-drop on the voltammograms is approximately exhibited in the form of the linearity of E_p to v^1/2 rather than log v.

The peak current normalized with the theoretically diffusion-controlled one decreased with an increase in the resistance, as shown in Figure 7. The decrease can be explained in terms of the following reasons: the shift of the peak potential delays the arrival at the peak; the delay is equivalent to a substantial decrease in the applied scan rate, v_ap, which leads a decrease in I_p so that I_p < I_p0 = k₁v_ap^1/2, where k₁ = 0.446Ac^*F(FD/RT)^1/2. By letting the effective scan rate, v_ef, be I_p = kv_ef^1/2, v_ef decreases with ρ, as shown on the right ordinate in Figure 7b. It varies linearly with ρ^1/2. The effective rate becomes less than a half for ρ > 10 or I_p0R_s > 0.25 V. The difference, 0.446– (I_p/Ac^*F)(RT/FDv_ap)^1/2, was approximately proportional to ρ^1/2. As a result, the peak current at any value of R_s can be approximated as

I_p/I_p0 = 1 – 0.09ρ^1/2

(7)

This is a new insight for the IR-drop effect that has not been reported yet [8,9]. Since ρ is proportional to v^1/2 through I_p0 (Equation (4)), the peak current for the IR-drop can be approximated as I_p = kv^1/2 – k₁v^3/4.

It is interesting to compare the effects of IR-drop on I_p and E_p with those of the negative capacitance (NC), C_rx. The E_p due to C_rx has varied linearly with log[(σ*/c*)²(v/FRTD)] [17] for E_p – E^o > 0.06 V, where σ* is the sum of the surface charge density of the oxidized and the reduced species, and c* is the bulk concentration of FcTMA. In contrast, its variation for E_p – E^o < 0.1 V is linear with (σ*/c*)(v/FRTD)^1/2, as shown in Figure 6 on the upper abscissa, which was scaled so that the line may be overlapped with the line for the IR-drop. The line can be approximated as 0.0217(σ*/c*)(v/FRTD)^1/2 + 0.03 in the unit of V for E_p – E^o > 0.06 V. The empirical equality, C_rx = 0.16(F/RT)σ*, in Equation (13) of ref. 17 can substitute C_rx for σ* to give

(E_p)_{NC, <0.10V} / V = 0.135(C_rx/Fc*)(RTv/FD)^1/2 + 0.03

(8)

When potential shift is small, the observed shift is the sum of the shifts for the IR-drop and the negative capacitance,

(E_p)_<0.10V / V = 0.135(C_rx/Fc*)(RTv/FD)^1/2 + 1.15I_p0R_s + 0.03

(9)

R_s varies with geometry of electrodes at given conductivity of solution. A disk electrode makes R_s be 1/4Λ[KCl]a = (π/A)^1/2/4Λ[KCl] [44] for the molar conductivity Λ of KCl. R_s at a wire electrode, on the other hand, is inversely proportional to A. Some variables that control E_p are summarized in Table 2 for comparing these two effects.

The IR-drop contribution normalized with I_p0 is

(I_p0 – I_p)/I_p0 ≡ X_IR = 0.09(I_p0R_sF/RT)^1/2

(10)

On the other hand, the negatively capacitive peak current is given by I_NC = I_p0 – AC_rxv. The normalized contribution is given by

X_NC = AC_rxv/ I_p0

(11)

Eliminating I_p0 from Equations (10) and (11) yields

X_IR = 0.09(AR_sC_rxvF/RTX_NC)^1/2

(12A)

Especially, the relation for the disk is given by

(X_IR)_disk = 0.06A^1/4(C_rxvF/RTΛ[KCl]X_NC)^1/2

(12B)

Figure 8 shows variations of (X_IR)_disk with X_NC for some electrode areas at conventionally experimental conditions of v, κ (=Λ[KCl]) and C_rx. The constant of X_IR²X_NC indicates that the effect of the negative capacitance is predominant to that of the IR-drop. Furthermore, high scan rates at large electrodes in low salt concentration should enhance both the IR-drop and the negative capacitance.

5. Discussion

Experimental values of E_p in the solution of 1 mM FcTMA + 5 mM KCl were plotted against ρ (= I_pR_sF/RT) in Figure 6e, where the value of R_s for ρ was obtained from the Nyquist plot. They were on a line, deviated upward from the theoretical curve Figure 6a of the IR-drop, and hence the potential shift cannot be ascribed only to the IR-drop. The plot of the experimental E_p vs. log v is shown in the inset of Figure 6, as is often plotted. The non-linearity indicates that the conventionally observed linearity should not explain any IR-drop effect. Addition of voltages does not agree with that of processes because of non-linear relation to voltages in electrochemical processes. The experimentally obtained ratio, I_p/I_p0, decreased with an increase in v, as shown in Figure 7c,d at two values of R_s. The variation of the triangular marks is close to that of the circles if ρ-axis is expanded. The experimental values (c,d) lower than the theory (a) were caused by the effect of the negative capacitance.

It is current that can make arithmetic operations possible through an addition of current at a parallel process or of inverse currents at a series one. In order to confirm the dependence of the IR-drop contribution on the scan rates, we plotted I_p0–I_p in solution of 1 mM FcTMA + 5 mM KCl against v logarithmically in Figure 9a. The plot shows a line with the slope, 0.71, which is close to the theoretical value, 3/4, in Table 2. The value of the slope suggests the control of the IR-drop. On the other hand, the currents at the 0.03 mm wire in 0.1 M KCl (in Figure 2d) did not contain any IR-drop effect, as demonstrated in Figure 3b. The logarithmic plots of I_p0–I_p vs. v shows a line with slope of 0.92 in Figure 9b. The value of the slope is close to unity, and hence it should belong to the negative capacitance.