Development of Galvanostatic Fourier Transform Electrochemical

Figure 3 shows: (a) ΔE(t) vs t1/2 when ΔI = 5.0 μA/cm2 is applied at the equilibrium ... The potential curve is linear after 0.15 s1/2 and shows th...
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Development of Galvanostatic Fourier Transform Electrochemical Impedance Spectroscopy Kwang-Mo Nam,† Dong-Hyup Shin,‡ Namchul Jung,§ Moon G. Joo,‡ Sangmin Jeon,§ Su-Moon Park,*,⊥ and Byoung-Yong Chang*,† †

Department of Chemistry and ‡Department of Information and Communications Engineering, Pukyong National University, 45 Yongso-ro, Nam-gu, Busan 608-739, Korea § Department of Chemical Engineering, Pohang University of Science and Technology, San 31 Namgu Hyojadong, Pohang, Korea ⊥ Interdisciplinary School of Green Energy, Ulsan National Institute of Science and Engineering, Ulsan 689-805, Korea S Supporting Information *

ABSTRACT: Here, we report development of the galvanostatic Fourier transform electrochemical impedance spectroscopy (FTEIS), which monitors impedance of electrochemical reactions activated by current steps. We first derive relevant relations for potential change upon application of a step current, obtain impedances theoretically from the relations by simulation, and verify them with experimental results. The validity of the galvanostatic FTEIS technique is demonstrated by measuring impedances of a semiconductive silicon wafer using the conventional frequency response analysis (FRA), the potentiostatic FTEIS, and the galvanostatic FTEIS methods, and the results are in excellent agreement with each other. This work is significant in that the galvanostatic FTEIS would allow one to record impedance changes during charge/discharge cycles of secondary batteries and fuel cells as well as electrochemically irreversible systems which may produce noise level chronoamperometric currents by potentiostatic techniques.

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have to resort to other supplementary experiments.7 Another powerful electrochemical technique capable of resolving the multiprocess mechanism is electrochemical impedance spectroscopy (EIS).8 EIS can analyze complex electrochemistry by resolving AC signals into its equivalent-circuit elements such as resistances of charge and mass transfers, capacitors of the electric double layer and adsorption, etc., rather than a single mixed DC current. Recently, EIS experiments are routinely made in the potentiostatic mode8b,9 while the galvanostatic modes are available.10 This is because, when a constant potential is applied for a long period, the current decays down to a steady value,11 which provides a stable condition for AC potential applications. However, when a current is applied for a long time, the steady potential is reached but is interrupted by a potential jump at the transition time.5b,6,12 The sudden change of potential activates additional reactions which are not of interest. The transition time also limits the number of applied AC frequencies. As each frequency application requires a finite time, time-drifts between different frequency applications should be considered. Thus, the frequency response analysis (FRA) method is not thought to offer the best conditions for the controlled current EIS experiments.

lectrochemical methods are frequently used in analytical chemistry because chemical changes are directly converted to electric signals without auxiliary transducers, reducing complexities of device and analysis processes, and they offer convenient and miniature platforms for use and development of, for example, point-of-care testing. Owing to such advantages, many electroanalytical methods have been developed in areas of chemical, environmental, energy, and medical research.1 Of many electrochemical techniques, the most fundamental and basic one is the controlled potential/current techniques where the controlled potential or current is applied to an electrochemical system and reactions are perturbed to give current or potential signals due to both the faradaic and nonfaradaic processes.2 Delahay derived an equation describing faradaic chronoamperometrics,2 while Sand reported one for chronopotentiometric curves,3 both based on rates of diffusion and electron transfer. Later, Delahay’s equation was completed by Chang and Park by taking into account the nonfaradaic current produced by the electric double layer.4 Another popular technique is voltammetry showing the electrochemical current behavior resulting from a linear scan of potential or vice versa,5 which allows dynamic information to be obtained from a simple experiment. These experiments, however, have limitations as simple E−I curves do not have enough resolution to deconvolute complex electrochemical processes such as the case when charge transfer, mass transfer, adsorption, and electric double layer formation are mixed.6 For those cases, we © 2013 American Chemical Society

Received: October 24, 2012 Accepted: January 20, 2013 Published: January 20, 2013 2246

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transformed to give out ΔI(ω) and ΔE(ω) in the frequency domain. Impedances, Z(ω), were then calculated by ΔV(ω)/ ΔI(ω). More details are described elsewhere.16 Technical Notes for FTEIS Instrumentation. FTEIS is conducted by applying a fast-rising step function as an activation signal to an electrochemical system. The reasons for applying the step signal are that: (1) time-dependent and transient responses as results of the electrochemical reaction provide rich information,4,17 (2) derivatives of the current and potential signal are regarded as sums of AC wave signals of different frequencies,18 and (3) successive step signals making a staircase waveform can allow one to record changing impedances in real-time while potential or current is scanned.16a,19 Thus, the measurement device should handle the signals with high speed and precision. To meet the requirement, a homemade galvanostat was constructed on the basis of the circuit diagrams in ref 6, Chapter 15. The operational amplifiers (Op. Amps) used here were OPA 627A, of which the slew rate is 55 mV/ns and the voltage noise is 4.5 nV/Hz1/2.20 Also, special care was taken in removing fluctuations and oscillations due to the high speed operation of the Op. Amps. Two parallel capacitors, 33 and 1 μF, were connected between the power lines and Op. Amps, and appropriate voltage dividers of high impedance were used at the junction between the homemade galvanostat and the data acquisition system installed in a PC.

There are two important reasons for developing a galvanostatic Fourier transform electrochemical impedance spectroscopy (FTEIS) technique. First, controlled currents are employed for charging/discharging processes of energy storage devices such as secondary batteries, fuel cells, and supercapacitors. However, due to the limitation of the FRA method mentioned above, it was impossible to monitor their impedances during charging/discharging processes and investigators and manufacturers always had to monitor the impedances ex situ; there have been and are dire needs to develop impedance monitoring instruments. Second, while the potentiostatic FTEIS method has been used to obtain dynamic impedance information during cyclic voltammetric experiments by employing small potential steps of 5−20 mV,5,8,12 this method provides reliable results only for electrochemically reversible and/or quasi-reversible systems. For irreversible cases, the currents produced by the potential steps are often too small to compete with nonfaradaic currents. Since a large overpotential is measured for an electrochemically irreversible system upon current steps, the galvanostatic FTEIS would provide much more reliable results. Irreversible electrochemical reactions include slow charge transfer reactions such as oxygen reduction reactions (ORRs), corrosion reactions,13 redox probes used in biosensing,14 and reactions taking place at electrocatalyst surfaces.15 Here, we report a way to directly convert a single chronopotentiometric curve to impedance spectrum by taking advantage of Fourier transform, which drastically reduces the experiment time and, thereof, prevents long application of controlled currents.

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RESULTS AND DISCUSSION Theoretical Considerations on Converting a Potential Curve upon a Current Step to Impedance Spectra. EIS displays electrochemical processes as a function of AC wave frequencies so that this technique basically uses the FRA to measure the AC signals. Recently, Park’s group developed FTEIS without using the FRA,16b,21 where the transient electrochemical signals obtained upon DC potential steps were directly transformed to AC current signals and impedance data were obtained employing Ohm’s law. It significantly reduced the experiment time enough to obtain a series of potentiodynamic impedance spectra while a conventional cyclic voltammogram was acquired. Here, the advantages for time reduction are applied to the galvanostatic FTEIS in order to make impedance measurements possible during currentcontrolled experiments. For a given electrochemical reaction, described by eq 1,

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EXPERIMENTAL SECTION Potassium ferricyanide (K3Fe(CN)6, 99%+) was purchased from Aldrich (Milwaukee,WI), and potassium chloride (KCl, 99.0%) was from Samchun Chemicals (Seoul, South Korea). The compounds were dissolved in doubly distilled, deionized water with a resistance of 18 MΩ. A gold disk electrode (area 0.020 cm2), platinum gauze, and homemade Ag|AgCl (in saturated KCl) electrodes were used as a working, counter, and reference electrodes, respectively. The working electrode was polished to a mirror finish with alumina powders of 14 μm sequentially down to 0.03 μm, followed by sonication in distilled water prior to each experiment. Silicon wafer provided by Siltron Inc. (Gumi, South Korea) was cut into pieces of 2.0 cm2 and was cleaned with ethanol and deionized water before experiments. Two copper wires were attached on both sides of the silicon wafer with silver paste, which provided connections to impedance measurement devices. A galvanostat capable of handling high speed signals with high precision was home-made and connected to a function generator, a signal digitizer, and an electrochemical cell.6 An NI PCI-5412 Arbitrary Waveform Generator (National Instruments, Austin, TX) was used as a high speed potential step generator, which was fed into the homemade galvanostat. The galvanostat converted potentials to currents and delivered them to the electrochemical cell. Here, the magnitudes of the converted step currents were adjusted to make the potentials change less than 10 mV in order to satisfy the linearity conditions between the current and potential. The current and responding potential data were acquired at 50 k samples/s using an NI PCI-5922 Digitizer installed in a PC. Impedances were calculated using a Labview program based on the algorithm published before. Briefly, currents and potentials recorded in the time domain were differentiated and Fourier

kf

O + e− ⇄ R kb

(1)

the faradaic process obtained upon a current step, ΔI, changes the potential as a function of time, ΔE(t). A general description of the faradaic process is depicted by electron transfer at the electrode/electrolyte interface and mass transfer in the diffusion layer as shown by the equivalent circuit in Figure 1, where impedances for electron transfer and mass transfer by diffusion

Figure 1. The equivalent circuit showing only the faradaic processes at the electrified interface and mass transfer within the diffusion layer, of which the rates are represented in terms of polarization resistance (Rp) and Warburg impedance (ZW). 2247

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circuit is presented by a simple parallel connection of Cd and Rp. The impedance of this case is expressed by 1/Z(s) = sCd + 1/Rp, and its inverse Laplace transform results in ΔE(t),

are represented by Rp and ZW, respectively. On the basis of the equivalent circuit, we can evaluate the faradaic impedance (Zfar) and the change of potential in the frequency domain by the following relations: Zfar(s) = R p + Z W

σ 2 = Rp + s

ΔE(s) = I(s) ·Zfar(s) =

ΔE(t ) = ΔI ·RP 2·Cd·(1 − e−t / CdR p)

Equations 6 and 8 describe the potential change of the fundamental faradaic and nonfaradaic processes, and impedance plots obtained from Fourier transform of each equation would be one with fundamental impedance elements, a straight line with 45°, and a semicircle on the Nyquist plane, respectively. Most electrochemical mechanisms, even complex, can be described as equivalent circuits formulated by combinations of those fundamental components. For example, if the electrode is polarized by the applied current and electron transfer and mass transfer are rendered with charging the electric double layer on the electrode surface, the total current is i = ifar + ic. Here, ifar is related to eq 6, whereas ic = −Cd(dE/ dt) is related to eq 8.6 The exact solution for ΔE(t) upon ifar + ic is almost impossible, but a numerical solution is obtained by computer simulation.23 Figure 2a shows a chronopotentio-

(2)

σ 2⎞ ΔI ⎛ ⎜R p + ⎟ s ⎝ s ⎠

(3)

and RT ⎛ 1 ⎜ + 2 ⎜ D · 2 F A ⎝ O CO, x = 0

σ=

⎞ 1 ⎟⎟ DR ·C R, x = 0 ⎠

(4)

where s = jω, DO, and DR are diffusion coefficients, CO,x=0 and CR,x=0 are surface concentrations of O and R, respectively, and R, T, F, and A are the gas constant, temperature, the faraday constant, and the area of electrode. Taking inverse Laplace transform of ΔE(s), the potential profile in the time domain is obtained. ΔE(t ) = L−1[ΔE(s)] = ΔI ·R p + ΔI ·R p +

=ΔI ·

∫0

t

(8)

σ 2 du = πu

ΔI ·σ 2 2 t π

(5)

⎡ RT RT ⎢ 2 t ⎛ 1 + ΔI · ⎜⎜ F ·ibias F ⎢⎣ FA π ⎝ DO ·CO, x = 0

+

⎞⎤ 1 ⎟⎟⎥ DR ·C R, x = 0 ⎠⎥⎦

(6)

This equation has the same form of the well-known current step equation (eq 7),22 η = i·

+

⎡ RT ⎢ 2 t ⎛ RT 1 + i· ⎜⎜ F ·i0 F ⎢⎣ FA π ⎝ DO ·CO, x = 0 ⎞⎤ 1 ⎟⎟⎥ DR ·C R, x = 0 ⎠⎥⎦

Figure 2. (a) A chronopotentiogram theoretically obtained by simulation using eqs 6 and 8 and plotted in the form of ΔE(t) vs t1/2. (b) An impedance spectrum calculated from the curve in (a) using the Fourier transform and Ohm’s law. The values of the parameters used in the simulation were ΔI = 1 μA, Rp = 1000 Ω, Rs = 200 Ω, Cd = * = Cred * = 1 mM, and Dox = Dred = 1 × 10−5 20 μF/cm2, A = 1 cm2, Cox cm2/s.

(7)

which is derived from the Sand’s equation and the concentration profiles changed along electrochemical reactions.2 Comparison of eqs 6 and 7 leads to the conclusion that Fourier transform of the potential recorded in the time domain upon a stepped DC current step creates impedance data in the frequency domain without using AC waves, i.e., ΔE(t) → Z(ω). Besides, eq 6 ensures extended applicability of eq 7 for the applied current from the exchange current (i0) to any bias current (ibias), which reveals the information of the polarization resistance at any electrochemical current.16a Not only faradaic processes but also nonfaradaic processes are generated by the controlled current step in the electrochemical system. As those processes take place independently at the electrified interface, an equivalent circuit can be built up by a parallel connection of a double layer capacitance (Cd) and the faradaic impedance as shown by the well-known Randles equivalent circuit.6,8a For simplicity, we presume a situation in which the rate of charge transfer is much slower than mass transfer, where Rp is dominant over ZW, and its equivalent

metric curve, ΔE(t), simulated with ΔI = 1 μA, Rp = 1000 Ω, Rs = 200 Ω, Cd = 20 μF/cm2, A = 1 cm2, C*ox = C*red = 1 mM, and Dox = Dred = 1 × 10−5 cm2/s, and Figure 2b presents the impedance data calculated from the potential curve through Fourier transform of the dE(t)/dt, which has the same shape of the well-known Randles circuit impedance. This simulation work concludes that impedance data can be obtained from a single potential curve changing with the time upon application of a current step without the requirement of stationary conditions which is necessary for AC signal measurements. Verification of the Current-Controlled EIS Measurement. An impedance spectrum can be divided into two regions according to the relative rates of charge and mass transfers: 2248

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the impedance spectrum obtained from ΔE(t). The Nyquist plot shows a typical EIS shape composed of a semicircle and a Warburg line which is similar to that of the simulation result shown in Figure 2b. The result is fitted to the Randles circuit, which gives Rs = 220 Ω, Rp = 8480 Ω, Cd = 34 μF/cm2, and ZW = 10.9 kΩ/Hz1/2. The potential curve is linear after 0.15 s1/2 and shows the same relationship of E − t1/2 of the Sand’s equation, which means that the potential change is controlled by diffusion. Since the signals at later time zone contribute to the lower frequency data in Fourier transform, the linear E − t1/2 line coincides with the Warburg impedance in the mass transfer-control region. On the other hand, the earlier signals contribute to impedances in the kinetic-controlled region. This experimental result ensures that the electrochemical impedance of charge transfer, mass transfer, and electric double layer can be obtained from a single chronopotentiometric curve perturbed by DC current without using multiple AC signals overlaid on a bias current. Figure 4 shows a chronopotentiometric curve when the current step (ΔI = 5.0 μA/cm2) is applied at 280 mV vs Ag|

kinetic and mass transfer controls. As the charge transfer takes place at the electrified interface, the parallel processes of Rp and Cd makes a semicircle in the kinetic-control region on a Nyquist plot. Normally, the mass transfer is prompted by concentration gradients generated as a result of electrode reaction by electron transfer so that the mass transfer-controlled region shows up at relatively lower frequencies than the kinetic-controlled region. The general shape of the impedance spectrum on Nyquist plot is thus a combination of the two regions as shown in Figure 2b. If the charge transfer rate is much slower than the mass transfer rate, impedance falls into the kinetic-controlled region. In that case, Rp is dominant over ZW, and only a semicircle is seen on the Nyquist plot (refer to eq 8). On the other hand, when the charge transfer is much faster than the other, it falls into the mass transfer-control region. Then, the value of Rp is small enough to be negligible when compared to that of ZW, and only a straight line with 45° can be observed on Nyquist plot (refer to eq 6).4,24 Those three cases, i.e., in which Rp and ZW are comparable, Rp ≫ ZW, and Rp ≪ ZW, are examined using our galvanostatic FTEIS in order to show the validity of our concept. The electrochemical reduction of Fe(CN)63− to Fe(CN)64− on Au is one of the most frequently used reactions as a reference redox pair in electrochemical impedance measurements. Figure 3 shows: (a) ΔE(t) vs t1/2 when ΔI = 5.0 μA/ cm2 is applied at the equilibrium (330 mV vs Ag|AgCl) and (b)

Figure 4. (a) A chronopotentiogram for reduction of Fe(CN)63− to Fe(CN)64− on Au by applying ΔI = 5.0 μA/cm2 at 280 mV vs Ag| AgCl, which corresponds to a higher overpotential for reduction by 60 mV and plotted vs t1/2. (b) An impedance spectrum calculated from the potentiometric curve in (a); fitting the data to the Randles circuit is shown in red with Rp = 235 Ω and ZW = 41.7 kΩ/Hz1/2. The linear relationship between ΔE(t) and t1/2 shows that the reduction is under mass transfer control from a very early t1/2 value, which agrees with the relatively large ZW than Rp in the Nyquist plot.

Figure 3. (a) A chronopotentiogram obtained during reduction of Fe(CN)63− to Fe(CN)64− on an Au electrode by applying ΔI = 5.0 μA/cm2 at 330 mV vs Ag|AgCl and plotted vs t1/2. The red line shows a linear relationship defined by the Sand’s equation. (b) Impedance spectrum obtained from the potentiometric curve in (a); fitting to the Randles equivalent circuit is shown in red with Rs = 220 Ω, Rp = 8480 Ω, Cd = 34 μF/cm2, and ZW = 10.9 kΩ/Hz1/2. 2249

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layer because the value of Rp is much larger than that of ZW. Fitting the data to the Randles equivalent circuit results in Rp = 42 kΩ, but the value of ZW is not determinable. Another observation is that the current applied until the activating potential is reached is consumed to drive nonfaradaic processes because little faradaic reaction is available. Thus, the electric double layer is sufficiently charged before the faradaic reaction, and the observed capacitance for the ORR by the current source is merely around 1 μF/cm2 due to small amounts of charges able to be accumulated at the electrified interface; Cd = ∂Qsurface/∂φ0, where Qsurface is the surface charge and φ0 is the surface potential on the electrode. To confirm it, a current step is reversed just before the ORR starts to discharge the electric double layer. During the discharging process, the charges are released, and the equivalent circuit is modeled as a parallel connection of a capacitor and a high resistor, of which ΔE(t) is expressed by eq 8. ΔE(t) and its impedance spectrum are shown in Figure 6. The observed ΔE(t) agrees well with eq 8, and its impedance is also well fitted to a parallel RC circuit reporting Cd = 1 μF/cm2.

AgCl and the impedance plot calculated from the curve in (a). The bias potential corresponds to a larger cathodic overpotential than the equilibrium potential by 60 mV, which is high enough for activating the facile faradaic reaction of Fe(CN)63−/4− on Au. Therefore, the charge transfer process is very fast with the mass transfer being the rate-determining process over the whole electrochemical reaction. On the impedance plot, only a Warburg line of ZW is found while Rp is negligibly small in the kinetic-control region. In order to quantitatively clarify it, the data is fitted to the Randles circuit, and it results in Rp = 235 Ω and ZW = 41.7 kΩ/Hz1/2, respectively. The other case, Rp ≫ ZW, was studied with the oxygen reduction reaction (ORR), which is one of the best known reactions of slow charge transfer kinetics and thus classified as an irreversible reaction. Due to its slow kinetics, it requires a high overpotential for activation so that a staircase of current steps (ΔI = 0.5 μA/cm2) was applied until the necessary overpotential is reached (1.29 V vs Ag|AgCl). Figure 5 shows ΔE(t) produced by ORR upon ΔI = 0.5 μA/cm2 and its impedance spectrum calculated form ΔE(t). Here, we only can see a semicircle of the electron transfer and the electric double

Figure 6. (a) A chronopotentiogram for discharging of the electric double layer charged just before the ORR activation is plotted vs t1/2 and fitted to eq 8 of a parallel RC circuit. (b) Impedance spectrum calculated from the potentiometric curve in (a) and fitted to a parallel RC circuit, which reports 1 μF/cm2 as the capacitance of the electric double layer for the 0.5 μA/cm2 discharging current.

Figure 5. (a) A chronopotentiogram of the oxygen reduction reaction on Au is obtained in aqueous solution by applying ΔI = 0.5 μA/cm2 at 1.29 V vs Ag|AgCl and plotted vs t1/2. (b) Impedance spectrum calculated from the potentiometric curve in (a), and the curve obtained by fitting the data to the Randles circuit is shown in red with Rp = 42 kΩ, and ZW is undeterminable because electron transfer at the electrified interface is very slow and the effect of mass transfer is hardly seen. (Here and in Figure 6, the oscillation signal comes from the 60 Hz environmental noise. As this noise is periodic, an outstanding plot appears in the impedance spectrum and this known noise is intentionally removed for clear display.)

Comparison of the Galvanostatic FTEIS Technique to Other Methods. To compare our EIS technique to other ones, we measure the impedances of a silicon wafer in place of a normal liquid system by three different methods: the conventional FRA using AC potential activation, the potentiostatic FTEIS using DC potential step (PS-FTEIS), and the galvanostatic FTEIS using DC current step (CS-FTEIS). For 2250

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CONCLUSION Electrochemical measurements have been made by either potentiostatic or galvanostatic methods. Electrochemical impedance spectroscopy is one of the most powerful techniques to sensitively monitor even minute electrochemical phenomena on the electrified interface but has been routinely used in potentiostatic modes because of the intrinsic restriction of the galvanostatic mode. Even though the galvanostatic impedance measurement is expected to provide a number of advantages, especially for energy storage and materials research, it has been hardly achieved due to the transition time, where chronopotentiometric curve abruptly increases and the long bias current cannot be applied. In this report, we overcome such obstacles by using Fourier transform impedance spectroscopy (FTEIS), which dramatically reduces the measurement time and avoids the time restriction. Theory for our new impedance technique is developed on the basis of the kinetic and mass transfer controlled processes under galvanostatic conditions, and its simulation results are verified by experiments. Even though the measured impedance spectra of high frequencies are somewhat noisy compared to theoretically expected ones, the concept and applications of the galvanostatic FTEIS are successfully proved. Additionally, this method is compared with the conventional FRA and the potentiostatic FTEIS methods, which give essentially the same results. The best benefit of our new method is the real-time impedance measurement. As the times of current application and impedance measurement are synchronized, we can monitor impedance changes of transient electrochemical reactions in real-time. Thus, rather than replacement of previous impedance methods, the purpose of our new method is aimed at monitoring electrochemical phenomena generated in the current-controlled experiments, such as charging/discharging of batteries25 and fuel cells27 and the electric double layer26 as shown in the context, corrosion,13 and biosensing14 and the cases where rates of faradaic reactions are very sluggish. The potentiostatic FTEIS can record real-time impedance change; our new CS-FTEIS method will be able to record real-time impedance changes while current is being scanned, which is expected to provide rich and dynamic information. The results of these experiments will be reported in due course.

the FRA measurements, EG&G’s 273 and Solartron SI 1260 are used, by applying sine waves from 50 kHz to 1 Hz with an amplitude of 5 mV rms. For the PS- and CS-FTEIS, a homemade fast rise potentiostat and galvanostat are used, and the data are acquired at 50 k samples/s for 1 s with a 5 mV potential step and 5 nA current step at 0.0 V and 0.0 A, respectively. Here, the reason for using a solid system is to match the measurement potentials of the potentiostatic and galvanostatic experiments. Generally, an electrochemical system has a potential changing with time under a constant current or vice versa, so that the exact matching potentials between the potentiostatic and galvanostatic experiments are hard to be determined. On the other hand, a silicon wafer has 0 mA at 0 mV, the impedance measured at E = 0 mV is the same as that at i = 0 mA. This silicon wafer, one of the semiconductive materials, has holes and electrons transferring through and charges accumulated at its interfaces, which resembles an equivalent circuit of electrochemistry. The results of the three different methods are shown in Figure 7. We see that the impedances obtained by the three

Figure 7. Impedance spectra data obtained from semiconductive silicon wafer by different methods, the conventional FRA, the potentiostatic FTEIS, and the galvanostatic FTEIS. The measured resistances, R’s, are 276, 285, and 270 kΩ, and capacitances, C’s, are 1.44, 1.38, and 1.36 nF/cm2, respectively.

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ASSOCIATED CONTENT

S Supporting Information *

methods are the same. First, there is no mass transfer due to the absence of molecules moving in and out of the electrode in the solid system. The charge carriers are only electrons flowing across the silicon wafer. Therefore, the equivalent circuit for the silicon sample would be a simple parallel connection of a resistance and a capacitance related to electron mobility and charge accumulation in the semiconductor, respectively. The R values are found to be 276, 285, and 270 kΩ and C’s are 1.44, 1.38, and 1.36 nF/cm2, by the conventional FRA, the PSFTEIS, and the CS-FTEIS, respectively. The observation proves that we can use any method for impedance measurements, and decisions for appropriate methods depend on given situations. For example, when current-controlled experiments are made such as in secondary battery charging and discharging,15,25 electrochemical reaction rates are very slow such as the ORR and corrosion, 13 and evaluation of performances of electrocatalysts26 are focused; the suitable method should be the galvanostatic FTEIS.

Bode plots of the impedance data and the high-frequency and the oscillation noises in detail. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (B.-Y.C.); [email protected] (S.-M.P.). Phone: +82-51-629-5597. Fax: +82-51-629-5583. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by Basic Science Research Program (2011-0009714) and the WCU program (R31-2008000-20012-0) granted to UNIST through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology. 2251

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dx.doi.org/10.1021/ac303108n | Anal. Chem. 2013, 85, 2246−2252