# 快速时时彩: Electrochemical impedance spectroscopy.pdf

Electrochemical impedance spectroscopy E. Ivers-Tiff′ee, A. Weber and H. Schichlein Universit¨at Karlsruhe, Karlsruhe, Germany 1 FUNDAMENTALS 1.1 Background Impedance spectroscopy has been established over many years as a powerful measurement technique for the electri- cal characterization of electrochemical systems. The power of the method lies in the fact that by small signal pertur- bation it reveals both the relaxation times and relaxation amplitudes of the various processes present in a dynamic system over a wide range of frequencies. Electrochemical impedance spectroscopy is especially useful if the system performance is governed by a num- ber of coupled processes each proceeding at a different rate. In these systems, steady-state polarization curves are not useful because static measurements can only identify relatively simple reaction mechanisms that are known to be clearly dominated by a single rate-determining step. However, fuel cells are prominent examples of complex dynamic materials systems that generally do not fulfill this requirement. The physical and chemical processes con- tributing to the internal resistance of the cell determine their dynamic behavior over a wide range of frequencies. The relaxation times typically span more than fifteen orders of magnitude, reaching from fast processes that sustain cell operation, e.g., current flow, gas conversion and charge transfer to long-term degradation processes limiting the life time of the cell (Figure 1). Due to practical factors limit- ing the frequency range of impedance measurements in fuel cells, the method is feasible for processes with relaxation times ranging from microseconds up to tens of seconds. Slower processes exhibiting time constants from several minutes to hundreds of hours are favorably observed in the time domain, e.g., by analyzing the response of the cell on a step function of the current with respect to the potential. Processes in fuel cell systems typically involve com- plex multi-step reactions that can proceed along sev- eral parallel reaction pathways, e.g., adsorption, heteroge- neous reactions at interfaces as well as bulk and surface transport. In the case of advanced electrodes, neither of these reaction steps dominates the overall cell character- istics. The extent to which the various steps contribute to cell performance is governed by their complex depen- dence on each other as well as on the particular operat- ing conditions and design of the cell. In order to clar- ify their contributions to the internal resistance of the cell, electrochemical impedance spectroscopy has to be used. The technique has therefore become widely used for characterizing the intrinsic loss factors of fuel cell operation. However, the relaxation times of the physical processes themselves cannot be observed directly from the measure- ment data if their impedance contributions overlap in the spectrum. Therefore, the impedance data has to be ana- lyzed with respect to the underlying dynamic processes. The physical interpretation of these kinetic information is the key to predicting fuel cell properties under different operating conditions and different materials configurations and thus to enable a well-directed improvement of fuel cell performance. Computer-controlled impedance analysis devices and data acquisition software enable the materials scientist to acquire large amounts of impedance data in a short time. Edited by Wolf Vielstich, Hubert A. Gasteiger, Arnold Lamm and Harumi Yokokawa. ? 2010 John Wiley how- ever, their choice is limited by the experimental conditions and the measurement equipment used. The frequency range should be scanned logarithmically, i.e., the samples are dis- tributed at equal distances along the logarithmic frequency scale, usually with about five to ten samples per frequency decade. The sample interval is then given by T = 1 N ln ω max ω min (2) with N as the number of data points. ω min and ω max are the lower and the upper bounds of the frequency range. The discrete impedance spectrum consists of a series of complex numbers representing the impedance data {Z k }={Z 1 ,Z 2 ,.,Z N },k= 1,.,N (3) Alternatively to sine wave excitation, step or noise func- tions may be used as the input perturbation. Input and 3 Electrochemical impedance spectroscopy response signals are then recorded in the time domain. In principle, the investigation of dynamic systems in the time-domain delivers the same amount of information as in the frequency domain because the step response of a linear system is related to its impedance by Fourier transformation. In practice, time-domain methods work considerably more rapidly than frequency-by-frequency measurements and need less instrumentation. They are ade- quate when a quick overview of the system dynamics is required. However, these methods are less accurate because the sample frequencies cannot be directly controlled but the impedance values at these frequencies have to be calculated from Fourier analysis of the data. To achieve a high reso- lution of fast processes, high sampling rates as well as an ideal step function in the perturbing signal are necessary. To cover a wide bandwidth of relaxation frequencies with a reasonable amount of data, a flexible sampling rate is required. Time-domain methods are, therefore, less suitable for a full-scale characterization of complex electrochemical systems like fuel cells. Figure 3 shows different representations of a typical fuel cell impedance spectrum. In the Nyquist-plot (Figure 3a), the imaginary part of the impedance is plotted vs. the real part of the impedance in the complex impedance plane. In this plot, the frequency is an implicit variable. Each data point corresponds to an impedance value at a distinct frequency. In the plot, the arrow shows the direc- tion of increasing frequency and the solid symbols denote decades of frequency. Because impedance data usually shows capacitive behavior rather than inductive behavior, it has become customary to plot the negative imaginary part rather than the positive imaginary part. Most of the curves then fall in the first quadrant of the complex plane. In the complex plane plot, physicochemical processes usually manifest themselves in semicircular arcs with time constants that directly correspond to the peak frequency of the semicircle (τ = 1/f peak ) and can be interpreted in terms of a parallel connection of a resistor and a capacitance. However, in complex electrochemical systems like fuel cells, a large number of several physicochemi- cal processes act in parallel, and the Nyquist plot often shows several overlapping arcs. Special data analysis pro- cedures then have to be used in order to identify the dynamic processes, which are represented in the com- plex plane plot (see Section 3.3). Although complex plane plots can reveal useful information for the identification of physicochemical processes, the frequency dependence of the data cannot be observed directly. It is useful to plot the real and imaginary part of the impedance sep- arately vs. frequency as shown in Figure 3(b). In this representation, physicochemical processes manifest them- selves as peaks in the imaginary parts and corresponding 0.2 0.1 0.0 ?0.2 10 ?2 10 ?1 10 0 10 1 f (Hz) 10 2 10 3 10 4 10 5 0.0 0.2 Z ( ? ) 0.4 0.6 0.0 5 4 3 21 0 ?1 ?2 0.1 (a) (b) R 0 R 0 R pol R 0 R pol Z pol R pol Re(Z ) (?) ? Im ( Z ) ( ? ) 0.2 0.3 0.4 0.5 0.6 0.7 T = 950 °C A = 1cm 2 0.25 slm H 2,dry 0.25 slm air Re(Z ) (?) Im(Z ) (?) Figure 3. A typical impedance spectrum of a planar SOFC element: (a) Nyquist plot of the impedance: Im{Z} vs. Re{Z} (solid symbols denote decades of frequency) and (b) Re{Z} and Im{Z} vs. the logarithmic frequency. 4 Methods in electrocatalysis slopes in the real part. It has also been suggested to use three-dimensional perspective plotting for impedance repre- sentation, revealing the inter-dependencies of the real part, imaginary part and frequency of the impedance at the same time. [1] The impedance curve comprises a frequency-independent ohmic part, R 0 , and a frequency-dependent polarization part, Z pol (ω). This is illustrated with the equivalent cir- cuit model shown in the inset in Figure 3(a). The total impedance of the system is Z(ω) = R 0 + Z pol (ω)(4) For low frequencies approaching the d.c. case, all polar- ization processes contribute to the impedance. With rising frequency, the processes drop out of the system response in the order of their relaxation frequencies. For frequencies above the highest relaxation frequency in the system, only the ohmic, i.e., instantaneous loss factor is still present. All systems with non-blocking electrodes like batteries and fuel cells exhibit pure ohmic behavior for high frequencies and the following holds Z(ω ???→ 0) = R 0 + R pol ,Z pol (ω ???→ 0) = R pol , Z(ω ???→∞) = R 0 ,Z pol (ω ???→∞) = 0 (5) Thus, with impedance spectroscopy the ohmic resistance, R 0 , can be discriminated from the total polarization loss, R pol . Every electrode/electrolyte system has a geometrical capacitance and a bulk resistance acting in parallel. These elements lead to a further relaxation process in the cell. However, its time constant is often so small, in particu- lar for the high operating temperatures of solid oxide fuel cells (SOFCs), that it is beyond the frequency measure- ment range and its semicircular arc does not show in the impedance curve. [1] Therefore, the electrolyte resistance can be treated as purely ohmic, and R 0 can usually be directly linked to the electrolyte conductivity of the cell when addi- tional ohmic losses in the electrodes and the wiring resis- tances are taken into account. Information on the dynamics of the physicochemical processes in the cell are contained in the polarization part, Z pol (ω). By appropriate data anal- ysis techniques (see Section 3), this information can be evaluated and employed for the understanding and further improvement of the electrochemical cell. 2 IMPEDANCE MEASUREMENT TECHNIQUES 2.1 Operating principles of frequency response analyzers In the past, bridge methods relying on the null detection of a balance condition have been used for impedance mea- surements. The frequency range was usually restricted from 20 Hz to 20 kHz. The determination of each impedance value involved cumbersome analog signal analysis in the frequency domain. A review of these basic measurement arrangements is given in Ref. [1]. Over the last three decades, automated impedance measurement equipment, termed “frequency response analyzers” have become avail- able to the researcher and are used almost exclusively except for special custom-made bridge set-ups for when a very high accuracy is required. Frequency response analyzers are digitally demodulated, computer-controlled, stepped-frequency impedance meters. Typical representa- tives of these devices are the Solartron 1260, Zahner IM6 or Autolab FRA2 with a system cost in the range of $20 000–40 000. [6, 7] Commercial suppliers typically pro- vide a full-scale suite of instruments including potentiostats, galvanostats, amplifiers and high current boosters as well as data acquisition and analysis software. These pack- ages allow convenient and high-precision measurement of impedance spectra over a wide range of frequencies. In frequency response analyzers, the impedance is deter- mined from the correlation of the cell response u(t) with two reference signals that are generated from the input sig- nal. One reference signal is in phase with the input while the other one is phase-shifted by π/2. As depicted in Figure 4, the output signal is correlated with each of the reference signals and the mean of the resulting signals is taken by Load I load i 0 sin ωtu 0 sin(ωt +?) ~ u 0 cos ? ~ Re(Z ) ~u 0 sin ? ~ Im(Z ) sin ωt cos ωt x x ∫ ∫ Figure 4. The orthogonal signal correlation principle of a frequency response analyzer for impedance determination. 5 Electrochemical impedance spectroscopy Electrolyte (a) Reference electrodeWorking electrode Counter electrode (c) Electrolyte Working cathode Working anode (d) Reference cathode Reference anode Electrolyte Working cathode Working anode (b) Working electrode Counter electrode Reference electrode Electrolyte Figure 5. Cross-sections of different electrode arrangements: (a) conventional three-electrode cell for the characterization of the working electrode, (b) refined circular symmetrical three-electrode pellet geometry providing uniform current distribution, (c) fuel cell element for testing under realistic operating conditions and (d) electrolyte-supported fuel cell element with additional reference electrodes for separate characterization of cathodic and anodic polarizations. integration over several cycle periods. The value of u(t) consists of the fundamental harmonic component, possible higher harmonic components due to non-linearity of the system and measurement noise, n(t). u(t) = u 0 sin(ωt + ?) + summationdisplay k A k sin(kωt + ? k ) + n(t) (6) If n(t) is stochastic, its mean value vanishes for a suffici- ently long measurement interval, T int .WhenT int is a full number of cycles, the higher harmonics are also cancelled by the integration. We, therefore, obtain at the outputs of the frequency response analyzer, a signal that is proportional to the real and the imaginary parts of the impedance (Figure 4). u 0 T int integraldisplay T int 0 sin(ωt + ?) sinωt dt = u 0 2 bracketleftbigg cos? ? 1 T int integraldisplay T int 0 cos(2ωt + ?) dt bracketrightbigg = 1 2i 0 |Z(ω)| cos?(ω) = 1 2i 0 Re{Z(ω)} u 0 T int integraldisplay T int 0 sin(ωt + ?) cosωt dt = u 0 2 bracketleftbigg sin? ? 1 T int integraldisplay T int 0 sin(2ωt + ?) dt bracketrightbigg = 1 2i 0 |Z(ω)| sin?(ω) = 1 2i 0 Im{Z(ω)} (7) 2.2 Impedance spectroscopy measurement arrangements for fuel cells Different electrode arrangements for the electrochemical characterization of SOFC electrodes and single cells are shown in Figure 5, some of them are also suitable for other types of fuel cells. To characterize the properties of a single electrode, three terminal configurations with a working electrode, reference electrode and counter electrode are used. In the case of high temperature solid electrolyte cells, the reference electrode and working electrode may consist of the same material composition and can be exposed to the same nominal atmosphere. Therefore the potential difference is zero under equilibrium conditions. [8–10] In the galvanostatic mode, a current from the working electrode to the counter electrode is set by the experimenter. As a consequence, the working electrode is polarized. Because there is no net current flow through the reference electrode, the reference electrode is not polarized and the potential difference between the working electrode and the reference electrode corresponds to the electrode polarization, the so- called overpotential. In the potentiostatic mode, the current flowing