# Electrochemical impedance spectroscopy as diagnostic tool.pdf

Electrochemical impedance spectroscopy as diagnostic tool S. H. Jensen, J. Hjelm, A. Hagen and M. Mogensen Technical University of Denmark, Roskilde, Denmark 1 INTRODUCTION Electrochemical impedance spectroscopy (EIS) has become a very popular tool for the study of electrochemical cells during the last three decades (see also Electrochemical impedance spectroscopy, Volume 2). [1] A single spectrum carries information about the characteristic time constants and the size and shape of the impedance arcs associated with the processes occurring in the object under study. An impedance spectrum can be analyzed with impedance mod- els. When matching the model impedance to the measured impedance, the variables in the model may converge to some values that may provide information about the stud- ied object. Several processes contribute to the impedance of a fuel cell. [2, 3] Unfortunately, this limits the information about the nature (physical, chemical, or electrochemical) of the processes we can derive from an impedance spectrum. This is discussed in a recent review addressing the difficul- ties in resolving the individual impedance contributions. [4] Further, the processes that contribute will in general vary from cell type to cell type, [2, 5] and the size, shape, and summit frequency of the impedance arcs may vary from cell to cell unless the cells are fabricated in a very repro- ducible manner. [2, 6] This limits very much the amount of information that can be derived from a single impedance spectrum. An equivalent circuit and a complex nonlinear least- square (CNLS) fitting routine [7–9] are often used to charac- terize the processes contributing to the cell impedance. The equivalent circuit should offer a good physical description of the electrochemical cell and provide a good model of the cell impedance. Typically, the equivalent circuit is not known and the main problem is to obtain such a circuit. [10] Different equivalent circuits (different combinations of processes) may have the same impedance at all frequencies. [11] In order to deal with this ambiguity, changes can be made to operating conditions in order to choose the circuit in which the changes are closest to theoretical expectations or possible knowledge about the physics of the system. [12] This approach, however, may fail if several processes contribute to the cell impedance since the impedance model or the equivalent circuit accordingly involves several variables. In general, a decreasing number of variables can be determined accurately as the total number of variables increases. This makes it interesting to try to break up the contributions to the cell impedance and model each contribution separately using (simple) equivalent circuits with a limited number of variables. However, a lot of information about process contribu- tions in full cells can be obtained directly, by carrying out systematic variations of the test conditions (temperature, gas composition, and flow) and observing the correspond- ing changes in the impedance spectrum. [13–17] Further, it is desirable to be able to analyze the obtained data sets using techniques that do not require initial assumptions regard- ing the equivalent circuit, or model, that may describe the measured system. If these techniques are able to provide information on the number, type, and magnitude of under- lying process contributions, the selection of an appropriate impedance model based on the physical–chemical under- standing of the system under study is greatly aided. Handbook of Fuel Cells – Fundamentals, Technology and Applications. Edited by Wolf Vielstich, Hubert A. Gasteiger, Arnold Lamm and Harumi Yokokawa. ? 2010 John Wiley (b) Nyquist plot of the impedance to be investigated by DIA; (c) temporal plots; and (d) spectral plots. [Reprinted from A. Barbucci, M. Viviani, P. Carpanese, D. Vladikova and Z. Stoynov, Electrochimica Acta, 51, 1641 (2006) with permission from Elsevier.] The resolution of DIA was compared to conventional CNLS analysis and was found to be significantly larger. [27] 2.4 Deconvolution of impedance spectra into distributions of relaxation times (DRT) A series of publications by Ivers-Tiff′ee and coworkers [33–38] describes the theory and application of a technique for deconvolution of a single impedance spectrum into a distribution of relaxation times, referred to hereafter as distributions of relaxation times (DRT) (see also Electro- chemical impedance spectroscopy, Volume 2). In analogy with the work carried out on analysis of dielectric relax- ation data, [39, 40] this technique also involves the calculation of a distribution function of relaxation times, γ(τ).Itcan be shown that γ(τ) is related to the complex impedance through: Z(ω) = R 0 + Z pol (ω) =R 0 +R pol integraldisplay ∞ 0 γ(t) 1 + jωt dτ with integraldisplay ∞ 0 γ(τ) dτ (1) where R 0 is the ohmic part and Z pol is the polarization part of the total impedance, Z(ω). R pol is the total dc polarization resistance and j is equal to (?1) 0.5 ,theimag- inary unit. An extension of the DRT technique that allows quantitative estimates of the resistances and capacitances of individual processes to be obtained was presented by Cros- bie and coworkers. [41] The complete algorithmic code (for MathCad) for calculation of the distribution function, the Kramers–Kronig transform, and power law extrapolations, with comments, were provided in the same publication. Key strengths of the DRT technique lie in its ability to resolve peaks that are closely overlapping in an impedance spectrum and that it allows calculation of the resistance associated with each peak, which means that both a resis- tance and time constant is obtained, which is valuable information for further modeling. Further, this technique can also help identify the type of elements that are present. An elegant application of DRT analysis carried out on full cells in conjunction with systematic variations of the test conditions was reported recently [38] (see Figure 3). Drawbacks of the DRT technique are that extrapolations of the low- and high-frequency ends of the spectrum to lower/higher frequencies are needed, and that filtering of the data is needed because of the enhancement of noise that the numerical procedure causes at high and low frequencies. Both the DIA and the DRT technique are based on the analysis of a single impedance spectrum at a time, and 4 Advanced diagnostics, models, and design 0.08 0.06 0.04 0.02 0.00 DRT, g (f) ( ? s) 10 0 10 1 10 2 10 3 10 4 10 5 Frequency, f (Hz) P 1A P 2A P 3A Warburg impedance pH 2 O (anode) 0.049 atm 0.072 atm 0.180 atm 0.625 atm Figure 3. A series of distribution curves generated from imped- ance spectra recorded at four different anode water partial pres- sures. [A. Leonide, V. Sonn, A. Weber and E. Ivers-Tiffee, J. Electrochem. Soc., 155, B36–B41 (2008) Reproduced by permis- sion of ECS - The Electrochemical Society.] they are applicable only in cases where it is possible to distinguish process contributions on the basis of a difference in their characteristic (distribution of) relaxation times. In contrast, analysis of difference in impedance spectra (ADIS) is based on the analysis of differences between pairs of spectra collected under different test conditions. This means that it is possible to distinguish contributions that may be completely overlapped with other processes, and assign them to, e.g., the anode or cathode side of a cell. 2.5 Analysis of difference in impedance spectra (ADIS) ADIS, like DIA and DRT, can provide information on the nature of the underlying process contributions and improve the resolution of overlapping processes in impedance spec- tra. Further, ADIS may help to reduce the number of variables (degrees of freedom) in CNLS modeling. It can also be viewed as a powerful graphical tool where no model assumptions, filtering, or extrapolations are needed. As mentioned in the introduction section, an often encountered fuel cell characterization strategy is to measure an impedance spectrum and subsequently analyze it. Usu- ally, this is difficult, because several processes contribute to the impedance spectra and the main task is therefore to resolve the individual contributions. A simple test strat- egy that may assist in this struggle is presented here. The strategy is based on subsequent changes in operating param- eters, such as gas composition at one of the electrodes, gas flow rate, or operating temperature. This affects parts of the processes contributing to the fuel cell impedance. EIS spectra are recorded before and after the change to measure the resulting impedance difference. An impedance model is then proposed to describe the impedance difference. Finally, the affected processes are characterized by CNLS model- ing of the impedance difference using a simple equivalent circuit with a limited number of variables. An equivalent circuit can be described as a network of impedance units such as capacitors, resistances, and inductances. To some extent these impedance units may form groups that are serially connected to all other units. Such a group is here defined as an impedance element. For instance, an equivalent circuit of a fuel cell would often involve three impedance elements describing the fuel cell anode, electrolyte, and cathode, each of which in turn may be broken down into smaller elements describing, e.g., single rate-limiting reaction steps. Hence, the impedance of an electrochemical cell can be represented as a series connection of impedance elements: Z(f n ) = summationdisplay i z i parenleftbig f n parenrightbig (2) where Z(f n ) is the impedance of the cell at the nth frequency f n and z i (f n ) is the impedance of the ith impedance element at the frequency f n . Each z i represents the impedance of either a single electrochemical process or a network of processes. Even when Z is known in a large frequency range, it may prove difficult, if not impossible, to determine the impedance of the individual processes from a single impedance spectrum. Suppose an operating parameter, Psi1 1 (flow rate, gas composition, temperature, etc.) is slightly changed from condition α to condition α prime . As a result, a number of impedance elements, z j , will be modified and a number, z k , stay constant. Hence, the change in Z can be written as Delta1Z = Z| α prime ? Z| α = summationdisplay i parenleftbig z i | α prime ? z i | α parenrightbig = summationdisplay j parenleftbig z j | α prime ? z j | α parenrightbig + summationdisplay k parenleftbig z k | α prime ? z k | α parenrightbig = summationdisplay j parenleftbig z j | α prime ? z j | α parenrightbig (3) where f n is omitted for simplicity. For a suitable choice of Psi1 1 , the index j may sum a significantly smaller number of impedance elements than the index i. This means the number of variables necessary to model the measured difference spectrum, Delta1Z, may be less than the number of variables necessary to model Z. Z is normally known only for a finite set of frequencies {f 1 ,f 2 ,.,f N },whereN denotes the number of frequen- cies, f 1 denotes the lowest frequency and f N denotes the highest frequency. For the nth frequency between 2 and N ? 1,Delta1 ˙ Z prime ,the derivative with respect to ln(f) of the real part of Delta1Z can Electrochemical impedance spectroscopy as diagnostic tool 5 be obtained as Delta1 ˙ Z prime parenleftbig f n parenrightbig ～ = [Z prime parenleftbig f n+1 parenrightbig ? Z prime parenleftbig f n?1 parenrightbig ] α ? [Z prime parenleftbig f n+1 parenrightbig ? Z prime parenleftbig f n?1 parenrightbig ] α prime ln parenleftbig f n+1 /Hz parenrightbig ? ln parenleftbig f n?1 /Hz parenrightbig (4) where Z prime is the real part of Z. Delta1 ˙ Z prime offers a reasonably good resolution of individual impedance contributions [16] and is fairly simple to obtain; i.e., no filtering, Fourier transformation, or aprioriassumptions are involved in the calculation. Delta1 ˙ Z primeprime , the derivative with respect to ln(f)ofthe imaginary part of Delta1Z can be found in a similar manner. While Delta1 ˙ Z prime is suitable for graphical visualization, it may be advantageous to model Delta1 ˙ Z, the derivative with respect to ln(frequency) of Delta1Z. This is because Delta1 ˙ Z contains both the real and imaginary part and thus involves twice as many measurement points as Delta1 ˙ Z prime and this will increase the modeling accuracy. We also model Delta1 ˙ Z rather than Delta1Z because Delta1 ˙ Z excludes any contribution to Delta1Z, being con- stant in the frequency interval [f 1 ,f N ]; i.e., it excludes contributions from impedance elements (including seri- ally connected resistances) having characteristic frequencies f s greatermuch f N . This will reduce the number of modeling vari- ables. Like usual CNLS fitting, [7–9] the modeling task is to minimize an error function, which in this case (with unity weighing [9] ) takes the form Err = N summationdisplay n=1 vextendsingle vextendsingle vextendsingleDelta1 ˙ Z parenleftbig f n parenrightbig ? parenleftBig ˙ Z (Eq) parenleftbig f n ,? α prime parenrightbig ? ˙ Z (Eq) parenleftbig f n ,? α parenrightbig parenrightBigvextendsingle vextendsingle vextendsingle 2 (5) where the vertical bracket denote the complex norm, f n is the frequency, and Delta1 ˙ Z(f n ) is the measured impedance dif- ference. ˙ Z (Eq) (f n ,? α prime) and ˙ Z (Eq) (f n ,? α ) are the derivatives with respect to ln(frequency) of the theoretical impedance of summationtext j z j (or the impedance of an equivalent circuit) at the operating conditions α prime and α, respectively. Err is mini- mized by variation of the two set of variables, ? α prime and ? α where each set contains M free variables. In an equiva- lent circuit, the variables in ? α prime and ? α are the resistances, capacitances, exponents, etc. Unfortunately, it often happens that the 2M variables are undetermined, i.e., the minimization of Err does not make the variables converge to a specific value. Instead, the average of a variable may converge, which is exemplified in the next section. In case of parallel connected processes, the admittance differences will only involve the processes affected by the change in the operating condition. ADIS is discussed in further detail elsewhere. [17] 3 EXAMPLES OF RESULTS 3.1 Equivalent circuit identification Anode-supported thin electrolyte Ni/YSZ|YSZ|LSM/YSZ cell has previously been studied by EIS in order to describe how the losses are distributed at a low current density. [42] An equivalent circuit consisting of an inductance, a serial resistance (R s ), and five arcs to describe the polarization resistance was suggested by Barfod and coworkers. [42] The equivalent circuit was based on studies of single electrodes in three-electrode and two-electrode symmetric cell setups in conjunction with extensive full cell studies in which the partial pressure of reactant gases on both the electrodes as w