Introduction
Impedance spectroscopy is an electrochemical characterization method, which measures a system under alternating current. It characterizes the conditions of solid materials or electrochemical systems by applying an alternating voltage of lower amplitude over a wide frequency range. The output signal is the impedance, which represents the electrical resistance of the system. Electrochemical impedance spectroscopy (EIS) is employed in nearly every electrochemical field to assess the material and system properties but especially in energy storage (batteries, fuel cells), the semiconductor industry, kinetics and catalysis and to study corrosion and surface coatings.
Principle
In electrochemical impedance spectroscopy, a potentiostat applies a sinusoidal potential (potentiostatic EIS) to an electrochemical system and a corresponding sinusoidal current is measured as an output signal. In galvanostatic EIS, a sinusoidal current is applied, which results in a sinusoidal output signal. Figure 1 shows the experimental arrangement of a potentiostatic impedance measurement in which an input voltage with amplitude \( E_0 \) and frequency \( \omega \) are applied to an electrochemical system. This results in an current with with an amplitude \( i_0 \) and while the angular frequency \( \omega \) stays the same, the input and output signals differ in a phase shift \( \delta \) ^{[1]}
.Figure 1: Principle of electrochemical impedance spectroscopy. The input and output signal differ in amplitude and phase shift (Figure: Riedlsperger Lisa).
This phase shift results from the capacitive and inductive effects of the electrochemical system when applying an alternating current. The impedance of such alternating current systems can be understood similarly to the resistance of direct current circuits, in which there is no phase shift and which can be described by Ohms law^{[2]}.
\[ I=\frac{V}{R} \]An expression analogous to Ohms law allows the calculation of the impedance \( Z \) for alternating current systems ^{[3]}.
\[ Z=\frac{E}{I}=\frac{E_0sin(\omega t)}{I_0sin(\omega t+\delta)} \]Instead of using the expression above, the current and voltage can be described as a complex function using Euler’s relationship \( e^{jx}=cos(x)+jsin(x) \) ( \( j \) is used to describe the imaginary unit in order to not confuse with current \( I \) )^{[3]}.
\[ Z=\frac{E}{I}=\frac{E_0e^{j\omega t}}{I_0e^{j(\omega t+\delta)}}=Z_0e^{j\delta} \]It is apparent that the impedance is not dependent on time, since the input and output signal have the same frequency. The amplitude \( Z_0 \) and phase shift \( \delta \) depend only on the frequency \( \omega \) and it is therefore possible to convert the timedependent signal into a frequencydependent domain using the Laplace transformation^{[1]}.
Graphical depiction and interpretation
Due to the fact that the impedance is expressed as a complex number, it can be divided into the inphase or “real” part \( Z_r \) and outofphase or “imaginary” part \( Z_i \) ^{[3]}.
\[ Z=Z_0e^{j\delta}=Z_0(cos\omega+jsin\omega)=Z_{r}+jZ_{i} \]Therefore, it is possible to depict the real and imaginary parts of impedance as a vector on a Nyquist plot. The real part of impedance \( Z_r \) is plotted on the xaxis and the imaginary part \( Z_i \) on the yaxis. The length of the vector is the absolute impedance value whereas the phase shift represents the angle between the vector and the xaxis (see Figure 2). So when scanning from low to high frequencies, the overall impedance gets smaller and the characteristic shape of the Nyquist plot forms^{[3]}.
Figure 2: Nyquist plot representing the EIS data (Figure: Riedlsperger Lisa).
Since the Nyquist plot does not give any information about the applied frequency, the Bode plot is also often used to describe electrochemical impedance spectroscopy. There, the magnitude of impedance and the phase angle is shown as a function of logarithm of frequency^{[1]}. In Figure 3, the difference between those two depiction methods is apparent. A Bode plot does not show imaginary impedance information whereas the the Nyquist plot does not include quantitative frequency information^{[1]}
.Figure 3: Graphical representation of Impedance using Nyquist and Bode plot (License: CC BYNB 4.0)^{[4]}
.Measurement Setup
Generally speaking the measurement setup for electrochemical impedance spectroscopy is a system with a working electrode and a counter electrode. Setups with 3 or 4 electrodes are also used where a reference electrode and a working sense electrode are used additionally. With these, the EIS measurement can be performed for liquid or solid samples by applying a potential between the working and the reference electrode and measuring the resulting current. A periodic perturbation signal with amplitude \( E_0 \) is applied between the working and reference electrode from high to low frequencies, the electrochemical response to this perturbation is then measured in the linear domain. The Impedance data can then be presented in Nyquist or Bode representations^{[5]}.
Which of the abovementioned setups is used for the measurement depends on the specific limitations. The twoelectrode system for example is used when a reference electrode cannot be easily inserted, which is the case for measurements of batteries and fuel cells. The missing reference electrode complicates the interpretation of the impedance response, but it is possible to measure the response as the sum of all contributions between the two electrodes. The classical setup used for almost all^{[6]} analytical electrochemical measurements is the threeelectrode setup, meaning a working electrode which is the sample of interest, a counter electrode and additionally a reference electrode. This enables to capture the response of the working electrode to the counter electrode independently. With a fourelectrode system electrolyte conductivity^{[7]}, freestanding films^{[8]} and for example embedded rebar in concrete^{[9]} can be measured as well as the interface between two immiscible electrolyte solutions^{[10]}. In this setup two reference electrodes are positioned on either side of the interface to measure the potential while the other electrodes allow for the current to flow^{[5]}.
Applications
Corrosion
EIS can be applied to analyze phase interfaces and thus is suitable to characterize corrosion. An alternating voltage with varying frequency is applied to the sample (e.g. industrial steel, pipes,...) and the resulting current is measured. In the experiment, the current and voltage must have a linear dependence and describe an ohmic relationship. The characteristic curve of the resulting impedance is usually plotted in a Nyquist or Bode diagram. The analysis of the spectra is based on fitting and interpreting an electrical equivalent circuit. Using basic electrical components such as resistance (R), capacitance (C), and inductance (L), the properties of the phase boundary can be described.^{[11]}
An example of the application of EIS in corrosion measurements can be seen in Figure 4. Ti6Al4V alloy samples with different roughness were exposed to 12 wt.% HCl at 35 °C. Electrochemical measurements were conducted with a standard threeelectrode setup in a frequency range of 1000000.01 Hz and with a sinusoidal potential of 10 mV. The results of the impedance measurement could then be fitted by an equivalent circuit, which represents the system with all parts. The Nyquist plot shows two capacitive loops for two time constants for every sample. The polished Ti6Al4V resulted in the largest capacitive loops, which indicates that it has the highest electrochemical resistance. The same can be seen in the Bode plot, in which the polished sample has the highest phase angle^{[12]}.
Figure 4: (a) Nyquist plot and (b) Bode plot of Ti6Al4V with different surface roughness exposed to 12 wt.% HCl at 35 °C. With help of these plots an (c) equivalent circuit could be drawn which represents the electrochemical system (License: CC BYNCND 4.0).^{[12]}
Batteries
The state of charge and state of health of batteries can be characterized by electrochemical impedance spectroscopy. The performance of batteries is dependent on the material properties, storage and cycling conditions. When batteries age, the impedance increases, which leads to power and energy decay. This can be measured by means of mass transport properties, doublelayer capacitance and ohmic resistance through EIS because the individual electrode materials can be distinctly analyzed. ^{[13]}
In order to measure the state of charge of the battery, it has to be charged and discharged for many cycles. After each charging, the battery has to be rested for a short amount of time to have minimal impact on the capacity determination. Then EIS is measured with the help of a potentiostat/galvanostat and after another rest time discharged to the desired state of charge. After being discharged to the set depth of discharge, the battery is charged again and a new cycle begins.^{[13]}To analyze the recorded data, Nyquist and Bode plots can be created and a correlation between the cell's imaginary impedance component and the state of charge can be established. To get information about the state of health of the battery is by monitoring the capacity. The capacitance decreases with cycle number due to aging can occur by virtue of loss or inactivation of material.^{[13]}
References
1. 
1
2
3
4
L. Murtomäki, T. Kallio, R. Lahtinen, Electrochemistry, Aaltoyliopiston teknillinen korkeakoulu, Espoo, 2010. 
2. 
1
J. LevaBueno, S.A. Peyman, P.A. Millner, A review on impedimetric immunosensors for pathogen and biomarker detection, Med Microbiol Immunol, 2020, 209, 343–362, (https://doi.org/10.1007/s00430020006680). 
3. 
1
2
3
4
V.F. Lvovich, Impedance Spectroscopy: Applications to Electrochemical and Dielectric Phenomena, John Wiley & Sons, Hoboken, 2012, (https://doi.org/10.1002/9781118164075open_in_new). 
4. 
1
W. Choi, H. Shin, J. M. Kim, J. Choi, W. Yoon, Modeling and Applications of Electrochemical Impedance Spectroscopy (EIS) for Lithiumion Batteries, Journal of Electrochemical Science and Technology, 2020, 11, 113, (https://doi.org/10.33961/jecst.2019.00528).

5. 
1
2
S. Wang, J. Zhang, O. Gharbi et al. Electrochemical impedance spectroscopy. Nat Rev Methods Primers, 2021, 1, 14 (https://doi.org/10.1038/s4358602100039w 
6. 
1
E. Engebretsen, G. Hinds, Q. Meyer, T. Mason, E. Brightman, L. Castanheira, P.R. Shearing, D.J.L. Brett, Localised electrochemical impedance measurements of a polymer electrolyte fuel cell using a reference electrode array to give cathodespecific measurements and examine membrane hydration dynamics. J. Power Sources, 2018, 382, 38–44, (https://doi.org/10.1016/j.jpowsour.2018.02.022). 
7. 
1
Y. Sone, P. Ekdunge, D. Simonsson, Proton conductivity of Nafion 117 as measured by a fourelectrode AC impedance method, J. Electrochem. Soc., 1996, 143, 1254–1259, (https://doi.org/10.1149/1.1836625). 
8. 
1
C. Deslouis, M. M. Musiani, B. Tribollet, Freestanding membranes for the study of electrochemical reactions occurring at conducting polymer/electrolyte interfaces. J. Phys. Chem., 1996, 100, 8994–8999, (https://doi.org/10.1021/jp953154n). 
9. 
1
M. Keddam, X. R. Novoa, V. Vivier, The concept of floating electrode for contactless electrochemical measurements: application to reinforcing steelbar corrosion in concrete. Corros. Sci., 2009, 51, 1795–1801, (https://doi.org/10.1016/j.corsci.2009.05.006). 
10. 
1
Z. Samec, Electrochemistry at the interface between two immiscible electolyte solutions (IUPAC technical report). Pure Appl. Chem., 2004, 76, 2147–2180, (https://doi.org/10.1351/pac200476122147). 
11. 
1
M. Mandel, L. Krüger, Elektrochemische Korrosion (Chapter 11) Moderne Methoden der Werkstoffprüfung (H. Biedermann, L. Krüger), WILEYVCH Verlag, Weinheim, 2014, (https://doi.org/10.1002/9783527670673.ch11). 
12. 
1
2
G. Chi, D. Yi, H. Liu, Effect of roughness on electrochemical and pitting corrosion of Ti6Al4V alloy in 12 wt.% HCl solution at 35 °C, Journal of Materials Research and Technology, 2020, 9, 11621174, (https://doi.org/10.1016/j.jmrt.2019.11.044). 
13. 
1
2
3
F. Cuibus, S. Ivanov, U. Schwalbe, M. Schilling, A. Bund, StateofCharge and StateofHealth Estimation of Commercial LiFePO_{4} Batteries by means of Impedance Spectroscopy (pp. 318). In Progress Reports on Impedance Spectroscopy: Measurements, Modeling, and Application (Eds. O. Kanoun), De Gruyter Oldenbourg, Berlin, Boston, 2017, (https://doi.org/10.1515/9783110449822001). 