Impedance of electrochemical systems

The impedance for the study of materials and electrochemical processes is of major importance. In principle, each property or external parameter that has an influence on the electrical conductivity of an electrochemical system can be studied by measurement of the impedance. The measured data can provide information for a pure phase, such as electrical conductivity, dielectrical constant or mobility of equilibrium concentration of charge carriers. In addition, parameters related to properties of the interface of a system can be studied in this way heterogeneous electron-transfer constants between ion and electron conductors, or capacity of the electrical double layer. In particular, measurement of the impedance is useful in those systems that cannot be studied with DC methods, e.g. because of the presence of a poor conductive surface coating. 

The simplest application of electrochemical impedance spectroscopy (EIS) is the determination of the conductivity of the electrolyte solution, where polarisation of the electrode surfaces is eliminated by choosing an appropriate frequency range for measurement of the conductivity31. 

The parameter impedance in electrical alternating-current circuits is the equivalent of resistance in direct-current circuits. If a linear and time-invariant system, L, is considered, then it can be said that  

The impedance is defined as the ratio of the alternating potential and the alternating-current signal  

This impedance can be presented as a vector in the complex plane with modulus Z =EJIm and argument o=a-( . As a consequence, it is expected to obtain a plane with axes having unit 1 for the real and j for the imaginary axis. However, mainly R2 is presented, so both axes are real33. The projection of the impedance vector at these axes results in the resistance Z and the reactance Z , also called the real and imaginary part of the impedance, respectively (Fig. 2.5)  

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Electrochemical methods include potentiometry, cyclic voltammetry and chronoamperometry. These methods as well as other voltammetric methods and the impedance of electrochemical systems are discussed in this chapter. 

Impedances of Electrochemical Systems Terminology, Nomenclature and Representation. Part I. Cells with Metal Electrodes and Liquid Solutions, (prepared for publicaUon by M. Sluyters-Rehbach) PureAppL Chem. 66 (1994) 1831. 

Impedance of an electrical circuit containing linear electrical elements R, C, and L can be calculated using the impedance of these elements and Ohm s and Kirchhoff s laws. The complex plane and Bode plots can be easily produced using programming in Excel, Zplot, Maple, Mathematica, etc., which are readily available. It should be stressed that these electrical elements are linear, that is, their impedance is independent of the applied ac amplitude. In subsequent chapters, we will see how the impedance of electrochemical systems can be described. 

Before discussing the impedance of electrochemical systems it is useful to recall briefly the alternating current response of electrical circuit elements. Three passive elements are normally present in an electrical circuit ... 

Sluyters-Rehbach M. Impedances of electrochemical systems terminology, nomenclature and representation, part I cells with metal electrodes and liquid solutions. Pure Appl Chem 1994 66 1831-91. 

With the noise techniques, both analogue and digital, no externally applied signal is required, and measurement of the fluctuations around the free corrosion potential provides all the information. Hie noise technique is useful in that it allows a fairly rapid estimation of the electrochemical impedance of the system being studied, whereas, with for instance, a.c. lnpedance techniques, very often the minimum frequency studied is still not low enough to provide sufficient information to allow an accurate estimation of the impedance. 

In contrast to kinetic studies, frequency resolved experiments analyze the response of electrochemical systems to periodic or sinusoidal perturbations of voltage or current.545 However, electrochemical impedance spectroscopy (EIS) is the only universally accepted electrochemical frequency resolved method because of the conceptual difficulty involved. Electrochemical perturbation and... 

Conductometric methods in conductometric methods, the conductivity of an electrolyte is assessed by measuring the impedance of this system using two identical electrodes, planarly positioned. However, much more can be done if the impedance is measured as a function of applied frequency, a method that is called electrochemical impedance spectroscopy more details about this method are given in section2.3. 

To carry out this type of study, a small AC amplitude voltage perturbation, AV ico,/), is applied, superimposed onto a DC bias voltage component, and the resulting alternating current response and its phase, A/(co,t), is measured [123,132], Then, the electrochemical impedance of the system is thus defined as... 

The equivalent circuit should be as simple as possible to represent the electrochemical system and it should give the best possible match between the model s impedance and the measured impedance of the system, whose equivalent circuit contains at least an electrolyte resistance, a double-layer capacity, and the impedance of the Faradaic or non-Faradaic process. Some common equivalent circuit elements for an electrochemical system are listed in Table 2.1. A detailed description of these elements will be introduced in Section 4.1. 

Popkirov GS, Schindler RN (1992) A new impedance spectrometer for the investigation of electrochemical systems. Rev Sci Instrum 63 5366-72... 

Gabrielli C, Keddam M, Takenouti LI (1990) New trends in the investigation of electrochemical systems by impedance techniques multi-transfer function analysis. Electrochim Acta 35 1553-7... 

The a.c. impedance technique [33,34] is used to study the response of the specimen electrode to perturbations in potential. Electrochemical processes occur at finite rates and may thus be out of phase with the oscillating voltage. The frequency response of the electrode may then be represented by an equivalent electrical circuit consisting of capacitances, resistances, and inductors arranged in series and parallel. A simplified circuit is shown in Fig. 16 together with a Nyquist plot which expresses the impedance of the system as a vector quantity. The pattern of such plots indicates the type and magnitude of the components in the equivalent electrical network [35]. 

The impedance response of electrochemical systems is often normalized to the effective area of the electrode. Such a normalization applies only if the effective area can be well defined, and is not used in this chapter on the impedance response of electrical circuits. The capacitance used in this chapter, therefore, has units of F rather than F/cm, the resistance has units of O rather than fl cm, and the inductance has units of H rather than H cm. ... 

As described in the subsequent chapters in Part m, models for the impedance response can be developed from proposed hypotheses involving reaction sequences (e.g., Chapters 10 and 12), mass transfer (e.g., Chapters 11 and 15), and physical phenomena (e.g.. Chapters 13 and 14). These models can often be expressed in the mathematical formalism of electrical circuits. Electrical circuits can also be used to construct a framework for accounting for the phenomena that influence the impedance response of electrochemical systems. A method for using electrical circuits is presented in this chapter. 

The EHD impedance is useful for analysis of electrochemical systems that are either partially or completely limited by mass transport. For a rotating disk electrode, the input quantities are, at least, one electrical quantity, e.g., overall current or electrode potential, and one nonelectrical quantity, i.e., the rotation speed of the rotating disk electrode Q. For EHD impedance, the input quantity is the rotation speed. Under galvanostatic regulation, the output quantity is the electrode potential under potentiostatic regulation, the output quantity is the overall current. To analyze this problem, the mass conservation equation must be considered with the normal velocity Vy near the electrode and the concentration of the involved species c, (0) as state quantities. 

While the line-shapes parameters may not be unequivocally associated with a set of deterministic or theoretical parameters for a given system, the measurement model approach has been shown to adequately represent the impedance spectra obtained for a large variety of electrochemical systems. The line-shape models represent the low-frequency stationary components of the impedance spectra (in a Fourier sense). Regardless of their interpretation, the measurement model representation can be used to filter and thus identify the nonstationary (drift) and high-frequency (noise) components contained in the same impedance spectrum. 

As mentioned in the introduction, the electrical nature of a majority of electrochemical oscillators turns out to be decisive for the occurrence of dynamic instahilities. Hence any description of dynamic behavior has to take into consideration all elements of the electric circuit. A useful starting point for investigating the dynamic behavior of electrochemical systems is the equivalent circuit of an electrochemical cell as reproduced in Fig. 1. The parallel connection between the capacitor and the faradaic impedance accounts for the two current pathways through the electrode/electrolyte interface the faradaic and the capacitive routes. The ohmic resistor in series with this interface circuit comprises the electrolyte resistance between working and reference electrodes and possible additional ohmic resistors in the external circuit. The voltage drops across the interface and the series resistance are kept constant, which is generally achieved by means of a potentiostat. 

The dc transient response of electrochemical systems is usually measured using potentiostats. In the case of EIS, an additional perturbation is added to the dc signal to obtain the frequency response of the system. The system impedance may be measured using various techniques ... 

Conventional kinetics is largely concerned with the description of dynamic processes in the time domain, and in consequence few conceptual problems are encountered in understanding time resolved experiments. By contrast, frequency resolved measurements often pose more of a challenge to understanding, in spite of the obvious correspondence between the time and frequency domains. This conceptual difficulty may explain why the only frequency resolved method to achieve universal acceptance in electrochemistry is electrochemical impedance spectroscopy (EIS) [27-29], which analyses the response of electrochemical systems to periodic (sinusoidal) perturbations of voltage or current. It is clear that EIS is a very powerful method, and there... 

To determine numerical values for the different elements of the equivalent circuit they have to be separated, for example, by electrochemical impedance spectroscopy (EIS). Similar to the above-described lock-in measurement a small ac signal of a few mV is superimposed to the electrode potential. The resulting current and its phase shift are then measured as a function of the frequency. Typical impedance spectra of thin oxide films on aluminum are shown in Fig. 17. At high frequencies (10 — lO Hz) the capacitors act as shorts and only the electrolyte resistor determines the impedance, which is typically 10 Ohm for concentrated electrolytes and independent of the electrode. At the lowest frequencies, for example, 10 Hz or below, current flow through the capacitors is impossible and the impedance of the system is given by the sum of the 3 resistors in the current path. The... 

Two plots are used to represent the results of impedance measurements of electrochemical systems. In the Nyquist plot the negative imaginary part (y-axis) is plotted versus the real part (jc-axis). The Nyquist plot of the equivalent circuits in Figures 4.9a and 4.10a is shown in Figures 4.9b and 4.10b. 

Many standard electrochemical techniques can be used, depending on the biological system to be studied. In the presence of redox markers in solution, modification of the electrode resulting from biomolecular interaction affects the impedance of the system, which can be measured by using electrochemical impedance spectroscopy (ElS). EIS is a very promising technique, in particular for the detection of DNA hybridization. 

There is a vast amount of literature on the subject of impedance measurements comprising a large number of different applications, such as corrosion, characterization of thin films and coatings, batteries, semiconductor electrodes, sensors, biological systems, and many more. It is beyond the scope of this article to cover all of these applications comprehensively. This chapter, therefore, concentrates on the description of the main principles and theories and selected applications of impedance methods. A more thorough treatment of the subject from the point of view of corrosion can be found in [1, 2], impedance spectroscopy of solid systems is described in [3]. The fundamentals of impedance spectroscopy of electrochemical systems are also explained in [4, 5]. 

Impedance analysis is used to study the response of electrochemical systems to sinusoidal perturbations about a steady state or equilibrium condition. In contrast to cyclic voltammetry which is a large amplitude technique, impedance measurements are carried out with small amplitude (voltage) perturbations. The voltage is typically 3-5 mV peak-to-peak about a d.c. voltage level so that the (current) response is linear. The frequency of perturbation is varied in order to separate the individual electrochemical relaxation processes which occur with different time constants. 

Conductivity is of course closely related to diffusion in a concentration gradient, and impedance spectroscopy has been used to determine diffusion coefficients in a variety of electrochemical systems, including membranes, thin oxide films, and alloys. In materials exhibiting a degree of disorder, perhaps in the hopping distance or in the depths of the potential wells, simple random walk treatments of the statistics are no longer adequate some modem approaches to such problems are introduced in Section 2.1.2.7. 

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