Circuit model, electrochemical cell

Figure 9.5 Basic equivalent circuit used to model electrochemical cell in Figure 9.4.

The impedance data have been usually interpreted in terms of the Randles-type equivalent circuit, which consists of the parallel combination of the capacitance Zq of the ITIES and the faradaic impedances of the charge transfer reactions, with the solution resistance in series [15], cf. Fig. 6. While this is a convenient model in many cases, its limitations have to be always considered. First, it is necessary to justify the validity of the basic model assumption that the charging and faradaic currents are additive. Second, the conditions have to be analyzed, under which the measured impedance of the electrochemical cell can represent the impedance of the ITIES. 

Modeling Electrochemical Phenomena via Markov Chains and Processes gives an introduction to Markov Theory, then discusses applications to electrochemistry, including modeling electrode surface processes, electrolyzers, the repair of failed cells, analysis of switching-circuit operations, and other electrochemical systems... 

Fig. 11. Differential capacitance versus voltage curves for alkyl monolayers of different chain lengths on Si(lll). The curves were obtained in an electrochemical cell with 0.1 M H2SO4 + 2% HF. The circuit model used to fit the observed behavior is also shown. Reprinted from [74],...

The second meaning of the word circuit is related to electrochemical impedance spectroscopy. A key point in this spectroscopy is the fact that any -> electrochemical cell can be represented by an equivalent electrical circuit that consists of electronic (resistances, capacitances, and inductances) and mathematical components. The equivalent circuit is a model that more or less correctly reflects the reality of the cell examined. At minimum, the equivalent circuit should contain a capacitor of - capacity Ca representing the -> double layer, the - impedance of the faradaic process Zf, and the uncompensated - resistance Ru (see -> IRU potential drop). The electronic components in the equivalent circuit can be arranged in series (series circuit) and parallel (parallel circuit). An equivalent circuit representing an electrochemical - half-cell or an -> electrode and an uncomplicated electrode process (-> Randles circuit) is shown below. Ic and If in the figure are the -> capacitive current and the -+ faradaic current, respectively. 

EIS data analysis is commonly carried out by fitting it to an equivalent electric circuit model. An equivalent circuit model is a combination of resistances, capacitances, and/or inductances, as well as a few specialized electrochemical elements (such as Warburg diffusion elements and constant phase elements), which produces the same response as the electrochemical system does when the same excitation signal is imposed. Equivalent circuit models can be partially or completely empirical. In the model, each circuit component comes from a physical process in the electrochemical cell and has a characteristic impedance behaviour. The shape of the model s impedance spectrum is controlled by the style of electrical elements in the model and the interconnections between them (series or parallel combinations). The size of each feature in the spectrum is controlled by the circuit elements parameters. 

When investigating an electrochemical system using EIS, the equivalent circuit model that has been constructed must be verified. An effective way to do so is to alter a single cell component and see if the expected changes in the impedance spectrum occur, or to keep adding components to the circuit to see if a suitable circuit can be achieved, until reaching a perfect fit. Nevertheless, empirical models should use as few components as possible. 

When we begin to investigate an electrochemical system, we normally know little about the processes or mechanisms within the system. Electrochemical impedance spectroscopy (EIS) can be a powerful approach to help us establish a hypothesis using equivalent circuit models. A data-fitted equivalent circuit model will suggest valuable chemical processes or mechanisms for the electrochemical system being studied. From Chapter 1, we know that a fuel cell is actually an electrochemical system involving electrode/electrolyte interfaces, electrode reactions, as well as mass transfer processes. Therefore, EIS can also be a powerful tool to diagnose fuel cell properties and performance. 

In summary, simple combinations of elements and basic equivalent circuits for electrochemical systems have been introduced in this section. Although these models are relatively simple, they are commonly employed in the investigation of electrochemical systems, including fuel cells. A real electrochemical system may be much more complicated. However, complicated electrochemical systems can still be constructed from these basic equivalent circuits. 

Ahn et al. have developed fibre-based composite electrode structures suitable for oxygen reduction in fuel cell cathodes (containing high electrochemically active surface areas and high void volumes) [22], The impedance data obtained at -450 mV (vs. SCE), in the linear region of the polarization curves, are shown in Figure 6.22. Ohmic, kinetic, and mass transfer resistances were determined by fitting the impedance spectra with an appropriate equivalent circuit model. 

While reaction parameters were not identified by regression to impedance data, the simulation presented by Roy et al. demonstrates that side reactions proposed in the literature can account for low-frequency inductive loops. Indeed, the results presented in Figures 23.4 and 23.5 show that both models can account for low-frequency inductive loops. Other models can also account for low-frequency inductive loops so long as they involve potential-dependent adsorbed intermediates. It is generally understood that equivalent circuit models are not unique and have therefore an ambiguous relationship to physical properties of the electrochemical cell. As shown by Roy et al., even models based on physical and chemical processes are ambiguous. In the present case, the ambiguity arises from uncertainty as to which reactions are responsible for the low-frequency inductive features. 

An electrochemical cell is a type of electrical circuit. As such, it may be modeled with an electrical analog circuit. The potentiometric cell can be considered to be an electrical potential applied to a capacitor and a resistor in series. The capacitor represents the interface between the electrode and the solution, the applied potential is the solution Eh, and the resistor represents the heterogeneous kinetics of the aqueous redox species. The term "heterogeneous kinetics" denotes electron transfer between different phases, in this case aqueous species and the noble-metal electrode. The time required for the capacitor to equilibrate with the applied potential depends on the size of the capacitor and the electrical current. 

With modern computerized frequency-analysis instrumentation and software, it is possible to acquire impedance data on cells and extract the values for all components of the circuit models of Figure 2.>7, This type of analysis, w hich is called electrochemical impedance spectroscopy, reveals the nature t>f the faradaic processes and often aids in the investigation of the mechanisms of electron-transfer reactions. In the section that follows, we explore the processes at the electrode-solution interface that give rise to the faradaic impedance. 

Any electrochemical cell can be represented in terms of an equivalent electrical circuit that comprises a combination of resistances, capacitances or inductances as well as mathematical components. At least the circuit should contain the doublelayer capacity, the impedance of the faradaic or non-faradaic process and the high-frequency resistance. The equivalent circuit has the character of a model, which more or less precisely reflects the reality. The equivalent circuit should not involve too many elements because then the standard errors of the corresponding parameters become too large (see Sect. II.5.7), and the model considered has to be assessed as not determined, i.e. it is not valid. 

Fig. 18 Model potentiostat and electrochemical cell circuit used in stability calculations.

Historically, the Warburg impedance, which models semi-infinite diffusion of electroactive species, was the first distributed circuit element introduced to describe the behavior of an electrochemical cell. As described above (see Sect. 2.6.3.1), the Warburg impedance (Eq. 38) is also analogous to a uniform, semi-infinite transmission line. In order to take account of the finite character of a real electrochemical cell, which causes deviations from the Warburg impedance at low frequencies. 

In order to correctly interpret the experimental information provided by EIS, conveyed in either Nyquist or Bode plots, the use of a sound physical model describing the relevant biophysicochemical processes taking place in the system is essential. A simple strategy to deal with the experimental information involves the implementation of the model into an equivalent circuit, which contains all the information of charge transport. In the equivalent circuit, the resistances and capacitances describe the charge loss and accumulation mechanisms that can take place in the system. In the following section, we first describe the most common circuit elements used in EIS data analysis, followed by the most common equivalent circuits used to describe typical electrochemical cells. 

---