Localized impedance

Here, Z(co) local is the magnitude of the local impedance, V((o)apPiied is the magnitude of the voltage between the working electrode and a distant reference electrode, and VXcoVbe is the AC voltage drop measured by the pseudoreference electrode pair. Again, it is implicitly assumed that the current density measured in solution is equal to the current density at the electrode surface. 

Brett DJL, Atkins S, Brandon NP, Vesovic V, Vasileiadis N, Kucemak A (2003) Localized impedance measurements along a single channel of a solid polymer fuel cell. Electrochem Solid-State Lett 6 A63-6... 

Figure 6.46. Local impedance spectra of the middle row of the straight channel cell [40], (Reprinted from Journal of Power Sources, 145(2), Elakenjos A, Elebling C. Spatially resolved measurement of PEM fuel cells, 307-11, 2005, with permission from Elsevier and the authors.)...

This chapter has examined a variety of EIS applications in PEMFCs, including optimization of MEA structure, ionic conductivity studies of the catalyst layer, fuel cell contamination, fuel cell stacks, localized impedance, and EIS at high temperatures, and in DMFCs, including ex situ methanol oxidation, and in situ anode and cathode reactions. These materials therefore cover most aspects of PEMFCs and DMFCs. It is hoped that this chapter will provide a fundamental understanding of EIS applications in PEMFC and DMFC research, and will help fuel cell researchers to further understand PEMFC and DMFC processes. 

Access to powerful computers and to commercial partial-differential-equation (PDE) solvers has facilitated modeling of the impedance response of electrodes exhibiting distributions of reactivity. Use of these tools, coupled with development of localized impedance measurements, has introduced a renewed emphasis on the study of heterogenous surfaces. This coupling provides a nice example for the integration of experiment, modeling, and error analysis described in Chapter 23. 

The methods described in this chapter and this book apply to electrochemical impedance spectroscopy. Impedance spectroscopy should be viewed as being a specialized case of a transfer-function analysis. The principles apply to a wide variety of frequency-domain measurements, including non-electrochemical measurements. The application to generalized transfer-function methods is described briefly with an introduction to other sections of the text where these methods are described in greater detail. Local impedance spectroscopy, a relatively new and powerful electrochemical approach, is described in detail. 

Local impedance measurements represent another form of generalized transfer-function analysis. In these experiments, a small probe is placed near tiie electrode surface. The probe uses either two small electrodes or a vibrating wire to allow measurement of potential at two positions. Under the assumption that the electrolyte conductivity between the two points of potential measurement is uniform, the current density at the probe can be estimated from the measured potential difference AVprobe by... 

A schematic representation of the electrode-electrolyte interface is given as Figure 7.10, where the block used to represent the local Ohmic impedance reflects the complex character of the Ohmic contribution to the local impedance response. The impedance definitions presented in Table 7.2 were proposed by Huang et al. ° for local impedance variables. These differ in the potential and current used to calculate the impedance. To avoid confusion with local impedance values, the symbol y is used to designate the axial position in cylindrical coordinates. 

Figure 7.10 The location of current and potential terms that make up definitions of global and local impedance.

Table 7.2 Definitions and notation for local impedance variables.

The term local impedance traditionally involves the potential of the electrode measured relative to a reference electrode far from the electrode surface. Thus, the local impedance is given by... 

The use of a lowercase letter signifies that z is a local value. The local impedance may have real and imaginary values designated as Zr and Zj, respectively. 

The global impedance can be expressed in terms of the local impedance as... 

The use of a lowercase letter again signifies that 2e is a local value, and the subscript e signifies that Ze represents a value associated only with the Ohmic character of the electrolyte. The local Ohmic impedance may have real and imaginary values designated as Zg,r and Zej, respectively. The local impedance... 

The representation of an Ohmic impedance as a complex number represents a departure from standard practice. As will be shown in subsequent sections, the local impedance has inductive features that are not seen in the local interfacial impedance. As the calculations assumed an ideally polarized blocking electrode, the result is not influenced by Faradaic reactions and can be attributed only to the Ohmic contribution of the electrolyte. 

A 3-D distribution of blocking components in terms of resistors and constant-phase elements is presented in Figure 13.1(b). Such a system will peld a local impedance with a CPE behavior, even in the absence of a 2-D distribution of surface properties. If the 3-D system shown schematically in Figure 13.1(b) is influenced by a 2-D distribution, the local impedance should reveal a variation along the surface of the electrode. Thus, local impedance measurements can be used to distinguish whether the observed global CPE behavior arises from a 2-D distribution, from a 3-D distribution, or from a combined 2-D and 3-D distribution. 

Following Hueing et al., ° a notation is presented in Section 7.5.2 that addresses the concepts of a global impedance, which involved quantities averaged over the electrode surface a local interfacial imgedance, which involved both a local current density and the local potential drop V — Oo(r) across the diffuse double layer a local impedance, which involved a local current density and the potential of the electrode V referenced to a distant electrode and a local Ohmic impedance, which involved a local current density and potential drop Oo(r) from the outer region of the diffuse double layer to the distant electrode. The corresponding list of symbols is provided in Table 7.2. 

The local impedance z can be represented by the sum of local interfacial impedance Zo and local Ohmic impedance z as... 

Huang et al. ° ° demonstrated for blocking disk electrodes that, while the local interfacial impedance represents the behavior of the system unaffected by the current and potential distributions along the surface of the electrode, the local impedance shows significant time-constant dispersion. The local and global Ohmic impedances were shown to contain the influence of the current and potential distributions. 

While the calculations presented here were performed in terms of solution of Laplace s equation for a disk geometry, the nature of the electrode-electrolyte interface can be imderstood in the context of the schematic representation given in Figure 13.5. Under linear kinetics, both Co and Rt can be considered to be independent of radial position, whereas, for Tafel kinetics, 1/Rf varies with radial position in accordance with the current distribution presented in Figure 5.10. The calculated results for global impedance, local impedance, local interfacial impedance, and both local and global Ohmic impedances are presented in this section. 

Figure 13.8 Calculated representation of the local impedance response for a disk electrode as a function of dimensionless frequency K under assumptions of Tafel kinetics with 7 = 1. (Taken from Huang et al. and reproduced with permission of The Electrochemical Society.)...

The calculated local impedance is presented in Figure 13.8 for Tafel kinetics with 7 = 1 and with radial position as a parameter. The impedance is largest at the center of the disk and smallest at the periphery, reflecting the greater accessibility of the periphery of the disk electrode. Similar results were also obtained for J = 0.1, but the differences between radial positions were much less sigiuficant. Inductive loops are observed at high frequencies, and these are seen in both Tafel and linear calculations for J = 0.1 and J = 1.0. ... 

The local Ohmic impedance Zg accounts for the difference between the loccil interfacial and the local impedances. The calculated local Ohmic impedance for Tafel kinetics with 7 = 1.0 is presented in Figure 13.9 in Nyquist format with normalized radial position as a pcirameter. The results obtained here for the local Ohmic impedance are very similar to those reported for the ideally polarized electrode and for the blocking electrode with local CPE behavior. ° ° At the periphery of the electrode, two time constants (inductive and capacitive loops) are seen, whereais at the electrode center only an inductive loop is evident. These loops are distributed around the asymptotic real value of 1/4. 

The local impedance is obtained by integration along the distance x from 0 to the coating thickness d, i.e.. 

Yovtng impedance, true CPE behavior is not fovmd. It should be noted that the CPE model corresponds to a specific distribution of time constants that may or may not correspond to a given physical situation. Local impedance measurements can give information about the nature of this distribution, whether 2-D, 3-D, or both. This example shows that not all depressed semicircles correspond to a CPE behavior. 

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