Equivalent Circuit Analogs

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. 

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Figure 28 A typical Nyquist plot obtained from a nickel electrode polarized to low potentials (0.2 V versus Li/Li+) in PC solutions (1 M LiBF4 in this case). The equivalent circuit analog of 4 R C circuits in series and their separate Nyquist plots (four semicircles) are also shown. The frame in the lower right represents a typical fitting between the experimental data and this equivalent circuit analog [34]. (With copyright from The Electrochemical Society Inc.)...

The last comment relates to the data analysis and the choice of appropriate models for impedance spectra. As shown by Orazem et al. [241], each single impedance spectrum can be fitted by a number of equivalent circuit analogs. Hence, the choice of a model has to be based on... 

Figure 18, taken from Ref. 77, describes several models proposed for the Li electrodes in solutions, their equivalent circuit analogs, and the expected impedance spectra (presented as Nyquist plots). Assuming parallel plate geometry for the solid electrolyte interface, as well as knowledge of the surface species involved from spectroscopy (and thus their dielectric constant, which is around 5 for many surface species formed on Li, including R0C02Li, Li2C03, LiF, ROLi, etc. [186]), it is possible to estimate the surface film s thickness from the electrode s capacitance (calculated from the model fitted to the spectra) ...

Figure 18 Various models proposed for the surface films that cover Li electrodes in nonaqueous solutions. The relevant equivalent circuit analog and the expected (theoretical) impedance spectrum (presented as a Nyquist plot) are also shown [77]. (a) A simple, single layer, solid electrolyte interphase (SEI) (b) solid polymer interphase (SPI). Different types of insoluble Li salt products of solution reduction processes are embedded in a polymeric matrix (c) polymeric electrolyte interphase (PEI). The polymer matrix is porous and also contains solution. Note that the PEI and the SPI may be described by a similar equivalent analog. However, the time constants related to SPI film are expected to be poorly separated (compared with a film that behaves like a PEI) [77]. (With copyrights from The Electrochemical Society Inc., 1998.)...

Figure 20 Scheme of the multilayer model of the Li-solution interphase, the division of the various layers, and the corresponding equivalent circuit analog, which can be fitted very well to the experimental data [49]. (With copyrights from The American Chemical Society, 1998.)... 

The equivalent circuit analog of this situation is a finite-length transmission line terminated with an open circuit. A constant activity or concentration is also a common condition for the interface removed from x = 0. In this case the finite-length transmission line would be terminated in a resistance, and the impedance is given by the expression... 

Scheme 1 A schematic illustration of layered inteiphase on active metal and equivalent-circuit analog of 4 RC circuits in series. Rq>rinted with copyright from The Electrochemical Society Inc.

In the equivalent circuit analog, resistors represent conductive pathways for ion and electron transfer. As such, they represent the bulk resistance of a material to charge transport such as the resistance of the electrolyte to ion transport or the resistance of a conductor to electron transport. Resistors are also used to represent the resistance to the charge-transfer process at the electrode surface. Capacitors and inductors are associated with space-charge polarization regions, such as the electrochemical double layer, and adsorption/ desorption processes at an electrode, respectively. 

The surface films formed on toe carbon electrodes may be similar to those formed on Li and noble metal electrodes (polarized to low potentials) in the same solutions, as discussed in sections 1 and 2 above. Hence, it is possible to describe Li-ion migration through toe surface films formed on carbon electrodes by electrical models (i.e., equivalent circuit analogs) similar to those used to describe toe behavior of Li and noble metal electrodes (covered by surface films) in Li salt solutions [25,31]. 

Figure 12 compares typical Nyquist plots obtained from a noble metal electrode covered by surface films (deposited at low potentials in a Li salt solution), Li metal, and lithiated graphite electrodes. The figure also shows the expected structure of surface films formed on these electrodes and toe relevant equivalent circuit analogs for toe impedance behavior of the three electrodes. The surface films formed on aU three types of electrodes should have a multilayer stmcture, and they comprise toe compact inner part and a porous outer (solution side) part. Hence, toe simplest analog for describing ion transport under an electrical field through different... 

Hence, we suggest a simple, serial equivalent circuit analog that describes the impedance behavior of carbon electrodes as seen in Figure 12. It contains a Voight-type analog in series with R-C, which reflects the charge transfer, a potential-dependent Warburg -type element (solid state diffusion of Li-ions), and, finally, a capacitive potential-dependent element that reflects the accumulation of lithium. This relatively simple model has already been discussed in depth [105-107]. 

Figure 12. A schematic illustration of impedance spectra, relevant equivalent circuit analogs, and the structure of the surface films for lithium, lithiated graphite electrodes, and noble metal electrodes polarized to low potentials In Li salt, nonaqueous solutions. Reprinted from reference 219 with permission from Bsevier Sdenoe.

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