Impedance Plots

Normally, the impedance plots are fitted to an often-complex equivalent circuit. Mathematically, this means searching for a global solution in R". However, problems arise if a complicated equivalent circuit is found which does not allow physical interpretation. Therefore, it is preferable to run a wide variety of experiments with different samples rather than trying to fit in detail the results of a single measurement in order to analyze the resulting impedance plots. 

Figure 15 shows a set of complex plane impedance plots for polypyr-rolein NaC104(aq).170 These data sets are all relatively simple because the electronic resistance of the film and the charge-transfer resistance are both negligible relative to the uncompensated solution resistance (Rs) and the film s ionic resistance (Rj). They can be approximated quite well by the transmission line circuit shown in Fig. 16, which can represent a variety of physical/chemical/morphological cases from redox polymers171 to porous electrodes.172... 

Figure 15. Complex plane impedance plots for polypyrrole at (A) 0.1, (B) -0.1, (C) -0.2, (D) -0.3, and (E) -0.4 V vs. Ag/AgCl in NaCl04(aq). The circled points are for a bare Pt electrode. Frequencies of selected points are marked in hertz. (Reprinted from X. Ren and P. O. Pickup, Impedance measurements of ionic conductivity as a probe of structure in electrochemi-cally deposited polypyrrole films, / Electmanal Chem. 396, 359-364, 1995, with kind permission from Elsevier Sciences S.A.)...

Figure 1.7 ORR impedance plots for a Pt electrode at several cathode DC currents, showing a low ftequency branch corresponding to slow increase of the rate (slow lowering of the faradaic impedance) following cathodic perturbation [Makharia et al., 2005].

FIG. 8 Complex impedance plot associated with the heterogeneous oxidation of Fc by ferri/ferro-cyanide at the water-nitrobenzene interface. The responses only in the presence of 0.1 M ferrocene ( ) are contrasted with ( ) those obtained upon addition of ImM K3Fe(CN)g and 0.1 mM K4Fe(CN)g. (Reprinted from Ref. 74 with permission from Elsevier Science.)... 

FIG. 6 Complex impedance plots for the electrode reaction of [Fe(CN)6] on bare (open circle) and DNA-modilied (filled circle) An electrodes. An equivalent circuit for the electrode system is shown in the inset and solid lines represent theoretical responses from the circuit. Parameters used in simulation are cited in the text. Electrode potential, + 205 mV (vs. Ag/AgCl) AC amplitude, 25 mV (p-p). Other conditions are the same as those in Fig. 5. 

This circuit is usually referred to as the Randles circuit and its analysis has been a major feature of AC impedance studies in the last fifty years. In principle, we can measure the impedance of our cell as a function of frequency and then obtain the best values of the parameters Rct,<7,C4i and Rso by a least squares algorithm. The advent of fast micro-computers makes this the normal method nowadays but it is often extremely helpful to represent the AC data graphically since the suitability of a simple model, such as the Randles model, can usually be immediately assessed. The most common graphical representation is the impedance plot in which the real part of the measured impedance (i.e. that in phase with the impressed cell voltage) is plotted against the 90° out-of-phase quadrature or imaginary part of the impedance. 

FIGU RE 1.41 Complex impedance plots of SSC cathodes on SSZ electrolytes with and without GDC interlayer [201]. 

To learn how to interpret an impedance plot for an electrochemical cell and be able to recognize the electrical components implied. 

A mathematician would say that a plot of Z" (as y ) against Z (as jc ) forms an Argand diagram (or Argand plane ). As electroanalysts, we will call such a set of axes a Nyquist plot or simply an impedance plot (see Figure 8.9). 

It is now time to look at the Nyquist impedance plot of a real cell. Figure 8.12(a) shows such an impedance plot for the all-solid-state cell, IT0/W03/PE0-H3P04/1T0(H), at 8°C. The two ITO layers are needed as transparent electronic conductors (cf. Section 8.1.2). 

The electrical components within the impedance plot are listed in Table 8.1. In summary, we see that a Nyquist plot of imaginary against real impedances can be dissected piece by piece, with each component representing a physical part of the cell or a kinetic phenomenon. We see that impedance analysis is a powerful and versatile tool which is capable of discerning the individual processes... 

It can be seen from Table 1 that all those specimens which could not support a stable rest potential, also showed impedance plots that were of a purely capacitive type. This impedance was also obtained from other specimens of both pigmentation, which exhibited stable rest potentials (A12 G6). The diameters of the semicircles (given as Rgc) large, classically Indicative of highly... 

Specimen Potential (SCE) Impedance plot (Ohms cm2) Rsc Capacitance pF/cm ... 

Coal tar epoxy and plasticized chlorinated rubber laquer coated on mild steel were studied by Scantlebury et al (28). Impedance plots show a gradual decrease in the value of R t and the onset of Warburg-type behavior with increasing Immersion time in 3 weight percent sodium chloride solution. Appearance of an inductive loop when the coal-tar epoxy had a pin-hole was clearly demonstrated. 

Table I shows the details of surface treatment and coating along with the calcxilated values of total resistance R and effective capacitance C. For specimens with Initial mechanical surface preparation, the Nyqulst Impedance plot shows the characteristic semicircular behavior with a resistance of the order of 1800 n cm and a capacitance of about 40 yF cm. As different surface treatments are Incorporated on a sequential basis, the complex plane diagram shows a gradual evolution.

Double layer paint provides additional protection since such coatings would be less porous than single layer paint. It Is also noted that In all specimens that are not rinsed there Is a tendency to show Inductive loops In the impedance plot. It is not clear If this Is due to the adsorption of Inhibitor on steel surface or due to the formation of oxides or due to Increased porosity (28). 

AC Impedance measurements taken on the same specimen at different temperatures In the range 25-90 C are shown In Table III. A specimen with no surface treatment other than mechanical polishing shows Cdi 40vF cm" at 25 C but the value Increases appreciably with Increasing temperature. The values of R t for different specimens ( 28, 29) show a systematic decrease with Increasing temperature whereas the values of C i show a systematic Increase. Figures 5 and 6 show the evolution of Impedance plots as a function of temperature. In addition to the variation In the values of Rgt Cjj, It Is noticed that the Warburg-... 

Figure 7 shows the Impedance plots as a function of temperature obtained using a specimen having the polyurethane paint coating. The resistance Is of the order of 10 S2 cm and the... 

Fig. 7 Complex plane impedance plot for p-type Si in 1% HF solution, in the dark, at bias potential 0 V vs. SCE (after Vanmaekelberg et al. [9]).

Other important characteristics of the converter are the reflected ripple attenuation and the turn-on characteristics. It is expected that the turn-on characteristics will be difficult to simulate because of the nonlinear characteristics of a saturating core. A nonsaturating core is simply described by Faraday s law, and it can be easily modeled by any of the SPICE simulators. The model used for the EMI filter is shown in Fig. 3.66, and the results of each of the simulators output and the measured impedance plots are shown in Figs. 3.67 to 3.70. 

Fig. 7.47. Cole-Cole impedance plot. (Reprinted from Southampton Electrochemistry Group, Instrumental Methods in Electrochemistry, Ellis Harwood, 1985, p. 268.)...

In the Cole-Cole (or complex impedance ) plot, one takes the real as ordinate and the Zimag part as abscissa. Each point on the resulting diagram is made up of a Z resolved into two components measured at a chosen frequency. There may be 20-30 points, each at different frequencies. Such plots tend to be semicircles (see Fig. 7.47), but even simple equivalent circuits have some structure (i.e., deviations from the semicircle), and these deviations provide information concerning events at the elec-trode/solution interface. 

Calculating Exchange Current Densities and Rate Constants from Impedance Plots. If one takes the Butler-Volmer equation (7.24) under the reversible condition, i.e that in which the overpotential, rj, tends to zero, then,... 

Of course, each C and R should bear an appropriate suffix that would indicate what particular element is to be understood (i.e., is it for electron transfer across the double layer, etc. ) (Fig. 7.51). It turns out that frequently the impedance plots due to the various elements maximize at characteristic frequencies, and if these maxima occur at... 

It is usually possible from Zmax to calculate C and R for the circuit element associated with the maximum. One of the many applications to which impedance plots can be put is that of determining the surface states of semiconductors. 

Fig. 17. (a) The components of the faradaic impedance plotted against to. (b) The components of the faradaic admittance plotted against to1/2. System parameters fict = lficm1, p = 0.03. The solid parts of the plots indicate the frequency range that is normally accessible for meaningful analysis of data with the simple theory described in this section... 

A remark has to be made that also applies to (b) anodic oxidation of a metal is often a very complex process, involving, for example, adsorption of intermediates in a multi-step mechanism, film formation of insoluble products, etc. In such cases, the impedance plots can be much more intricate and the straightforward relation to the corrosion rate can be obscured. 

Fig. 6-14. Potentiostatic EHD impedance plots, in Bode representation (reduced amplitude A(pSc /3)/A(0) and phase shift, versus dimensionless frequency pSc / ) for the oxidation of hydroquinone on a 360 nm thick poly(TV-ethylcarbazole) film at E - 0.7 V (diffusion plateau).

---