Nyquist-Warburg plot

When an electtochemical process is controlled by diffusion or film adsorption, the electrochemical system can be modeled using the ideal circuit shown in Figure 3.8b. In this case, a diffusion impedance (Zp) is included in the circuit series and it is known as Warburg impedance. Notice that Zo and Rp are connected in series. An ideal Nyquist-Warburg plot is shown in Figure 3.13... 

Figure 3.14 shows an experimental Nyquist-Warburg plot for a precipitation hardening (aged) 2195 Al-Li alloy. Notice that the semicircle is depressed due to a diffusion process, which is confirmed by the 45° line. The polarization resistance for this alloy is = 140 ohm in an aerated solution containing 3.5% NaCl at 21.5°C. 

Figure 3.14 Nyquist-Warburg plot for Wledalite 2195 Al-Li alloy.

In many cases, the Nyquist plot for SEI electrodes consists of only one, almost perfect, semicircle whose diameter increases with storage time (and a Warburg section at low frequencies). For these cases the following can be concluded the SEI consists of only one sublayer, 7 GT), / GB... 

Figure 8.14 Huggins analysis of a Warburg element in a Nyquist plot such as that shown in Figure 8.12(a), for the diffusion of Li" ions through solid-state WO3. The traces for Z and Z" against will not be parallel for features other than that of the Warburg. From Ho, C., Raistrick, I. D. and Huggins, R. A., Application of AC techniques to the study of lithium diffusion in tungsten trioxide thin films , J. Electrochem. Soc., 127, 343-350 (1980). Reproduced by permission of The Electrochemical Society, Inc.

Flash Rusting (Bulk Paint and "Wet" Film Studies). The moderate conductivity (50-100 ohm-cm) of the water borne paint formulations allowed both dc potentiodynamic and ac impedance studies of mild steel in the bulk paints to be measured. (Table I). AC impedance measurements at the potentiostatically controlled corrosion potentials indicated depressed semi-circles with a Warburg diffusion low frequency tail in the Nyquist plots (Figure 2). These measurements at 10, 30 and 60 minute exposure times, showed the presence of a reaction involving both charge transfer and mass transfer controlling processes. The charge transfer impedance 0 was readily obtained from extrapolation of the semi-circle to the real axis at low frequencies. The transfer impedance increased with exposure time in all cases. 

Fig. 5.6 Equivalent electrical circuit of electrochemical cell (top) and corresponding Nyquist plot containing Warburg impedance W (bottom)...

The same consideration applies to the impedance measurement according to Fig. 8.1b. It is a normal electrochemical interface to which the Warburg element (Zw) has been added. This element corresponds to resistance due to translational motion (i.e., diffusion) of mobile oxidized and reduced species in the depletion layer due to the periodically changing excitation signal. This refinement of the charge-transfer resistance (see (5.23), Chapter 5) is linked to the electrochemical reaction which adds a characteristic line at 45° to the Nyquist plot at low frequencies (Fig. 8.2)... 

The impedance plot in the Nyquist plane as presented in Figure 1.20 shows two different parts a loop at high frequency and a line at low frequency, also named Warburg line, at 45° angle with the real axis [10]. 

In the plot in Figure 8.23, it is evident that the ohmic factors are independent of frequency, after this, the ideal activation processes display a semicircular conduct with a frequency which is typical of the corresponding relaxation processes (see Equation 8.88 and Figures 8.20 and 8.21) finally, the concentration processes exhibit a diagonal conduct characteristic of diffusion processes (see Figure 8.22) often referred to as the Warburg behavior [124,129,130] (to see a real Nyquist plot related to an EIS test of a battery, see Section 8.9.1). 

In contrast, Fig. 11.6 shows a typical Nyquist plot for the layer after switching between the oxidised and reduced forms in background electrolyte for several days (Fig. 11.4(c)). A pronounced semicircular region, Warburg 45° line and vertical capacitive region can clearly be seen. We have fitted these data to the transmission line circuit (Fig. 11.1). The value of Cs obtained is found to vary with dc potential (Fig. 11.7) and with the... 

In a situation where a charge transfer is also influenced by diffusion to and from the electrode, the Warburg impedance will be seen in the impedance plot. This circuit model presents a cell in which polarization is controlled by the combination of kinetic and diffusion processes. The equivalent circuit and the Nyquist and Bode plots for the system are all shown in Figure 2.40. It can be seen that the Warburg element is easily recognizable by a line at an angle of 45° in the lower frequency region. 

Figure 4.8. a Resistor and Warburg element in series (Model D7) b Simulated Nyquist plot of resistor and Warburg element in series over the frequency range 1 MHz to 1 mHz (Model... 

The Nyquist plot is presented in Figure 4.9b. At high frequencies (real axis at a value of R. At low frequencies ( 0), it intercepts the real axis at a value of R+R0. Note that the bounded Warburg impedance is easily recognized from its Nyquist plot. At high frequencies, this circuit element looks like a traditional Warburg impedance and shows a 45° line on the Nyquist plot. At low frequencies, it looks like the semicircle of a Randles cell,... 

Figure 4.156 shows the simulated Nyquist plot of the modified Randles cell with a combination of kinetic and diffusion processes plus infinite thickness. As in the example, the Warburg coefficient is assumed to be er = 5 Qs 12. Other... 

Figure D.23. Nyquist plot of a resistor and a bounded Warburg in series over the frequency range 1 MHz to 0.1 mHz (R = 50 2 )...

Figure D.46. Nyquist plot of Randles cell with a Warburg element in series with R over the frequency range from 1 MHz to 1 mHz (Rel = 100, Cdl = 0.0001 F, the unit of...

Figure D.57. Nyquist plot of modified Randles cell with a bounded Warburg element over the frequency range 1 MHz to 1 mHz (Rel = 50 Q, Rct = 100 Q,, Cjj = 0.00001 F,...

By taking into account the double-layer capacity, Q, and the electrolyte resistance, Re, one obtains the Randles equivalent circuit [150] (Fig. 10), where the faradaic impedance Zp is represented by the transfer resistance Rt in series with the Warburg impedance W. It can be shown that the high-frequency part of the impedance diagram plotted in the complex plane (Nyquist plane) is a semicircle representing Rt in parallel with Cd and the low-frequency part is a Warburg impedance. 

A line of 45° versus the coordinate axis represents the Warburg impedance in the complex plain presentation (Nyquist plot. Figure 5.7a). The representation in the Bode diagram is shown in Figure 5.7b. The phase shift has a constant value of 45°, whereby the modulus of the impedance, IZI is linearly decreasing with increasing frequency. 

Figure 10a shows the Nyquist plot for a situation when diffusion impedance is much larger compared to the charge transfer resistance. The 45° Warburg line dominates the... 

Figure 5.7 Warburg impedance for semi-infinite diffusion, (a) Nyquist plot and (b) Bode plot.

Figure 5.10 Representation of the impedance spectrum of the equivalent circuit in Figure 5.8 for when Warburg impedance is much larger than the charge transfer resistance = 1000 Mil, IZ I = 1 Mil s , Cj, = 100 nF, = 10 il. (a) Nyquist plot and (b) Bode plot.

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