Warburg impedance Nyquist plot

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)... 

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. 

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,... 

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.

When a range of frequencies is applied to the DUT, both El and ECI techniques are called spectroscopies, i.e., electrical impedance spectroscopy and electrochemical impedance spectroscopy. Electrochemical impedance spectroscopy (EIS) profiles, measured as a function of the interrogating frequency, can be presented by two popular plots complex plane impedance diagrams, sometimes called Nyquist or Cole-Cole plots, and Bode (I Z I and 6) plots (Fig. 2). As the impedance, Z, is composed of a real and an imaginary part, the Nyquist plot shows the relationship of the imaginary component of impedance, Z" (on the Y-axis), to the real component of the impedance, Z (on the X-axis), at each frequency. A diagonal line with a slope of 45° on a Nyquist plot represents the Warburg... 

An overview on the topic of IS, with emphasis on its application for electrical evaluation of polymer electrolytes is presented. This chapter begins with the definition of impedance and followed by presenting the impedance data in the Bode and Nyquist plots. Impedance data is commonly analyzed by fitting it to an equivalent circuit model. An equivalent circuit model consists of elements such as resistors and capacitors. The circuit elements together with their corresponding Nyquist plots are discussed. The Nyquist plots of many real systems deviate from the ideal Debye response. The deviations are explained in terms of Warburg and CPEs. The ionic conductivity is a function of bulk resistance, sample... 

Impedance spectroscopy (IS) Nyquist plot Warburg impedance... 

FIGURE 15.6 Complex plane (or Nyquist) plot of the impedance spectrum for the equivalent circuit shown. An example impedance vector at some arbitrary frequency is illustrated by the dashed arrow. Frequency increases in the direction shown by the solid curved arrow. Circuit elements uncompensated solution resistance J s double layer capacitance Cji polarization resistance Rp and diffiasional (Warburg) impedance Z -... 

FIGURE 10.17 Characterization by two techniques of the Randles circuit plot of the current measured on imposition of a potential ramp (a) and Cole-Cole (Nyquist) plot of the complex impedance (b). The Warburg impedance is supposed to be small in comparison with the parallel capacitance. 

Figure 5. Nyquist-plot of impedance data and the finite Warburg Modd.

Figure 11.14. Illustration of equivalent circuit and the corresponding Nyquist plot for an electrochemical reaction involving two resolvable processes. Rs = li2 Ri = lfi Ci = 10F R2 = 2 il C2 = 0.001 F negligible Warburg impedance.

This model contains lumped circuit elements and one Warburg impedance, and its parameter set has six elements P= [ R2,R i,R, Ci,C, Q. In the Nyquist plot this model results in two symmetric semicircles and a -45° diffusion branch shifted on real axis with the value of the series resistance R. Figure 5 shows the electrical equivalent circuit and the impedance is given... 

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. (a) Nyquist plot and (b) Bode plot, obtained from the equivalent circuit of Figure 2. The impedance spectra were theoretically determined by arbitrarily taking Rq=5 Q, Rj=20 Q, Cj=10 jjF, Rc=35 Q, and C =2 mF. The diffusion impedance is expressed as Zji AjJa>) hanh[5(ja)f ] (where, Sis defined as L/D , a is the angular frequency, and A is the Warburg coefficient expressed as Ri/5). Rd=400 Q, L=I0 /am, and D=10 cm /s were taken for the calculation of The elemental resistance r and capacitance c in the TML were estimated to be 4x10 and 2.5x10 s C2 m respectively.

On a Nyquist plot, the infinite Warburg impedance appears as a diagonal line with a slope of 1. On a Bode plot, the Warburg impedance exhibits a phase shift of 45°. 

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