Warburg straight line

In Fig. 11, the oscillation above the Warburg straight hne (at frequencies from 0.4 to 0.14 Hz)) is related to the Nemst hypothesis. A numerical calculation taking into account the convection term in the transport equation showed that the impedance diagram is below the Warburg straight line. 

For a constant phase process, as a diffusion process, the plot is represented as a straight line with one slope (see Figure 8.22) [75], This is evident from Equation 8.86, because when Z, is plotted versus Zr, the Warburg component is represented as a straight line with a unitary slope. 

This low-frequency limit is a straight line of unit slope, which extrapolated to the real axis gives an intercept of (Ra + Rct - 2o2Cd). The line corresponds to a reaction controlled solely by diffusion, and the impedance is the Warburg impedance, the phase angle being jt/4, see Fig. 11.6. 

The Warburg impedance has a minimum at 1/2. The mass-transfer impedance is a vector containing real and imaginary components that are identical, that is, the phase angle (p = atan(Z" v/Z w) = atan(-l) = 5°. The faradaic impedance is shown in Fig. 11(b) (dashed line). On the complex plane plot, it is a straight line with a slope of 1 and intercept The total electrode impedance consists of the solution resistance, R, in series with the parallel connection of the double-layer capacitance, Qi,... 

As Fig. 10 indicates, the extrapolation of the 45° straight line, which represents the Warburg impedance, intercepts the real axis at ... 

The Warburg impedance for hemispherical diffusion corresponds to that of planar diffusion however, with a resistance par in parallel [10]. 1/ par is inversely proportional to the mean size of the active centres. 1/ par can be determined by representing l// par vs. This gives a straight line with the intersection point for... 

The Constant Phase Element and Its Simple Combinations. Although Warburg and open-ended diffusion effects frequently appear in supported situations and sometimes in unsupported ones and exhibit characteristic 6 = 45° lines in the Z or plane, one often finds approximate straight-line behavior over a limited frequency range with 0 45° (e.g. McCann and Badwal [1982]). Then the frequency response of Z and Z is no longer proportional to but to some other power of (0. To describe such response it is convenient to write, as in Eq. (7) in Section 1.3, at the admittance level,... 

The phase shift is thus equal to 0 = - 45°. In the Nyquist diagram, the Warburg impedance is therefore represented by a straight line at a 45° angle. 

The Warburg region (that means the straight line at 45° in the medium-low frequencies) is only observed when Ri is not too high, that is, for E E°. When the potential is greater, in absolute value, than the standard potential, this domain, relative to the semi infinite diffusion, is hidden by the RiffCi adsorption-dipole. 

Figure 4.7 shows plots of the impedance in the Warburg region against (o, as required by Eqn. 28. It is satisfactory to obtain good straight lines. 

For a semi-infinite diffusion process at cathode represented by Warburg impedance, the Nyquist plot appears as a straight line with a slope of 45°. The impedance increases linearly with decreasing frequency. The infinite diffusion model is only valid for infinitely thick diffusion layer. For finite diffusion layer thickness, the finite Warburg impedance converges to infinite Warburg impedance at high frequency. At low frequencies or for small... 

The fuel cell cathode can be represented by a series electrolyte resistance, a parallel double-layer capacitance, a charge transfer impedance, and finite Warburg impedance for diffusion process. This circuit model polarization is due to a combination of kinetic and diffusion processes. The Nyquist plot for this shows a semicircle with a 45° straight line. 

The Faradaic resistance or polarization resistance Rp is inversely proportional to the corrosion rate. It is evident from the Nyquist plot that the solution resistance. Rn, measured at high frequency can be subtracted from the sum of Rp and Rn at low frequency to give the value of Rp corrected for ohmic interferences from solution resistance. For processes controlled by diffusion in the electrolyte (concentration polarization) or in a surface film or coating, an additional resistive element called the Warburg impedance, W, must be included in the circuit. The Warburg impedance appears at low frequencies on the Nyquist plot as a straight line superimposed at 45° (slope = 1) to both axes, as shown in Fig. 31.5. 

An extrapolation of the -45° straight line to high frequencies (co ) representing the Warburg impedance in the complex plane intersects the real axis at the value allowing us to calculate diffusion coefficients and from known rate constants and (Figure 5-lOA) ... 

One should note that the real and the imaginary parts of the Warburg impedance in Eq. (16.22) depend on frequency in the same way. Therefore the phase shift generated by the Warburg impedance is independent of frequency. Plotted in the complex-plane impedance format, this leads to a straight line with a slope of unity, as shown in Figure 16.6. 

The Nyquist plot of this spatially restricted diffusion impedance, calculated for Ri = 0.05 cm s and D = 63 x 10 cm s , is presented in Fig. 20 for various values of the electrolyte film thickness e. The diagrams exhibit a classical Warburg behavior at high frequency (straight line with a 45° slope) followed by a capacitive behavior at lower frequencies. Electrically, the low-ffequency behavior is equivalent to a resistor-capacitor series coimection. The value of this low-frequency resistor decreases as the electrolyte film thickness decreases. An increase of the characteristic frequencies (corresponding to the transition frequency between the Warburg and the capacitive behavior, for example) is also observed as the electrolyte film thickness decreases. 

The diffusion with reflecting boundary produces a transmission line terminated by an open circuit (as = °o), revealing a straight -90° capacitive line at low frequencies following the high frequency -45° Warburg line (Fig-... 

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