Warburg element

Under this electrochemical configuration, it is commonly accepted that the system can be expressed by the Randles-type equivalent circuit (Fig. 6, inset) [23]. For reactions on the bare Au electrode, mathematical simsulations based on the equivalent circuit satisfactorily reproduced the experimental data. The parameters used for the simulation are as follows solution resistance, = 40 kS2 cm double-layer capacitance, C = 28 /xF cm equivalent resistance of Warburg element, W — R = 1.1 x 10 cm equivalent capacitance of Warburg element, IF—7 =l.lxl0 F cm (

charge-transfer resistance, R = 80 kf2 cm. Note that these equivalent parameters are normalized to the electrode geometrical area. On the other hand, results of the mathematical simulation were unsatisfactory due to the nonideal impedance behavior of the DNA adlayer. This should... 

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.

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 bounded Warburg element (BW) describes linear diffusion in a homogeneous layer with finite thickness. Its impedance is written as... 

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 Warburg impedance is only valid if the diffusion layer has an infinite thickness. If the diffusion layer is bounded, the impedance at lower frequencies no longer obeys Equation 4.32. Instead, the bounded Warburg element (BW) should be used to replace the Warburg. The impedance of the series connection between the resistance and the BW, shown in Figure 4.9a, can be calculated by adding their impedances ... 

Figure 5.34. Electric equivalent circuit for the impedance spectra in Figure 5.37. Ref. ohmic resistance Rct charge-transfer resistance CPE constant phase element IV Warburg element. The subscripts a and c denote anode and cathode, respectively [36]. (Modified from Boillot M, Bonnet C, Jatroudakis N, Carre P, Didierjean S, Lapicque F. Effect of gas dilution on PEM fuel cell performance and impedance response. Fuel Cells 2006 6 31-7. 2006 John Wiley Sons Limited. Reproduced with permission, and with the permission of the authors.)...

Figure D.17. A resistor and a Warburg element in series (Figure 4.8a)...

Figure D.43. Modified Randles cell with a Warburg element in series with Rct. (Figure 4.15a)...

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,...

When the polymer flhn is oxidized, its electronic conductivity can exceed the ionic conductivity due to mobile counterions. Then, the film behaves as a porous metal with pores of limited diameter and depth. This can be represented by an equivalent circuit via modified Randles circuits such as those shown in Figure 8.4. One Warburg element, representative of linear finite restricted diffusion of dopants across the film, is also included. The model circuit includes a charge transfer resistance, associated with the electrode/fllm interface, and a constant phase element representing the charge accumulation that forms the interfacial double... 

Macdonald 18 introduced a generalized finite-length Warburg element described as... 

Greszczuk et al. [252] employed the a.c. impedance measurements to study the ionic transport during PAn oxidation. Equivalent circuits of the conducting polymer-electrolyte interfaces are made of resistance R, capacitance C, and various distributed circuit elements. The latter consist of a constant phase element Q, a finite transmission line T, and a Warburg element W. The general expression for the admittance response of the CPE, Tcpr, is [253]... 

Constant phase elements (CPEs) have been used in bioimpedance models since the late 1920s. A CPE can be modeled by a resistor and capacitor, both having frequency-dependent values, in such a way that the phase angle is frequency independent. A CPE is mathematically simple, but not so simple as to realize with discrete, passive components in the real world. A particular type of CPE is the Warburg element, known from electrochemistry and solid state physics. It is diffusion controlled with a constant phase angle of 45° (Warburg 1899). 

In addition to capacitors and resistors, equivalent circuit models include elements that do not have electrical analogs, i.e., as the Warburg (W) element and the constant phase element (CPE). These elements can explain the deviations from theoretical predictions of the models. The Warburg element is frequency-dependent, and its impedance may be represented by following equation ... 

The fraction a has values between 0 and 1. When a =0.5, the CPE is called the Warburg element, W. The Warburg element is used to describe ionic diffusion (Macdonald, 1992) and the impedance is termed as Warburg impedance. In the case of intercalation electrodes, ionic species can diffuse at the interfaces, in the electrolyte or electrode and charge transfer can occur across the interfaces with resistance R... 

There are several other, more complicated elements available to describe the various processes that can occur in a photoelectrochemical cell, such as the Warburg element (to model diffusion), the Constant Phase Element (CPE, used to describe processes that have a distribution of time constants or activation energies), and transmission lines (to model porous electrodes [47]). Porous electrodes and CPE elements that represent nonideal capacitive elements are briefly discussed below. For more detailed information, the reader is referred to the literature [48, 49]. 

CPE is used in a model in place of a capacitor to compensate for non-homogeneity in the system. A rough or porous surface can cause double-layer capacitance to appear as CPE and Warburg element [116, 117]. 

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