Warburg element, impedance

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... 

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... 

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.)...

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]... 

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... 

The finite-length transmissive impedance may be found in ZView as Warburg Element (short) Ws and the finite-length reflective impedance as Warburg Element (open) Wq. Examples of simulations in Mathematica are shown in files in the exercises. 

In practical applications of EIS it is often found that the experimental data for the finite-length diffusion cannot be approximated by Eq. (4.72) or Eq. (4.83). For example, in the case of hydrogen absorption in Pd the low-frequency reflective impedance is not strictly capacitive, or in the transmissive case the complex plane plot is slightly depressed [154-156]. In such cases one should use a so-called generalized finite-length Warburg element for transmissive... 

In a paper by Kemer and Pajkossy [2002], anion adsorption rates have been measured by impedance spectroscopy at Au (111), for SO4, Cr, Br and T. Cr ion adsorption is very fast, so the equivalent circuit for the process not only has to include a Faradaic pseudocapacitance and its corresponding reaction resistance, but also a Warburg element for the anion diffusion. The interfacial capacitance is then plotted in the Nyquist plane as real vs. imaginary capacitance components, C and... 

The shape of the ac impedance plots may deviate from that expected for the simple RC and Warburg elements. There are different reasons for deviations. Typical reasons are rough siufaces, constriction resistance, and distribution of elements with different characteristic parameters, mainly in the bulk. The constriction resistance is due to a smaller contact area of the electrode than the nominal electrode area. At low frequencies the capacitance reflects the actual contact area, while at high frequencies the capacitance reflects the area of the electrode material which may be larger. Thus the contact caimot be described by a single capacitance. It has also been shown that for a MIEC electrode the impedance of transfer of oxygen from the gas phase into the MIEC and the impedance of diffusion inside the MIEC, though coupled in series, do not yield separated parts in the Cole-Cole plot. 

Warburg impedance — Warburg impedance is a transport-related interfacial impedance element originated from the diffusion of the electroactive species taking part in the electrode process. In general,... 

Figure 3.14b. In this figure, is the resistance of the solution located between the electrodes, R and R+ are the resistances of the interfaces of the negatively and positively charged electrodes, and Cedl(-) and Cedl(+) are the interfacial EDL a adtances of the negatively and positively charged electrodes, respectively. and 2w(+> are called Warburg elements, which are impedances related to diffusion of the eleclrochemically active species in the EDL region. One of the nsefnl applications of the EIS is extracting the solution resistance from the total impedance of the condnctivity cell [1].

An ideal electrode-electrolyte interface with an electron-transfer process can be described using Randle equivalent circuit shown in Fig. 2.7. The Faradaic electron-transfer reaction is represented by a charge transfer resistance and the mass transfer of the electroactive species is described by Warburg element (W). The electrolyte resistance R is in series with the parallel combination of the double-layer capacitance Cdi and an impedance of a Faradaic reaction. However, in practical application, the impedance results for a solid electrode/electrolyte interface often reveal a frequency dispersion that cannot be described by simple Randle circuit and simple electronic components. The interaction of each component in an electrochemical system contributes to the complexity of final impedance spectroscopy results. The FIS results often consist of resistive, capacitive, and inductive components, and all of them can be influenced by analytes and their local environment, corresponding to solvent, electrolyte, electrode condition, and other possible electrochemically active species. It is important to characterize the electrode/electrolyte interface properties by FIS for their real-world applications in sensors and energy storage applications. 

Diffusion resistance Zp,yy to current flow carried by electroactive species can create impedance, frequently known as the Warburg element [23, p. 376]. If the diffusion layer Lp is assumed to have an unlimited thickness within the experimental AC frequency range, than a "semi-infinite" diffusion may become the rate-determining step in the Faradaic kinetic process. In the "semiinfinite" diffusion model the diffusion layer thickness Lp is assumed to be always much smaller than the total thickness of the sample d (Lp d. The equation for the "semi-infinite" Warburg impedance Z m) is a function of concentration-driven potential gradient dV/rfC. The "semi-infinite" diffusion limitation is modeled by characteristic resistance and a Warburg infinite diffusion component Z that can be derived [8] as ... 

Features of the impedance spectra of Fig. 3.15a may be modeled by a simple modified Randles-Ershler equivalent circuit shown in Fig. 3.15c. In this model, is the solution resistance, and is the charge-transfer resistance at the electrode/eIectrol e interface. A constant phase element (CPE) was used instead of a doublelayer capacitance to take into account the surface roughness of the particle. Qn is the insertion capacitance, and Zw is the Warbui impedance that corresponds to the solid-state diffusion of the Li-ion into the bulk anode. The Warburg element was used only for impedance data obtained at the tenth charge. The electrical components of the surface film which is likely formed on the electrode were disregarded, because no time constant related to this process could be seen in the electrochemical impedance spectroscopy (EIS) spectra. It was also checked that their inclusion in the model of Fig. 3.15c does not improve the fit. 

Each of these layers behaves just like an RC element (that is, a capacitor and resistor in parallel) within the equivalent circuit (see Figure 8.13). The respective values o/R, and C, will be unique to each RC element since each layer has a distinct value of [H ]. In order to simplify the equivalent circuit, this infinite sum ofRC elements is given the symbol Zw or -W and is termed a Warburg impedance, or just a Warburg . The Warburg in Figure 8.12 extends from about 50 down to 15 Hz. 

Figure 8.13 Schematic representation of a Warburg impedance Zw as an infinite sum of RC elements, which is commonly employed in an equivalent circuit as a simple model for a concentration gradient.

How does the simplest electrochemical interface look, in terms of an equivalent circuit The appropriate circuit element is shown in Fig. 7.49. It is worth noting that the famous Warburg impedance has been left out The reason is that for most situations in which relatively fast electrode reactions occur, it is negligible. 

In studies of these and other items, the impedance method is often invoked because of the diagnostic value of complex impedance or admittance plots, determined in an extremely wide frequency range (typically from 104 Hz down to 10 2 or 10 3 Hz). The data contained in these plots are analyzed by fitting them to equivalent circuits constructed of simple elements like resistances, capacitors, Warburg impedances or transmission line networks [101, 102]. Frequently, the complete equivalent circuit is a network made of sub-circuits, each with its own characteristic relaxation time or its own frequency spectrum. 

Fig 29. A simple equivalent circuit for the artificial permeable membrane. Physical meaning of the elements C, membrane capacitance (dielectric charge displaceme-ment) R, membrane resistance (ion transport across membrane) f pt, Phase transfer resistance (ion transport across interface) Zw, Warburg impedance (diffusion through aqueous phase) Ctt, adsorption capacitance (ion adsorption at membrane side of interface) Cwa, aqueous adsorption capacitance (ion adsorption at water side of interface). From ref. 109. 

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