Warburg diffusion element

EIS data analysis is commonly carried out by fitting it to an equivalent electric circuit model. An equivalent circuit model is a combination of resistances, capacitances, and/or inductances, as well as a few specialized electrochemical elements (such as Warburg diffusion elements and constant phase elements), which produces the same response as the electrochemical system does when the same excitation signal is imposed. Equivalent circuit models can be partially or completely empirical. In the model, each circuit component comes from a physical process in the electrochemical cell and has a characteristic impedance behaviour. The shape of the model s impedance spectrum is controlled by the style of electrical elements in the model and the interconnections between them (series or parallel combinations). The size of each feature in the spectrum is controlled by the circuit elements parameters. 

The Warburg diffusion element is a common diffusion circuit element used to fit this line and is described by the following impedance ... 

The Warburg diffusion element is a particular constant phase element (CPE) of 45°. The CPE is a nonintuitive element created to fit the Nyquist plot of real-world systems. In some systems (e.g., batteries or fuel cells), it was expected to be... 

However, although powerful numerical analysis software, e.g., Zview, is available to fit the spectra and give the best values for equivalent circuit parameters, analysis of the impedance data can still be troublesome, because specialized electrochemical processes such as Warburg diffusion or adsorption also contribute to the impedance, further complicating the situation. To set up a suitable model, one requires a basic knowledge of the cell being studied and a fundamental understanding of the behaviour of cell elements. 

The EIS response depends on the flhn thickness and morphology, applied potential, and, obviously, the nature of the components of the hybrid system. The hydro-phobic nature of the polymer, the level of doping within the film, and the size of ions in contact with the polymer surface are factors to be considered for studying the response of such materials. In short, the kinetics of the overall charge transfer process should take into account (1) electron hopping between adjacent redox sites (Andrieux et al., 1986) usually described in terms of a Warburg diffusion impedance element (Nieto and Tucceri, 1996) and (2) double-layer charging at the metal-flhn interface, represented in terms of a double-layer capacitance element. 

Determination of Parameters from Randles Circuit. Electrochemical three-electrode impedance spectra taken on electrochromic materials can very often be fitted to the Randles equivalent circuit (Randles [1947]) displayed in Figure 4.3.17. In this circuit R /denotes the high frequency resistance of the electrolyte, Ra is the charge-transfer resistance associated with the ion injection from the electrolyte into the electrochromic film and Zt, is a Warburg diffusion impedance of either semi-infinite, or finite-length type (Ho et al. [1980]). The CPEdi is a constant phase element describing the distributed capacitance of the electrochemical double layer between the electrolyte and the film having an impedance that can be expressed as... 

Historically, it should be noted that double-layer capacitance measurements were first reliably made (at Hg) by Bowden and Rideal [1928] using the dc charging-current method and by Proskumin and Frumkin [1935] by means of ac modulation. Randles [1947,1952] pioneered the examination of impedance of an electrode process (e.g. redox reactions), using phase-sensitive detector instrumentation to record the frequency dependence of the separated real and imaginary components of Z. Under diffusion-control, the dependence of Z" and Z were found due to the Warburg impedance element. 

Figure 11-lOA [75, 76] shows the Nyquist spectra at +1.00 V at TiOj electrodes in solution containing electroactive [Fe(CN)ions (pH 4.7) with and without the addition of HSA into the bulk solution. In the absence of the protein, the equivalent circuit used to fit the experimental data is reduced to a typical Randles circuit containing Warburg diffusion and [Fe(CN) ] charge-transfer resistance elements in parallel with a double-layer capacitance (Figure 11-lOA). In the presence of a full protein layer, additional processes related... 

Figure 10.6 shows that the overall impedance of the system decreases after addition of plasticizer. The data are in agreement with the increase observed in ionic conductivity. From the parameters obtained by fitting the experimental data shown in Fig. 10.6, the apparent diffusion coefficient can be estimated using equation 10.7,where 4 is the thickness of the electrolyte film and 5 is a parameter related to the element O in the equivalent circuit proposed, which accounts for a finite-length Warburg diffusion (Zd), which represents a kind of resistance to mass transfer. 

At low frequencies, a Warburg -type element is seen in the spectra, related to the slow, solid-state (potential-dependent) diffusion of... 

Hence, we suggest a simple, serial equivalent circuit analog that describes the impedance behavior of carbon electrodes as seen in Figure 12. It contains a Voight-type analog in series with R-C, which reflects the charge transfer, a potential-dependent Warburg -type element (solid state diffusion of Li-ions), and, finally, a capacitive potential-dependent element that reflects the accumulation of lithium. This relatively simple model has already been discussed in depth [105-107]. 

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.

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. 

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)... 

Gerischer impedance — The Gerischer impedance is a transport-related interfacial impedance element which differs from the Warburg impedance in that the electroactive species taking part in the electrode process is chemically generated in a spatially homogeneous way prior to diffusing to the interface. It has the form ... 

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 ... 

For properly describing electrochemical processes, additional impedance elements have been introduced. The Warburg impedance (Raistrick and Huggins, 1982 Honders and Broers, 1985) is representative of diffusive constraints, being defined, for the case of linear diffusion, as a frequency-dependent impedance given by ... 

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... 

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). 

---