Impedance Warburg

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°. 

In Table 8.3, the elements used in common equivalent circuit models along with equations for admittance and impedance are given for each element. 

The impedance of a resistor is independent of frequency and has only a real component. Because there is no imaginary impedance, the current through a resistor is always in phase with the voltage. The impedance of an inductor increases as frequency increases. Inductors have only an imaginary 

The common equivalent circuit models used to interpret simple EIS data are shown in Table 8.4 along with fuel cell representative components and Nyquist plots. 

Since for a purely resistor the imaginary components of the resistance is zero, the Nyquist plot for a resistor is a single point on the real axis with a value R. A purely capacitive coating can be represented by a series of a capacitor and a resistor. The Nyquist plot for a series RC circuit is a vertical line where the intercept of the line with the real axis gives an estimate of the resistance value. The imaginary component of the impedance (contributed by the capacitor) dominates the response of the circuit. One limitation of this Nyquist plot is that the value of the capacitance cannot be determined from the plot. It can be determined by a curve fit or from an examination of the data points. Also, the plot does not indicate which frequency was used to take each data point. 

In Section 4.2.1 the method of impedance spectroscopy was introduced. The polarization of an electrode with an alternating potential of small amplitude is also influenced by restricted mass transport. The Warburg impedance for semi-infinite diffusion describes the diffusion process. The Warburg impedance is a complex quantity with real and imaginary parts of equal magnitude. The impedance is given by the equation 

Details are given in the special literature, e.g., by J.R. MacDonald or Sluyters-Rehbach and Sluyters.  

The impedance is represented in the Nyquist and Bode diagrams for two different situations in Figmes 5.9 and 5.10. 

A comparison of these cases shows that the two representations of impedance measurements provide different information. While low-frequency components dominate in the Nyquist plot, high-frequency components dominate the character of the Bode diagram. 

So far unlimited diffusion has been assumed. When the diffusion layer has finite dimension, with a boundary of constant concentration of the diffusing species and permeable for these species, however, a different expression for the diffusion impedance is derived which is the so-called Nemst impedance. This impedance is given by the equation 

Transport by diffusion is the movement of material under the influenee of the gradient of chemical potential, from more eoneentrated media toward less eoneentrated ones. Its limitation ean be expressed in terms of eleetroehemieal impedance, which is often ealled the Warburg impedanee. 

60 In chemistry, the term adsorption refers to a surface phenomenon whereby atoms or molecules of gases or hquids affix to a sohd surface. Adsorption must not be confused with absorption, whereby atoms, molecules or ions enter into the pores of a solid. 

Three scenarios are generally to be considered, depending on the thickness of the diffusion layer semi-infinite thickness, finite thickness in the presence of convection (or Nemst s hypothesis) and finite thickness through a thin film in an open circuit. 

Diffuse layer External layer Internal layer 

Depending on the case in question, the diffusion impedance can be calculated in different ways  

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Fig. 7. (a) Simple battery circuit diagram where represents the capacitance of the electrical double layer at the electrode—solution interface, W depicts the Warburg impedance for diffusion processes, and R is internal resistance and (b) the corresponding Argand diagram of the behavior of impedance with frequency, for an idealized battery system, where the characteristic behavior of A, ohmic B, activation and C, diffusion or concentration (Warburg... 

FIGURE 12.15 Electrode impedance with kinetic (a), diffusional (b), and combined (c) reaction control (W is the Warburg impedance). 

The impedance with its components R and C is known as the Warburg diffusion impedance, and constant as the Warburg constant. In the equivalent circuits for electrochemical reactions, a Warburg impedance is represented by the symbol -W- as shown in the lower part of Fig. 12.15b. 

In the Warburg impedance, parameters and C are not constant but depend on frequency according to Eq. (12.28). Figure 12.16ft shows plots of the values of... 

In the case of reactions that are not completely irreversible (or not completely reversible), we must account for both the kinetic factors (e.g., the polarization resistance R and the concentration changes (the Warburg impedance). The simplest equivalent circuit for this case is shown in Fig. 12.15c, while Fig. 12.17c shows the impedance diagram for this circuit (AJS = 10 = 1 Q the other parameters... 

Since the ion transfer is a rather fast process, the faradaic impedance Zj can be replaced by the Warburg impedance Zfy corresponding to the diffusion-controlled process. Provided that the Randles equivalent circuit represents the plausible model, the real Z and the imaginary Z" components of the complex impedance Z = Z —jZ " [/ = (—1) ] are given by [60]... 

Returning to the fundamental ac harmonic in Fig. 3.42, we wish to establish the relationship between I and the faradaic impedance Z( instead of considering a combination of a series resistance Rs and a pseudo-capacity C8, the alternative is to separate a pure resistance of charge transfer Rct and a kind of resistance to mass transfer Zw, the Warburg impedance the derivation of the polarogram39 then (for AEtc < 8/remV) leads to the equation... 

Fig. 5.22 Equivalent circuit of an electrode with diffusing el-ectroactive substances. W is the Warburg impedance (Eq. 5.5.21)...

Here W is the Warburg impedance corresponding to the diffusion process... 

The Warburg impedance is related to the concentration overpotential and applied AC by... 

Very often, the electrode-solution interface can be represented by an equivalent circuit, as shown in Fig. 5.10, where Rs denotes the ohmic resistance of the electrolyte solution, Cdl, the double layer capacitance, Rct the charge (or electron) transfer resistance that exists if a redox probe is present in the electrolyte solution, and Zw the Warburg impedance arising from the diffusion of redox probe ions from the bulk electrolyte to the electrode interface. Note that both Rs and Zw represent bulk properties and are not expected to be affected by an immunocomplex structure on an electrode surface. On the other hand, Cdl and Rct depend on the dielectric and insulating properties of the electrode-electrolyte solution interface. For example, for an electrode surface immobilized with an immunocomplex, the double layer capacitance would consist of a constant capacitance of the bare electrode (Cbare) and a variable capacitance arising from the immunocomplex structure (Cimmun), expressed as in Eq. (4). 

At high frequencies diffusion of the reactants to and from the electrode is not so important, because the currents are small and change sign continuously. Diffusion does, however, contribute significantly at lower frequencies solving the diffusion equation with appropriate boundary conditions shows that the resulting impedance takes the form of the Warburg impedance ... 

Impedances of Real Cells Quantification of Diffusion Phenomena and the Warburg Impedance... 

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.

The interface impedance for a case such as Ag/Ag4Rbl5 will consist of a capacitance (derived from the Helmholtz formula) in parallel with i et so that in the complex plane impedance a semi-circle will be found from which Qi and can be evaluated. Rq will cause this semicircle to be offset from the origin by a high frequency semicircle due to the bulk impedance between the interface and the reference electrode (Fig. 10.12). There can be no Warburg impedance (a line at 45° to the real axis generally due to diffusion effects) in this case. 

At high frequencies, a semicircle is expected as a result of a parallel combination of R and Cg. At low frequencies a Warburg impedance may be found as part of the interfacial impedance. In some cases it may dominate the interfacial impedance as in Fig. 10.13(a), in which case only the diffusion coefficient of the salt will be determinable. It should be noted that, in the absence of a supporting electrolyte, the electroactive species, in this case Li, cannot diffuse independently of the anions. 

In other cases as in Fig. 10.13(b) the interfacial impedance will show a semicircle due to / <., and Cj, in parallel, with the Warburg impedance becoming apparent at significantly lower frequencies. In such cases R can be evaluated without difficulty. 

Fig. 10.13 Impedance plane diagrams for metal non-blocking electrodes with two mobile species in the electrolyte, (a) Interfacial impedance is only a Warburg impedance. (b) Interfacial impedance shows a charge transfer resistance semicircle.

In general it will be necessary to measure via impedance measurements using a four electrode cell. A schematic diagram of the cell which would be used for such measurements is shown in Fig. 10.15. The expected behaviour will be as described in Eqn (10.3) except that Warburg impedances can arise from either or both phases. An example of an impedance spectrum of the H2O/PVC interface is shown in Fig. 10.16. The application of a constant overpotential will, in general, lead to a slowly decaying current with time due to the concentration changes which occur in both phases, so that steady state current potential measurements will be of limited use. 

In the second case (limit of fast kinetics at the gas-solid interface), the film becomes entirely bulk transport limited, corresponding to the limit of Hebb— Wagner polarization. Since electronic conduction is fast, this situation yields a Warburg impedance for finite length diffusion ... 

Many specimens ( 1, 6, 8, 10 and 12) also show the Initiation of Warburg Impedance behavior at the lower end of the frequency range covered In this study. 

Warburg impedance is a well-known term in the field of impedance spectroscopy because of the early date at which it was published, the formulation came before the rest of the properties of the interface were known. In fact, for nearly all real situations examined in electrochemistry, the Warburg impedance is relatively small. Thus, for a concentration of 1 mol liter and a frequency of 1 kilocycle s l, and using the normal parameters for room temperature, the resistance is in the milliohm cm-2 range. 

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. 

This is a form of Sand s equation and dates from the remarkably early time of 1901. It reminds us that there was much progress made in the early days of electrochemistry with diffusion-related situations one might call it the macrophase of the subject. (Hie Warburg impedance was derived even earlier it deals with impedance due to diffusion. See Section 7.5.13.5.) Consideration on a microscale (the things that happen during charge transfer over 5 A at the interface) had another 50 years before it became a funded research topic. 

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