The Warburg Impedance

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). 

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... 

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. 

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. 

Obviously, the faradaic impedance equals the sum of the two contributions f ct, the charge transfer resistance, and Zw = aco-1/2 (1 — i), the Warburg impedance. Again, the meaning of the parameters Rct and a is still implicit at this stage of the treatment and explicit expressions have to be deduced from an explicit rate equation, e.g. the expressions given in eqns. (51). 

In the more explicit classical theories, the Warburg impedance was... 

So, the term [A0a0 + AR aR ] co-1/2 (1 — i) resembles the Warburg impedance corresponding to diffusional mass transport of A, O and R, with a mobile equilibrium between A and 0, i.e. kQ -> °°, whereupon the term in g = kQ /co would vanish. If, however, kQ has a finite value, the faradaic impedance is enlarged by the Gerischer impedance expressed by the term containing g. 

No simple combination of resistors correctly models the phase angle and current response of Zf at all frequencies, but it can be modeled at any single frequency by a resistor and capacitor arranged either in series or in parallel. This effective impedance, now termed the Warburg impedance (Zw), is a function of frequency and can be extracted from experimental data either numerically or... 

The Warburg impedance has been substituted by ZD, the diffusion impedance ... 

One possible equivalent circuit of a battery is shown in Figure 8.18, in which Csc is the capacitance of the electrical double layer, W the Warburg impedance for diffusion processes, Rt the internal resistance, and ZA and Zc the impedances of the electrode reactions [124,130],... 

In a simple case, the electrochemical reaction at the electrode-electrolyte interface of one of the electrodes of the battery can be represented by the so-called Randles circuit (Figure 8.19), which is composed of [129] a double layer capacitor formed by the charge separation at the electrodeelectrolyte interface, in parallel to a polarization resistor and the Warburg impedance connected in series with a resistor, which represents the resistance of the electrolyte. 

The Warburg impedance, which is important at low frequencies, is related to the transport of the active species in the electrochemical reaction. The expression for the Warburg impedance in an infinite medium is given by [129,130]... 

If in the equivalent circuit represented in Figure 8.19 the Warburg impedance is included, the whole impedance of the circuit is given by [131,132]... 

Subdivision into a resistance measuring the resistance to charge transfer, Rct, and an impedance that measures the difficulty of mass transport of the electroactive species, called the Warburg impedance, Zw ... 

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. 

For wider pores it becomes possible to detect the influence of diffusive transport within the pores. Under these circumstances, the use of the Warburg impedance (Zw) is required to fit the data, where... 

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 ... 

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