Warburg impedance coefficient

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. 

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

The diffusion coefficient of lithium ions in the intercalation electrodes also can be measured, using EIS, by the analysis of Warburg impedance representing diffusion through the electrode with I.C. in Equation (5.28), impermeable B.C. in Equation (5.29), and sinusoidal oscillation B.C. 

The analysis above has shown that it is possible to derive kinetic parameters from the impedance spectra of a redox system. The charge-transfer resistance is directly related to the exchange current (Eq. 27) and a medium diffusion coefficient for oxidized and reduced species can be calculated from the coefficient a of the Warburg impedance (Eq. 33). 

At low frequencies, a diffusion-limited region appears, which, if fitted to a Warburg impedance, can be used to determine diffusion coefficients of charge carriers in the conducting polymer. At... 

The terms in o /Vm correspond to the normal Warburg impedance they do not contain the heterogeneous rate constants. The more general treatments, however, indicate that the Warburg impedance does in general contain coefficients that depend on the rate constants and their potential dependence. It is only on the basis of absolute rate theory that these coefficients cancel out of the final expression. 

If an analyte affects one or more of these equivalent circuit parameters and these parameters are not affected by interfering species, then impedance methods can be used for analyte detection. arises primarily from the electrolyte resistance and is analytically useful mainly in conductivity sensors. The Warburg impedance, which can be used to measure effective diffusion coefficients, is seldom useful for analytical applications. The equivalent circuit elements in Figure 14.1 that are most often useful for analyte detection are and Qi. 

Dq being the diffusion coefficient of the oxidized species and Dg that of the reduced species. Treatment of this system assumes that we can write the equivalent circuit of kinetic and diffusion control as shown in Fig. 2.32, where the diffusion component of the impedance is given by the Warburg impedance W. It should also be noted that the derivation applies to a planar electrode only. Electrodes with more complex geometries such as porous electrodes require a transmission-line analysis. 

Fig. 1 The equivalent circuit of the interfacial impedance in the present of adstxption, wherea , is the coefficient of Warburg impedance,, defined as Z(Wai)= and Cm, Cad, and Rad, stand for double layer and...

An extrapolation of the -45° straight line to high frequencies (co ) representing the Warburg impedance in the complex plane intersects the real axis at the value allowing us to calculate diffusion coefficients and from known rate constants and (Figure 5-lOA) ... 

In systems where diffusion phenomena are of significance, the mechanistic study is facilitated by using the general expression for Impedance Z (26). This equation shows for instance how the Warburg coefficient can be evaluated by conducting impedance studies at very low frequencies. These coefficients in turn enable the evaluation of diffusion coefficients for the diffusing species. 

Cadmium atomic layer electrodeposition above reversible Cd2+/Cd potential (underpotential deposition, upd) on bulk tellurium and Te atomic layer predeposited on gold has been characterised with potentiodynamic electrochemical impedance spectroscopy (PDEIS) by variations, with the electrode potential E, of double layer pseudocapacitance Q,u, charge transfer resistance Rrt and Warburg coefficient Aw of diffusion impedance. 

Ret is the charge transfer resistance while cra / (l — i) is the Warburg, or mass transfer, impedance. Eqs.5 and 6 can now be used to study the electrode reaction in detail. For example, a cot (f> versus dc potential plot will aid in finding the charge transfer coefficient, while a cot (j> versus plot serves to find the heterogeneous rate constant. Details on these examples can be found in the literature. ... 

It is evident that the shape of the impedance spectra varies with the potential since the values of the charge transfer resistance (Ret), the low frequency (redox) capacitance (Cl) and the Warburg coefficient change with the potential more exactly, they depend on the redox state of the polymer. In many cases D is also potential-dependent. The double-layer capacitance (Cdi) usually shows only slight changes with potential. The ohmic resistance (Rq) is the sum of the solution resistance and the film resistance, and the latter may also be a function of potential due to the potential-dependent electron conductivity, the sorption of ions, and the swelling of the film. In Fig. 3.9 three spectra are displayed, which were constructed using the data obtained for a PTCNQ electrode at three different potentials near its equilibrium potential [23]. 

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