Diffusion impedance response

Diffusion through a stagnant layer of finite thickness can also yield a uniformly accessible electrode. The diffusion impedance response of a coated (or film-covered) electrode, imder the condition that the resistance of the coating to diffusion is much larger than that of the bulk electrol5M e, is approximated by the diffusion impedance of file coating. This problem is also analyzed in Section 15.4.2. 

Figure 31.33 indicates an unstable electrochemical system for D,e/dp/s in NaCl that tends to show a diffusion impedance response at initial immersion. And this system shows a gradual depressed semicircle, implying low surface coverage on the exposed steel surface. 

A finite-length diffusion impedance of charged particles is represented by Zq parameter. The resulting "finite length" diffusion-impedance response does not have the -45° line, instead displaying a depressed semicircle or a vertical -90° line. The circuit is representeid by a parallel combination of a CPE and an ideal resistor which also strongly depends on the electrochemical potential. The universal expression for finite diffusion impedance (/to) was... 

Further information on this subject can be obtained by frequency response analysis and this technique has proved to be very valuable for studying the kinetics of polymer electrodes. Initially, it has been shown that the overall impedance response of polymer electrodes generally resembles that of intercalation electrodes, such as TiS2 and WO3 (Ho, Raistrick and Huggins, 1980 Naoi, Ueyama, Osaka and Smyrl, 1990). On the other hand this was to be expected since polymer and intercalation electrodes both undergo somewhat similar electrochemical redox reactions, which include the diffusion of ions in the bulk of the host structures. One aspect of this conclusion is that the impedance response of polymer electrodes may be interpreted on the basis of electrical circuits which are representative of the intercalation electrodes, such as the Randles circuit illustrated in Fig. 9.13. The figure also illustrates the idealised response of this circuit in the complex impedance jZ"-Z ) plane. 

As expected, the impedance responses obtained in practice do not fully match that of the model of Fig. 9.13. However, as shown by the typical case of Fig. 9.14 which illustrates the response obtained for a 5 mol% ClO -doped polypyrrole electrode in contact with a LiC104-propylene carbonate solution (Panero et al, 1989), the trend is still reasonably close enough to the idealised one to allow (possibly with the help of fitting programmes) the determination of the relevant kinetics parameters, such as the charge transfer resistance, the double-layer capacitance and the diffusion coefficient. 

It must be expected that a polymer material having a much lower conductivity than polyaniline will give impedance responses revealing the effect of the three time constants obtained in the model. The system investigated was chosen for this reason since pECBZ conductivity and redox capacity [94] correspond to DE = 10 7cm2,s 1 and therefore, for the same layer thickness (500nm), the diffusion time constant would be 0.025 s. Electron diffusion should therefore be detectable in the a.c. and even in the EHD frequency domain. 

Alternatively, an equally powerful visualization of impedance data involves Bode analysis. In this case, the magnitude of the impedance and the phase shift are plotted separately as functions of the frequency of the perturbation. This approach was developed to analyze electric circuits in terms of critical resistive and capacitive elements. A similar approach is taken in impedance spectroscopy, and impedance responses of materials are interpreted in terms of equivalent electric circuits. The individual components of the equivalent circuit are further interpreted in terms of phemonenological responses such as ionic conductivity, dielectric behavior, relaxation times, mobility, and diffusion. 

The classic problem of diffusion in an infinite medium can be solved by use of a similarity transformation. The associated impedance response is discussed in Section 11.3. A general method for finding the time-dependent concentration profile is presented here in the form of an example. 

The uncertainty associated with the interpretation of the impedance response can be reduced by using an electrode for which the current and potential distribution is uniform. There are two types of distributions that can be used to guide electrode design. As described in Section 5.6.1, the primary distribution accounts for the influence of Ohmic resistance and mass-transfer-limited distributions account for the role of convective diffusion. The secondary distributions account for the role of kinetic resistance which tends to reduce the nonuniformity seen for a primary distribution. Thus, if the primary distribution is uniform, the secondary... 

The product Mg " " diffuses through a porous layer o/Mg (OH) 2 that has a thickness of S. Find the impedance response for this reaction sequence. 

Example 11.1 Diffusion with First-Order Reaction Develop an expression for the impedance response for the reduction of ferricyanide at potentials sufficiently cathodic to allow the anodic reaction to be ignored, yet sufficiently anodic to avoid reduction of oxygen (as a side reaction). Under these conditions, the reaction... 

Develop an expression for the impedance response, taking into account diffusion of the reactant Zn(OH) J toward the electrode and of the product OH" away from the electrode. 

Equation (11.70) can be considered to be a finite-length diffusion impedance. As tanh(oo) = 1, the impedance response asymptotically approaches the response for an infinite domain at high frequencies, i.e.. 

Equation (11.71) is the Warburg impedance. The impedance response for a stagnant layer is often imposed incorrectly for situations where convective diffusion takes place. The more correct approach is to accovmt explicitly for the role of convection. 

The quantitative and qualitative analysis presented in Section 20.2.1 demonstrates that the finite-diffusion-layer model provides an inadequate representation for the impedance response associated with a rotating disk electrode. The presentation in Section 20.2.2 demonstrates that a generic measurement model, while not providing a physical interpretation of the disk system, can provide an adequate representation of the data. Thus, an improved mathematical model can be developed. 

The three-term convective-diffusion model provides the most accurate solution to the one-dimensional convective-diffusion equation for a rotating disk electrode. The one-dimensional convective-diffusion equation applies strictly, however, to the mass-transfer-limited plateau where the concentration of the mass-transfer-limiting species at the surface can be assumed to be both uniform and equal to zero. As described elsewhere, the concentration of reacting species is not uniform along the disk surface for currents below the mass-transfer-limited current, and the resulting nonuniform convective transport to the disk influences the impedance response. ... 

At first glance, it may not be obvious that such an approach should work. It is well known, for example, that the impedance spectrum associated with an electrochemical reaction limited by the rate of diffusion through a stagnant layer (either the Warburg or the finite-layer diffusion impedance) can be approximated by an infinite number of RC circuits in series (the Voigt model). In theory, then, a measurement model based on the Voigt circuit should require an infinite number of parameters to adequately describe the impedance response of any electrochemical system influenced by mass transfer. 

Nernst applied the electrical bridge invented by Wheatstone to the measurement of the dielectric constants for aqueous electrolytes and different organic fluids. Nemst s approach was soon employed by others for measurement of dielectric properties and the resistance of galvanic cells. Finkelstein applied the technique to the analysis of the dielectric response of oxides. Warburg developed expressions for the impedance response associated with the laws of diffusion, developed almost 50 years earlier by Fick, and introduced the electrical circuit analogue for electrolytic systems in which the capacitance and resistance were functions of frequency. The concept of diffusion impedance was applied by Kruger to the capacitive response of mercury electrodes. ... 

Several approximations for the calculation of the effect of 02-transport limitations on the complex impedance response of the electrode have been studied [144]. In one of these approximations, only oxygen diffusion limitations were considered, whereas proton transport limitations were neglected (equipotential electrolyte phase), cf. Sect. 8.2.3.4.4. The stationary solution for this case provides the definition of a gas-diffusion-limited reaction-penetration depth,... 

In the case where only oxygen diffusion transport limitations are considered and proton transport limitations are neglected, the linear impedance response prescribes a perfect semicircle in the complex plane without showing a linear branch in the high-frequency limit. The response given by Eq. (110) is, thus, an exclusive feature of proton transport limitations, which, thereby, provides a feasible tool for their characterization. 

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. 

Euclidean geometry fails to describe disordered surfaces such as real solid surfaces. Fractal geometry, which has been developed to overcome this obstacle, covers surface, mass, and pore fractality. It has been pointed out that the diffusion process can be used to characterize the fractal dimension of a rough surface. The impedance response of a rough electrode could be used for the characterization of the roughness and. 

Sapoval, B, J,-N, Chazalviel, and J. Peyridre, Electrical response of fractal and porous interfaces. Physical Review A, 1988. 38(11) pp. 5867-5887 Reiser, H, K.D, Beccu, and M.A. Gutjahr, Electrochimica Acta, 1976, 21 p. 539 Diard, J.R, B, Le Gorrec, and C. Montella, Linear diffusion impedance. General expression and applications. Journal of Electroanalytical Chemistry, 1999. 471 pp, 126-131 Deslouis, C, C, GabrieUi, M. Keddam, A, Khalil, R. Rosset, B. TriboUet, and M. Zidoune, Impedance techniques at partially blocked electrodes by scale deposition. Electrochimica Acta, 1997, 42(8) pp, 1219-1233... 

The deviations of the impedance responses [23,28, 30,32, 59,64,66,69,71, 76,120,123,132,144-146] predicted by the theories have been explained by taking into account different effects, such as interactions between redox sites [30, 136], ionic relaxation processes [95], distributions of diffusion coefficients [28], migration [65, 118, 125, 132], film swelling [64, 137], slow reactions with solution species [22,138], nonuniform film thickness [23], inhomogeneous oxida-tion/reduction processes [123], etc. 

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