Warburg, Capacitance

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... 

Under this electrochemical configuration, it is commonly accepted that the system can be expressed by the Randles-type equivalent circuit (Fig. 6, inset) [23]. For reactions on the bare Au electrode, mathematical simsulations based on the equivalent circuit satisfactorily reproduced the experimental data. The parameters used for the simulation are as follows solution resistance, = 40 kS2 cm double-layer capacitance, C = 28 /xF cm equivalent resistance of Warburg element, W — R = 1.1 x 10 cm equivalent capacitance of Warburg element, IF—7 =l.lxl0 F cm (

charge-transfer resistance, R = 80 kf2 cm. Note that these equivalent parameters are normalized to the electrode geometrical area. On the other hand, results of the mathematical simulation were unsatisfactory due to the nonideal impedance behavior of the DNA adlayer. This should... 

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). 

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. 

For given values of double layer capacitance solution resistance Rjj and Warburg coefficient a, plots of -Z versus Z have been made for selected values of charge transfer resistance. Ret (26). It is observed that at smaller values of R t ("10 S2 cm ) relaxation due to Rct dl Warburg diffusion behavior are both clearly seen. 

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. 

This materials-specific term is proportional to the inverse of the thermodynamic factor and measures the increase of particle number density with chemical potential (while the electrical capacitance measures the increase of charge with electrical potential). For short times at which the profile near one electrode does not yet perceive the influence of the second one, the result is a 4t -law, and obviously differs from the heuristic approach. Thus more correctly one has to replace Cs by a Warburg-type capacitance as already discussed above (for a more exact description cf. Part I2, Section VI.7). Figure 45 shows a kinetic analysis for YBa2Cu306+r for the short- and the long-time behavior in the time domain yielding identical D5 values. (Note that in these figures different symbols have been used for Lf)... 

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 contrast, Fig. 11.6 shows a typical Nyquist plot for the layer after switching between the oxidised and reduced forms in background electrolyte for several days (Fig. 11.4(c)). A pronounced semicircular region, Warburg 45° line and vertical capacitive region can clearly be seen. We have fitted these data to the transmission line circuit (Fig. 11.1). The value of Cs obtained is found to vary with dc potential (Fig. 11.7) and with the... 

Figure 19 Schematic Bode plots from EIS measurements and equivalent circuits that could be used to fit them for various possible corrosion product deposit structures (A) nonporous deposit (passive film) (B) deposit with minor narrow faults such as grain boundaries or minor fractures (C) deposit with discrete narrow pores (D) deposit with discrete pores wide enough to support a diffusive response (to the a.c. perturbation) within the deposit (E) deposit with partial pore blockage by a hydrated deposit (1) oxide capacitance (2) oxide resistance (3) bulk solution resistance (4) interfacial capacitance (5) polarization resistance (6) pore resistance (7) Warburg impedance (8) capacitance of a hydrated deposit.

The detailed derivation leads to an impedance function, having an equivalent circuit where serially to the adsorption capacitance there are a resistance and a -> Warburg impedance [1]. Important points are as follows ... 

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. 

Z(a)i,ak) = Zre(a)i,ak) +jZim(cai,ak) is the model function, which can be altered using the adjustable parameters the model function can often be presented by an equivalent circuit, involving such elements as resistance, capacitance, and Warburg in series and/or in parallel ... 

Warburg resistance represents the resistance related to mass transfer in an electrochemical process. The resistance is frequency dependent, and consists of both resistance and capacitance. As discussed in Chapter 3, the impedance of the Warburg resistance (Zw (co)) is written as follows... 

The Warburg impedance can be considered as a resistance (Rf=craf 2) connected with a capacitance (C = aof 11) in series. Since the impedance of the... 

Equivalent circuits for the catalyst layer are similar to those for porous electrodes, where charge-transfer resistance, capacitance, and Warburg resistance should be considered. The catalyst layer can be conceived of as a whole uniform unit or as a non-uniform circuit. In the case of a uniform unit, the equivalent circuits are similar to the modified ones discussed in Section 4.2.2 2, and the equations in that section apply. In many cases, such as in the presence of adsorbents, the surface is covered by the adsorbed species. For example, in direct methanol fuel cells and in H2/air fuel cells, CO adsorption should be considered. One example is illustrated in Ciureanu s work [7], as shown in Figure 4.31. 

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. ... 

Figure 8.6. Complete representation of a conductivity cell. Re - ohmic resistance, Cs - double layer capacitance, Rw - Warburg s resistance, Cw - Warburg s capacitance, - resistive component due to the finite rate of electrode reaction, Cq — stray capacitance.

The Warburg s impedance, the resistive and capacitive parts of which are defined according to Randles (1947) by the relations... 

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