Warburg diffusion resistance

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. 

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. 

Determination of Parameters from Randles Circuit. Electrochemical three-electrode impedance spectra taken on electrochromic materials can very often be fitted to the Randles equivalent circuit (Randles [1947]) displayed in Figure 4.3.17. In this circuit R /denotes the high frequency resistance of the electrolyte, Ra is the charge-transfer resistance associated with the ion injection from the electrolyte into the electrochromic film and Zt, is a Warburg diffusion impedance of either semi-infinite, or finite-length type (Ho et al. [1980]). The CPEdi is a constant phase element describing the distributed capacitance of the electrochemical double layer between the electrolyte and the film having an impedance that can be expressed as... 

The impedance of CPE is described according to eq. 3 (Zoltowski 1998). When a is close to 0, the CPE describes a resistance, close to -1 it describes an inductance, close to 1 it describes a capacity and finally, for the value of 0.5, the result is equivalent to the Warburg diffusion impedance. 

The electrode layers formed using die physical loading method are usually relatively thicker (more than 10 pm in thickness), and the composite layers are composed of nanoparticles of the electrode material and the ionic polymer. These layers are both electronically and ionically conductive. The impedance for such electrodes is assumed to be similar to diat of porous electrodes. Levie (1963, 1964) was the first to develop a transmission line circuit (TLC) model of the porous electrode consisting of the electrolyte resistance and the double-layer capacitance. Subsequently, a number of authors proposed modified TLC models for the impedance of porous electrodes on the basis of Levie s model. Bisquert (2000) reviewed the various impedance models for porous electrodes. The composite electrode layers prepared by the physical loading method could be successfully represented by the impedance model for porous electrodes, as shown in Fig. 6d this model is composed of the double-layer capacitance, Cj, the Warburg diffusion capacitance, W and the electrolyte resistance, 7 (Liu et al. 2012 Cha and Porfiri 2013). 

Diffusion resistance Zp,yy to current flow carried by electroactive species can create impedance, frequently known as the Warburg element [23, p. 376]. If the diffusion layer Lp is assumed to have an unlimited thickness within the experimental AC frequency range, than a "semi-infinite" diffusion may become the rate-determining step in the Faradaic kinetic process. In the "semiinfinite" diffusion model the diffusion layer thickness Lp is assumed to be always much smaller than the total thickness of the sample d (Lp d. The equation for the "semi-infinite" Warburg impedance Z m) is a function of concentration-driven potential gradient dV/rfC. The "semi-infinite" diffusion limitation is modeled by characteristic resistance and a Warburg infinite diffusion component Z that can be derived [8] as ... 

Diffusion and charge-transfer kinetics are usually coupled. A typical electrochemical (or "Faradaic") reaction is composed of both mass-transport processes of charged species to the electrode surface and their redox discharge at the interface. The Faradaic impedance can be represented by a series combination of Warburg diffusion impedance and charge-transfer resistance ... 

In this expression the parameter R, Q( ) essentially represents a ratio of diffusion and capacitive dispersion contributions to the overall impedance process. This type of process can be represented by a parallel combination of a CPE and diffusion resistance R. At high frequencies a familiar -45° semiinfinite-diffusion Warburg impedance line is observed as a function of ... 

Figure 11-lOA [75, 76] shows the Nyquist spectra at +1.00 V at TiOj electrodes in solution containing electroactive [Fe(CN)ions (pH 4.7) with and without the addition of HSA into the bulk solution. In the absence of the protein, the equivalent circuit used to fit the experimental data is reduced to a typical Randles circuit containing Warburg diffusion and [Fe(CN) ] charge-transfer resistance elements in parallel with a double-layer capacitance (Figure 11-lOA). In the presence of a full protein layer, additional processes related... 

Figure 10.6 shows that the overall impedance of the system decreases after addition of plasticizer. The data are in agreement with the increase observed in ionic conductivity. From the parameters obtained by fitting the experimental data shown in Fig. 10.6, the apparent diffusion coefficient can be estimated using equation 10.7,where 4 is the thickness of the electrolyte film and 5 is a parameter related to the element O in the equivalent circuit proposed, which accounts for a finite-length Warburg diffusion (Zd), which represents a kind of resistance to mass transfer. 

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... 

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). 

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. 

The same consideration applies to the impedance measurement according to Fig. 8.1b. It is a normal electrochemical interface to which the Warburg element (Zw) has been added. This element corresponds to resistance due to translational motion (i.e., diffusion) of mobile oxidized and reduced species in the depletion layer due to the periodically changing excitation signal. This refinement of the charge-transfer resistance (see (5.23), Chapter 5) is linked to the electrochemical reaction which adds a characteristic line at 45° to the Nyquist plot at low frequencies (Fig. 8.2)... 

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],... 

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.

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