Resistance charge-transfer

When the polarization depends only on the charge transfer kinetics, the Butler-Volmer equation is given as (Equation 5.78) 

When the overpotential, r, is very small and the electrochemical system is at equilibrium, the expression for the charge transfer resistance changes into (Equation 5.80) 

From this equation, the exchange current density can be calculated when Re, is known. 

For very small overpotential E Eq a linear dependence of the current density on the potential is observed, usually for an overpotential below 10 mV. From the slope of the linear dependence the exchange current density can be determined. 

The influence of the charge transfer resistance on the current-potential curves is shown in 

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Figure Bl.28.8. Equivalent circuit for a tliree-electrode electrochemical cell. WE, CE and RE represent the working, counter and reference electrodes is the solution resistance, the uncompensated resistance, R the charge-transfer resistance, R the resistance of the reference electrode, the double-layer capacitance and the parasitic loss to tire ground.

F r d ic Current. The double layer is a leaky capacitor because Faradaic current flows around it. This leaky nature can be represented by a voltage-dependent resistance placed in parallel and called the charge-transfer resistance. Basically, the electrochemical reaction at the electrode surface consists of four thermodynamically defined states, two each on either side of a transition state. These are (11) (/) oxidized species beyond the diffuse double layer and n electrons in the electrode and (2) oxidized species within the outer Helmholtz plane and n electrons in the electrode, on one side of the transition state and (J) reduced species within the outer Helmholtz plane and (4) reduced species beyond the diffuse double layer, on the other. 

Figure 13. Schematic presentation of a small segment of polyheteromicrophase SEI (a) and its equivalent circuit (b) A, native oxide film B, LiF or LiCl C, non conducting polymer D, Li2CO, or LiCO, R GB, grain boundary. RA,/ B,RD, ionic resistance of microphase A, B, D. Rc >Rqb charge-transfer resistances at the grain boundary of A to B or A to D, respectively. CA, CB, CD SEI capacitance for each of the particles A to D.

Impedance spectroscopy is best suited for the measurement of electronic conductivities in the range 10 -7to 10 2S cm 1.145 In principle, it is perhaps the best method for this range, but it is often difficult to interpret impedance data for conducting polymer films. The charge-transfer resistance can make measurements of bulk film resistances inaccurate,145 and it is often difficult to distinguish between the film s ionic and electronic resistances.144 This is even more of a problem with chronoamperometry146 and chronopotentiometry,147 so that these methods are best avoided. 

Figure 15 shows a set of complex plane impedance plots for polypyr-rolein NaC104(aq).170 These data sets are all relatively simple because the electronic resistance of the film and the charge-transfer resistance are both negligible relative to the uncompensated solution resistance (Rs) and the film s ionic resistance (Rj). They can be approximated quite well by the transmission line circuit shown in Fig. 16, which can represent a variety of physical/chemical/morphological cases from redox polymers171 to porous electrodes.172... 

From an analysis of data for polypyrrole, Albery and Mount concluded that the high-frequency semicircle was indeed due to the electron-transfer resistance.203 We have confirmed this using a polystyrene sulfonate-doped polypyrrole with known ion and electron-transport resistances.145 The charge-transfer resistance was found to decrease exponentially with increasing potential, in parallel with the decreasing electronic resistance. The slope of 60 mV/decade indicates a Nemstian response at low doping levels. 

By comparing impedance results for polypyrrole in electrolyte-polymer-electrolyte and electrode-polymer-electrolyte systems, Des-louis et alm have shown that the charge-transfer resistance in the latter case can contain contributions from both interfaces. Charge-transfer resistances at the polymer/electrode interface were about five times higher than those at the polymer/solution interface. Thus the assignments made by Albery and Mount,203 and by Ren and Pickup145 are supported, with the caveat that only the primary source of the high-frequency semicircle was identified. Contributions from the polymer/solution interface, and possibly from the bulk, are probably responsible for the deviations from the theoretical expressions/45... 

Analysis of Zje and Z- as a function of the frequency of potential modulation (Randles plots) provides the phenomenological ET rate constant [63,74]. It should be noted that the extrapolation of Z at high frequency gives effectively the sum 7 ct + where 7 ct is the charge transfer resistance. 

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 (

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... 

Here, i ct.hare and i ct.oNA indicate the charge transfer resistance for the redox couple on the bare and the DNA-modilied Au electrodes, respectively and 6 was calculated to be 80%. 

The appearance of a semicircle at high frequency after cycling (Figure 3b) suggests a parallel existence of a charge transfer resistance, which is related with the already mentioned pseudofaradaic processes, i.e. with the formation of the isolating phase. 

The partial derivative on the right-hand side represents, in essence, the electrochemical rate constant the larger the change in current with potential the easier charge transfer is and we can define RCT, the charge-transfer resistance, as 1 /(dJ/dE)CoCfl. Given that (1 /i)112 = (1 — i)/J2 and defining the transport part of (2.106) ... 

Evidently, the impedance of the interface consists of two components a charge-transfer resistance Rex, which will depend on the electrochemical rate constants, and a more unusual element arising from the diffusion of the redox couple components to and from the interface. The magnitude of this element... 

Radhakrishnan R, Virkar AV, and Singhal SC. Estimation of charge-transfer resistivity of Pt cathode on YSZ electrolyte using patterned electrodes. J. Electrochem. Soc. 2005 152 A927-A936. 

The capacitance CH and the charge transfer resistance Rct representing the Helmholtz layer. 

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. 

Fig. 10.12 Expected complex plane impedance diagram for an electrolyte with one mobile species which is contacted by two non-blocking metal electrodes, e.g. Ag/Ag Rblj/Ag. is the bulk resistance of the electrolyte and R is the charge transfer resistance for the Ag/Ag Rbls interface.

Fig. 10.13 Impedance plane diagrams for metal non-blocking electrodes with two mobile species in the electrolyte, (a) Interfacial impedance is only a Warburg impedance. (b) Interfacial impedance shows a charge transfer resistance semicircle.

There will always be a charge transfer resistance (R i) associated with the ion exchange across the interface. Where there are very small Debye lengths in each phase (compared with the size of an ion) the exchange current I o can be evaluated from the relationship... 

Figure 3.13 (a) Values of charge-transfer resistance of different systems based on carbon, using the redox probe Fe(CN)6 . (b) Nyquist plot of different carbon nanotube composites in the presence of the redox couple, (c) Table with the electron-transfer rate constants calculated from cyclic voltammet data by using Nicholson method. Adapted with permission from Ref [103]. Copyright, 2008, Elsevier. 

Figure 10. Resistor-network representation of porous-electrode theory. The total current density, i, flows through the electrolyte phase (2) and the solid phase (1) at each respective end. Between, the current is apportioned on the basis of the resistances in each phase and the charge-transfer resistances. The charge-transfer resistances can be nonlinear because they are based on kinetic expressions.

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