Coupled Diffusion Impedance

The (ice)-1/2 frequency dependence appears also in the impedance function of unsupported systems, that is, when the transport of the electroactive species proceeds via coupled diffusion and migration. 

Solution The issue is that the condition at the interface between the two diffusion layers is not treated correctly when diffusion impedances, such as presented in equation (11.70), are added. The solutions to the two sets of equations are coupled by the continuity of concentration at the interface, as seen in Figure 11.1. 

When the a.c. signal is first impressed, a time-dependent diffusion layer is created. As no net current flows, a steady state is set up after a few cycles. This coupled diffusion and electrochemical problem has been solved with the following result [8]. Assuming that the impedance can be expressed as a series combination of a resistor and a capacitor ... 

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 ... 

FIGURE 5-12 Transmission-line representation of diffusion impedance A. simple diffusion with reflecting boundary B. reflecting boundary diffusion coupled with homogeneous reaction C. simple diffusion with absorbing boundary D. absorbing boundary diffusion coupled with homogeneous reaction... 

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... 

When the characteristic time for charge diffusion is lower than the experiment timescale, not all the redox sites in the film can be oxidized/reduced. From experiments performed under these conditions, an apparent diffusion coefficient for charge propagation, 13app> can be obtained. In early work choroamperometry and chronocoulometry were used to measure D pp for both electrostatically [131,225] and covalently bound ]132,133] redox couples. Laviron showed that similar information can be obtained from cyclic voltammetry experiments by recording the peak potential and current as a function of the potential scan rate [134, 135]. Electrochemical impedance spectroscopy (EIS) has also been employed to probe charge transport in polymer and polyelectrolyte-modified electrodes [71, 73,131,136-138]. The methods... 

While eqn. (211) is a bit complex, the similarity to the more familiar diffusion equation [e.g. eqns. (43), (44), (158) or (197)] is apparent. The diffusion coefficient has to be replaced by a position-dependent tensor which couples (or connects) the motion of one of the particles, e.g. the particle k, with that of the other particles, e.g. a particle j. When these particles are a long way apart, the solvent between them can be squeezed out easily. As the particles approach, this is no longer true because the two particles block certain directions for escape of the solvent as the particles approach. Increasingly, the solvent has to be squeezed out of the way in a direction perpendicular to that of the approach of the particles and this causes the solvent to impede the particle approach more and more effectively. When j = k, the effect on the same particle of its own motion is negligible. Hence, the diffusion coefficient tensor elements, Tjj = kBT/%, are the same as the diffusion coefficient for the particle... 

EHD impedances have been measured on the diffusion plateau at 0.7 V/SCE. The mass transport time constant of the redox couple in solution, which is one of the terms implied in the impedance expression is independent of the interface nature. The Schmidt number Sc was first determined on a bare electrode, and this value of 1540 is further used as a fixed parameter in the analysis of the diagrams obtained on pECBZ films at different Q (Fig. 6-14). The different diagrams are analyzed in the light of the theoretical model predicted by expression (6-34). 

The response to the applied perturbation, which is generally sinusoidal, can differ in phase and amplitude from the applied signal. Measurement of the phase difference and the amplitude (i.e. the impedance) permits analysis of the electrode process in relation to contributions from diffusion, kinetics, double layer, coupled homogeneous reactions, etc. There are important applications in studies of corrosion, membranes, ionic solids, solid electrolytes, conducting polymers, and liquid/liquid interfaces. 

The impedance of a more complex process involving coupling between adsorption and diffusion was studied by Armstrong and co-... 

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