Diffusion impedance Frequency dependence

Thus, the diffusion impedance expressions depend on the electrode separations d at low frequencies. One way to identify the finite Warburg impedance is to use measurements at various values of the electrodes separation d. When LpCO / D 3 (at oo °o), the tank term approaches imity, the diffusion length is negligible compared to the whole region avaUable for diffusion d, and Zpjj-j-approaches infinite length Warburg Zy ... 

Other more complicated forms of impedance-frequency dependencies can be revealed when a combination of diffusion process and homogeneous bulk solution reaction or recombination of electroactive species is considered [34]. In the case of homogeneous bulk solution and electrochemical reaction, the corresponding charge-transfer impedance Z can be simplified by a kinetic resistance R. Another parameter, —critical frequency of reaction or recom-... 

It should be noted that the presence of diffusion controlled corrosion processes does not invalidate the EIS method but does require extra precaution. In the case of a finite diffusional impedance added in series with the usual charge transfer parallel resistance shown in Fig. 3b, the frequency-dependent diffusional impedance can be described as (21)... 

In general, the frequency dependence of the dif-fusional impedance and the geometry of diffusion are correlated. The (ice)-1/2 frequency dependence corresponds to the semi-infinite planar diffusion such a frequency dependence is valid only if the characteristic length, (D/ce)1/2 of diffusion is much shorter than any size of the electrode or the thickness of the electrolyte layer from which diffusion proceeds. Otherwise spherical, or bounded diffusion with different frequency dependence is observed. 

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. 

However, real electrochemical systems exhibit much more complex behaviours. They are not simply resistive. The electrochemical double layer adds a capacitive term. Other electrode processes, such as diffusion, are time and/or frequency dependent. Therefore, for an actual electrochemical system, impedance is used instead of resistance. The impedance of an electrochemical system (defined as Ziot)) is the AC response of the system being studied to the application of an AC signal (e.g., sinusoidal wave) imposed upon the system. The form of the current-voltage relationship of the impedance in an electrochemical system can also be expressed as... 

From Equations 3.92 and 3.93, we can see that the frequency dependence at low frequencies comes from Warburg impedance terms hence, the linear correlation of Zm versus Zre features a diffusion-controlled electrode process. As the frequency rises, the presupposition for Equation 3.94 will not exist. The linear correlation will deviate with increasing frequencies. 

For properly describing electrochemical processes, additional impedance elements have been introduced. The Warburg impedance (Raistrick and Huggins, 1982 Honders and Broers, 1985) is representative of diffusive constraints, being defined, for the case of linear diffusion, as a frequency-dependent impedance given by ... 

Randles model are used to describe the frequency dependence of diffusion and the capacitive impedance observed in the intermediate and low frequency ranges. A dual transmission line model has been proposed by including ionic and electronic resistance rails connected in parallel with a capacitance Cp (Fig. 6.5). The model has been used to define the electrochemical behavior of polyaniline, and the capacitance was explained as a result of oxidation and reduction of the pol5mier. Ionic (i i) and electronic (Rg) resistances are used to describe hindered motion of ions and electrons in the system, respectively. The impedance behavior has been found to be dependent on the ratio of the two resistances. 

Another method for determining chemical diffusion coefficients is to use the frequency dependence of the cell impedance, which is obtained by ac measurements. This will not be treated here. 

The function describing the frequency dependence of such an impedance element is equivalent to that of diffusion in Eq. (28), bnt the meaning of the parameters must be redefined as Rd = Rm, Rp = Ret, Cd = Cdi, and Cp = Cad- A complete equivalent circuit of the lead acid battery will involve serially connected elements described in previous paragraphs for the electrolyte and anode and cathode, with the addition of series resistance and inductance. 

It can be seen that changes in the charge transfer resistance (high-frequency depressed semicircle) and in the chemical diffusion coefficients defining the low frequency dependence involve Li intercalation phase changes. The resistance Ra and diffusion hindrance increase in areas where change from one phase to the next occurs. The value of impedance spectra for correlation with state of charge is problematic for the Li-ion battery because impedance increases and decreases in the... 

Historically, it should be noted that double-layer capacitance measurements were first reliably made (at Hg) by Bowden and Rideal [1928] using the dc charging-current method and by Proskumin and Frumkin [1935] by means of ac modulation. Randles [1947,1952] pioneered the examination of impedance of an electrode process (e.g. redox reactions), using phase-sensitive detector instrumentation to record the frequency dependence of the separated real and imaginary components of Z. Under diffusion-control, the dependence of Z" and Z were found due to the Warburg impedance element. 

Here, Z is the absolute magnitude of the impedance. What allows us to separate different electrochemical processes using EIS is the fact that capacitors have a frequency-dependent ability to be charged or discharged - a property called capacitive reactance. As a consequence, different capacitances within the system will respond at different frequencies, allowing us to separate temporally the electrochemical processes in EIS experiments on the basis of their capacitive reactance. Similarly, diffusion limitations usually also have a characteristic low frequency where they predominantly govern the system response and thus are easily distinguishable. 

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