Nyquist resistance

An important early paper on fluctuation processes is that of Harry Nyquist (1928), who suggested an equation linking the mean-square amplitude of thermal noise in an electrical circuit to the resistance R of the noise EMF (or current) generator ... 

The resulting dependence of Z" on Z (Nyquist diagram) is involved but for values of Rp that are not too small it has the form of a semicircle with diameter Rp which continues as a straight line with a slope of unity at lower frequencies (higher values of Z and Z"). Analysis of the impedance diagram then yields the polarization resistance (and thus also the exchange current), the differential capacity of the electrode and the resistance of the electrolyte. 

Application of the F-D theorem produced [122] several significant results. Apart from the Nyquist formula these include the correct formulation of Brownian motion, electric dipole and acoustic radiation resistance, and a rationalization of spontaneous transition probabilities for an isolated excited atom. 

Figure 8.11 Nyquist plots for circuits comprising more than one electrical component (a) pure resistive impedances in series (b) pure resistive impedances in parallel (c) an RC element (d) an RC element in series with a resistance.

Region (i). First, we note that there is a slight offset near the origin of the Nyquist plot, as caused by the resistances of the ITO layers, leads and contacts. We will call this an ohmic resistance because we do not know any further details about the features in the gap. Note that it is merely a gap because the frequencies... 

In fact, many semicircular features in Nyquist plots represent layers that have both resistive and capacitive components, i.e. they behave as RC elements. 

The movement of the electron is activated, i.e. it requires energy. Stated another way, movement is restricted - there is a resistance. We call this the resistance to charge transfer, Rcr. The latter parameter and C i behave as though being in parallel, and hence the appearance of the semicircular arc in the Nyquist plot. 

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. 

To better understand the diffusion-limited school of thought mentioned above, it is worth digressing momentarily on another noble -metal electrode system silver on YSZ. Kleitz and co-workers conducted a series of studies of silver point-contact microelectrodes, made by solidifying small (200—2000 //m) silver droplets onto polished YSZ surfaces. Following in-situ fabrication, the impedance of these silver microelectrodes was measured as a function of T (600-800 °C), P02 (0.01-1.0 atm), and droplet radius. As an example. Figure 9a shows a Nyquist plot of the impedance under one set of conditions, which the authors resolve into two primary components, the largest (most resistive) occurring at very low frequency (0.01—0.1 Hz) and the second smaller component at moderately low frequency ( 10 Hz). 

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)... 

Equations 2.37-2.40 result in the commonly used presentation of the impedance, e.g. the Nyquist and the Bode plots. The first one shows the total impedance vector point for different values of co. The plane of this figure is a complex plane, as shown in the previous section. Electrochemical-related processes and effects result in resistive and capacitive behaviour, so it is common to present the impedance as ... 

Figure 2.7 shows a Nyquist plot corresponding to the electrical equivalent circuit of Fig. 2.6. The slope of the impedance can be explained by a circuit, consisting of different resistive and capacitive components37. The...

In this section, the behaviour of the textile electrodes when used for a longer period in the electrochemical cell is investigated. It is expected that this behaviour can change as a function of time because of uptake of electrolyte solution by the textile electrodes and possible corrosion reactions that can occur. Additionally in this case, the data and results obtained for the textile electrodes will be compared with those obtained for palladium electrodes. Bode and Nyquist plots are recorded for the four types of electrodes and the electrolyte resistance was measured as a function of time for electrolyte concentrations of 1 xlCT1,1 xlO 2,1 xlO 3 and 1 xl(T4moll The values for A and d are 180 mm2 and 103 mm, respectively. For all these concentrations, the resistances are summarised in Tables9.9-9.12. 

Figure 13. (a) Nyquist plots of the impedance spectra and (b) plots of reduced capacitance Cred vs. frequency CO experimentally measured on carbon specimens A (-o-), B (- -), and C (-A-) at an applied potential of 0.2 V (vs. SCE) in a 30 wt.% H2S04 solution. Here, the solution resistance was subtracted from the measured impedance spectra. The reduced capacitance in (b) was determined from the normalization of the capacitance with respect to the value of the capacitance calculated from the impedance spectra at 10 Hz. Reprinted from Ref. 17, Copyright (2006), with permission from Elsevier. 

When the rate determining step of the electrode reaction is the charge transfer process (kinetic control), the faradic impedance ZF in Figure 1.18 can be described as RCJ, the charge transfer resistance [7,8], The impedance plot in the Nyquist plane describes a semicircle, as shown in Figure 1.19. 

We will consider here the case where the mass transfer limitation is due to the diffusion of the species. When the mass transfer becomes predominant in the low-frequency range, experimental plots obtained in the Nyquist plane shift from the semicircular shape. In that case, indeed, the impedance ZFcan no longer be described as only a charge transfer resistance, but as a combination of Rcv with the impedance of diffusion ZD. ZD changes with the frequency it takes into account the relaxation processes inside the diffusion layer. Different cases can be described depending on the diffusion layer thickness. 

A basic electric equivalent circuit to describe an EDLC is presented in Figures 1.22a and b, which shows the Nyquist (Figure 1.22b) plot of an ideal capacitor C, in series with a resistance... 

Rs (Figure 1.22a). The double layer capacitance is represented by the capacitance C, and Rs is the series resistance of the EDLC, also named the equivalent series resistance (ESR). This series resistance shows the nonideal behavior of the system. This resistance is the sum of various ohmic contributions that can be found in the system, such as the electrolyte resistance (ionic contribution), the contact resistance (between the carbon particles, at the current collector/carbon film interface), and the intrinsic resistance of the components (current collectors and carbon). Since the resistivity of the current collectors is low when A1 foils or grids are used, it is generally admitted that the main important contribution to the ESR is the electrolyte resistance (in the bulk and in the porosity of the electrode) and to a smaller extent the current collector/active film contact impedance [25,26], The Nyquist plot related to this simple RC circuit presented in Figure 1.22b shows a vertical line parallel to the imaginary axis. 

So far, the ionic conductivity of most ILs has been measured by the complex impedance method [116], In this method, charge transfer between carrier ions and electrode is not necessary. Therefore platinum and stainless steel are frequently used as blocking electrodes. However, it is often difficult to distinguish the resistance and dielectric properties from Nyquist plots obtained by the impedance measurement. In order to clarify this, additional measurements using non-blocking electrodes or DC polarization measurement are needed. 

We now analyze the Nyquist plots corresponding to some circuits. In this regard, for a parallel network involving a resistance, Rp, and a capacitance, (the impedance is given by... 

It is only a small step from here to the equivalent statement, known as the Nyquist theorem, for current noise of thermal fluctuations in a circuit with resistance R.4... 

Some typical Nyquist plots for an electrochemical system are shown in Figure 2.38. The usual result is a semicircle, with the high-frequency part giving the solution resistance (for a fuel cell, mainly the membrane resistance) and the width of the semicircle giving the charge-transfer resistance. 

The simplest and most common model of an electrochemical interface is a Randles circuit. The equivalent circuit and Nyquist and Bode plots for a Randles cell are all shown in Figure 2.39. The circuit includes an electrolyte resistance (sometimes solution resistance), a double-layer capacitance, and a charge-transfer resistance. As seen in Figure 2.39a, Rct is the charge-transfer resistance of the electrode process, Cdl is the capacitance of the double layer, and Rd is the resistance of the electrolyte. The double-layer capacitance is in parallel with the charge-transfer resistance. 

The Nyquist plot of a Randles cell is always a semicircle. At high frequencies the impedance of Cdl is very low, so the measured impedance tends to Reh At very low frequencies the impedance of Cm becomes extremely high, and thus, the measured impedance tends to Rct + Rd. Accordingly, at intermediate frequencies, the impedance falls between Rd and Rct + Rd. Therefore, the high-frequency intercept is associated with the electrolyte resistance, while the low-frequency intercept corresponds to the sum of the charge-transfer resistance and the electrolyte resistance. The diameter of the semicircle is equal to the charge-transfer resistance. 

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