Phase element

Dispersion of the measured complex dielectric constant is known from dielectric relaxation experiments. The complex dielectric constant e may be represented as 

Cole and Cole described the observed distribution of relaxation times as 

In the presence of the faradaic reaction, assuming that the faradaic impedance can be expressed as a simple equivalent resistance, the complex plane plots represent a rotated semicircle p ig. 28(b)], instead of a semicircle centered on the Z axis. Similarly, the double-layer capacitance in the presence of the faradaic reaction may be obtained as 

VanderNoottried to extract the distribution function G(x) from the CPE model. He found that the Fourier inversion method is not suitable, but that the maximum entropy deconvolution works relatively well. However, because this is an ill-posed problem, the results obtained are very sensitive to the experimental errors (noise). 

Historically, the CPE phenomenon was usually attributed to surface roughness. Pajkossy recently studied the origins of the CPE. 

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Recently, a constant-phase element has been found607 to be present at pc-Pb/KF + HaO interfaces by impedance measurements. The Pb electrode was cathodically reduced before use. The assumption has been made that the CPE is due to the inhomogeneity of the metal surface. Frequency-... 

This equation is valid regardless of solution properties (the values of 8 and tj), surface properties (the value of 0, and the size of the disperse-phase elements. All parameters of this equation can be determined by independent measurements. The validity of Eq. (31.12) was demonstrated by such measurements. This result is an additional argument for the claim that all four of the electrokinetic processes actually obey the same laws and have the same physical origin. 

Departures of the electrokinetic behavior of real systems from that described by the equations reported occurs most often because of breakdown of two of the assumptions above because of marked surface conductivity (particularly in dilute solutions, where the bulk conductivity is low) and because of a small characteristic size of the disperse-phase elements (e.g., breakdown of the condition of bg <5 r in extremely fine-porous diaphragms). A number of more complicated equations allowing for these factors have been proposed. 

The interaction between the dispersed-phase elements at high volume fractions has an impact on breakup and aggregation, which is not well understood. For example, Elemans et al. (1997) found that when closely spaced stationary threads break by the growth of capillary instabilities, the disturbances on adjacent threads are half a wavelength out of phase (Fig. 43), and the rate of growth of the instability is smaller. Such interaction effects may have practical applications, for example, in the formation of monodisperse emulsions (Mason and Bibette, 1996). 

Surface roughness is also expected to result in depression of the capacitance semi-circle. This phenomenon, which is indeed apparent in both Figures 1 and 2, is, however, unrelated to surface area. Rather, it is attributable to surface heterogeneity, i.e. the surface is characterized by a distribution of properties. Macdonald (16) recently reviewed techniques for representing distributed processes. A transmission line model containing an array of parallel R/C units with a distribution of values is physically attractive, but not practical. An alternative solution is introduction of an element which by its very nature is distributed. The Constant Phase Element (CPE) meets such a requirement. It has the form P = Y0 wn... 

We found an equivalent electrical circuit that fits best the LixC6 electrode behavior at high frequency. The circuit consists of a resistor R in parallel with a constant phase element (CPE). The latter is defined with a pseudo-capacitance Q and a parameter a with 0< a <1 [6], The impedance of... 

A constant phase element (CPE) rather than the ideal capacitance is normally observed in the impedance of electrodes. In the absence of Faradaic reactions, the impedance spectrum deviates from the purely capacitive behavior of the blocking electrode, whereas in the presence of Faradaic reactions, the shape of the impedance spectrum is a depressed arc. The CPE shows... 

Sweep rate (mV s 1) Electrolyte Resistance (Q cm2) Surface/species Parameter (106fi Van 2) Constant.phase/ element exponent Oxide filmResist ance (k 2 cm2) Ox. film thickness (nm)... 

Andrade and Molina [46] have performed electrochemical impedance studies of mercury electrodes with hematite particles adhered at different electrode potentials. Adhesion of such particles was strong and the decrease in the impedance was accompanied by an increase in the number of attached particles. Experimental results were analyzed in terms of an equivalent circuit including the constant phase element (CPE), the magnitude of which appeared to be directly related to the electrode coverage. A pore model for the metal/hematite particles interface has been proposed. 

The second most abundant crystalline compound (estimated volume fraction 0.1) was gadolinium titanate, Gd2Ti20y. There is evidence that U, Zr, Y, and possibly Sm and Eu are in solid solution with this gadolinium phase. Elemental silicon crystals remaining in the waste were found to be surrounded by Si02. 

This is the phenomenon of the so-called constant phase element (CPE) as it follows from eqn. (115) that Yc is composed of an imaginary and a real component, with a frequency-independent phase angle cor/2. Though the phenomenon is most clearly discovered in impedance or admittance analyses, its effect in time-domain methods should not be ignored. 

Constant-phase elements were first used to explain dielectrical properties of polar liquids and solids, and were attributed to the presence of the investigated material properties as a partitioning between extreme conditions, rather than as constant or uniform parameters40. Furthermore in the... 

In this book, an explanation of capacitive behaviour in similar and comparable systems is not directly possible with constant-phase elements because such a comparison is only possible if n values are equal, particularly in the study of surfaces covered with polymer coatings where a unification of the envisaged parameters is necessary. The impedances measured match with a relatively large amount of samples, of which the structure can be complex, showing many sources of non-idealities (e.g. variations in thickness of the membrane, pore size and pore density42 7). A good indication if such non-idealities occur can be found in the values of n. If they are not comparable, non-idealities occur. 

A solution to this problem is the use of electrical equivalent circuits without constant-phase elements (note that a good numerical simulation of the experiments can be obtained only by inserting constant-phase elements) only pure capacities are used. This method, although not convincing, results in comparable capacities. 

Another possibility is based on the fact that constant-phase elements represent processes that are not purely conservative (charging and discharging of a capacitor is a conservative process), first because of the fact that n is not equal to 1. From such an element, one can isolate the conservative contribution, which can then be further treated as a conservative process and simulated by a pure capacitor. The starting point for this separation is the existence of an angle frequency, to, for which48 ... 

Finally, it can be seen from Fig. 9.9a that the real impedance does not remain constant at low frequencies for the textile electrode, and this effect is more pronounced at higher electrolyte concentrations. Probably, Zr is influenced by other effects only occurring in the low-frequency range. This effect is frequently observed and described in the literature and is caused by non-uniformity of surfaces at the micro-scale, which in fact is the case for the textile electrodes. It is also not possible to explain this effect by a pure resistor or a pure capacitor in the electrical equivalent circuit. For this purpose, constant-phase elements are implemented as described in the theoretical discussion of electrochemical impedance spectroscopy (presented in Chapter 2, section 2.4). 

Due to the complexity of BN-structures and atomic bonding situations, the characterization of BN-phases by spectroscopic methods (e.g., IR and Raman) is difficult. It is not possible to identify BN phases using only one analytical method. For example, the X-ray diffraction peaks of c-BN correspond to those of Cu, Ni, and many other cubic phases. Elemental composition must be known or measured to be sure that no other phases are present. 

Carbon on Zr-C Reverse phase Elemental carbon on zirconia that is useful in the... 

The electrochemical impedance of a real electrode is frequently represented by an equivalent circuit containing constant phase element (CPE) showing power-law frequency dependence as follows... 

Process design modifications usually have a bigger impact on operability (dynamic resilience). Dynamic resilience depends on controller structure, choice of measurements, and manipulated variables. Multivariable frequency-response techniques have been used to determine resilience properties. A primary result is that closed-loop control quality is limited by system invertability (nonmin-imum phase elements). Additionally, it has been shown that steady-state optimal designs are not necessarily optimal in dynamic operation. 

The first resistance Rs is the resistance of the electrolyte outside the pores the R, elements are the electrolyte resistances inside the pores of the electrode and are the double layer capacitances along the pores. This model is called the Transmission Line Model (TLM) by De Levie. A careful selection of a set of Rv C values allows to calculate back the experimental plot such as the one presented in Figure 1.23 [28]. It can be noted that constant phase element (CPE) can be used to replace the capacitance C for better fitting, the CPE impedance ZCPE being ZCPE = l//(Cco) . 

Fig. 10. Equivalent circuits of electrode (a) general circuit of a thin-film electrode (b) the Randles circuit and (c) circuit with a constant phase element.

The characteristic frequency dependence shown in Fig. 12b implies the presence of a so-called constant phase element (CPE) in the electrode s equivalent circuit. The CPE impedance equals to [65]... 

Rugosity and porosity give rise to the so-called constant phase element (CPE), which can be described by groups of parallel or branched transmission lines. The CPE is manifested in real systems by an impedance spectrum altered from the expected shape, especially in the... 

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