Response constant phase

Keywords Electrochemical Impedance Spectroscopy, modeling, Li-ion cell, inner resistance, voltage step response, Constant Phase Element... 

If a system is disturbed by periodical variation of an external parameter such as temperature (92), pressure, concentration of a reactant (41,48,65), or the absolute configuration of a probe molecule (54,59), then all the species in the system that are affected by this parameter will also change periodically at the same frequency as the stimulation, or harmonics thereof (91). Figure 24 shows schematically the relationship between stimulation and response. A phase lag <)) between stimulation and response occurs if the time constant of the process giving rise to some signal is of the order of the time constant Inim of the excitation. The shape of the response may be different from the one of the stimulation if the system response is non-linear. At the beginning of the modulation, the system relaxes to a new quasi-stationary state, about which it oscillates at frequency cu, as depicted in Fig. 24. In this quasi-stationary state, the absorbance variations A(v, t) are followed by measuring spectra... 

One characteristic feature of theories that incorporate DD interactions is a density-dependent dielectric constant s(p ) > Ss, induced by a varying concentration of dipolar ion pairs. Friedman [3] suggested such a variation of the dielectric constant to be responsible for phase separation. As a byproduct, WS theory provides a generalization of the well-known Onsager expression for the dielectric permittivity of dipolar Quids to a system comprising free ions [221],... 

The term piezoelectric nonlinearity is used here to describe relationship between mechanical and electrical fields (charge density D vs. stress a, strain x vs. electric field E) in which the proportionality constant d, is dependent on the driving field, Figure 13.1. Thus, for the direct piezoelectric effect one may write D = d(a)a and for the converse effect x = d(E)E. Similar relationships may be defined for other piezoelectric coefficients (g, h, and e) and combination of electro-mechanical variables. The piezoelectric nonlinearity is usually accompanied by the electro-mechanical (D vs. a or x vs. E) hysteresis, as shown in Figure 13.2. By hysteresis we shall simply mean, in the first approximation, that there is a phase lag between the driving field and the response. This phase lag may be accompanied by complex nonlinear processes leading to a more general definition of the hysteresis [2],... 

For coatings with no pitting, the generalized model must be amended to account for that fact that all current must flow through the barrier coating. The coating resistivity, Rt, is on the order of 100 to 1000 MQ cm2 and behaves essentially as an open circuit under near-DC conditions (f = 0). The EIS response over the typically measured frequency domain is that of a constant phase element (CPE) in series with a solution resistance (Fig. 22b). 

This method of estimating Rc is useful when it can be applied, since the determination is not based on any presumed model of the corrosion damage process or any of the assumptions that come with assignment of an equivalent circuit model. This method is particularly helpful when there is more than one time constant in the spectrum, or the impedance spectrum is particularly complicated. Caution is warranted however. This method of estimation can be in serious error for samples with large capacitance-dominated low-frequency impedances. As a general rule, for this estimation method to be reasonably accurate, the impedance function must exhibit a clear DC limit, or a diffusional response that can be modeled by a constant phase element in equivalent circuit analysis (75). 

EIS data analysis is commonly carried out by fitting it to an equivalent electric circuit model. An equivalent circuit model is a combination of resistances, capacitances, and/or inductances, as well as a few specialized electrochemical elements (such as Warburg diffusion elements and constant phase elements), which produces the same response as the electrochemical system does when the same excitation signal is imposed. Equivalent circuit models can be partially or completely empirical. In the model, each circuit component comes from a physical process in the electrochemical cell and has a characteristic impedance behaviour. The shape of the model s impedance spectrum is controlled by the style of electrical elements in the model and the interconnections between them (series or parallel combinations). The size of each feature in the spectrum is controlled by the circuit elements parameters. 

The constant phase element (CPE) is a non-intuitive circuit element that was discovered in the course of investigations into responses from real systems. In general, a Nyquist plot (also called a Cole-Cole plot or complex impedance plane plot) should be a semicircle with the centre on the x-axis. However, the observed plot for some real systems was indeed the arc of a circle but with the centre located somewhere below the x-axis. Figure 4.1 shows the impedance spectra of a circuit of a resistor and a constant phase element connected in parallel. The centre of the semicircle is located at (l-n)x90° below the real axis. 

Figure 5.34. Electric equivalent circuit for the impedance spectra in Figure 5.37. Ref. ohmic resistance Rct charge-transfer resistance CPE constant phase element IV Warburg element. The subscripts a and c denote anode and cathode, respectively [36]. (Modified from Boillot M, Bonnet C, Jatroudakis N, Carre P, Didierjean S, Lapicque F. Effect of gas dilution on PEM fuel cell performance and impedance response. Fuel Cells 2006 6 31-7. 2006 John Wiley Sons Limited. Reproduced with permission, and with the permission of the authors.)...

The impedance response of electrodes rarely show the ideal response expected for single electrochemical reactions. The impedance response typically reflects a distribution of reactivity that is commonly represented in equivalent electrical circuits as a constant phase element (CPE). ° For a blocking electrode, the impedance can be expressed in terms of a CPE as... 

In the 1920s, impedance was applied to biological systems, including the resistance and capacitance of cells of vegetables and the dielectric response of blood suspensions. ° Impedance was also applied to muscle fibers, skin tissues, and other biological membranes. " The capacitance of the cell membranes was found to be a function of frequency, and Fricke observed a relationship between the frequency exponent of the impedance and the observed constant phase angle. In 1941, brothers Cole and Cole showed that the frequency-dependent complex... 

We use a standard tapping/intermittent contact mode set-up, which is assembled as described in Sect. 3.2.1 (see also hands-on example 31). The operation frequency and amplitude are adjusted to v at 0.85 A0 and 100 nm, respectively. The surface is tracked most accurately by utilizing a high setpoint amplitude ratio (close to 0.9). Care must be taken that the tip does not lose track of the surface when traversing elevated features, therefore the phase (and the amplitude) signal is carefully monitored. Halos in the fast scan direction that are devoid of contrast (i.e. show a constant phase shift or amplitude response), are indicative of absent tip-sample interaction. In this case the setpoint can be lowered slightly. 

Analogously, the generalized Warburg equation, representative of the response of constant phase elements in electrochemical impedance spectroscopy experiments, becomes (Nyikos and Pajkossy, 1990 Dassas and Duby, 1995) ... 

Figure 3. Simulation of the kinetic scheme from Figure 2 using the constants from Table 3. The top figure represents the kinetics of Ca dissociation which has a biphasic response (fast phase 663 s and slow phase 9 s ). The middle figure represents the titration of calmodulin by Ca ". The signal rising between 0 and 2-3 Ca " ions is associated with the occupancy of the sites from the COOH terminus and the other signal is associated with the occupancy of the N-terminal sites. The bottom figure is a Scatchard representation of the direct calcium binding isotherm.

Most of the literature on the dynamic properties of solid interfaces is based on the analogy between membrme and interfacial kinetics tnd the electric response of rough electrodes, for which an anomalous impedance scaling with frequency w, referred to as Constant Phase Angle behavior, Z[oj) = / 4- 0 < w <, i / has been... 

Greszczuk et al. [252] employed the a.c. impedance measurements to study the ionic transport during PAn oxidation. Equivalent circuits of the conducting polymer-electrolyte interfaces are made of resistance R, capacitance C, and various distributed circuit elements. The latter consist of a constant phase element Q, a finite transmission line T, and a Warburg element W. The general expression for the admittance response of the CPE, Tcpr, is [253]... 

Presa, M.J.R., R.I. Tucceii, M.I. Floiit, and D, Posadas, Constant phase element behavior in the poly(o-toluidine) impedance response. Journal of Electroanalytical Chemistry, 2001. 502 pp. 82-90... 

Kerner, Z, and T, Pajkossy, Impedance of rough capacitive electrodes the role of surface disorder. Journal of Electroanalytical Chemistry, 1998. 448 pp. 139-142 Ldng, G, and K.E, Heusler, Remarks of the energetics of interfaces exhibiting constant phase element behaviour. Journal of Electroanalytical Chemistry, 1998. 457 pp, 257-260 Liu, S.H, Eractal model for the ac response of a rough interface. Physical Review Letters, 1985, 55 pp, 529-532... 

In all real systems, some deviation from ideal behavior can he observed. If a potential is applied to a macroscopic system, the total current is the sum of a large number of microscopic current filaments, which originate and end at the electrodes. If the electrode surfaces are rough or one or more of the dielectric materials in the system are inhomogeneous, then all these microscopic current filaments would be different. In a response to a small-amplitude excitation signal, this would lead to frequency-dependent effects that can often be modeled with simple distributed circuit elements. One of these elements, which have found widespread use in the modeling of impedance spectra, is the so-called constant phase element (CPE). A CPE is defined as... 

Attempts to explore this complicated interaction and to model the response of the eddy viscosity and turbulent shear stresses to the time variation of pressure gradients in turbulent air flow over a solid wavy surface have been made by Thorsness et al. [85] and Abrams and Hanratty [89]. Large variations of the amplitude and phase angle of the surface shear stress with the dimensionless wave number were predicted (Figure 3). The analysis shows that the surface shear stress fluctuation is shifted upstream with respect to the wave elevation and the phase shift varies in the range of 0-80° in comparison to the constant phase shift predicted by Benjamin [84],... 

The disadvantages of IS are primarily associated with possible ambiguities in interpretation. An important comphcation of analyses based on an equivalent circuit (e.g. Bauerle [1969]) is that ordinary ideal circuit elements represent ideal lumped-constant properties. Inevitably, aU electrolytic cells are distributed in space, and their microscopic properties may be also independently distributed. Under these conditions, ideal circuit elements may be inadequate to describe the electrical response. Thus, it is often found that Z/to) cannot be well approximated by the impedance of an equivalent circuit involving only a finite number of ordinary lumped-constant elements. It has been observed by many in the field that the use of distributed impedance elements [e.g. constant-phase elements (CPEs) (see Section 2.2.2.2)] in the equivalent circuit greatly aids the process of fitting observed impedance data for a cell with distributed properties. 

As was pointed out by Cole and Cole, dielectric response corresponding to the function of Fq. (39) may be decomposed into the circuit shown in Figure 2.1.7, which contains a constant-phase element (CPF). The CPF is an empirical impedance function of the type... 

The Constant Phase Element and Its Simple Combinations. Although Warburg and open-ended diffusion effects frequently appear in supported situations and sometimes in unsupported ones and exhibit characteristic 6 = 45° lines in the Z or plane, one often finds approximate straight-line behavior over a limited frequency range with 0 45° (e.g. McCann and Badwal [1982]). Then the frequency response of Z and Z is no longer proportional to but to some other power of (0. To describe such response it is convenient to write, as in Eq. (7) in Section 1.3, at the admittance level,... 

Jonscher [1974,1975a,b, 1980,1983], in an extensive series of papers, working primarily in the dielectric area, independently emphasized the importance and ubiquity of constant phase response, and proposed and demonstrated the utility of three different empirical frequency response functions in IS data fitting. These three equations, termed universal dielectric response by Jonscher, were originally... 

FIGURE 13 Lines of constant phase for a planar cholesteric liquid crystal texture between two substrates (shown in cross section). The ripple arises in response to an applied field when the anisotropy is positive. 

Sapoval B. Fractal electrodes and constant phase angle response exact examples and counter examples. Solid State Ionics 1987 23(4) 253-9. 

ABSTRACT State determination of Li-ion cells is often accomplished with Electrochemical Impedance Spectroscopy (EIS). The measurement results are in frequency domain and used to describe the state of a Li-ion cell by parameterizing impedance-based models. Since EIS is a costly measurement method, an alternative method for the parameterization of impedance-based models with time-domain data easier to record is presented in this work. For this purpose the model equations from the impedance-based models are transformed from frequency domain into time domain. As an excitation signal a current step is applied. The resulting voltage step responses are the model equations in time domain. They are presented for lumped and derived for distributed electrical circuit elements, i.e. Warburg impedance, Constant Phase Element and RCPE. A resulting technique is the determination of the inner resistance from an impedance spectrum which is performed on measurement data. 

Figure 14.8 (A) FIS responses of a Pt/PANI-PV-SO anti-OTA immunosensor to standard OTA solutions in PBS (a, b, c, d, e, and f are 0, 2, 4, 6, 8, and lOng/ml OTA antigen, respectively). fE = 0 V). (B) equivalent circuit of Pt/PANI-PV-SO (R =solution resistance, CPE = constant phase element = charge-transfer resistance). From Ref [29] with permission. Copyright 2011 Wiley-VCH Verlag CmbH Co. KCaA...

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