Relaxation impedance theory

Impedance theory and relaxation theory often do not include resonance phenomena, as these are usually not found in macro tissue samples in the frequency range from pHz to MHz. 

In EHD impedance studies, the relaxation times for the different transport processes are obtained by variation of the modulation frequency of the flow. The utility of the technique relies on being able to separate out individual transport relaxation times. This is possible because, with EHD, each relaxation time will have a different functional dependence upon the perturbation. The theory and methodology were first developed for sinusoidal modulation of the flow velocity in a tube [20, 22]. The results... 

The experimental situation is inconclusive and sometimes even the same experimental techniques used by different groups give contrary results. Especially for the compounds k-(ET)2Cu(NCS)2 and K-(ET)2Cu[N(CN)2]Br many different techniques have been employed to measure A(T). Evidence for non BCS-like behavior has been obtained by complex ac susceptibility [220], radio-frequency penetration depth [221], muon spin relaxation (//SR) [222], and microwave surface impedance measurements [223]. In contrast, results consistent with conventional BCS theory, sometimes revealing a tendency towards strong coupling, are reported for measurements of the //SR [224], microwave surface impedance [225, 226], and dc magnetization [227]. 

The models that consider this approach are largely based on the assumption of effectively homogeneous local relaxation processes related to transport in each of the phases and electrical charge exchange between them. Thus, the complex problem of an uneven distribution of electrical current and potential inside the electrode can be described analytically, and impedances can be calculated. Furthermore the models may be conveniently pictured as a double-channel transmission line (Fig. 3.5). In several papers, the theory of the impedance of porous electrodes has been extended to cover those cases in which a complex frequency response arises in the transport processes [100] or at the inner surface [194,203]. 

A single pulse or a step function excitation is the basis of relaxation theory. Power dissipation and temperature rise may for instance impede the use of repeated waveforms, and single pulse excitation is necessary. A single pulse is a pulse waveform with repetition interval oo, it has a continuous frequency spectrum as opposed to a line spectrum. The unit impulse (delta function) waveform is often used as excitation waveform. It is obtained with the pulse width 0 and the pulse amplitude oo, keeping the product = 1. The frequency spectrum consists of equal contributions of all frequencies. In that respect, it is equal to white noise (see the following section). Also, the infinite amplitude of the unit pulse automatically brings the system into the nonlinear region. The unit impulse is a mathematical concept a practical pulse applied for the examination of a system response must have limited amplitude and a certain pulse width. 

The Cole single-dispersion impedance model (Eq. 9.26) is based upon an ideal conductance as a dependent variable and a characteristic time constant as an independent variable. Usually, however, the characteristic time constant of tissue or cell suspensions is a function of conductance according to relaxation theory (see Section 3.4). The Cole model is therefore not in accordance with relaxation theory (Figures 9.21 and 9.22). 

Figure 9.21 Dispersion model in accordance with relaxation theory (Eq. 9.43). (a) Impedance model and (b) admittance model.

The deviations of the impedance responses [23,28, 30,32, 59,64,66,69,71, 76,120,123,132,144-146] predicted by the theories have been explained by taking into account different effects, such as interactions between redox sites [30, 136], ionic relaxation processes [95], distributions of diffusion coefficients [28], migration [65, 118, 125, 132], film swelling [64, 137], slow reactions with solution species [22,138], nonuniform film thickness [23], inhomogeneous oxida-tion/reduction processes [123], etc. 

Electrochemistry in general and the EIS in particular are often used to analyze both bulk sample conduction mechanisms and interfacial processes, where electron transfer, mass transport, and adsorption are often present. EIS analysis has often treated the bulk and interfacial processes separately [4]. The analysis is achieved on the basis of selective responses of bulk and interfacial processes to sampling AC frequencies. The features appearing in the impedance AC frequency spectmm can be described according to the theory of impedance relaxations. Again, as in the case of any other spectroscopy method, the subject of the EIS analysis is the detection and interpretation of these spectrum features. 

The concentration of rigid rodlike molecules cannot be increased very much before spontaneous separation occurs into a concentrated ordered and a very dilute disordered phase, as required by thermodynamic relations, although the separation may occur at higher concentrations if there is some degree of flexibility, and may be delayed by steric hindrance so that a metastable disordered state may persist. The rotation of rigid rods at finite concentration is impeded by collisions and the rotatory relaxation time should increase rapidly with concentration. The theory has been treated by Doi and Doi and Edwards, who conclude that the relaxation time is approximately proportional to c L, where L is the molecular length. 

Historically, the bulk lubricant has been studied by dielectric spectroscopy and interpreted according to the Debye relaxation theory [3,4]. In impedance terms the system can also be represented according to a theory of colloidal dispersions or polycrystalline media composed of spheres of vastly different conductivities, where the contaminants become a more conductive phase suspended inside the less conductive additive/base oil matrix [6, 34]. Alternatively, when the contaminants are absent, the polar additives can be considered as a conductive discontinuous phase suspended inside insulating continuous base oil. Initially the description of the impedance representation of the fresh, uncontaminated oil will be provided, and then the effects of oxidative degradation and contaminants will be discussed. 

The complex impedance measurements are the basis for the evaluation of ji and n and the complex impedance measurements can actually be directly employed to determine As another test measurements on aqueous solutions were made using Kl. Figure 20 shows the graph of imaginary part of the complex impedance Z " against the real part Z for 3><1CH M solution at different temperatures. The relaxation time, Th was obtained from the frequency corresponding to the minimum in Z shown in Figure 20. This T2 value coincides with the frequency of the maximum in tan as described in the theory section. 

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