Constant phase-angle model

For the frequently applied constant phase-angle model (Bomer, 1991, 1992, 2006 Dissado and Hill, 1984 Jonscher, 1981) complex conductivity is ... 

The constant phase-angle model was also applied to other problems of fluid-solid interface reactions in rocks such as ... 

The constant phase-angle model does not give the best fit to experimental results in all cases. Based on the Cole-Cole model, Cosenza et al. (2009) found that the position of the maximum phase peak in the phase versus frequency plot shows a proportionality to the grain size and that the exponent is related to the width of the grain-size distribution. 

Scott and Barker (2003) investigated the correlatiOTi between a dominant pore-throat diameter and the frequency at which a peak in the phase angle versus frequency plot occurs in Triassic sandstones. This does not follow the constant phase-angle model. 

Constant phase elements (CPEs) have been used in bioimpedance models since the late 1920s. A CPE can be modeled by a resistor and capacitor, both having frequency-dependent values, in such a way that the phase angle is frequency independent. A CPE is mathematically simple, but not so simple as to realize with discrete, passive components in the real world. A particular type of CPE is the Warburg element, known from electrochemistry and solid state physics. It is diffusion controlled with a constant phase angle of 45° (Warburg 1899). 

The important feature of these lines is that they produce a constant phase angle, like a Warburg impedance, but with the phase angle not restricted to ti/4. This is exactly the behavior often found at the electrode-electrolyte interface and has been termed a constant-phase element (CPE). It appears to be true that roughness is an important contributing factor to the observed frequency dispersion. Scheider s model, however, remains qualitative, and the microscopic link between the topology and the circuit is absent. 

Complexities associated with the simple equivalent circuit models described above include phenomena (some interrelated) associated with depressed arcs and constant phase angles [99], porous electrodes and electrodes with rough surfaces 1100-102], pseudoinduction behavior associated with electrosorbed intermediates [103], and nonidealities associated with (polymer) coatings on electrode surfaces. We now briefly explore this... 

Fig. 2. Orientational probability distributions of the molecular axes in (a) a-nitrogen and (b) y-nitrogen. Contours of constant probability for the molecule in the origin, calculated in the mean field model, are plotted as functions of the polar angles (0, ) with respect to the crystal axes (Fig. 1). The angle 0 increases linearly with the radius of the plots from 0 (in the center) to tt72 (at the boundary) d> is the phase angle.

In addition to comparing the sum of squares, the experimental and simulated data should be compared by using complex plane and Bode plots. The phase-angle Bode plot is particularly sensitive in detecting time constants. Boukamp proposed to study the residual sum of squares after subtracting the assumed model values from the total impedance data. If the model is valid, the residuals should behave randomly. If they display regular tendencies, it may mean that the model is not correct and further elements should be added. However, the variations of the residuals should be statistically important. 

For the static part of the solution, T, f h zero, otherwise the actual temperature modulation frequency, /, is used to calculate the amplitude and the phase angle of the complex temperature distribution, TM( ) The considered thermal model of the sensor was verified by comparison with the measured temperature distribution of a real sensor. The factor G relates the volume source of heat q with the geometry of the modulation heater. K is, in general, the decay constant - due to the fact that it is a complex number, a decaying thermal wave is the result. 

If the bias potential and I Ip I, and therefore Cd/ are kept constant, then changes in Cm can be calculated from I once Cd is determined. In practice, calibrated capacitors (to model Cm) can be inserted into the circuit prior to formation of the membrane. Calibration curves of the modelled Cm vs. I and vs. ( >,where ( > is the phase angle between I and the LED current (light intensity), can then be used to determine Cm once the membrane is formed. 

The general phase-space theory is regarded as difficult to understand, because it was first presented [1,5] for mechanical models that may have internal constraints (constant bond lengths and/or constant bond angles). The attendant notational and mathematical complexity tends to obscure some of the physical ideas. 

The resistance of membranes can be measured by AC impedance methods [85,86], using the four-point-probe technique. The test membrane is placed in a cell consisting of two Pt-foil electrodes, spaced 3 cm apart, to feed the current to a sample of 3 x 1 cm and two platinum needles placed 1 cm apart, to measure the potential drop (see Fig. 4.3.26). The cell is placed in a vessel maintained at constant temperature by circulating water. The impedance measurements are then carried out at 1-10 kHz using a frequency-response analyzer (e.g., Solatron Model 1255HF frequency analyzer). After ensuring that there are no parasitic processes (from the phase angle measurements, which should be zero), one can measure the resistance directly. The membrane resistance can also be obtained directly from the real part of the impedance (see typical data in Fig. 4.3.27). 

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],... 

Although the CPE and fractal systems give the same impedance in the absence of redox reactions, a comparison of Eq. (8.9) for the CPE model with Eq. (8.17) for a fractal system in the presence of a redox reaction shows that they are structurally different. In fact, they produce different complex plane and Bode plots. This is clearly visible from Fig. (8.9), which can be compared with Fig. 8.4 for the CPE model. With a decrease in the value of , an asymmetry on the complex plane plot occurs that is also visible oti the phase angle Bode plots. This is related to the different topology of the equivalent circuits they are compared in Fig. 8.10. In the CPE model, only the impedance of the double-layer capacitance is taken to the power while in the fractal model the whole electrode impedance is taken to the power (p. The asynunetiy of the complex plane and Bode plots for fractal systems arises from the asymmetric distribution function of time constants in Eq. (8.4) according to the equation [298, 347]... 

Effective modeling of complex electrochemical processes of corrosion in the systems based on iron required the use of a more complex equivalent electrical circuit, i.e., circuit containing CPE - constant phase elements. Constant phase element (CPE) is characterized by a constant angle of phase shift. Impedance of the CPE is described by the following expression Zcpe = l/Yo(jffl)", where Yo and n are parameters related to the phase angle. The more heterogeneous the corrosion processes occurring on the metal surface the smaller value of the parameter n. 

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