Cole Complex dielectric constant

Fig. 4.2. Imaginary part e" of the complex dielectric constant versus real part with frequency as a parameter (Cole-Cole plot) at different temperatures. Arrows indicate the frequency of 10 Hz in each case. Insert shows thermal activation energy plot. (See Text)...

The symmelrical loss-frequency curve predicted by this simple theory is commonly observed for simple substances, but its maximum is usually lower and broader because of the existence of more than one relaxation time. Various functions have been proposed to represent the distribution of relaxation times. A convenient representation of dielectric behavior is obtained, according to the method of Cole and Cole, by writing the complex dielectric constant as... 

Fig. 9.4. Cole-Cole plot of the complex dielectric constant of a single crystal of ice at —10-9 °C. e O—O fij., — . The small numbers indicate measurement frequency in kHz. (From Humbel, Jona Scherrer, 1953 Helv. Phys. Acta. a6, 17-3Z, fig. 6 Birkhkuser Verlag.)...

Cole and Cole [3] have shown how the real and the imaginary components of the complex dielectric constant may be evaluated by an Argand diagram in which each point corresponds to one frequency. This diagram plots e" as the ordinate and e the abscissa (Figure 5.4). The shape of the plot is given by... 

Other empirical distributed elements have been described, which can be expressed as a combination of a CPE and one or more ideal circuit elements. Cole and Cole found that frequency dispersion in dielectrics results in an arc in the complex e plane (an alternative form of presentation) with its center below the real axis (Fig. 10a) [16]. They suggested the equivalent circuit shown in Fig. 10(b), which includes a CPE and two capacitors. For ft) —> 0, the model yields capacitance Co and for ft) —> oo the model yields capacitance Coo- The model can be expressed with the following empirical formula for the complex dielectric constant... 

Similar property distributions occur throughout the frequency spectrum. The classical example for dielectric liquids at high frequencies is the bulk relaxation of dipoles present in a pseudoviscous liquid. Such behavior was represented by Cole and Cole [1941] by a modification of the Debye expression for the complex dielectric constant and was the first distribution involving the important constant phase element, the CPE, defined in Section 2.1.2.3. In normalized form the complex dielectric constant for the Cole-Cole distribution may be written... 

Now, as already mentioned in Section 1.3, the h of that section s Eq. (6) is of just the same form as the well-known Cole-Cole dielectric dispersion response function (Cole and Cole [1941]). In its normalized form, the same / function can thus apply at either the impedance or the complex dielectric constant level. We may generalize this result (J. R Macdonald [1985a,c,d]) by asserting that any IS response... 

In addition to an examination of the frequency response of series and parallel components of the circuit impedance/admittance, another approach may be particularly valuable. This analysis method involves plotting the real versus imaginary parts of some such complex quantity as admittance or impedance as parametric functions of frequency. Such Argand or "circle diagrams" have been used for many years in electrical engineering when complex dielectric constant is the quantity considered, they are known as Cole-Cole plots. ... 

The dielectric relaxation in nematics can be conveniently described by the so-called Cole-Cole equation for the complex dielectric constant e (to)... 

For the qualitative evaluation of distribution of relaxation time, Cole-Cole plot is drawn between the real part (e ) and imaginary part (e") of the complex dielectric constant (Parab et al. 2013). How well the e and e" are fitted to form a semicircle, is an indication of the nature of relaxation behavior. The highest point of semi-circle curve of gives the information about the relaxation frequency/, of the orientational... 

The Complex Dielectric Constant and the Cole-Cole Function... 

Figure 5.27 The imaginary versus the real components of the complex dielectric constant according to the Cole-Cole function, Eq. 5.60 showing the meaning of the angle theta.

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... 

In the standard description of the dielectric properties of the chiral tilted smectics worked out by Carlssonet al. [152], four independent modes are predicted. In the smectic C the collective excitations are the soft mode and the Goldstone mode. In the SmA phase the only collective relaxation is the soft mode. Two high frequency modes are connected to noncollective fluctuations of the polarization predicted by the theory. These two modes become a single noncollective mode in the smectic A phase. There is no consensus [153] as yet as to whether these polarization modes really exist. Investigations of the temperature dependence of the relaxation frequency for the rotation around the long axis show that it is a single Cole-Cole relaxation on both sides of the phase transition between smectic A and smectic C [154]. The distribution parameter a of the Cole-Cole function is temperature-dependent and increases linearly (a=a-pT+bj) with temperature. The proportionality constant uj increases abruptly at the smectic A to SmC transition. This fact points to the complexity of the relaxations in the smectic C phase. 

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