Cole plot

Another coordinate system, plots of capacitive component of impedance X, against the resistive component R was proposed in 1941 by K. S. Cole and R. H. Cole for electric circuits. In 1963 this system (called Cole-Cole plots) was used by M. Sluyters-Rehbach and J. H. Sluyters in electrochemistry for extrapolation of the experimental data. In the case discussed, the resulting impedance diagram has the typical form of a semicircle with the center on the horizontal axis (Fig. I2.I7a). This is readily understood when the term coCp is eliminated from the expressions for R and in Eq. (12.25). Then we obtain, after simple transformations. 

To better understand the nature of the two types of dynamics, the Cole-Cole plots were plotted, and they clearly indicate the evolution from FR to SR with the changing temperature. Strikingly, the plots can be nicely fitted by the sum of two modified Debye functions (Figure 3.8b, inset), which is used to explain such a complex relaxation process. Here, the two separated relaxation processes are most likely associated with distinct anisotropic centres, that is, two Dy ions with... 

The phenomenological parameter, a, is typically estimated from fitting of Cole-Cole plots (x vs x f°r a fixed temperature) [9]. Small values of a are expected for SMMs having, ideally, one characteristic time. Larger values of a... 

Fig. 13 Cole-Cole plots (10 Hz to 10 MHz frequencies) of alternating currents of the DNA aligned film placed perpendicular to the comb-type electrodes (5 xm distance), a The film prepared from the long DNA molecules (10 xm, 30000 bps) and b the film prepared from the short DNA molecules (0.2 p.m, 500 bps)...

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)...

Figure 12.9d shows the dielectric function of several metals that either have been discussed in Chapter 9 or will be discussed in connection with small particle extinction in Section 12.4. The energy dependence of the dielectric function is given in the form of trajectories in the complex e plane, similar to ihe Cole-Cole plots (1941) that are commonly used for polar dielectrics the numbers indicated on the trajectories are photon energies in electron volts. 

Figure 12.9 Contour plots of constant dimensionless cross section for spheres (a), needles (b), and disks (c). Cole-Cole plots are shown in (d) for various metals.

Chronopotentiometry, galvanostatic transients, 1411 as analytical technique, 1411 activation overpotential, 1411 Clavilier, and single crystals, 1095 Cluster formation energy of, 1304 and Frumkin isotherm, 1197 Cobalt-nickel plating, 1375 Cold combustion, definition, 1041 Cole-Cole plot, impedance, 1129, 1135 Colloidal particles, 880, 882 and differential capacity, 880 Complex impedance, 1135 Computer simulation, 1160 of adsorption processes, 965 and overall reaction, 1259 and rate determining step, 1260... 

The Impedance (or Cole-Cole) Plot, in the preceding discussion, the attitude was that one should calculate Z, the impedance of the circuit one thinks best represents events at the interface, as a function of to and find if the Z-to plot52 from the model of the interface fits the plot from experiment. 

However, the Cole-Cole plot usually makes calculations (done automatically within the impedance bridge concerned) of the two components into which any measured impedance can be resolved. The components consist of the real part of the impedance, which is in phase with the applied voltage, and the imaginary" part, which is 90° out of phase. One can plot either of these quantities against log to to obtain mechanism-indicating plots. 

The deviation from a semicircle on the right provides information. One can obtain from its slope the value of i interface. There are two values in the real axis at which the plot (or its extrapolation) intercepts with the axis. The one at the high-frequency side of Fig. 7.47 turns out to give the solution resistance the low-frequency one gives the solution resistance together with the interfacial resistance (which can be determined later). The maximum of the semicircle must be associated with a certain value of and this value is 1 / DL intcrfaee. One can see that Cole-Cole plots provide a lot of information. 

Impedance spectroscopy a single interface. Draw the equivalent circuits for the following electrode/electrolyte interfaces, then derive their impedance expression and explain what their Cole-Cole plot will look like (a) An ideally polarizable interface between electrode and electrolyte, (b) An ideally nonpolarizable interface between electrode and electrolyte, (c) A real-life electrode/... 

Fig. 6 The Cole-Cole plot of the contribution from water to the frequency dependent dielectric function. The reduced real part [c (u )] is plotted against the reduced imaginary part (c (u>)]. Note the non-Debye character in the micellar solution.

Fig. 9. Reduced Cole-Cole plot for the asymptotic linear array. Dispersion parameters are a = 0.10, / = 0.54...

Fig. 26. Cole-Cole plots at some selected temperatures T (in °C) for the 60/40 copolymer...

Figure 10. Cole-Cole plot for n-GaAs in 0.8 1 melt, bias 0.0 V (0) experimental data (------) theory for circuit insert of Figure 8 (-) single RC hemisphere...

Dilute polyelectrolyte solutions, such as solutions of tobacco mosaic virus (TMV) in water and other solvents, are known to exhibit interesting dynamic properties, such as a plateau in viscosity against concentration curve at very low concentration [196]. It also shows a shear thinning at a shear strain rate which is inverse of the relaxation time obtained from the Cole-Cole plot of frequency dependence of the shear modulus, G(co). 

The complex modulus components E and E" (or G and G" or their compliance counterparts), are functions of the loading frequency or angular frequency co. To try to identify these functions, it is usual to determine E and E" experimentally in a wide interval of temperature and frequency and to build E" = f(E ) (Cole-Cole) plots. 

Actually, polymer relaxations are complex processes characterized by a relaxation spectrum (coexistence of many relaxation time). Their Cole-Cole plots are generally nonsymmetric (Fig. 11.11). 

Figure 11.11 Cole-Cole plots for networks resulting from the condensation of diglycidyl ether of bisphenol A (DGEBA) and diethyltoluenediamine (ETHA), with various amine/epoxide molar ratios (numbers on the figures). Reprinted from Tcharkhtchi et at. 1998. Copyright 2001 with permission from Elsevier Science.

Models proposed by Havriliak and Negami (1966) and by Perez (1992) usually give a good lit of Cole-Cole plots, provided there is no overlapping between a and (1 relaxations (Fig. 11.12). 

Figure 11.12 Shape of the Cole-Cole plots in the case of overlapping of a and P transitions.

When Cole-Cole plots are established from the usual DMTA experiments, care must be taken on storage modulus measurements. As a matter of fact, these systems are relatively accurate in phase measurements (tan 8), but modulus values (E ) can be less precise. Since E" is determined from E" = E tan 5,... 

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