Cole equations

The parameters of this expansion, as well as the number N of Lorentzian functions, are determined (from the experimental data) by a non-linear least squares fit along with statistical tests. It can be noticed that this expansion has no physical meaning but is merely a numerical device allowing for smoothing and interpolation of the experimental data. Nevertheless, this procedure proves to be statistically more significant than the Cole-Cole equation and thus to account much better for the representation of experimental data. The two physically meaningful parameters, i.e., C(0) and (Xo), can then be easily deduced from the quantities involved in (71)... 

The dielectric dispersion for some solvents is poorly modeled by a multiple Debye form. Alternative, e(cu) distributions such as the Davidson-Cole equation or the Cole-Cole equation are often more appropriate. 

From the Cole-Cole equation [equation (13)] one may derive equation (49), and when equations (48) and (49) are combined the result is equation (50). Therefore the observed quantities tb and a lead to unique... 

Microwave measurements have established with some high degree of probability that a small spread of relaxation times is exhibited by pure water. According to the Cole-Cole equation, the region between the static permittivity and the infinite frequency permittivity is spanned by the complex permittivity e, as defined in equation (13), where t is the principal relaxation time and a the spread parameter w = 2itv is the radial e = (Co - c )/tl + (j OT) -"] (13)... 

Returning to anomalous dielectric relaxation, it appears that a significant amount of experimental data on disordered systems supports the following empirical expressions for dielectric loss spectra, namely, the Cole-Cole equation... 

As far as the physical mechanism underlying the Cole-Cole equation is concerned, we first remark that Eq. (9) arises from the diffusion limit of a continuous-time random walk (CTRW) [17] (see Section II.A). In this context one should recall that the Einstein theory of the Brownian motion relies on the diffusion limit of a discrete time random walk. Here the random walker makes a jump of a fixed mean-square length in a fixed time, so that the only random... 

The second model of Debye or the Debye-Frohlich model may also be generalized to fractional diffusion [8,25] (including inertial effects [26]). Moreover, it has been shown [25] that the Cole-Cole equation arises naturally from the solution of a fractional Fokker-Planck equation in the configuration space of orientations derived from the diffusion limit of a CTRW. The broadening of the dielectric loss curve characteristic of the Cole-Cole spectrum may then be easily explained on a microscopic level by means of the relation [8,24]... 

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

The complex susceptibility is given by the Cole-Cole equation [Eq. (9)] with the static susceptibility for fixed-axis rotators... 

Figure 42. Broad-band dielectric loss spectrum of 10% v/v solution of probe molecule CH2CI2 in glassy decalin at 110 K. Filled circles are the experimental data [98], Curve 1 is the best fit for the anomalous diffusion in the double-well cosine potential (a - 1.5, t,v = 8, and 7 = 0.003) curve 2 is the best fit for the normal diffusion ( J. = 1, v = 7, and 7 — 0.001) in the double-well cosine potential. Dashed line (curve 3) is the Cole-Cole equation [Eq. (9)] with a = 2 — a.

Young s modulus relaxation to occur between a value Eq at zero frequency, and a value E o at infinite frequency. The difference E o — Eq is the magnitude of the relaxation process. An empirical modification of the Zener model, known as the Cole-Cole equation, has the complex modulus equation... 

The specific case a=l, 1 gives the Debye relaxa tion law, P= 1, a 1 corresponds to the so-called Cole-Cole equation, whereas the case a= 1, P 1 corresponds to the Cole-Davidson formula. Recently, some progress in the understanding of the physical meaning of the empirical parameters (a, P) has been made (7, 8). Using the conception of a self-similar relaxation process it is possible to understand thenature of a nonexponential relaxation of the Cole-Cole, Cole-Davidson, or Havriliak-Negami type. 

Dispersion is therefore a hroad concept, and many types of DRTs are possible. The Cole brothers proposed a certain DRT corresponding to the apparently simple Cole—Cole equation (Section 9.2). The Cole—Cole equation presupposes a CPE. However, other distributions than the Cole—Cole type are also found to be in agreement with measured tissue values. The Cole—Cole model is attractive however because the mathematical expressions are so simple. Dispersion models as described below therefore pertain to many types of dispersion mechanisms, among those are also Cole—Cole systems. 

Figure 4.4 shows Cole—Cole plots for glycine (amino acid), glycylglycine (peptide), and albumin (protein). As a biomolecule becomes more and more complicated and large (amino acid—peptide—protein), the frequency exponent 1-a of the Cole—Cole equation becomes higher (a lower), indicating a broader distribution of time constants. 

Immittance is the dependent variable in the Cole equations. For most biological systems, it is observed that the center of the impedance circular arc locus is situated below the real axis in the Wessel diagram. This was clear from the late 1920s, and Cole and Fricke published diagrams and equations based upon a frequency-independent phase angle. But in 1940, Kenneth S. Cole proposed the following empirical equation z = Zoo + (ro — rtissue impedance. The basis was findings... 

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