Davidson-Cole

Relation [1] Is the frequency-dependent analogue of a formula proposed by Chasset and Thirion (2, 3) which has since been applied very frequently to relaxation measurements on cured rubbers. The next three equations are Inspired by similar relations In dielectrics (they are not derived from these) Equation [2] by the Cole-Cole and Equation [3] by the Davidson-Cole relation (15, 16). Both are special cases of the most general Equation [4] which contains five parameters (17). 

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. 

Employing the Davidson-Cole model for propylene carbonate and the Cole-Cole model for propionitrile with the appropriate dielectric parameters from the literature, we have predicted C(t) for these polar aprotic solvents according to the dielectric continuum model. The agreement between the predicted and observed C(t) is not nearly as good as the alcohol examples (see below). 

P = 0.5. The Cole-Cole function forms a symmetric arc, which approaches the intercepts with finite slope and has a maximum e" value less than (er — eu)/2. The Williams-Watts function also forms a flattened arc, but is asymmetric. The shape of the Davidson-Cole function is very similar to the Williams-Watts function, as discussed by Lindsey and Patterson33). The evaluation of the Williams-Watts function requires numerical methods 33,34). Computer programs implementing this function from published tabular values are readily available35). 

They thus calculate Pit) in the absence of significant conductivity for three forms of e(s) behaviour, Debye, Cole-Cole, and Davidson-Cole types. In... 

Figure 2 For packed samples of crushed ice particles, (a) Temperature dependence of the dielectric relaxation time rand (h) Cole-Cole plots of the sample (after 400 h of annealing at -1 °C, 2-4 mm particle size, 536 kg/m packed density) at -10 °C. The dielectric dispersion is of the Davidson-Cole type (a=0.97, f=0.85).

The results described above indicate two interesting facts many samples consisting of packed ice particles exhibit a dielectric dispersion of the Davidson-Cole type, and ice samples grown from the vapor phase exhibit a shorter relaxation time r and a lower activation energy of rthan ordinary ice samples grown from the liquid phase. 

A sample consisting of packed ice particles has numerous cavities and ice particle surfaces inside, and the number of dangling bonds oriented with a hydrogen nucleus at the surface of ice tends to increase. Therefore we can expect that the number of L-defects near the ice particle surfaces also tends to increase and protons near the ice particle surface can easily move. The dispersion of the Davidson-Cole type may be caused by proton behaviors near the ice particle surfaces in a sample consisting of packed ice. 

It was also found that the variations of l/Ti required a Davidson-Cole distribution function for best fit. In the mixed alkali glasses the MT plots at high temperatures become symmetrical and broader. The activation energies determined from NMR, Enmr, and the E from conductivity measurements have also been compared. Since Li NMR senses only the lithium ion and not the other alkali, the Enmr in mixed alkali regions Eire observed to be lower than Ea. 

In most liquids and to a good approximation, y0= (1-n). Therefore n can be evaluated by evaluating p. P itself is given by the ratio WJW, where and W are the Debye and actual widths of the relaxation spectrum. Even when loss curves (dielectric, mechanical or any other) are fitted to other standard analytical functions such as Cole-Cole (CC), Davidson-Cole (DC) or Haveriliac-Negami expressions, (see earlier section) one can determine p using the empirical relations... 

Note The exponent in the HN equation is determined by acc and aoc, which are the exponents of the Cole-Cole and the Davidson-Cole functions respectively and the constants in equation 9.02 have been redesignated as (1 - a) =acc and P =aoc)- P is temperature dependent and the values of P are identified as >9= 0 at Tg and y9= 1 at T. The validity of this approach has been verified in a number of materials and a typical example given by Rault (2000) based on the measurements done in PIBMA (Poly isobutylene methacrylate) by Dhinojwala et al.(1992) using the data of the decay of chromophore orientation in a poled film is shown in Figure 9.05. In the region between Tg and T, >9 is found to vary... 

Cole-Cole fit) (Davidson-Cole fit) (Haveriliac-Negami fit)... 

Many materials display non-Debye dielectric behavior by a broader asymmetric loss peak. This non-Debye a.c response can be described by a combination of Cole-Cole [23] and Davidson-Cole [24] functions, and an empirical expression proposed by Havrilink-Negami [25]. The natural gum Arabica is found to obey a non-Debye type of response [25,26] and may be described [27] by the Havrilink-Negami function. 

In general, plant tissue immittance data are not in accordance with Cole—Cole system models. Nonsymmetrical models of the Davidson Cole and Havriliak—Negami types have... 

Equations (1.23a), (1.23b) and (1.23c) are, respectively, Cole-Cole (C-C) (0Havriliak-Negami (0empirical laws. The calculations of permittivity on the base of Eq. (1.22) with relaxation function corresponding to KWW law (see Eq. 1.20) yield Eq. (1.23c) with y8 = a - [30]. Expression (1.23c) delivers pretty good description of experimental data obtained by dielectric spectroscopy, radiospectroscopy and quasielastic neutron scattering. It can be shown, that the physical mechanism, underlying the expressions (1.23) is the distribution of relaxation times in a system. Namely, Equation (1.23) can be derived by the averaging of simple Debye response (1.21) with properly tailored distribution function of relaxation times F(x) ... 

Fig. 1.9 Relaxation time distribution functions calculated on the basis of Debye law (D), Cole-Cole law (CC), Davidson-Cole law (DC) and...

The expression (1.24) allows obtaining the distribution function of relaxation times for all empirical laws (1.23). In Fig. 1.9, we show the relaxation time distribution functions, obtained in Ref. [31] with the help of Eq. (1.24). The distribution functions have been obtained for the laws of Cole-Cole k = 0.2), Davidson-Cole (P = 0.6) and Havriliak-Nagami at a = 0.42 when it corresponds to KWW law. It is seen that only C-C law leads to symmetric dishibution function while DC and KWW laws correspond to essentially asymmetric one. The physical mechanisms responsible for different forms of distribution functions in the disordered ferroelechics had been considered in Ref. [32]. It has been shown that random electric field in the disordered systems alters the relaxational barriers so that the distribution of the field results in the barriers distribution, which in turn generates the distribution of relaxation times. Nonlinear contributions of random field are responsible for the functions asymmetry, while the linear contribution gives only symmetric C-C function. 

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