Distribution function of relaxation times

Unfortunately the computation of the Z)i values is difficult and the final expression for the distribution function of relaxation times contains some rather arbitrary parameters. 

The viscoelastic property may be expressed by another function H(t), the distribution function of relaxation time t or the relaxation spectrum. The relaxation spectrum is related to the complex modulus by... 

Equations (1.23a), (1.23b) and (1.23c) are, respectively, Cole-Cole (C-C) (0Davidson-Cole (D-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) ... 

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. 

For pure glassy state the distribution function of relaxation times can be approximately represented by horizontal line in the range tq < x < Xmax, where xq is very small (around 10 " s) and Xn x can achieve macroscopic values like 10 s, see above. This actually means that any value (even infinite) of relaxation time exists... 

The dielectric loss can be written using the distribution function of relaxation time /(logx) (or fix)) assuming the Debye-type relaxations for simplicity of the kernel ... 

The energy absorption of a sample is given by the integral over the E" curve. From this, we obtain the idea that, for a material suitable for low temperature use, no sharp distribution function of relaxation times must exist The broader this function, the more mechanisms exist to distribute mechanical energy over the entire sample and convert it into ordinary heat. In the region of 4 K, there is only a small possibility of location changes... 

Fig. 19 Distribution function of relaxation times H calculated from the G and G" master curves in Figs. 15 and 16. (From Ref. 16.)...

Comparison between Cole-Cole plots for the Debye, Cole-Cole and Davidson-Cole equations is made in Fig. 4.2. The arc corresponding to the Cole-Cole equation is symmetrical and forms a portion of a circle, the centre of which is below the 6 -axis. The corresponding distribution function of relaxation times is symmetric, although there is no closed expression for F t) which would give the Cole-Cole equation. The Davidson-Cole arc is a skewed one, and reflects strongly asymmetric distribution of relaxation times. The distribution is peaked at the critical relaxation time Tq, with a decaying tail of shorter relaxation times. There is an exact expression for the autocorrelation function leading to the Davidson-Cole equation. 

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