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2 Classical relaxation models

[1283.4.1] The Debye relaxation model describes the electric relaxation of dipoles after switching an applied electric field [21]. [1283.4.2] The normalized relaxation function f⁢t, which corresponds to the polarization, obeys the Debye law

τ⁢dd⁢t+1⁢f⁢t=0,(1)

[page 1284, §0]    with the relaxation time τ and initial condition f⁢0=1. [1284.0.1] The response function, i.e. the dynamical dielectric susceptibility, χ is related to the relaxation function via [17]

χ⁢t=-dd⁢t⁢f⁢t.(2)

[1284.0.2] In the following discussions we focus on Laplace transformed quantities. [1284.0.3] We use the Laplace transformation of f⁢t

L⁢f⁢t⁢u=∫0∞e-u⁢t⁢f⁢t⁢d⁢t,(3)

where u=i⁢ν and ν is the frequency. [1284.0.4] Rewriting equation (2) in frequency space and using f⁢0=1 leads to

χ⌃⁢u=L⁢χ⁢t⁢u=1-u⁢L⁢f⁢t⁢u=11+u⁢τ,(4)

the well known Debye susceptibility. [1284.0.5] In experiments one measures not the normalized quantity χ⌃⁢u, but instead

ε⁢u=ε0-ε∞⁢χ⌃⁢u+ε∞,(5)

where ε0 and ε∞ are the dynamical susceptibilities at low, respectively high frequencies.

[1284.1.1] The Debye model is not able to describe the experimental data well, because experimental relaxation peaks are broader and asymmetric. [1284.1.2] For this reason other fitting functions were proposed such as the Cole-Cole [22], Cole-Davidson [23] and Havriliak-Negami [16] expressions. [1284.1.3] Their normalized forms have typically 2 or 3 parameters (see table 1) and they were introduced purely phenomenologically to fit the data. [1284.1.4] This can be considered as a drawback. [1284.1.5] These functions with three parameters are able to fit the data over a range of at most 5 decades (Havriliak-Negami). [1284.1.6] To fit a broader frequency range several functions are commonly added. [1284.1.7] A combination of one Havriliak-Negami expression plus one Cole-Cole form results in 6 fit parameters.

functional

form χ⌃⁢u

number of

parameters

Cole-Cole 1/1+u⁢τα 2
Cole-Davidson 1/1+u⁢τα 2
Havriliak-Negami 1/1+u⁢ταγ 3
Table 1: List of traditional fit functions for dielectric spectroscopy data of glass forming materials.
Figure 1: Simultaneous fits of real and imaginary part with model A (left figure) and model B (right figure) for 5-methyl-2-hexanol at 155.4⁢K. Both models show an excellent fitting capability. The data are from [6].
Figure 2: Simultaneous fits of real and imaginary part with model A (left figure) and model B (right figure) for methyl-m-toluate at 179.2⁢K. Model A can fit the data over the whole spectral range of 7 decades, which is more than with Havriliak-Negami which uses the same number of fit parameters. With this data we obtain the broadest fit with model A. The data are from [12].