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2 Relaxation functions and dielectric susceptibilities

[123.3.1] Let f(t) denote the normalized, i.e. f(0)=1, electrical dipolar polarisation or a similar relaxation function. [123.3.2] Then the complex frequency dependent dielectric susceptibility is \varepsilon(u)=1-u\widehat{f}(u), where \widehat{f}(u) denotes the Laplace transform of f(t), u=-2\pi\text{i}\nu, \text{i}^{{2}}=-1, and \nu is the frequency [16, p. 402, eq. (18)]. [123.3.3] Time honoured functional expressions for f(t) and \varepsilon(u) are the exponential (Debye) relaxation [6, ch.III,§10]

\displaystyle f(t) \displaystyle=\exp\left(-\frac{t}{\tau}\right) \displaystyle\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Debye)}, (1a)
\displaystyle\varepsilon(u) \displaystyle=\frac{1}{1+u\tau}, (1b)

or stretched exponential Kohlrausch relaxation [26, 27], revived in [35] (KWW),

\displaystyle f(t) \displaystyle=\exp\left(-\left[\frac{t}{\tau}\right]^{\alpha}\right) \displaystyle\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (KWW)}, (2a)
\displaystyle\varepsilon(u) \displaystyle=1-H^{{11}}_{{11}}\left([u\tau]^{\alpha}\left|\begin{array}[]{l}{(1,1)}\\ {(1,a)}\end{array}\right.\right), (2b)

where \tau>0 is the relaxation time and 0<\alpha\leq 1 is the stretching exponent. [123.3.4] Remarkably, while formula (2a) for the relaxation function has been used since 1854, formula (2b) for the dielectric susceptibility was discovered only in 2001 and published in [18]. [123.3.5] It is given in terms of inverse Mellin-transforms of \Gamma-functions, also known as H-functions [5]. [123.3.6] A brief definition is can be found in the Appendix below.

[123.4.1] A popular alternative to stretching time is to stretch frequency. [123.4.2] In this case a stretching exponent \alpha is introduced into eq. (1b) rather than into eq. (1a). [123.4.3] This leads to the Cole-Cole (CC) relaxation [3]

\displaystyle f(t) \displaystyle=\mathrm{E}_{\alpha}\left(-\left[\frac{t}{\tau}\right]^{\alpha}\right), (3a)
\displaystyle\varepsilon(u) \displaystyle=\frac{1}{1+(u\tau)^{\alpha}} \displaystyle\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (CC)}, (3b)

where

E_{a}(x)=\sum _{{k=0}}^{\infty}\frac{x^{k}}{\Gamma(ak+1)} (4)

is the Mittag-Leffler function [31]. [123.4.4] It is by now well known, that the relaxation function f(t) for Cole-Cole relaxation is intimately related to fractional calculus [15]. [123.4.5] Unfortunately, the Cole-Cole form (3b) exhibits a symmetric \alpha-peak, while asymmetric \alpha-peaks are observed experimentally for many materials [28]. [123.4.6] Therefore, a second way to introduce the stretching exponent \alpha into the Debye function (1b), known as the Cole-Davidson (CD) form, was introduced in [4]

\displaystyle f(t) \displaystyle=\displaystyle\frac{\Gamma(\alpha,t/\tau)}{\Gamma(\alpha)}, (5a)
\displaystyle\varepsilon(u) \displaystyle=\frac{1}{(1+u\tau)^{\alpha}}\varepsilon(u) \displaystyle\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (CD)}, (5b)

where

\Gamma(a,x)=\int _{a}^{\infty}y^{{a-1}}e^{{-y}}\mathrm{d}y (6)

denotes the complementary incomplete Gamma function, [123.4.7] Finally, the CC-form and CD-form are combined into the popular Havriliak-Negami (HN) form given as

\displaystyle f(t) \displaystyle=1-\displaystyle\frac{1}{\Gamma(\beta)}H^{{11}}_{{12}}\left(\left[\frac{t}{\tau}\right]^{\alpha}\left|\begin{array}[]{l}{(1,1)}\\ {(\beta,1)(0,\alpha)}\end{array}\right.\right), (7a)
\displaystyle\varepsilon(u) \displaystyle=\frac{1}{\left(1+[u\tau]^{\alpha}\right)^{\beta}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (HN)}. (7b)

[123.4.8] Formula(7a) for the Havriliak-Negami relaxation function was first given in [18].

[123.5.1] The functional forms (1), (3), (5), and (7) are used universally almost without exception to fit broadband dielectric data. [123.5.2] A quantitative comparison of the different forms is shown for propylene carbonate at T=193K in Figure 2. [123.5.3] It is found, that all of the functional forms (1), (3), (5), and (7) deviate from the experimetal data at high ferquency or give an unsatisfactory fit. [123.5.4] Therefore, a combination of two or more of these functional forms is routinely used to fit the excess wing in glass forming materials (see e.g. Figure 3.5 in [28, p.66]).