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 ε⁢u=1-u⁢f^⁢u, where f^⁢u denotes the Laplace transform of f⁢t, u=-2⁢π⁢i⁢ν, i2=-1, and ν is the frequency [16, p. 402, eq. (18)]. [123.3.3] Time honoured functional expressions for f⁢t and ε⁢u are the exponential (Debye) relaxation [6, ch.III,§10]

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

where τ>0 is the relaxation time and 0<α≤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 Γ-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 α is introduced into eq. (1b) rather than into eq. (1a). [123.4.3] This leads to the Cole-Cole (CC) relaxation [3]

where

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 α-peak, while asymmetric α-peaks are observed experimentally for many materials [28]. [123.4.6] Therefore, a second way to introduce the stretching exponent α into the Debye function (1b), known as the Cole-Davidson (CD) form, was introduced in [4]

where

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

[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=193⁢K 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]).