Analytical representations for relaxation functions of glasses

Analytical representations in the time and frequency domains are derived for the most frequently used phenomenological fit functions for non-Debye relaxation processes. In the time domain the relaxation functions corresponding to the complex frequency dependent Cole-Cole, ColeDavidson and Havriliak-Negami susceptibilities are also represented in terms of H-functions. In the frequency domain the complex frequency dependent susceptibility function corresponding to the time dependent stretched exponential relaxation function is given in terms of H-functions. The new representations are useful for fitting to experiment. PACS: 77.22.Gm,61.20.Lc,71.55.Jv, 67.40.Pf, 78.30.Ly published in: Journal of Non-Crystalline Solids, vol. 305 (2002), page 122 Analytical representaions of relaxation functions in the time domain and susceptibilities in the frequency domain are important to fit experimental data in a broad variety of experiments on glasslike systems. Dielectric spectroscopy, viscoelastic modulus measurements, quasielastic light scattering, shear modulus and shear compliance as well as specific heat measurements all show strong deviations from the normalized exponential Debye relaxation function f(t) = exp(−t/τ) (1) where τ is the relaxation time [1]. All relaxation functions in this paper are normalized to f(0) = 1. Relaxation in the frequency domain is described in terms of the normalized complex susceptibility χ̂(u) = χ(ø)− χ∞ χ0 − χ∞ = 1− uL{f(t)} (u) (2) where u = −ıø, ø is the frequency, χ(ø) is a dynamic susceptibility normalized by the corresponding isothermal susceptibility, χ0 = limø→0 Reχ(ø) is the static susceptibility, χ∞ = limø→∞ Reχ(ø) gives the “instantaneous” response, and L{f(t)} (u) is the Laplace transform of the relaxation function f(t). For the exponential relaxation function this leads to χ̂(ø) = 1 1 + iøτ , (3) i.e. the well known Debye susceptibility. Most generalizations of equations (1) and (3) for glasses and other complex materials are obtained by the method of introducing a fractional “stretching” exponent. In the time domain this method leads to the “stretched exponential” or Kohlrausch relaxation function given as f(t) = exp[−(t/τβ) β ] (4) with fractional exponent β [2]. Of course all formulae obtained by the method of stretching exponents are constructed such that they reduce to the exponential Debye expression when the stretching exponent becomes unity. Extending the method of stretching exponents to the frequency domain one obtains the ColeCole susceptibility [3] χ̂(ø) = 1 1 + (iøτα) , (5) the Davidson-Cole expression [4] χ̂(ø) = 1 (1 + iøτγ) (6) or the combined Havrialiak-Negami form [5] χ̂(u) = 1 (1 + (uτH)) (7) as empirical expressions for the experimentally observed broadened relaxation peaks. Most surprisingly, the analytical transformations between the time and frequency domain for general values of the parameters in these simple analytical expressions seem to be largely unknown [6], and authors working in the time 1 domain usually employ the stretched exponential function while authors working in the frequency domain use the stretched susceptibilities. An exception are the results in [7] where the real and imaginary part of the elastic modulus were obtained for Kohlrausch relaxation. Note however that there is a sign error in the real part in the results of [7]. It is therefore the purpose of this short communication to rederive expressions for the Kohlrausch susceptibility in the frequency domain. Secondly the same methods are used to obtain for the first time the relaxation function corresponding to the Havriliak-Negami susceptibility (and hence also the Cole-Davidson and Cole-Cole susceptibilities) in the time domain. It is hoped that these expressions will be useful for facilitating the fitting of experimental data. The objective of this paper is achieved by employing a method based on so called H-functions [8]. The H-function of order (m,n, p, q) ∈ N and with parameters Ai ∈ R+(i = 1, . . . , p), Bi ∈ R+(i = 1, . . . , q), ai ∈ C(i = 1, . . . , p), and bi ∈ C(i = 1, . . . , q) is defined for z ∈ C, z 6= 0 by a contour integral in the complex plane [8, 9] H p,q ( z ∣∣∣∣ (a1, A1), . . . , (ap, Ap) (b1, B1), . . . , (bq, Bq) ) = 1 2πi ∫ L η(s)z ds (8) where the integrand is