Various empirical and semi-empirical equations have been suggested in the literature for the description of dielectric relaxation phenomena. As the majority of the spectra represent a sum of various relaxation processes the best description of the spectra can be achieved by a superposition of various equations.
1.3.1 Debye equation
The simplest approach to model dielectric spectra of liquids is the Debye equation.42 The basic assumption is a decrease of the orientational polarization in the absence of an external electric field directly proportional to the polarization itself43 resulting in a time law of the first order,
where τ represents the relaxation time, characteristic for the dynamics of the system.
Solution of the first order differential equation yields the step response function, FP(t) = exp(−t/τ): The pulse response function can be obtained according to eq. 1.43
fP(t) = 1 By application of the Laplace transformation, the pulse response function in the time-domain is converted to the frequency time-domain according to eq. 1.44 yielding finally the response function of the Debye equation in its complex presentation
Fj(ω) = 1
1 + iωτ (1.48)
The dispersion curve, ε0 =ε0(ln(ω)), is a monotonically decreasing point-symmetric func-tion and the absorpfunc-tion curve, ε00 =ε00(ln(ω)), an axis-symmetric band reaching its maxi-mum at ω= 1/τ.
1.3.2 Extensions of the Debye equation
The dielectric spectra of a considerable number of condensed systems deviate from the mono-exponential relaxation of the Debye equation. Description of these spectra can be improved by assuming a continuous relaxation time distribution, G(τ).34 For practical
1.3. EMPIRICAL DESCRIPTION OF DIELECTRIC RELAXATION 13
reasons, a logarithmic distribution, G(lnτ), is preferred. The complex permittivity can then be expressed as Unfortunately, G(lnτ) cannot be obtained from the experimental data in a straightfor-ward manner. Nevertheless, empirical parameters can be introduced that account for the broadness and shape of the relaxation time distribution. Some of the common empirical equations yielding relaxation time distributions will be presented in the following section.
Cole-Cole equation
A symmetrical relaxation time distribution around a principal relaxation time τ0 is mod-elled by introducing an additional empirical parameter α ∈ [0..1]. Thus, the Cole-Cole equation44,45 is
Fj = 1
1 + (iωτ0)1−α (1.50)
which results in flatter dispersion curves and broadened absorption spectra. For α = 0, the Cole-Cole equation reduces to the Debye equation.
Cole-Davidson equation
The Cole-Davidson equation46,47 with an empirical parameter β ∈ [0..1], describes an asymmetric relaxation time distribution around the center of gravity τ0
Fj = 1
(1 + iωτ0)β (1.51)
The modification results in asymmetric dispersions and absorption curves. For β = 1, eq. 1.51 becomes equivalent to the Debye equation.
Havriliak-Negami equation
Introduction of both parameters α ∈ [0..1] and β ∈ [0..1], yields the Havriliak-Negami equation describing a broad asymmetric relaxation time distribution:48
Fj = 1
(1 + (iωτ0)1−α)β (1.52)
Both the dispersion and absorption curves are asymmetric. For α = 0 and β = 1 this equation simplifies to the Debye equation.
For time-domain dielectric data, the Kohlrausch-Williams-Watts49 (KWW) model is gen-erally preferred as it can be derived from mode coupling theory. There is no exact equiv-alent of KWW in the frequency domain but to a good approximation it corresponds to a Havriliak-Negami model with restrictions on the values of α and β.34,50
Modified Cole-Cole equation
Contributions of a relaxation function at frequencies where librational modes (see section below) occur are physically unreasonable, because a relaxation process evolves from libra-tional fluctuations.51 This means that the assumption of linearly independent processes is not valid anymore. The Debye equation, and even worse due to the broadening, the Cole-Cole and Cole-Davidson equations contribute considerably at THz and far-infrared frequencies. To avoid this unphysical behavior the relaxation equations have to be ter-minated at high frequencies by including an upper boundary. Following the approach suggested by Turton and Wynne,52 this can be realized by subtracting a fast exponential decay characterized by an initial rise rate, γlib, determined by the librational frequencies, yielding:
According to Turton and Wynne,52 γlib is higher than than the measured librational frequencies, because it represents a rise time rather than a full oscillation. Therefore, γlib ≈ hωlibi/(2π), where hωlibi is the average resonance angular frequency of the libra-tional modes (see below).
If the relaxation process is dependent on a lower frequency mode, Turton and Wynne,52 additionally introduced a low frequency termination, deduced from the relaxation behavior of glass forming liquids. This “α-termination” is described in detail elsewhere.52
1.3.3 Damped harmonic oscillator
Many vibrational dielectric processes (intermolecular and intramolecular vibrations as well as librations) can be modelled satisfactorily by harmonic oscillation.39 Assuming a har-monic oscillator driven by a harhar-monically oscillating fieldE(t) =E0eiωt which is subject to a damping force that is linearly dependent upon the velocity∂x(t)/∂t, the time-dependent motion x(t) of an effective charge q can be obtained from the solution of the differential equation
m∂2
∂t2x(t) +mγ ∂
∂tx(t) +kx(t) = qE(t) = qE0eiωt (1.54) derived from Newton’s equation, eq. 1.5. With a characteristic angular resonance frequency ω0 =p where τD is the characteristic damping time and C =qE0/m a constant.
1.3. EMPIRICAL DESCRIPTION OF DIELECTRIC RELAXATION 15
Solving eq. 1.55 yields
x(t) = C
ω02−ω2+ iωτD−1eiωt (1.56) The oscillation of the effective chargeqresults in a time dependent polarization, defined by the total overall dipole moment per unit volume, P(t) =ρqx(t), where ρ is the resonator number density.
From an extension of eq. 1.37 to a polarization caused by a resonance type dielectric dispersion, the complex permittivity can be expressed by eq. 1.57.
ˆ
ε(ω)−ε∞= P(t)
ε0E(t) = ρqx(t)
ε0E0eiωt (1.57)
Taking eq. 1.56 into account, the damped oscillation can be written in terms of complex permittivity ε(ω):ˆ For a static field (eq. 1.6) the low frequency limit,
ε= ˆε(0) =ε∞+ ρq2
ε0mω02 (1.59)
is obtained. Combination of eq. 1.58 and 1.59 yields the frequency dependent response function of the system:
Fj(ω) = ω20
(ω02−ω2) + iωτD−1 = ν02 ν02− 2πω2
+ i2πωγ (1.60)
In the limit ofτDω0−1, i.e. the damping time constant is much shorter than an oscillation period, the response reproduces the Debye equation.
1.3.4 Combination of models
For many real systems the complex permittivity spectrum is composed of several relax-ation processes. In these cases the complex permittivity spectrum can be modelled by a superposition ofn single relaxation processes: Each of the processes is treated as linearly independent with its own response function, Fj(ω), and dispersion amplitude, Sj: