0
fPor(t)dt = 1. (1.44) The complex permittivity, ε(ω), can than be calculated asˆ 60
ˆ
ε(ω) =ε(ω)−iε(ω) =ε∞+ (ε−ε∞)· Liω[fPor(t)]. (1.45)
1.3 Empirical description of dielectric relaxation
To characterize the behavior of the orientational polarization, a number of equations are used to describe the experimental data.
1.3.1 Debye equation
The Debye (D) equation can be obtained by assuming that the decrease of the polarization in the absence of an electric field is directly proportional to the polarization itself.62,63 Then, the polarization is described by
∂
∂t
Pμ(t) =−1 τ
Pμ(t), (1.46)
where τ is the relaxation time. From the solution of Eq. 1.46, Pμ(t) = Pμ(0) exp
the pulse response function (Eq. 1.48) is obtained as fPor(t) = 1
Applying Eq. 1.45, the Debye equation for the complex dielectric permittivity is obtained as
ˆ
ε(ω) = ε∞+ ε−ε∞
1 + iωτ. (1.49)
The dispersion and loss curves are
ε(ω) =ε∞+ ε−ε∞
1 +ω2τ2 and (1.50)
ε(ω) =ωτ ε−ε∞
1 +ω2τ2. (1.51)
On a logarithmic scale, the real part is a monotonically decreasing point-symmetric function and the imaginary part is a symmetric peak with a maximum value at ω= 2πν = 1/τ.
1.3.2 Distribution functions
A distribution of relaxation times on a linear, g(τ), or logarithmic scale, G(lnτ), may be used for the description of dielectric relaxation.60 The complex permittivity is written as
ˆ As the distribution functions cannot be determined from the experimental spectra in a straightforward way,60 empirical extensions of the Debye equation have been introduced.
Cole-Cole equation. A symmetrically-broadened loss curve in combination with a flat-ter dispersion curve is modelled by a Cole-Cole (CC) equation,64,65
ˆ
ε(ω) =ε∞+ ε−ε∞
1 + (iωτ)1−α. (1.53)
The CC parameter, α ∈ [0..1[, describes a symmetric relaxation time distribution of the principal relaxation time,τ. For α= 0, Eq. 1.53 reduces to the Debye equation.
Modified Cole-Cole equation. To account for the inertial rise of the dipole reorienta-tion, a modified Cole-Cole (CCm) equation,
ˆ may be used.66 Eq. 1.54 avoids unphysical contributions of the CC equation at high (THz to far-infrared) frequencies, where librational and/or vibrational modes contribute to the spectra. The inertial rise constant, γlib, is in the order of the resonance frequency of the librational/vibrational mode(s). Note that for α = 0 the corresponding modified Debye equation (Dm) is obtained.
Cole-Davidson equation. An asymmetrical relaxation time distribution is described by the Cole-Davidson (CD) equation,67,68
ˆ
ε(ω) = ε∞+ ε−ε∞
(1 + iωτ)β, (1.55)
with the empirical CD parameter, β ∈]0..1]. The Debye equation is obtained for β = 1.
Havriliak-Negami equation. For the representation of broadened and asymmetrically shaped dispersion and loss curves both parameters α ∈ [0..1[ and β ∈]0..1] are combined in the Havriliak-Negami (HN) equation,69
ˆ
ε(ω) = ε∞+ ε−ε∞
[1 + (iωτ)1−α]β (1.56) Forα = 0 and β = 1, Eq. 1.56 is equal to the Debye equation.
1.3.3 Damped harmonic oscillator
Resonant absorptions, like vibrations and librations in the THz or far-infrared regions, can be modelled as a damped harmonic oscillator (DHO). Assuming a harmonic oscillator subjected to a damping force and driven by a harmonically oscillating field, one obtains
ˆ
ε(ω) =ε∞+ (ε−ε∞)ω20
(ω02−ω2) + iωτD−1. (1.57) as the solution of the differential equation describing the time-dependent motion, x(t), of an effective charge, q.70,71 In Eq. 1.57, ω0 =
k/m = 2πν0 and γ = 1/(2πτD) are the angular resonance frequency and damping constant of the oscillator, respectively. For τDω0−1, Eq. 1.57 reduces to the Debye equation.
1.3.4 Combination of models
In real systems, the DR spectrum may be the result of a superposition of distinct relaxation modes. Therefore, Eq. 1.52 is written as a sum of j = 1. . . n separate processes: Each process is characterized by its own relaxation time,τj, and dispersion amplitude, Sj, defined via:
This leads to the general expression for superpositions of HN, CCm and DHO equations:
ˆ
To extract physical information from complex permittivity spectra, an appropriate math-ematical description of the measured complex permittivity data has to be found. As men-tioned above, the dielectric response may be created by more than one relaxation process.
In the ideal case, each process can be unambiguously modelled by one empirical function.
However, due to the broad nature of the relaxations and technological limitations,38 a decomposition is rarely trivial. Therefore, more than one relaxation model can possibly describe the experimental spectra and thus, the choice of the ‘true’ relaxation model has to follow some rules. First of all, the parameters obtained have to be physically meaningful.
Second, the normalized variance of the fit, χ2r, defined as χ2r = 1 of the spectra presented here. Figure 1.1 shows the effect of a superposition of three (3D model) and four (4D model) Debye equations to describe a spectrum that was simulated by a combination of a CC and a D equation (CC + D model). Apart from model-dependent fluctuations, the relative percentage deviations, δεfit = 100·(εCC+D−ε3D,4D)/εCC+D and δεfit = 100·(εCC+D−ε3D,4D)/εCC+D, for the real and imaginary parts are well below the probable experimental uncertainties of ca. ±2 % in ε.ˆ38 Thus, consideration of χ2r alone will not necessarily provide meaningful physical insights.
Additionally, the number of parameters should be as small as possible and the relaxation model should not change in a concentration series, except for specific physical reasons.
The overall fitting procedure performed in the present PhD thesis was the following: ex-perimental η(ν)ˆ data were corrected for the conductivity contribution. Then, different relaxation models were tested by simultaneously fitting ε(ν) and ε(ν) using the MW-FIT program. A non-linear least-squares routine based on the method of Levenberg and Marquardt is implemented in this program.72 To minimize systematic deviations at low ν, the conductivity was slightly varied. The origin of the small deviations of the resulting corrected values from conventionally measured (dc) conductivities are well understood.37 Due to the nonlinear nature of the fitting process, it is not possible to assign statistically meaningful standard uncertainties to the individual fit parameters, but the square root of the diagonal elements of the covariance matrix can be used as a measure for the certainty of the resulting parameters.72,73