for specific physical reasons. Additionally, it is desirable to find a physical interpretation for the relaxation processes.
For many systems, more than one relaxation model with similar χ2r can be used. The model also depends on the measured frequency range and also the precision of the data, the density of the spot frequency and the empirical conductivity correction, which is not always exactly identical with the measured Ohmic conductivity. For the evaluation of the data the MWFIT program based on the method of Levenberg and Marquardt40 was used.
For the purpose of this study, the program was extended by a term describing a damped harmonic oscillator. It determines the best set of relaxation parameters of the chosen relaxation model for the experimentally obtained ε and ε data. The quality of the fit is described by the variance
In Eq.(2.73) m represents the number of value triples, n the number of fit parameters, ˆεi the experimental and ˆεi,calc the calculated dielectric permittivity.
2.4 Microscopic models of dielectric relaxation
2.4.1 Onsager equation
The Onsager model27,41 describes dielectric polarization of dipole mixtures. It uses a continuum description of the material which provides an environment for the dipole reori-entation. Specific interactions and the anisotropy of the surrounding field are not taken into account.
From this picture, Onsager deduced Eq. 2.74 for the interpretation of dielectric properties.
ε0(ε−1)E =Eh· Here, ρj represents the charge density, αj the polarizability, fj the reaction field factor of the speciesj and µef f,j the effective dipole moment of species j.
Eh stands for the cavity field which can be calculated by Eh = 3ε
2ε+ 1
E (2.75)
under the conditions of a sphere-like cavity in a continuum dielectric material of permit-tivity ε.
A combination of Eq.(2.75) and Eq.(2.74) leads to the general form of the Onsager equation, (ε−1)(2ε+ 1)ε0
16 CHAPTER 2. THEORETICAL BACKGROUND
Furthermore, for a simple dipolar liquid, Eq.(2.76) can be converted to (ε−ε∞)(2ε+ε∞)
ε(ε∞+ 2)2 = ρµ2ef f
9ε0kBT. (2.77)
Statistical mechanics provides a possibility of taking into account the influence of specific intermolecular interactions. In this way molecular characteristics can be interpreted in terms of macroscopic properties. The theory42,43 is based on a model of a dipole whose orientation correlates with its neighbors and results in the Kirkwood-Fr¨ohlich equation,
(ε−ε∞)(2ε+ε∞)
ε(ε∞+ 2)2 = ρµ2
9ε0kBT ·g, (2.78)
whereg is the Kirkwood factor, representing the interactions between the particles. If the orientation between the neighbors is preferentially parallel, a value g >1 is found whereas g <1 indicates an antiparallel orientation correlation.
2.4.2 Cavell equation
The Cavell equation44 represents the extension of the Onsager equation (2.76) to systems with more dispersion steps, caused by different dipolar species. According to this theory, the dispersion amplitude, Sj =εj −εj+1, of a relaxation process j can be obtained from the concentration cj of the species, its effective dipole moment, µef f,j and polarizability, αj, by
2ε+ 1
ε ·(εj −ε∞,j) = NAcj
kBT ε0 · µ2ef f,j
(1−fjαj)2, (2.79) wherefj is a reaction field factor. For a sphere-like cavity of radius aj, this factor is given as27
fj = 1
4πε0a3j · 2ε−2
2ε+ 1. (2.80)
For the evaluation of dispersion amplitudes Eq.(2.79) is used in the following form cj = (εj −ε∞,j)ε0(2ε+ 1)
ε ·(1−αjfj)2 µ2j ·kBT
NA (2.81)
However, since spherical cavities cannot always be assumed, Eq. 2.79 was extended for ellipsoidal particles with half-axes aj > bj > cj27,45:
ε+Aj(1−ε)
2.4. MICROSCOPIC MODELS OF DIELECTRIC RELAXATION 17
2.4.3 Debye model of rotational diffusion
Debye tried to predict the relaxation time of a simple system consisting of an aggregation of sphere-like inelastic dipoles which do not interact with each other. Microscopically, colliding dipolar particles cause a reorientation of the dipole. Thus, this mechanism is regarded as a diffusion of dipole orientation.
However, the picture only holds as long as the moment of inertia and the dipole-dipole interaction can be neglected. Therefore, the applicability of this theory is limited to non-associated systems.
Within these limitations and by describing the inner field with a Lorentz field, Debye obtained the dipole correlation function,
γ(t) = exp
where the relaxation time, τs, can be calculated from the friction factor,ζ, τs = ζ
2kBT. (2.86)
Using hydrodynamic laws for the rotation of a sphere in viscous media, namely the Stokes-Debye-Einstein equation,
τs= 3V η
kBT (2.87)
is obtained, whereV represents the volume of the sphere andηthe dynamic viscosity of the environment of the sphere (the so-called microscopic viscosity). However the application of this theory has its drawbacks as the relation between macroscopic and microscopic viscosity is not clear.
This problem can be solved by introducing various parameters into the equation46 that lead to the term
τs = 3V η
kBTfstickC+τs0. (2.88)
The shape factor, fstick, describes the deviation of the shape of the molecule from the ideal form of a sphere. For a sphere with stick boundary conditions of rotational diffusion fstick = 1. The friction parameter, C, represents a correction of the macroscopic viscosity and its value in the case of stick conditions isC = 1. For slipmotion, C= 1−fstick−2/3. τs0 is an empirical value which can be interpreted as a correlation time of the free rotor.
18 CHAPTER 2. THEORETICAL BACKGROUND
2.4.4 Microscopic and macroscopic relaxation time
The relation between the experimentally measurable relaxation time, τ, and the micro-scopic relaxation time, τs, plays an important role in the interpretation of the dielectric spectra and there are various theoretical approaches that address this problem.
Debye suggested the expression47
τ = ε+ 2
ε∞+ 2 ·τs, (2.89)
derived under the assumption of a Lorentz field as inner field. However, this approach is not accurate enough for polar dielectrics and so it can be used for non-polar systems only.
For the case of pure rotational diffusion Powles and Glarum combined the macroscopic and microscopic relaxation time48,49 in the following manner
τ = 3ε
2ε+ε∞ ·τs (2.90)
Using the corresponding macro-micro correlation theorem of statistical mechanics50–52, a generalized form
τ = 3ε 2ε+ε∞ · g
˙
g ·τs (2.91)
can be obtained, whereg is the Kirkwood correlation factor and ˙g the dynamic correlation factor. When g/g˙ = 1, expression (2.91) turns into the Powles-Glarum equation (2.90).