Broadband Dielectric Spectroscopy

The interaction of electromagnetic fields with matter is described by Maxwell’s equations (Eqs. 1.1–1.4) and

The complex dielectric1 function \( \mathop {\lim }\limits_{x \to \infty } \) can be measured in the extraordinary broad frequency regime [1-13] from 10^-6 Hz up to 10¹² Hz (in wavelength\( 3 \times {10^{16}}cm - 0.03cm \)). To span this dynamic range different measurement systems based on different measurement principles have to be combined (Fig. 2.1) From 10^-6 to 10⁷ Hz lumped circuit methods are used in which the sample is treated as a parallel or serial circuit of an ideal capacitor and an ohmic resistor. Effects of the spatial extent of the sample on the electric field distribution are neglected. With increasing frequency the geometrical dimensions of the sample capacitor become more and more important limiting this approach to about 10 MHz. In addition parasitic impedances caused by cables, connectors, etc. become important at frequencies >100 kHz.

The complex dielectric function ɛ*(ω) in its dependence on angular frequency (\( \omega = 2\pi v \) (v-frequency of the outer electrical field) and temperature originates from different processes: (i) microscopic fluctuations of molecular dipoles [1] (rotational diffusion1), (ii) the propagation of mobile charge carriers (translational diffusion of electrons, holes or ions), and (iii) the separation of charges at interfaces which gives rise to an additional polarization. The latter can take place at inner dielectric boundary layers (Maxwell/Wagner/Sillars-polarization [2, 3] ) on a mesoscopic scale and/or at the external electrodes contacting the sample (electrode polarization) on a macroscopic scale. Its contribution to the dielectric loss can be orders of magnitude larger than the dielectric response due to molecular fluctuations.

Despite the fact that glasses are materials which have been available since the rise of mankind and despite the fact that they play an essential role in modern technology their physical understanding is still controversial and remains an unresolved problem of condensed matter physics [1, 2,3]. The most prominent features observed when a glass-forming liquid or polymer melt coos down is the rapid increase of the characteristic relaxation time and the strong non-Debye behaviour of the relaxation function. This has been observed by a manifold of different experimental methods including mechanical-dynamical spectroscopy [4], ultrasonic attenuation [5] light [6] and neutron scattering [8](see Chap. 18), NMR spectroscopy [8] (see Chap. 17) and especially broadband dielectric spectroscopy [9–[13].

Glassy materials have been used by man since prehistoric times [1,2] and nowadays are ubiquitous in our daily live. There are of course the classical technical applications in, e.g. architecture (windows),for containers or for optical components. However, recently glassy materials have also gained increasing importance in quite different fields, e.g. communication technique (optical fibres) or medicine (bioactive implants). The modern definition of glass as a non-crystalline solid includes also the large group of the polymers and glass ceramics, but also more exotic materials as amorphous metals, which are believed to have a great technological future. Glasslike behaviour is also found in some crystalline materials, the so-called plastic crystals and orientational glasses, which at low temperatures are characterized by static disorder with respect to the orientational degrees of freedom of the translationally ordered molecules [3]. These materials are often considered as model systems for “conventional” glass formers and they are much simpler to treat in theoretical and simulation approaches to the glassy state.

The molecular and collective dynamics in confining space is determined by the counter balance between surface- and confinement effects [1]. The former results from interactions of a host system with guest molecules which take place at the interface between both, the latter originates from the inherent length scale on which the underlying molecular fluctuations take place. Surface effects cause a decrease while confinement effects are characterised by an increase of the molecular dynamics with decreasing spatial dimensions of the confining space (Fig.6.1). Hence in glass-forming systems [2–7]for the calorimetric glass transition temperature an increase resp. a decrease is observed. It is evident that this counterbalance must depend sensitively on the type of confined molecules (glass-forming liquids, polymers, liquid crystals), on the properties of the (inner) surfaces (wetting, non-wetting) and on the architecture of the molecules with respect to the walls (grafted, layered or amorphous systems).

It has been proven (see for instance [1–11] that dielectric spectroscopy is a useful tool to study the molecular dynamics of polymers. This is due to the fact that a broad dynamical range from the milli- to the Giga-Hertz region can be covered by this method in its modern form (see Chap. 2). Therefore motional processes which take place for polymeric systems on extremely different time scales can be investigated in a broad frequency and temperature range. Moreover the motional processes depend on the morphology and micromorphology of the system under investigation. Therefore information on the structural state of the material can be indirectly extracted by taking the molecular mobility as a probe for the structure.

There are very good reasons for the current surge of interest in fundamental and applied aspects of dielectric spectroscopy (DS) of polymeric materials. Fundamental investigations of the dielectric response yield a wealth of information about different molecular motions and relaxation processes. A unique characteristic of DS is the wide frequency range, from 10^-5 Hz to 10¹¹ Hz, over which polymers respond to an applied electric field. This remarkable breadth is the key feature that enables one to relate the observed dielectric response to slow (low frequency) and/or fast (high frequency) molecular events. Complementary information to DS studies can be obtained from nuclear magnetic resonance (NMR), dynamic mechanical analysis (DMA), quasi-elastic light scattering (QELS), quasi-elastic neutron scattering (QENS), transient fluorescence depolarization, and ultrasonic measurements, but none of those techniques can cover as wide a frequency range. A strong industrial interest in dielectric and electrical properties of polymers reflects the growing use of these materials in electronic interconnect devices, optoelectronic switches, printed board circuitry, microwave assemblies for radar, batteries, fuel cells, and so on.

Thermotropic liquid crystals form a class of matter which is located between the crystalline and the isotropic state [1,2]: In crystals molecules have maximal (positional and translational) order and minimal mobility while in liquids the reverse is the case. In liquid crystals (LC) aspects of both states are combined and — so-called — mesomorphic phases are developed in which order and mobility compete. The former results in anisotropic (optical) properties, the latter enables one to modify (to “switch”) the orientation of the molecules by use of external electric or magnetic fields. Broadband dielectric spectroscopy has turned out to be a versatile tool to study the dynamics in these systems.

The structural and dynamic properties of thin supported and freely standing polymer films are in the focus of scientific discussion [1–4]. Up to now, broadband dielectric spectroscopy has turned out to be a convenient experimental access which enables one to measure directly the molecular fluctuations of polar moieties in these systems in an extraordinary broad frequency and temperature range. A further benefit results from the fact that its sensitivity increases with decreasing sample thickness and hence with decreasing amount of sample material.

In Maxwell’s equations the current density J and the time derivative of the electric displacement \( \frac{{\partial D}}{{\partial t}} \) are additive quantities. Hence for sinusoidal electric fields the complex conductivity σ* and the complex dielectric function ɛ* are related by \( {\sigma^{*}} = i\omega {\varepsilon_0}{\varepsilon^{*}} \) (ɛ0 being the permittivity of the free space). Both quantities σ* and ɛ* are key features of (semi)-conducting (disordered) materials. Measured over a wide enough frequency — and temperature — range it enables one to analyse the underlying mechanisms of charge transport. Thereby it reflects a continuous process. At high frequencies (≥1010 Hz) the charge carriers are driven by the external electric field over distances corresponding to atomic length scales, while in the direct current (d.c.) limit of ω→ 0 they propagate on some percolation path from one side of the sample to the other. Thus with decreasing frequency a length scale is involved going from microscopic to macroscopic dimensions.

Inhomogeneous media present an interesting class of materials for dielectric research. Differences in conductivity of the phases of an inhomogeneous medium give rise to interfacial polarization, the build-up of space charges near the interfaces between the various phases. Such a polarization usually occurs at frequencies lower than the time scales typical of dipolar polarizations. Moreover, the contribution of interfacial polarization to the dielectric properties of a material is often much larger than the dipolar contributions.

Nowadays dielectric relaxation measurements can be carried out almost routinely in broad frequency ranges continuously covering ten to fifteen decades and more [1. This allows one to track the time scale on which dipolar motions occur in a wide temperature interval and to investigate in detail the shape of permittivity and dielectric loss spectra. Early on it has become clear that at a given temperature and pressure the molecular motion in most dielectric materials cannot be characterized by a unique time constant. In order to describe the experimentally observed relaxations intrinsic non-exponential as well as distributed processes have been considered. For a long time the distribution concept was quite popular in the description of dielectric phenomena [2–[4]. It was often based on the assumption that environments differing from site to site lead to locally varying time constants. However, the alternative option which starts from the consideration of nonexponential responses [5] has also gained considerable attention [6]–[9]. To justify these approaches theoretically it has been pointed out that the interactions which exist between dipolar molecules should render a description in terms of a simple distribution of relaxation times at least questionable.

This chapter is concerned with a method of measuring dielectric relaxation phenomena locally, in order to complement the information regarding the dynamics of molecules inferred from the various macroscopic dielectric techniques outlined in previous chapters. In a simplified picture, these solvation dynamics experiments measure the dielectric relaxation of a liquid as a response to a step in the dielectric displacement of a molecule rather than the macroscopic effects following a field step applied to a capacitor. The time dependent dielectric polarization in the immediate vicinity of a probe molecule gives rise to a Stokes-shift of the luminescence. This tendency of the emission wavenumbers v to shift towards the red is monitored by recording the emission spectra I (v) as a function of time using straightforward techniques of optical spectroscopy. The key quantity for assessing the dynamics of the liquid is the time dependent average emission energy 〈v (t)〉1.

Mechanical spectroscopy is a technique of material characterization in which material deformation and flow behavior is analyzed by means of dynamic mechanical methods. As in dielectric spectroscopy, advantage is taken of the material reaction to periodic variation of the external field, however with the difference that the applied field is mechanical instead of the electrical one. Materials respond to the applied field (stress or strain) by dissipating the input energy in a viscous flow (non-reversible response), by storing the energy elastically (reversible response), or through a combination of both of these two extremes. Dynamic mechanical method makes it possible to detect variation of both contributions as a function of temperature or deformation rate and to determine in this way the spectra of relaxation processes which control the viscoelastic behavior of a given material. The mechanical behavior is often a key criterion for a possibility of application of the materials even if some other properties (e.g., optical or electrical) are effectively used. As a result mechanical spectroscopy is a widely used experimental technique both as a material testing method and as a tool for analysis of the dynamics in complex polymer systems.

Broadband dielectric spectroscopy (BDS) and solid-state nuclear magnetic resonance (NMR) are two of the working horses in modern materials research. Both methods are well suited to study the dynamics in disordered condensed matter in an extremely broad range of more than 15 decades in time and/or frequency. However, the strategies employed to retrieve information from a given substance are quite different for NMR as compared to BDS. To the practitioner of conventional dielectric spectroscopy, who is used to the relatively simple sampling of the electrical polarization in the frequency or time domains, the wealth of NMR techniques sometimes appears bewildering. Another difference between the two methods is that quantum effects are rarely needed explicitly in order to analyze a dielectric experiment, while in NMR at least some elementary knowledge of quantum mechanics is required from the outset. The benefits of the seeming complexity inherent in theory and experiment of NMR, as compared to BDS, are numerous: NMR is element specific and the selectivity to the dynamics of certain sites or groups within a molecule or to certain components within a mixture can often be enhanced further by spectral filtering or isotope labeling. Some of the more sophisticated NMR techniques, e.g., when performing multidimensional spectroscopy, due to limitations imposed by the requirement to achieve an acceptable signal to noise ratio, can be relatively time consuming as compared to experiments carried out using BDS. This advantage of the latter method may be viewed as arising at the expense of the fact that usually only an integral response of the subset of those degrees of freedom is measured which couple to charge carriers or dipole moments in the sample. Hence systematic (as a function of composition, say) BDS studies are often required to obtain firm assignments of relaxation processes as seen in, e.g., dielectric loss spectra. The combined application of experimental methods, and particularly of the two complementary ones dealt with in the present chapter, has greatly advanced our understanding of the molecular dynamics in disordered materials.

Polymeric systems show a very rich variety of dynamical processes which manifest themselves in different frequency ranges at a given temperature. Between vibrations taking place at time scales faster than the picosecond range and rep-tation at very long times, a number of dynamical processes can been detected in such systems. Some of them can be specific for the particular microstructure of the polymer, like, for instance, rotations of methyl groups. However, it is well established that two relaxation processes are present in all glass-forming polymers (see, e.g., [1, 2]): the primary or structural α-relaxation and the secondary or β-relaxation, also known as the Johari-Goldstein process [3]. The two relaxations coalesce in what we will call α/β-process in a temperature range 10%–20% above the glass transition temperature T _g , The α-relaxation is commonly believed to be related to segmental relaxations of the main chain. The temperature dependence of its relaxation time shows a dramatic increase around T _g , leading to the glassy state at lower temperatures. The β-relaxation is active above as well as below T _g , and occurs independently of the existence of side groups in the polymer. This relaxation has traditionally been attributed to local relaxation of flexible parts, e.g., side groups, and, in main chain polymers, to twisting or crankshaft motion in the main chain [1]. On the other hand, the a-relaxation relates to the structural relaxation of the material and is necessarily of intermolecular nature [4]. However, the molecular nature of the secondary relaxation and its relationship with the primary relaxation are still poorly understood.