The dielectric relaxation process

The basic theory of dielectric relaxation behaviour, pioneered by Debye, begins with a macroscopic treatment of frequency dependence. This treatment rests on two essential premises exponential approach to equilibrium and the applicability of the superposition principle. In outline, the argument is as follows. 

Consider the application at time t = 0 of a steady field E0 across a dielectric. The resulting electric displacement D t) at subsequent times t will then comply with the equation  

The fust term on the right-hand side, eqe Eq, represents the instantaneous response of the material to the field. The second term, o( s — Eoo O Oj represents the slower contribution from polarisation of dipoles, with the factor F(t) describing the time development of the underlying orientation process. By this definition T(0) = 0 and f (oo) = 1. We further assume that the rates at which dipolar polarisation DP(t) progresses towards its equilibrium value DP(oo) = e0(es - oo)Eq is proportional to its degree of departure from equilibrium, i.e. 

Here r is a characteristic time constant, usually called the dielectric relaxation time. To conform to analogous theories of visco-elastic behaviour we should really use the term dielectric retardation time, because it refers to a gradual change in a strain (the polarisation or resulting electric displacement) following an abrupt change in stress (the applied field). Dielectric relaxation time is, however, still most commonly used in spite of this inconsistency. By integration of Equation (3.12)  

If polarisation of dipoles is a linear function of applied field, then a higher field Eo + Eu applied at time / = 0, will produce a proportional increase in electric displacement at all subsequent times  

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The third relaxation process is located in the low-frequency region and the temperature interval 50°C to 100°C. The amplitude of this process essentially decreases when the frequency increases, and the maximum of the dielectric permittivity versus temperature has almost no temperature dependence (Fig 15). Finally, the low-frequency ac-conductivity ct demonstrates an S-shape dependency with increasing temperature (Fig. 16), which is typical of percolation [2,143,154]. Note in this regard that at the lowest-frequency limit of the covered frequency band the ac-conductivity can be associated with dc-conductivity cio usually measured at a fixed frequency by traditional conductometry. The dielectric relaxation process here is due to percolation of the apparent dipole moment excitation within the developed fractal structure of the connected pores [153,154,156]. This excitation is associated with the selfdiffusion of the charge carriers in the porous net. Note that as distinct from dynamic percolation in ionic microemulsions, the percolation in porous glasses appears via the transport of the excitation through the geometrical static fractal structure of the porous medium. 

The conversion of the initially formed Si np state to the Si ct state by intramolecular electron transfer is very fast and varies in a way that parallels but does not exactly correspond to the dielectric relaxation time for the solvent used. This is because the local environment around the excited-state molecule is different from that surrounding a solvent molecule [120, 340]. That is, the ICT process is to a large extent determined by the dielectric relaxation processes of the solvent surrounding the ANS molecule. Thus, solvent motion seems to be the controlling factor in the formation and decay of the ICT excited state of ANS and other organic fluorophores [120, 340]. A detailed mechanism for fast intramolecular electron-transfer reactions of ANS and 4-(dimethylamino)benzonitrile, using two simplified molecular-microscopic models for the role of the solvent molecules, has been given by Kosower [340] see also reference [116]. 

Thermally stimulated depolarization currents are detected in a sample first cooled to low temperature in a capacitor with shorted electrodes, then warmed slowly with the electrodes connected to a sensitive d.c. electrometer. In this way the dielectric relaxation processes occurring in the sample are displayed separately, according to their activation energies and barrier heights, during the scan over temperature. 

The dielectric relaxation process of ice can be understood in terms of proton behavior namely, the concentration and movement of Bjerrum defects (L- and D-defect) and ionic defects (HaO and OH ), which are thermally created in the ice lattice. We know that ice samples highly doped with HE or HCl show a dielectric dispersion with a short relaxation time r and low activation energy of The decreases in the relaxation time and... 

The dielectric relaxation processes of matter can be analyzed with an empirical model of dielectric dispersion, for example, the one described by Havriliak-Negami s equation. " We analyzed dielectric data obtained for our samples using a model of complex permittivity k with two dispersions (the main and the low-frequency dispersion of a space charge effect) and conductivity ao (caused by electrode discharge), as follows ... 

To explain the bimodal dielectric relaxation in aqueous protein solutions, Nandi and Bagchi proposed a similar dynamic exchange between the bound and the free water molecules [21]. The bound water molecules are those that are attached to the biomolecule by a strong hydrogen bond. Their rotation is coupled with that of the biomolecule. The water molecules, beyond the solvation shell of the proteins, behave as free water molecules. The free water molecules rotate freely and contribute to the dielectric relaxation process, whereas the rotation of the doubly hydrogen-bonded bound water molecules is coupled with that of the biomolecule and hence is much slower. The free and bound water molecules are in a process of constant dynamic exchange. The associated equilibrium constant, K, can be written as... 

The dielectric relaxation process can be described formally by the following relations ... 

Studies of the dielectric relaxation process in the region of Eg show that the frequency-temperature dependence is not of Arrrhenius type (Figure 7.7). The process being observed corresponds to the rotation of the ester group around the polymer backbone and is designated the a process ... 

Maintaining polar order in a poled pol5uner is of great importance for second-order applications (88,89). The dielectric relaxation process leading to decay in the orientation of ordered polymers has been studied extensively and is the subject of another article (see Dielectric Relaxation). Several models that describe the chromophore reorientation for NLO materials have been proposed, including the Kohlrausch-Williams-Watts (KWW) model (90,91), biexponential and triexponential decay models (92), time-dependent Debye relaxation time models (93), and the Liu-Ramkrishna-Lackritz (LRL) model (94). For further information on... 

Einfiled, J., Meipner, D., and Kwasniewski, A. (2003). Contribution to the molecular origin of the dielectric relaxation processes in polysac-charidesr The high temperature range. Journal of Non Crystalline Solids 320, 40-55. 

Hence the dielectric relaxation process cannot possibly be assigned to the relaxation of the transport of charges their presence either as free ions or as triplets accounts only to a minute minority of the total solute population around csQ.lM. 

Sato, T., Chiba, A., and Nozaki, R. (1999). Dynamical aspects of mixing schemes in ethanol-water mixtures in terms of the excess partial molar activation free energy, enthalpy, and entropy of the dielectric relaxation process. J. Chem. Phys., 110, 2508-2521. 

The results of the D-B and H-H models, as shown in a comparison of Figures 4-7 with Figures 32-35 are very similar. In both cases a = 0.6 and approximately independent of either the K or the shape parameter The )3 parameter changes in a sigmoidal pattern from about 0.35 to 1.0 for the D-B model and from 0.35 to about the 0.7-0.8 range for the H-H model. In other words, both the D-B and the H-N models predict the shape of the dielectric relaxation process to have nearly the same dependence on sphere diameter or segment asymmetry and both models have very different starting points. 

In the case of DiMarzio and Bishop, they solved the hydrodynamic equations for the Debye model and the non-Newtonian case exactly. The important result of their analysis is that the dielectric response is no longer a Debye type but depends explicitly on how the local viscosity depends on time. In other words the nature of the viscoelastic properties surrounding the sphere determines the shape of the dielectric relaxation process. This result is in marked contrast to the results of the model... 

The main conclusion of this part is that the dielectric response of a nematic material is strongly influenced by the dielectric relaxation processes. One of the consequences is that the sharp front of an applied voltage pulse can be perceived by the NLC as a high-frequency field for which the anisotropy of dielectric permittivity is different than that for the (low) frequency of the driving field, or even be of the opposite sign. Of course, the effect should manifest itself not only for DFN, but for all nematic materials the difference would be only in the time/frequency domain where the effect is most pronounced. For example, if would be of interest to verify whether the delay effects observed by Clark s group in Ref. [8] at the scale of tens of nanoseconds are caused by the dielectric dispersion effect described above. 

Dielectric constants and refractive indices, as well as electrical conductivities of liquid crystals, are physical parameters that characterize the electronic responses of liquid crystals to externally applied fields (electric, magnetic, or optical). Because of the molecular and energy level stractures of nematic molecules, these responses are highly dependent on the direction and the frequencies of the field. Accordingly, we shall classify om studies of dielectric permittivity and other electro-optical parameters into two distinctive frequency regimes (1) dc and low frequency, and (2) optical frequency. Where the transition from regime (1) to (2) occurs, of course, is governed by the dielectric relaxation processes and the dynamical time constant typically the Debye relaxation frequencies in nematics is on the order of 10 ° Hz. 

Fig. 4.8. The mean relaxation rate of the dielectric relaxation process as a function of temperature for (a) heptylcyanobiphenyl, and (b) pentylcyano-biphenyl. Open circles and the solid line correspond to experimental data, and broken lines to theoretical predictions. (From Refs 32 and 36).

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