Permittivities frequency effects

In this section we wish to consider all the possible contributions to the electric permittivity of liquid crystals, regardless of the time-scale of the observation. Conventionally this permittivity is the static dielectric constant (i.e. it measures the response of a system to a d.c. electric field) in practice experiments are usually conducted with low frequency a.c. fields to avoid conduction and space charge effects. For isotropic dipolar fluids of small molecules, the permittivity is effectively independent of frequency below 100 MHz, but for liquid crystals it may be necessary to go below 1 kHz to measure the static permittivity polymer liquid crystals can have relaxation processes at very low frequencies. 

The dielectric constant (permittivity) tabulated is the relative dielectric constant, which is the ratio of the actual electric displacement to the electric field strength when an external field is applied to the substance, which is the ratio of the actual dielectric constant to the dielectric constant of a vacuum. The table gives the static dielectric constant e, measured in static fields or at relatively low frequencies where no relaxation effects occur. 

Crystals with one of the ten polar point-group symmetries (Ci, C2, Cs, C2V, C4, C4V, C3, C3v, C(, Cgv) are called polar crystals. They display spontaneous polarization and form a family of ferroelectric materials. The main properties of ferroelectric materials include relatively high dielectric permittivity, ferroelectric-paraelectric phase transition that occurs at a certain temperature called the Curie temperature, piezoelectric effect, pyroelectric effect, nonlinear optic property - the ability to multiply frequencies, ferroelectric hysteresis loop, and electrostrictive, electro-optic and other properties [16, 388],... 

Fig. 1.7 Effect of frequency and temperature on the complex dielectric permittivity (s is the real part and the imaginary part or the dielectric loss) [12].

In the equation s is the measured dielectric constant and e0 the permittivity of the vacuum, M is the molar mass and p the molecular density, while Aa and A (po2) are the isotope effects on the polarizability and the square of the permanent dipole moment respectively. Unfortunately, because the isotope effects under discussion are small, and high precision in measurements of bulk phase polarization is difficult to achieve, this approach has fallen into disfavor and now is only rarely used. Polarizability isotope effects, Aa, are better determined by measuring the frequency dependence of the refractive index (see below), and isotope effects on permanent dipole moments with spectroscopic experiments. 

Dispersions of nanoparticles in ferroelectric liquid crystals (FLCs) predominantly focused on induced or altered electro-optic effects, but also on the alignment of FLCs. Raina and co-workers reported on a gradual decrease of the dielectric permittivity, e, by doping with SiC>2 nanoparticles at frequencies up to 1 kHz and a rather minor increase of as well as an increase in optical transmission at frequencies above 2 kHz [279]. Liang et al. used BaTiC>3 nanoparticles (31 nm in diameter after grinding commercially available 90 nm nanoparticles Aldrich) and showed, perhaps expectably, a twofold increase in the spontaneous polarization... 

We have introduced the effective complex susceptibility x ( ) = X,( )+ X ) stipulated by reorienting dipoles. This scalar quantity plays a fundamental role in subsequent description, since it connects the properties and parameters of our molecular models with the frequency dependences of the complex permittivity s (v) and the absorption coefficient ot (v) calculated for these models. 

In spite of numerous studies, the properties of liquid water are still far from been understood at a molecular level. For instance, large isotope effects are seen in some properties, such as the temperature of maximum density, which occur at 277.2 K in liquid H20 and 284.4 K in D20. The isotope shift 7.4 K will be used below with the purpose to employ the Liebe et. al. formula [17] for calculation of the low-frequency dielectric permittivity of D20 in analogous way as it used for H20. 

Figure 35. Frequency dependence in the submillimeter wavelength region of the real (a, b) and imaginary (c, d) parts of the complex permittivity. Solid lines Calculation for the composite HC-HO model. Dashed lines Experimental data [51]. Dashed-and-dotted lines show the contributions to the calculated quantities due to stretching vibrations of an effective non-rigid dipole. The vertical lines are pertinent to the estimated frequency v b of the second stochastic process. Parts (a) and (c) refer to ordinary water, and parts (b) and (d) refer to heavy water. Temperature 22.2°C.

Notation. relative permittivity, K -imaginary relative permittivity, A-surface conduction, subscript "elT" denotes an effective quantity, f frequency, vv vvater content... 

Dielectric permittivity and loss for both polymers under study can be observed on Figs. 2.17 and 2.18. In both figures a prominent peak corresponding to the dynamic glass transition temperature can be observed, which at low frequencies is overlapped by conductivity effects. Moreover, in both polymers a broad secondary peak is observed at about -50°C. This peak is more prominent in P2tBCHM which is in good... 

Figures 2.37 and 2.38, show the isochronal curves of the permittivity and loss factor for P2NBM and P3M2NBM as a function of temperature at fixed frequencies. A prominent relaxation associated with the dynamic glass transition is observed in both polymers. Clearly the effect of the methyl substitution in position 3 of the norbornyl group is to decrease the temperature of this relaxational process.

Fig. 2.39 Dielectric permittivity (A) and loss ( ) for P3M2NBM in the frequency domain at 105°C. The discontinuous straightline represents the conductive effects. Circular black points ( ) represent the resulting loss curve after subtraction showing the dipolar a - relaxation. The continuous line correspond to the Havriliak-Negami curve fit [35,70], (From ref. [35])...

Many of the different susceptibilities in Equations (2.165)-(2.167) correspond to important experiments in linear and nonlinear optics. x<(>> describes a possible zero-order (permanent) polarization of the medium j(1)(0 0) is the first-order static susceptibility which is related to the permittivity at zero frequency, e(0), while ft> o>) is the linear optical susceptibility related to the refractive index n" at frequency to. Turning to nonlinear effects, the Pockels susceptibility j(2)(- to, 0) and the Kerr susceptibility X(3 —to to, 0,0) describe the change of the refractive index induced by an externally applied static field. The susceptibility j(2)(—2to to, to) describes frequency doubling usually called second harmonic generation (SHG) and j(3)(-2 to, to, 0) describes the influence of an external field on the SHG process which is of great importance for the characterization of second-order NLO properties in solution in electric field second harmonic generation (EFISHG). 

The dispersion interaction between an atom and a metal surface was first calculated by Lennard-Jones in 1932, who considered the metal as a perfect conductor for static and time-dependent fields, using a point dipole for the molecule [44], Although these results overestimate the dispersion energy, the correct l/d3 dependence was recovered (d is the metal-molecule distance). Later studies [45 17] extended the work of Lennard-Jones to dielectrics with a frequency-dependent dielectric constant [48] (real metals may be approximated in this way) and took into account electromagnetic retardation effects. Limiting ourselves to small molecule-metal distances, the dispersion interaction of a molecule characterized by a frequency-dependent isotropic polarizability a embedded in a dielectric medium with permittivity esol (note that no cavity is built around the molecule) reads ... 

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