Dielectric permittivity, nematics

Physical properties of liquid crystals are generally anisotropic (see, for example, du Jeu, 1980). The anisotropic physical properties that are relevant to display devices are refractive index, dielectric permittivity and orientational elasticity (Raynes, 1983). A nematic LC has two principal refractive indices, Un and measured parallel and perpendicular to the nematic director respectively. The birefringence An = ny — rij is positive, typically around 0.25. The anisotropy in the dielectric permittivity which is given by As = II — Sj is the driving force for most electrooptic effects in LCs. The electric contribution to the free energy contains a term that depends on the angle between the director n and the electric field E and is given by... 

Anisotropic fluids, of which nematic liquid crystals are the most representative and simplest example, are characterized by an anisotropic dielectric permittivity. The nematic phase has D,yuh symmetry, and in a laboratory frame with the Z axis parallel to the C , symmetry axis (the director) the permittivity tensor has the form ... 

The principal components of the dielectric permittivity of nematics are related to the average dipole moment of the constituting molecules as ... 

The nematic mean-field U, the molecule-field interaction potential, WE, and the induced dipole moment, ju d, are evaluated at different orientations using Equation (2.263), and then the coefficients of their expansion on a basis of Wigner rotation matrices can be calculated, according to Equation (2.268). The permittivity is obtained by a self-consistency procedure, because the energy WE and the induced dipole moment / md, as well as the reaction field contribution to the nematic distribution function p( l), themselves depend on the dielectric permittivity. 

A. Ferrarini, Dielectric permittivity of nematics with a molecular based continuum model, Mol. Cryst. Liq. Cryst., 395 (2003) 233-252. 

This anisotropy, illustrated by refractive index, extends to other properties, and common properties of interest would be the anisotropy in linear polarizability (Aa), dielectric permittivity (Ae), and diamagnetism (Ax). In the nematic phase, these properties are quite strongly temperature dependent the order parameter, S, increases as samples cool away from the N-I transition. This is illustrated in Figure 19 where it is also seen that the parallel component has the stronger temperature dependence as it is the orientational correlations that increase on cooling. 

Abstract A systematic overview of various electric-field induced pattern forming instabilities in nematic liquid crystals is given. Particular emphasis is laid on the characterization of the threshold voltage and the critical wavenumber of the resulting patterns. The standard hydrodynamic description of nematics predicts the occurrence of striped patterns (rolls) in five different wavenumber ranges, which depend on the anisotropies of the dielectric permittivity and of the electrical conductivity as well as on the initial director orientation (planar or homeotropic). Experiments have revealed two additional pattern types which are not captured by the standard model of electroconvection and which still need a theoretical explanation. 

POM), the Schlieren and homeotropic defect textures of the nematic phase, and the focal conic and homeotropic defect textures of the smectic A phase could be observed (Fig. 4.3). In addition, the parallel and perpendicular dielectric permittivities of mixtures of the GNPs dispersed in host nematic LCs were determined by using one-cell method , which was reported by Clark et al. [47 9]. 

Fig. 7.5 Typical temperature behavior of principal dielectric permittivities for two nematic liquid crystals, one with positive solid lines) and the other with negative (dash lines) dielectric anisotropy...

Fig. 7.8 Typical temperature behavior of the principal dielectric permittivities in the nematic and SmA phases...

Fig. 7.15 Spectra of principal dielectric permittivities for nematic phase. Qiaracteristic dispersion ranges correspond to relaxation modes with frequencies coi, CO2 and CO3 illustrated by Fig. 7.14...

If a nematic liquid crystal has negligible conductivity the results of Sections 11.2.1-11.2.5 for the Frederiks transition induced by a magnetic field may be directly applied to the electric field case. To this effect, it suffices to substitute H by E and all components of magnetic susceptibility tensor Xij hy correspondent components of dielectric permittivity tensor s,y. From the practical point of view the electrooptical effects are much more important and further on we discuss the optical response of nematics to the electric field. 

To show this, it is necessary to insert the Fourier components E(q) of the dielectric permittivity tensor e( ) of the cholesteric into the general formula for the scattering cross section a oc (r s(q) f) as already discussed for nematics in Section 11.1.3. Here f and r are polarization vectors for the incident and reflected light, q is the wavevector of scattering coinciding in this simple geometry with the wavevector of the reflected wave [2]. 

Let us simulate an appearance of the higher harmonics and optical properties of the cholesteric structure with the following parameters typical of chiral materials based on the well-known nematic mixture E7 helical pitch 0.4 pm, elastic modulus K22 = 5 X 10 dyn (or 5 pN) principal dielectric permittivity values Sn = 20,... 

Here, f, Fx and are local field factors, a is a molecular parameter including the dipole moment, second order and third order hyperpolarizabilities. Under condition F = Fx = the formula does not agree with the experiment for 8CB. Indeed, it follows that rji should increase at the transition from the nematic to smectic A phase according to the increase in (F2) However, in the experiment, Fn markedly decreases. The similar temperature behavior was earlier observed for dielectric permittivity of 8CB. In the latter case the decrease in is due to the antiparallel correlation of molecular dipoles in the smectic A phase, which results in a decrease in the effective dipole moment /x Thus, the decrease... 

As a rule, the phase transition from the isotropic phase into the nematic phase is a weak first-order transition [6] with a small jump in the order parameter 5 (Fig. 1.3 [7]) and other thermodynamic properties. The so-called clearing point corresponds to this first-order transition temperature Tni. At the same time, in the pretransitional region of the isotropic phase we can observe the temperature divergence in some physical parameters, such as heat capacity, dielectric permittivity, etc., according to the power law (T — T i) where T j is the other, virtual, second-order phase transition point, (Tni — T 0.1 K) and t) is an exponent, depending on the physical property under consideration. 

Recently, Sharma has proposed some extension of the Maier-Meier approach to the case of nematogens with antiparallel dipole-dipole correlations of the molecules. He treated a polar LC material as a mixture of unpaired molecules with a finite dipole moment /u. and antiparallel pairs with zero dipole moment. The molecules interact with each other through a combination of the generalized Maier-Saupe pseudopotential for nematic mixtures and a reaction field energy term calculated from an extension of the Maier-Meier theory. Additionally, it was assumed that a dipole with dipole moment fi is embedded in a spherical cavity of dielectric permittivity n, which is surrounded by a medium of average dielectric permittivity e. In that case the expressions for the cavity field factor h and the reaction field factor / are given by h + n ), /= (e - rt")/[2rre a (2e-i-n )] and the left sides of... 

Equation (4) is valid with high accuracy because induced magnetic dipoles interact only weakly. By contrast, interaction between the induced electric dipoles is strong and produces substantial local field effects which do not allow one to express the dielectric permittivity of the nematic phase in terms of the molecular parameters in a simple way (see, for example, [4]). 

Experimentally, it is indeed observed that the complex dielectric permittivity of a nematic liquid crystal has two dispersion regions one when the measuring external electric field is parallel and one when it is perpendicular to the director. A typical experimental dielectric spectra of a nematic liquid crystal 4-pentylphenyl-4-propoylbenzoate is shown in Figure 8.6. 

A. Jakli, A. Buka, Rotational Brownian Motion and dielectric permittivity in nematic liquid crystals. Reprint KFKI-1984-23 (HU ISSN 0368 5330, ISBN 963... 

Figure 10-2. A typical frequency dependence of real and imaginary parts of the components of the dielectric permittivity tensor for a nematic material...

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. 

The static dielectric permittivity is an important parameter that characterises the response of a medium to the application of an electric field. Its value is determined by the distribution of the electric charges in molecules (polar and non-polar compounds) as well as by the intermolecular interactions (for example anisotropy of the medium and intermolecular correlations). In nematics the dielectric permittivity is a tensorial quantity. The value and the sign of the dielectric anisotropy play an important role in the application of nematics in display technologies. 

For an anisotropic medium, like the nematic phase, the dielectric permittivity is a tensorial quantity. Due to the axial symmetry of the nematic phase we are dealing with two principal components of the permittivity, and 8j., which define the dielectric anisotropy Ae = - 8j. is measured if the electric field E is parallel to the director n, E n, whereas 8j corresponds to the perpradicular geometiy E n (see FIGURE 1). 

FIGURE 2 Dielectric permittivities and the refractive indices squared in the nematic and isotropic phases of typical nematogens (a) non-polar [33], (b) polar with P 68° [34], and (c) strsmall angle from the effective symmetry axis of the molecule (P 30°) [35]. In case (c) the parallel component was measured frx two kinds of (vioiting fields B = 0.7 T (fiill points) and E = 2500 V/cm (op i points). For 6CX2B the values of n are not shown as th are considerably smaller than the pomittivities. 

Liquid crystals in the nematic phase are dielectrically anisotropic, and the real (s ) and imaginary (s ) parts of the dielectric permittivity have two independent components, corresponding to the parallel and perpendicular orientations, respectively, of the applied electric field with respect to the nematic director. The imaginary part of the permittivity [12] is given by the phenomenological Havriliak-Negami equation... 

Due to the uniaxial or symmetry of a nematic phase, the dielectric permittivity of a nematic is represented by a second rank tensor with two principal elements, 8 and 8 The component 8 is parallel to the macroscopic symmetry axis, which is along the director, and is perpendicular to this. According to a molecular field theory, they are approximated by... 

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