Flexoelectric effect

Binary molecular co-crystals of 2,5-bis(3-pyridyl)-l,3,4-oxadiazole and 2,5-bis-(4-pyridyl)-l,3,4-oxadiazole with benzene-1,3,5-tricarboxylic and benzene-1,2,4,5-tetracarboxylic acids were studied by X-ray and thermogravimetric analysis of mass loss <2005MI1247>. Dipole moments were used to study the flexoelectric effect in guest-host mixtures of 2,5-(4-pentylbenzene)-l,3,4-oxadiazole with commercial liquid crystal hosts <2005CM6354>. The luminescence properties of many other copolymers were also investigated (see Section 5.06.12.3). 

A series of model nematic liquid crystals (among them oxadiazole derivatives) with transverse dipole moments were used to study the flexoelectric effect in guest-host mixtures with a commercial liquid crystal host <2005CM6354>. 

Converse flexoelectric effects (i.e. voltage-generated curving) have been demonstrated in uranyl-acetate-stabilized phosphatidylserine BLMs by real-time stroboscopic interferometric measurements the obtained satisfactory agreement between the converse and the direct (i.e. curvature-generated voltage) flexoelectric coefficients have been in accord with the Maxwell relationship [8]. 

The enhanced chirality by doping SmC with BSMs can be explained qualitatively in the same way as in the N phase. However, the situation is more complicated in SmC because of spontaneous polarization and flexoelectric effect, and (3) must be replaced by an equation including such effects. Actually, the contribution of flexoelectric effect has been discussed by Gorecka et al. [4]. The other important effect is caused by the fact that the BSMs are in the tilted smectic phase. As mentioned above, the tilt of BSMs induces chirality as observed in the B2 phase. 

The third electro-optical effect using calamitic nematic liquid crystals makes use of a flexoelectric effect manifested by a curved asymmetrical nematic medium. This corresponds to piezoelectricity in crystals. The existence of flexoelectricity in a nematic phase under certain boundary conditions was predicted in the late 1960s and then confirmed experimentally several years later. However, LCDs using this effect, such as bistable nematic displays are only in the development stage and as such they will not be discussed in this monograph. 

As possible explanations, several ideas have been proposed a hand-waving argument based on destabilization of twist fluctuations" [52], a possibility of an isotropic mechanism based on the non-uniform space charge distribution along the field [53] and the flexoelectric effect [55-57]. 

The centro-symmetry may, however, be broken by the application of an applied dc electric field [12] such symmetry can also be broken on a surface either as freely suspended film or by surface alignment modification technique that induces flexoelectric effect [36]. More recently, Sukhov and Timashev have shown that the centro-symmetry can also be broken optically [46]. The main obstacle in getting efficient harmonic generation then Is the phase matching of the fundamental and the second harmonic s wave vectors. 

Figure 6.26 The flexoelectric effect in BaTiO (a) the evolution of the potential energy curve under a homogeneous stress and in a strain gradient (b) domain switching via mechanical stress imposed by a probe (Original data from Lu et al. (2012))...

In this book the flexoelectric effect is mainly considered from the phenomenological point of view. At the same time it is very interesting and important to reveal the molecular origin of flexoelectricity and, in particular, to consider different types of intermolecular interactions that may be responsible for the dipolar ordering in a deformed liquid crystal, and to study the effects of intermolecular correlations and the molecular structure. This problem can only be solved using a molecular-statistical theory, which eventually allows us to express the flexoelectric coefficients in terms of molecular model parameters using various approximations. 

It should be noted, however, that the flexoelectric effect is not necessarily related to the ordering of molecular dipoles. Frost and Marcerou proposed another microscopic mechanism of the flexoelectric effect, which requires neither the asymmetry of the molecular shape nor the permanent molecular dipole. The macroscopic polarization may simply appear in the direction of the gradient of average density of the molecular quadrupole moments. The quadrupole mechanism of flexoelectricity is more general because, in principle, it should manifest itself in any anisotropic material with a non-zero quadrupole density including solid crystals d and elastomers. 

Finally note that the flexoelectric effect is also important in the smectic phase although the corresponding molecular theory is at a rudimentary stage. Recently a molecular model for the conventional and the so-called discrete flexoelectric effect in tilted smectic phases has been proposed... 

At the same time experimental facts indicate that the difference between the flexocoefficients is non-zero and even rather large for a number of nematic materials, and it strongly depends on the absolute value and the orientation of the permanent dipole within the molecular structure.Moreover, the difference between the flexocoefficients determines the flexoelectro-optic effect, which has been extensively studied experimentally. There exist also some other experimental data which, in principle, allowed us to distinguish between dipolar and quadrupolar flexoelectricity. This can be considered as an argument in favour of the dipolar interpretation of the flexoelectric effect. On the other hand, the actual ratio of the dipolar and quadrupolar contributions to the flexocoefficients of particular nematic materials remains unknown. It is only possible to speculate... 

A molecular-statistical theory of the flexoelectric effect in the nematic phase can be derived in a general way using the density-functional approach to the theory of liquid crystals. In this approach, the free energy of a liquid crystal, F, is a functional of the density po(a ) = Po/(w) where /(w) is the orientational distribution fimction. The general structure of the functional F p) is not known, but the functional derivatives are known and are related to the direct correlation functions of the nematic phase. 

In the present section we consider only the flexoelectric effect relating to the polar molecular shape, and we therefore assume that the pair attraction interaction potential V(xi,X2,ri2) is even in ai,a2,bi,b2. In this case the first term in Eq. (1.30) does not contribute to the flexoelectric coefficients, which are determined mainly by steric dipoles. 

As mentioned above, the first expressions for the flexoelectric coefficients were obtained by Helfrich and Petrov and Derzhanski while a systematic molecular-statistical theory was developed later by Straley. The results of these two approaches were compared by Marcerou and Prost who concluded that the theories of Helfrich and Petrov and Derzhanski and of Straley describe different mechanisms for the dipolar flexoelectric effect because Straley s theory 5nelds values for the flexocoefficients that are two orders of magnitude smaller than the experimental ones, and which therefore can be neglected. 

Let us now discuss the approximate expressions for the flexoelectric coefficients, Eq. (1.31), in more detail. Firstly, note that the expressions for both coefficients and es contain terms proportional to both S and S. It has been assumed in the literature that the dipolar contribution to the flexoelectric coefficients is always proportional to while the quadrupole contribution is proportional to S, and even the method of separation between the dipolar and quadrupolar flexoelectric effect has been proposed based on these preliminary results. The results of the consistent molecular theory presented in this section allow us to conclude that the relation e S for the dipolar contribution is due to the shortcomings of the semi-phenomenological approach. The results of this section also cast some doubt on the quantitative ratio of the dipolar and quadrupole contributions based on a comparison of the two terms in the expression e = eoS + C2S. At the same time, the absence of the linear term in S in the dependence e S) for a number of nematic materials stUl points to the predominant role of the quadrupole flexoeffect for those materials. 

Secondly, it follows from Eqs (1.31) and (1.32) that the longitudinal molecular dipole d provides a much smaller contribution to the flexocoef-ficients than the transverse dipole d , since A/k 10. Thus we conclude that the dipole flexoeffect is expected to be important only for molecules with large transverse dipoles. Note that the significant dipole flexoeffect has indeed been determined for nematics composed of molecules with large transverse dipoles. For cyanobiphenyl liquid crystals Marcerou and Frost did not find any dipolar flexoelectric effect, which may be determined not only by the tendency to form dimers with antiparallel dipoles but also by a relatively small contribution from transverse molecular dipoles to the flexoelectric coefficients. 

M.A. Osipov, Molecular theory of flexoelectric effect in nematic liquid crystals, Sov. Phys. JETP 58(6), 1167-1171, (1983). 

Y. Singh and U.R Singh, Density-functional theory of the flexoelectric effect in nematic liquids, Phys. Rev. A 39(8), 4254-4262, (1989). 

J. Stelzer, R. Berardi and C. Zannoni, Flexoelectric effects in liquid crystals formed by pear-shaped molecules. A computer simulation study, Chem. Phys. Lett. 299(1), 9-16, (1999). doi 10.1016/S0009-2614(98)01262-7... 

A.V. Emelyanenko and M.A. Osipov, Theoretical model for the discrete flexoelectric effect and a description for the sequence of intermediate smectic phases with increasing periodicity, Phys. Rev. E 68(5), 051703/1-16, (2003). doi 10.1103/PhysRevE.68.051703... 

This chapter is concerned with experimental measurements of flexo-electric coefficients. After a brief introduction to flexoelectricity in nematic liquid crystaJs, some applications exploiting the flexoelectric effect and the influence of this effect on electrohydrodynamic instabilities are pointed out. Flexoelectricity axises in samples with a splay-bend distortion in the director field and as such its measurement is not as direct as for dielectric constants. The theoretical background needed to analyse electro-optic experiments and extract the flexocoefficients is outlined in Section 2.2. Various experimental techniques that have been developed are described in Section 2.3. These involve cells in which the alignment of the nematic director is homeotropic, or planar or hybrid. In the first case, the interdigitated electrode technique is particularly noteworthy, as it has been used to establish several features of flexoelectricity (1) the effect can arise purely from the quadrupolar nature of the medium, and (2) the dipolar contribution relaxes at a relatively low frequency. 

Since the flexoelectric effect is associated with curvature distortions of the director field it seems natural to expect that the splay and bend elastic constants themselves may have contributions from flexoelectricity. The shape polarity of the molecules invoked by Meyer will have a direct mechanical influence independently of flexoelectricity and can be expected to lower the relevant elastic constants.The flexoelectric polarization will generate an electrostatic self-energy and hence make an independent contribution to the elastic constants. In the absence of any external field, the electric displacement D = 0 and the flexoelectric polarization generates an internal field E = —P/eo, where eq is the vacuum dielectric constant. Considering only a director deformation confined to a plane, and described by a polar angle 9 z), and in the absence of ionic screening, the energy density due to a splay-bend deformation reads as ... 

Though this type of periodic structure with multiple arches of the director is difficult to generate in a nematic, it is already present in a cholesteric liquid crystal when viewed in a plane whose normal makes an oblique angle with the helical axis. The flexoelectric effect changes the periodicity of this structure under a DC field applied normal to the helical axis, effectively rotating the latter. This can be used in tmn to measure (ei — 63). ... 

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