The Flexoelectric Effect

In this section we will consider the orientational deformations of a nematic director which are flexoelectric in nature, i.e., induced due to the interaction of the flexoelectric polarization (see (4.2)) with the external electric field. Our consideration will be limited to spatially uniform fields E the case when E depends on coordinates is discussed in Chapter 5. We also discuss semiphenomenological approaches for the determination of nematic flexoelectric moduli ei and 63. Different types of electrooptical phenomena, where flexoelectric distortion plays a dominant role will be considered. Some of them are promising for potential applications. 

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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>. 

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]. 

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. 

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

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). ... 

Fig. 2.4. A transverse electric field, indicated by the arrow pointed towards the right at the top of the figiu-e, tilts the apolar director as shown by double headed arrows in a specific direction due to the flexoelectric effect on a 90° twisted nematic cell. The tilting direction reverses if the field direction is reversed. The transmitted intensity mesisured with a polarized light beam traversing the cell vertically as indicated by the dashed line will be identical in the two cases. On the other hand, with an oblique beam, the transmitted intensities for the two tilted director structures will be different, and can be used to me siu-e the flexocoefficient (adapted from Kischfai et cU. ).

S.A. Jewell and J.R. Sambles, Fully leaky guided mode study of the flexoelectric effect and surface polarization in hybrid aligned nematic cells, J. Appl. Phys. 92(1), 19-24, (2002). doi 10.1063/1.1483392... 

In addition to the pear-shaped molecules, bent-shaped molecules were used to illustrate the dipolar origin of the flexoelectric effects in nematic liquid crystals. It was assumed that the constituent molecules of the nematic liquid crystals are free to rotate around their axes, and in the absence of electric fields, their dipole moments average out so the net polarization of the material is zero. However, when liquid crystals made from polar pear- or banana-shaped molecules are subjected to splay or bend deformations, respectively, they can become macroscopically polar, because the polar structures correspond to a more efficient packing of the molecules. It follows from symmetry considerations that the deformation-induced fiexo-electric polarization Pa can be written as ... 

The flexoelectric effect is a phenomenon where a space variation of the order parameter induces polarization. Chiral polar smectics are liquid crystals formed of chiral molecules and organized in layers. All phases in tilted chiral polar smectic liquid crystals have modulated structures and they are therefore good candidates for exhibiting the flexoelectric effect. The flexoelectric effect is less pronounced in the ferroelectric SmC phase and in the antiferroelectric SmC. The flexoelectric effect is more pronounced in more complex phases the three-layer SmCpu phase, the four-layer SmCFi2 phase and the six-layer SmCe a phase. 

In this chapter we consider several important aspects of the flexoelectric effect for chiral polar smectic liquid crystals and for the variety of phases. First, we discuss the reason for indirect interlayer interactions, which extend to more distant layers, and the lock-in to multi-layer structures. Second, although it was believed for a long time that polarization in tilted chiral smectics is always perpendicular to the tilt with the smectic layer normal, a component in the direction of the tilt may exist. And third, in multi-layer structures, the flexoelectricaUy induced polarization can be extremely large but is difficult to measure. 

The chapter is organized as follows The second section discusses the prototype polar smectics the ferroelectric liquid crystals. We discuss the structure of the ferroelectric phase, the theoretical explanation for it and we introduce the flexoelectric effect in chiral polar smectics. Next we introduce a new set of chiral polar smectics, the antiferroelectric liquid crystals, and we describe the structures of different phases found in these systems. We present the discrete theoretical modelling approach, which experimentally consistently describes the phases and their properties. Then we introduce the discrete form of the flexoelectric effect in these systems and show that without flexoelectricity no interactions of longer range would be significant and therefore no structures with longer periods than two layers would be stable. We discuss also a few phenomena that are related to the complexity of the structures, such as the existence of a longitudinal, i.e. parallel to the... 

Which of the phases are important for flexoelectricity As we shall see below, the flexoelectric effect is the main reason for the large variety of phases. The flexoelectric interaction is actually the reason for significant interactions with the more distant layers. In addition, phases with larger phase differences are a source of another phenomenon the local polarization can also have a component parallel to the tilt direction of the polarization. However, to understand the richness of the phenomena, let us first focus on the phenomenological model, which describes all the phases above, their properties and the phase sequence. 

In the previous section flexoelectric interactions were not considered in the free energy. We have also seen that only three of the structures found in antiferroelectric liquid crystals can be explained with the form of the free energy presented in the previous section. Let us first consider the discrete form of the flexoelectric effect and its influences on the theoretical description of the structures. We shall see that the flexoelectric effect is a source of indirect interactions between more distant layers and consequently the reason for all structures that cannot be expressed by the single phase difference. 

The historical theoretical explanation of the structures found in antifer-roelectric liquid crystals was actually different. Initially it was believed that anticlinic tilts in next nearest layers, which are the source of the competing interactions, are present because polar interactions with next nearest layers are strong enough. It was shown later that the flexoelectric effect is much more important and that these interactions are most probably the main source of the competition. 

The term is the result of the electrostatic free energy of the flexoelectrically induced part of the polarization. It can clearly be seen that this interaction originates in the flexoelectric effect and is always positive. The next nearest layers therefore favour an anticlinic orientation. The basic period of the structure, if this is the only interaction, would consist of four layers (Fig. 5.6, second row). In addition, if the flexoelectrically induced polarization is comparable to the piezoelectrically induced polarization, the indirect interaction with the next nearest layers can be surprisingly strong. 

In the expression Eq. (5.33) we can see that the flexoelectric effect when it is combined with the piezoelectric effect (the second part of the coefficient /i) has similar effects as direct chiral interactions due to the van der Waals field having chiral s mimetry around the chiral molecules given by /i. We cannot distinguish between the two components as the piezoelectric coefficient Cp and the coefficient describing direct chiral interactions /i probably depend equally (they are proportional) on the enantiomeric excess. 

As the field is the same in all layers, it can clearly be seen that the contributions remain only for surface layers, where no neighbouring layers of the same material exist. The bulk contribution cancels pairwise out and is exactly zero. This result is not new, as early papers on flexoelectricity mentioned that the flexoelectric effect is limited to a contribution at the surfaces. ... 

What are the other possibilities for measuring the flexoelectric effect We could consider dielectric coupling in a high-frequency electric field. 

In more complex chiral polar smectics, antiferroelectric liquid crystals, there are many consequences of the flexoelectric effect. It influences interlayer interactions and causes indirect interactions between more distant layers to appear and become important. The phenomenon is the reason for the appearance of commensurate structures that extend up to six layers. In addition, longitudinal polarization, i.e. the polarization that has a component parallel to the tilt, exists in more complex structures such as the SmCpi2, the SmC jj and the SmC phases. Unfortunately it seems that flexoelectric polarization cannot be detected separately from other phenomena by simple means. A way of measuring the flexoelectric contribution in tilted polar smectics still seems to be an open question. 

A lamellar lyotropic phase, essentially a stack of lipid bilayers, is a nano-heterogeneous medium and any attempt to comprehend its flexoelectricity in terms of bulk polarization and a bulk flexocoefficient would run into problems. The flexoelectric effect of the lamellar phase can most easily be described by summing up the flexovoltages of all consecutive curved bilayers that are connected in series by the inter bilayer electrolyte. 

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