Flexoelectric effect direct

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]. 

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. 

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. 

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 ... 

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. ).

The 63 values obtained from the measm-ements on ClPbislOBB using Eq. (3.27) are plotted in Fig. 3.12. It can be seen that 63 deduced from these measm-ements is in satisfactory agreement with that obtained from the direct flexoelectric effect based on polarization current detection imder an apphed cm-vatm-e strain (also plotted in Fig. 3.12). 

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. 

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 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. 

The theory and experiments of lyotropic and biomembrane flexoelectricity are reviewed. Flexoelectricity is a reciprocal relation between electricity and mechanics in soft lyotropic systems, i.e., between curvature and polarization. Experimental evidence of model and biomembrane flexoelectricity (including the direct and the converse flexoelectric effects) is reported. The biological implications of flexoelectricity are underlined. Flexoelectricity enables membrane structures to function like soft micromachines and nanomachines, sensors and actuators, thus providing important input to nanoionics apphcations. Nanobio examples include membrane transport, membrane contact, mechanosensitiv-ity, electromotility, hearing, nerve conduction, etc. 

A model using the direct flexoelectric effect for the transformation of mechanical into electrical energy in the hearing process in stereocilia has been proposed." ... 

A. Todorov, Experimental Investigations of Direct and Converse Flexoelectric Effect in BUayer Lipid Membranes, PhD Thesis, Syracuse University, 1993. 

Another bistable device is the Binem display, which does not involve flexoelectric effects but is sufficiently similar for a brief mention. This device uses a chirally doped nematic in a cell with parallel alignment directions... 

Fig. 11.29 Conversed flexoelectric effect in cells with homeotropic (a) and homogeneous (b) director alignment and electric field applied along the cell normal. Weak anchoring energy at the bottom plate allows the flexoelectric deflection of the director 3 at the surface propagating up in the vertical direction (e = 0)...

It is worthwhile to point out two characteristics of the flexoelectric effect. First, there is no threshold for the applied field, which is different from Freedericksz transition, where there is a threshold below which no deformation occurs. Deformation of the director configuration occurs under any field. Second, the direction of the bend depends on the polarity of the applied field, which is also different from Freedericksz transition where the deformation is independent of the polarity of the applied field. 

The most direct method of finding the coefficient en would be to fill the space between the metallic coaxial cylinders with a liquid crystal having pear-shaped molecules, with the surfaces of the cylinders having been pretreated for homeotropic orientation and to measure the potential difference between the cylinders. In fact, because of the difference in radii of the cylinders, the nematic liquid crystal structure proves to be splay deformed, and if the molecules have even a small longitudinal dipole moment the plates of the coaxial capacitor would be charged. However, despite its apparent simplicity, this experiment is, in fact, complicated because of the screening of the potential caused by the flexoelectric effect by firee charges from the liquid crystal and the atmosphere. 

It is well known that nematic liquid crystals are nonpolar. However, for a certain asymmetrical shape of the molecules, splay or bend deformations of the director field lead to an electrical polarization [87]. This feature is known as the flexoelectric effect. Theoretically, the influence of an electric field on CLCs for the case where the helical axis is oriented parallel to the plane of the sample was first considered by Goossens [88]. Experimentally, the flexoelectric electro-optic effect in CLCs can be observed in conventional sandwich cells with transparent electrodes when the helix axis of the CLC lies parallel to the glass surfaces [89]. In the absence of an electric field, the CLC behaves as a uniaxial material with its optic axis perpendicular to the director and parallel to the helix axis. When an electric field is applied normal to the pitch axis, the helix distorts, as shown in Figure 6.6. Thus, the optical axis is reoriented and the medium becomes biaxial. The deviation direction... 

We now turn to the changes that occur in the macroscopic structure of a liquid crystal due to a destabilization and reorientation of the director under direct action of an electric or magnetic field. The external field might be coupled either to the dielectric (diamagnetic) anisotropy (magnetically or electrically driven uniform Frederiks transition and periodic pattern formation) or to the macroscopic polarization (flexoelectric effect and ferroelectric switching) of the substance. The fluid is considered to be nonconductive. 

Figure 37. An electric field E applied perpendicular to the helix axis of a cholesteric will turn the director an angle ) and thereby the optic axis by the same amount. The director tilt is coupled to the periodic splay—bend director pattern shown below, which is generated in all cnts perpendicular to the new optic axis. In this inverse flexoelectric effect, splay and bend will cooperate if and have the same sign. The relation between E and 0 is shown for a positive helical wave vector k (right-handed helix) and a positive average flexoelectric coefficient e= (Cs+ b)-When the sign ofE is reversed, the optic axis tilts in the opposite direction (0—>-0).

Figure 41. Model of the director configuration in a helical smectic C. To the left is shown a single layer. When such layers are successively added to each other, with the tilt direction shifted hy the same amount every time, we obtain a space-filling twist-bend structure with a bend direction rotating continuously from layer to layer. This bend is coupled, by the flexoelectric effect, to an equally rotating dipole.

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