Flexoelectric effect, ferroelectric

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. 

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... 

We finally estimate the flexoelectric effect value. For ferroelectric PbTiOs at room temperature we have SI units, which is comparable... 

Among different (like flexoelectric, flexomagnetic etc.) flexoeffects, the influence of flexoelectric effect on the nanosystem properties had been studied in most details. One can conclude that even rather moderate flexoelectric effect significantly renormalizes all the polar, piezoelectric and dielectric properties and the correlation radius in particular. The effect also suppresses the size-induced phase transition from ferroelectric to paraelectric phase and thus stabilizes the ordered phase in ferroic nanoparticles. 

The divergences of dielectric permittivity and correlation radius at the critical value of the flexoelectric coefficient (related to the critical radius) give new possibilities to control the physical properties of ferroelectric materials. The effect of the correlation radius renormalization by the flexoelectric effect alters the intrinsic width of domain walls. The predicted effects are useful for design of ferroelectric nanowires with radius up to several nanometers, which have ultra-thin domain walls and reveal polar properties close to those in bulk samples. 

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. 

Although the nematic phase is nonpolar, there are very interesting and important polar effects in this phase, in a sense analogous to piezoelectric effects in solid crystals. This was recognized by Meyer [18] in 1969. These so-called flexoelectric effects are discussed in Sec. 2.4 of this Chapter. Meyer also recognized [61] in 1974 that all chiral tilted smectics would be truly polar and the first example of this kind, the helielectric smectic C, was presented [62] in 1975. Out of Meyer s discovery grew the whole research area of ferroelectric and antiferroelectric liquid crystals, which is today a major part of liquid crystal physics and chemistry. 

As noted earlier, the incorporation of chiral groups in the liquid crystal moieties can have the effect of inducing non-linear properties, which include thermochromism, ferroelectricity, antiferroelectricity, electrostriction, and flexoelectricity. In a now classical study, Hult [82] demonstrated that it was possible for supermolecular material 34 to exhibit two-state ferroelectric switching. The remarkable material he investigated, shown in Fig. 30, was found to exhibit two hitherto unclassified mesophases between the smectic... 

Converse flexoelectric studies of lyotropic liquid crystals, such as vesicles, is still an active subject. Notably, the sensory mechanism of outer hair cell composite membranes " can be understood by the flexoelectric properties of the lipid bilayer. The converse of this effect, i.e., a voltage-generated curvature, has also been observed and was discussed by Todorov et Another related phenomenon is the ferroelectricity which results from the tilted layered structures of chiral molecules, which has been discussed extensively since the 1980s.Ferroelectric phases are called... 

The improper ferroelectricity origin can be roughly explained in the following way, see Fig. 4.31a. Elastic strain induces the polarization vector F, dil]uki via the flexoelectric and piezoelectric effects. Since the strain... 

Catalan, G., Sinnamon, L.J., Gregg, J.M. The effect of flexoelectricity on the dielectric properties of inhomogeneously strained ferroelectric thin films. J. Phys. Condens. Matter 16, 2253-2264 (2004)... 

It should be noted that phenomenologically this effect is analogous to the deformed helix electrooptical mode observed in ferroelectric liquid crystals where coupling of an electric field with the spontaneous (instead of flexoelectric) polarization is used [83, 84]. 

This book was conceived as a renewed version of the earlier published original book, Electro-Optical and Magneto-Optical Properties of Liquid Crystals (Wiley, Chichester, 1983) written by one of us (L.M. Blinov). That book was first published in Russian (Nauka, Moscow, 1978) and then was modified slightly for the English translation. Since then new information on electrooptical effects in liquid crystals has been published. Novel effects have been discovered in nematics and cholesterics (such as the supertwist effect), and new classes of liquid crystalline materials, such as ferroelectric liquid crystals, appear. Recently, polymer liquid crystals attracted much attention and new electrooptical effects, both in pure polymer mesophases and polymer dispersed liquid crystals, were studied. An important contribution was also made in the understanding of surface properties and related phenomena (surface anchoring and bistability, flexoelectricity, etc.). 

This effect is observed in a geometry where the cholesteric axis h is homogeneously oriented in the plane of the cell (along x) and an electric field is applied to the electrodes of a sandwich cell along the z axis [137,138]. In this case, the helical structure, even the ideal one, is incompatible with the planar boundary conditions, and splayed and bended regions form near the boundaries. Thus, according to Eq. (38), the flexoelectric polarization arises in those regions which can interact with the electric field. The distortion is very similar to that observed in the ferroelectric smectic C phase (see Fig. 24) for a so-called deformed helix ferroelectric effect [139]. 

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