Flexoelectricity ferroelectrics

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 presence of clusters in BC nematics is now well established from various measmements. Recent studies " have in fact indicated a ferroelectric or an antiferroelectric response to an applied electric field, and an unusual low-frequency (presumably collective) mode has been detected in the dielectric spectra of bent-core nematics, which might also be related to clusters. In spite of the intense studies, however, the exact structure and the physical properties of the clusters are still unknown. Therefore, not surprisingly, a precise physical model for the role of polar clusters in the flexoelectric response of BC nematics and a quantitative estimation of the resulting increment of the flexocoefiicients has not yet been worked out. 

Finally, it is worth mentioning that a phenomenon analogous to the difference between the normal and giant flexoelectricity of calamitic and bent-core nematics, respectively, exists in crystals, ceramics and polymers too. The flexoelectric response (defined in Eq. (3.1)) of perovskite-type ferroelectrics, " of relaxor ferroelectric ceramics and polyvinylidene fluoride (PVDF) films are four orders of magnitude larger than the flexoelectricity of dielectric crystals. In those sohd ferroelectric materials the polarization induced by flexing is evidently of piezoelectric origin. 

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... 

In the most simple chiral polar tilted smectics, ferroelectric liquid crystals, the flexoelectric phenomenon influences the structure of the SmC phase only quantitatively. It affects the elastic and chiral couplings and consequently slightly changes the transition temperature to the tilted phase and the pitch of the helicoidal modulation. 

The simple expression for / means that the response time will be inversely proportional to the field, r oc 1/ PE) when there is flexoelectric coupling. This is the same dynamics as in ferroelectric liquid crystal (FLC) switching. In the cholesteric case the expression for / is somewhat more complicated and, as we will see below, it turns out that r is independent of E. 

I. Dozov, I. Penchev, Ph. Martinot-Lagarde and G. Durand, On the sign of flexoelectric coefficients in nematic liquid crystals, Ferroelectrics Letters 2(4), 135-141, (1984). doi 10.1080/07315178408202434... 

The distribution of relative vertical displacement Uzlh caused by flexoeffect for different values of pill thickness h and flexocoefficient/i 1 is shown in Fig. 4.24b-e for an example of ferroelectric PbTi03. One can see that relative displacement decreases with pill thickness increase (compare curves 1 in parts b and d) and increases with flexoelectric coefficient increase (compare curves in parts c and e). The displacement profile is parabolic. For p = 0 the displacement... 

The divergences of correlation radius could be achieved only for T = Tcr R,/aa) or at R = Rcr, corresponding to the paraelectric-ferroelectric phase transition point as one can see from Eq. (4.24). One can see from Fig. 4.28, that these conditions can be met at fixed radius R for arbitrary f A or for arbitrary R at given temperature T. Since the same fixed values of R or f A correspond to the divergence (or maxima for finite electric field value) of dielectric permittivity x (because R y/guJ.) these values of the radius and flexoelectric coefficient... 

It is clear from the Fig. 4.28a, b that in ferroelectric phase (i.e. at R>Rcr) the correlation radius monotonically decreases with the increase of the flexoelectric coefficient/44. At the same time, in paraelectric phase correlation radius increases with the increase of the flexoelectric coefficient, since the critical temperature (see Eq. (4.22)) increases with the increase of the flexoelectric coefficient. This opens the possibility to control the phase diagram and polar properties (e.g. via influence on domain wall width) by the choice of the material with necessary flexoelectric coefficient at given temperature or nanoparticle radius. 

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. 

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]. 

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