Response flexoelectric

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. 

N.T. Kirkman, T. Stirner and W.E. Hagston, Continuum modelling of hybrid-aligned nematic liquid crystal cells optical response and flexoelectricity-induced voltage shift, Liq. Cryst. 30(9), 1115-1122, (2003). doi 10.1080/02678290310001594562... 

Although most of the scientists working on liquid crystals may think that flexoelectricity is a special property of liquid crystals, it was actually first discussed in 1964 for crystals as a response to strain (stress) gradients. The direct and converse flexoelectric coupling constants Cijki were described with a fourth-rank tensor, as... 

The direct method introduced in the previous section was first employed for studying the flexoelectric response of a bent-core nematic liquid crystal. 

Homeotropic cells offer another way to detect the flexoelectric response via observing the bend director distortions induced by an electric field parallel to the substrates (the Helfrich method ). Takezoe s group applied it to ClPbislOBB and found 63 20 pCm (which is the order of flexocoef-... 

The flexoelectro-optic effect in cholesterics (which is described in more detail in Chapter 7 by Rudquist and Lagerwall ) offers another way to measure 61 — 63. A tight-pitch helical structure can easily be induced in nematics by adding a few per cent of chiral dopant. Recently this technique has been adopted to test the flexoelectric response of another BC nematic, 4-cyano-l,3-phenylene bis [4-[4 -(hexyl) benzoyloxy] benzoate] (C6ban). ... 

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. 

Kumar et al. proposed that a chain formed by intercalated bent-core molecules with alternating dipoles may have a considerable quadrupolar moment and they assumed that this idea is transferable to antiferroelectric layers. They estimated an increment in the quadrupole moment proportional to the number of molecules (m) in such clusters. As the typical cluster size indicated by X-ray measurements is about 4-5 layers of 20-30 nm, m may be of the order of 1,000. Therefore, on the one hand, the presence of clusters with an antiferroelectric SmCP structure may cause a huge increase in the quadrupolar contribution to the flexoelectric response compared to that of individual molecules. 

This discrepancy clearly indicates that the BC nematics characterized by the presence of polar clusters cannot be regarded - at least from the point of view of the flexoelectric response - as a homogeneous continuum instead the behaviour of the clusters in and their interaction with the surrounding nematic matrix should be handled separately. This challenging problem still waits for a solution. 

Summarizing, experimental observations suggest that the giant (direct or converse) flexoelectricity of bent-core nematics is related to the polar smectic clusters occurring in them. In order to explore the exact mechanism for how clusters contribute to the flexoelectric response, further experimental and theoretical studies are needed. 

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 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. 

Despite all these problems, flexoelectricity has, from time to time, been found wanting when the results of an experiment could not fully be interpreted by the dielectric interaction alone though its relevance or responsibility could not always be proved. 

Bent-core liquid crystal elastomers have shown to exhibit large values of flexoelectricity as many as three orders of magnitude larger than liquid crystal elastomers containing rod-shaped molecules [44]. These high responses are attributed to a piezoelectric phenomenon. Liquid crystal elastomers combine elasticity and flexibility inherent to rubbers and the optical and electrical properties of liquid crystals, and are promising materials for applications such as electrooptics, flexible electronics, and actuator technologies for biomedical applications. 

In both the cases considered, an optical contrast of the patterns observed in isotropic liquids is very small. Certainly, the anisotropy of Uquid crystals brings new features in. For instance, the anisotropy of (helectric or diamagnetic susceptibility causes the Fredericks transition in nematics and wave like instabilities in cholesterics (see next Section), and the flexoelectric polarizaticm results in the field-controllable domain patterns. In turn, the anisotropy of electric conductivity is responsible for instability in the form of rolls to be discussed below. All these instabilities are not observed in the isotropic liquids and have an electric field threshold controlled by the corresponding parameters of anisotropy. In addition, due to the optical anisotropy, the contrast of the patterns that are driven by isotropic mechanisms , i.e. only indirectly dependent on anisotropy parameters, increases dramatically. Thanks to this, one can easily study specific features and mechanisms of different instability modes, both isotropic and anisotropic. The characteristic pattern formation is a special branch of physics dealing with a nonlinear response of dissipative media to external fields, and liquid crystals are suitable model objects for investigation of the relevant phenomena [39]. 

The surface flexoelectric energy, which is found from (4.2) and (4.3). Attaining the minimum of the nematic free energy, (4.5) or (4.6), it is possible to derive the equilibrium director distribution in a static case. To find the response times, we have to solve the equations of nematodynamics in the electric field. The corresponding analysis shows that the director reorientation is always accompanied by the macroscopic flow, the so-called backflow [5]. (The only exclusion is the pure twist rotation of the director [1].) Backflow considerably affects the characteristic times of the electrooptical effects in uniform structures, especially in the case of strong deformations of the initial director orientation [3, 5]. 

Due to the above-mentioned results [188, 189] the flexoelectric response is considerable in a narrow frequency region limited by the frequency of the structural relaxation fs / d where K is the average elastic co-... 

A liquid crystal (LC) in which the electric dipoles point in the same direction as the respective LC directors should exhibit not only a nonuniform strain but also a piezoelectric response when it undergoes one or more of the three nonuniform deformation modes that are identified as splay, bend, and twist. Accordingly, three different modes of piezoelectricity from nonuniform strain distributions were postulated for liquid crystals (Meyer 1969), but it was not clear whether the resulting piezoelectric effects were large enough to be observed in real experiments (Helfrich 1971). In the meantime, since the early concepts, a whole new field - flexoelectricity in liquid crystals (Buka and Eber 2013) - has developed from the pioneering work of Meyer and Helfrich on splay and bend deformation in liquid crystals. 

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