Deformation splay-bend

Here K, K2 and iTs are elastic moduli associated with the three elementary types of deformations splay, twist and bend, respectively. Though the three elastic moduli are of the same order of magnitude the ordering K2 < K < K3 holds for most nematics. As a consequence of the orientational elasticity a local restoring torque (later referred to as elastic torque) acts on the distorted director field which tends to reduce the spatial variations. 

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 ... 

According to Eq. (7.1) P is zero for the two cases of uniform director fields and pure twist. Hence both cases can serve as a zero state as far as flexoelectric excitations are concerned. It is important to note that a twist is not associated with a polarization (i.e. C2 is identically zero, cf. Fig. 7.2). An imstrained nematic has a centre of symmetry (centre of inversion). On the other hand, none of the elementary deformations - splay, twist or bend have a centre of symmetry. According to Curie s principle they could then be associated with the separation of charges analogous to the piezoeffect in solids. This is true for splay and bend but not for twist because of an additional symmetry in that case if we twist the adjacent directors in a nematic on either side of a reference point, there is always a two-fold symmetry axis along the director of the reference point. In fact, any axis perpendicular to the twist axis is such an axis. Due to this symmetry no vectorial property can exist perpendicular to the director. In other words, a twist does not lead to the separation of charges. This is the reason why twist states appear naturally in liquid crystals and are extremely common. It also means that an electric field cannot induce a twist just by itself in the bulk of a nematic. If anything it reduces the twist. A twist can only be induced in a situation where a field turns the director out of a direction that has previously been fixed by boundary conditions (which, for instance, happens in the pixels of an IPS display). 

The fiexoelectric coupling is not chiral, so what is the role of chirality in this case The answer is that the helically twisted state is the only one that is fiexoelectrically neutral (there is no local polarization related to twist) and therefore the only state from where a splay-bend deformation can increase continuously from zero in a symmetric fashion independent of the direction of E, while allowing for a homogeneously space-filling splay-bend. How it increases is illustrated in Fig. 7.5. 

Fig. 7.5. Increasing curvature in response to an increasing electric field the thresholdless field-induced periodic splay-bend deformation. The pattern in the director field is shown to the left (in any oblique cut perpendicular to the optic axis) and the optic axis deflection with increasing field is shown to the right. (From Rudquist et reproduced with kind permission of Taylor Francis, http //www.tandfonline.com.)...

FIGURE 2.17. (a) S deformation (splay), (b) B deformation (bend), and (c) T deformation (twist) in an oriented layer of a nematic liquid crystal. 

The only curvature strains of the director field which must be considered correspond to the splay, bend, and twist distortions (Fig. 2.17). Other types of deformation either do not change the elastic energy (e.g., the above mentioned pure shears) or are forbidden due to the symmetry. In nematic liquid crystals the cylindrical symmetry of the structure, as well as the absence of polarity (head to tail symmetry) must be taken into account. 

Figure 10.2. (a)-(c) Basic deformation modes of a nematic director field (a) splay deformation (divn + 0) (b) twist deformation (n curln 7 0) and (c) bend deformation (n X curln + 0). (d)-(f) The same deformations of the director field in a smectic phase. Only the splay deformation of the director field (d) is compatible with the constant layer spacing. A twist deformation (e) [bend deformation (f)j is only possible if screw dislocations [edge dislocations] appear. 

The three (positive) elastic constants Kn (splay), K22 (twist), and K33 (bend) are associated to the three principal deformations. In the surface term, fs is the contribution of the two anchorings, k is the unit vector normal to the surface and directed outward, K13 is the splay-bend constant, and K24 is the saddle-splay constant. The two last surface terms play only for thin films the mere existence of the splay-bend constant K13 is a matter of debate. In the framework of Landau-de Gennes analysis, = K33 and the elastic... 

The bending elasticity of fluid membranes is closely related to the director field elasticity of liquid crystals. Of the three elastic deformations in nematics, which are splay, bend, and twist, only splay remains as it does in the case of smectics. In fact, a membrane is like an isolated smectic layer and this is why membrane curvature is sometimes expressed in terms of splay and saddle splay. 

The elastic constants kn, 22, and 33 pertain to the three basic deformations splay, twist, and bend, respectively. For typical nematics with prolate molecules one has hi > hi > k22 and hi 10 " N. 

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. 

In discotic systems, the roles of ATu and K22 are reversed, because in such phases the bend deformations require the lowest energy [181]. Measurements of splay and bend constants in a homologuous series of discotic n-hexa(alkanoylox)truxenes [76] revealed that K22 is always smaller than A, . The splay/bend ratio approached unity at the high temperature transition to the columnar phase. Qualitatively different results have been obtained, however, by Raghunathan et al. [75], who found K22>Kn in a disco-tic nematic phase enclosed between two columnar phases. The authors interpreted this unexpected result as being a consequence of short-range columnar order. 

Figure 31. The three elementary deformations splay, twist, and bend. None of them possesses a center of symmetry.

The periodic deformation in Fig. 34 b cannot be observed in a nonchiral nematic because it does not allow for a space-filling splay-bend structure. Instead, such apattem would require a periodic defect structure. However, we can continually generate such a space-filling structure without defects in a cholesteric by rotating the director everywhere in a plane containing the helix axis. 

The magnetic field-induced director deformations involving bend or splay are illustrated in Figure 7.5b. 

Let us consider the possible deformations (splay, twist and bend) of anisotropic materials. As illustrated in Figure C.l, without loosing generality we choose the coordinate system so tirat the z-axis be parallel to the imdistorted director field (iiIIz ). 

FIGURE 2.21 The three different modes of liquid crystal deformation (a) bend, (b) splay, and (c) twist. 

Figure 3.1. (a) Twist deformation in a nematic liquid crystal (b) splay deformation (c) bend deformation. 

Consider the three typical aligned nematic liquid crystal cells as depicted in Figures 6.6a-6.6c corresponding to planar, homeotropic, and twisted NLCs of positive anisotropy. With the applied electric field shown, they correspond to the splay, bend, and twist deformations in nematic liquid crystals. Strictly speaking, it is only in the third case that we have an example of pure twist deformation, so that only one elastic constant K22 enters into the free-energy calculation. In the first and second cases, in general, substantial director axis reorientation involves some combination of splay (S) and bend (B) deformations pure S and B deformations, characterized by elastic constants Kii and respectively, occur only for small reorientation. 

---