Splay-twist mode

Here V n and V x n are the divergence and the curl of n. The three contributions to Wd are associated with the three independent modes of distortion splay, twist, and bend, depicted in Fig, 10-6. Terms of higher order than quadratic in Vn are only required if spatial distortions become severe. The Frank constants K, K2, and are of the order u /a,... 

The way in which 6 R) varies with position depends on the form of the torsional deformation applied to the liquid crystal, and in order to calculate the principal elastic constants, it makes sense to calculate the free energy density for normal mode deformations, i. e. those which correspond to splay, twist and bend. These can be achieved easily by confining the director to a plane, and assuming the undisturbed director at the origin to be along the z-axis. g is the wave-vector of the deformation, and for q constrained to the X, z plane, the components of the director as a function of position become ... 

Here, Eq is the amplitude of the incident optical field, cOo the frequency of the incident light, V the scattering volume and R the distance between the scattering volume and the detector. The scattered light consists of two modes, splay/bend (a=l) and twist/bend (a=2), as shown in Fig. 2. It can be seen from Fig. 3 (a) that the mode I fluctuations can contain only contributions from the bend and splay distortions. Mode 2 fluctuations are in the perpendicular plane... 

The de Gennes formulae [28, 29] establish a relation between the elastic coefficients and the scattering intensities. Small thermal director fluctuations can be expressed in terms of two eigenmodes, the splay-bend mode Srii and the twist-bend mode Sn2. The equipartition theorem gives the intensities... 

From the expression (5) one can see that waves propagating perpendicular to the nematic director (q -O) can be either pure splay or pure twist modes. For the splay mode drii=dnx 0 (see Fig. 2a) whereas for the twist mode 5ri2 = 8nyjt0 (Fig. 2 b). Alternatively, waves propagating along the director (qx=0) can be only pure bend modes (Fig. 2c). For a general direction of the wave-vector we have a mixture of both polarizations bend-splay or bend-twist modes. 

Pretransitional behavior of twist and splay elastic constants was also measured in CBOOA by Chu and McMillan [193]. The splay is not renormalized, whereas the twist elastic constant shows a mean-field like divergence. They also report slowing down of the twist mode near Tg, which is in apparent disagreement with both mean-field and helium-like models. A similar slowing down of the twist mode was observed by Delaye [118]. Pretransitional bend mode behavior is reported by Birecki and Litster... 

Fig. XV-10. Illustration of a bilayer membrane and two of its deformation modes (a) twist (b) splay. (From Ref. 75.)...

Figure 6.9. The director fluctuation causes light scattering (a) two independent fluctuation modes 8n and <5ri2 (b) two components in 8n splay and bend and (c) two components of <5n2 bend and twist. (Modified from DuPre, 1982.)...

First detailed dynamic light scattering (DLS) experiments using bulk liquid crystal samples have confirmed the theoretically predicted existence of two dissipative fluctuation eigenmodes in the nematic liquid crystalline phase the first mode being a combination of splay and bend distortion and the second one a combination of twist and bend fluctuations [55,56]. Both modes are overdamped and the relaxation rate 1/r of each mode depends on the fluctuation wave vector q and viscoelastic properties of the sample [57] ... 

Here, a = 1,2 denotes the splay-bend and twist-bend mode, respectively, i i,2,3 are the Prank elastic constants, 771 2 are the rotational viscosities, is the component of the fluctuation wave vector parallel to the director and q the component perpendicular to it. 

In Fig. 8.10b, we see that the fluctuation mode i(q) is a mixture of the splay and bend distortions, and the component 2(q) is a mixture of twist and bend distortions. This may be clarified as follows the splay-bend (SB) mode on the left side of Fig. 8.10b corresponds to realignment of the molecules within the, z-plane as q evolves and there is no twist here. In contrast, on the right side of the same figure the molecules are deflected from the q z-plane of the figure therefore, the twist and bend are present but the splay is absent (TB mode). 

Fig. 8.10 New coordinate axes, ej and 62 appropriate to the normal modes of director fluctuations in a nematic liquid crystal (a) and the structure of the normal modes, namely splay-bend (SB) and twist-bend modes (TB)...

To fix the twist direction of the twisted nematic (TN) display mode, several methods are used. One is to add a small amount of a chiral agent. A typical chiral pitch for the 5 pm gap TN mode is 100 pm. The other method involves the rubbing direction. The rubbing direction determines the inclination direction of the LC, and it is selected so as to avoid a splay alignment as shown in Fig. 2.32. If the pretilt... 

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. 

Frank elastic constants were introduced giving the energy cost of individual deformation modes for splayed, K22 for twisted, and K33 for bent di-... 

This expression can be diagonalized by transforming (rix ny) —> (ni,n2) to yield two uncoupled modes ni( and ri2( [6.28]. ni( lies on the (g,no) plane and describes a periodic distortion involving splay and bend deformations, while ri2 is perpendicular to the q ho) plane and describes a periodic distortion involving twist and bend. Defining Q =... 

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. 

For each in a uniaxial phase there are two normal modes corresponding to a splay-bend distortion n q) and a twist-bend distortion ri2(q) biaxial liquid crystal phases have five normal modes for each value of q. The free energy density can be written in terms of the normal coordinates for torsional displacement in a uniaxial nematic as ... 

Figure 2. Modes of pure splay (a), twist (b) and bend (c).

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

---