Surface elastic constants splay-bend

A. N. Shalaginov, Influence of the liquid crystal splay-bend surface elastic constant on director fluctuations and light scattering, Phys. Rev. E1994, 2472-2475. 

The effect of surface elastic constants on the nematic director configurations is of basic interest for the elastic theory of liquid crystals and plays a critical role in those device applications where the nematic is confined to a curved geometry. The saddle-splay surface elastic constant, K24, and the splay-bend surface elastic constant, K13, defied measurement for more than sixty years, since the pioneering work of Oseen, who made the first steps toward the elastic theory of liquid crystals. 

Measurements of the saddle-splay surface elastic constant, K24, and the splay-bend surface elastic constant, K13, were first introduced by Oseen [1] in 1933 from a phenomenological viewpoint, and later by Nehring and Saupe [2] from a molecular standpoint. These constants tend to be neglected in conventional elastic continuum treatments for fixed boundary conditions because they do not ent the Euler-Lagrange equation for bulk equilibrium. Experimental determination of tiie two surface elastic constants is undoubtedly a difficult task, since their effects are hard to discriminate from those of ordinary sur ce anchoring [3]. 

H e Ki, K2 and K3 correspond to the splay, twist, and bend bulk elastic constants, respectively. Further, the terms K24 and K13, known as the saddle-splay surface elastic constant and mixed splay-bend surface elastic c( istant, respectively, are called surface elastic constants because they enter EQN (3) as divergences of a volume integral, which are converted to surface integrals via Green s theorem. It should be noted that the second derivative of the K24 term in EQN (3) is apparent however, this is... 

For a nematic LC, the preferred orientation is one in which the director is parallel everywhere. Other orientations have a free-energy distribution that depends on the elastic constants, K /. The orientational elastic constants K, K22 and K33 determine respectively splay, twist and bend deformations. Values of elastic constants in LCs are around 10 N so that free-energy difference between different orientations is of the order of 5 x 10 J m the same order of magnitude as surface energy. A thin layer of LC sandwiched between two aligned surfaces therefore adopts an orientation determined by the surfaces. This fact forms the basis of most electrooptical effects in LCs. Display devices based on LCs are discussed in Chapter 7. 

The spatial and temporal response of a nematic phase to a distorting force, such as an electric (or magnetic) field is determined in part by three elastic constants, kii, k22 and associated with splay, twist and bend deformations, respectively, see Figure 2.9. The elastic constants describe the restoring forces on a molecule within the nematic phase on removal of some external force which had distorted the nematic medium from its equilibrium, i.e. lowest energy conformation. The configuration of the nematic director within an LCD in the absence of an applied field is determined by the interaction of very thin layers of molecules with an orientation layer coating the surface of the substrates above the electrodes. The direction imposed on the director at the surface is then... 

A definition of these angles is given in Fig, 1, The deformation profile is dependent on the dielectric constants C and Ej., the elastic constants for splay, twist and bend Kn, K22> 33 the total twist (90-23q), the tilt angle at the surface of the substrates ao, and the applied voltage. The optical response depends in addition on the refractive indices ne and viq and the ratio of wavelength to cell thickness. For display applications a finite tilt at the surfaces is required to avoid areas of opposite tilt. Therefore the deformation profiles are calculated for various combinations of K33/K11, Ae/ej and using Berreman s program. All calculations are performed for 10 im cells and a pretilt ao=l°. 

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 strain tensor must conform to the symmetry of the liquid crystal phase, and as a result, for nonpolar, nonchiral uniaxial phases there are ten nonzero components of kij, of which four are independent ( i i, 22> A 33 and 24)- These material constants are known as torsional elastic constants for splay (k, 1), twist ( 22) bend ( 33) and saddle-splay ( 24) terms in 24 do not contribute to the free energy for configurations in which the director is constant within a plane, or parallel to a plane. The simplest torsional strains considered for liquid crystals are one dimensional, and so neglect of 24 is reasonable, but for more complex director configurations and at surfaces, k24 can contribute to the free energy [7]. In particular 24 is important for curved interfaces of liquid crystals, and so must be included in the description of lyotropic and membrane liquid crystals [8]. Evaluation of Eq. (16) making the stated assumptions, leads to [9] ... 

The A" and K22, terms, which vanish identically in the apolar nematic and cholesteric phases, are not considered here. As discussed above, the volume integrals of the free energy terms containing the splay-bend elastic constant and the saddle-splay elastic constant K24 can be transformed into integrals over the nematic surface... 

Kralj and Zumer [47], in their derivation, have re-expressed of Eqs. (14)-(16). In addition, they included expressions for the surface-like elastic coefficients. In a one-constant approximation (only Lf O), all constants are equal except for /f]3=0 in a second-order approximation (only L O), there is still a splay-bend degeneracy, - 11 = 33 22 24 13 re ains zero. 

It is energetically unfavorable to alter the layer thickness of the smectic phase, therefore any process that relies on this feature is very unlikely to occur. However, bending of the smectic layers is possible, because this need not cause a change in the layer thickness thus the molecules can undergo a splay (A, ) distortion. In a smectic film, any point distortion of the layer, e.g., arising from a surface feature, propagates for some distance into the film this feature is important for electrically addressed dynamic scattering devices. Bend ( 33) and twist ( "22) elastic constants approach infinity at the nematic (N) to smectic transition. 

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