Splay torsional elasticity

Here the unit vector n is the director, the elastic moduli Ku, K22, and K33 describe, respectively, transverse bending (splay), torsion (twist), and longitudinal bending (bend)... 

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 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 ... 

Finally, there are some unique measurements of the mechanical torque connected with an elastic deformation of the nematic. Faetti et al. [63] determined the splay and bend elastic constants by means of such torsion measurements, and Grupp [ 139] made measurements of the twist elastic constant 22-... 

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