Euler nematics

The optical-field-induced Freedericksz transition for a twist deformation by a normally incident laser beam in a planar-aligned nematic liquid crystal is studied. The Euler equation for the molecular director and the equations describing the evolution of the beam polarization in the birefringent medium are solved simultaneously in the small-perturbation limit. The stability of the undistorted state is investigated. An alternate series of stable and unstable bifurcations is found. This phenomenon has no analog in the Freedericksz transition induced by dc electric and magnetic external fields. 

Reference [189] examined theoretically the behavior of a nematic layer (with a geometry similar to Fig. 4.29(c)) in an alternating field with allowance for the flexoelectric eflFect. Euler s equation for the volume of a sample (see (4.29)) is written in the hnearized form (sin 0 0, cos 0 1)... 

The molecular theory is similar to Cauchy s description of the elastic theory of solids [1] and utilizes additive local molecular pair interactions to describe elasticity. The latter approach was taken by Oseen [2], who was the first to establish an elastic theory of anisotropic fluids. Oseen assumed short-range intermolecular forces to be the reason for the elastic properties, and he derived eight elastic constants in the expression for the elastic free energy density of uniaxial nematic phases. Finally, he retained only five of them, which enter the Euler-Lagrange equations describing equilibrium deformation states of the nematic mesophase, and omitted the other three. 

For a chiral nematic with positive dielectric anisotropy and an electric field E orthogonal to the helix axis (z), then we observe a chiral nematic to unwound nematic transition for d >p. The analysis [132,139] of the unwinding mechanism is exactly the same as for the magnetic field case with the coupling -YiAxin H) replaced by - AnAe n E) in the free energy density expression. The equilibrium condition (see Eq. 127) or Euler s equation then becomes... 

Fig. 1—(a) Schematic representation of the structure of a nematic liquid crystal, (b) The Euler angles required to describe the orientation of a molecule in a nematic liquid. 

If the anisotropies of the elastic constants and optical field propa tion characteristics in the birefringent nematic film are taken into account, the resulting equations for the optical field, elastic and optical torques, and so on become more complicated. The starting point of the analysis is the Euler-Lagrange equation associated with a small director axis reorientation angle 0(z) ... 

To see the nature of the orientational order parameters needed to characterise the biaxial nematic we start with the singlet orientational distribution function. Since the molecules are taken to be rigid the distribution is a function of the three Euler angles, a 3y, which we denote by Q connecting the director... 

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