Torque elastic

Fig. 13 Schematic representation of the orientation-dependent elastic torque on a nickel nanowire with longitudinal anchoring in 5CB (a) in equilibrium, and (b) in the presence of a magnetic field, (c) Levitation of the nanowire in a twisted nematic cell [446]. (Copyright 2004, American Association for the Advancement of Science)...

The wire comes to rest at an angle 6 when the magnetic torque and elastic torque balance (see Fig. 13b). 

When n r) is disturbed, say by an external magnetic field, an elastic torque arises ... 

The evolution of ( ) is governed by a balance between viscous and elastic torques ... 

Electric or magnetic fields acting on the anisotropy of the electric or magnetic susceptibility exert torques within a liquid crystal which may compete with the elastic torques determining its internal structure (55). Equations w ich describe the liquid crystalline structure can be derived from molecularly uniaxial liquid crystals on the basis of the curvature-elasticity theory (54). In doing so, tl structure is determined so as to minimize the total free energy of the system, and this method is applied to the cholesteric structure (55, 55). 

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. 

Figure 33 shows that the RPA can be an advanced, roiorless curcnietcr. Both the S elastic torque and S " viscous torque are shown during cure. The S " cure curve is more sensitive to variations in oil and carbon black than the S cure curve [123]. 

Variational Problem and Elastic Torques 8.3.1 Euler Equation... 

Mathematically the molecular field vector h can be found using the Euler-Lagrange approach by a variation of the elastic and magnetic (or electric) parts of the free energy with respect to the director variable n(r) (with a constraint of = 1). For the elastic torque, in the absence of the external field, the splay, twist and bend terms of h are obtained [9] fi-om the Frank energy (8.16) ... 

As during relaxation the external field is switched off, the molecular field includes only the elastic torque. For the pure twist distortion and our geometry, the molecular field vector h is opposite to the magnetic field, i.e. directed opposite to X- axis. Its absolute value is (Section 8.3.3). The torque (n x h) is... 

The reason for this is a flow of the nematic, which is lunched by the director rotation. The flow arises in the beginning of the director relaxation process when the elastic torque exerted on the director is very high near both interfaces due to a strong curvature of the director field. However, the curvature at the two interfaces has different sign, see Fig. 11.19a, where 89(z) = (ti/2) — 9. Therefore, the flow of nematic fluid coupled to the director rotation (backflow) at the two interfaces is... 

At a certain critical voltage the destabilising shear-induced torque My = a.i dvj dx), which comes from the interaction of a field driven charged volume of a liquid with the director, becomes large enough to equalise the stabilising elastic torque. This balance of the elastic and hydrodynamic torques is the condition for the onset of instability ... 

At a low field with ignored term in Eq. (13.10), the dynamics of the director soft mode can be investigated using the Landau-Khalatnikov equation, see Section 6.5.1. The corresponding equation describing the balance of the viscous and elastic torques reads ... 

As to dynamics of the response of a SSFLC cell to the alternating field, it is controlled by Eq. (13.38) with the viscous j dipldt torque added. When the helical stmcture of the SmC is unwound in a thin cell (a typical case) one can neglect the elastic torque d

anchoring energies Wf and W are reasonably weak, the electric field torque would solely be balanced by the viscous torque ... 

Santamato et ai assumed that with homeotropic boundary conditions the normal component of the optical torque cannot be balanced by an elastic torque, hence the molecules are set into rotation. is balanced by a viscous torque arising from the precession of the director. They established a simple relation between the angular velocity of the director rotation and the total change of the Stokes parameter, S3. This relation was experimentally verified. 

The stationary configuration can be determined in general from the balance between the electric and elastic torques. In our case there is always a trivial solution for the balance the unperturbed configuration, in which L = (0,0,1). To calculate the threshold we follow the usual procedure looking for the value of the light intensity at which the linearized equation of the balance of torques has a non-trivial solution. 

In general, the free-energy density of the system consists of the terms from the three elastic torques (bend, splay, and twist) and the optional torque. The free-energy density term F associated with the bending distortion is given by... 

As we mentioned earlier, as a result of the significant elastic torque from molecules surrounding the laser beam (besides the boundary elastic torque), the field required to create finite molecular reorientation (which we shall denote 6th) is in general, larger than that associated with infinite-beam-size lasers. Figure 4 shows a plot of the value of 6th for which nonzero reorientation r) occurred. As a function of Wo/d, we note that 6th for d>... 

The elastic torque supports the initial director orientation, fixed by the boundary conditions on the surface (S), unchanged... 

The most important geometries of electrically controlled birefringence (ECB) are shown in Fig. 4.1. A compromise between dielectric and elastic torques results in director reorientation from the initial alignment 0(z) with the maximum deviation 0m at the center of the layer (the Prederiks transition). The effect occurs when the electric field exceeds a certain threshold value... 

A theoretical investigation of the stability of nematic liquid crystals with homeotropic orientation requires a three-dimensional approach. Helfrich s one-dimensional theory predicts the dependence of the threshold of the instability on the magnitude of Ae, as shown by curve 2 in Fig. 5.8, according to which the electrohydrodynamic instability should be observed when either Ae < 0 (and consequently the bend Frederiks effect reorientation will not take place), or when small Ae > 0. In Helfirich s model the destabilizing torque as dvzjdx is responsible for this instability, which replaces the destabilizing torque a dvzjdx in the equation for the director rotations (5.27). Although the torque is small ( a3 -C o 2 ) it is not compensated for (e.g., when Ae = 0) by anything else apart from the elastic torque. 

The instability is caused by the torque induced by the electrical conductivity acting against the elastic torque of the cholesteric, and although the cause is different from the purely dielectric case, the result obtained is the same— the appearance of a two-dimensional periodic pattern for the distribution of the director. 

The switching time of the director tilt angle in the electroclinic effect is independent of the electric field, and is defined only by the rotational viscosity je and the elastic modulus A. The corresponding switching times are derived from the Landau-Khalatnikov equation for the balance of the viscous and elastic torques... 

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