Freedericksz state

Figure 9. Snapshots of electroconvection patterns superposed on the Freedericksz state in case C. a oblique rolls, b normal rolls.

Upon increasing the voltage the first transition is usually to the homogeneous (q = qp = 0) splay Freedericksz state with a frequency-independent threshold voltage Upi (Fig. 10a), i.e the absolute minimum of the neutral curve is at = 0. However, very near to e = 0 the absolute minimum appears at a finite q. Thus we have the planar counterpart of the a-induced FC described in case C. The Freedericksz threshold smoothly transforms into an FC threshold Ua at eo/ej = 0.0057 (see Fig. lOc-d). Below this, a direct transition to FC is predicted with q growing continuously from zero and remaining extremely small. This transition seems to be a better candidate for experimental observation than its homeotropic counterpart, because both Ua and qa are substantially lower than in case C. 

For a/e > 0.0057 EC occurs superimposed onto the Freedericksz state (secondary instability) at a higher voltage Uc > Upi- The standard model can also be applied here by carrying out numerical linear stability analysis of the Freedericksz distorted state, and one is faced with similar modifications and difficulties as mentioned in case C before. The ea-dependent Uc and Qc, presented by the dotted hnes in Fig. lO(a-b), have been calculated numerically. It should be noted that the convection rolls are now oriented parallel to the initial director alignment, contrary to the normal rolls in case A or C. 

Preedericksz transition in planar geometry is uniform in the plane of the layer and varies only in the z direction. However, in some exceptional cases, when the splay elastic constant Ki is much larger than the twist elastic constant K2 (e.g., in liquid crystal polymers), a spatially periodic out-of-plane director distortion becomes energetically favourable. The resulting splay-twist (ST) Freedericksz state is manifested in experiments in the form of a longitudinal stripe pattern running parallel to the initial director alignment no x. 

The dependence of Uc and qc for flexodomains on the strength of the flexocoefficients, on the frequency w and on the elastic constants was discussed by Krekhov et al In general both Uc (see Fig. 4.5 below) and qc rise very steeply as functions of w. Thus for /it > 0 (e > 0) the bifurcation to the homogeneous Freedericksz state (with an w independent critical voltage) will prevail for very small w. The competition with standard EC will be discussed in Section 4.4. In view of the discontinuities of Uzit) and... 

In the following, we first discuss the situations where EC occurs as a primary forward bifurcation and where the standard model is directly apphcable (cases A and B). Then we discuss configurations where EC sets in as a secondary instability upon an already distorted Freedericksz ground state and compare it with experiments (cases C and D). Note that in this case the linear analysis based on the standard model already becomes numerically demanding. Finally, we address those combinations of parameters where a direct transition to EC is not very robust, since it is confined to a narrow Ca range around zero. For cases E and H this range may be accessible experimentally while for cases F and G it is rather a theoretical curiosity only. 

Case C homeotropic alignment, Ca < 0 0. In this combination of the material parameters, the linear stability analysis of the basic state does not predict a direct transition to EC since the resulting expression for U q) in Eq. (8) is negative for all q 0 (except for e in the immediate vicinity of zero, see below). The reason is that the two terms in the denominator act differently compared to the case B ca > 0, aa < 0) described in the previous subsection. The Carr-Helfrich torque is now stabilizing while the dielectric torque (oc is destabilizing. At q = qp = 0, this term dominates and describes, at the threshold Ups (see Eig. 7a), the continuous bifurcation to the Freedericksz distorted state of homogeneous (along the x direction) bend (see Fig. 8a). 

Figure 7. Threshold voltages Uth/Co and the critical wavenumber qc versus the relative dielectric anisotropy eajei. calculated from Eq. 8. Homeotropic alignment with <7a > 0. The upper (a b ) and lower (c d ) plots differ only in the axis scales. Dashed lines correspond to the Freedericksz transition, solid lines correspond to the direct transition to an ("a-induced") EC patterned state, dotted lines represent a secondary transition to EC.

Figure 8. Schematic director profile in case C. a Freedericksz distorted state, b with superposed electroconvection pattern.

The subcritical nature of the Freedericksz transition can be explained as follows. When the director settles to the precession state, light becomes el-liptically polarized inside the nematic. On the other hand, it is known that the Freedericksz transition for elliptically polarized light depends on ellipticity and... 

Besides the elastic and the electric torques the so-called flexoelectric (or flexo) torques on the director play an important role as well. Their effect on pattern-forming instabilities in nematics is the main issue of this chapter. Flexotorques originate from the fact that typically (in some loose analogy to piezoelectricity) any director distortion is accompanied by an electric flexopolarization Pa (characterized by the two ffexocoefScients ei, 63). From a microscopic point of view, finite ei and 03 naturally arise when the nematic molecules have a permanent dipole moment. But also for molecules with a quadrupolar moment, finite ei and 63 are possible (see also Chapter 1 in this book ). Flexopolarization has to be incorporated into the free energy P n) for finite E. It is not surprising that this leads to quantitative modifications of phenomena, which exist also for ci = 63 = 0. Though, for example, the Freedericksz threshold field Ep is not modified, the presence of flexoelectricity leads to considerable modifications of the Freedericksz distorted state for E > Ep- ... 

Much more exciting is the possibility of qualitatively new phenomena, which are generically related to flexopolarization. A prominent example is provided by the so-called flexodomains. They appear as the result of an equilibrium transition from the basic planar state if the applied electric field strength exceeds a certain threshold, Er. Flexodomains are stripe patterns parallel to the imposed preferred direction no x, i.e. with a wave vector qc T In contrast to the standard Freedericksz transition, the sign of... 

We consider the dynamics of the Freedericksz transition in the splay geometry upon the removal of the applied field [24-27]. Initially the liquid crystal director is aligned vertically by the applied field, as shown in Figure 5.17(a). When the applied field is removed, the liquid crystal relaxes back to the homogeneous state. The rotation of the molecules induces a macroscopic translational motion known as the backflow effect. The velocity of the flow is... 

From Figure 8.19, an excellent dark state is observed at normal incidence. As the applied voltage exceeds the Freedericksz threshold voltage (V,ft-2.1 V s), LC directors are reoriented by the applied electric field, resulting in light transmission fi om the crossed analyzer. At -6 Vrms, the normalized transmittance reaches 100% for the green light (X = 550 nm). 

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. 

When the intensity 7 gradually increases from zero, the system remains undistorted with =( —1,0,0) and =0 until the first branching point 7=7q is reached. At this point, a second-order optical twist Freedericksz transition occurs and the output ellipticity begins to deviate from zero, following the solid line in Fig. 4. To reach the second steady-state branch in Fig. 4, the intensity 7 should be suddenly switched to the stable region and then... 

These configurations define Freedericksz transitions from undeformed to deformed states for which the threshold fields are sep-... 

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