Preedericksz transition

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. 

F. Lonberg and R.B. Meyer, New ground state for the splay-Preedericksz transition in a polymer nematic liquid crystal, Phys. Rev. Lett. 55(7), 718-721, (1985). doi 10.1103/PhysRevLett.55.718... 

A. Buka and L. Kramer, Theory of nonlinear transient patterns in the splay Preedericksz transition, Phys. Rev. A 45(8), 5624-5631, (1992). 

W. Zimmermann and L. Kramer, Periodic splay-twist Preedericksz transition in nematic liquid crystals, Phys. Rev. Lett. 56(24), 2655-2655, (1986). 

This Section will present our first example of how a magnetic field can influence the orientation of the director n in a nematic liquid crystal. This example also introduces the idea of a magnetic coherence length. The equations presented here will be refashioned in a natural way, in the next Section, to obtain the equations required for the study of Preedericksz transitions. 

There are therefore two solutions available for H > He satisfying the boundary conditions (3.105), namely, the undistorted solution = 0 from (3.112) and the distorted solution 0 z) provided by (3.118) and (3.119), or its equivalent form given via (3.122) and (3.123). To verify that He is indeed the critical threshold and that a Preedericksz transition occurs, we need to check that the distorted solution is energetically favoured over the undistorted solution for H > He, the only solution available for 0 < H < He is the undistorted solution. For H > Hey consider the difference in energies per unit area of the plates... 

A similar calculation to that involved in deriving the equilibrium equation (3.111) for the first classical Preedericksz transition shows that the equilibrium equation for this example is... 

Preedericksz transitions also occur under the influence of electric fields. All of the critical threshold magnetic field magnitudes derived in the previous Section have their analogues for electric fields when the electric energy is approximated by that given by, for example, equation (2.87). The simple identification... 

Figure 3.14 The Preedericksz transition when the director is strongly anchored parallel to the boundaries and the electric field is applied as shown. The FVeedericksz transition threshold occurs at Vc = Ecd = when Co > 0. For the post-...

As an elementary example of a weak Preedericksz transition consider the same situation first encountered in Section 3.4.1 for a nematic but now with the director weakly anchored to both boundary plates, so that n satisfies... 

Figure 3.16 The Preedericksz transition when the director is weakly anchored parallel to the boundary plates. The critical field strength is given by (3.263). For H < the director remsdns parallel to the plates throughout the sample, but...

The solution = 0 is a solution to the bulk equilibrium equation (3.248) and the boundary condition (3.252) when Oq = 0. However, as in the classical Preedericksz transitions, there is also the possibility of a distorted solution for H 0. Integrating (3.248) and using the conditions (3.243) and (3.247) provides the solution... 

The identification (3.207) allows the critical threshold value for the Preedericksz transition to be calculated in terms of the analogous electric field problem. Por instance, for a > 0, the critical voltage is given by K = dEe and so from (3.314) we have... 

These inequalities define constraints on the possible values of the elastic constants which validate the above analysis for showing that K is the critical threshold they are clearly always satisfied in the one-constant approximation Ki = K2 = K3 = K. Further, if K3 — 2K2 > 0 then knowing that the condition (3.321) is satisfied is sufficient to validate VJ. mathematically as the critical threshold for a Preedericksz transition on the other hand, if K3 — 2K2 < 0 then both the inequalities in (3.315)2 and (3.321) need to be fulfilled mathematically to justify Vc as the critical threshold. 

We do not pursue a derivation of these particular viscosities here, but introduce them for reference they originate in the dynamics of Preedericksz transitions which are discussed later in Section 5.9, where it is also shown that they are necessarily non-negative (see pages 228 and 235). Table 4.1 gives a summary of the viscosities discussed in this Section in terms of the Leslie viscosities. The viscosities t/i, 7/2, 3, 7712 and 7i constitute a canonical set of five independent viscosities for nematics. 

Section 5.9 introduces the dynamics of the Preedericksz transition in the classical geometries described in Chapter 3 in Section 3.4.1. The switch-on and switch-off times will be defined when flow is considered to be negligible in the usual twist geometry, as detailed in Section 5.9.1. In some instances, however, flow turns out to be quite influential and leads to the phenomena of backflow and kickback y as to be discussed in detail in the case of the splay geometry in Section 5.9.2. Backflow in the bend geometry is discussed in Section 5.9.3. 

The switch-off time can be determined by similar reasoning. Initially, it is assumed that H > He and that the post-Preedericksz transition static solution (t>o z) is known, for example via the results in Section 3.4.1. Prom equation (5.405) (or (3.136)) this post-threshold static solution for H > He satisfies... 

We now turn our attention to the bend homeotropic to planar Preedericksz transition geometry of Fig. 3.8 on page 78. First note that if flow is considered to be negligible then when H is near He an approximation for Ton is given analogously to (5.427) by... 

Section 6.2 introduces the basic mathematical description of SmC liquid crystals and proceeds to discuss the associated elastic energy, magnetic and electric energies, equilibrium equations and an elementary Preedericksz transition. The dynamic theory will be summarised in Section 6.3. It is worth emphasising now that in this theory the viscous stress is identical for both SmC and SmC liquid crystals and... 

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