Frederiks transitions

Fig. 24. Three principal types of orientational effects induced by electric (E) and magnetic (H) fields in nematic low molecular liquid crystals. At the top of the figure the initial geometries of molecules are shown. Below the different variants of the Frederiks transition — splay-, bend- and twist-effects are represented...

Liquid-Crystalline Polymers 5.1.2.1 S-effect (Frederiks Transition)... 

Figure 1.18. The Frederiks transitions I, II and III. The arrows represents the direction of the applied magnetic held H.

The elastic constants of liquid crystalline polymers can be measured in terms of the Frederiks transitions under the presence of a magnetic or electric field. Raleigh light scattering is also a method for measuring the elastic constants. Those techniques successfully applied to small molecular mass liquid crystals may not be applicable to liquid crystalline polymers. This is why very few experimental data of elastic constants are available for liquid crystalline polymers. 

Three kinds of Frederiks transitions can be used to measure the three elastic constants of nematic polymers. The magnetic threshold fields Hci are functions of Ku(i = 1,2,3) respectively,... 

Fig. 11.15 Three basic configurations of the director and the magnetic field for the Frederiks transition onset, namely, splay (a), bend (b) and twist (c)...

Now the free energy density has a form (11.45) wherein, due to a large the field E becomes dependent oti coordinates. In this case, one should operate with electric displacement D. For example, in the case of the Frederiks transition and the splay geometry of Fig. 11.15a the field strength is ... 

Here EsJEp represents the ratio of the saturation field E at = Eb to the Frederiks transition field (electric or magnetic). As EsJEp k, dib 1, the left part of Eq. (11.62) is close to 1. Therefore, assuming Kn = 33 = K, we turn back to Eqs. (11.60) and (11.61). In the next chapter we shall meet the break of anchoring effect when discussing bistable devices. 

If a nematic liquid crystal has negligible conductivity the results of Sections 11.2.1-11.2.5 for the Frederiks transition induced by a magnetic field may be directly applied to the electric field case. To this effect, it suffices to substitute H by E and all components of magnetic susceptibility tensor Xij hy correspondent components of dielectric permittivity tensor s,y. From the practical point of view the electrooptical effects are much more important and further on we discuss the optical response of nematics to the electric field. 

The oscillations of I (U) are well seen in the experimental plot. Fig. 11.21. The measurements were made at 27°C on 55 nm thick cell filled with a mixture having ta = 22. From the I (U) curve, the field dependence of the phase retardation 8(17) and the Frederiks transition threshold Uc were obtained. In mm, from Ec = UJd and Fq. (11.56) the splay elastic constant Ku was found. The bend modulus "33 was calculated from the derivative dbldU. The same material parameters may be found for the whole temperature range of the nematic phase. 

Fig. 11.21 The oscillating experimental curve I(U) right axis) is voltage dependent intensity of the light transmitted by the 50 pm thick planar nematic cell placed between crossed polarizers (the logarithmic voltage scale for /([/ j is the bottom axis). The pointed curve is the voltage dependence of phase retardation 5 calculated from curve I(U) with a Frederiks transition threshold at Uc (the scale for 5(1/) is on the top axis and its argument i.e. voltage is on the left axis)...

This effect is a version of the splay-bend Frederiks transition, but it is observed in liquid crystals doped with dyes. The liquid crystalline matrix (the host) is subjected to the influence of a field the function of the dye (the guest) is to enable the effect to be seen with only one polarizer or even without any. 

Such an antisymmetric distortion differs from the symmetric distortion characteristic of the Frederiks transition. It is instructive to compare these two cases. In Fig. 11.28 the space distributions of the director n and its x-projection rix = sinfl 9 are pictured for the Fredericks transition (a) and flexoelectric effect (b) the anchoring energy at both surfaces is infinitely strong in case (a) and finite in case (b). 

Fig. 11.28 Comparison of the distortion profile (molecular picture below and angle 3(z) above) for the Frederiks transition with infinite anchoring energies (a) and flexoelectric effect with finite anchoring energies (b) (homeotropic initial director alignment in both cases)...

The dependencies 6 ocd and 8 ocE agree well with experiment [30]. Therefore, in principle, we can find from the measured value of the cell retardation because usually 33 is known from the Frederiks transition threshold. However, in a real experiment it is almost impossible to have zero anchoring energy. For the finite anchoring energy, we can only find ratio ej,/W and the accuracy of determination... 

In Chapter 11 we have found that, for the Frederiks transition in nematics, the threshold field coherence length is determined by the cell thickness, = din, see Eq.(11.53). Now we shall briefly discuss another type of instability with a threshold determined by the geometrical average of the two parameters mentioned [17],... 

From Eq. (13.29) is seen that, at (p = 0, there is no electric torque exerted on the director. Thus, there should be a threshold for the distortion as in the case of the Frederiks transition in nematics. We can find the threshold field Ec, considering a small distortion (p 0. The equation... 

Here, the first term describes the nematic-like elastic energy in raie crmstant approximation (K in 9). This term allows a discussion of distortions below the AF-F threshold (a kind of the Frederiks transition as in nematics in a sample of a finite size). In fact, the most important specific properties of the antiferroelectric are taken into account by the interaction potential W between molecules in neighbour layers the second term in the equation corresponds to interaction of only the nearest layers (/) and (/ + 1). Let count layers from the top of our sketch (a) then for odd layers i, i + 2, etc. the director azimuth is 0, and for even layers / + 1, / + 3, etc. the director azimuth is n. The third term describes interactimi of the external field with the layer polarization Pq of the layer / as in the case of ferroelectric cells. Although for substances with high Pq the dielectric anisotropy can be neglected, the quadratic-in-field effects are implicitly accounted for by the highest order terms proportiOTial to P. ... 

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