Elastic Frederiks transitions

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,... 

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. 

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. ... 

Elastic moduli could be measured from the electro- (and magneto)-optical characteristics of the Frederiks transition, such as the threshold voltage C/p oi the threshold magnetic field ifp, according to the relationships... 

To find the value of the elastic moduli ratio Kss/Ku it is also possible to measure either the birefringence 6 = And/X And is the optical path difference and the A is the wavelength of light), or the capacitance C versus voltage U curve in the electrically induced Frederiks transition [55]. 

The field-induced reorientation of nematic polymers caused by the coupling of the electric field with their dielectric anisotropy was studied in a variety of papers [229, 231-238]. Unfortunately, only in a few papers (e.g., [237]) is a certain preliminary orientation of a polymer specified and we can speak of the true Frederiks transition with a well-defined threshold voltage. Nevertheless, the general opinion is that the Prank elastic moduli of both comb-like [232-235, 238] and linear-chain [236, 237] nematic polymers are of the same order of magnitude as of their low-molecular mass counterparts. 

The moduli were calculated from the threshold of the Frederiks transition ((4.9) induced by a magnetic (Ax > 0) and electric (Ae < 0)) field in homeotropically oriented liquid crystal layers. The same order of magnitude (10 -10 dyn), which is typical of conventional nematics, has been found for elastic moduli Kn and for other nematic polymers [233, 234]. Unwinding of the helical structure of chiral nematic polymers allowed the elastic constant K22 to be calculated K22 10" dyn for an arylic comb-like copolymer with cholesterol and cyanobiphenyl side-chair mesogens [229]). 

As in the case of the Frederiks transition (discussed in Section 9.3.1.1 of this Chapter), the theoretical interpretation of the twist effect is based on the minimization of the free energy of the system. In this case, however, the problem is two dimensional, since both the azimuthal angle 0(z) and the tilt angle G(z) are considered to be dependent on the z coordinate. In the case of the infinitely strong anchoring, the threshold for the distortion includes all three elastic moduli of a nematic liquid crystal [111] ... 

It is also possible to use turbidity measurements to evaluate elastic constants (Fig. 5, though elastic constants measured in this way for alkylcyanobiphenyls (nCBs) [25] and mixtures [26] are consistently higher than those determined by Frederiks transition studies. Hakemi [27] measured j, 22 and 33 for 8CB (octylcyano biphenyl) close to the nematic-smectic A phase transition and deduced a value of the critical exponent from the divergence of 22 and 33. 

The anchoring transition considered above is driven by repulsive interactions as controlled by the grafting density. Related transitions may be driven by geometrical confinementwhich can induce a tilting transition. The transition is reminiscent of the Frederiks transition in single component nematics, but is driven by the elasticity of the LCP rather than by an external field. 

A procedure is described that can measure optical phase retardations of birefrigent materials with a resolution 2x10-4 radians. The method relies on phase modulation with alternate right and left handed circularly polarized light. Phase sensitive detection is employed to reduce noise and thermal fluctuations in the optics and light source. The method is useful in Frederik s transition measurements to determine the elastic constants of weakly birefringent polymer liquid crystals with long equilibration times. 

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