Temperature director fluctuations

The local order in a cholesteric may be expected to be very weakly biaxial. The director fluctuations in a plane containing the helical axis are necessarily different from those in an orthogonal plane and result in a phase biaxiality . Further, there will be a contribution due to the molecular biaxiality as well. It turns out that the phase biaxiality plays a significant role in determining the temperature dependence of the pitch. Goossens has developed a general model taking this into account. The theory now involves four order parameters the pitch depends on all four of them and is temperature dependent. However, a comparison of the theory with experiment is possible only if the order parameters can be measured. 

It is considerably larger in the confined liquid crystals above Tni than in the bulk isotropic phase. The additional relaxation mechanism is obviously related to molecular dynamics in the kHz or low MHz frequency range. This mechanism could be either order fluctuations, which produce the well-known low-frequency relaxation mechanism in the bulk nematic phase [3], or molecular translational diffusion. Ziherl and Zumer demonstrated that order fluctuations in the boundary layer, which could provide a contribution to are fluctuations in the thickness of the layer and director fluctuations within the layer [36]. However, these modes differ from the fluctuations in the bulk isotropic phase only in a narrow temperatnre range of about IK above Tni, and are in general not localized except in the case of complete wetting of the substrate by the nematic phase. As the experimental data show a strong deviation of T2 from the bulk values over a broad temperature interval of at least 15K (Fig. 2.12), the second candidate, i.e. molecular translational diffusion, should be responsible for the faster spin relaxation at low frequencies in the confined state. 

Along the second-order part of the transition line one would expect 3D-XY critical behavior at least in the limit of the critical point. In real experiments, one observes crossover between XY critical and tricritical behavior, resulting in a ff values where CXxY<0Ceft<0tYc because the temperature range for most experimental data is limited to 10 director fluctuations and the smectic order parameter intervenes as a second source of deviation from isotropic behavior and influences the behavior of the smectic susceptibility and the correlation lengths (parallel and perpendicular to the director) much more than the specific heat. Here also a broad crossover should be ob-... 

First, unlike superconductors, SmA order is not long range (Peierls argument). This introduces an additional length, 1. As a result, correlations in director fluctuations and SmA order parameter fluctuations have different lengths. Where director orientational fluctuations are correlated on a length, (, hermo> SmA order parameter fluctuations are correlated on a length where l/4ff= 1/ thermo+l/l- The temperature dependence (i.e. critical exponents) of 1 depends on splay fluctuations which in turn depend on Xj. 

The temperature dependence of the director fluctuation induced Tf is given by S TIK) j]IKy and should be weak because KocS, where S is the magnitude of the nematic order parameter 5= (3cos 0-1). This has been indeed observed (Fig. 11) in the proton T in 5CB at 20 kHz [94]. 

The above result was confirmed by Yethiraj et al [56, 66] using an optical technique that quantified intensity fluctuations in the liquid crystal. Since the intensity fluctuations in the nematic phase are caused by nematic director fluctuations, which in turn are suppressed in the smectic-A phase, this is a very sensitive probe of the phase transition. In addition, the temperature dependence of the phase transition was obtained optically by placing the sample in a well-calibrated temperature gradient. Indeed the existence of a sharp static interface is itself a qualitative indicator of first-order character. Not only was the transition in the 8CB-10CB system always first order, the strength of the discontinuity was larger not smaller than the HLM prediction. This result is consistent with a subsequent calculation by Herbut et al which takes into account tp fluctuations. 

As in the case of electric field induced structural transition, the temperature induced structural transition is also associated with critical slowing down of the director fluctuations. However, as reorientation of the droplet director can occur along any direction in the xy plane, both 6n ey and 6n ex fundamental modes exhibit a critical behaviour of 1/ro in the vicinity of Tc and consequently a slowing down detected in the dynamic light scattering experiments is more pronounced than in case of electric field induced transition. 

The atoms in crystals are not always fixed in a perfectly ordered position except at the absolute temperature Due to the thermal movement, atoms more or less deviate from their equilibrium position. For the same reason, the orientational order in liquid crystals is not perfect either. Because of the thermal fluctuation, the orientation and position of liquid crystals vary constantly. If the positions and orientations of liquid crystal molecules are frozen at a moment in time, the picture should look like that shown in Figure 1.5. The molecules tend to align along a preferred direction, but imperfectly. This preferred direction is defined as the director n. Because the molecules are moving all the time, they are not fixed at a constant... 

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