Orientational director fluctuations

In a uniaxial nematic liquid crystal, the spatial orientation of the optical axis is determined by the orientation of the director. Due to thermally excited orientational director fluctuations, the spatial direction of the optical axis is not constant in time. As a result, any light illuminating the sample is... 

Dynamic processes that are specific for liquid crystalline phases are the spatially correlated collective orientational fluctuations. These collective fluctuations include the director fluctuations, describing the local fluctuations in the orientation of the director, and the order parameter fluctuations, describing the fluctuations in the magnitude of S. However, in most cases (except in the vicinity of a phase transition), the latter can be neglected and the orientational director fluctuations are dominant. 

Dynamic properties of a liquid crystal and spectrum of orientational director fluctuations, discussed previously for bulk samples, are strongly influenced by the presence of a surface. As the system is no longer infinite and homogeneous, the fluctuation modes are no longer plane waves with an arbitrary wave vector q. 

The orientational director fluctuation modes have been described and discussed for thin planar cells and liquid crystal droplets. Now experimental observations for these two special geometries will be presented and the implementation of... 

It is often preferable to evaluate 6q by EMD methods because the director fluctuates around the preferred orientation in a shear flow simulation, which makes it hard to obtain accurate estimates. If one performs such a simulation one must fix the director at several alignment angles and calculate the antisymmetric pressure tensor, which, according to Eq. (4.10e), is a linear function of cos 26. One can fit a straight line to the data points and the zero gives... 

Dielectric relaxation spectroscopy can monitor molecular and collective modes for motion of liquid crystalline molecules. In ferrolectric liquid crystals based on chiral tilted smectic phases the complex dielectric permittivity e has, in addition to molecular orientational modes, two contributions from the director fluctuations. 

Thus, director fluctuations in the small angle approximation can contribute to the spectral density Ji uj) through the Qi q (r) term where Sq = refers to ordering of the molecule with respect to the local director. This small angle approximation is probably not warranted for low orientational orders, where might be as large as 40°. While g o( )... 

The detection of director fluctuations in liquid crystals by means of deuteron relaxation in mesogens at conventional frequencies has not met with success [6.45, 6.46]. This can be attributed to two factors (1) the numerical value of Adf is small because most C- H bonds are oriented at an unfavorable angle to the long molecular axis this is especially true for... 

The SmA liquid crystalline phase results from the development of a one-dimensional density wave in the orientationally ordered nematic phase. The smectic wave vector q is parallel to the nematic director (along the z-axis) and the SmA order parameter i/r= i/r e is introduced by P( ) = Po[1+R6V ]- Thus the order parameter has a magnitude and a phase. This led de Gennes to point out the analogy with superfluid helium and the normal-superconductor transition in metals [7, 59]. This would than place the N-SmA transition in the three-dimensional XY universality class. However, there are two important sources of deviations from isotropic 3D-XY behavior. The first one is crossover from second-order to first-order behavior via a tricritical point due to coupling between the smectic order parameter y/ and the nematic order parameter Q. The second source of deviation from isotropic 3D-XY behavior arises from the coupling between director fluctuations and the smectic order parameter, which is intrinsically anisotropic [60-62]. 

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 simplest model. The physical mechanism of the instability could be described as follows [236], A homogeneously oriented nematic liquid crystal is stabilized by the elastic torque (due to surface anchoring). The dielectric torque is considered to be negligible ( a=0)- Now, let us imagine a small incident director fluctuation with a period of the order of cell thickness d ... 

In-depth treatments of the topic are available in several books [1-6] and in a large number of review articles. The monograph by Dong [6], for example, focuses on aspects like the dynamics of nuclear spins, orientational order, molecular field theories, nuclear spin relaxation theory, director fluctuation and spin relaxation, rotational and translational dynamics, internal dynamics of flexible mesogens, and multiple-quantum and two-dimensional NMR, topics that will be touched upon very briefly here. Re-... 

The characteristic frequency dependence is a direct result of the gapless Goldstone mode type nature of the director fluctuations. The constants C[ and C2 depend on the magnitude of the nematic order parameter, the viscoelastic constants, the molecular geometry of the spin positions and the orientation of the director with respect to the external magnetic field. 

For a nuclear probe consisting of two protons with a fixed separation distance r, the dipole-dipole interactions of which are modulated by fluctuations in the orientation of intemuclear vector r with respect to the external magnetic field H due to nematic order director fluctuations, the spin-lattice relaxation rate is... 

In contrast to nematics, a helical twist of the molecular director takes place in the chiral nematic phase. Studies of the spin-lattice relaxation in chiral nematics have shown that the relaxation mechanisms are essentially the same as in pure nematics [141, 142]. At high Larmor frequencies the relaxation is diminished by molecular self-diffusion and by local molecular rotations, whereas director fluctuations determine the relaxation rate at low Larmor frequencies. This can be easily understood because the spin-lattice relaxation rate in the MHz region is dominated by orientational fluctuations with wavelength much smaller than the period of the helix. The influence upon the rotating frame spin-lattice relaxation time Tip of the rotation of the molecules due to diffusion along the helix, an effect specific for twisted structures, has not been observed in COC [143]. 

Director fluctuations In a droplet, the wavelengths of director fluctuations cannot be larger than the dimension of the droplet. This results in a cut-off frequency where q i =Kl2R. This result has been derived previously for bulk nematics where was introduced to take into account the finite dimension of the uniformly oriented domain [140]. The consequence of the small wave vector cut-off is a leveling-off of the relaxation disper-... 

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