Order parameter director field

An aligned monodomain of a nematic liquid crystal is characterized by a single director n. However, in imperfectly aligned or unaligned samples the director varies tlirough space. The appropriate tensor order parameter to describe the director field is then... 

The above spectral densities can be modified for the occurence of chain flexibility, and for the director being oriented at dLD w.r.t. the external BQ field in the L frame. For CD bonds located in the flexible chain, the effect of DF is reduced due to an additional averaging of the time dependent factor (/f g) by conformational transitions in the chain. Consequently, the spectral densities given in Eqs. (60)-(62) are modified by replacing Soc%0(Pm,q) by the segmental order parameter YCD of the C-D bond at a particular carbon site on the chain.146,147 As observed experimentally,148,149 the spectral densities in a flexible chain show a SqD dependence when DF dominate the relaxation rates. The general expression of Jm(co 0LD) due to DF in uniaxial nematic phases is given by... 

We write the solution as the vector X = (6,(j),u,vx,vy,i ,P,) consisting of the angular variables of the director, the layer displacement, the velocity field, the pressure, and the modulus of the (nematic or smectic) order parameter. For a spatially homogeneous situation the equations simplify significantly and the desired solution Xo can directly be found (see Sect. 3.1). To determine the region of stability of Xq we perform a linear stability analysis, i.e., we add a small perturbation Xi to... 

Equation (39) shows that nematic degrees of freedom couple to simple shear, but not the smectic degrees of freedom the modulus of the nematic order parameter has a non-vanishing spatially homogeneous correction (see (39)), whereas the smectic order parameter stays unchanged. The reason for this difference lies in the fact that J3 and /3 include h and p, respectively, which coupled differently to the flow field (see (22) and (23)). Equation (38) gives a well defined relation between the shear rate y and the director tilt angle 9o, which we will use to eliminate y from our further calculations. To lowest order 0O depends linearly on y ... 

The application of an electric field between the electrodes results in a realignment of the nematic liquid crystal mixture and the dichroic dye molecules parallel to the electric field resulting in a lower optical density (absorption) and, theoretically, the disappearance of colour assuming an ideal order parameter (S = 1) of the nematic liquid crystal director and the dye molecules. A residual absorption in this state gives rise to a display with a strongly coloured background and weakly coloured information. 

Because nematic liquid-crystalline polymers by definition are both anisotropic and polymeric, they show elastic effects of at least two different kinds. They have director gradient elasticity because they are nematic, and they have molecular elasticity because they are polymeric. As discussed in Section 10.2.2, Frank gradient elastic forces are produced when flow creates inhomogeneities or gradients in the continuum director field. Molecular elasticity, on the other hand, is generated when the flow is strong enough to shift the molecular order parameter S = S2 from its equilibrium value 5 . (Microcrystallites, if present, can produce a third type of elasticity see Section 11.3.6.)... 

Although we expect for dimensional reasons that the average magnitude of h, and hence the magnitude of ([nh]), will be proportional to pv, the tensorial form of ([nh]) is unknown. To obtain ([nh]), without having to revert back to (an almost impossible) microscopic calculation of the director field, Larson and Doi (1991 Kawaguchi 1996) assumed Aat ([nh]) is a function of the mesoscopic order parameter S— that is, that ([nh]) = Ka f(S). Dimensional reasoning then leads to the ansatz that... 

In equations (5)-(8), i is the molecule s moment of Inertia, v the flow velocity, K is the appropriate elastic constant, e the dielectric anisotropy, 8 is the angle between the optical field and the nematic liquid crystal director axis y the viscosity coefficient, the tensorial order parameter (for isotropic phase), the optical electric field, T the nematic-isotropic phase transition temperature, S the order parameter (for liquid-crystal phase), the thermal conductivity, a the absorption constant, pj the density, C the specific heat, B the bulk modulus, v, the velocity of sound, y the electrostrictive coefficient. Table 1 summarizes these optical nonlinearities, their magnitudes and typical relaxation time constants. Also included in Table 1 is the extraordinary large optical nonlinearity we recently observed in excited dye-molecules doped liquid... 

In spatially constrained liquid crystals the director field is space and time dependent (from the view of a molecule that diffuses between regions with different local director orientations). The orientational order parameter usually varies within the sample. In a local region of the cavity, represented by the position vector r, a selectively deuterated compoimd yields a spectrum of two sharp lines separated by... 

Fig. 8.6. Profiles of the lowest (thick line) and of one of the highly excited modes of biaxial (P i,n) and director (P 2,n) fluctuations in the LC heterophase system in contact with (a) ordering and (b) disordering substrates. Dfished lines correspond to the mean-field scalar order parameter. In all cases T —5- Tni-...

In the bent-director structure the biaxiality can be neglected and the scalar order parameter corresponds to its bulk value at the temperature 0eff — 0+ (37t /4) due to elastic deformation of the director field the temperature of the hybrid system effectively increases [10]. The increase of the ef-... 

Due to the effect of external fields, the order can vary in space and gradient terms have to be added to the Landau expansion (8.9). Usually, only the terms up to the quadratic order are considered. There are many symmetry allowed invariants related to gradients of the tensorial order parameter [29]. However, in the vicinity of the phase transition, one is not interested in elastic deformations of the nematic director but rather in spatial variations of the degree of nematic order. Therefore, the pretransitional nematic system is described adequately within the usual one-elastic-constant approximation. 

The diagonal componeirts of the tensor Q represent the degree of order with respect to the axes x, y, and z. The off-diagonal components, represent the bending of the director field in the plane (a,/3). The parameter coupling the two preferred directions x and 2 ,... 

These two examples illustrate that it is important whether the structure already possesses a polarization P or if P is field-induced. If we ignore the possibility of a spatially non-uniform order parameter (which is likely only very close to a surface) the contribution to the free energy density in the presence of a field is, in the first case, / = — P E, and the torque on the director is P = P x E, where... 

Unoriented poly (p-hydroxybenzoic acid-co-2,6-hydroxynaphthoic acid) exhibited smoothly wandering director fields in three dimensions. Alignment with a 1.1 T magnetic field for 30 min at 300 C transformed this structure to domains with an anisotropic shape within which the polymer was highly oriented, and the global order parameter amounted to 0.85 [110]. Boundaries were of the splay-bend type and involved a 180 director rotation. At lower field strengths, the domains were less... 

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