The Nematic Liquid Crystal Case

We now investigate the Couette flow of a nematic liquid crystal in precisely the same type of experiment discussed above for an anisotropic fluid described in Fig. 5.8 and in essence follow the work of Atkin and Leslie [6]. The one-constant approximation for the nematic elastic energy will be assumed in order to simplify the presentation  

The constraints (4.118) are again satisfied. As before, the boundary conditions on (jj are given by 

Firstly, notice that the expressions for Ui, n, Ay, Wij, Ni and i are exactly the same as those derived earlier in equations (5.243) to (5.247), the only difference being that 0 will turn out not to be constant. Prom formulae (C.8) to (C.IO) in Appendix C and the form for the director given by (5.279), the non-zero physical components of [Vn] = riij in cylindrical coordinates are 

The Lagrange multiplier A can be eliminated from these two equations in a similar way to that used to obtain equation (5.250) earlier, resulting in the equation 

If desired, the Lagrange multiplier can be evaluated directly by taking the scalar product of equations (5.283) with n to find, with the aid of equations (5.247) and (5.287) to (5.289), 

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As in the nematic liquid crystal case, it is clear that (6,78) represents a balance of forces at equilibrium. This can be seen by applying arguments that parallel those in Remark (i) on page 40. Similarly, it can be shown that the equilibrium equations (6.79) and (6.80) actually arise from a balance of moments, as has been demonstrated by Stewart and McKay [267], by means of an appropriate extension to the Ericksen identity (B.6) (cf. [181]). For the present, however, we restrict our attention to the derivation of the couple stress tensor Uj and its associated couple stress vector Analogous to the discussion for the nematic case in Remark (i) on page 40 when the balance of moments was discussed, consider an arbitrary, infinitesimal, rigid body rotation a for which... 

In the case of the nematic liquid crystal MBBA, the viscosity in the nematic phase is lower than in the isotropic phase. At 50 C the nematic phase reappears under 0.2 MPa of pressure then, the viscosity decreases and rises again exponentially as pressure increases. At 90°C the nematic state reappears at 1,2 MPa. 

As discussed abfeve, ellipsometry is directly sensitive only to the interfacial variations of the nematic order parameter, which is connected to the optical refraction indices. The interfacial smectic order, which has no direct influence on the optical properties, can only be observed due to its coupling to the nematic (orientational) order. The same experimental setup as described in Sect. 4.1.3 has been used to study the interface between smectic liquid crystal dodecylcyanobiphenyl (12CB) in the isotropic phase and the silanated glass. Although only orientational order is observed, the temperature dependence of pb is in this case quite different from the case with the nematic liquid crystal, as evident from Fig. 4.4. 

With the aim of quantitatively predicting the orientational order of rigid solutes of small dimensions dissolved in the nematic liquid crystal solvent, 4-n-pentyl-4 cyanobiphenyl (5CB), an atomistic molecular dynamics (MD) computer simulation has been applied. It is found that for the cases examined the alignment mechanism is dominated by steric and van der Waals dispersive forces. A computer simulation of the deuterium NMR spectra of molecules in a thin nematic cell has been carried out and the director distribution in the cell has been studied. An experiment for the direct estimation of an element of the order matrix from H NMR spectra of strongly dipolar coupled spins that is based on the multiple quantum spin state selected detection of single quantum transitions has been proposed. The experiment also enables obtaining nearly accurate starting dipolar... 

The case of homeotropic alignment is quite easy to envisage and is identical to the situation described for the nematic liquid crystal phase except for the lamellar nature of the phase. However, the focal-conic formation is more complex and more commonly... 

The reorienution of the director of a nematic liquid crystal induced by the field of a light wave is considered. An oblique (with respect to the director) extraordinary wave of low intensity yields the predicted and previously observed giant optical nonlinearity in a nematic liquid crystal. For normal incidence of the light wave on the cuvette with a homeotropic orientation of the nematic liquid crystal, the reorientation appears only at light intensities above a certain threshold, and the process itself is similar to the Fredericks transition. The spatial distribution of the director direction is calculated for intensities above and below threshold. Hysteresis of the Fredericks transition in a light field, which has no analog in the case of static fields, is predicted. 

A particularly interesting case occurs when the static value of exceeds that of . In this case, as a result of the low-frequency dispersion in at a certain frequency /o, a change in sign of the dielectric anisotropy of the nematic liquid crystals can occur. Sometimes this frequency is low. 

It is well known that a nematic liquid crystal is nonpolar as a result of the free or hindered rotation of its constituent molecules around their axes. In the absence of an external field the distribution of the dipoles in an undistorted nematic liquid crystal has a nonpolar cylindrical symmetry. This is shown schematically in Fig. 4.29(a). However, as Meyer [183] has shown, a polar axis can arise in a liquid crystal made up of polar pear-shaped molecules when it is subjected to splay deformations, or in a liquid crystal made up of banana-shaped molecules subjected to bend deformations. In this case, the polar structure corresponds to closer packing of the molecules (Fig. 4.29(b)). Thus, the external mechanical deformation of the nematic liquid crystal results in the occurrence of a charge at electrodes perpendicular to the polar axis, i.e., there is a similarity to the piezoelectric effect in solid crystals. 

Domain patterns arise in sufficiently strong fields with both a homogeneous [80, 81] and a homeotropic [82] initial orientation of the nematic liquid crystal. The period of these domains is always of the order of the thickness of the layer (but not of the separation between electrodes). The Kapustin-Williams domains also occur in twist cells in this case, the strips are oriented at an angle of 45° to the direction of rubbing of the electrodes [87]. 

The term surface means here the surface of the other phase in contact with the nematic liquid crystal. The symmetry of this surface (and of 7s) is independent of the orientation taken by the nematic phase at the surface. In contrast, the symmetry of the interface depends on this orientation it is the subgroup of the surface symmetry group containing the symmetry elements which leave invariant the anchoring direction effectively taken by the liquid crystal. If the other phase is a solid or liquid substrate, the surface is simply the surface of this substrate. In the case when the other phase is the gas or isotropic phase, the surface is not a physical entity. However, one can still, in principle, distinguish this isotropic surface (C symmetry) from the interface with the nematic phase, the symmetry of which is C , Cjv, and Cjj, for homeotropic, planar, and tilted anchoring, respectively. 

If the director is already parallel to the electric field, the free energy decreases as increases. This means an ordering of the anisotropic state, which in the isotropic phase is equivalent to a field-induced transition to the nematic liquid crystal phase. This is similar to the case when the pressure induces a transition to the lower symmetric phase. Such a situation is described by the Clausius-Clapeiron equation that relates the increment of the phase transition temperature, to the pressure. In our case this is equivalent to the field-induced increase of the isotropic-nematic transition as ... 

Closely related phenomenon to the piezoelectricity in liquid crystals is the flexoelectricity introduced by R.B. Meyer. Flexoelectricity means a linear coupling between the distortion of the director and the electric polarization. The constituent molecules of the nematic liquid crystals are rotating around their axes, and in absence of electric fields they are nonpolar. However, polar axes can arise in a liquid crystal made up from polar pear- or banana-shape molecules when they are subjected to splay or bend deformations, respectively. In these cases, the polar structures correspond to more efficient packing of the molecules (see Figure 8.17). 

The stability of the nematic liquid-crystal phase arises from the existence of strong interactions between pairs of the constituent molecules. As in normal liquids, the potential of interactions will have attractive contributions to provide the cohesion of the fluid, and repulsive contributions which prevent the interpenetration of the molecules. In the case of the rod-like molecules of nematics, however, these interactions are highly anisotropic. That is, the forces acting between such molecules depend not only on their separation but also, and most importantly, on their mutual orientations. From the symmetry and structure of the nematic phase, we see that the rod-like molecules in fact interact in a manner that favors the parallel alignment of neighboring molecules. 

Instead of a 90° twist of the nematic liquid crystal phase, STN displays have a twist angle between 180 and 240° (occasionally 270° is also used). The twist in these cases... 

The viscous stress need not be symmetric in this nematic liquid crystal case, in contrast to that in equation (5.256) for the purely anisotropic case. The function... 

The present appendix represents a detailed derivation of the kinetic equations of the fluctuating liquid cage model in the classical formalism. A natural generalization is done for the case of partially ordered media, e.g. nematic liquid crystals. One of the simplest ways to take into account the back reaction is demonstrated, namely to introduce friction. 

If the molecules of a liquid crystal are optically active Ichiral), then the nematic phase is not formed. Instead of the director being locally constant as is the case for nematics, the director rotates in helical fashion throughout the sample, Within any plane perpendicular to the helical axis the order is nematic-like. In other words, as in a nematic there is only orienlalional order in chiral nematic liquid crystals, and no positional order. 

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