Director equilibrium

Fig. 6. The two generic shapes of molecules which exhibit flexoelectric polarisation under distortion of the equilibrium director distribution...

Figure 8.29 Illustration showing how all-anticlinic bilayer smectic should be ferroelectric is given. In case of covalent dimers (bent-core mesogens), equilibrium tilt of director combined with anti clinic layer interfaces in bow plane provides SmCsPF ferroelectric banana structure.

Figure 15 A lamellar block copolymer phase is reoriented through external shear. The initial phase has the direction of the lamellae parallel to the shear gradient direction. The most stable state would be to orient the director parallel to the shear and shear gradient direction. However, the reorientation process gets stuck before true equilibrium is reached. The stuck orientation is relatively stable, because the lamellae have to be broken up before they can further align with respect to the shear flow. Reprinted with permission from Ref. 56.

Most of the results presented in this section, including Eqs. (4.15)—(4.17), are not valid when the equilibrium state of the fluid exhibits global orientational order, for example, a global director. However, they do apply to an isotropic suspension of locally anisotropic objects, such as vesicles or liposomes, which may exhibit a local director, provided that long-range orientational correlations do not extend over a significant fraction of the volume sampled in the experiment. 

Tumbling regime At very low shear rates, the birefringence axis (or the director) of the nematic solution tumbles continuously up to a reduced shear rate T < 9.5. While the time for complete rotation stays approximately equal to that calculated from Eq. (85), the scalar order parameter S,dy) oscillates around its equilibrium value S. Maximum positive departures of S(dy) from S occur at 0 n/4 and — 3n/4, and maximum negative departures at 0 x — k/4 and — 5it/4, while the amplitude of oscillation increases with increasing T. 

By adding the function u(x) to the phase factor in (4) one can describe departures from the planar (lamellar, one-dimensional) layer arrangement, which is characteristic for the 2D structures. The first term in (3) is the smectic layer compressibility energy. It is zero when layers are of the equilibrium thickness. If cx(T) > 0, the second term in (3) requires the director to be along the smectic layer normal (the smectic-A phase). If c (T) < 0, this term would prefer the director to lie in the smectic plane. So the last term in (3) is needed to stabilize a finite tilt of the director with respect to the smectic layer normal. In addition this term gives the energy penalty for the spatial variation of the smectic layer normal. 

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

The spatial and temporal response of a nematic phase to a distorting force, such as an electric (or magnetic) field is determined in part by three elastic constants, kii, k22 and associated with splay, twist and bend deformations, respectively, see Figure 2.9. The elastic constants describe the restoring forces on a molecule within the nematic phase on removal of some external force which had distorted the nematic medium from its equilibrium, i.e. lowest energy conformation. The configuration of the nematic director within an LCD in the absence of an applied field is determined by the interaction of very thin layers of molecules with an orientation layer coating the surface of the substrates above the electrodes. The direction imposed on the director at the surface is then... 

Ilya Prigogine (b. 1917 in Moscow, Russia) is Director of the International Solvay Institutes of Chemistry and Physics, Brussels, Belgium, and of the 1. Prigogine Center for Statistical Mechanics and Complex Systems, The University of Texas at Austin. He is a Belgian citizen. He received the Nobel Prize in Chemistry in 1977 for his contributions to non-equilibrium thermodynamics, particularly the theory of dissipative structures. ... 

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.)... 

These results can be cross checked by performing a simulation where the director is constrained to lie in the vorticity plane but leaving it free to select the alignment angle. The angular distribution of the director is shown in Fig. 5 In these simulations only 256 particles were used. Therefore the distribution is fairly wide. As the system size increases the distribution becomes narrower and it is completely sharp in the thermodynamic limit. The maximum of the distribution appears ai 6 - 20° which is in agreement with the zero of the antisymmetric pressure tensor. A similar value of 6q was also found by using the equilibrium fluctuation relations (4.13) and (4.14). One can consequently conclude that the liquid crystal is flow stable. 

One also finds that fixing the director generates a new equilibrium ensemble where the Green-Kubo relations for the viscosities are considerably simpler compared to the conventional canonical ensemble. They become linear functions of time correlation function integrals instead of rational functions. The reason for this is that all the thermodynamic forces are constants of motion and all the thermodynamic fluxes are zero mean fluctuating phase functions in the constrained ensemble. 

We can identify four pairs of thermodynamic forces and fluxes, the symmetric traceless strain rate (Vu) and the symmetric traceless pressure tensor, the director angular velocity relative to the background, (l/2)Vxu-I2 and the torque density X, the streaming angular velocity relative to the background (l/2)Vxu- and the torque density and the trace of the strain rate V-u and difference between the trace of the pressure tensor and the equilibrium... 

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... 

Let us first consider the set of equidistant diffuse streaks (c) perpendicular to the director [la,b, 30]. These streaks are very similar to those observed on the X-ray scattering patterns of the nematic phases of main-chain LCPs discussed in Sect. 3.2 but they must be interpreted differently because the SmA phase shows (quasi) long-range positional order. These diffuse streaks also correspond to the intersection with the Ewald sphere of a set of equidistant diffuse planes. But here this set represents the Fourier transform of uncorrelated rows of side-chains displaced along the director from their equilibrium position inside the layers (Fig. 13). 

The thermodynamical equilibrium of nematics would correspond to a spatially uniform (constant n(r)) director orientation. External influences, like boundaries or external fields, often lead to spatial distortions of the director field. This results in an elastic increment, fd, of the volume/ree energy density which is quadratic in the director gradients [2, 3] ... 

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