Distortions in nematics

Figure 10.6 Three types of distortion in nematics. (From de Gennes and Prost, Copyright 1993, by Oxford University Press, Inc. Used by permission of Oxford University Press, Inc.)...

Figure 1. Distortions in nematics, (a) Pure splay, (b) pure twist and (c) pure bend [19],...

Fig. 8.6 Splay, bend and twist distortions in nematics confined between two glasses that align liquid crystal at the surfaces either homogeneously (for splay and twist) or homeotropically (for bend)...

As discussed in Sec. 2.2.2.1, the foundations of the continuum model were laid by Oseen [61] and Zocher [107] some seventy years ago, and this model was reexamined by Frank [65], who introduced the concept of curvature elasticity to describe the equilibrium free energy. This theory is used, to this day, to determine splay, twist, and bend distortions in nematic materials. The dynamic models or how the director field behaves in changing from one equilibrium state to another have taken much longer to evolve. This is primarily due to the interdependency of the director it (r, t) and v (r, /) fields, which in the case of chiral nematics is made much more complex due to the long-range, spiraling structural correlations. The most widely used dynamic theory for chiral... 

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... 

Not only reaction rates, but also the positions of chemical equilibria can be influenced by liquid crystals as reaction media. A nice example is the ionization equilibrium of chloro-tris(4-methoxyphenyl)methane according to Ar3C-Cl Ar3C+ -I-C1 , which is more shifted in favour of the nearly planar triarylcarbenium ion in nematic liquid crystals as compared to in an isotropic reaction medium [868], Obviously, the discshaped carbenium ion fits better into the rod-Hke nematic phase than the tetrahedral covalent ionogen, which distorts the internal structure of the nematic liquid crystal. 

Here u is the position of a layer plane and z is the position coordinate locally parallel to the director n, where n is parallel to the average molecular axis, which is assumed to remain normal to the layer plane, du/dz = e is the compressional (or dilational) strain. Thus, layer bending and layer compression are characterized by a splay (or layer-bend) modulus K and a compression modulus B. Other kinds of distortion present in nematics, such as bend or twisting of the director n, are not compatible with layers that remain nearly parallel, and hence are forbidden. Equation (10-36) is not invariant to rotations of frame, and its validity is limited to weak distortions a rotationally invariant expression has been given by -Grinstein and Pelcovits (1981).---------------------------------------------------------... 

We first consider the moderate anchoring strengths. The average distortions in the nematic director field are roughly given by V 1 /, where stands... 

Distortions and Defects Around Colloidal Inclusions in Nematic Liquid Crystal... 

From such a study, it appears that the sulfur atom acts as a 71-donor. The spectrum of benzo [b]thiophene in a nematic solvent is consistent with a planar molecule without significant distortion in the geometry of the benzene and thiophene rings.71... 

Electric field effects are more complicated because of conduction. The Carr-Helfrich instability, which occurs in nematics of negative dielectric anisotropy (see 3.10.2), may be expected to take place in this case too, only the bend and twist distortions are now coupled. Moreover, the fluid motion along z can occur only by the process of permeation ( 4.5.1). 

The stratified structure of a smectic liquid crystal imposes certain restrictions on the types of deformation that can take place in it. A compression of the layers requires considerable energy - very much more than for a curvature elastic distortion in a nematic - and therefore only those deformations are easily possible that tend to preserve the interlayer spacing. Consider the smectic A structure in which each layer is, in effect, a two-dimensional fluid with the director n normal to its surface. Assuming the layers to be incompressible, the integral... 

Chen, F. L., and Jamieson, A. M., Molecular weight-dependent behavior of the twist distortion in a nematic monodomain containing a main-chain hquid crystal polymer. Macromolecules, 27, 1943-1948 (1994). 

The first interpretation of this newly identified short-time breakup of thin nematic films was put forward by van Effenterre et al. [73] and is rooted in basic thermodjmamics, i.e., in the biphase coexistence of the two (meta)stable phases taking part in the discontinuous transition. However, as pointed out in detail by Ziherl and Zumer [74], the explanation misses two critical points First, that the isotropic-nematic coexistence can only exist in the temperature range where both phases are thermodynamically metastable or stable, and second, that the nematic director is only distorted in thick enough films (i.e., above the critical thickness dc = [Ai — A2I). Therefore, the dewetting process in thin LC films is still an open question for both theoretical and experimental studies. 

This chapter is concerned with experimental measurements of flexo-electric coefficients. After a brief introduction to flexoelectricity in nematic liquid crystaJs, some applications exploiting the flexoelectric effect and the influence of this effect on electrohydrodynamic instabilities are pointed out. Flexoelectricity axises in samples with a splay-bend distortion in the director field and as such its measurement is not as direct as for dielectric constants. The theoretical background needed to analyse electro-optic experiments and extract the flexocoefficients is outlined in Section 2.2. Various experimental techniques that have been developed are described in Section 2.3. These involve cells in which the alignment of the nematic director is homeotropic, or planar or hybrid. In the first case, the interdigitated electrode technique is particularly noteworthy, as it has been used to establish several features of flexoelectricity (1) the effect can arise purely from the quadrupolar nature of the medium, and (2) the dipolar contribution relaxes at a relatively low frequency. 

Fig. 2.2. Schematic diagram of a hybrid-aligned nematic cell. The field-free director (shown by the continuous curved line) has a splay-bend curvature distortion in the xz plane. A DC field applied along the y axis rotates the polarization and the director (shown by the curved dashed line) acquires a 4>(z) profile. (Reproduced from Dozov et al. with the permission of EDP Sciences, http //publications.edpsciences.org.)...

Besides the elastic and the electric torques the so-called flexoelectric (or flexo) torques on the director play an important role as well. Their effect on pattern-forming instabilities in nematics is the main issue of this chapter. Flexotorques originate from the fact that typically (in some loose analogy to piezoelectricity) any director distortion is accompanied by an electric flexopolarization Pa (characterized by the two ffexocoefScients ei, 63). From a microscopic point of view, finite ei and 03 naturally arise when the nematic molecules have a permanent dipole moment. But also for molecules with a quadrupolar moment, finite ei and 63 are possible (see also Chapter 1 in this book ). Flexopolarization has to be incorporated into the free energy P n) for finite E. It is not surprising that this leads to quantitative modifications of phenomena, which exist also for ci = 63 = 0. Though, for example, the Freedericksz threshold field Ep is not modified, the presence of flexoelectricity leads to considerable modifications of the Freedericksz distorted state for E > Ep- ... 

More frequently, instead of the equilibrium pattern sketched so far, one observes electroconvection (EC) patterns in nematics, which present dissipative structmes characterized by director distortions, space charges and material flow. A necessary requirement for their existence is the presence of charge carriers in the nematic. In a distorted nematic, where n is neither parallel nor perpendicular to E, the generation of a non-zero space charge, pei, by charge separation is then inevitable. The resulting Coulomb force in the flow equations (generalized Navier-Stokes equations) drives a... 

The theoretical analysis of ffexodomains in planar nematics, which is briefly reviewed in the following, exploits the balance of torques by minimizing the total free energy E n). If not otherwise stated, an assumption of strong anchoring of the director n at the confining plates is used i.e. the director at the boundaries remains parallel to no x, irrespective of director distortions in the bulk of the nematic layer. 

G. Durand, Flexoelectricity in nematics Continuous distortions and instabilities, Mol. Cryst. Liq. Cryst. 113, 237-246, (1984). 

For nematic liquid crystals, the synunetry is reduced and we need additional variables. The nematic is degenerate in the sense that all equilibrium orientations of the director are equivalent. According to the Goldstone theorem the parameter of degeneracy is also a hydrodynamic variable for a long distance process 0 and the relaxation time should diverge, x—>oo. In nematics, this parameter is the director n(r), the orientational part of the order parameter tensor. For a finite distortion of the director over a large distance (L—>oo), the distortion wavevector 0 and the... 

As has been shown, the splay and bend distortions of a nematic create electric polarization. There is also a converse effect the external electric field causes a distortion due to the flexoelectric mechanism. For example, if the banana-shape molecules with transverse dipoles are placed in the electric field, the dipoles are partially aligned along the field and their banana shape induces some bend. This effect takes place even in nematics with zero dielectric anisotropy. 

From Eq. (13.29) is seen that, at (p = 0, there is no electric torque exerted on the director. Thus, there should be a threshold for the distortion as in the case of the Frederiks transition in nematics. We can find the threshold field Ec, considering a small distortion (p 0. The equation... 

For instance, for T(p = 10 dyn,ii = 2 x 10 " cm,Fs = 300 statC/cm (1 mC/m ), the threshold field is 0.1 statV/cm, i.e. 3 kV/m. Due to a high value of the Frederiks type distortion in SmC can be observed at extremely low voltage across the cell Uc = dE 30 mV for 10 pm thick cell). However, independently of the field magnitude, after switching the field off, the distortion relaxes to the initial uniform structure, cp(x) = 0. The relaxation time of the distorted structure is owed to pure elastic, nematic-like torque and for small distortion only fundamental Fourier harmonic is important. 

Here, the first term describes the nematic-like elastic energy in raie crmstant approximation (K in 9). This term allows a discussion of distortions below the AF-F threshold (a kind of the Frederiks transition as in nematics in a sample of a finite size). In fact, the most important specific properties of the antiferroelectric are taken into account by the interaction potential W between molecules in neighbour layers the second term in the equation corresponds to interaction of only the nearest layers (/) and (/ + 1). Let count layers from the top of our sketch (a) then for odd layers i, i + 2, etc. the director azimuth is 0, and for even layers / + 1, / + 3, etc. the director azimuth is n. The third term describes interactimi of the external field with the layer polarization Pq of the layer / as in the case of ferroelectric cells. Although for substances with high Pq the dielectric anisotropy can be neglected, the quadratic-in-field effects are implicitly accounted for by the highest order terms proportiOTial to P. ... 

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