Free energy cholesteric

The Helfrich-Prost model was extended in a pair of papers by Ou-Yang and Liu.181182 These authors draw an explicit analogy between tilted chiral lipid bilayers and cholesteric liquid crystals. The main significance of this analogy is that the two-dimensional membrane elastic constants of Eq. (5) can be interpreted in terms of the three-dimensional Frank constants of a liquid crystal. In particular, the kHp term that favors membrane twist in Eq. (5) corresponds to the term in the Frank free energy that favors a helical pitch in a cholesteric liquid crystal. Consistent with this analogy, the authors point out that the typical radius of lipid tubules and helical ribbons is similar to the typical pitch of cholesteric liquid crystals. In addition, they use the three-dimensional liquid crystal approach to derive the structure of helical ribbons in mathematical detail. Their results are consistent with the three conclusions from the Helfrich-Prost model outlined above. 

A very different model of tubules with tilt variations was developed by Selinger et al.132,186 Instead of thermal fluctuations, these authors consider the possibility of systematic modulations in the molecular tilt direction. The concept of systematic modulations in tubules is motivated by modulated structures in chiral liquid crystals. Bulk chiral liquid crystals form cholesteric phases, with a helical twist in the molecular director, and thin films of chiral smectic-C liquid crystals form striped phases, with periodic arrays of defect lines.176 To determine whether tubules can form analogous structures, these authors generalize the free-energy of Eq. (5) to consider the expression... 

However, further analysis of the behavior of the system in LC cells cast doubt on this interpretation. First, while intuitively attractive, the idea that relaxation of the polarization by formation of a helielectric structure of the type shown in Figure 8.20 would lower the free energy of the system is not correct. Also, in a thermodynamic helical LC phase the pitch is extremely uniform. The stripes in a cholesteric fingerprint texture are, for example, uniform in spacing, while the stripes in the B2 texture seem quite nonuniform in comparison. Finally, the helical SmAPF hypothesis predicts that the helical stripe texture should have a smaller birefringence than the uniform texture. Examination of the optics of the system show that in fact the stripe texture has the higher birefringence. 

Electric or magnetic fields acting on the anisotropy of the electric or magnetic susceptibility exert torques within a liquid crystal which may compete with the elastic torques determining its internal structure (55). Equations w ich describe the liquid crystalline structure can be derived from molecularly uniaxial liquid crystals on the basis of the curvature-elasticity theory (54). In doing so, tl structure is determined so as to minimize the total free energy of the system, and this method is applied to the cholesteric structure (55, 55). 

Because of the layered structure, defects in the cholesteric can be likened in many respects to those in smectic A. Both of them exhibit focal conic textures and both allow for the existence of screw and edge dislocations. To discuss these similarities we employ a coarse-grained approximation in which the cholesteric distortions are considered to be small and to vary slowly over a pitch. In this approximation the free energy of distortion may be expressed in terms of layer displacement u parallel to the twist axis ... 

The rotation exists because of the correlations <.s Sy which are nonvanishing in the case of a cholesteric. The averages may be evaluated on the basis of de Gennes s model (see 2.5). To allow for the non-centrosymmetric ordering in the cholesteric, we include an additional term of the form s-V x s in the free energy of the isotropic phase... 

Topologically, it turns out that the helical structure of the cholesteric cannot be deformed continuously to produce a cubic lattice without creating defects. Thus BP I and BP II are unique examples in nature of a regular three-dimensional lattice composed of disclination lines. Possible unit cells of such a disclination network, arrived at by minimizing the Oseen-Frank free energy, are shown in fig. 4.8.3. The tubes in the diagram represent disclination lines, whose cores are supposed to consist of isotropic (liquid) material. Precisely which of these configurations represents the true situation is a matter for further study. 

The threshold field can be calculated thermodynamically by comparison of the free energy of the helical and uniform structures in the presence of the field. In our geometry, the free energy density of a cholesteric in a magnetic field is... 

If we intend to calculate precisely the threshold field for the two-dimensional distortion we should write the Frank free energy with the director compcments (12.34) and the field term (Ea/47t)(En) and then make minimization of the free energy with respect to the two variables cp and 9 [18]. For a qualitative estimation of the threshold we prefer to follow the simple arguments by Helfrich [17]. We consider a one-dimensional (in layer plane xy) periodic distortion of a cholesteric... 

For weak anchoring and 0 by analogy with a nematic (see Eq. 10.77), the free energy of the distortion includes the elastic term due to the bend-distortion (we assume K = Kn = K s) and the flexoelectric term with an average coefficient e. The second elastic term is due to the cholesteric helical structure (modulus K22). ... 

A Landau theory for blue phase was proposed by Brazovskii, Dmitriev, Homreich, and Shtrik-man [7-10]. In this theory, the free energy of the blue phase is expressed in terms of a tensor order parameter which is expanded in Fourier components. The free energy is then minimized with respect to the order parameter with the wave vector in various cubic symmetries. In a narrow temperature region below the isotropic transition temperature, the stmctures with certain cubic symmetries have free energy lower than both the isotroic and cholesteric phases. 

Now we consider a uniaxial cholesteric liquid crystal where //to(0) = and d Q. The average free energy density is... 

The formation of this instability can be understood qualitatively from scaling rules. Consider a block copolymer material between two plane parallel electrodes, where the lamellar microstructure is aligned parallel to the electrodes. The electrostatic energy can be reduced by a tilt in the lamellar layers, for example, by a ripple distortion as shown for a cholesteric in Figure 13. If the ripple has amplitude a and transverse wavelength A, the electrostatic free energy is reduced by an amount that scales with the square of the lamellar tilt away from horizontal ... 

Liquid crystalline phases formed by chiral molecules (i.e. molecules differing from their mirror image) show unique macroscopic properties. The best-known example is the cholesteric phase which is termodynamically equivalent to the nematic phase. In the later phase the free-energy of the system corresponds to a uniform director distribution in the whole sample. On the other hand in cholesterics the molecules tend to form a helical structure the helical axis being perpendicular to the director. A similar helical structure develops in the smectic C phase when the molecules are chiral. In this case the helical axis is parallel to the layer normal the tilt angle is constant while the azimuthal angle is rotating in space. The pitch of the helix in these systems is typically in the order of a micron. 

The equations governing the interaction of external fields with LC s can be derived by minimizing the free energy SSFdV = 0. The free energy density F [erg/cm ] of a nematic or cholesteric LC in a light, field is given by... 

The threshold field for the formation of periodic distortion has been calculated [22, 23] based on the expression for the free energy of a cholesteric liquid crystal... 

The process can be described within the framework of the elasticity theory of cholesterics. Let us start with (2.24) and with an initial helical distribution of the director. The free energy of cholesterics in an electric field is... 

Berreman and Heffner [59] considered the cholesteric Grandjean texture with tilted director orientation on the boundaries, Fig. 6.16. In the absence of the tilt, the free energy g is minimum at the following thicknesses d of the Cano wedge ... 

FIGURE 6.16. Bistable switching in long-pitch cholesterics with a tilt of the director o- (a) Tilted states with n/2 turns in zero field, = 55 . (b) Free energies g as functions of thickness to pitch ratio d/Po at zero field, dlPoY = 0.89 is the operating point, (c) g(d/PoY in an electric field versus reduced volteige U/Ufi Uf is the Frederiks transition threshold, Ae > 0. 

When the deviations of layers from the flat geometry are substantial, the deformations are more appropriately characterized by the principal curvatures a = jR and cholesteric layers [10]. The elastic free energy density can be cast in the form... 

The expression (6.3) for Ecn was calculated for infinitely thick films without taking into account the boundary conditions. However, the cholesteric to nematic phase transition was investigated for different thickness [67], [68]. The influence of the surface orientation was taken into account [68] by introducing a surface free energy per unit area F which leads to the following expression for Vcn -... 

The peculiar structure of the TGBa phase can be considered as a compromise between the incompatible properties of the cholesteric phase which appears at higher temperatures and the SmA phase which appears at lower temperatures. The cholesteric phase is characterized by a helical director field and shows no positional order of the molecules. Thus, its director field can be deformed due to surface effects or external fields. The increase of the Gibbs free energy due to the elastic deformations of nematic and cholesteric phases is given by... 

When an electric or magnetic field is applied to a liquid crystal cell, a texture transition occurs to minimize the free energy of the system. These texture changes in cholesteric liquid crystals are physically similar to the Frederiks transition in a nematic liquid crystal and result in a significant change in the optical properties of the layer. Texture transitions have been reviewed previously [8, 9] with allowance made for the sign of the dielectric or diamagnetic anisotropy, the initial texture, and the direction of the applied field. Here, we consider only the instability of the planar cholesteric texture, which has been widely discussed in recent literature. 

The theory for the threshold of the instability in cells with thicknesses considerably exceeding the equilibrium pitch (1>Pq) has been considered by analogy with the case of dielectric instability [121, 266], but with allowance being made for the additional, destabilizing term in the free energy which is caused by the space charge. The frequency dependence of the threshold field for <0 has been shown to be similar to that caleulated for nematics. For a cholesteric liquid crystal with >0 the presence of electrical conductivity is revealed by a lowering of the threshold of the instability at low frequencies. 

The A" and K22, terms, which vanish identically in the apolar nematic and cholesteric phases, are not considered here. As discussed above, the volume integrals of the free energy terms containing the splay-bend elastic constant and the saddle-splay elastic constant K24 can be transformed into integrals over the nematic surface... 

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