Free energy smectic

As for crystals, tire elasticity of smectic and columnar phases is analysed in tenns of displacements of tire lattice witli respect to the undistorted state, described by tire field u(r). This represents tire distortion of tire layers in a smectic phase and, tluis, u(r) is a one-dimensional vector (conventionally defined along z), whereas tire columnar phase is two dimensional, so tliat u(r) is also. The symmetry of a smectic A phase leads to an elastic free energy density of tire fonn [86]... 

Here Fq is tire free energy of the isotropic phase. As usual, tire z direction is nonnal to tire layers. Thus, two elastic constants, B (compression) and (splay), are necessary to describe tire elasticity of a smectic phase [20,19, 86]. 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

Using this order parameter, the free energy in the nematic phase close to a transition to the smectic phase can be shown to be given by [20, 88, 89, 91]... 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

It is also possible that a membrane might have an even lower symmetry than a chiral smectic-C liquid crystal in particular, it might lose the twofold rotational symmetry. This would occur if the molecular tilt defines one orientation in the membrane plane and the direction of one-dimensional chains defines another orientation. In that case, the free energy would take a form similar to Eq. (5) but with additional elastic constants favoring curvature. The argument for tubule formation presented above would still apply, but it would become more mathematically complex because of the extra elastic constants. As an approximation, we can suppose that there is one principal direction of elastic anisotropy, with some slight perturbations about the ideal twofold symmetry. In that approximation, we can use the results presented above, with 4) representing the orientation of the principal elastic anisotropy. 

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... 

Since P must remain normal to z and n, the polarization vector forms a helix, where P is everywhere normal to the helix axis. While locally a macroscopic dipole is present, globally this polarization averages to zero due to the presence of the SmC helix. Such a structure is sometimes termed a helical antiferroelectric. But, even with a helix of infinite pitch (i.e., no helix), which can happen in the SmC phase, bulk samples of SmC material still are not ferroelectric. A ferroelectric material must possess at least two degenerate states, or orientations of the polarization, which exist in distinct free-energy wells, and which can be interconverted by application of an electric field. In the case of a bulk SmC material with infinite pitch, all orientations of the director on the tilt cone are degenerate. In this case the polarization would simply line up parallel to an applied field oriented along any axis in the smectic layer plane, with no wells or barriers (and no hysteresis) associated with the reorientation of the polarization. While interesting, such behavior is not that of a true ferroelectric. 

The free energy density is written as a sum of the nematic-type contribution (/n), the general smectic contribution (fs), the energy due to the polarization selfinteraction (/,), and the term specific for the bent-core systems (/be) ... 

The free energy density terms introduced so far are all used in the description of the smectic phases made by rod-like molecules, the electrostatic term (6) being characteristic for the ferroelectric liquid crystals made of chiral rod-like molecules. To describe phases made by bent-core molecules one has to add symmetry allowed terms which include the divergence of the polar director (polarization splay) and coupling of the polar director to the nematic director and the smectic layer normal ... 

According to Oseen, the smectic state corresponds to the vanishing of all the moduli except k22 and 33 (our notation). The (free) energy is then minimized when... 

In thermotropic (solvent-free) smectic-A phases, two types of distortion are permitted, namely, splaying of the director (which corresponds to bending of the layers) and layer compression. Note The material itself is assumed to remain incompressible only the layers compress.) For weak distortions, the free energy cost of these is given by (de Gennes and Frost 1993)... 

When the layers of a lamellar block copolymer are distorted, the free energy density is augmented by a distortional term that can, like the smectic-A phase, be described as the sum of layer compression/dilation and layer-bending energies ... 

The form of free energy for smectic liquid crystals is different. If there are no defects in the smectic liquid crystals, the curl of n, V x n, must be zero. Thus, no twist and bend deformations exist in the smectic liquid crystals. In addition, there is an energy penalty associated with the translational deformation. For example, the displacement of smectic layer u will cause an additional term of elastic energy... 

For k<1/V2 a normal phase is found, but for k> H2 a dislocation-stabilized Abrikosov phase becomes possible. Thus, the smectic A free energy density... 

The smectic A free energy reduces to the Landau-Ginsburg superconductor free energy provided that = 0 for splay and kt = kb for twist and bend. The full extent of the relationships between liquid crystals and superconductors in the de Gennes analogy are summarized in Table 1 [23],... 

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 ... 

A more complete description of smectic A needs to take into account the compressibility of the layers, though, of course, the elastic constant for compression may be expected to be quite large. The basic ideas of this model were put forward by de Gennes. > We consider an idealized structure which has negligible positional correlation within each smectic layer and which is optically uniaxial and non-ferroelectric. For small displacements u of the layers normal to their planes, the free energy density in the presence of a magnetic field along z, the layer normal, takes the form... 

To discuss the critical behaviour of the twist and bend elastic constants in the nematic phase, we observe that the Frank free energy expression should include the contribution due to smectic short-range order ... 

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