Elasticity of smectics

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]... 

Stenull O, Lubensky TC. 2005. Phase transitions and soft elasticity of smectic elastomers. Phys Rev Lett 94 018304. 

The theory of the elasticity of smectic liquid crystals has its own features. Deformations related to a change in the spacing between the layers are common to all smectic phases. The deformations is, in general, not related to a change in director orientation, and here an additional modulus of elasticity B occurs. 

F.M. Leslie, Elasticity of smectic C liquid crystals, in Contemporary Research in the Mechanics and Mathematics of Materials, R.C. Batra and M.F. Beatty (Eds.), 226-235, CIMNE, Barcelona, 1996. 

R. Ribotta, Experimental Study of the Elasticity of Smectic Liquid Crystals Undulation Instability of Layers, in Molecular Fluids, R. Balian and G. Weill (Eds.), 353-371, Gordon and Breach, London, 1976. 

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]. 

Leadbetter AJ, Norris EK (1979) Molec Phys 38 669. There are different contributions which give rise to a broadening a of the molecular centre of mass distribution function f(z). The most important are the long-wave layer displacement thermal fluctuations and the individual motions of molecules having a random diffusive nature. The layer displacement amplitude depends on the magnitude of the elastic constants of smectics ... 

When there are no defects in the sample, A 0 and B is then the viscosity of the smectic in orientation b. Both A and B were found to increase as the number of defects in the sample increases. A is an elastic yield stress of a defect-containing smectic. The magnitude of A can can be estimated from the density of smectic defects as... 

The time-dependent rheological properties of disordered smectics are also peculiar. Figure 10-37 shows apparent G and G" data (labeled MD ) measured for 8CB after a quench from the isotropic state. Quenched samples contain a very high density of smectic defects, which produce elastic-like behavior in G at low frequencies of oscillation. The apparent moduli G and G" in Fig. 10-37 are not true linear viscoelastic moduli, since they were... 

The elasticity of multilamellar vesicles can be discussed in reference to that of emulsion droplets. The crystalline lamellar phase constituting the vesicles is characterized by two elastic moduli, one accounting for the compression of the smectic layers, B, and the second for the bending of the layers, K [80]. The combination has the dimension of a surface tension and plays the role of an effective surface tension when the lamellae undergo small deformations [80]. This result is valid for multilamellar vesicles of arbitrary shapes [81, 82]. Like for emulsion droplets, the quantity a/S is the energy scale that determines the cost of small deformations. 

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... 

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... 

An extension of rubber elasticity (i.e. of the description of large, static and incompressible deformations) to nematic elastomers has been given in a large number of papers [52, 61-66]. Abrupt transitions between different orientations of the director under external mechanical stress have been predicted in a model without spatial nonuniformities in the strain field [52,63]. The effect of electric fields on rubber elasticity of nematics has been incorporated [65]. Finally the approach of rubber elasticity was also applied recently to smectic A [67] and to smectic C [68] elastomers. Comparisons with experiments on smectic elastomers do not appear to exist at this time. Recently a rather detailed review of the model of an-... 

Structural forces due to long-range positional order are quite easily observed in the smectic A liquid crystals. SFA measurements have been performed on lamellar lyotropic smectics [42,43] and in thermotropic smectics [44-46]. These works extend to a nanometer scale the early studies on elasticity, viscoelastic response and layers instability of smectic A, observed in macroscopic wedge-shaped piezoelectric cells [47,48]. 

Because the layers of LCs transmit elastic energy, they can be used to do mechanical work, such as to assemble nanomaterials and induce topological defects in LCs. The feasibility of this scheme has been illustrated in the lab scale. For instance, linear and hexagonal arrays of nanoparticles were elastically trapped at various sites of smectic defects. Single wall carbon nanotubes could be organized into 2D parallel sheets between smectic layers due to specific interactions between n and n interactions between the hexagonal rings of the carbon nanotubes and the aromatic moieties of LC molecules [62, 67, 71]. 

SmA is a one-dimensional lamellar crystal with the interlayer distance almost rigidly fixed. In order to discuss elasticity we need an additional variable that would describe the lamellar structure. Consider a small distortion of smectic layers [17]. In Fig. 8.23 dash and solid lines indicate undisturbed and distorted layers, respectively. Short rods perpendicular to the lowest solid line indicate local directors, which are always perpendicular to the layers. Now, we introduce a layer displacement along the z-axis, u = Uz.ln fact, it is a scalar field y, z), depending generally on aU the three co-ordinates. Its derivatives describe two types of elasticity ... 

In Section 5.7.2 we discussed a general problem of stability of one, two- and three-dimensional phases. Here, we shall analyze stability of the smectic A liquid crystal, which is three-dimensional structure with one-dimensional periodicity. The question of stability is tightly related to the elastic properties of the smectic A phase. Consider a stack of smectic layers (each of thickness Z) with their normal along the z-direction. The size of the sample along z is L, along x and y it is L, the volume is V = Lj L. Fluctuations of layer displacement u(r) = u(z, r i) along z and in bofli directions perpendicular to z can be expanded in the Fourier series with wavevec-tors q and q (normal modes) ... 

The elastic theory of SmC has beep considered by the Orsay groupand by Rapini. Both approaches use the Oseen description of smectic A, neglecting all changes in internal parameters such as density, interlayer distance, and tilt angle. The Orsay group used the Lagrai an description for the elastic strains with a vector IKF) describing the local rotation of the director. In order to be consistent with the elastic theory for NLC s, we shall use the Eulerian description developed by Rapini based on the director. 

In the smectic A phase the director is always perpendicular to the plane of the smectic layers. Thus, only the splay distortion leaves the interlayer distance unchanged [7], and only the elastic modulus K i is finite while K22 and Kzz diverge when approaching the smectic A phase from the nematic phase. On the other hand, the compressibility of the layered structure and the corresponding elastic modulus B is taken into account when discussing the elastic properties of smectic phases. The free energy density for the smectic A phase, subjected to the action of an external electric field, is... 

Stannarius R, Kohler R, Dietrich U, Losche M, Tolksdorf C, Zentel R (2002) Structure and elastic properties of smectic liquid crystalline elastomer films. Phys Rev E 65(4) 11. doi 04170710.1103/PhysRevE.65.041707... 

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