Elastic constants smectic

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 ... 

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. 

For small-molecule thermotropic smectic-A phases, typical values of two elastic constants are K 10 dyn and B 10 dyn/cm (Ostwald and Allain 1985). For lyotropic smectics, such as those made from surfactants in oil or water solvents, the layer compression modulus B can be much lower (see Chapter 12). From B and K, a length scale A. = ( 1 /B) 1 nm is defined it is called the permeation depth and its magnitude... 

The radius a of the onions in the intermediate shear-rate regime of lyotropic smectics depends on shear rate, scaling roughly as a A similar texture size scaling rule is found in nematics (see Section 10.2.7) there it reflects a balance of shear stress r y against Frank elastic stress. In smectics, the two important elastic constants B and Ki have differing... 

F. Jahnig and F. Brochard, Critical Elastic Constants and Viscosities above a Nematic-Smectic A Transition of Second Order, J. Phys. 35 (1974) 301 ... 

An important source of error in these calculations is the neglect of short-range order. In particular, the theory fails for the bend and twist elastic constants when smectic-like short-range order is present in the nematic liquid crystal. Under such circumstances these two constants exhibit a critical divergence as the temperature approaches the smectic-nematic point and the light scattering also shows a marked temperature dependence. The present treatment is then inadequate and more elaborate models have been proposed. The phenomenological theory of this aspect of the problem will be discussed in chapter 5. 

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... 

Fig. 5.3.10. Temperature dependence of the elastic constants of the smectic A phase of diethyl 4,4 -azoxydibenzoate determined by ultrasonic velocity measurements open circles, 2 MHz filled circles, 5 MHz triangles, 12 MHz crosses, 20 MHz. (After Miyano and Ketterson. )...

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 ... 

Here the A terms describe curvature distortions of the smectic planes, the B terms the distortions of the director when the smectic planes are unperturbed, and the C terms the coupling between these two types of distortions. All the coefficients are approximately of the same order of magnitude as the nematic elastic constants. A term of the type B(du/dzY may also be included to allow for the compression of the layers, but we shall neglect it in the present discussion. 

The Gay-Berne potential has successfully been used for many liquid crystal simulations, and (depending on the parameterisation used and the state points studied) can be used to simulate nematic, smectic-A and smectic-B phases. Variants of the GB potential have also been used to study the biaxial nematic phase (biaxial GB potential) [21] and the smectic C phase (GB with quadrupole) [22]. The GB model has been used also to provide predictions for key material properties, such as elastic constants [23] and rotational viscosities [24], which have an important role in determining how a nematic liquid crystal responds in a liquid crystal display (LCD). 

Fig. 19. Elastic constants ku and (i.e. splay and bend) as a function of concentration for a polysiloxane smectic copolymer of the type shown in Fig. 3(b) dissolved in a low molar mass cyanobiphenyl liquid crystal host.

Slant was determined, which turned out to be for OCB near to the actual elastic constants. The temperature dependence of eff followed well the temperature dependence of Kz in particular the same pretransitional increase was found near to the nematic-smectic A transition. The wavelength dependence of the threshold was found to be weak, in agreement with the theory. 

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