Chirality elasticity theory

In smectic C materials, the relative twist of the planes is uncoupled to the layer thickness, giving rise to strong, nematic-like scattering [106]. There are relatively few light scattering studies of either achiral or chiral SmC phases, despite the technological importance of the ferroelectric SmC phase. This is in part due to the few discussions of the elastic theories of these phases, and in particular descriptions that include dynamic behaviour. Indeed it is only relatively recently that Leslie et al. [107] de-... 

Continuum shell models used to study the CNT properties and showed similarities between MD simulations of macroscopic shell model. Because of the neglecting the discrete nature of the CNT geometry in this method, it has shown that mechanical properties of CNTs were strongly dependent on atomic structure of the tubes and like the curvature and chirality effects, the mechanical behavior of CNTs cannot be calculated in an isotropic shell model. Different from common shell model, which is constmcted as an isotropic continuum shell with constant elastic properties for SWCNTs, the MBASM model can predict the chirality induced anisotropic effects on some mechanical behaviors of CNTs by incorporating molecular and continuum mechanics solutions. One of the other theory is shallow shell theories, this theory are not accurate for CNT analysis because of CNT is a... 

Twenty-four arm-Anisotropic elastic shell chair, zigzag and using Flugge shell theory chiral SWCNTs have... 

One optical feature of helicoidal structures is the ability to rotate the plane of incident polarized light. Since most of the characteristic optical properties of chiral liquid crystals result from the helicoidal structure, it is necessary to understand the origin of the chiral interactions responsible for the twisted structures. The continuum theory of liquid crystals is based on the Frank-Oseen approach to curvature elasticity in anisotropic fluids. It is assumed that the free energy is a quadratic function of curvature elastic strain, and for positive elastic constants the equilibrium state in the absence of surface or external forces is one of zero deformation with a uniform, parallel director. If a term linear in the twist strain is permitted, then spontaneously twisted structures can result, characterized by a pitch p, or wave-vector q=27tp i, where i is the axis of the helicoidal structure. For the simplest case of a nematic, the twist elastic free energy density can be written as ... 

The foundations of continuum theory were first established by Oseen [61] and Zocher [107] and significantly developed by Frank [65], who introduced the concept of curvature elasticity. Erickson [17, 18] and Leslie [15, 16] then formulated the general laws and constitutive equations describing the mechanical behavior of the nematic and chiral nematic phases. 

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... 

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