Abstract
Liquid crystals (LC) are molecular dielectrics encompassing several properties of both liquids and solids; in particular, they are often characterized by an order parameter which can be employed to distinguish among possible LC phases. In the nematic phase, liquid crystals show a significant degree of orientational order, their elongated organic molecules being aligned in a mean direction in space, as described by a vectorial field n called director. Sincemost nematics are derivative of benzene, they feature “cigar-like” molecules; hence, the macroscopic system can be regarded as an optically uniaxial crystalline fluid. The dielectric tensor \( \overleftrightarrow{\varepsilon \left( r \right)} \), describing the optical polarization of the medium, can be expressed as \( \overleftrightarrow{\varepsilon} = \overleftrightarrow{R}^\dag \cdot \overleftrightarrow{\varepsilon} _{{\rm NLC}} \cdot \overleftrightarrow{R} \), with \( \overleftrightarrow{\varepsilon}_{{\rm NLC}} = \left[ \varepsilon_\bot , \varepsilon_\bot , \varepsilon_\| \right] I, I_{ij} = \delta_{ij }\) (\(\delta_{ij }\) is the Kronecker delta) and \( \overleftrightarrow{R}(n) \) a rotation tensor. The steady-state director configuration is obtained as an extremal point of the action integral \( \mathcal{I} = \smallint \mathcal{L} d\rm{x}d\rm{y}d\rm{z} \), whose density \( \mathcal{L} \) defines the energy spent by the molecular system to hold a specific director configuration (Frank freeenergy formulation) [1]. The energy density \( \mathcal{L} \) can be further expanded into elastic \( \mathcal{L}_{\rm{el}} \) and electromagnetic \( \mathcal{L}_{\rm{em}} \) terms: \( \mathcal{L} = \mathcal{L}_{\rm{el}} + \mathcal{L}_{\rm{em}} \). The contribution \( \mathcal{L}_{\rm{el}} \) can be evaluated in the framework of the elastic continuum theory and, in the single constant approximation [2], reads:
with K accounting for elastic deformations ([K] = N). The electromagnetic contribution can be calculated by considering that the electric field induces dipoles on the nematic liquid crystal (NLC) molecules; the latter are then subjected to a torque and change their angular orientation towards a minimum energy configuration (e.g., parallel to the applied field). The contribution describing such reorientation process is [2]:
being \( \Delta \varepsilon = \varepsilon_{\|} - \varepsilon_{\bot} \) the NLC birefringence and \( \left\langle \ldots \right\rangle \) denoting a square time average. The balance between field-induced reorientation and elastic interactions gives rise to the steady state distribution n, found as an extremal of the action integral \( \delta \mathcal{L} = 0 \).
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Assanto, G., Fratalocchi, A. (2010). Nonlinear Optical Waves in Liquid Crystalline Lattices. In: Denz, C., Flach, S., Kivshar, Y. (eds) Nonlinearities in Periodic Structures and Metamaterials. Springer Series in Optical Sciences, vol 150. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02066-7_2
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