The Frank twist elastic constant is calculated through the use of a novel orientational perturbation method. By measurement of a system's energetic response to external twist deformation of its director field, the relationship between the change in Helmholtz free energy and pitch distance is determined. From Frank's equation for elastic free energy density, we are able to estimate the twist elastic constant. We apply this method to a model achiral liquid-crystal fluid
Among the many properties of nematic liquid crystals, the Frank elastic constants [1] are particularly useful in characterising their large-scale behaviour. These constants represent free energy penalties for different modes of elastic deformation of the nematic ordering director. There are three main constants: splay, twist and bend. A fourth constant, named saddle-splay, concerns bidirectional modes of elastic deformation. A complete [2] expression for the total elastic energy free energy of an achiral, non-polar, nematic may be written in terms of these four constants. Of these coefficients, it is the twist elastic constant that is the focus of the current work. This is of interest for a number of practical reasons, not least of which concerns the performance of liquid-crystal display applications. Pixel response times in twisted-nematic displays are directly linked to the value of the twist elastic constant. As such, knowledge of this constant is important for design and selection of molecules for use in these displays.
Previous simulation studies to calculate the twist elastic constant have featured a variety of approaches. Allen and Frenkel [3] used fluctuation expressions for the Frank elastic constants, in the thermodynamic limit. By expressing these constants as components of a wave-vector-dependent order matrix, the elastic constants for hard prolate ellipsoids and hard spherocylinders from molecular dynamics were estimated using an orientational fluctuation method. The same method has since been applied to fluids of Gay-Berne molecules [4], as well as thin [5] and thick [6] hard-platelets. The molecular dynamics study of Allen and Masters [7] simulated achiral liquid crystals, with a director field distortion created by twisted periodic boundaries. Measurement of the torque exerted between all molecule pairs allowed evaluation of the twist elastic constant. A more recent development by Joshi et al. [8] is a first principles method capable of calculating the saddle-splay constant, in addition to bend, twist and splay constants.
In the present work, we utilise a perturbative approach, based on a thermodynamic expression. In its application to a model liquid crystal fluid, we will also demonstrate how to overcome issues related to system size, and the use of periodic boundaries.
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[3] Allen and Frenkel, Phys. Rev. A 37, 1813 (1988).
[4] Allen, et al., J. Chem. Phys. 105, 2850 (1996).
[5] O’Brien, et al., Phys. Rev. E 78, 051705 (2008).
[6] O’Brien, et al., Soft Matter 7, 153 (2011).
[7] Allen and Masters, Mol. Phys. 79, 277 (1993).
[8] Joshi, et al., Soft Matter 10, 882 (2014).
