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The phase behaviour of shape-changing spheroids

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The Mathematics of Liquid Crystals

Low-molecular-weight liquid crystals are typically modelled as collections of either hard rods or hard discs. However, small,flexible molecules known as tetrapodes also exhibit liquid crystalline phases, including the elusive biaxial nematic phase [1,2]. This is a consequence of the interplay between conformational and packing entropies: the molecules are able to adopt an anisometric stable conformation that allows then to pack more efficiently into orientationally ordered mesophases. Previous theoretical studies of such systems have been presented [3], but in order to capture the essential physics of the process, we introduce a minimal model which permits a clear detailed analysis. In our model a particle can exist in one of two states, corresponding to a prolate and an oblate spheroid. The energies of these two states differ by a prescriamount ε, and the two conformers are in chemical equilibrium. The interactions between the particles are described by the Gaussian Overlap Model [4] and we investigate the phase behaviour using a second-virial (Onsager) approach, which has been successfully applied to binary mixtures of plate-like and rod-like particles [5]. Depending on conditions these mixtures may exhibit biaxial nematic phases and N+—N co-existence. We use both bifurcation analysis and a numerical minimisation of the free energy to show that, in the L2 approximation: (u) there is no stable biaxial phase even for ε=0 (although there is a metastable biaxial phase in the same density range as the stable uniaxial phase); (ii) the isotropic-to-nematic transition is into either one of two degenerate uniaxial phases, rod-rich or disc-rich.

References: [1] K. Merkel et al., Phys. Rev. Lett. 93, 237801 (2004). [2] J. L. Figueirinhas et al., Phys. Rev. Lett. 94, 107802 (2005). [3] A. G. Vanakaras et al., Mol. Cryst. Liq. Cryst. 362, 67 (2001). [4] B. J. Berne and P. Pechukas, J. Chem. Phys. 56, 4213 (1972). [5] P. J. Camp et al., J.

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