Advances in generalised van der Waals approaches for the isotropic–nematic fluid phase equilibria of thermotropic liquid crystals–an algebraic equation of state for attractive anisotropic particles with the Onsager trial function

Abstract

In this contribution we provide a review and reformulation of perturbation theories (generalised van der Waals approaches) for the description of the fluid phase behaviour and orientational ordering transitions of thermotropic nematic liquid crystals. Free-energy functionals of the basic Onsager form are used as the platform for the development of a general formulation that reduces to the specific forms of the various theories that have found common use. A novel closed analytical description of the thermodynamic properties and degree of nematic order is then obtained by employing the Onsager trial function to represent the orientational distribution function in terms of a single parameter. The latter essentially constitutes an algebraic equation of state for the nematic phase appropriate for use in engineering applications. The description of ordering transitions with scaled-Onsager theories and suitable trial functions has already been illustrated by its application to systems of hard spherocylinders (HSCs), indicating that the approach provides an excellent representation of the orientational order of the hard-core (athermal) system [Mol. Phys. 106, 649 (2008)]. Here, the hard-body model is extended to account for attractive interactions (treated at the van der Waals level) of a general isotropic/anisotropic form (e.g., Lennard-Jonesium (LJ), square-well (SW), Maier–Saupe (MS), etc.). The adequacy of our generalised van der Waals–Onsager theory is exemplified by an analysis of the vapour–liquid, liquid–nematic, and vapour–nematic phase equilibria for hard spherocylinders with attractive square-well interactions (HSC-SW). The effect of the potential range and molecular aspect ratio on the vapour–liquid–nematic equilibria and orientational ordering transitions is examined to investigate the van der Waals limit (corresponding states) of the ordering phase behaviour. In the case of systems with an aspect ratio of ∼5 the corresponding-states limit is reached when the range is about 16 times the molecular diameter. For progressively longer molecules with an attractive range that follows their long dimension, the fluid–nematic equilibrium is enhanced to the point that the vapour–liquid boundary becomes metastable relative to fluid–nematic equilibria. In the case of molecules of very large aspect ratio (∼50), an additional region of nematic–nematic coexistence is exhibited by the system.

Keywords:

• perturbation theory
• statistical mechanics
• liquid crystals

Acknowledgements

We are very grateful to Jean-Pierre Hansen for useful discussions and suggestions during the PhD examination of MFM that greatly improved the presentation of the work. MFM would also like to thank the Consejo Nacional de Ciencia y Tecnología (CONACYT) and the Secretaría de Educación Pública (SEP) of Mexico for funding a PhD studentship. We acknowledge further support from the Engineering and Physical Sciences Research Council (EPSRC) of the UK (grant numbers GR/N03358, GR/N20317, GR/N35991, GR/R09497, and EP/E016340/1), the Joint Research Equipment Initiative (JREI) for computer hardware (GR/M94427), and the Royal Society–Wolfson Foundation for the award of a refurbishment grant.