A generalisation of the Onsager trial-function approach: describing nematic liquid crystals with an algebraic equation of state

The molecular theory of Onsager [L. Onsager, Ann. N.Y. Acad. Sci. 51, 627 (1949)] for liquid crystals is developed and extended to describe ordering transitions in systems of generic cylindrically symmetrical molecules. A number of new analytical results are discussed for particles characterised by a general form of the excluded-volume interaction. Our description makes use of the Onsager trial function (OTF) to represent the orientational distribution and degree of anisotropy. Algebraic expressions for the thermodynamic properties, which provide a particularly tractable description of the isotropic-nematic equilibria, are also presented. The degree of orientational order can be represented by a simple cubic equation in the molecular parameters (molecular diameter and aspect ratio) and thermodynamic variables (temperature and number density). Onsager's theory was originally developed at the level of the second virial coefficient; here the Parsons-Lee decoupling approximation is used to describe the higher-body contributions in a straightforward manner. The adequacy of treating the scaled Onsager (Parsons-Lee) free-energy functional within the OTF formalism to describe anisotropic states is illustrated by examining systems of hard spherocylinders. An excellent representation of the equation of state of the isotropic and nematic phases and the ordering transition is demonstrated for molecules of moderate aspect ratio (L/D = 5). Algebraic equations of state of the type developed here are suitable for practical engineering applications involving anisotropic fluids particularly in the case of multicomponent systems; our general analytical results for the averages of orientational functions will turn out to be useful in the development of a description of molecules with more realistic attractive and Maier-Saupe interactions.