We develop an equation of state for describing the isotropic- and nematic fluid-phase behavior of a prototypical model for (thermotropic) liquid crystal (LC) molecules of prolate type. The molecular model assumes a LC molecule as a chain of equally sized tangentially-bonded spheres, the interaction of which is described by a Lennard-Jones pair potential. To model the typical structure of real LC molecules, one part of the chain is arranged in a rigid linear conformation (referred to as ‘rod’), while the remaining part is freely jointed (referred to as ‘coil’). By changing the ratio ‘rod’ to ‘coil’, the flexibility of a molecule can be varied.
The equation of state is developed by applying the perturbation theory of Barker and Henderson to a reference fluid of hard chain molecules. The reference fluid's Helmholtz energy is described by a rescaled Onsager theory. The perturbational Helmholtz energy contribution of an orientationally ordered fluid is in our approach calculated as that of a (hypothetical) randomly oriented fluid of same density, plus an anisotropic part, namely the perturbational Helmholtz energy contribution for the transition from the randomly oriented fluid to the ordered fluid at constant density.
Comparison of calculated phase equilibria to results obtained from Monte Carlo simulations indicates the anisotropic part of the perturbation contribution is small, and can be neglected. We consider this as the main insight of this work. Specifically, this result suggests that a reliable description of attractive dispersion interactions of real nematic LCs can be obtained from a theoretical approach designed for isotropic fluids. Given the relative maturity of such isotropic-fluid theories (e.g. SAFT), this could lead to a considerable simplification of the description of nematic fluids.
Using the equation of state, we systematically investigate the influence of molecular shape (chain length), intra-molecular flexibility, and intermolecular attractions on the phase behavior. A key result of this analysis is that incorporation of intra-molecular flexibility drives calculated phase equilibrium properties (e.g. density at phase transition, density difference of phase change) towards values of real (thermotropic) liquid crystals. Compared to the ubiquitously used rigid molecular models for LCs, this is considered a significant step forward. Detailed comparison to experimental results reveals the molecular model is still too simplistic, however.
