The isotropic-nematic phase transition: the Onsager theory revisited

Abstract

A simple Monte Carlo annealing technique is proposed for the minimisation of the Onsager-Helmholtz free energy functional at the level of the second virial coefficient. Random changes are made to vary a discrete representation of the single particle distribution function f( θ) in order to determine the minimum of the free energy surface. The annealing technique gives results of comparable accuracy to other commonly used minimisation methods. The main advantage of the annealing technique is that it is not necessary assume a particular functional form for f( θ). It is also easy to extend the method to include higher virial coefficients and to mixtures. We anticipate that this type of technique could be of quite general use for a number of problems involving the minimisation of functionals. The Onsager theory is extended to include higher virial coefficients in the expansion of the Helmholtz free energy by introducing anisotropy into an accurate equation of state for isotropic phase of the hard rod system using a Parsons-type scaling. A brief summary of other attempts to improve the accuracy of the Onsager theory in this manner is given and the results of all these variations are compared with recent simulation results. The original Parsons scaling is found to be by far the most accurate equation of state for both the isotropic and nematic phases and the results of this extension are studied in some detail. Finally, the Onsager theory is extended to a system of molecules made up of tangentially bonded linear hard sphere chains (RHSC). The Parsons-type scaling is used in conjuction with an accurate equation of state proposed by Jackson et al. The results of this theory are compared with simulation results for semi-flexible sphere chains. The agreement is poor, although preliminary simulation results for rigid linear chains suggest that the theory works reasonably well.