Generalized van der Waals theory of the isotropic–nematic phase transition

Abstract

We extend the generalized van der Waals (GVDW) theory to fluids of anisotropic molecules interacting through angle-dependent potentials which can be broken up into strong short-ranged repulsions and weak long-ranged attractions. This approach provides a description of the gas–liquid phase transition in which corresponding states behavior breaks down with increasing anisotropy of the molecular (hard rod) shape. Allowing for the orientational order characterizing nematic phases of liquid crystals, we propose a GVDW form for the Helmholtz free energy A as a functional of the orientational distribution f (Ω). The molecular shape repulsions 𝓋rep are treated within the approximation of scaled particle theory. The long-range attractions 𝓋attr enter directly through a mean field average which explicitly excludes the relative positions denied to a pair of molecules because of their anisotropic cores. For the special case in which 𝓋rep→𝓋hardsphere and 𝓋attr → 𝓋dispersion ? −Ciso/r6ij - Can cos2ϑij/r6ij (here ϑij is the angle between the long axes of molecules i and j and rij is the distance between their centers), minimization of A[f (Ω)] leads to an equation for f (Ω) which is identical in form to that of the Maier–Saupe (MS) theory of isotropic–nematic phase transitions. Instead of a phenomenological parameter specifying the strength of the mean field potential, we obtain an explicit expression involving Can, the density ρ, temperature T, and molecular size σ. We show further that for nonspherical molecular shapes the effective one-body potential still assumes (to a very good approximation) the familiar form Aoρ+A2ρη P2 (cosϑ), where Ao and A2 are independent of T and ρ, and η is the usual order parameter. However, as a result of the angle dependence in the core-excluded volume which defines the effective attraction as a mean field average over 𝓋attr, the dominant contribution to A2 is determined almost exclusively by the −Ciso/r6ij term rather than by the much smaller anisotropic term −Can cos2ϑij/r6ij. Similarly, the free energy A[f (Ω)] includes a coupling between contributions from the hard rod (length L and diameter σ, say) repulsions and from the dispersional attractions. This result (with no adjustable parameters) is argued to comprise a consistent molecular theory of the Can, Ciso, ρ, T, and shape (L,σ) dependence of the orientational phase transition and order in simple liquid crystals.