Stability of the order-order critical points of Heisenberg and nematic model fluids

Recently it was found that Heisenberg and nematic model fluids exhibit, for a range of (negative) values of the ratio $R$ of anisotropic to isotropic interaction strengths, a new type of critical point corresponding to the terminus of a first-order phase transition between two orientationally ordered liquids with different densities. In this paper we present a more systematic and detailed study of these order-order critical points (OOCPs). We start by deriving the equations for the OOCPs and solve them numerically, within a mean-field (MF) approximation for the free energies of either model. We then investigate local stability by expanding the free energies in powers of the order parameter, about the lines of OOCPs. In addition, we examine the stability of the OOCPs with respect to the ordered-liquid–ordered-solid transition (global stability), by bifurcation analysis. We conclude that the MF OOCPs are locally and globally stable over a range of $R<\mathrm{}0\mathrm{}$ which is much broader in the case of the Heisenberg fluid, where the ordering transition is continuous. Here the line of OOCPs ends at a fourth-order critical point on the Curie line, whereas that of the model nematic ends at a critical end point on the nematic-isotropic coexistence curve. Finally, we discuss the relationship between our approach to the stability of critical points and Landau theory.