Minimal coupling model of the biaxial nematic phase

A minimal coupling model exhibiting isotropic, uniaxial, and biaxial nematic phases is analyzed in detail and its relation to existing models known in the literature is clarified. Its intrinsic symmetry properties are exploited to restrict the relevant ranges of coupling constants. Further on, properties of the model are thoroughly investigated by means of bifurcation theory as proposed by Kayser and Raveché [Phys. Rev. A 17, 2067 (1978)] and Mulder [Phys. Rev. A 39, 360 (1989)]. As a first step toward this goal, the bifurcation theory is applied to a general formulation of density functional theory in terms of direct correlation functions. On a general formal level, the theory is then analyzed to show that the bifurcation points from the reference, high-symmetry equilibrium phase to a low-symmetry structure depend only on the properties of the one-particle distribution function and the direct pair correlation function of the reference phase. The character of the bifurcation (whether spinodal, critical, tricritical, isolated Landau point, etc.) depends, in addition, on a few higher-order direct correlation functions. Explicit analytical results are derived for the case when only the leading L=2 terms of the potential (mean-field analysis) or of the direct pair correlation function expansion in the symmetry-adapted basis are retained. Formulas are compared with the numerical calculations for the mean-field, momentum L=2 potential model, in which case they are exact. In particular, bifurcations from the isotropic and uniaxial nematic to the biaxial nematic phases are discussed. The possibility of the recently reported nematic uniaxial-nematic biaxial tricritical point [A. M. Sonnet, E. G. Virga, and G. E. Durand, Phys. Rev. E 67, 061701 (2003)] is analyzed as well.