*Phase and Bifurcation Analysis of a Mean-Field Model for Biaxial Nematic Liquid Crystals*

**Eugene Gartland**

Kent State University

Department of Mathematical Sciences

`gartland@math.kent.edu`

**Abstract**: In recent years, experimental evidence has been
presented of the existence of stable biaxial-nematic liquid-crystal bulk
phases in newly synthesized thermotropic liquid-crystalline materials. This
has led to a renewed interest in models of biaxiality. We report on the
exploration of one such model, a mean-field model originally due to Straley
[1], which was reconsidered by Sonnet, Virga, and Durand in [2]. The
mean-field free-energy model is based upon a quadrupolar particle-particle
interaction potential that depends on two dimensionless coupling parameters.
The phase and bifurcation behavior, as a function of these two
material-dependent parameters, is quite rich and involves temperature-driven
isotropic-nematic-biaxial sequences as well as direct isotropic-biaxial
transitions, tricritical points, secondary bifurcations, and multiple symmetry
breaking. The complete exploration of this model necessitated the development
of new criteria to characterize the metastability of phases of such a model in
regions of the parameter space in which the free be positive definite [3] and
enabled the derivation of the universal mean-field phase diagram for all
quadrupolar interactions between nematogenic biaxial molecules [4]. We will
summarize our mathematical and numerical results.

In collaboration with Fulvio Bisi, Department of Mathematics, University of Pavia, Italy, fulvio.bisi@unipv.it and Epifanio G. Virga, Department of Mathematics, University of Pavia, Pavia, Italy, eg.virga@unipv.it

1. J.P. Straley, Ordered phases of a liquid of biaxial particles, Phys. Rev. A, 10 (1974) 1881-7.

2. A.M. Sonnet, E.G. Virga, & G.E. Durand, Dielectric shape dispersion and biaxial transitions in nematic liquid crystals, Phys. Rev. E, 67 (2003) 061701-1-7.

3. E.C. Gartland, Jr. & E.G. Virga, Minimum principle for indefinite mean-field free energies, Arch. Rational Mech. Anal. 196 (2010), 143-189. 4. F. Bisi, E.G. Virga, E.C. Gartland, Jr., G. DeMatteis, A.M. Sonnet, G.E. Durand, Universal mean-field phase diagram for biaxial nematics obtained from a minimax principle, Phys. Rev. E, 73 (2006) 051709-1-9.