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Carnegie Mellon Center for Nonlinear Analysis
Phase and Bifurcation Analysis of a Mean-Field Model for Biaxial Nematic Liquid Crystals


Eugene Gartland
Kent State University
Department of Mathematical Sciences

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, and Epifanio G. Virga, Department of Mathematics, University of Pavia, Pavia, Italy,

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.