The Landau-de Gennes theory of nematic liquid crystals:\ud Uniaxiality versus Biaxiality

We study small energy solutions within the Landau-de Gennes theory for nematic liquid crystals, subject to Dirichlet boundary conditions. We consider two-dimensional and threedimensional domains separately and study the correspondence between Landau-de Gennes theory and Ginzburg-Landau theory for superconductors. We treat uniaxial and biaxial cases separately. In the uniaxial case, topological defects correspond to the zero set and we obtain results for the location and dimensionality of the defect set, the solution profile near and away from the defect set. In the three-dimensional case, we establish the C^1,a-convergence of uniaxial small energy solutions to a limiting harmonic map, away from the defect set, for some 0 < a < 1, in the vanishing core limit. Generalizations for biaxial small energy solutions are also discussed, which include physically relevant estimates for the solution and its scalar order parameters. This work is motivated by the study of defects in liquid crystalline systems and their applications

The radial-hedgehog solution in Landau-de Gennes' theory

We study the radial-hedgehog solution on a unit ball in three dimensions, with homeotropic boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. The radial-hedgehog solution is a candidate for a globally stable configuration in this model framework and is also a prototype configuration for studying isolated point defects in condensed matter physics. We use a combination of Ginzburg-Landau techniques, perturbation methods and stability analyses to study the qualitative properties of the radial-hedgehog solution, the structure of its defect core, its stability and instability with respect to biaxial perturbations. Our results complement previous work in the field, are rigorous in nature, give information about the role of geometry, elastic constants and temperature on the properties of the radial-hedgehog solution and the associated biaxial instabilities

Symmetry of uniaxial global Landau-de Gennes minimizers in the\ud theory of nematic liquid crystals

We extend the recent radial symmetry results by Pisante [23] and Millot & Pisante [19] (who show that all entire solutions of the vector-valued Ginzburg-Landau equations in superconductivity theory, in the three-dimensional space, are comprised of the well-known class of equivariant solutions) to the Landau-de Gennes framework in the theory of nematic liquid crystals. In the low temperature limit, we obtain a characterization of global Landau-de Gennes minimizers, in the restricted class of uniaxial tensors, in terms of the well-known radial-hedgehog solution. We use this characterization to prove that global Landau-de Gennes minimizers cannot be purely uniaxial for sufficiently low temperatures

Equilibrium order parameters of liquid crystals in the Landau-De Gennes theory

We study nematic liquid crystal configurations in confined geometries within the continuum Landau--De Gennes theory. These nematic configurations are mathematically described by symmetric, traceless two-tensor fields, known as \Qvec-tensor order parameter fields. We obtain explicit upper bounds for the order parameters of equilibrium liquid crystal configurations in terms of the temperature, material constants, boundary conditions and the domain geometry. These bounds are compared with the bounds predicted by the statistical mechanics definition of the \Qvec-tensor order parameter. They give quantitative information about the temperature regimes for which the Landau-De Gennes definition and the statistical mechanics definition of the \Qvec-tensor order parameter agree and the temperature regimes for which the two definitions fail to agree. For the temperature regimes where the two definitions do not agree, we discuss possible alternatives.Comment: Submitted to SIAM Journal on Applied Mathematic

Equilibrium order parameters of nematic liquid\ud crystals in the Landau-De Gennes theory

We study equilibrium liquid crystal configurations in three-dimensional domains, within the continuum Landau-De Gennes theory. We obtain explicit bounds for the equilibrium scalar order parameters in terms of the temperature and material-dependent constants. We explicitly quantify the temperature regimes where the Landau-De Gennes predictions match and the temperature regimes where the Landau-De Gennes predictions don’t match the probabilistic second-moment definition of the Q-tensor order parameter. The regime of agreement may be interpreted as the regime of validity of the Landau-De Gennes theory since the Landau-De Gennes theory predicts large values of the equilibrium scalar order parameters - larger than unity, in the low-temperature regime. We discuss a modified Landau-De Gennes energy functional which yields physically realistic values of the equilibrium scalar order parameters in all temperature regimes

Biaxial escape in nematics at low temperature

In the present work, we study minimizers of the Landau-de Gennes free energy in a bounded domain $\Omega\subset \mathbb{R}^3$. We prove that at low temperature minimizers do not vanish, even for topologically non-trivial boundary conditions. This is in contrast with a simplified Ginzburg-Landau model for superconductivity studied by Bethuel, Brezis and H\'elein. Merging this with an observation of Canevari we obtain, as a corollary, the occurence of biaxial escape: the tensorial order parameter must become strongly biaxial at some point in $\Omega$. In particular, while it is known that minimizers cannot be purely uniaxial, we prove the much stronger and physically relevant fact that they lie in a different homotopy class