Apala Majumdar

Landau-de Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond

We study global minimizers of a continuum Landau–De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W1,2, to a global minimizer predicted by the Oseen–Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen–Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau–De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau–De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.

Nematic liquid crystals : from Maier-Saupe to a continuum theory

We define a continuum energy functional that effectively interpolates between the mean-field Maier-Saupe energy and the continuum Landau-de Gennes energy functional and can describe both spatially homogeneous and inhomogeneous systems. In the mean-field approach the main macroscopic variable, the Q-tensor order parameter, is defined in terms of the second moment of a probability distribution function. This definition imposes certain constraints on the eigenvalues of the Q-tensor order parameter, which may be interpreted as physical constraints. We define a thermotropic bulk potential which blows up whenever the eigenvalues of the Q-tensor order parameter approach physically unrealistic values. As a consequence, the minimizers of this continuum energy functional have physically realistic order parameters in all temperature regimes. We study the asymptotics of this bulk potential and show that this model also predicts a first-order nematic-isotropic phase transition, whilst respecting the physical constraints. In contrast, in the Landau-de Gennes framework the Q-tensor order parameter is often defined independently of the probability distribution function, and the theory makes physically unrealistic predictions about the equilibrium order parameters in the low-temperature regime.

Equilibrium order parameters of nematic 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

Symmetry of Uniaxial Global Landau--de Gennes Minimizers in the Theory of Nematic Liquid Crystals

\Qvec

The radial-hedgehog solution in Landau–de Gennes' theory for nematic liquid crystals

-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

Order reconstruction patterns in nematic liquid crystal wells

\Qvec

Topology and bistability in liquid crystal devices.

-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

Morphological instability of a non-equilibrium ice–colloid interface

\Qvec

Elastic energy of liquid crystals in convex polyhedra

-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.

Uniaxial versus biaxial character of nematic equilibria in three dimensions

We extend the recent radial symmetry results by Pisante [J. Funct. Anal., 260 (2011), pp. 892--905] and Millot and Pisante [J. Eur. Math. Soc.