Xavier Lamy

Some properties of the nematic radial hedgehog in the Landau–de Gennes theory

Abstract We consider, in the Landau–de Gennes theoretical framework of a Q -tensor description of nematic liquid crystals, a radial hedgehog defect with strong anchoring conditions in a ball B ⊂ R 3 . We show that the scalar order parameter is monotonic, and we prove uniqueness of the minimizing hedgehog below the spinodal temperature T ∗ .

Bifurcation Analysis in a Frustrated Nematic Cell

Using Landau-de Gennes theory to describe nematic order, we study a frustrated cell consisting of nematic liquid crystal confined between two parallel plates. We prove the uniqueness of equilibrium states for a small cell width. Letting the cell width grow, we study the behavior of this unique solution. Restricting ourselves to a certain interval of temperature, we prove that this solution becomes unstable at a critical value of the cell width. Moreover, we show that this loss of stability comes with the appearance of two new solutions: there is a symmetric pitchfork bifurcation. This picture agrees with numerical simulations performed by Palffy-Muhoray, Gartland and Kelly, and also by Bisi, Gartland, Rosso, and Virga. Some of the methods that we use in the present paper apply to other situations, and we present the proofs in a general setting. More precisely, the paper contains the proof of a general uniqueness result for a class of perturbed quasilinear elliptic systems, and general considerations about symmetric solutions and their stability, in the spirit of Palais’ Principle of Symmetric Criticality.

Uniaxial symmetry in nematic liquid crystals

Abstract Within the Landau–de Gennes theory of liquid crystals, we study theoretically the equilibrium configurations with uniaxial symmetry. We show that the uniaxial symmetry constraint is very restrictive and can in general not be satisfied, except in very symmetric situations. For one- and two-dimensional configurations, we characterize completely the uniaxial equilibria: they must have constant director. In the three dimensional case we focus on the model problem of a spherical droplet with radial anchoring, and show that any uniaxial equilibrium must be spherically symmetric. It was known before that uniaxiality can sometimes be broken by energy minimizers. Our results shed a new light on this phenomenon: we prove here that in one or two dimensions uniaxial symmetry is always broken, unless the director is constant. Moreover, our results concern all equilibrium configurations, and not merely energy minimizers.

Analytical description of the Saturn-ring defect in nematic colloids.

We derive an analytical formula for the Saturn-ring configuration around a small colloidal particle suspended in nematic liquid crystal. In particular we obtain an explicit expression for the ring radius and its dependence on the anchoring energy. We work within Landau-de Gennes theory: Nematic alignment is described by a tensorial order parameter. For nematic colloids this model had previously been used exclusively to perform numerical computations. Our method demonstrates that the tensorial theory can also be used to obtain analytical results, suggesting a different approach to the understanding of nematic colloidal interactions.

Minimizers of the Landau–de Gennes Energy Around a Spherical Colloid Particle

We consider energy minimizing configurations of a nematic liquid crystal around a spherical colloid particle, in the context of the Landau–de Gennes model. The nematic is assumed to occupy the exterior of a ball Br0, and satisfy homeotropic weak anchoring at the surface of the colloid and approach a uniform uniaxial state as

Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime

{|x|\to\infty}

Vortex structure in p-wave superconductors

|x|→∞. We study the minimizers in two different limiting regimes: for balls which are small