Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEquilibrium configurations of nematic liquid crystals on a torus
131
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our aim is to show how a different modeling of the effect of extrinsic curvature acts as a selection principle among equilibria of the classical energy, and how new configurations emerge. In particular, our analysis predicts the existence of new stable equilibria with complex windings.
rate research
Read More
We analyze an elastic surface energy which was recently introduced by G. Napoli and L.Vergori to model thin films of nematic liquid crystals. We show how a novel approach that takes into account also the extrinsic properties of the surfaces coated by the liquid crystal leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; ii) we prove, by a Ginzburg-Landau approximation, the well-posedness of the gradient flow of the energy; iii) in the case of a parametrized axisymmetric torus we obtain a stronger characterization of global and local minimizers, which we supplement with numerical experiments.
In the first part of this paper, we will consider minimizing configurations of the Oseen-Frank energy functional $E(n, m)$ for a biaxial nematics $(n, m):Omegato mathbb S^2times mathbb S^2$ with $ncdot m=0$ in dimension three, and establish that it is smooth off a closed set of $1$-dimension Hausdorff measure zero. In the second part, we will consider a simplified Ericksen-Leslie system for biaxial nematics $(n, m)$ in a two dimensional domain and establish the existence of a unique global weak solution $(u, n, m)$ that is smooth off at most finitely many singular times for any initial and boundary data of finite energy. They extend to biaxial nematics of earlier results corresponding to minimizing uniaxial nematics by Hardt-Kindelerherer-Lin cite{HKL} and a simplified hydrodynamics of uniaxial liquid crystal by Lin-Lin-Wang cite{LLW10} respectively.
We develop a rigorous, field-theoretical approach to the study of spontaneous emission in inertial and dissipative nematic liquid crystals, disclosing an alternative application of the massive Stueckelberg gauge theory to describe critical phenomena in these systems. This approach allows one not only to unveil the role of phase transitions in the spontaneous emission in liquid crystals but also to make quantitative predictions for quantum emission in realistic nematics of current scientific and technological interest in the field of metamaterials. Specifically, we predict that one can switch on and off quantum emission in liquid crystals by varying the temperature in the vicinities of the crystalline-to-nematic phase transition, for both the inertial and dissipative cases. We also predict from first principles the value of the critical exponent that characterizes such a transition, which we show not only to be independent of the inertial or dissipative dynamics, but also to be in good agreement with experiments. We determine the orientation of the dipole moment of the emitter relative to the nematic director that inhibits spontaneous emission, paving the way to achieve directionality of the emitted radiation, a result that could be applied in tuneable photonic devices such as metasurfaces and tuneable light sources.
We consider a monomer-dimer system with a strong attractive dimer-dimer interaction that favors alignment. In 1979, Heilmann and Lieb conjectured that this model should exhibit a nematic liquid crystal phase, in which the dimers are mostly aligned, but do not manifest any translational order. We prove this conjecture for large dimer activity and strong interactions. The proof follows a Pirogov-Sinai scheme, in which we map the dimer model to a system of hard-core polymers whose partition function is computed using a convergent cluster expansion.
We consider the simplified Ericksen-Leslie model in three dimensional bounded Lipschitz domains. Applying a semilinear approach, we prove local and global well-posedness (assuming a smallness condition on the initial data) in critical spaces for initial data in $L^3_{sigma}$ for the fluid and $W^{1,3}$ for the director field. The analysis of such models, so far, has been restricted to domains with smooth boundaries.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
