250 research outputs found
Landau-De Gennes theory of nematic liquid\ud 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 W 1,2 , 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.\ud
\ud
\ud
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
Function spaces for liquid crystals
We consider the relationship between three continuum liquid crystal theories:
Oseen-Frank, Ericksen and Landau-de Gennes. It is known that the function space
is an important part of the mathematical model and by considering various
function space choices for the order parameters , , and ,
we establish connections between the variational formulations of these
theories. We use these results to derive a version of the Oseen-Frank theory
using special functions of bounded variation. This proposed model can describe
both orientable and non-orientable defects. Finally we study a number of
frustrated nematic and cholesteric liquid crystal systems and show that the
model predicts the existence of point and surface discontinuities in the
director
Orientability and energy minimization in liquid crystal models
Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory
through a unit vector field . This theory has the apparent drawback that it
does not respect the head-to-tail symmetry in which should be equivalent to
-. This symmetry is preserved in the constrained Landau-de Gennes theory
that works with the tensor .We study
the differences and the overlaps between the two theories. These depend on the
regularity class used as well as on the topology of the underlying domain. We
show that for simply-connected domains and in the natural energy class
the two theories coincide, but otherwise there can be differences
between the two theories, which we identify. In the case of planar domains we
completely characterise the instances in which the predictions of the
constrained Landau-de Gennes theory differ from those of the Oseen-Frank
theory
Line defects in the small elastic constant limit of a three-dimensional Landau-de Gennes model
We consider the Landau-de Gennes variational model for nematic liquid
crystals, in three-dimensional domains. More precisely, we study the asymptotic
behaviour of minimizers as the elastic constant tends to zero, under the
assumption that minimizers are uniformly bounded and their energy blows up as
the logarithm of the elastic constant. We show that there exists a closed set S
of finite length, such that minimizers converge to a locally harmonic map away
from S. Moreover, S restricted to the interior of the domain is a locally
finite union of straight line segments. We provide sufficient conditions,
depending on the domain and the boundary data, under which our main results
apply. We also discuss some examples.Comment: 71 pages, 5 figure
Constant-angle surfaces in liquid crystals
We discuss some properties of surfaces in R3 whose unit normal has constant angle with an assigned direction field. The constant angle condition can be rewritten as an Hamilton-Jacobi equation correlating the surface and the direction field. We focus on examples motivated by the physics of interfaces in liquid crystals and of layered fluids, and discuss the properties of the constant-angle surfaces when the direction field is singular along a line (disclination) or at a point (hedgehog defect
Blow up criterion for compressible nematic liquid crystal flows in dimension three
In this paper, we consider the short time strong solution to a simplified
hydrodynamic flow modeling the compressible, nematic liquid crystal materials
in dimension three. We establish a criterion for possible breakdown of such
solutions at finite time in terms of the temporal integral of both the maximum
norm of the deformation tensor of velocity gradient and the square of maximum
norm of gradient of liquid crystal director field.Comment: 22 page
- …