M Zyskin

Topology and bistability in liquid crystal devices.

We study nematic liquid crystal configurations in a prototype bistable device -- the post aligned bistable nematic (PABN) cell. Working within the Oseen-Frank continuum model, we describe the liquid crystal configuration by a unit-vector field n , in a model version of the PABN cell. First, we identify four distinct topologies in this geometry. We explicitly construct trial configurations with these topologies which are used as initial conditions for a numerical solver, based on the finite-element method. The morphologies and energetics of the corresponding numerical solutions qualitatively agree with experimental observations and suggest a topological mechanism for bistability in the PABN cell geometry.

Elastic energy of liquid crystals in convex polyhedra

We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants. For a right rectangular prism and a large class of topologies, we derive upper bounds by introducing test configurations constructed from local conformal solutions of the Euler–Lagrange equation. The ratio of the upper and lower bounds depends only on the aspect ratios of the prism. As the aspect ratios are varied, the minimum-energy conformal state undergoes a sharp transition from being smooth to having singularities on the edges.

Classification of unit-vector fields in convex polyhedra with tangent boundary conditions

A unit-vector field n on a convex three-dimensional polyhedron is tangent if, on the faces of , n is tangent to the faces. A homotopy classification of tangent unit-vector fields continuous away from the vertices of is given. The classification is determined by certain invariants, namely edge orientations (values of n on the edges of ), kink numbers (relative winding numbers of n between edges on the faces of ), and wrapping numbers (relative degrees of n on surfaces separating the vertices of ), which are subject to certain sum rules. Another invariant, the trapped area, is expressed in terms of these. One motivation for this study comes from liquid crystal physics; tangent unit-vector fields describe the orientation of liquid crystals in certain polyhedral cells.

Lower bound for energies of harmonic tangent unit-vector fields on convex polyhedra

AbstractWe derive a lower bound for energies of harmonic maps of convex polyhedra in

Energies of S2-valued harmonic maps on polyhedra with tangent boundary conditions

{\mathbb R}^{3}

TANGENT UNIT-VECTOR FIELDS

to the unit sphere S2 with tangent boundary conditions on the faces. We also establish that C∞ maps satisfying tangent boundary conditions are dense with respect to the Sobolev norm in the space of continuous tangent maps of finite energy.

Quasi-stable configurations of liquid crystals in polyhedral geometries

Abstract A unit-vector field n : P → S 2 on a convex polyhedron P ⊂ R 3 satisfies tangent boundary conditions if, on each face of P, n takes values tangent to that face. Tangent unit-vector fields are necessarily discontinuous at the vertices of P. We consider fields which are continuous elsewhere. We derive a lower bound E P − ( h ) for the infimum Dirichlet energy E P inf ( h ) for such tangent unit-vector fields of arbitrary homotopy type h. E P − ( h ) is expressed as a weighted sum of minimal connections, one for each sector of a natural partition of S 2 induced by P. For P a rectangular prism, we derive an upper bound for E P inf ( h ) whose ratio to the lower bound may be bounded independently of h. The problem is motivated by models of nematic liquid crystals in polyhedral geometries. Our results improve and extend several previous results.

XV CONFERENCE ON LIQUID CRYSTALS

Abstract Let O be a closed geodesic polygon in S 2 . Maps from O into S 2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S 2 , we compute the infimum Dirichlet energy ɛ(H) for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H . The expression for ɛ(H) involves a topological invariant – the spelling length – associated with the (non-abelian) fundamental group of the n -times punctured two-sphere, π 1 ( S 2 − { s 1 ,…, s n }, *). The lower bound for ɛ(H) is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O , are alternatively locally conformal or anticonformal. For conformal and anticonformal classes (classes containing wholly conformal and anticonformal representatives respectively), the expression for ɛ(H) reduces to a previous result involving the degrees of a set of regular values s 1 , …, s n in the target S 2 space. These degrees may be viewed as invariants associated with the abelianization of π 1 ( S 2 - { s 1 ,…, s n }, *). For nonconformal classes, however, ɛ(H) may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees. This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy (one-constant Oseen-Frank energy) of reflection-symmetric tangent unit-vector fields in a rectangular prism.

Quasi-Stable configurations of liquid crystals in polyhedral geometries +

Bistable director configurations are of great interest in liquid crystal display technologies, offering the possibility of higher resolution combined with reduced power consumption. One way to achieve such bistability is to use the cell geometry. As part of an ongoing programme to analyze quasi-stable configurations of liquid crystals in polyhedral geometries, we construct a topological classification scheme of unit-vector fields in convex polyhedra subject to tangential boundary conditions and obtain a general lower bound on the energy of these configurations. We also study the specific case of a unit cube, where we obtain lower and upper bounds for the energies of a family of topological types.

Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy

Bistable director configurations are of great interest in liquid crystal display technologies, offering the possibility of higher resolution combined with reduced power consumption. One way to achieve such bistability is to use the cell geometry. As part of an ongoing programme to analyze quasi-stable configurations of liquid crystals in polyhedral geometries, we construct a topological classification scheme of unit-vector fields in convex polyhedra subject to tangential boundary conditions and obtain a general lower bound on the energy of these configurations. We also study the specific case of a unit cube, where we obtain lower and upper bounds for the energies of a family of topological types.