Homotopy classification of director fields on polyhedral domains with tangent and periodic boundary conditions, with applications to bi-stable post-aligned liquid crystal displays

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

A unit-vector field n on a convex three-dimensional polyhedron P is tangent if, on the faces of P, n is tangent to the faces. A homotopy classification of tangent unit-vector fields continuous away from the vertices of P is given. The classification is determined by certain invariants, namely edge orientations (values of n on the edges of P), kink numbers (relative winding numbers of n between edges on the faces of P), and wrapping numbers (relative degrees of n on surfaces separating the vertices of P), which are subject to certain sum rules. Another invariant, the trapped area, is expressed …

Energies of S^2-valued harmonic maps on polyhedra with tangent boundary conditions

A unit-vector field n:P \to S^2 on a convex polyhedron P \subset 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^inf_P(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 de…

轴对称向列膜最小化器的存在性和唯一性

Existence and uniqueness of minimizers for axisymmetric nematic films

本文研究了轴对称向列膜的变分模型，证明了最小化器的存在性和唯一性，并通过数值模拟验证了结果。

Bistable curvature potential at hyperbolic points of nematic shells

Nematic shells are colloidal particles coated with nematic liquid crystal molecules which may freely glide and rotate on the colloid's surface while keeping their long axis on the local tangent plane. We describe the nematic order on a shell by a unit director field on an orientable surface. Equilibrium fields can then be found by minimising the elastic energy, which in general is a function of the surface gradient of the director field. We learn how to extract systematically out of this energy a fossil component, related only to the surface and its curvatures, which expresses a curvature pote…

Liquid crystals and harmonic maps in polyhedral domains

Unit-vector fields $\nvec$ on a convex polyhedron $P$ subject to tangent boundary conditions provide a simple model of nematic liquid crystals in prototype bistable displays. The equilibrium and metastable configurations correspond to minimisers and local minimisers of the Dirichlet energy, and may be regarded as $S^2$-valued harmonic maps on $P$. We consider unit-vector fields which are continuous away from the vertices of $P$. A lower bound for the infimum Dirichlet energy for a given homotopy class is obtained as a sum of minimal connections between fractional defects at the vertices of $P$…

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

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, Ecal(H), for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H. The expression for Ecal(H) involves a topological invariant - the spelling length - associated with the (nonabelian) fundamental group of the n-times punctured two-sphere, pi_1(S^2 - {s_1,..., s_n},*). The lower bound for Ecal(H) is obtained from c…