Physics – Mathematical Physics
Scientific paper
2009-05-11
Physics
Mathematical Physics
21 pages, 8 figures; in "Analysis and Stochastics of Growth Processes and Interface Models", P Morters et al. eds., Oxford Uni
Scientific paper
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 $P$. In certain cases, this lower bound can be improved by incorporating certain nonabelian homotopy invariants. For a rectangular prism, upper bounds for the infimum Dirichlet energy are obtained from locally conformal solutions of the Euler-Lagrange equations, with the ratio of the upper and lower bounds bounded independently of homotopy type. However, since the homotopy classes are not weakly closed, the infimum may not be realised; the existence and regularity properties of continuous local minimisers of given homotopy type are open questions. Numerical results suggest that some homotopy classes always contain smooth minimisers, while others may or may not depending on the geometry of $P$. Numerical results modelling a bistable device suggest that the observed nematic configurations may be distinguished topologically.
Majumdar Arka
Robbins JM
Zyskin M.
No associations
LandOfFree
Liquid crystals and harmonic maps in polyhedral domains does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Liquid crystals and harmonic maps in polyhedral domains, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Liquid crystals and harmonic maps in polyhedral domains will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-701199