Introduction to Q-tensor theory

This paper aims to provide an introduction to a basic form of the ${\bf Q}$-tensor approach to modelling liquid crystals, which has seen increased interest in recent years. The increase in interest in this type of modelling approach has been driven by investigations into the fundamental nature of defects and new applications of liquid crystals such as bistable displays and colloidal systems for which a description of defects and disorder is essential. The work in this paper is not new research, rather it is an introductory guide for anyone wishing to model a system using such a theory. A more complete mathematical description of this theory, including a description of flow effects, can be found in numerous sources but the books by Virga and Sonnet and Virga are recommended. More information can be obtained from the plethora of papers using such approaches, although a general introduction for the novice is lacking. The first few sections of this paper will detail the development of the ${\bf Q}$-tensor approach for nematic liquid crystalline systems and construct the free energy and governing equations for the mesoscopic dependent variables. A number of device surface treatments are considered and theoretical boundary conditions are specified for each instance. Finally, an example of a real device is demonstrated

An operational system for subject switching between controlled vocabularies: A computational linguistics approach

The NASA Lexical Dictionary (NLD), a system that automatically translates input subject terms to those of NASA, was developed in four phases. Phase One provided Phrase Matching, a context sensitive word-matching process that matches input phrase words with any NASA Thesaurus posting (i.e., index) term or Use reference. Other Use references have been added to enable the matching of synonyms, variant spellings, and some words with the same root. Phase Two provided the capability of translating any individual DTIC term to one or more NASA terms having the same meaning. Phase Three provided NASA terms having equivalent concepts for two or more DTIC terms, i.e., coordinations of DTIC terms. Phase Four was concerned with indexer feedback and maintenance. Although the original NLD construction involved much manual data entry, ways were found to automate nearly all but the intellectual decision-making processes. In addition to finding improved ways to construct a lexical dictionary, applications for the NLD have been found and are being developed

A simple derivation of Kepler's laws without solving differential equations

Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple reconsideration of Newton's figure naturally leads to en explicit expression of the velocity and to the equation of the trajectory. This derivation, which can be fully apprehended by beginners at university (or even before) can be considered as a first application of mechanical concepts to a physical problem of great historical and pedagogical interest

Mesh-free simulation of complex LCD geometries

We use a novel mesh-free simulation approach to study the post aligned bistable nematic (PABN) cell. By employing the Qian-Sheng formalism for liquid crystals along with a smooth representation of the surface posts, we have been able to identify two distinct stable configurations. The three-dimensional order field configurations of these states and their elastic free energies are consistent with both experimental results and previous simulation attempts. However, alternative states suggested in previous studies do not appear to remain stable when finite post curvature is considered.</p

Resonance enhanced turbulent transport

The effect of oscillatory shear flows on turbulent transport of passive scalar fields is studied by numerical computations based on the results provided by E. Kim [Physics of Plasmas 13, 022308 (2006)] . Turbulent diffusion is found to depend crucially on the competition between suppression due to shearing and enhancement due to resonances, depending on the characteristic time and length scales of shear flow and turbulence. Enhancements in transport occur for turbulence with finite memory time either due to Doppler or parametric resonances. Scalings of turbulence amplitude and transport are provided in different parameter spaces. The results suggest that oscillatory shear flows are not only less efficient in regulating turbulence, but also can enhance the value of turbulent diffusion, accelerating turbulent transport

Remarkable symmetries in the Milky Way disk's magnetic field

Using a new, expanded compilation of extragalactic source Faraday rotation measures (RM) we investigate the broad underlying magnetic structure of the Galactic disk at latitudes $|b|$ $\lesssim 15^{\circ}$ over all longitudes $l$, where our total number of RM's in this low-latitude range of the Galactic sky is comparable to those in the combined Canadian Galactic Plane Survey(CGPS) at $|b| < 4^{\circ}$ and the Southern Galactic Plane (SGPS) $|b| < 1.5^{\circ}$ survey. We report newly revealed, remarkably coherent patterns of RM at $|b|$ $\lesssim 15^{\circ}$ from $l \sim 270^{\circ}$ to $\sim 90^\circ$ and RM($l$) features of unprecedented clarity that replicate in $l$ with opposite sign on opposite sides of the Galactic center. They confirm a highly patterned bisymmetric field structure toward the inner disc, an axisymmetic pattern toward the outer disc, and a very close coupling between the CGPS/SGPS RM's at $|b| \lesssim 3^{\circ}$ ("mid-plane") and our new RM's up to $|b| \sim 15^{\circ}$ ("near-plane"). Our analysis also shows the approximate $z$-height -- the vertical height of the coherent component of the disc field above the Galactic disc's mid-plane -- to be $\sim 1.5$kpc out to $\sim 6$ kpc from the Sun. This identifies the approximate height of the transition layer to the halo field structure. We find no RM sign change across the plane within $|b| \sim 15^{\circ}$ in any longitude range. The prevailing {\it disc} field pattern, and its striking degree of large scale ordering confirm that our side of the Milky Way has a very organized underlying magnetic structure, for which the inward spiral pitch angle is $5.5^{\circ}\, \pm 1^{\circ}$ at all $|b|$ up to $\sim 12^{\circ}$ in the inner semicircle of Galactic longitudes. It decreases to $\sim 0^{\circ}$ toward the anticentre.Comment: 7 pages, 5 figures, Version 3. Accepted 2011 for publication in Publications of the Astronomical Society of Australia(PASA

Completed cohomology of Shimura curves and a p-adic Jacquet-Langlands correspondence

We study indefinite quaternion algebras over totally real fields F, and give an example of a cohomological construction of p-adic Jacquet-Langlands functoriality using completed cohomology. We also study the (tame) levels of p-adic automorphic forms on these quaternion algebras and give an analogue of Mazur's `level lowering' principle.Comment: Updated version. Contains some minor corrections compared to the published versio

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, in a model version of the PABN cell. Firstly, 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

Adaptive grid methods for Q-tensor theory of liquid crystals : a one-dimensional feasibility study

This paper illustrates the use of moving mesh methods for solving partial differential equation (PDE) problems in Q-tensor theory of liquid crystals. We present the results of an initial study using a simple one-dimensional test problem which illustrates the feasibility of applying adaptive grid techniques in such situations. We describe how the grids are computed using an equidistribution principle, and investigate the comparative accuracy of adaptive and uniform grid strategies, both theoretically and via numerical examples