arXiv subjects

S. J. Bedford

Publications and source records attributed to S. J. Bedford.

4 recordsLinked to original sources

arXiv2014

Discontinuous order parameters in liquid crystal theories

The paper is concerned with various issues surrounding the mathematical description of defects in models of liquid crystals, drawing on experience from solid mechanics. The roles played by a suitable choice of function space and by the growth properties of the free-energy density are highlighted. Models in which the director can jump across surfaces are formulated, and their relevance for nematic elastomers, order reconstruction and smectic A thin films discussed.

math.AP↗

arXiv2014

Global minimisers of cholesteric liquid crystal systems

In this paper we examine the modelling and minimisation of cholesteric liquid crystals systems within the Oseen-Frank theory. We focus on a cuboid domain with the frustrated boundary conditions ${\bf n}(0,x,y)=(1,0,0)$ and ${\bf n}(1,x,y)=(0,0,1)$. With general elastic constants, we find the unique global minimisers amongst functions of one variable and prove that these are global minimisers of the entire problem if the cholesteric pitch is sufficiently long. Finally we show that our analysis easily translates over the study the global stability of the constant state ${\bf n}(x,y,z) = (0,0,1)$ with unfrustrated boundary conditions.

math.AP↗

arXiv2014

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 $s$, ${\bf n}$, and ${\bf Q}$, 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.

math.AP↗

arXiv2014

Analysis of local minima for constrained minimization problems

We consider vectorial problems in the calculus of variations with an additional pointwise constraint. Our admissible mappings ${\bf n}:\mathbb{R}^k\rightarrow \mathbb{R}^d$ satisfy ${\bf n}(x)\in M$, where $M$ is a manifold embedded in Euclidean space. The main results of the paper all formulate necessary or sufficient conditions for a given mapping ${\bf n}$ to be a weak or strong local minimizer. Our methods involve using projection mappings in order to build on existing, unconstrained, local minimizer results. We apply our results to a liquid crystal variational problem to quantify the stability of the unwound cholesteric state under frustrated boundary conditions.

math.AP↗