Steady state solutions in a model of a cholesteric liquid crystal sample
- PDF / 1,374,287 Bytes
- 28 Pages / 439.37 x 666.142 pts Page_size
- 33 Downloads / 159 Views
Steady state solutions in a model of a cholesteric liquid crystal sample Fernando P. da Costa1,2 · Michael Grinfeld3 · Nigel J. Mottram4 · João T. Pinto2,5 · Kedtysack Xayxanadasy6 Received: 29 July 2020 / Accepted: 7 October 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract Motivated by recent mathematical studies of Fréedericksz transitions in twist cells and helix unwinding in cholesteric liquid crystal cells [(da Costa et al. in Eur J Appl Math 20:269– 287, 2009), (da Costa et al. in Eur J Appl Math 28:243–260, 2017), (McKay in J Eng Math 87:19–28, 2014), (Millar and McKay in Mol Cryst Liq Cryst 435:277/[937]–286/[946], 2005)], we consider a model for the director configuration obtained within the framework of the Frank-Oseen theory and consisting of a nonlinear ordinary differential equation in a bounded interval with non-homogeneous mixed boundary conditions (Dirichlet at one end of the interval, Neumann at the other). We study the structure of the solution set using the depth of the sample as a bifurcation parameter. Employing phase space analysis techniques, time maps, and asymptotic methods to estimate integrals, together with appropriate numerical evidence, we obtain the corresponding novel bifurcation diagram and discuss its implications for liquid crystal display technology. Numerical simulations of the corresponding dynamic problem also provide suggestive evidence about stability of some solution branches, pointing to a promising avenue of further analytical, numerical, and experimental studies. Keywords Non-homogeneous two-points boundary value problems · Bifurcations · Asymptotic evaluation of integrals · Cholesteric liquid-crystal cells · Fréedericksz transition · Nonlinear pendulum Mathematics Subject Classification Primary 34B15 · 34C23; Secondary 41A60 · 76A15
1 Introduction 1.1 Motivation: cholesteric liquid crystals The strong anisotropic nature of the constitutive molecules in a liquid crystal, and their preference to align with each other, lead to a local macroscopic orientational order [7]. This orientational order allows us to define an axis of local rotational symmetry, the average
B
Fernando P. da Costa [email protected]
Extended author information available on the last page of the article
123
F. P. da Costa et al.
molecular orientation at location x = (x1 , x2 , x3 ). This local symmetry axis defines a macroscopic variable, and is usually represented by a unit vector n = n(x), called the director. The director may vary with the location x in space as a result of internal or external influences, to create director distortion structures that may change the stored elastic energy of the system. In this paper we consider liquid crystals that have only orientational order, termed nematics, and we do not consider those liquid crystals that also show positional ordering, which are termed smectics. In nematic liquid crystals consisting of chiral molecules, often called cholesterics or chiral nematics, the molecular alignment ma
Data Loading...
