This special issue of the European Journal of Applied Mathematics is devoted to the ‘Liquid Crystal Theory and Modelling: Discussion Meeting’ that was held in Oxford on the 29th-30th October 2009. This international, interdisciplinary meeting was organised jointly by Dr Apala Majumdar (University of Oxford) and Prof. Nigel Mottram (University of Strathclyde) and was funded by OCCAM (Oxford Centre for Collaborative Applied Mathematics), the London Mathematical Society, Taylor and Francis and Cambridge University Press.

Basic methods from bifurcation theory are applied to derive a criterion that predicts when a symmetric tricritical point may occur in a transition between condensed ordered phases described by any finite number of scalar order parameters. At such a point, a change of order takes place in the phase transition, which passes from first to second order, or vice versa.

The excluded volume of a pair of molecules is proportional to the second virial coefficient in hard-core models that represent, for instance, the reference model in perturbation approaches to statistical theories of fluids [see, e.g. Chap. 5 of Kalikmanov, V. (2001), Statistical Physics of Fluids. Texts and Monograph in Physics, Springer, Berlin]. In three space dimensions, there exist exact results for convex molecules and in fact lack of convexity has been a major obstacle in applying the mathematical techniques employed in the convex case. In this paper, we illustrate how a mixed—analytical and numerical—method can be used to obtain exact expressions of the excluded volume for a pair of non-convex molecules conceived as aggregates of hard spheres; these can model van der Waals regions associated to the atoms forming each molecule. To compute the excluded volume for molecules endowed with C2v symmetry, modelled as chains of tangent hard spheres, we adapt a numerical code available to the scientific community. Because the result is a rather cumbersome expression in term of the relative orientation of the interacting molecules, we expand it over a suitable set of symmetry adapted Wigner functions to build up approximate, but faithful expressions, and we also prove analytical results announced elsewhere [Bisi, F., Durand, G., Rosso, R. & Virga, E. (2008), Polar steric interactions for v-shaped molecules. Phys. Rev. E, 78, 011705].

We study the radial-hedgehog solution in a three-dimensional spherical droplet, with homeotropic boundary conditions, within the Landau–de Gennes theory for nematic liquid crystals. The radial-hedgehog solution is a candidate for a global Landau–de Gennes minimiser in this model framework and is also a prototype configuration for studying isolated point defects in condensed matter physics. The static properties of the radial-hedgehog solution are governed by a non-linear singular ordinary differential equation. We study the analogies between Ginzburg–Landau vortices and the radial-hedgehog solution and demonstrate a Ginzburg–Landau limit for the Landau–de Gennes theory. We prove that the radial-hedgehog solution is not the global Landau–de Gennes minimiser for droplets of finite radius and sufficiently low temperatures and prove the stability of the radial-hedgehog solution in other parameter regimes. These results contain quantitative information about the effect of geometry and temperature on the properties of the radial-hedgehog solution and the associated biaxial instabilities.

In this paper, we review a number of uses of conformal mapping techniques for obtaining director profiles of liquid crystals in confined and semi-confined geometries. In particular, we will consider geometries which allow more than one stable state, some of which are of use in bistable displays. These solutions also allow the investigation of the energy of stable states and enable conclusions to be reached as to how such geometries may be optimised for bistable display applications. Such techniques are also able to provide initial configurations for the solution of more complicated situations where numerical methods are used to investigate switching characteristics.

Liquid crystal elastomers present features not found in ordinary elastic materials, such as semi-soft elasticity and the related stripe domain phenomenon. In this paper, the two-dimensional Bladon–Terentjev–Warner model and the one-constant Oseen–Frank energy expression are combined to study the liquid crystal elastomer. We also impose two material constraints, the incompressibility of the elastomer and the unit director norm of the liquid crystal. We prove existence of minimiser of the energy for the proposed model. Next we formulate the discrete model, and also prove that it possesses a minimiser of the energy. The inf-sup values of the discrete linearised system are then related to the smallest singular values of certain matrices. Next the existence and uniqueness of the Lagrange multipliers associated with the two material constraints are proved under the assumption that the inf-sup conditions hold. Finally numerical simulations of the clamped-pulling experiment are presented for elastomer samples with aspect ratio 1 or 3. The semi-soft elasticity is successfully recovered in both cases. The stripe domain phenomenon, however, is not observed, which might be due to the relative coarse mesh employed in the numerical experiment. Possible improvements are discussed that might lead to the recovery of the stripe domain phenomenon.

A two-dimensional computational study is performed on the texturing of fibre-filled nematic liquid crystals using the Landau-de Gennes model describing the spatio-temporal evolution of the second moment of the orientation distribution function or quadrupolar tensor order parameter. The investigation is performed on a consistent computational domain comprising a square array of four circular fibres embedded within a unit square containing a uniaxial low molar mass calamitic liquid crystal. Interest is focused on the role of temperature, boundary conditions and their effect on the nucleation and evolution of defect structures. Thermal effects are characterised below and above the temperature at which the nematic state is stable. Simulations in the stable nematic state serves as a scenario for investigating the effect of imposing different external boundary conditions, namely periodic and Dirichlet; the former describes a square lattice array of fibres embedded in a nematic liquid crystal, and the latterdescribes a four-fibre arrangement in an aligned nematic material. In each case, the influence of temperature is characterised, with defect structures forming and either remaining or splitting into lower strength defects. For fibre lattices, splitting transitions of defects at the centre of the domain occur at a critical temperature, but for the four-fibre arrangement, defect transitions occur continuously over a temperature range. The discontinuous defect splitting transition in fibre arrays occurs at lower temperatures than the continuous defect transformation in the four-fibre arrangement. At sufficiently low temperatures, the four-fibre arrangement and the fibre lattice give the same texture consisting of two disclination lines close to each fibre. The evolution of the texture with respect to temperature can be characterised as a change from single-fibre mode at low temperature to a collective mode with a centre-located heterogeneity at higher temperature. At higher temperatures, in the stable isotropic state, it is shown that surface-induced ordering arising from the fibre/liquid crystal interaction propagates into the bulk forming thin disclination lattices around the four-fibre configuration.

We present an asymptotic theory that includes in a perturbative expansion the coupling effects between the director dynamics and the velocity field in a nematic liquid crystal. Backflow effects are most significant in the presence of defect motion, since in this case the presence of a velocity field may strongly reduce the total energy dissipation and thus increase the defect velocity. As an example, we illustrate how backflow influences the speeds of opposite-charged defects.