Dimension reduction for the Landau-de Gennes model on curved nematic thin films

We use the method of $\Gamma$-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film attached to a general fixed surface in the limit of vanishing thickness. This paper generalizes the approach that we used previously to study a similar problem for a planar surface. Since the anchoring energy dominates when the thickness of the film is small, it is essential to understand its influence on the structure of the minimizers of the limiting energy. In particular, the anchoring energy dictates the class of admissible competitors and the structure of the limiting problem. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and strong Dirichlet boundary conditions on the lateral boundary of the film when the surface is not closed. We establish a general convergence result to an energy defined on the surface that involves a somewhat surprising remnant of the normal component of the tensor gradient. Then we exhibit one effect of curvature through an analysis of the behavior of minimizers to the limiting problem when the substrate is a frustrum

Nematic liquid crystals on curved surfaces - a thin film limit

We consider a thin film limit of a Landau-de Gennes Q-tensor model. In the limiting process we observe a continuous transition where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. Main properties of the thin film model, like uniaxiality and parameter phase space, are preserved in the limiting process. For the derived surface Landau-de Gennes model, we consider an L2-gradient flow. The resulting tensor-valued surface partial differential equation is numerically solved to demonstrate realizations of the tight coupling of elastic and bulk free energy with geometric properties.Comment: 20 pages, 4 figure

Solution landscapes of the simplified Ericksen--Leslie model and its comparison with the reduced Landau--de Gennes model

We investigate the solution landscapes of a simplified Ericksen--Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau--de Gennes (rLdG) models can be viewed as Ginzburg--Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare with the solution landscapes of the corresponding rLdG models. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviors, and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solution $C\pm$ with interior defects, high-index "fake defects" solutions, novel tessellating solutions, and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics