Dimension Reduction for the Landau-de Gennes Model in Planar Nematic Thin Films

Abstract

We use the method of Γ-convergence to study the behavior of the Landau-1 de Gennes model for a nematic liquid crystalline film in the limit of vanishing thickness. In this asymptotic regime, surface energy plays a greater role and we take particular care in understanding its influence on the structure of the minimizers of the derived two-dimensional energy. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and the strong Dirichlet boundary conditions on the lateral boundary of the film. The constants in the weak anchoring conditions are chosen so as to enforce that a surface-energy-minimizing nematic Q-tensor has the normal to the film as one of its eigenvectors. We establish a general convergence result and then discuss the limiting problem in several parameter regimes.

Phase transitions in nematics: textures with tactoids and disclinations

“…In this scaling, the potential W given by (4.9) is now minimized by any symmetric traceless matrix with eigenvalues −1, −1, 2, and the global minimum value of W is equal to zero. In our simulations we consider a simplified form of a Q-tensor that can be obtained via a dimension reduction procedure for thin nematic films [17]. In the corresponding ansatz, one eigenvector of admissible Q-tensors must be perpendicular to the plane of the film.…”

Order Reconstruction for Nematics on Squares with Isotropic Inclusions: A Landau--De Gennes Study

“…These solutions necessarily have three degrees of freedom. For the model problem considered here, as heuristically explained by the analysis in [39], we do not expect to have stable critical points with full five degrees of freedom, with the exception of perhaps very small isotropic square inclusions. It would be interesting to study the LdG critical points on a three-dimensional rectangular box, where the vertical dimension is much smaller than the cross-sectional dimension, and then gradually increase the vertical dimension to check when the out-of-plane perturbations destabilise the WORS or BD solutions.…”

“…Finally, we comment on why the WORS and BD-configurations are stable with respect to all out-of-plane perturbations for this model problem. It is rigorously proven in [39] that for certain thin geometries (where the vertical dimension is much smaller than the lateral dimensions) and for certain surface energies consistent with tangent boundary conditions, the LdG energy minimization problem reduces to a variational problem on the two-dimensional cross-section (such as the square domain in our case) and energy minimizers indeed only have three degrees of freedom. The energy minimizers have a fixed eigenvector in the z-direction; one degree of freedom describes the inplane alignment of the NLC molecules and two scalar order parameters account for the in-plane ordering and the ordering about the z-direction.…”

An electro‐elastic phase‐field model for nematic liquid crystal elastomers based on Landau‐de‐Gennes theory

“…The order parameters S 3 and S 1 are related to the eigenvalues via www.gamm-mitteilungen.org [27]. Consequently, the biaxial Q-tensor has five independent coefficients so that it can be described by the representation (see, for example, SONNET, KILIAN AND HESS [51], JAMES ET AL.…”