Gradient theory of domain walls in thin, nematic liquid crystals films

Thin Film Liquid Crystals with Oblique Anchoring and Boojums

We study a two-dimensional variational problem which arises as a thin-film limit of the Landau-de Gennes energy of nematic liquid crystals. We impose an oblique angle condition for the nematic director on the boundary, via boundary penalization (weak anchoring.) We show that for strong anchoring strength (relative to the usual Ginzburg-Landau length scale parameter,) defects will occur in the interior, as in the case of strong (Dirichlet) anchoring, but for weaker anchoring strength all defects will occur on the boundary. These defects will each carry a fractional winding number; such boundary…

微流控通道中向列序重构的多方面研究

A Multi-Faceted Study of Nematic Order Reconstruction in Microfluidic Channels

本文在Beris-Edwards框架下研究微流控通道中被动与主动向列流的序重构解，证明其存在性条件、唯一对称非奇异剖面，并通过渐近展开与数值模拟揭示畴壁内部结构及OR型结构的普适性。

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…

A thin-film limit in the Landau-Lifshitz-Gilbert equation relevant for the formation of Néel walls

We consider an asymptotic regime for two-dimensional ferromagnetic films that is consistent with the formation of transition layers (Néel walls). We first establish compactness of S2-valued magnetizations in the energetic regime of Néel walls and characterize the set of accumulation points. We then prove that Néel walls are asymptotically the unique energy minimizing configurations. We finally study the corresponding dynamical issues, namely the compactness properties of the magnetizations under the flow of the Landau-Lifshitz-Gilbert equation.

Analysis of Nematic Liquid Crystals with Disclination Lines

We investigate the structure of nematic liquid crystal thin films described by the Landau--de Gennes tensor-valued order parameter with Dirichlet boundary conditions of nonzero degree. We prove that as the elasticity constant goes to zero a limiting uniaxial texture forms with disclination lines corresponding to a finite number of defects, all of degree 1/2 or all of degree -1/2. We also state a result on the limiting behavior of minimizers of the Chern-Simons-Higgs model without magnetic field that follows from a similar proof.

铁磁薄膜中静态奈耳壁的非线性稳定性

Nonlinear Stability of Static Néel Walls in Ferromagnetic Thin Films

本文研究铁磁薄膜中180°静态奈耳壁的非线性稳定性，通过证明其线性化算子谱位于稳定复半平面，得出小扰动收敛于静态壁生成流形上的平移轨道的结论。

Asymmetric domain walls of small angle in soft ferromagnetic films

We focus on a special type of domain walls appearing in the Landau-Lifshitz theory for soft ferromagnetic films. These domain walls are divergence-free $S^2$-valued transition layers that connect two directions in $S^2$ (differing by an angle $2θ$) and minimize the Dirichlet energy. Our main result is the rigorous derivation of the asymptotic structure and energy of such "asymmetric" domain walls in the limit $θ\to 0$. As an application, we deduce that a supercritical bifurcation causes the transition from symmetric to asymmetric walls in the full micromagnetic model.

One-dimensional in-plane edge domain walls in ultrathin ferromagnetic films

We study existence and properties of one-dimensional edge domain walls in ultrathin ferromagnetic films with uniaxial in-plane magnetic anisotropy. In these materials, the magnetization vector is constrained to lie entirely in the film plane, with the preferred directions dictated by the magnetocrystalline easy axis. We consider magnetization profiles in the vicinity of a straight film edge oriented at an arbitrary angle with respect to the easy axis. To minimize the micromagnetic energy, these profiles form transition layers in which the magnetization vector rotates away from the direction of…

One-dimensional domain walls in thin ferromagnetic films with fourfold anisotropy

We study the properties of domain walls and domain patterns in ultrathin epitaxial magnetic films with two orthogonal in-plane easy axes, which we call fourfold materials. In these materials, the magnetization vector is constrained to lie entirely in the film plane and has four preferred directions dictated by the easy axes. We prove the existence of $90^\circ$ and $180^\circ$ domain walls in these materials as minimizers of a nonlocal one-dimensional energy functional. Further, we investigate numerically the role of the considered domain wall solutions for pattern formation in a rectangular s…