发表机构
University of Strathclyde（斯特拉斯克莱德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过渐近分析揭示 Landau-de Gennes 框架中 Oseen-Frank 极限的锚定诱导倾斜软模式，该模式由线性 Jacobi 方程控制，仅通过表面项影响能量，为系统降能提供途径。
AI 中文摘要
对 Landau-de Gennes 框架进行了渐近分析，以计算在适当缩放的表面锚定能量下，有界三维区域中 Oseen-Frank 极限的高阶修正。将 $\mathbf{Q}$ 张量系统地分解为三个相互正交的子空间——单轴标量、几何倾斜向量和横向各向异性张量——揭示了 Oseen-Frank 指向矢场 $\mathbf{n}_0(\mathbf{x})$ 的首阶 $\mathcal{O}(\varepsilon)$ 修正由非零倾斜场 $\mathbf{p}_1(\mathbf{x})$ 主导，其中 $\varepsilon$ 表示向列相干长度与特征区域尺寸之比。这种宏观变化是一种软模式，当表面锚定能量保持在其物理尺度而非被驱动至无限强度时，从边界释放出来。我们证明该倾斜场由线性 Jacobi 方程 $\mathcal{J}_{\mathbf{n}_0}(\mathbf{p}_1)=\mathbf{0}$ 控制，并受非平凡、锚定驱动的 Dirichlet 边界条件约束，其中 $\mathcal{J}_{\mathbf{n}_0}$ 是调和映射 $\mathbf{n}_0$ 在 $\mathbb{S}^2$ 上的在壳 Jacobi 算子。这两个场在 $\mathcal{O}(\varepsilon^2)$ 阶伴随有对单轴标量和横向各向异性张量的离壳修正，这些修正由弹性非均匀性 $(\nabla\mathbf{n}_0\neq\mathbf{0})$ 和边界驱动的倾斜 $(\mathbf{p}_1\neq\mathbf{0})$ 被动诱导。通过 $\mathcal{O}(\varepsilon^2)$ 阶，倾斜仅通过表面项进入能量，而非体项，为系统降低能量提供了途径。在刚性 Dirichlet 条件的常规基准下，这种响应被完全消除，表明基于无限能量壁垒的修正掩盖了锚定驱动倾斜模式的潜在物理机制。
英文摘要
An asymptotic analysis of the Landau--de Gennes framework is performed to compute higher-order corrections to the Oseen--Frank limit in a bounded three-dimensional domain under appropriately scaled surface anchoring energy. A systematic decomposition of the $\mathbf{Q}$-tensor into three mutually orthogonal subspaces-the uniaxial scalar, geometric tilt vector, and transverse anisotropy tensor-reveals that the leading $\mathcal{O}(\varepsilon)$ correction to the Oseen--Frank director field $\mathbf{n}_0(\mathbf{x})$ is dominated by a non-vanishing tilt field $\mathbf{p}_1(\mathbf{x})$, where $\varepsilon$ represents the ratio of the nematic coherence length to the characteristic domain size. This macroscopic variation constitutes a soft mode released from the boundary once the surface anchoring energy is retained at its physical scaling rather than driven to an infinite strength. We show that this tilt field is governed by the linear Jacobi equation, $\mathcal{J}_{\mathbf{n}_0}(\mathbf{p}_1)=\mathbf{0}$, subject to a non-trivial, anchoring-driven Dirichlet boundary condition, where $\mathcal{J}_{\mathbf{n}_0}$ is the on-shell Jacobi operator of the harmonic map $\mathbf{n}_0$ on $\mathbb{S}^2$. The two fields are accompanied at $\mathcal{O}(\varepsilon^2)$ by an off-shell correction to both the uniaxial scalar and the transverse anisotropy tensor, passively induced by the elastic non-uniformity $(\nabla\mathbf{n}_0\neq\mathbf{0})$ and the boundary-driven tilt $(\mathbf{p}_1\neq\mathbf{0})$. Through $\mathcal{O}(\varepsilon^2)$, the tilt enters the energy only through surface terms, not the bulk, providing a pathway for the system to lower its energy . Under the conventional benchmark of rigid Dirichlet conditions, this response is annihilated outright, demonstrating that corrections built upon infinite energy barriers obscure the underlying physics of anchoring-driven tilt modes.
