Stability Analysis in the Twist-Bend Nematic Liquid Crystal Model

The recently discovered twist-bend nematic liquid crystal (LC) phase is characterized by a nanoscale helical modulation of the nematic director n, forming a conical helix along the z-axis at an oblique angle θ. While many models assume a constant cone angle and equal elastic constants K11 = K22 = K33, this dissertation removes both assumptions by considering a fully anisotropic elastic energy with K11 ≠ K22 ≠ K33, and allowing θ to vary spatially. We analyze the stability of this system under frustrated and free boundary conditions using variational methods. The free energy is formulated as an integral functional on Sobolev vector fields constrained by |n| = 1. Using constrained variations, we derive the Euler–Lagrange equations and compute the second variation to establish both necessary and sufficient conditions on the material parameter η for local and global stability. We identify critical thresholds for η that distinguish stable configurations across the nematic (N), cholesteric (N*), and twist-bend nematic (NTB) phases. A coercive lower bound on the second variation is obtained via Poincaré-type inequalities. Additionally, we prove Γ-convergence of a perturbed family of functionals to a limiting energy, providing a rigorous framework for studying minimizers and phase transitions. Numerical simulations based on the truncated Newton–Conjugate Gradient method are implemented to compute equilibrium states under the magnet field. Together, these results offer both analytical and computational insight into the stability structure of the NTB phase and contribute to the broader understanding of variational models in liquid crystal theory.