Phase Transition With Rapini-Papoular Surface Anchoring
Shun Li, Yong Yu
TL;DR
The paper addresses the P-HAN transition in nematic liquid crystals under a uniform magnetic field with Rapini-Papoular weak anchoring. It reduces the 3D Ericksen-Leslie dynamics to a coupled $(\mathbf{u},\phi)$ system on a slab $\Omega = \mathbb{T}^2 \times (0,d)$ and analyzes both stationary and dynamical aspects via a sine-Gordon framework; a key result is the explicit thickness threshold $d_c = \frac{1}{h}\tan^{-1}\left(\frac{h}{L_H}\right)$ that separates trivial and nontrivial equilibria, with a least-energy SG solution $\phi_*$ for $d>d_c$ that depends only on the normal coordinate. The analysis combines a generalized Steklov-Dirichlet eigenproblem, the Łojasiewicz-Simon inequality, and energy-dissipation methods to show that any global suitable weak solution converges, as $t\to\infty$, to an SG equilibrium $\phi_\infty$ (which is $0$ for $d\le d_c$ and $\phi_*$ for $d>d_c$), with convergence rates governed by a Łojasiewicz exponent $\theta$. When $d\neq d_c$, the least-energy solution is strongly stable and yields exponential convergence of the flow to the equilibrium; the paper also proves boundary partial regularity for weak solutions under small dissipation energy, ensuring the P-HAN transition is rigorously justified in 3D. Overall, the work bridges PDE analysis with LC physics by providing rigorous criteria and rates for the P-HAN transition in the RP weak-anchoring setting.
Abstract
We analyze the dynamical (in)stability of nematic liquid crystals in the presence of external magnetic fields and Rapini-Papoular surface potential. The P-HAN transition is investigated using a simplified 3D Ericksen-Leslie system. We find the thickness threshold of the P-HAN transition. If the thickness of the nematic layer exceeds this threshold, there is a global-in-time suitable weak solution converging exponentially to a nontrivial equilibrium state as time tends to infinity. If the thickness is no more than the threshold, the global-in-time suitable weak solution has a trivial long-time asymptotic limit. Our results rigorously justify the P-HAN transition discussed in the physics literature.