Phase Transition With Rapini-Papoular Surface Anchoring

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.

💡 Research Summary

The paper investigates the well‑known planar‑to‑hybrid (P‑HAN) transition in nematic liquid crystal cells when one substrate imposes a homeotropic (vertical) anchoring and the opposite substrate imposes a unidirectional planar anchoring. The authors consider a three‑dimensional simplified Ericksen–Leslie system, originally introduced by Lin and Liu, and augment it with an external magnetic field of constant strength (h) and a Rapini‑Papoular weak anchoring condition on the homeotropic substrate. By assuming the magnetic field points in the vertical direction ((H^\circ = h,\mathbf e_3)) and parametrising the director field as (\mathbf n = (\cos\varphi,0,\sin\varphi)), the coupled fluid–director dynamics reduce to a system for the velocity (u) and the director angle (\varphi): \