Phase Transition With Rapini-Papoular Surface Anchoring

TL;DR

This paper rigorously analyzes the stability and phase transition behavior of nematic liquid crystals under external magnetic fields and surface anchoring, identifying a critical thickness for the P-HAN transition.

Contribution

It provides a mathematical framework for the P-HAN transition in nematic liquid crystals, including the threshold thickness and long-term solution behavior.

Findings

Identified the critical thickness for the P-HAN transition.

Proved exponential convergence to equilibrium for thicker layers.

Established trivial long-term limits for thinner layers.

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.