The Saturn Ring Effect in Nematic Liquid Crystals with External Field: Effective Energy and Hysteresis

In this work we consider the Landau–de Gennes model for liquid crystals with an external magnetic field to model the occurrence of the Saturn ring effect under the assumption of rotational equivariance. After a rescaling of the energy, a variational limit is derived. Our analysis relies on precise estimates around the singularities and the study of a radial auxiliary problem in regions, where a continuous director field exists. Studying the limit problem, we explain the transition between the dipole and Saturn ring configuration and the occurence of a hysteresis phenomenon, giving a rigorous explanation of what was derived and simulated previously by [H. Stark, Eur. Phys. J. B 10, 311–321 (1999)].