Polar Phase of an Egg-Yolk Particle in a Rotating Magnetic Field

Abstract

The orbit selected by a magnetic particle in a viscous liquid under a rotating field is commonly described in the instantaneous Stoner--Wohlfarth limit, in which the magnetic moment always occupies a minimum of the anisotropy energy. In this limit, precession out of the field plane exists only for reduced fields $B_0/B_a<1/\sqrt2$, and stronger fields admit only planar synchronous rotation or libration. We study a macroscopic egg-yolk particle, a printed shell containing two freely rotating permanent magnets in fluid-filled cavities, driven by a compensated rotating field and tracked optically. In addition to synchronous rotation, precession, and libration, the particle exhibits a polar phase: above the instantaneous bound and at high frequency, the shell axis leaves the field plane and locks onto a narrow cone about the rotation axis. We extend the model by a finite viscous friction between the magnets and the shell. The extension introduces a single dimensionless parameter, the ratio $κ$ of inner to outer rotational friction, and reduces to the instantaneous theory as $κ\to0$. Numerical phase diagrams show that increasing $κ$ tilts the bistable band to higher fields at higher frequencies and extends stable precession above $B_0/B_a=1/\sqrt2$, whereas for large $κ$ planar libration returns at high frequency. With a calibrated anisotropy field, relaxation rate, and $κ$ of order unity, the model reproduces the terminal state of all measured records and their tilt angles, including the polar cone. In the model the polar state is maintained by a steady rotation of the magnets relative to the shell, driven by the field and anisotropy torques and dissipated by the inner friction; it therefore exists only for intermediate $κ$.