Dynamics of magnetic spheroids in time periodic magnetic fields

Abstract

Magnetic nano and micro particles are used for a lot of novel applications, like mixing in inherently laminar microscale systems, bio-rheological measurements, drug delivery, and proctored surgeries. A number of studies have elucidated the movement of magnetic particles of different shapes and magnetic natures in the presence of different types of magnetic fields, analytically and experimentally. In the current study, a fundamental approach to understand the motion of spheroids in the presence of time periodic fields is adopted by accounting for the different magnetic natures of the particles. The non-hysteretic superparamagnetic particles can be modeled by the signum, linear, or the Langevin moment models. The moment of the hysteretic soft ferromagnetic particles is modeled by the Stoner- Wohlfarth model. The hard ferromagnetic materials’ moment is modeled as a permanent dipole. The hydrodynamic torque acting on the particle counters the magnetic torque applied by the field. In the presence of a rotating field, magnetic particles corotate with the field. But as the field frequency increases beyond the breakdown frequency, the particle slips relative to the field. If the motion is in the field’s plane, then it is called parallel; else, it is precessed motion. In the current work we understand this from a dynamical systems perspective in terms of non-dimensional numbers. For the simpler non-hysteretic models, the dynamics is completely defined by ω†, the ratio of the field frequency and the particle viscous relaxation rate. The more practical models, the Langevin and the SW models require one more material parameter. It is the ratio of the magnetic saturation and the product of the magnetic susceptibility and the field strength (ms/(χH)) for the Langevin model, and h, the ratio of the Zeeman and anisotropy energies for the SW model. The dynamics of the two-parameter models can be mapped onto the one-parameter models, which broadly depict the behaviours of parallel corotation and slip and precessed corotation and slip. However, the experiments show initial condition dependent stable states of precessed corotation and parallel slip at higher field frequencies. The SW model is able to capture this for 0.5 < h < 1/ √ 2. The experimental setup of an oscillating magnetic field is easier than that of a rotating field. Hence, in the next part of the study, we look at the effect of an oscillating magnetic field on magnetic spheroids. For superparamagnetic materials, the particles eventually align along the field, and hence no steady hydrodynamic torque can be imparted by them. For hard ferromagnetic spheroids (permanent dipole), the spheroid oscillates with the field, and the trajectories are initial condition dependent. In the ω† ≪ 1, the torque scaled by the product of the magnetic saturation and the field amplitude, is proportional to ω†1/2 and saturates to a constant dependent on the initial condition for ω† ≫ 1. For soft ferromagnetic materials (SW model), for h0 < 0.5 (h0 is the h calculated with Zeeman energy in terms of the field amplitude), the behaviour is similar to that of the permanent dipolar particle. For high h0, the moment switches between the two poles of the orientation, leading to small-amplitude oscillations and reduced torque fluctuations. For intermediate values of h0, either of the extremities is possible based on the initial condition. In the next part, the effect of simple shear on a permanent dipolar spheroid in the presence of an oscillating magnetic field is studied. The relevant nondimensional numbers are ω∗, the ratio of field frequency and strain rate, Σ, the ratio of magnetic and hydrodynamic torques, and rotation number, which is the ratio of the particle angular velocity and the field frequency. The different types of particle behaviour are mapped onto the Σ−ω∗ plane. These bpundaries form Arnold tongues for ω∗ < 1/2, with downward cusps at ω∗ = 1/(2n0), where n0 is an odd integer, in the limit Σ ≪ 1. For Σ ≫ 1, the particle rotation number is one as the particle gets phase-locked and rotates in the shear plane. For ω∗ < 1/2, as Σ decreases, the Arnold tongues merge to form cusps that give rise to strips of constant odd rotation numbers, in which the particle is phase locked. The mean and root mean square torques change discontinuously as these boundaries are crossed. For non-integer rotation numbers, the particle shows quasi-periodic out-of-shear plane rotation. For ω∗ ≫ 1, the boundary of the transition from quasiperiodic to phase-locked rotations increases as exp (1/ω†). On extending this understanding to a spheroid, the presence of Arnold tongues was observed for ω∗ < ω∗ J , where ω∗ J is the ratio of Jeffery frequency and strain rate. The boundaries of the Arnold tongues for the different values of B, shape factor of spheroid, can be mapped onto a universal curve with different scalings with respect to B across the values of ω∗ < ω∗ J . As B decreases the effective area of the Arnold tongue reduces. For B = 1, a thin rod, Arnold tongues are not observed. For ω∗ ≫ 1, the boundary scales in the same way as that of the sphere, and is independent of B.