Ph.D Thesis Colloquium : Isha Misra

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 [1]. Several 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 [2, 3, 4, 5, 6]. 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 modelled by the signum, linear, or the Langevin moment models. The moment of the hysteretic soft ferromagnetic particles is modelled by the Stoner-Wohlfarth (SW) model. The hard ferromagnetic materials’ moment is modelled as a permanent dipole. The hydrodynamic torque acting on the particle counters the magnetic torque applied by the field [7].

In the presence of a rotating field, magnetic particles corotate with the field at small field frequency. When the field frequency exceeds the breakdown frequency, the particle slips relative to the field. The trajectory is termed parallel if it is in the same plane as the field; 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. For the non-hysteretic Langevin model, this is the ratio of the magnetic saturation and the product of the magnetic susceptibility and the field strength, and for the hysteretic SW model, the second parameter is h, the ratio of the Zeeman and anisotropy energies. 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. The SW model captures experimental results of initial condition dependent stable states of precessed corotation and parallel slip at higher field frequencies for 0.5<h<1⁄√2.

In the next part of the study, the dynamics of a magnetic spheroid in an oscillating magnetic field is examined. For superparamagnetic materials, the particles align along the field, in the long time limit. 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 the torque saturates to a constant dependent on the initial condition for ω^†≫1. For soft ferromagnetic materials (SW model), for h_0<0.5 (h_0 is the parameter h calculated based on the amplitude of the magnetic field), the behaviour is similar to that of the permanent dipolar particle. For high h_0, the moment switches between the two poles of the orientation, leading to small-amplitude oscillations and reduced torque fluctuations. For intermediate values of h_0, there could be either slipping or switching based on the initial condition [8].

The effect of simple shear on a permanent dipolar sphere in the presence of an oscillating magnetic field is studied. The relevant non-dimensional numbers are ω^, the ratio of field frequency and strain rate, and Σ, the ratio of magnetic and hydrodynamic torques, and rotation number, which is the ratio of the particle angular velocity and the field frequency. Different types of particle behaviour are mapped onto the Σ-ω^ plane. These boundaries form Arnold tongues for ω^ <1⁄2, with downward cusps at ω^ =1⁄((2n_0 ) ), where n_0 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^† ) [9].

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 of oscillation in the absence of the magnetic field and the 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.