| dc.description.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. | en_US |
