Equilibrium shapes and stability of magnetic filaments.

Equilibrium shapes of magnetic rods and their stability under the action of an applied field are analyzed. The family of shapes is characterized by two magnetoelastic numbers due to the remanent magnetization and paramagnetic susceptibility of the rod. Since in experiments with flexible magnetic rods the ends are usually unfixed and unclamped, their stability is analyzed under these conditions. Solutions of the corresponding eigenvalue problems for particular cases show that under these conditions the equilibrium shapes are unstable.

Rotating-field-driven ensembles of magnetic particles.

Vortex patterns in ensembles of magnetic particles driven by a rotating field are studied. The driving arises due to the lubrication forces between the rotating particles acting in the direction perpendicular to the radius vector between the particles. Since the lubrication forces cannot be equilibrated by the radial forces due to the dipolar attraction and steric repulsion, the ensemble is in a nonequilibrium state. Different regimes are found for the dynamics of the driven ensembles-solid-body rotation at low frequency of the rotating field and stick-slip motion of the external layers of the aggregate with respect to the internal structure as the frequency is increased. The relation obtained for describing the angular velocity of the solid-body rotation is in good agreement with existing experimental data.

Synchronized rotation in swarms of magnetotactic bacteria.

Self-organizing behavior has been widely reported in both natural and artificial systems, typically distinguishing between temporal organization (synchronization) and spatial organization (swarming). Swarming has been experimentally observed in systems of magnetotactic bacteria under the action of external magnetic fields. Here we present a model of ensembles of magnetotactic bacteria in which hydrodynamic interactions lead to temporal synchronization in addition to the swarming. After a period of stabilization during which the bacteria form a quasiregular hexagonal lattice structure, the entire swarm begins to rotate in a direction opposite to the direction of the rotation of the magnetic field. We thus illustrate an emergent mechanism of macroscopic motion arising from the synchronized microscopic rotations of hydrodynamically interacting bacteria, reminiscent of the recently proposed concept of swarmalators.

Hydrodynamics with spin in bacterial suspensions.

We describe a kind of self-propelling motion of bacteria based on the cooperative action of rotating flagella on the surface of bacteria. Describing the ensemble of rotating flagella in the framework of the hydrodynamics with spin, the reciprocal theorem of Stokesian hydrodynamics is generalized accordingly. The velocity of the self-propulsion is expressed in terms of the characteristics of the vector field of flagella orientation and it is shown that the unusually high velocities of Thiovulum majus bacteria may be explained by the cooperative action of the rotating flagella. The expressions obtained enable us to estimate the torque created by the rotary motors of the bacterium and show quantitative agreement with the existing experimental data.

Relaxation of polar order in suspensions with Quincke effect.

The Quincke effect--spontaneous rotation of dielectric particles in a liquid with low conductivity under the action of an electric field--is considered. The distribution functions for the orientation of particle rotation planes are introduced and a set of nonlinear kinetic equations is derived in the mean field approximation considering the dynamics of their orientation in the flow induced by rotating particles. As a result the nonequilibrium phase transition to the polar order, if the concentration of the particles is sufficiently high, is predicted and the condition of the synchronization of particle rotations is established. Two cases are considered: the layer of the Quincke suspension with one free boundary and the ensemble of the particles rolling on the solid wall under the action of a torque in an electric field. It is shown that in both cases the synchronization of particle rotations occurs due to the hydrodynamic interactions. In the limit of small spatial nonhomogeneity a set of nonlinear partial differential equations for the macroscopic variables--the concentration and the director of the polar order--is derived from the kinetic equation. Its properties are analyzed and compared with available recent experimental results.

Parametric excitation of bending deformations of a rod by periodic twist.

A model of a semiflexible magnetic filament with magnetization frozen in the direction perpendicular to the tangent of its center line is formulated. It is shown that if the rod is magnetized at its ends in opposite directions, an AC magnetic field causes parametric excitation of bending deformations. Neutral curves of parametric excitation are calculated both analytically and numerically. The shapes arising upon parametric excitation of bending deformations are chiral. Periodic rotation of the chiral filament due to nonhomogeneous twist in a nonhomogeneous AC field causes its unidirectional motion.

Ferromagnetic microswimmer.

The self-propelling motion of the flexible ferromagnetic swimmer is described. Necessary symmetry breaking is achieved by the buckling instability at field inversion. The characteristics of self-propulsion are in good agreement with the numerical calculations of the Floquet multipliers for the ferromagnetic filament under the action of ac magnetic field. In the low frequency range the power stroke of self-propelling motion is similar to that used by the unicellular green algae chlamydomonas and in the high frequency region the self-propulsion is due to the undulation waves propagating from the free ends perpendicularly to ac magnetic field.

Nonlinear dynamics of semiflexible magnetic filaments in an ac magnetic field.

Flexible spontaneously magnetized filaments exist in the living world (magnetotactic bacteria) and arise in magnetic colloids with large magnetodipolar interaction parameter. We demonstrate that these filaments possess variety of novel nonlinear phenomena in an ac magnetic field: orientation of the filament in the direction perpendicular to the field and the development of the oscillating U-like shapes, which presumably can lead to the formation of rings of magnetic filaments. It is found that these phenomena are determined by the development of the localized boundary modes of the filament deformation. We have illustrated by qualitative estimates that the phenomena found may be useful for insight into the complex pattern formation phenomena in ensembles of magnetic particles under the action of an ac magnetic field.

Dynamic fluctuations of dipolar semiflexible filaments.

On the basis of the model of a flexible magnetic filament, the characteristics of their thermal fluctuations are considered. The crossover of the time dependence of the mean quadratic displacement from t(3/4) to t(1/2) at the magnetic field increase is found. Two characteristic mechanisms of the magnetization relaxation time distribution--straightening of the thermal undulations and excitation of the bending modes of the free ends under the action of an ac magnetic field--are described. In both cases, the characteristic scaling law omega(-3/4) of the magnetic susceptibility in a high-frequency range is found.