Bending of magnetic filaments under a magnetic field

Magnetic beads and superparamagnetic (SP) colloid particles have successfully been employed for micromechanical manipulation of soft material, in situ probing of elastic properties, and design of smart materials (ferrogels). Here we derive analytical expressions for the equilibrium shape of magnetic fibers, considering two end-member cases, (a) SP or single-domain particles concentrated at the free end of cantilevered rods or tubes, and (b) filaments consisting of SP particles, with this case being mathematically equivalent to tubes containing SP particles. Our analysis yields also metastable equilibrium states (MES's), which only exist above a critical filament length, but become more stable with increasing magnetic field. The MES's for case (a) are, like the ground state, circular arcs, but more strongly bent. The multiform MES's in case (b), which comprise hairpin, sinuous, or even closed shapes, have recently been observed in experiments, too. We also study the effect of gravity on the balance between bending and magnetic energy, which leads to curves with inflection point if the influence of gravity is stronger than that of the magnetic field. Because of their simple experimental realization, case (a) magnetic filaments are deemed highly suitable for micromechanical experiments on long chains of polymer molecules. Another potential application of cantilevered magnetic filaments with magnetic material attached to the free end is in scanning probe microscopes. Because the magnetic field due to the magnetic tip is comparatively weak, the magnetization structure of the sample to be investigated would not be affected by the probe. Thus, for the examination of magnetically soft materials, probes in the form of magnetic filaments may hold advantages over tips usually employed in magnetic force microscopy.