Paramagnetic filaments in a fast precessing field: Planar versus helical conformations

Abstract

We examine analytically equilibrium conformations of elastic chains of paramagnetic beads in the presence of a precessing magnetic field. Conformations of these filaments are determined by minimizing their total energy, given in the harmonic approximation by the sum of the bending energy, quadratic in its curvature, and the magnetic dipolar interaction energy, quadratic in the projection of the vector tangent to the filament onto the precession axis. In particular, we analyze two families of open filaments with their ends aligned along the precession axis and described by segments of planar curves and helices. These configurations are characterized in terms of two parameters encoding their features such as their length, separation between their ends, as well as their bending and magnetic moduli, the latter being proportional to the magnitude and precession angle of the magnetic field. Based on energetic arguments, we present the set of parameter values for which each of these families of curves is probable to occur.

Crystalline membrane morphology beyond polyhedra

“…The versatility of magnetoelastic filaments, which consist of super-paramagnetic particles connected by elastic linkers, has also been demonstrated experimentally [12][13][14] and numerically [15,16].…”

Actuation of magnetoelastic membranes in precessing magnetic fields

“…S7). This analysis yields the total energy of the membrane given by H = Lo dl H , where the linear energy density H coincides with the energy density of a paramagnetic filament (28). The Euler-Lagrange equations arising from H lead to the profile curves of the ground state.…”

Flexible paramagnetic membranes in fast precessing fields

“…We know that helices minimize the sum of bending and magnetic linear energies [14,15], and since the helicoid can be foliated by helices, we might expect it to be a critical point of H. The helicoid is a ruled minimal surface with glide rotational symmetry characterized by its pitch 2πp; see Fig 6. For a minimal surface, with K = 0, the EL Eq. (7) reduces to E MS = mK ab e z a e z b = 0.…”

“…They are highly versatile. For instance, depending on their rigidity and the magnetic field strength, arrays of paramagnetic filaments may collapse into hairpins, loops, sheets, or pillars [13], and in a fast precessing magnetic field, depending on the precession angle they may adopt planar or helical conformations [14,15]. An obvious extension in the study of this kind of systems would be the consideration of the two-dimensional counterparts of paramagnetic filaments, that is, membranes consisting of two-dimensional arrays of paramag- * Electronic address: pablo.vazquez@correo.uady.mx † m-olvera@northwestern.edu netic beads linked by elastic polymers.…”

“…Here we study theoretically such paramagnetic membranes in a fast precessing magnetic field, whose energy is associated with their bending and the interaction between the induced dipoles on the beads. To address the latter contribution, we introduce an energy density describing the continuum limit of the dipolar interactions between nearest neighbors, analogous to the linear magnetic energy density for paramagnetic filaments [9,15,18]. Although general frameworks for studying elastic membranes in a magnetic field have been developed using different approaches, for instance magnetoelastic theory [19,20], here we determine their equilibrium configurations from a variational principle that exploits the geometric character of the membrane's energy.…”