Flexible paramagnetic membranes in fast precessing fields

Elastic membranes composed of paramagnetic beads offer the possibility of assembling versatile actuators operated autonomously by external magnetic fields. Here we develop a theoretical framework to study shapes of such paramagnetic membranes under the influence of a fast precessing magnetic field. Their conformations are determined by the competition of the elastic and magnetic energies, arising as a result of their bending and the induced dipolar interactions between nearest-neighbors beads. In the harmonic approximation, the elastic energy is quadratic in the surface curvatures. To account for the magnetic energy we introduce a continuum limit energy, quadratic in the projections of the surface tangents onto the precession axis. We derive the Euler-Lagrange equation governing the equilibria of these membranes, as well as the corresponding stresses. We apply this framework to examine paramagnetic membranes with quasiplanar, cylindrical, and helicoidal geometries. In all cases we found that their shape, energy, and stresses can be modified by means of the parameters of the magnetic field, mainly by the angle of precession.

Figure 1

The surface $\mathrm{\Sigma }$ is parametrized by local coordinates $u^1$ and $u^2$; the tangent and unit normal vectors are ${\mathbf{e}}_a$ and $\mathbf{n}$.

Figure 2

The axis of precession of the magnetic field is chosen as the $Z$ axis and the angle of precession is $\vartheta$.

Figure 3

(a) Ground state and (b–d) first excited states resulting from the compression of a planar membrane.

Figure 4

Cylinders oriented (a) along and (b) orthogonally to the axis of precession.

Figure 5

Deformations of a circular cylinder ($\mathfrak{m}=0$) oriented orthogonally to the precession axis $Z$ (dashed line). For $\mathfrak{m}>0$ its cross section elongates along the precession axis (black line), whereas for $\mathfrak{m}<0$ it gets squeezed in the orthogonal direction (gray line). The amplitude of the deformation has been exaggerated for illustration purposes.

Figure 6

A helicoid of pitch $2\pi p$ bounded by two straight lines of length $L$ and two helices of radius $L/2$.