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

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.

Figure 1

The filament is described by a space curve $\mathrm{\Gamma }:s\to \mathbf{Y}\left(s\right)$ parametrized by arc length $s$. The adapted FS basis is formed by the tangent vector $\mathbf{t}$, the principal normal $\mathbf{n}$ and the binormal $\mathbf{b}$.

Figure 2

(a) The magnetic field $\mathbf{H}$ precesses with an angle $\vartheta$ about the $Z$ axis. (b) The critical angle ${\vartheta }_m$, at which the magnetic modulus vanishes, divides the precession parameter space in two regions, the first one denoted by I, with $0<\vartheta <{\vartheta }_m$ and $\mathcal{M}>0$, and the second one denoted by II with ${\vartheta }_m<\vartheta <\pi /2$ and $\mathcal{M}<0$.

Figure 3

The planar curve can described by the angle $\mathrm{\Theta }$, formed by $\mathbf{t}$ and $\widehat{\mathbf{z}}$.

Figure 4

Eigenvalues corresponding to the first deformation modes. Only the state $n=1$ is stable in the perturbative regime.

Figure 5

Planar configurations with $n=1$ in regime I $\left(\mathcal{M}>0\right)$ for different values of the separation between boundaries: $\xi =0.8$ (top row), 0.457 (middle row), and 0.1 (bottom row). The Euler elastica for which $\gamma =0$ is shown with a dashed line in the fist column. The local energy density is color-coded in these figures.

Figure 6

Planar configurations with $n=2$ in regime I $\left(\mathcal{M}>0\right)$ for different values of the separation between boundaries: $\xi =0.8$ (top row), 0.457 (middle row), and 0.1 (bottom row). The local energy density is color-coded in these figures.

Figure 7

Planar configurations with $n=1$ in regime II $\left(\mathcal{M}<0\right)$ for different values of the separation between boundaries: $\xi =0.8$ (top row), 0.457 (middle row), and 0.1 (bottom row). The local energy density is color-coded in these figures.

Figure 8

Planar configurations with $n=2$ in regime II $\left(\mathcal{M}<0\right)$ for different values of the separation between boundaries: $\xi =0.8$ (top row), 0.457 (middle row), and 0.1 (bottom row). The local energy density is color-coded in these figures.

Figure 9

Magnitude of the scaled force $\overline{F}$ for the first two planar states (a) $n=1$ and (b) $n=2$. Configurations with vanishing force $F=0$ are represented with a solid black line; elastic curves with $\gamma =0$ are represented with a dashed black line.

Figure 10

Scaled total energy $\overline{H}$ for the first two planar states (a) $n=1$ and (b) $n=2$, the solid black line represents configurations with $H=0$ and the dashed line elastic curves with $\gamma =0$. (c) Difference between the total energies, $\mathrm{\Delta }{H}_1=H_{n=2}-H_{n=1}$, as a function of the boundary separation $\xi$ and the magnetoelastic parameter $\gamma$. $\mathrm{\Delta }{H}_1\ge 0$ everywhere.

Figure 11

The helix is characterized by its radius $\varrho$ and pitch $p=2\pi \varrho tan\psi$, with $\psi$ the angle formed by $\mathbf{t}$ and $\widehat{\mathbf{\phi }}$.

Figure 12

Magnitude of the scaled force $\overline{F}$ for the first two helical states (a) $n=1$ and (b) $n=2$. The solid black line represents configurations with $F=0$, the dashed black line elastic curves with $\gamma =0$.

Figure 13

Total energy of helices with (a) $n=1$ and (b) $n=2$, the solid black line represents configurations with $H=0$ and the dashed line elastic curves with $\gamma =0$. (c) Difference between the total energies, $\mathrm{\Delta }{H}_1=H_{n=2}-H_{n=1}$, as a function of the boundary separation $\xi$ and the magnetoelastic parameter $\gamma$. $\mathrm{\Delta }{H}_1\ge 0$ everywhere.

Figure 14

(a) Difference between the total energy of the helices and planar curves with $n=1$ and same values of $\xi$ and $\gamma$, the black line indicates points where $\mathrm{\Delta }H=0$ and the dotted line $\gamma =-{\mathrm{\Phi }}^2$. (b) Zoom of the region where $\mathrm{\Delta }H$ becomes negative, red regions correspond to $\mathrm{\Delta }H>0$ (planar curves have lower energy) and blue regions correspond to $\mathrm{\Delta }H<0$, (helices have lower energy).

Figure 15

(a) Scaled total energy as a function of $t$ of the sequence of homotopic configurations interpolating the planar and helical states with $n=1$, and parameter values $\xi =0.5$ and $\gamma =-500$. Sequence with $\omega =\pi /2$ $\left(-\pi /2\right)$ exhibits the smallest (largest) amplitude (b) Zoom of the total energy of sequence with $\omega =\pi /2$.

Figure 16

Sequence of homotopic configurations starting with (a) the planar curve and ending with (f) the helix, both with $n=1$ and parameters $\xi =0.5$ and $\gamma =-500$.

Figure 17

Force and total bending energy of planar Euler elastica as a function of the separation of the ends, $\xi$, both increase monotonically as $\xi$ decreases.

Figure 18

Potential for $\mathcal{M}$ (a) positive and (b) negative. For $\chi >1$, there is only one minimum (black lines) and for $\chi <1$ two additional critical points are developed (black dots on gray lines).