Tunable dynamics of flexible magnetic microcrosses: synchronous rotation, breathing and out-of-plane arm overtaking

TL;DR

This work introduces flexible magnetic microcrosses composed of four paramagnetic elastic arms attached to a central sphere, actuated by an in-plane rotating magnetic field. Combining soft-lithography fabrication with colloidal self-assembly and a four-filament theoretical model based on Kirchoff rod theory and resistive-force dynamics, the authors uncover tunable dynamical regimes: a planar breathing mode at modest field amplitudes and a 3D arm-overtaking gyroscopic rotation at higher amplitudes, with a critical frequency $f_c$ described by an Adler-like relation and a magnetoelastic number $Cm$ governing arm deformation. Experiments and simulations show that slight asymmetries in arm magnetization drive the transition from in-plane to out-of-plane motion, and the model successfully reproduces the key transitions and phase diagram in the $B$-$f$ parameter space. The crosses offer potential as programmable micro-stirrers and microrheology probes in microfluidic and biomedical contexts, with prospects for collective dynamics in dense suspensions. All mathematical relations are presented with explicit $...$ notation to support reproducibility and integration into broader analyses.

Abstract

We combine colloidal self-assembly and soft-lithography techniques to realize flexible magnetic microcrosses that can be manipulated via external, time dependent magnetic fields. The crosses are characterized by a central domain connected via four flexible arms. When subjected to an in-plane, rotating magnetic field, the crosses transit from a synchronous to an asynchronous spinning motion where their average rotation decreases with the driving frequency. In the asynchronous regime and at low field amplitudes, the crosses display a breathing mode, characterized by relative oscillations between the arms, while remaining localized in the two dimensional plane. In contrast, for high field amplitudes, we observe an arm overtaking regime where two opposite filaments surpass the remaining ones forcing the cross to perform a three-dimensional gyroscopic-like rotation. Using slender body theory and balancing the effect of magnetic and elastic interactions, we recover the experimental findings and show that the overtaking regime occurs due to different arm magnetizations. Our engineered microscopic colloidal rotors characterized by multiple flexible filaments may find potential applications for precise lab-on-a-chip operations or as stirrers dispersed within microfluidic or biological channels.

Figures (6)

• Figure 1: (a) Schematic showing the fabrication method to realize the flexible magnetic microcrosses, combining soft lithography and colloidal self-assembly. (b) Microscope image showing three separate microcrosses within the PDMS block after being filled by the suspension of paramagnetic colloids ($1 \, \rm{\mu m}$ diameter). (c) Image of one microcross with the embedded paramagnetic particles visible under an optical microscope.
• Figure 2: Position of the magnetic filament in the ($x,y$) plane when subjected to a static magnetic field $B$ and averaged over different microscope images. Scattered symbols are experimental data, continuous lines are non-linear regressions using Eq.34 in the Appendix. Bottom inset shows the image of one cross with a filament of length $L= 6.2\rm{\mu m}$ under a magnetic field $B=7$mT. The dashed line denotes the location of the starting point ($0,0$). Top inset shows the extracted magnetoelastic number $Cm$ versus field square $B^2$ with a regression, $Cm=\delta B^2$, where $\delta = 0.067 \pm 0.003 \, \rm{mT^{-2}}$.
• Figure 3: (a) Sequence of optical microscopy images showing a microcross rotating in the $(\hat{\bm{x}},\hat{\bm{y}})$ plane due to a circularly polarized, rotating magnetic field with amplitude $B=3$mT and frequency $f= 10$ Hz. The sequence proceeds left to right then down, and each image is $0.3$ seconds apart. (b) Normalized mean rotational frequency of the microcross, $\bar{f}_r/f$ as a function of the driving frequency $f$. Scattered symbols are experimental data, continuous lines are non linear regressions following Eq. \ref{['adler']} in the text. Inset shows the determined critical frequency $f_c$ versus square of the field amplitude $B^2$ with a linear regression.
• Figure 4: (a,c) Sequence of optical microscope images showing the arm overtaking (a, time interval between images $\Delta t = 0.75$s) and the breathing (c, $\Delta t = 0.05$s) regimes. Two nearest arms of the crosses are highlighted by a red and a green disk. (b) Diagram in the $(f,B)$ plane showing the regions where overtaking (red) and breathing (blue) regimes are observed. In the first one, the cross rotation is synchronous (pink-red region), while the second one occurs when the crosses rotate asynchronously with the field. The boundaries of the diagram are guides to eye to distinguish between the two regimes. The corresponding locations of the sequence of images in (a) and (c) are shown by the open black circles in the diagram. (d) Left: schematic showing a cross with four arms and the relative angles $\Delta \theta_{12}$ and $\Delta \theta_{13}$ between nearest and opposite filaments. Right: time evolution of the angles $\Delta \theta_{12}$ (blue) and $\Delta \theta_{13}$ (green) for the breathing (top) and overtaking (bottom) regimes obtained for rotating field with amplitude $B=5$mT and frequencies $f=5$Hz (breathing) and $f=0.1$Hz (overtaking).
• Figure 5: Simulation results: (a,c) Snapshots from numerical simulations showing the relaxation to synchronous out-of-plane ($B=17.3$ mT, $f=5.2$ Hz) (a) and asynchronous in-plane breathing ($B=13.4$mT , $f=5.2$ Hz) (c) regimes. The white arrows in the crosses indicate the magnetic field direction, and the time is written on each snapshot. The blue and green arms have a $10\%$ smaller $Cm$ than the red and yellow arms. See also Movie S3 for (c) and Movie S4 for (a) in the Supporting Information. (b) Diagram in the ($B,f$) plane showing the regions where the synchronous out-of-plane motion (pink) and the asynchronous in-plane breathing (light blue) is observed. The red crosses and the blue disks show simulation results. The dark red and dark blue arrows show which points correspond to the (a) and (c) plots. (d) Time evolution of the angle $\Delta\theta_{12}$ between two adjacent arms as seen from the direction of the field rotation for the two different regimes. The field rotates with $f=5.2$ Hz. The dashed line shows where this angle is $0$, and thus where the cross tilts as illustrated in (a). In both regimes the angle $\Delta\theta_{13}=180^{\circ}$ is constant up to machine precision, thus it is not shown. (e) Inverse period $T^{-1}$ of the oscillation of $\Delta \theta_{12}$ versus $B^2$ measured along the transition path given by the dashed line in graph (b). Scattered disks are simulation results while continuous line is a non linear regression using power law with exponent $\alpha= 0.39$. The inset shows the log-log plot of $T^{-1}$ versus $B_0^2-B^2$, where $B_0^2 = 261.5\rm{ mT^2}$. (f) The polar angle $\vartheta$ of the cross as a function of time for the synchronous out-of-plane regime ($B=17.3\rm{ mT}, f=5.2$Hz). Three different lines show the cases when all four arms have the same $Cm$ (blue), one opposite pair of arms has a $10\%$ smaller $Cm$, and the less magnetic pair is initially aligned with the field (orange, it is also illustrated in (a)), when one opposite pair of arms has a $10\%$ smaller $Cm$, but the more magnetic pair is now initially aligned with the field (green). The dashed line shows $\vartheta=90^\circ$, where the cross is aligned perpendicular to the field rotation plane. See also Movie S5 in the Supporting Information.