Quantifying hydrodynamic collective states of magnetic colloidal spinners and rollers

Ferromagnetic microparticles energized by an alternating magnetic field exhibit fascinating collective behavior ranging from the emergent self-assembled spinners to a variety of self-organized rolling states. Despite their simplicity, quantifying their essentially multibody collective behavior remains elusive due to a multitude of relevant interactions, from short-range collisions to long-range magnetic and hydrodynamic forces. Here we develop a high-performance computational algorithm based on smoothed particle hydrodynamics to quantify the role of individual interactions in the emergent collective state. The computational model provides insight into the role of hydrodynamic interaction on the onset of collective behavior and allows characterization of dynamic regimes that are hard to access experimentally. Comparison with high-resolution experimental data allows validation of the algorithm. Our work expands the scope of modern computational tools for predictive modeling of microscopic active systems and provides insight into the intricate role of hydrodynamic interactions on the onset of collective behavior in living and synthetic active matter.

Figure 1

Schematics of the experimental setups. (a) In the 2D scenario, magnetic floaters are suspended at the water-air interface, energized by an in-plane ac magnetic field [6]. (b) In the 3D scenario, magnetic rollers are at the bottom of a concave container, energized by a vertical ac magnetic field [4].

Figure 2

Comparison of experiment [5, 6] and numerical simulations of the SPH model for 2D spinners and wires. For spinners, streamlines are shown from (a) experiment and (b) simulations and the fluid velocity magnitude from (c) experiment and (d) simulations. (e) Spinners with the fluid velocity vector field from simulations. (f) Wires with the fluid velocity vector field from simulations.

Figure 3

(a) Imbalance in the number of spinners with different sense of rotation as a function of time. Within the reported 2000 periods, the spinner imbalance parameter fluctuates significantly. (b) Histogram of the spinner imbalance parameter values. Both plots indicate the absence of directional synchronization between spinners.

Figure 4

(a) Single-particle trajectory and (b) fluid velocity vector field generated by a single particle.

Figure 5

Flocking behavior obtained in a system of magnetic rollers. The direction of particle motion is color coded in visualization using the following rule: $\left[\mathrm{red},\mathrm{green},\mathrm{blue}\right]=\left\{\right[1+sin\left(\phi \right)]/2,(1+cos\left(\phi \right)]/2,0\}$, where $\phi$ is the angle of the in-plane velocity vector with respect to the $x$ axis. Thus, a group of particle with the same color identifies one flock. (a) Computational results obtained for frequency $f=35$ Hz and magnitude $H_0=50$ Oe. (b) Experimental result for $f=50$ Hz. Correlated motion of intermittent flocks of particles is illustrated by color patches.

Figure 6

Vortex obtained with the vertical ac magnetic field with frequency $f=40$ Hz and magnitude $H_0=50$ Oe. (a) Fluid velocity magnitude (experiment). (b) Fluid velocity magnitude superimposed over the fluid vector field (computation).

Figure 7

(a) Velocity spatial correlation functions for different phases as obtained from simulations; $a$ stands for the particle radius. (b) Rotational order parameter ${\varphi }_R$ for different phases (simulations).

Figure 8

(a) Comparison of the vortex velocity profiles versus distance from the center; the solid line corresponds to the experimental data and squares correspond the simulations. (b) Ratio between the peak velocity $V_p$ and normalized particle velocity $V_0=\omega a$, which is used to determine the slipping parameter ${\alpha }_s$; experimental data are shown by diamonds and simulation data by circles.

Figure 9

(a) Vertical center of mass position of a colloidal roller vs time; $D_c$ and $R$ stand for the location and radius of a colloid, respectively. (b) In-plane streamlines and (c) vertical cross section of the fluid velocity field in the vortex phase. The cross section is taken through the eye of the vortex.

Figure 10

Time evolution of the roller vortex horizontal center of mass from (a) experiment and (b) simulations. Blue and red lines correspond to $x$ and $y$ components of vortex center of mass, respectively. Also shown is Fourier analysis of the vortex eye oscillations for (c) experiment and (d) simulations.

Figure 11

Stroboscopic plots of the vertical components of the colloids' magnetic orientation $cos\left(\theta \right)$ in three different dynamic phases. The angle $\theta$ is the angle between the particle magnetic moments and the vertical direction. Different colors correspond to a different (a) gas, (b) low-frequency flocking, and (c) vortex.