Windmilling clusters of active quadrupoles

TL;DR

This study addresses pattern formation with orthogonal alignment in active matter by coupling active Brownian propulsion to magnetic quadrupoles on dumbbell-shaped particles. Using overdamped 2D Brownian Dynamics and a detailed dipole-dipole interaction framework, the authors map how activity and quadrupolar attraction compete to generate diverse microstructures, including a ground-state triangular motif at $N=3$ and windmill-like rotating clusters arising from polarity. The phase behavior reveals four regimes—Active-gaseous, magnetically dominated, and two triangular-motif phases—with a pronounced peak at cluster size $n=3$ in the distribution $p(n)$, tunable by the magnetic coupling $\\lambda$ and Péclet number $\\text{Pe}$ and modulated by density $\\phi$. These insights demonstrate a controllable spectrum of microstructures, offering potential experimental realization and avenues for magnetic-field-directed self-assembly in active systems.

Abstract

Active matter has thrived in recent years, driven both by the insight that it underlies fundamental processes in nature, and by its vast potential for applications. This allows for innovation both inspired by experimental observations, and by construction of novel systems with desired properties. In this paper, we develop a novel system in the search for a new kind of pattern formation: microstructural motifs with orthogonal alignment. Taking a simple active Brownian particle (ABP) model applied to dumbbell-shaped particles, we add a quadrupolar interaction by positioning two antiparallel magnetic dipolar moments on each particle. We find that the phase behavior is determined by the competition between active motion and the orthogonal alignment favored by quadrupolar attraction. By varying these quantities, we are able to tune both the internal structure of the aggregates, and find a surprising stability of triangular aggregates, to the point of clusters of size $N=3$ being strongly overrepresented. Although none of the component particles are chiral, the resulting structures spin in a random, fixed direction due to combination of the polarity of the active motion. This results in an ensemble of windmilling (randomly spinning in a circular motion) aggregates with windmill-like shape (due to the three- or four core component dumbbells). Ultimately, this simple model shows an interesting range of microstructural motifs, with great potential for experimental implementations.

Figures (6)

• Figure 1: Left to right: one, two, three and four quadrupolar active particles in their respective ground states. Grey denotes the steric repulsive particle shape, while the orientation of the two magnetic dipoles is shown in blue and the direction of the activity is shown in orange. To aid the eye in distinguishing which particles belong together, an additional black circle is added behind the connection point where the two dumbbell components meet.
• Figure 2: Simulation snapshots from a quadrupolar dumbbell system with Péclet=0 and $\lambda =10$, for densities $\phi =0.05$ (left) and $\phi =0.3$ (right).
• Figure 3: The phase diagrams of active quadrupolar dumbbells, in dependence of the magnetic coupling constant $\lambda$ and the Péclet number at different area fractions $\phi$. We distinguish between 4 key phases. Active-gaseous, in green, shows little aggregation except at high densities and no characteristic structures or ordering. Magnetically dominated, in red, shows aggregation typical of magnetic systems, albeit with what seems to be more orthogonal alignment between particles. At intermediate values of $\lambda$ and the Péclet number, we find phases with more triangular motifs: either triangular-active (blue), which most closely resembles the active-gaseous system except for a preponderance of triangles, and triangular-magnetic (orange), which similarly features a more magnetic aggregation pattern, except for a peak in aggregation at $n=3$. Representative simulation snapshots are inset at a few key values: in these cases, the frame color denotes the phase. The dumbbells inside these snapshots are colored based on the number of particles in their aggregate, with a coloring pattern that repeats for large $n$, shown with an additional fictitious connecting particle in the middle as a visual aid. It should be noted that the snapshots cannot fully represent the system due to size constraints, and many of the clusters shown are truncated.
• Figure 4: The percentage of particles in a cluster of size $n$, truncated at $n=20$ and $n=30$ in order to highlight the $n=3$ behavior. Colors also correspond to the phases indicated in Figure \ref{['fig:phase']}. Top: $\phi = 0.05$, Bottom: $\phi =0.3$. Note the difference in $p(n)$ axis, as at high densities, many particles are contained within less frequent, larger clusters.
• Figure 5: Histograms showing the count $p(\theta)$ of angles $\theta$ occurring within a cluster of size 2 (top), 3 (middle) and 4 (lowest), normalized by the count of angles. For pairs, we see characteristic angles close to 90, and a symmetry with respect to 90 degrees. For the triangles, we see the combined effect of activity in the break of symmetry, although much less so for the magnetically dominated systems. Finally, for the larger aggregates, we see the effect of increasing magnetic strength. While the aggregation distribution of triangle-magnetic at high Péclet follows more magnetic patterns, the internal structuring of the clusters is still altered by the activity.