Clustering of self-propelled particles

Many experimental realizations of self-propelled particles exhibit a strong tendency to aggregate and form clusters,^{[1][2][3][4][5]} whose dynamics are much richer than those of passive colloids. These aggregates of particles form for a variety of reasons, from chemical gradients to magnetic and ultrasonic fields.^[6][7] Self-propelled enzyme motors and synthetic nanomotors also exhibit clustering effects in the form of chemotaxis.^[7][8] Chemotaxis is a form of collective motion of biological or non-biological particles toward a fuel source or away from a threat, as observed experimentally in enzyme diffusion^[9][10][11] and also synthetic chemotaxis^[12][13][14] or phototaxis.^[14] In addition to irreversible schooling, self-propelled particles also display reversible collective motion, such as predator–prey behavior and oscillatory clustering and dispersion.^{[15][16][17][18][19][20]}

Phenomenology

This clustering behavior has been observed for self-propelled Janus particles, either platinum-coated gold particles^[1] or carbon-coated silica beads,^[2] and for magnetically or ultrasonically powered particles.^[5][6] Clustering has also been observed for colloidal particles composed of either an embedded hematite cube^[3] or slowly-diffusing metal ions.^{[4][15][16][17][18]} Autonomous aggregation has also been observed in anatase TiO₂ (titanium dioxide) particles.^[21] Clustering also occurs in enzyme molecule diffusion.^{[9][10][11][22]} Recently, enzymes such as hexokinase and alkaline phosphatase were found to aggregate in the presence of their substrates.^[23] In all these experiments, the motion of particles takes place on a two-dimensional surface and clustering is seen for area fractions as low as 10%. For such low area fractions, the clusters have a finite mean size^[1] while at larger area fractions (30% or higher), a complete phase separation has been reported.^[2] The dynamics of the finite-size clusters are very rich, exhibiting either crystalline order or amorphous packing. The finite size of the clusters comes from a balance between attachment of new particles to pre-existing clusters and breakdown of large clusters into smaller ones, which has led to the term "living clusters".^{[3][4][15][16][17][18]}

Mechanism for synthetic systems

The precise mechanism leading to the appearance of clusters is not completely elucidated and is a current field of research for many systems.^[24] A few different mechanisms have been proposed, which could be at play in different experimental setups.

Self-propelled particles can accumulate in a region of space where they move with a decreased velocity.^[25] After accumulation, in regions of high particle density, the particles move more slowly because of steric hindrance. A feedback between these two mechanisms can lead to the so-called motility induced phase separation.^[26] This phase separation can, however, be arrested by chemically-mediated inter-particle torques^[27] or hydrodynamic interactions,^[28][29] which could explain the formation of finite-size clusters. Self-propelled particles spinning at intermediate Reynolds numbers exhibit flocking, chiral active phases, and clustering, highlighting the role of fluid inertia and rotational motion in active matter behavior.^[30]

Alternatively, clustering and phase-separation could be due to the presence of inter-particle attractive forces, as in equilibrium suspensions. Active forces would then oppose this phase separation by pulling apart the particles in the cluster,^[31][32] following two main processes. First, single particles can exist independently if their propulsion forces are sufficient to escape from the cluster. Secondly, a large cluster can break into smaller pieces due to the build-up of internal stress: as more and more particles enter the cluster, their propulsive forces add up until they break down its cohesion.

Diffusiophoresis is also a commonly cited mechanism for clustering and collective behavior, involving the attraction or repulsion of particles to each other in response to ion gradients.^{[4][15][16][17][18]} Diffusiophoresis is a process involving the gradients of electrolyte or non-electrolyte concentrations interacting with charged (electrophoretic interactions) or neutral (chemophoretic interactions) particles in solution and with the double layer of any walls or surfaces (electroosmotic interactions).^[17][18]

In experiments, arguments have been put forward in favor of any of the above mechanisms. For carbon-coated silica beads, attractive interactions are seemingly negligible and phase-separation is indeed seen at large densities.^[2] For other experimental systems, however, attractive forces often play a larger role.^{[1][3][17][18]}

See also

• Active Brownian particle – Model of self-propelled motion in a dissipative environment
• Self-propelled particles – Type of autonomous agent