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] Self-propelled enzyme motors and synthetic nanomotors also exhibit clustering effects in the form of chemotaxis. 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[7][8][9] and also synthetic chemotaxis[10][11][12] or phototaxis.[12] In addition to irreversible schooling, self-propelled particles also display reversible collective motion, such as predator–prey behavior and oscillatory clustering and dispersion.[13][14][15][16]


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][13][14][15][16] Clustering also occurs in enzyme molecule diffusion.[7][8][9][17] 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][13][14][15][16]

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.[18] 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.[19] 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.[20] This phase separation can, however, be arrested by chemically-mediated inter-particle torques[21] or hydrodynamic interactions,[22][23] which could explain the formation of finite-size clusters.

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,[24][25] 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][13][14][15][16] 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).[15][16]

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][4][13][14][15][16]


Clustering behavior in self-propelled particles and enzyme motors is discussed in great detail in sections on Collective Behavior, Chemotaxis, and/or Diffusiophoresis within several reviews by leading researchers in the self-propelled particles and nanomotors fields.[26][27][28][29][30][31][32]


  1. Theurkauff, I.; Cottin-Bizonne, C.; Palacci, J.; Ybert, C.; Bocquet, L. (26 June 2012). "Dynamic Clustering in Active Colloidal Suspensions with Chemical Signaling". Physical Review Letters. 108 (26): 268303. arXiv:1202.6264. Bibcode:2012PhRvL.108z8303T. doi:10.1103/PhysRevLett.108.268303. PMID 23005020.
  2. Buttinoni, Ivo; Bialké, Julian; Kümmel, Felix; Löwen, Hartmut; Bechinger, Clemens; Speck, Thomas (5 June 2013). "Dynamical Clustering and Phase Separation in Suspensions of Self-Propelled Colloidal Particles". Physical Review Letters. 110 (23): 238301. arXiv:1305.4185. Bibcode:2013PhRvL.110w8301B. doi:10.1103/PhysRevLett.110.238301. PMID 25167534.
  3. Palacci, Jeremie; Sacanna, Stefano; Steinberg, Asher Preska; Pine, David J.; Chaikin, Paul M. (31 January 2013). "Living Crystals of Light-Activated Colloidal Surfers". Science. 339 (6122): 936–40. Bibcode:2013Sci...339..936P. doi:10.1126/science.1230020. ISSN 0036-8075. PMID 23371555.
  4. Ibele, Michael; Mallouk, Thomas E.; Sen, Ayusman (20 April 2009). "Schooling Behavior of Light-Powered Autonomous Micromotors in Water". Angewandte Chemie. 121 (18): 3358–3362. doi:10.1002/ange.200804704. ISSN 1521-3757.
  5. Kagan, Daniel; Balasubramanian, Shankar; Wang, Joseph (10 January 2011). "Chemically Triggered Swarming of Gold Microparticles". Angewandte Chemie International Edition. 50 (2): 503–506. doi:10.1002/anie.201005078. ISSN 1521-3773. PMID 21140389.
  6. Wang, Wei; Castro, Luz Angelica; Hoyos, Mauricio; Mallouk, Thomas E. (24 July 2012). "Autonomous Motion of Metallic Microrods Propelled by Ultrasound". ACS Nano. 6 (7): 6122–6132. doi:10.1021/nn301312z. ISSN 1936-0851. PMID 22631222.
  7. Muddana, Hari S.; Sengupta, Samudra; Mallouk, Thomas E.; Sen, Ayusman; Butler, Peter J. (24 February 2010). "Substrate Catalysis Enhances Single-Enzyme Diffusion". Journal of the American Chemical Society. 132 (7): 2110–2111. doi:10.1021/ja908773a. ISSN 0002-7863. PMC 2832858. PMID 20108965.
  8. Sengupta, Samudra; Dey, Krishna K.; Muddana, Hari S.; Tabouillot, Tristan; Ibele, Michael E.; Butler, Peter J.; Sen, Ayusman (30 January 2013). "Enzyme Molecules as Nanomotors". Journal of the American Chemical Society. 135 (4): 1406–1414. doi:10.1021/ja3091615. ISSN 0002-7863. PMID 23308365.
  9. Dey, Krishna Kanti; Das, Sambeeta; Poyton, Matthew F.; Sengupta, Samudra; Butler, Peter J.; Cremer, Paul S.; Sen, Ayusman (23 December 2014). "Chemotactic Separation of Enzymes". ACS Nano. 8 (12): 11941–11949. doi:10.1021/nn504418u. ISSN 1936-0851. PMID 25243599.
  10. Pavlick, Ryan A.; Sengupta, Samudra; McFadden, Timothy; Zhang, Hua; Sen, Ayusman (26 September 2011). "A Polymerization-Powered Motor". Angewandte Chemie International Edition. 50 (40): 9374–9377. doi:10.1002/anie.201103565. ISSN 1521-3773. PMID 21948434.
  11. Hong, Yiying; Blackman, Nicole M. K.; Kopp, Nathaniel D.; Sen, Ayusman; Velegol, Darrell (26 October 2007). "Chemotaxis of Nonbiological Colloidal Rods". Physical Review Letters. 99 (17): 178103. Bibcode:2007PhRvL..99q8103H. doi:10.1103/PhysRevLett.99.178103. PMID 17995374.
  12. Chaturvedi, Neetu; Hong, Yiying; Sen, Ayusman; Velegol, Darrell (4 May 2010). "Magnetic Enhancement of Phototaxing Catalytic Motors". Langmuir. 26 (9): 6308–6313. doi:10.1021/la904133a. ISSN 0743-7463. PMID 20102166.
  13. Hong, Yiying; Diaz, Misael; Córdova-Figueroa, Ubaldo M.; Sen, Ayusman (25 May 2010). "Light-Driven Titanium-Dioxide-Based Reversible Microfireworks and Micromotor/Micropump Systems". Advanced Functional Materials. 20 (10): 1568–1576. doi:10.1002/adfm.201000063. ISSN 1616-3028.
  14. Ibele, Michael E.; Lammert, Paul E.; Crespi, Vincent H.; Sen, Ayusman (24 August 2010). "Emergent, Collective Oscillations of Self-Mobile Particles and Patterned Surfaces under Redox Conditions". ACS Nano. 4 (8): 4845–4851. doi:10.1021/nn101289p. ISSN 1936-0851. PMID 20666369.
  15. Duan, Wentao; Liu, Ran; Sen, Ayusman (30 January 2013). "Transition between Collective Behaviors of Micromotors in Response to Different Stimuli". Journal of the American Chemical Society. 135 (4): 1280–1283. doi:10.1021/ja3120357. ISSN 0002-7863. PMID 23301622.
  16. Altemose, Alicia; Sánchez-Farrán, Maria A.; Duan, Wentao; Schulz, Steve; Borhan, Ali; Crespi, Vincent H.; Sen, Ayusman (2017). "Chemically-Controlled Spatiotemporal Oscillations of Colloidal Assemblies". Angew. Chem. Int. Ed. 56 (27): 7817–7821. doi:10.1002/anie.201703239. PMID 28493638.
  17. Zhao, Xi; Palacci, Henri; Yadav, Vinita; Spiering, Michelle M.; Gilson, Michael K.; Butler, Peter J.; Hess, Henry; Benkovic, Stephen J.; Sen, Ayusman (18 December 2017). "Substrate-driven chemotactic assembly in an enzyme cascade". Nature Chemistry. 10 (3): 311–317. doi:10.1038/nchem.2905. ISSN 1755-4330. PMID 29461522.
  18. Ball, Philip (11 December 2013). "Focus: Particle Clustering Phenomena Inspire Multiple Explanations". Physics. 6. Retrieved 22 September 2015.
  19. Schnitzer, Mark J. (1 October 1993). "Theory of continuum random walks and application to chemotaxis". Physical Review E. 48 (4): 2553–2568. Bibcode:1993PhRvE..48.2553S. doi:10.1103/PhysRevE.48.2553. PMID 9960890.
  20. Cates, Michael E.; Tailleur, Julien (1 January 2015). "Motility-Induced Phase Separation". Annual Review of Condensed Matter Physics. 6 (1): 219–244. arXiv:1406.3533. Bibcode:2015ARCMP...6..219C. doi:10.1146/annurev-conmatphys-031214-014710.
  21. Pohl, Oliver; Stark, Holger (10 June 2014). "Dynamic Clustering and Chemotactic Collapse of Self-Phoretic Active Particles". Physical Review Letters. 112 (23): 238303. arXiv:1403.4063. Bibcode:2014PhRvL.112w8303P. doi:10.1103/PhysRevLett.112.238303. PMID 24972234.
  22. Matas-Navarro, Ricard; Golestanian, Ramin; Liverpool, Tanniemola B.; Fielding, Suzanne M. (18 September 2014). "Hydrodynamic suppression of phase separation in active suspensions". Physical Review E. 90 (3): 032304. arXiv:1210.5464. Bibcode:2014PhRvE..90c2304M. doi:10.1103/PhysRevE.90.032304. PMID 25314443.
  23. Zöttl, Andreas; Stark, Holger (18 March 2014). "Hydrodynamics Determines Collective Motion and Phase Behavior of Active Colloids in Quasi-Two-Dimensional Confinement". Physical Review Letters. 112 (11): 118101. arXiv:1309.4352. Bibcode:2014PhRvL.112k8101Z. doi:10.1103/PhysRevLett.112.118101. PMID 24702421.
  24. Redner, Gabriel S.; Baskaran, Aparna; Hagan, Michael F. (26 July 2013). "Reentrant phase behavior in active colloids with attraction". Physical Review E. 88 (1): 012305. arXiv:1303.3195. Bibcode:2013PhRvE..88a2305R. doi:10.1103/PhysRevE.88.012305. PMID 23944461.
  25. Mognetti, B. M.; Šarić, A.; Angioletti-Uberti, S.; Cacciuto, A.; Valeriani, C.; Frenkel, D. (11 December 2013). "Living Clusters and Crystals from Low-Density Suspensions of Active Colloids". Physical Review Letters. 111 (24): 245702. arXiv:1311.4681. Bibcode:2013PhRvL.111x5702M. doi:10.1103/PhysRevLett.111.245702. PMID 24483677.
  26. Sánchez, Samuel; Soler, Lluís; Katuri, Jaideep (26 January 2015). "Chemically Powered Micro- and Nanomotors". Angewandte Chemie International Edition. 54 (5): 1414–1444. doi:10.1002/anie.201406096. ISSN 1521-3773. PMID 25504117.
  27. Sengupta, Samudra; Ibele, Michael E.; Sen, Ayusman (20 August 2012). "Fantastic Voyage: Designing Self-Powered Nanorobots". Angewandte Chemie International Edition. 51 (34): 8434–8445. doi:10.1002/anie.201202044. ISSN 1521-3773. PMID 22887874.
  28. Duan, Wentao; Wang, Wei; Das, Sambeeta; Yadav, Vinita; Mallouk, Thomas E.; Sen, Ayusman (1 January 2015). "Synthetic Nano- and Micromachines in Analytical Chemistry: Sensing, Migration, Capture, Delivery, and Separation". Annual Review of Analytical Chemistry. 8 (1): 311–333. Bibcode:2015ARAC....8..311D. doi:10.1146/annurev-anchem-071114-040125. PMID 26132348.
  29. Wang, Wei; Duan, Wentao; Ahmed, Suzanne; Mallouk, Thomas E.; Sen, Ayusman (1 October 2013). "Small power: Autonomous nano- and micromotors propelled by self-generated gradients". Nano Today. 8 (5): 531–554. doi:10.1016/j.nantod.2013.08.009.
  30. Yadav, Vinita; Duan, Wentao; Butler, Peter J.; Sen, Ayusman (1 January 2015). "Anatomy of Nanoscale Propulsion". Annual Review of Biophysics. 44 (1): 77–100. doi:10.1146/annurev-biophys-060414-034216. PMID 26098511.
  31. Wang, Wei; Duan, Wentao; Ahmed, Suzanne; Sen, Ayusman; Mallouk, Thomas E. (21 July 2015). "From One to Many: Dynamic Assembly and Collective Behavior of Self-Propelled Colloidal Motors". Accounts of Chemical Research. 48 (7): 1938–1946. doi:10.1021/acs.accounts.5b00025. ISSN 0001-4842. PMID 26057233.
  32. Dey, Krishna Kanti; Wong, Flory; Altemose, Alicia; Sen, Ayusman (1 February 2016). "Catalytic Motors—Quo Vadimus?". Current Opinion in Colloid & Interface Science. 21: 4–13. doi:10.1016/j.cocis.2015.12.001.
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