Clustering of self-propelled particles

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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[edit]

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. ^ a b c d 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. ^ a b c d 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. ^ a b c d 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. ^ a b c d e 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. ^ a b 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. ^ a b 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. ^ a b 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. ^ a b 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. ^ a b Dey, Krishna Kanti; Das, Sambeeta; Poyton, Matthew F.; Sengupta, Samudra; Butler, Peter J.; Cremer, Paul S.; Sen, Ayusman (2014-12-23). "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. ^ a b 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. ^ a b c d e 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. ^ a b c d e 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. ^ a b c d e f 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. ^ a b c d e f 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 (2017-12-18). "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 (2013-12-11). "Focus: Particle Clustering Phenomena Inspire Multiple Explanations". Physics. 6. Retrieved 2015-09-22.
  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.
  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. (2015-07-21). "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 (2016-02-01). "Catalytic Motors—Quo Vadimus?". Current Opinion in Colloid & Interface Science. 21: 4–13. doi:10.1016/j.cocis.2015.12.001.