View source for Clustering of self-propelled particles

Many experimental realizations of [[Self-propelled particles|self-propelled particles]] exhibit a strong tendency to aggregate and form clusters,<ref name=":0">{{Cite journal|title = Dynamic Clustering in Active Colloidal Suspensions with Chemical Signaling|journal = Physical Review Letters|date = 26 June 2012|pages = 268303|volume = 108|issue = 26|doi = 10.1103/PhysRevLett.108.268303|first1 = I.|last1 = Theurkauff|first2 = C.|last2 = Cottin-Bizonne|first3 = J.|last3 = Palacci|first4 = C.|last4 = Ybert|first5 = L.|last5 = Bocquet|bibcode=2012PhRvL.108z8303T|pmid=23005020|arxiv = 1202.6264| s2cid=4890068 }}</ref><ref name=":1">{{Cite journal|title = Dynamical Clustering and Phase Separation in Suspensions of Self-Propelled Colloidal Particles|journal = Physical Review Letters|date = 5 June 2013|pages = 238301|volume = 110|issue = 23|doi = 10.1103/PhysRevLett.110.238301|first1 = Ivo|last1 = Buttinoni|first2 = Julian|last2 = Bialké|first3 = Felix|last3 = Kümmel| author4-link=Hartmut Löwen |first4 = Hartmut|last4 = Löwen|first5 = Clemens|last5 = Bechinger|first6 = Thomas|last6 = Speck|bibcode=2013PhRvL.110w8301B|pmid=25167534|arxiv = 1305.4185| s2cid=17127522 }}</ref><ref name=":2">{{Cite journal|title = Living Crystals of Light-Activated Colloidal Surfers|journal = Science|date = 31 January 2013|issn = 0036-8075|pmid = 23371555|pages = 936–40|doi = 10.1126/science.1230020|first1 = Jeremie|last1 = Palacci|first2 = Stefano|last2 = Sacanna|first3 = Asher Preska|last3 = Steinberg|first4 = David J.|last4 = Pine|first5 = Paul M.|last5 = Chaikin|volume=339|issue = 6122|bibcode = 2013Sci...339..936P| s2cid=1974474 }}</ref><ref name=":3">{{Cite journal|title = Schooling Behavior of Light-Powered Autonomous Micromotors in Water|journal = Angewandte Chemie|date = 20 April 2009|issn = 1521-3757|pages = 3358–3362|volume = 121|issue = 18|doi = 10.1002/ange.200804704|language = en|first1 = Michael|last1 = Ibele|first2 = Thomas E.|last2 = Mallouk|first3 = Ayusman|last3 = Sen| bibcode=2009AngCh.121.3358I }}</ref><ref name=":4">{{Cite journal|title = Chemically Triggered Swarming of Gold Microparticles|journal = Angewandte Chemie International Edition|date = 10 January 2011|issn = 1521-3773|pages = 503–506|volume = 50|issue = 2|doi = 10.1002/anie.201005078|language = en|first1 = Daniel|last1 = Kagan|first2 = Shankar|last2 = Balasubramanian|first3 = Joseph|last3 = Wang|pmid=21140389}}</ref> 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.<ref name=":5">{{Cite journal|title = Autonomous Motion of Metallic Microrods Propelled by Ultrasound|journal = ACS Nano|date = 24 July 2012|issn = 1936-0851|pages = 6122–6132|volume = 6|issue = 7|doi = 10.1021/nn301312z|pmid = 22631222|first1 = Wei|last1 = Wang|first2 = Luz Angelica|last2 = Castro|first3 = Mauricio|last3 = Hoyos|first4 = Thomas E.|last4 = Mallouk|url = https://figshare.com/articles/Autonomous_Motion_of_Metallic_Microrods_Propelled_by_Ultrasound/2502772}}</ref> Self-propelled enzyme motors and synthetic [[Engineering:Nanomotor|nanomotors]] also exhibit clustering effects in the form of chemotaxis. [[Physics:Chemotaxis|Chemotaxis]] is a form of [[Collective motion|collective motion]] of biological or non-biological particles toward a fuel source or away from a threat, as observed experimentally in enzyme diffusion<ref name=":6">{{Cite journal|title = Substrate Catalysis Enhances Single-Enzyme Diffusion|journal = Journal of the American Chemical Society|date = 24 February 2010|issn = 0002-7863|pmc = 2832858|pmid = 20108965|pages = 2110–2111|volume = 132|issue = 7|doi = 10.1021/ja908773a|first1 = Hari S.|last1 = Muddana|first2 = Samudra|last2 = Sengupta|first3 = Thomas E.|last3 = Mallouk|first4 = Ayusman|last4 = Sen|first5 = Peter J.|last5 = Butler}}</ref><ref name=":7">{{Cite journal|title = Enzyme Molecules as Nanomotors|journal = Journal of the American Chemical Society|date = 30 January 2013|issn = 0002-7863|pages = 1406–1414|volume = 135|issue = 4|doi = 10.1021/ja3091615|pmid = 23308365|first1 = Samudra|last1 = Sengupta|first2 = Krishna K.|last2 = Dey|first3 = Hari S.|last3 = Muddana|first4 = Tristan|last4 = Tabouillot|first5 = Michael E.|last5 = Ibele|first6 = Peter J.|last6 = Butler|first7 = Ayusman|last7 = Sen}}</ref><ref name=":11">{{Cite journal|last1=Dey|first1=Krishna Kanti|last2=Das|first2=Sambeeta|last3=Poyton|first3=Matthew F.|last4=Sengupta|first4=Samudra|last5=Butler|first5=Peter J.|last6=Cremer|first6=Paul S.|last7=Sen|first7=Ayusman|date=2014-12-23|title=Chemotactic Separation of Enzymes|journal=ACS Nano|volume=8|issue=12|pages=11941–11949|doi=10.1021/nn504418u|pmid=25243599|issn=1936-0851|doi-access=free}}</ref> and also synthetic chemotaxis<ref>{{Cite journal|title = A Polymerization-Powered Motor|journal = Angewandte Chemie International Edition|date = 26 September 2011|issn = 1521-3773|pages = 9374–9377|volume = 50|issue = 40|doi = 10.1002/anie.201103565|language = en|first1 = Ryan A.|last1 = Pavlick|first2 = Samudra|last2 = Sengupta|first3 = Timothy|last3 = McFadden|first4 = Hua|last4 = Zhang|first5 = Ayusman|last5 = Sen|pmid=21948434| s2cid=6325323 }}</ref><ref>{{Cite journal|title = Chemotaxis of Nonbiological Colloidal Rods|journal = Physical Review Letters|date = 26 October 2007|pages = 178103|volume = 99|issue = 17|doi = 10.1103/PhysRevLett.99.178103|first1 = Yiying|last1 = Hong|first2 = Nicole M. K.|last2 = Blackman|first3 = Nathaniel D.|last3 = Kopp|first4 = Ayusman|last4 = Sen|first5 = Darrell|last5 = Velegol|bibcode=2007PhRvL..99q8103H|pmid=17995374}}</ref><ref name="Chaturvedi 6308–6313">{{Cite journal|title = Magnetic Enhancement of Phototaxing Catalytic Motors|journal = Langmuir|date = 4 May 2010|issn = 0743-7463|pages = 6308–6313|volume = 26|issue = 9|doi = 10.1021/la904133a|pmid = 20102166|first1 = Neetu|last1 = Chaturvedi|first2 = Yiying|last2 = Hong|first3 = Ayusman|last3 = Sen|first4 = Darrell|last4 = Velegol}}</ref> or phototaxis.<ref name="Chaturvedi 6308–6313"/> In addition to irreversible schooling, [[Self-propelled particles|self-propelled particles]] also display reversible collective motion, such as predator–prey behavior and oscillatory clustering and dispersion.<ref name=":8">{{Cite journal|title = Light-Driven Titanium-Dioxide-Based Reversible Microfireworks and Micromotor/Micropump Systems|journal = Advanced Functional Materials|date = 25 May 2010|issn = 1616-3028|pages = 1568–1576|volume = 20|issue = 10|doi = 10.1002/adfm.201000063|language = en|first1 = Yiying|last1 = Hong|first2 = Misael|last2 = Diaz|first3 = Ubaldo M.|last3 = Córdova-Figueroa|first4 = Ayusman|last4 = Sen| s2cid=51990054 }}</ref><ref name=":9">{{Cite journal|title = Emergent, Collective Oscillations of Self-Mobile Particles and Patterned Surfaces under Redox Conditions|journal = ACS Nano|date = 24 August 2010|issn = 1936-0851|pages = 4845–4851|volume = 4|issue = 8|doi = 10.1021/nn101289p|pmid = 20666369|first1 = Michael E.|last1 = Ibele|first2 = Paul E.|last2 = Lammert|first3 = Vincent H.|last3 = Crespi|first4 = Ayusman|last4 = Sen|url = https://figshare.com/articles/Emergent_Collective_Oscillations_of_Self_Mobile_Particles_and_Patterned_Surfaces_under_Redox_Conditions/2740561}}</ref><ref name=":10">{{Cite journal|title = Transition between Collective Behaviors of Micromotors in Response to Different Stimuli|journal = Journal of the American Chemical Society|date = 30 January 2013|issn = 0002-7863|pages = 1280–1283|volume = 135|issue = 4|doi = 10.1021/ja3120357|first1 = Wentao|last1 = Duan|first2 = Ran|last2 = Liu|first3 = Ayusman|last3 = Sen|pmid=23301622}}</ref><ref name=":12">{{Cite journal|last1=Altemose|first1=Alicia|last2=Sánchez-Farrán|first2=Maria A.|last3=Duan|first3=Wentao|last4=Schulz|first4=Steve|last5=Borhan|first5=Ali|last6=Crespi|first6=Vincent H.|last7=Sen|first7=Ayusman|date=2017|title=Chemically-Controlled Spatiotemporal Oscillations of Colloidal Assemblies|journal=Angew. Chem. Int. Ed.|volume=56|issue=27|pages=7817–7821|doi=10.1002/anie.201703239|pmid=28493638|doi-access=}}</ref><ref>{{Cite journal |last1=Zhang |first1=Jianhua |last2=Laskar |first2=Abhrajit |last3=Song |first3=Jiaqi |last4=Shklyaev |first4=Oleg E. |last5=Mou |first5=Fangzhi |last6=Guan |first6=Jianguo |last7=Balazs |first7=Anna C. |last8=Sen |first8=Ayusman |date=2023-01-10 |title=Light-Powered, Fuel-Free Oscillation, Migration, and Reversible Manipulation of Multiple Cargo Types by Micromotor Swarms |url=https://pubs.acs.org/doi/10.1021/acsnano.2c07266 |journal=ACS Nano |language=en |volume=17 |issue=1 |pages=251–262 |doi=10.1021/acsnano.2c07266 |pmid=36321936 |s2cid=253257444 |issn=1936-0851}}</ref>

== Phenomenology ==
This clustering behavior has been observed for self-propelled [[Physics:Janus particles|Janus particles]], either platinum-coated gold particles<ref name=":0" /> or carbon-coated silica beads,<ref name=":1" /> and for magnetically or ultrasonically powered particles.<ref name=":4" /><ref name=":5" /> Clustering has also been observed for colloidal particles composed of either an embedded hematite cube<ref name=":2" /> or slowly-diffusing metal ions.<ref name=":3" /><ref name=":8" /><ref name=":9" /><ref name=":10" /><ref name=":12" /> Autonomous aggregation has also been observed in anatase TiO<sub>2</sub> (titanium dioxide) particles.<ref>{{Cite journal|date=2021-02-26|title=Titania-Based Micro/Nanomotors: Design Principles, Biomimetic Collective Behavior, and Applications|journal=Trends in Chemistry|language=en|doi=10.1016/j.trechm.2021.02.001|issn=2589-5974|doi-access=free|last1=Zhang |first1=Jianhua |last2=Song |first2=Jiaqi |last3=Mou |first3=Fangzhi |last4=Guan |first4=Jianguo |last5=Sen |first5=Ayusman |volume=3 |issue=5 |pages=387–401 }}</ref> Clustering also occurs in enzyme molecule diffusion.<ref name=":6" /><ref name=":7" /><ref name=":11" /><ref>{{Cite journal|last1=Zhao|first1=Xi|last2=Palacci|first2=Henri|last3=Yadav|first3=Vinita|last4=Spiering|first4=Michelle M.|last5=Gilson|first5=Michael K.|last6=Butler|first6=Peter J.|last7=Hess|first7=Henry|last8=Benkovic|first8=Stephen J.|last9=Sen|first9=Ayusman|date=2017-12-18|title=Substrate-driven chemotactic assembly in an enzyme cascade|journal=Nature Chemistry|language=En|volume=10|issue=3|pages=311–317|doi=10.1038/nchem.2905|pmid=29461522|issn=1755-4330|bibcode=2018NatCh..10..311Z}}</ref> Recently, enzymes such as hexokinase and alkaline phosphatase were found to aggregate in the presence of their substrates.<ref>{{Cite journal |last1=Gentile |first1=Kayla |last2=Bhide |first2=Ashlesha |last3=Kauffman |first3=Joshua |last4=Ghosh |first4=Subhadip |last5=Maiti |first5=Subhabrata |last6=Adair |first6=James |last7=Lee |first7=Tae-Hee |last8=Sen |first8=Ayusman |date=2021-09-22 |title=Enzyme aggregation and fragmentation induced by catalysis relevant species |url=https://pubs.rsc.org/en/content/articlelanding/2021/cp/d1cp02966e |journal=Physical Chemistry Chemical Physics |language=en |volume=23 |issue=36 |pages=20709–20717 |doi=10.1039/D1CP02966E |pmid=34516596 |bibcode=2021PCCP...2320709G |s2cid=237507756 |issn=1463-9084}}</ref> 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<ref name=":0" /> while at larger area fractions (30% or higher), a complete phase separation has been reported.<ref name=":1" /> 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".<ref name=":2" /><ref name=":3" /><ref name=":8" /><ref name=":9" /><ref name=":10" /><ref name=":12" />

== 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.<ref>{{Cite journal|title = Focus: Particle Clustering Phenomena Inspire Multiple Explanations|journal = Physics|volume = 6|url = https://physics.aps.org/articles/v6/134|access-date = 2015-09-22|date = 2013-12-11|last1 = Ball|first1 = Philip| page=134 |doi = 10.1103/physics.6.134}}</ref> 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.<ref>{{Cite journal|title = Theory of continuum random walks and application to chemotaxis|journal = Physical Review E|date = 1 October 1993|pages = 2553–2568|volume = 48|issue = 4|doi = 10.1103/PhysRevE.48.2553|pmid = 9960890|first = Mark J.|last = Schnitzer|bibcode = 1993PhRvE..48.2553S}}</ref> 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.<ref>{{Cite journal|title = Motility-Induced Phase Separation|journal = Annual Review of Condensed Matter Physics|date = 1 January 2015|pages = 219–244|volume = 6|issue = 1|doi = 10.1146/annurev-conmatphys-031214-014710|first1 = Michael E.|last1 = Cates|first2 = Julien|last2 = Tailleur|arxiv = 1406.3533|bibcode = 2015ARCMP...6..219C| s2cid=15672131 }}</ref> This phase separation can, however, be arrested by chemically-mediated inter-particle torques<ref>{{Cite journal|title = Dynamic Clustering and Chemotactic Collapse of Self-Phoretic Active Particles|journal = Physical Review Letters|date = 10 June 2014|pages = 238303|volume = 112|issue = 23|doi = 10.1103/PhysRevLett.112.238303|first1 = Oliver|last1 = Pohl|first2 = Holger|last2 = Stark|bibcode=2014PhRvL.112w8303P|pmid=24972234|arxiv = 1403.4063| s2cid=15305058 }}</ref> or hydrodynamic interactions,<ref>{{Cite journal|title = Hydrodynamic suppression of phase separation in active suspensions|journal = Physical Review E|date = 18 September 2014|pages = 032304|volume = 90|issue = 3|doi = 10.1103/PhysRevE.90.032304|pmid = 25314443|first1 = Ricard|last1 = Matas-Navarro|first2 = Ramin|last2 = Golestanian|first3 = Tanniemola B.|last3 = Liverpool|first4 = Suzanne M.|last4 = Fielding|arxiv = 1210.5464|bibcode = 2014PhRvE..90c2304M| s2cid=34233710 }}</ref><ref>{{Cite journal|title = Hydrodynamics Determines Collective Motion and Phase Behavior of Active Colloids in Quasi-Two-Dimensional Confinement|journal = Physical Review Letters|date = 18 March 2014|pages = 118101|volume = 112|issue = 11|doi = 10.1103/PhysRevLett.112.118101|first1 = Andreas|last1 = Zöttl|first2 = Holger|last2 = Stark|bibcode=2014PhRvL.112k8101Z|pmid=24702421|arxiv = 1309.4352| s2cid=12399192 }}</ref> 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,<ref>{{Cite journal|title = Reentrant phase behavior in active colloids with attraction|journal = Physical Review E|date = 26 July 2013|pages = 012305|volume = 88|issue = 1|doi = 10.1103/PhysRevE.88.012305|pmid = 23944461|first1 = Gabriel S.|last1 = Redner|first2 = Aparna|last2 = Baskaran|first3 = Michael F.|last3 = Hagan|bibcode = 2013PhRvE..88a2305R|arxiv = 1303.3195| s2cid=6919624 }}</ref><ref>{{Cite journal|title = Living Clusters and Crystals from Low-Density Suspensions of Active Colloids|journal = Physical Review Letters|date = 11 December 2013|pages = 245702|volume = 111|issue = 24|doi = 10.1103/PhysRevLett.111.245702|first1 = B. M.|last1 = Mognetti|first2 = A.|last2 = Šarić|first3 = S.|last3 = Angioletti-Uberti|first4 = A.|last4 = Cacciuto|first5 = C.|last5 = Valeriani|first6 = D.|last6 = Frenkel|bibcode=2013PhRvL.111x5702M|pmid=24483677|arxiv = 1311.4681| s2cid=46111375 }}</ref> 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.<ref name=":3" /><ref name=":8" /><ref name=":9" /><ref name=":10" /><ref name=":12" /> 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).<ref name=":10" /><ref name=":12" />

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.<ref name=":1" /> For other experimental systems, however, attractive forces often play a larger role.<ref name=":0" /><ref name=":2" /><ref name=":10" /><ref name=":12" />

== See also ==
* [[Active Brownian particle]]

==References==
{{reflist|32em}}

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[[Category:Multi-agent systems]]

{{Sourceattribution|Clustering of self-propelled particles}}