{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T05:01:38Z","timestamp":1785301298246,"version":"3.55.0"},"reference-count":69,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T00:00:00Z","timestamp":1784851200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Micromachines"],"abstract":"<jats:p>The dynamics of active particles are increasingly being leveraged to design and control micro-robotic swarms. Local interactions play a crucial role in the phase transitions of active particles; how the combined effects of alignment, short-range repulsion, and boundary interactions regulate their packing structure and collective order with different confinement scales remains less systematically explored. In this study, we investigate the packing of active particles within a confined region, focusing on the role of local interaction rules in shaping both the packing structure and the polar order parameter. The effects of key controlling variables related to local interaction rules, including interaction radius, repulsion radius, confined boundary radius, and noise strength, are numerically studied. Specifically, by comparing systems with and without velocity\u2013alignment interactions, we reveal the role of alignment in dictating both structural and dynamical properties of the ensemble. To quantify the packing structure, we employ Voronoi tessellation to evaluate both local and global packing densities. The results show that strong confinement induces a jammed state in which alignment effects are suppressed, resulting in high global packing density and low polar order, regardless of the noise amplitude. Upon increasing the boundary radius beyond a critical threshold, the system unjams, enabling alignment interactions to significantly enhance both the polar order parameter and packing density. Interestingly, the relationship between global packing density and micro-structural parameters, such as coordination number and Voronoi tessellation metrics, is similar in the systems with and without alignment. Our results demonstrate that collective packing and phase behaviour of active matter are governed by the nontrivial interplay between alignment, confinement, and noise, with alignment interactions driving the transition from disordered to ordered states as geometric constraints are relaxed, offering critical insights for the design of targeted micro-robotic swarms and active microfluidic sorting systems.<\/jats:p>","DOI":"10.3390\/mi17080884","type":"journal-article","created":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T16:11:56Z","timestamp":1785168716000},"page":"884","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Effect of Velocity Alignment on the Packing of Active Particles"],"prefix":"10.3390","volume":"17","author":[{"given":"Jigarkumar","family":"Modi","sequence":"first","affiliation":[{"name":"School of Engineering, Western Sydney University, Sydney 2747, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ruizhi","family":"Jin","sequence":"additional","affiliation":[{"name":"School of Engineering, Western Sydney University, Sydney 2747, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0520-1739","authenticated-orcid":false,"given":"Kejun","family":"Dong","sequence":"additional","affiliation":[{"name":"School of Engineering, Western Sydney University, Sydney 2747, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3723-1665","authenticated-orcid":false,"given":"Gu","family":"Fang","sequence":"additional","affiliation":[{"name":"School of Engineering, Western Sydney University, Sydney 2747, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.physrep.2012.03.004","article-title":"Collective motion","volume":"517","author":"Vicsek","year":"2012","journal-title":"Phys. Rep."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1938","DOI":"10.1021\/acs.accounts.5b00025","article-title":"From one to many: Dynamic assembly and collective behavior of self-propelled colloidal motors","volume":"48","author":"Wang","year":"2015","journal-title":"Acc. Chem. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"7666","DOI":"10.1021\/jacs.7b02347","article-title":"Chemically propelled molecules and machines","volume":"139","author":"Dey","year":"2017","journal-title":"J. Am. Chem. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.physrep.2007.02.018","article-title":"Active behavior of the cytoskeleton","volume":"449","author":"Juelicher","year":"2007","journal-title":"Phys. Rep."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"098103","DOI":"10.1103\/PhysRevLett.93.098103","article-title":"Self-concentration and large-scale coherence in bacterial dynamics","volume":"93","author":"Dombrowski","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"10813","DOI":"10.1073\/pnas.0702027104","article-title":"Regulation of the protein disulfide proteome by mitochondria in mammalian cells","volume":"104","author":"Yang","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"15988","DOI":"10.1073\/pnas.0705062104","article-title":"Collective migration of an epithelial monolayer in response to a model wound","volume":"104","author":"Poujade","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Frewen, T.A., Couzin, I.D., Kolpas, A., Moehlis, J., Coifman, R., and Kevrekidis, I.G. (2011). Coarse Collective Dynamics of Animal Groups, Springer.","DOI":"10.1007\/978-3-642-14941-2_16"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"054002","DOI":"10.1088\/1742-5468\/aa6bc5","article-title":"Active matter","volume":"2017","author":"Ramaswamy","year":"2017","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1038\/s42254-022-00445-3","article-title":"Topological active matter","volume":"4","author":"Shankar","year":"2022","journal-title":"Nat. Rev. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"045006","DOI":"10.1103\/RevModPhys.88.045006","article-title":"Active particles in complex and crowded environments","volume":"88","author":"Bechinger","year":"2016","journal-title":"Rev. Mod. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1226","DOI":"10.1103\/PhysRevLett.75.1226","article-title":"Novel type of phase transition in a system of self-driven particles","volume":"75","author":"Vicsek","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"040301","DOI":"10.1103\/PhysRevE.84.040301","article-title":"Active jamming: Self-propelled soft particles at high density","volume":"84","author":"Henkes","year":"2011","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear Soft Matter Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"026113","DOI":"10.1103\/PhysRevE.79.026113","article-title":"Consensus of self-driven agents with avoidance of collisions","volume":"79","author":"Peng","year":"2009","journal-title":"Phys. Rev. E\u2014Stat. Nonlinear Soft Matter Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"129307","DOI":"10.1016\/j.physleta.2024.129307","article-title":"Cohesion and polarization of active agent with visual perception","volume":"495","author":"Meng","year":"2024","journal-title":"Phys. Lett. A"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Degond, P., Dimarco, G., Mac, T.B.N., and Wang, N. (2014). Macroscopic models of collective motion with repulsion. arXiv.","DOI":"10.4310\/CMS.2015.v13.n6.a12"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"064122","DOI":"10.1103\/9rdd-tbn7","article-title":"Role of random interaction connection in the order transition of active matter based on the Vicsek model","volume":"111","author":"Jin","year":"2025","journal-title":"Phys. Rev. E"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1146\/annurev-conmatphys-031214-014710","article-title":"Motility-induced phase separation","volume":"6","author":"Cates","year":"2015","journal-title":"Annu. Rev. Condens. Matter Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2064","DOI":"10.1103\/PhysRevLett.84.2064","article-title":"Is random close packing of spheres well defined?","volume":"84","author":"Torquato","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Mani, E., and L\u00f6wen, H. (2015). Effect of self-propulsion on equilibrium clustering. arXiv.","DOI":"10.1103\/PhysRevE.92.032301"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"8865","DOI":"10.1039\/C9SM01691K","article-title":"Tuning the self-organization of confined active particles by the steepness of the trap","volume":"15","author":"Rana","year":"2019","journal-title":"Soft Matter"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"238104","DOI":"10.1103\/PhysRevLett.112.238104","article-title":"Self-induced polar order of active Brownian particles in a harmonic trap","volume":"112","author":"Hennes","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1386","DOI":"10.1038\/s41598-018-36824-z","article-title":"Activity induced delocalization and freezing in self-propelled systems","volume":"9","author":"Caprini","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1312","DOI":"10.1039\/D2SM01402E","article-title":"Collective behavior of chiral active particles with anisotropic interactions in a confined space","volume":"19","author":"Lei","year":"2023","journal-title":"Soft Matter"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"023196","DOI":"10.1103\/PhysRevResearch.5.023196","article-title":"Geometry-induced dynamics of confined chiral active matter","volume":"5","author":"Negi","year":"2023","journal-title":"Phys. Rev. Res."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Becton, M., Hou, J., Zhao, Y., and Wang, X. (2024). Dynamic clustering and scaling behavior of active particles under confinement. Nanomaterials, 14.","DOI":"10.3390\/nano14020144"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"You, L., Duan, H., and Yuan, Y. (2026). A Phase Transition Control Framework for UAV Swarms Inspired by Pigeon Roosting Behavior. Drones, 10.","DOI":"10.3390\/drones10050326"},{"key":"ref_28","unstructured":"German, R.M. (1989). Particle Packing Characteristics, Metal Powder Industry."},{"key":"ref_29","unstructured":"Hansen, A., and Bideau, D. (1993). Disorder and Granular Media, North Holland."},{"key":"ref_30","first-page":"479","article-title":"Random packings and the structure of simple liquids. I. The geometry of random close packing","volume":"319","author":"Finney","year":"1970","journal-title":"Proc. R. Soc. Lond. A Math. Phys. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1038\/23819","article-title":"Jamming is not just cool any more","volume":"396","author":"Liu","year":"1998","journal-title":"Nature"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1007\/s10035-013-0410-3","article-title":"Packing of fine particles in an electrical field","volume":"15","author":"Yang","year":"2013","journal-title":"Granul. Matter"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"117853","DOI":"10.1016\/j.powtec.2022.117853","article-title":"Ordered packing of uniform spheres via random packing protocol","volume":"409","author":"Amirifar","year":"2022","journal-title":"Powder Technol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"9856","DOI":"10.1039\/C8SM01763H","article-title":"Self-assembly of granular spheres under one-dimensional vibration","volume":"14","author":"Amirifar","year":"2018","journal-title":"Soft Matter"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"6137","DOI":"10.1039\/C8SM00222C","article-title":"Phase separation in binary mixtures of active and passive particles","volume":"14","author":"Dolai","year":"2018","journal-title":"Soft Matter"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2704","DOI":"10.1038\/ncomms3704","article-title":"Pushing the glass transition towards random close packing using self-propelled hard spheres","volume":"4","author":"Ni","year":"2013","journal-title":"Nat. Commun."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"e7","DOI":"10.1017\/pma.2023.6","article-title":"Programmable active matter across scales","volume":"1","author":"Yu","year":"2023","journal-title":"Program. Mater."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Fu, Y., Yu, H., Zhang, X., Malgaretti, P., Kishore, V., and Wang, W. (2022). Microscopic swarms: From active matter physics to biomedical and environmental applications. Micromachines, 13.","DOI":"10.3390\/mi13020295"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4282","DOI":"10.1103\/PhysRevE.51.4282","article-title":"Social force model for pedestrian dynamics","volume":"51","author":"Helbing","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"27505","DOI":"10.1038\/s41598-024-77219-7","article-title":"Aggregate morphing of self-aligning soft active disks in semi-confined geometry","volume":"14","author":"Chugh","year":"2024","journal-title":"Sci. Rep."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"044611","DOI":"10.1103\/PhysRevE.106.044611","article-title":"Emergent synchronization and flocking in purely repulsive self-navigating particles","volume":"106","author":"Casiulis","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"8267","DOI":"10.1039\/D4SM00848K","article-title":"Particle\u2013wall alignment interaction and active Brownian diffusion through narrow channels","volume":"20","author":"Bag","year":"2024","journal-title":"Soft Matter"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1016\/j.ces.2016.07.013","article-title":"Voronoi analysis of the packings of non-spherical particles","volume":"153","author":"Dong","year":"2016","journal-title":"Chem. Eng. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1016\/j.powtec.2020.04.020","article-title":"Experimental study on 3D vibrated packing densification of mono-sized dodecahedral particles","volume":"367","author":"Li","year":"2020","journal-title":"Powder Technol."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/j.powtec.2021.04.079","article-title":"On the relationships between structural properties and packing density of uniform spheres","volume":"388","author":"An","year":"2021","journal-title":"Powder Technol."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1007\/s10035-016-0612-6","article-title":"Quasi-universality in the packing of uniform spheres under gravity","volume":"18","author":"An","year":"2016","journal-title":"Granul. Matter"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1016\/j.ces.2014.12.059","article-title":"A novel method based on orientation discretization for discrete element modeling of non-spherical particles","volume":"126","author":"Dong","year":"2015","journal-title":"Chem. Eng. Sci."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"6477","DOI":"10.1039\/C4SM00927D","article-title":"Aggregation and segregation of confined active particles","volume":"10","author":"Yang","year":"2014","journal-title":"Soft Matter"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"062314","DOI":"10.1103\/PhysRevE.88.062314","article-title":"Collective behavior of penetrable self-propelled rods in two dimensions","volume":"88","author":"Abkenar","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1038\/s42005-025-02466-7","article-title":"Dynamics of active swarms at the edge of disorder","volume":"9","author":"Iyer","year":"2026","journal-title":"Commun. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"30008","DOI":"10.1209\/0295-5075\/103\/30008","article-title":"Microscopic theory for the phase separation of self-propelled repulsive disks","volume":"103","author":"Speck","year":"2013","journal-title":"Europhys. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2132","DOI":"10.1039\/C3SM52469H","article-title":"Freezing and phase separation of self-propelled disks","volume":"10","author":"Fily","year":"2014","journal-title":"Soft Matter"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.powtec.2020.10.023","article-title":"An assessment of the mathematical models for estimating the coordination number of the packing of multisized particles","volume":"379","author":"Yi","year":"2021","journal-title":"Powder Technol."},{"key":"ref_54","first-page":"301","article-title":"Surface energy and the contact of elastic solids","volume":"324","author":"Johnson","year":"1971","journal-title":"Proc. R. Soc. Lond. A Math. Phys. Sci."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1680\/geot.1979.29.1.47","article-title":"A discrete numerical model for granular assemblies","volume":"29","author":"Cundall","year":"1979","journal-title":"Geotechnique"},{"key":"ref_56","unstructured":"Wakao, N., and Kagei, S. (1982). Heat and Mass Transfer in Packed Beds, Yokohama National University."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1002\/cjce.5450680110","article-title":"A new experimental method to specify the diffusing component in a reacting particulate packing","volume":"68","author":"Hao","year":"1990","journal-title":"Can. J. Chem. Eng."},{"key":"ref_58","unstructured":"Pinson, D. (1997). Powder entrapment in a multi-phase flow packed bed. Fuel and Energy Abstracts, The Minerals, Metals & Materials Society."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1232","DOI":"10.1073\/pnas.0711437105","article-title":"Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study","volume":"105","author":"Ballerini","year":"2008","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Gautrais, J., Ginelli, F., Fournier, R., Blanco, S., Soria, M., Chat\u00e9, H., and Theraulaz, G. (2012). Deciphering interactions in moving animal groups. PLoS Comput. Biol., 8.","DOI":"10.1371\/journal.pcbi.1002678"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"062203","DOI":"10.1103\/PhysRevE.92.062203","article-title":"Structural characterization of the packings of granular regular polygons","volume":"92","author":"Wang","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"025702","DOI":"10.1103\/PhysRevLett.92.025702","article-title":"Onset of collective and cohesive motion","volume":"92","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"46003","DOI":"10.1209\/0295-5075\/86\/46003","article-title":"Critical states and phase diagram in the packing of uniform spheres","volume":"86","author":"Dong","year":"2009","journal-title":"Europhys. Lett."},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Bouffanais, R. (2016). Design and Control of Swarm Dynamics, Springer.","DOI":"10.1007\/978-981-287-751-2"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1177\/10597123231163948","article-title":"A unifying method-based classification of robot swarm spatial self-organisation behaviours","volume":"31","author":"Peillard","year":"2023","journal-title":"Adapt. Behav."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Pan, Z., Wen, G., Yin, H., Yin, S., and Tan, Z. (2024). Crystallization-Inspired Design and Modeling of Self-Assembly Lattice-Formation Swarm Robotics. Sensors, 24.","DOI":"10.3390\/s24103081"},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Wang, C., Wang, D., Gu, M., Huang, H., Wang, Z., Yuan, Y., Zhu, X., Wei, W., and Fan, Z. (2022). Bioinspired environment exploration algorithm in swarm based on levy flight and improved artificial potential field. Drones, 6.","DOI":"10.3390\/drones6050122"},{"key":"ref_68","first-page":"031043","article-title":"Freezing a flock: Motility-induced phase separation in polar active liquids","volume":"9","author":"Geyer","year":"2019","journal-title":"Phys. Rev. X"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"7235","DOI":"10.1038\/s41467-025-62626-9","article-title":"Self-sustained frictional cooling in active matter","volume":"16","author":"Antonov","year":"2025","journal-title":"Nat. Commun."}],"container-title":["Micromachines"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-666X\/17\/8\/884\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T04:25:45Z","timestamp":1785299145000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-666X\/17\/8\/884"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,24]]},"references-count":69,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2026,8]]}},"alternative-id":["mi17080884"],"URL":"https:\/\/doi.org\/10.3390\/mi17080884","relation":{},"ISSN":["2072-666X"],"issn-type":[{"value":"2072-666X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,24]]}}}