{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T22:40:08Z","timestamp":1784155208946,"version":"3.55.0"},"reference-count":50,"publisher":"AIP Publishing","issue":"15","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["416229255 (SFB1411)"],"award-info":[{"award-number":["416229255 (SFB1411)"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2025,10,21]]},"abstract":"<jats:p>The hard disk model is a foundational system for understanding two-dimensional phase behavior. Its extension to binary mixtures introduces additional structural complexity and kinetic constraints that hinder the formation of ordered phases. Although geometric constructions predict a variety of binary crystals at high densities, these phases often fail to self-assemble in simulations due to slow dynamics and competing phase separation. Here, we investigate the spontaneous formation of the H2 hexagonal binary hard disk crystal in additive mixtures using event-driven molecular dynamics. We show that crystallization is highly sensitive to thermodynamic conditions, requires long equilibration, and is facilitated by particle swap moves. Despite its high packing fraction, the H2 crystal forms spontaneously at equimolar composition and near-ideal size ratio but is kinetically hindered at intermediate densities by the formation of competing one-component hexagonal phases. Our results identify the conditions under which H2 binary crystals spontaneously form and show that their emergence can proceed through non-classical nucleation pathways involving metastable intermediates. These findings offer new insights into entropy-driven self-assembly in two-dimensional mixtures and inform strategies for realizing complex order in colloidal and granular materials.<\/jats:p>","DOI":"10.1063\/5.0293041","type":"journal-article","created":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T12:19:41Z","timestamp":1760530781000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Formation of hexagonal binary crystals in additive hard disk mixtures"],"prefix":"10.1063","volume":"163","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-8627-5270","authenticated-orcid":false,"given":"Appala Naidu","family":"Rayavarapu","sequence":"first","affiliation":[{"name":"Department of Chemical Engineering, National Institute of Technology Warangal 1 , Telangana 506004,","place":["India"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-0758-2943","authenticated-orcid":false,"given":"Navid","family":"Panchi","sequence":"additional","affiliation":[{"name":"Institute for Multiscale Simulation, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg 2 , 91058 Erlangen,","place":["Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7031-3825","authenticated-orcid":false,"given":"Michael","family":"Engel","sequence":"additional","affiliation":[{"name":"Institute for Multiscale Simulation, Friedrich-Alexander-Universit\u00e4t Erlangen-N\u00fcrnberg 2 , 91058 Erlangen,","place":["Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6580-4769","authenticated-orcid":false,"given":"Praveen K.","family":"Bommineni","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, National Institute of Technology Warangal 1 , Telangana 506004,","place":["India"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2025,10,15]]},"reference":[{"key":"2025101508193776100_c1","doi-asserted-by":"publisher","first-page":"11220","DOI":"10.1021\/acs.chemrev.6b00196","article-title":"Self-assembly of colloidal nanocrystals: From intricate structures to functional materials","volume":"116","year":"2016","journal-title":"Chem. Rev."},{"key":"2025101508193776100_c2","doi-asserted-by":"publisher","first-page":"14791","DOI":"10.1021\/acsnano.3c10201","article-title":"Nanocrystal assemblies: Current advances and open problems","volume":"18","year":"2024","journal-title":"ACS Nano"},{"key":"2025101508193776100_c3","doi-asserted-by":"publisher","first-page":"045003","DOI":"10.1103\/revmodphys.96.045003","article-title":"Colloidal hard spheres: Triumphs, challenges, and mysteries","volume":"96","year":"2024","journal-title":"Rev. Mod. Phys."},{"key":"2025101508193776100_c4","doi-asserted-by":"publisher","first-page":"1208","DOI":"10.1063\/1.1743957","article-title":"Phase transition for a hard sphere system","volume":"27","year":"1957","journal-title":"J.\u00a0Chem. Phys."},{"key":"2025101508193776100_c5","doi-asserted-by":"publisher","first-page":"1207","DOI":"10.1063\/1.1743956","article-title":"Preliminary results from a recalculation of the Monte Carlo equation of state of hard spheres","volume":"27","year":"1957","journal-title":"J.\u00a0Chem. Phys."},{"key":"2025101508193776100_c6","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1103\/physrev.127.359","article-title":"Phase transition in elastic disks","volume":"127","year":"1962","journal-title":"Phys. Rev."},{"key":"2025101508193776100_c7","doi-asserted-by":"publisher","first-page":"1020","DOI":"10.1038\/35059035","article-title":"Prediction of absolute crystal-nucleation rate in hard-sphere colloids","volume":"409","year":"2001","journal-title":"Nature"},{"key":"2025101508193776100_c8","doi-asserted-by":"publisher","first-page":"340","DOI":"10.1038\/320340a0","article-title":"Phase behaviour of concentrated suspensions of nearly hard colloidal spheres","volume":"320","year":"1986","journal-title":"Nature"},{"key":"2025101508193776100_c9","doi-asserted-by":"publisher","first-page":"3801","DOI":"10.1103\/physrevlett.68.3801","article-title":"Superlattice formation in binary mixtures of hard-sphere colloids","volume":"68","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c10","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1038\/365035a0","article-title":"Entropy-driven formation of a superlattice in a hard-sphere binary mixture","volume":"365","year":"1993","journal-title":"Nature"},{"key":"2025101508193776100_c11","doi-asserted-by":"publisher","first-page":"218003","DOI":"10.1103\/physrevlett.124.218003","article-title":"Spontaneous crystallization in systems of binary hard sphere colloids","volume":"124","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c12","doi-asserted-by":"publisher","first-page":"128005","DOI":"10.1103\/physrevlett.122.128005","article-title":"Complex crystals from size-disperse spheres","volume":"122","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c13","doi-asserted-by":"publisher","first-page":"208001","DOI":"10.1103\/physrevlett.116.208001","article-title":"Hiding in plain view: Colloidal self-assembly from polydisperse populations","volume":"116","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c14","doi-asserted-by":"publisher","first-page":"191101","DOI":"10.1063\/1.5028279","article-title":"Communication: From close-packed to topologically close-packed: Formation of Laves phases in moderately polydisperse hard-sphere mixtures","volume":"148","year":"2018","journal-title":"J.\u00a0Chem. Phys."},{"key":"2025101508193776100_c15","doi-asserted-by":"publisher","first-page":"10269","DOI":"10.1073\/pnas.1525358113","article-title":"Generic phase diagram of binary superlattices","volume":"113","year":"2016","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2025101508193776100_c16","doi-asserted-by":"publisher","first-page":"155704","DOI":"10.1103\/physrevlett.107.155704","article-title":"Two-step melting in two dimensions: First-order liquid-hexatic transition","volume":"107","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c17","doi-asserted-by":"publisher","first-page":"042134","DOI":"10.1103\/physreve.87.042134","article-title":"Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods","volume":"87","year":"2013","journal-title":"Phys. Rev. E"},{"key":"2025101508193776100_c18","doi-asserted-by":"publisher","first-page":"158001","DOI":"10.1103\/physrevlett.118.158001","article-title":"Two-dimensional melting of colloidal hard spheres","volume":"118","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c19","doi-asserted-by":"publisher","first-page":"115702","DOI":"10.1103\/physrevlett.119.115702","article-title":"Disappearance of the hexatic phase in a binary mixture of hard disks","volume":"119","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c20","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1080\/13642819308215284","article-title":"Complex alloy phases for binary hard-disc mixtures","volume":"68","year":"1993","journal-title":"Philos. Mag. B"},{"key":"2025101508193776100_c21","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/s00454-005-1172-4","article-title":"Compact packings of the plane with two sizes of discs","volume":"35","year":"2006","journal-title":"Discrete Comput. Geom."},{"key":"2025101508193776100_c22","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1007\/s00454-019-00166-y","article-title":"Compact packings of the plane with three sizes of discs","volume":"66","year":"2021","journal-title":"Discrete Comput. Geom."},{"key":"2025101508193776100_c23","doi-asserted-by":"publisher","first-page":"787","DOI":"10.1007\/s00454-021-00348-7","article-title":"Density of binary disc packings: The nine compact packings","volume":"67","year":"2022","journal-title":"Discrete Comput. Geom."},{"key":"2025101508193776100_c24","doi-asserted-by":"publisher","first-page":"105690","DOI":"10.1016\/j.jcta.2022.105690","article-title":"Density of binary disc packings: Playing with stoichiometry","volume":"194","year":"2023","journal-title":"J.\u00a0Comb. Theory, Ser. A"},{"key":"2025101508193776100_c25","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1080\/10586458.2022.2061651","article-title":"Density of binary disc packings: Lower and upper bounds","volume":"33","year":"2024","journal-title":"Exp. Math."},{"key":"2025101508193776100_c26","doi-asserted-by":"publisher","first-page":"061503","DOI":"10.1103\/physreve.65.061503","article-title":"Kinetic bottleneck to the self-organization of bidisperse hard disk monolayers formed by random sequential adsorption","volume":"65","year":"2002","journal-title":"Phys. Rev. E"},{"key":"2025101508193776100_c27","doi-asserted-by":"publisher","first-page":"43604","DOI":"10.5488\/cmp.15.43604","article-title":"Towards frustration of freezing transition in a binary hard-disk mixture","volume":"15","year":"2012","journal-title":"Condens. Matter Phys."},{"key":"2025101508193776100_c28","doi-asserted-by":"publisher","first-page":"1317","DOI":"10.1080\/08927022.2016.1139106","article-title":"Hard sphere simulation in statistical physics\u2014Methodologies and applications","volume":"42","year":"2016","journal-title":"Mol. Simul."},{"key":"2025101508193776100_c29","doi-asserted-by":"publisher","first-page":"145701","DOI":"10.1103\/physrevlett.117.145701","article-title":"Applicability of dynamic facilitation theory to binary hard disk systems","volume":"117","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c30","doi-asserted-by":"publisher","first-page":"4559","DOI":"10.1063\/1.478337","article-title":"Glass transition in hard disc mixtures","volume":"110","year":"1999","journal-title":"J.\u00a0Chem. Phys."},{"key":"2025101508193776100_c31","doi-asserted-by":"publisher","first-page":"5721","DOI":"10.1103\/physreve.59.5721","article-title":"Stability and structure of a supercooled liquid mixture in two dimensions","volume":"59","year":"1999","journal-title":"Phys. Rev. E"},{"key":"2025101508193776100_c32","doi-asserted-by":"publisher","first-page":"225502","DOI":"10.1103\/physrevlett.96.225502","article-title":"Do binary hard disks exhibit an ideal glass transition?","volume":"96","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c33","doi-asserted-by":"publisher","first-page":"015701","DOI":"10.1103\/physrevlett.104.015701","article-title":"Observation of the disorder-induced crystal-to-glass transition","volume":"104","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c34","doi-asserted-by":"publisher","first-page":"414001","DOI":"10.1088\/1361-648x\/ab9b52","article-title":"Dense hard disk ordering: Influence of bidispersity and quenched disorder","volume":"32","year":"2020","journal-title":"J.\u00a0Phys.: Condens. Matter."},{"key":"2025101508193776100_c35","doi-asserted-by":"publisher","first-page":"028202","DOI":"10.1103\/physrevlett.133.028202","article-title":"Irreversible Monte Carlo algorithms for hard disk glasses: From event-chain to collective swaps","volume":"133","year":"2024","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c36","unstructured":"V.\n            Bolton-Lum\n          , R. C.Dennis, P.Morse, and E.Corwin, \u201cThe ideal glass and the ideal disk packing in two dimensions,\u201d arXiv:2404.07492 (2024)."},{"key":"2025101508193776100_c37","doi-asserted-by":"publisher","first-page":"204901","DOI":"10.1063\/5.0008230","article-title":"Infinite-pressure phase diagram of binary mixtures of (non)additive hard disks","volume":"152","year":"2020","journal-title":"J.\u00a0Chem. Phys."},{"key":"2025101508193776100_c38","doi-asserted-by":"publisher","first-page":"268002","DOI":"10.1103\/physrevlett.127.268002","article-title":"Topographic control of order in quasi-2D granular phase transitions","volume":"127","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c39","doi-asserted-by":"publisher","first-page":"2654","DOI":"10.1039\/d3sm00179b","article-title":"Self-assembly of dodecagonal and octagonal quasicrystals in hard spheres on a plane","volume":"19","year":"2023","journal-title":"Soft Matter"},{"key":"2025101508193776100_c40","doi-asserted-by":"publisher","first-page":"8303","DOI":"10.1021\/acsnano.2c12668","article-title":"Nonadditive interactions unlock small-particle mobility in binary colloidal monolayers","volume":"17","year":"2023","journal-title":"ACS Nano"},{"key":"2025101508193776100_c41","doi-asserted-by":"publisher","first-page":"5675","DOI":"10.1039\/d5sm00433k","article-title":"Frustrated crystallization kinetics in marginally mismatched binary self-assembly driven by depletion interactions","volume":"21","year":"2025","journal-title":"Soft Matter"},{"key":"2025101508193776100_c42","doi-asserted-by":"publisher","first-page":"244902","DOI":"10.1063\/5.0210034","article-title":"Inverse design of crystals and quasicrystals in a non-additive binary mixture of hard disks","volume":"160","year":"2024","journal-title":"J.\u00a0Chem. Phys."},{"key":"2025101508193776100_c43","doi-asserted-by":"publisher","first-page":"054509","DOI":"10.1063\/5.0268711","article-title":"Self-assembly and non-equilibrium phase coexistence in a binary granular mixture","volume":"163","year":"2025","journal-title":"J.\u00a0Chem. Phys."},{"key":"2025101508193776100_c44","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1038\/s41567-023-02364-1","article-title":"Quasi-crystalline order in vibrating granular matter","volume":"20","year":"2024","journal-title":"Nat. Phys."},{"key":"2025101508193776100_c45","doi-asserted-by":"publisher","first-page":"062101","DOI":"10.1103\/physreve.102.062101","article-title":"Phase behavior of Lennard-Jones particles in two dimensions","volume":"102","year":"2020","journal-title":"Phys. Rev. E"},{"key":"2025101508193776100_c46","doi-asserted-by":"publisher","first-page":"048202","DOI":"10.1103\/physrevlett.132.048202","article-title":"Quasicrystal of binary hard spheres on a plane stabilized by configurational entropy","volume":"132","year":"2024","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c47","doi-asserted-by":"publisher","first-page":"107275","DOI":"10.1016\/j.cpc.2020.107275","article-title":"freud: A software suite for high throughput analysis of particle simulation data","volume":"254","year":"2020","journal-title":"Comput. Phys. Commun."},{"key":"2025101508193776100_c48","doi-asserted-by":"publisher","first-page":"174104","DOI":"10.1063\/5.0194873","article-title":"Simple and efficient methods for local structural analysis in polydisperse hard disk systems","volume":"160","year":"2024","journal-title":"J.\u00a0Chem. Phys."},{"key":"2025101508193776100_c49","doi-asserted-by":"publisher","first-page":"198201","DOI":"10.1103\/physrevlett.134.198201","article-title":"Entropic trapping of hard spheres in spherical confinement","volume":"134","year":"2025","journal-title":"Phys. Rev. Lett."},{"key":"2025101508193776100_c50","doi-asserted-by":"publisher","first-page":"188001","DOI":"10.1103\/physrevlett.128.188001","article-title":"Tuning stoichiometry to promote formation of binary colloidal superlattices","volume":"128","year":"2022","journal-title":"Phys. Rev. Lett."}],"container-title":["The Journal of Chemical Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/jcp\/article-pdf\/doi\/10.1063\/5.0293041\/20754713\/154503_1_5.0293041.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/jcp\/article-pdf\/doi\/10.1063\/5.0293041\/20754713\/154503_1_5.0293041.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T12:19:43Z","timestamp":1760530783000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/jcp\/article\/163\/15\/154503\/3367915\/Formation-of-hexagonal-binary-crystals-in-additive"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,15]]},"references-count":50,"journal-issue":{"issue":"15","published-print":{"date-parts":[[2025,10,21]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0293041","relation":{},"ISSN":["0021-9606","1089-7690"],"issn-type":[{"value":"0021-9606","type":"print"},{"value":"1089-7690","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2025,10,21]]},"published":{"date-parts":[[2025,10,15]]},"article-number":"154503"}}