{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T01:21:58Z","timestamp":1773796918751,"version":"3.50.1"},"reference-count":66,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2018,12,11]],"date-time":"2018-12-11T00:00:00Z","timestamp":1544486400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Israel Ministry of Immigrant Absorption","award":["ooooo"],"award-info":[{"award-number":["ooooo"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Voronoi entropy is a mathematical tool for quantitative characterization of the orderliness of points distributed on a surface. The tool is useful to study various surface self-assembly processes. We provide the historical background, from Kepler and Descartes to our days, and discuss topological properties of the Voronoi tessellation, upon which the entropy concept is based, and its scaling properties, known as the Lewis and Aboav\u2013Weaire laws. The Voronoi entropy has been successfully applied to recently discovered self-assembled structures, such as patterned microporous polymer surfaces obtained by the breath figure method and levitating ordered water microdroplet clusters.<\/jats:p>","DOI":"10.3390\/e20120956","type":"journal-article","created":{"date-parts":[[2018,12,12]],"date-time":"2018-12-12T03:27:49Z","timestamp":1544585269000},"page":"956","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":49,"title":["Characterization of Self-Assembled 2D Patterns with Voronoi Entropy"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1356-2486","authenticated-orcid":false,"given":"Edward","family":"Bormashenko","sequence":"first","affiliation":[{"name":"Department of Chemical Engineering, Biotechnology and Materials, Engineering Sciences Faculty, Ariel University, Ariel 407000, Israel"}]},{"given":"Mark","family":"Frenkel","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, Biotechnology and Materials, Engineering Sciences Faculty, Ariel University, Ariel 407000, Israel"}]},{"given":"Alla","family":"Vilk","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, Biotechnology and Materials, Engineering Sciences Faculty, Ariel University, Ariel 407000, Israel"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6384-7570","authenticated-orcid":false,"given":"Irina","family":"Legchenkova","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, Biotechnology and Materials, Engineering Sciences Faculty, Ariel University, Ariel 407000, Israel"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6595-3927","authenticated-orcid":false,"given":"Alexander A.","family":"Fedorets","sequence":"additional","affiliation":[{"name":"University of Tyumen, 6 Volodarskogo St., Tyumen 625003, Russia"}]},{"given":"Nurken E.","family":"Aktaev","sequence":"additional","affiliation":[{"name":"University of Tyumen, 6 Volodarskogo St., Tyumen 625003, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6290-019X","authenticated-orcid":false,"given":"Leonid A.","family":"Dombrovsky","sequence":"additional","affiliation":[{"name":"University of Tyumen, 6 Volodarskogo St., Tyumen 625003, Russia"},{"name":"Joint Institute for High Temperatures, 17A Krasnokazarmennaya St., Moscow 111116, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0980-3670","authenticated-orcid":false,"given":"Michael","family":"Nosonovsky","sequence":"additional","affiliation":[{"name":"University of Tyumen, 6 Volodarskogo St., Tyumen 625003, Russia"},{"name":"Mechanical Engineering, University of Wisconsin\u2014Milwaukee, 3200 North Cramer St., Milwaukee, WI 53211, USA"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1038\/369387a0","article-title":"Self-organized honeycomb morphology of star-polymer polystyrene films","volume":"369","author":"Widawski","year":"1994","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/s100510050685","article-title":"Formation of ordered micro-porous membranes","volume":"8","author":"Pitois","year":"1999","journal-title":"Eur. Phys. J. B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"6071","DOI":"10.1021\/la0001732","article-title":"Water-assisted formation of micrometer-size honeycomb patterns of polymers","volume":"16","author":"Karthaus","year":"2000","journal-title":"Langmuir"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bormashenko, E. (2017). Breath-figure Self-assembly, a Versatile Method of manufacturing membranes and porous structures: physical, chemical and technological aspects. Membranes, 73.","DOI":"10.20944\/preprints201706.0130.v1"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1515\/crll.1908.134.198","article-title":"Nouvelles applications des param\u00e8tres continus \u00e0 la th\u00e9orie des formes quadratiques. Deuxi\u00e8me m\u00e9moire. Recherches sur les parall\u00e9lo\u00e8dres primitifs","volume":"134","author":"Voronoi","year":"1908","journal-title":"Reine Angew. Math."},{"key":"ref_6","unstructured":"Descartes, R. (1644). Principia Philosophiae, Ludovicus Elzevirius."},{"key":"ref_7","first-page":"419","article-title":"Voronoi diagrams and Delaunay triangulations: Ubiquitous Siamese Twins","volume":"Extra Volume ISMP","author":"Liebling","year":"2012","journal-title":"Doc. Math."},{"key":"ref_8","unstructured":"Snow, J. (1855). Report on the Cholera Outbreak in the Parish of St. James, Westminster: during the autumn of 1854."},{"key":"ref_9","first-page":"209","article-title":"\u00dcber die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen","volume":"40","author":"Dirichlet","year":"1850","journal-title":"J. Reine Angew. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"114501","DOI":"10.1063\/1.2011390","article-title":"Voronoi cell volume distribution and configurational entropy of hard-spheres","volume":"123","author":"Kumar","year":"2005","journal-title":"J. Chem. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2010.11.002","article-title":"Spatial networks","volume":"499","year":"2011","journal-title":"Phys. Rep."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1080\/00107518408210979","article-title":"Soap, cells and statistics\u2014random patterns in two dimensions","volume":"25","author":"Weaire","year":"1984","journal-title":"Contemporary Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1080\/08893110412331323170","article-title":"Voronoi\u2013Dirichlet polyhedra in crystal chemistry: theory and applications","volume":"10","author":"Blatov","year":"2004","journal-title":"Crystallography Reviews"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3762","DOI":"10.1103\/PhysRevLett.76.3762","article-title":"Evidence for convective effects in breath figure formation on volatile fluid surfaces","volume":"76","author":"Limaye","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1002\/ar.1090380305","article-title":"The correlation between cell division and the shapes and sizes of prismatic cell in the epidermis of Cucumis","volume":"38","author":"Lewis","year":"1928","journal-title":"Anat. Rec."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1002\/ar.1090470108","article-title":"A volumetric study of growth and cell division in two types of epithelium-the longitudinally prismatic cells of Tradescantia and the radially prismatic epidermal cells of Cucumis","volume":"47","author":"Lewis","year":"1930","journal-title":"Anat. Rec."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1002\/j.1537-2197.1943.tb10328.x","article-title":"The geometry of growth and cell division in epithelial mosaics","volume":"30","author":"Lewis","year":"1943","journal-title":"Am. J. Bot."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1002\/j.1537-2197.1944.tb12173.x","article-title":"The geometry of growth and cell division in columnar parenchyma","volume":"31","author":"Lewis","year":"1944","journal-title":"Am. J Bot."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/1044-5803(94)00081-U","article-title":"Aboav-Weaire\u2019s and Lewis\u2019 laws\u2014A review","volume":"34","author":"Chiu","year":"1995","journal-title":"Mater. Charact."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"L143","DOI":"10.1088\/0305-4470\/15\/3\/012","article-title":"On the correlation between sizes and shapes of cells in epithelial mosaics","volume":"15","author":"Rivier","year":"1982","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"77","DOI":"10.15252\/embj.201592374","article-title":"Fundamental physical cellular constraints drive self-organization of tissues","volume":"35","author":"Tozluoglu","year":"2016","journal-title":"EMBO J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1080\/09500830902720925","article-title":"Polygonal networks on the surface of Mars; applicability of Lewis, Desch and Aboav\u2013Weaire laws","volume":"89","author":"Saraiva","year":"2009","journal-title":"Phil. Mag. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1086","DOI":"10.1103\/PhysRevB.42.1086","article-title":"Two-dimensional ordering during droplet growth on a liquid surfaceth on a liquid surface","volume":"42","author":"Steyer","year":"1990","journal-title":"Phys Rev. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2188","DOI":"10.1039\/B814061H","article-title":"Nano- and micro-sized honeycomb patterns through hierarchical self-assembly of metal-loaded diblock copolymer vesicles","volume":"5","author":"Pietsch","year":"2009","journal-title":"Soft Matter"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5347","DOI":"10.1021\/la035915g","article-title":"Breath figure patterns prepared by spin coating in a dry environment","volume":"20","author":"Park","year":"2004","journal-title":"Langmuir"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"14163","DOI":"10.1021\/la403086w","article-title":"Revisiting the fine structure of the triple line","volume":"29","author":"Bormashenko","year":"2013","journal-title":"Langmuir"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3517","DOI":"10.1021\/la703363a","article-title":"Breath figures in polymer and polymer blend films spin-coated in dry and humid ambience","volume":"24","author":"Madej","year":"2008","journal-title":"Langmuir"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1080\/13642818508240637","article-title":"Statistical crystallography structure of random cellular networks","volume":"52","author":"Rivier","year":"1985","journal-title":"Phil. Mag.B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/0026-0800(70)90038-8","article-title":"The arrangement of grains in a polycrystal","volume":"3","author":"Aboav","year":"1970","journal-title":"Metallography"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/0026-0800(74)90004-4","article-title":"Some remarks on the arrangement of grains in a polycrystal","volume":"7","author":"Weaire","year":"1974","journal-title":"Metallography"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"598","DOI":"10.1103\/PhysRevE.48.598","article-title":"Mitosis and growth in biological tissues","volume":"48","author":"Mombach","year":"1993","journal-title":"Phys.Rev. E"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1016\/j.physa.2007.07.063","article-title":"On the size distribution of Poisson Voronoi cells","volume":"385","author":"Zoltan","year":"2007","journal-title":"Phys. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2765","DOI":"10.1080\/01418610010032364","article-title":"The geometrical properties of irregular two-dimensional","volume":"81","author":"Zhu","year":"2001","journal-title":"Phil. Mag. A"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/0097-8493(93)90100-N","article-title":"Generating fractals from Voronoi diagrams","volume":"17","author":"Shirriff","year":"1998","journal-title":"Comput. Graph."},{"key":"ref_35","first-page":"793","article-title":"Sur la sph\u00e8re vide","volume":"6","author":"Delaunay","year":"1934","journal-title":"Bulletin de l\u2019Acad\u00e9mie des Sciences de l\u2019URSS, Classe des Sciences Math\u00e9matiques et Naturelles"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"031801","DOI":"10.1103\/PhysRevE.81.031801","article-title":"Structure of void space in polymer solutions","volume":"81","author":"Sung","year":"2010","journal-title":"Phys Rev E Stat Nonlin. Soft Matter Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0022-5096(01)00060-6","article-title":"Three-dimensional micromechanical modeling of voided polymeric materials","volume":"50","author":"Danielsson","year":"2002","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1038\/nmat1611","article-title":"Kinetically driven self assembly of highly ordered nanoparticle monolayers","volume":"5","author":"Bigioni","year":"2006","journal-title":"Nat. Mater."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"5646","DOI":"10.1021\/cm0618953","article-title":"Highly Ordered Arrays of Nanoparticles in Large Areas from Diblock Copolymer Micelles in Hexagonal Self-Assembly","volume":"18","author":"Yun","year":"2006","journal-title":"Chem. Mater."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1126\/science.1193369","article-title":"Block Copolymer Self-Assembly\u2013Directed Single-Crystal Homo- and Heteroepitaxial Nanostructures","volume":"330","author":"Arora","year":"2010","journal-title":"Science"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3915","DOI":"10.1039\/c0sm01066a","article-title":"Block copolymer self-assembly in chemically patterned squares","volume":"7","author":"Xu","year":"2011","journal-title":"Soft Matter."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"25735","DOI":"10.1039\/C6CP04297J","article-title":"Self-assembly of like-charged nanoparticles into Voronoi diagrams","volume":"18","author":"Suzuno","year":"2016","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"116103","DOI":"10.1103\/PhysRevLett.99.116103","article-title":"Controlling Pattern Formation in Nanoparticle Assemblies via Directed Solvent Dewetting","volume":"99","author":"Martin","year":"2007","journal-title":"Phys. Lett."},{"key":"ref_44","unstructured":"Lim, J.S. (1990). Two Dimensional Signal and Image Processing, Prentice Hall."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"022720","DOI":"10.1103\/PhysRevE.88.022720","article-title":"Automatic sorting of point pattern sets using Minkowski functionals","volume":"88","author":"Parker","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"P12015","DOI":"10.1088\/1742-5468\/2008\/12\/P12015","article-title":"Utilizing Minkowski functionals for image analysis: A marching square algorithm","volume":"2008","author":"Mantz","year":"2008","journal-title":"J. Stat. Mech. Theor. Exp."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1002\/macp.200700552","article-title":"Mesoscopic patterning in evaporated Polymer solutions: Poly (ethylene glycol) and room-temperature-vulcanized Polyorganosilanes\/-siloxanes Promote formation of honeycomb structures","volume":"209","author":"Bormashenko","year":"2008","journal-title":"Macromol. Chem. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1017\/S0370164600048434","article-title":"Breath Figures","volume":"20","author":"Aitken","year":"1895","journal-title":"Proc. R. Soc. Edinb."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"516","DOI":"10.1038\/086516a0","article-title":"Breath figures","volume":"86","author":"Aitkek","year":"1911","journal-title":"Nature"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1038\/086416d0","article-title":"Breath figures","volume":"86","author":"Rayleigh","year":"1911","journal-title":"Nature"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1038\/090436c0","article-title":"Breath figures","volume":"90","author":"Rayleigh","year":"1912","journal-title":"Nature"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"9604","DOI":"10.1021\/la0518492","article-title":"Mesoscopic patterning in evaporated polymer solutions: new experimental data and physical mechanisms","volume":"21","author":"Bormashenko","year":"2005","journal-title":"Langmuir"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1016\/j.jcis.2005.11.025","article-title":"Self-assembly in evaporated polymer solutions: Influence of the solution concentration","volume":"297","author":"Bormashenko","year":"2006","journal-title":"J. Colloid Interface Sci."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"19056","DOI":"10.1021\/jp040386q","article-title":"Morphology of Voids in Molecular Systems. A Voronoi-Delaunay analysis of a simulated DMPC membrane","volume":"108","author":"Alinchenko","year":"2004","journal-title":"J. Phys. Chem. B"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"40444","DOI":"10.1038\/srep40444","article-title":"Rules of tissue packing involving different cell types: human muscle organization","volume":"7","author":"Paradas","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1888","DOI":"10.1038\/s41598-017-02166-5","article-title":"Self-assembled levitating clusters of water droplets: pattern-formation and stability","volume":"7","author":"Fedorets","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"5599","DOI":"10.1021\/acs.jpclett.7b02657","article-title":"Small levitating ordered droplet clusters: stability, symmetry, and Voronoi Entropy","volume":"8","author":"Fedorets","year":"2017","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1134\/1.1772434","article-title":"Droplet cluster","volume":"79","author":"Fedorets","year":"2004","journal-title":"JETP Lett."},{"key":"ref_59","first-page":"437","article-title":"On the Mechanism of non-coalescence in a droplet cluster","volume":"81","author":"Fedorets","year":"2005","journal-title":"JETP"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"988","DOI":"10.1134\/S1063785012110077","article-title":"Mechanism of stabilization of location of a droplet cluster above the liquid\u2013gas interface","volume":"38","author":"Fedorets","year":"2012","journal-title":"Tech. Phys. Lett."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1268","DOI":"10.1016\/j.ijheatmasstransfer.2016.09.087","article-title":"Generation of levitating droplet clusters above the locally heated water surface: A thermal analysis of modified installation","volume":"104","author":"Fedorets","year":"2017","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1017\/jfm.2011.510","article-title":"Three-dimensional Lagrangian Voronoi analysis for clustering of particles and bubbles in turbulence","volume":"693","author":"Tagawa","year":"2012","journal-title":"J. Fluid Mech."},{"key":"ref_63","unstructured":"Tolman, R.C. (1934). Relativity, Themodynamics and Cosmology, Oxford University Press."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"113","DOI":"10.3390\/e9030113","article-title":"Entropy of relativistic mono-atomic gas and temperature relativistic transformation in thermodynamics","volume":"9","author":"Bormashenko","year":"2007","journal-title":"Entropy"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"804","DOI":"10.1103\/PhysRev.43.804","article-title":"On the Constitution of metallic Sodium","volume":"43","author":"Wigner","year":"1933","journal-title":"Phys. Rev."},{"key":"ref_66","unstructured":"Ashcroft, N.W., and Mermin, N.D. (1976). Solid State Physics, Holt, Rinehart and Winston."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/12\/956\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:33:17Z","timestamp":1760196797000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/12\/956"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,12,11]]},"references-count":66,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["e20120956"],"URL":"https:\/\/doi.org\/10.3390\/e20120956","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201811.0535.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,12,11]]}}}