{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T04:00:36Z","timestamp":1782446436450,"version":"3.54.5"},"reference-count":38,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T00:00:00Z","timestamp":1682294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Hunan Province","award":["2022JJ40081"],"award-info":[{"award-number":["2022JJ40081"]}]},{"name":"Natural Science Foundation of Hunan Province","award":["6142A05QN22003"],"award-info":[{"award-number":["6142A05QN22003"]}]},{"name":"National Defense Science and Technology Key Laboratory Foundation","award":["2022JJ40081"],"award-info":[{"award-number":["2022JJ40081"]}]},{"name":"National Defense Science and Technology Key Laboratory Foundation","award":["6142A05QN22003"],"award-info":[{"award-number":["6142A05QN22003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Solidification patterns during nonequilibrium crystallization are among the most important microstructures in the natural and technical realms. In this work, we investigate the crystal growth in deeply supercooled liquid using the classical density functional-based approaches. Our result shows that the complex amplitude expanded phase-field crystal (APFC) model containing the vacancy nonequilibrium effects proposed by us could naturally reproduce the growth front nucleation (GFN) and various nonequilibrium patterns, including the faceted growth, spherulite, symmetric and nonsymmetric dendrites among others, at the atom level. Moreover, an extraordinary microscopic columnar-to-equiaxed transition is uncovered, which is found to depend on the seed spacing and distribution. Such a phenomenon could be attributed to the combined effects of the long-wave and short-wave elastic interactions. Particularly, the columnar growth could also be predicted by an APFC model containing inertia effects, but the lattice defect type in the growing crystal is different due to the different types of short-wave interactions. Two stages are identified during the crystal growth under different undercooling, corresponding to diffusion-controlled growth and GFN-dominated growth, respectively. However, compared with the second stage, the first stage becomes too short to be noticed under the high undercooling. The distinct feature of the second stage is the dramatic increments of lattice defects, which explains the amorphous nucleation precursor in the supercooled liquid. The transition time between the two stages at different undercooling is investigated. Crystal growth of BCC structure further confirms our conclusions.<\/jats:p>","DOI":"10.3390\/e25050708","type":"journal-article","created":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T02:06:35Z","timestamp":1682388395000},"page":"708","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Pattern Formation under Deep Supercooling by Classical Density Functional-Based Approach"],"prefix":"10.3390","volume":"25","author":[{"given":"Kun","family":"Wang","sequence":"first","affiliation":[{"name":"College of Materials Science and Engineering, Hunan University, Changsha 410082, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenjin","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Materials Science and Engineering, Hunan University, Changsha 410082, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shifang","family":"Xiao","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Hunan University, Changsha 410082, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jun","family":"Chen","sequence":"additional","affiliation":[{"name":"Institute of Applied Physics and Computational Mathematics, Beijing 100088, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wangyu","family":"Hu","sequence":"additional","affiliation":[{"name":"College of Materials Science and Engineering, Hunan University, Changsha 410082, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1038\/nmat5021","article-title":"Additively manufactured hierarchical stainless steels with high strength and ductility","volume":"17","author":"Wang","year":"2018","journal-title":"Nat. Mater."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/0022-0248(88)90169-8","article-title":"Bulk crystallization of liquid selenium Primary nucleation, growth kinetics and modes of crystallization","volume":"87","author":"Ryschenkow","year":"1988","journal-title":"J. Cryst. Growth"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3143","DOI":"10.1023\/A:1017974016928","article-title":"Review Spherulites: A personal perspective","volume":"36","author":"Magill","year":"2001","journal-title":"J. Mater. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1146\/annurev.physchem.51.1.99","article-title":"Spatially Heterogeneous Dynamics in Supercooled Liquids","volume":"51","author":"Ediger","year":"2000","journal-title":"Annu. Rev. Phys. Chem."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2727","DOI":"10.1103\/PhysRevLett.81.2727","article-title":"Length Scale of Dynamic Heterogeneities at the Glass Transition Determined by Multidimensional Nuclear Magnetic Resonance","volume":"81","author":"Tracht","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1038\/nmat815","article-title":"Growth of \u2018dizzy dendrites\u2019 in a random field of foreign particles","volume":"2","author":"Pusztai","year":"2003","journal-title":"Nat. Mater."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"042802","DOI":"10.1103\/PhysRevE.65.042802","article-title":"Nonequilibrium pattern formation in the crystallization of polymer blend films","volume":"65","author":"Ferreiro","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"011605","DOI":"10.1103\/PhysRevE.72.011605","article-title":"Growth and form of spherulites","volume":"72","author":"Pusztai","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1038\/nmat1190","article-title":"A general mechanism of polycrystalline growth","volume":"3","author":"Pusztai","year":"2004","journal-title":"Nat. Mater."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1694","DOI":"10.1007\/s11661-013-1988-0","article-title":"Phase-Field Modeling of Polycrystalline Solidification: From Needle Crystals to Spherulites\u2014A Review","volume":"45","author":"Korbuly","year":"2014","journal-title":"Metall. Mater. Trans. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"195502","DOI":"10.1103\/PhysRevLett.106.195502","article-title":"Faceting and Branching in 2D Crystal Growth","volume":"106","author":"Tegze","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"245701","DOI":"10.1103\/PhysRevLett.88.245701","article-title":"Modeling Elasticity in Crystal Growth","volume":"88","author":"Elder","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1007\/s00339-018-1800-5","article-title":"Phase-field crystal simulation facet and branch crystal growth","volume":"124","author":"Chen","year":"2018","journal-title":"Appl. Phys. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"012405","DOI":"10.1103\/PhysRevE.89.012405","article-title":"Phase-field-crystal simulation of nonequilibrium crystal growth","volume":"89","author":"Tang","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"21858","DOI":"10.1039\/D0CP03901B","article-title":"Phase-field crystal modeling of crystal growth patterns with competition of undercooling and atomic density","volume":"22","author":"Muhammad","year":"2020","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"175702","DOI":"10.1103\/PhysRevLett.107.175702","article-title":"Amorphous Nucleation Precursor in Highly Nonequilibrium Fluids","volume":"107","author":"Pusztai","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"225504","DOI":"10.1103\/PhysRevLett.96.225504","article-title":"Phase-Field Crystals with Elastic Interactions","volume":"96","author":"Stefanovic","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"051110","DOI":"10.1103\/PhysRevE.79.051110","article-title":"Phase-field-crystal and Swift-Hohenberg equations with fast dynamics","volume":"79","author":"Galenko","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"052801","DOI":"10.1103\/PhysRevE.95.052801","article-title":"Hydrodynamic theory of freezing: Nucleation and polycrystalline growth","volume":"95","author":"Podmaniczky","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"011601","DOI":"10.1103\/PhysRevE.74.011601","article-title":"Renormalization-group theory for the phase-field crystal equation","volume":"74","author":"Athreya","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"020601","DOI":"10.1103\/PhysRevE.72.020601","article-title":"Renormalization group approach to multiscale simulation of polycrystalline materials using the phase field crystal model","volume":"72","author":"Goldenfeld","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physd.2015.06.002","article-title":"Traveling wave profiles for a crystalline front invading liquid states: Analytical and numerical solutions","volume":"308","author":"Galenko","year":"2015","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"062802","DOI":"10.1103\/PhysRevE.102.062802","article-title":"Traveling waves of the solidification and melting of cubic crystal lattices","volume":"102","author":"Ankudinov","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"064113","DOI":"10.1103\/PhysRevB.83.064113","article-title":"Marginal stability analysis of the phase field crystal model in one spatial dimension","volume":"83","author":"Galenko","year":"2011","journal-title":"Phys. Rev. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"103386","DOI":"10.1016\/j.ijplas.2022.103386","article-title":"Exploring atomic mechanisms of microstructure evolutions in crystals under vacancy super- or undersaturation states by a kinetic amplitude-expanded phase-field-crystal approach","volume":"157","author":"Wang","year":"2022","journal-title":"Int. J. Plast."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1080\/00018732.2012.737555","article-title":"Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: An overview","volume":"61","author":"Emmerich","year":"2012","journal-title":"Adv. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Hansen, J.P., and Mcdonald, I.R. (2013). Theory of Simple Liquids: With Applications to Soft Matter, Academic press. [4th ed.].","DOI":"10.1016\/B978-0-12-387032-2.00012-X"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1103\/PhysRevA.15.319","article-title":"Hydrodynamic fluctuations at the convective instability","volume":"15","author":"Swift","year":"1977","journal-title":"Phys. Rev. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1080\/14786430903164572","article-title":"Density-amplitude formulation of the phase-field crystal model for two-phase coexistence in two and three dimensions","volume":"90","author":"Yeon","year":"2010","journal-title":"Philos. Mag."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"065105","DOI":"10.1103\/PhysRevE.80.065105","article-title":"Nonlinear elasticity of the phase-field crystal model from the renormalization group","volume":"80","author":"Chan","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"104106","DOI":"10.1103\/PhysRevB.88.104106","article-title":"Complex order parameter phase-field models derived from structural phase-field-crystal models","volume":"88","author":"Stolle","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"023301","DOI":"10.1103\/PhysRevE.96.023301","article-title":"Controlling the energy of defects and interfaces in the amplitude expansion of the phase-field crystal model","volume":"96","author":"Salvalaglio","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1145\/1268776.1268779","article-title":"deal.II\u2014A general-purpose object-oriented finite element library","volume":"33","author":"Bangerth","year":"2007","journal-title":"ACM Trans. Math. Softw."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"660","DOI":"10.1038\/nmat1693","article-title":"Orientation selection in dendritic evolution","volume":"5","author":"Haxhimali","year":"2006","journal-title":"Nat. Mater."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1080\/09506608.2018.1537090","article-title":"Progress in modelling solidification microstructures in metals and alloys: Dendrites and cells from 1700 to 2000","volume":"64","author":"Kurz","year":"2019","journal-title":"Int. Mater. Rev."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"035702","DOI":"10.1103\/PhysRevLett.103.035702","article-title":"Diffusion-Controlled Anisotropic Growth of Stable and Metastable Crystal Polymorphs in the Phase-Field Crystal Model","volume":"103","author":"Tegze","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"015501","DOI":"10.1103\/PhysRevLett.88.015501","article-title":"Crystallization Kinetics of Hard Spheres in Microgravity in the Coexistence Regime: Interactions between Growing Crystallites","volume":"88","author":"Cheng","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"205402","DOI":"10.1088\/0953-8984\/22\/20\/205402","article-title":"Extended phase diagram of the three-dimensional phase field crystal model","volume":"22","author":"Jaatinen","year":"2010","journal-title":"J. Phys. Condens. Matter"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/5\/708\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:22:40Z","timestamp":1760124160000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/5\/708"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,24]]},"references-count":38,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["e25050708"],"URL":"https:\/\/doi.org\/10.3390\/e25050708","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,24]]}}}