Journal of Materials Science

, Volume 52, Issue 10, pp 5544–5558 | Cite as

Phase-field modeling of eutectic structures on the nanoscale: the effect of anisotropy

  • László Rátkai
  • Gyula I. Tóth
  • László Környei
  • Tamás Pusztai
  • László Gránásy


A simple phase-field model is used to address anisotropic eutectic freezing on the nanoscale in two (2D) and three dimensions (3D). Comparing parameter-free simulations with experiments, it is demonstrated that the employed model can be made quantitative for Ag–Cu. Next, we explore the effect of material properties and the conditions of freezing on the eutectic pattern. We find that the anisotropies of kinetic coefficient and the interfacial free energies (solid–liquid and solid–solid), the crystal misorientation relative to pulling, the lateral temperature gradient play essential roles in determining the eutectic pattern. Finally, we explore eutectic morphologies, which form when one of the solid phases are faceted, and investigate cases, in which the kinetic anisotropy for the two solid phases is drastically different.


Anisotropy Liquid Interface Eutectic Structure Free Energy Density Regular Solution Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work has been supported by the National Agency for Research, Development, and Innovation (NKFIH), Hungary under contract OTKA-K-115959, and by the EU FP7 EU FP7 projects “ENSEMBLE” (Grant Agreement NMP4-SL-2008-213669) and “EXOMET” (contract No. NMP-LA-2012-280421, co-funded by ESA).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interests.


  1. 1.
    He G, Eckert J, Löser W, Schultz L (2003) Novel Ti-base nanostructure–dendrite composite with enhanced plasticity. Nat Mater 2:33–37CrossRefGoogle Scholar
  2. 2.
    Louzguine DV, Kato H, Inoue A (2004) High-strength hypereutectic Ti–Fe–Co bulk alloy with good ductility. Philos Mag Lett 84:359–364CrossRefGoogle Scholar
  3. 3.
    Park JM, Kim DH, Kim KB, Kim WT (2007) Deformation-induced rotational eutectic colonies containing length-scale heterogeneity in an ultrafine eutectic Fe83 Ti7 Zr6 B4 alloy. Appl Phys Lett 91:131907–1–131907–3Google Scholar
  4. 4.
    Park JM, Mattern N, Kühn U, Eckert J, Kim KB, Kim WT, Chattopadhyay K, Kim DH (2009) High-strength bulk Al-based bimodal ultrafine eutectic composite with enhanced plasticity. J Mater Res 24:2605–2609CrossRefGoogle Scholar
  5. 5.
    Samal S, Biswas K (2013) Novel high-strength NiCuCoTiTa alloy with plasticity. J Nanopart Res 15:1783-1–1783-11CrossRefGoogle Scholar
  6. 6.
    Pawlak DA, Kolodziejak K, Turczynski S, Kisielewski J, Rozniatowski K, Diduszko R, Kaczkan M, Malinowski M (2006) Self-organized, rodlike, micrometer-scale micro-structure of Tb3Sc2Al3O12–TbScO3: Pr eutectic. Chem Mater 18:2450–2457CrossRefGoogle Scholar
  7. 7.
    Pawlak DA, Kolodziejak K, Diduszko R, Rozniatowski K, Kaczkan M, Malinowski M, Kisielewski J, Lukasiewicz T (2007) The PrAlO3–Pr2O3 eutectic, its microstructure, instability, and luminescent properties. Chem Mater 19:2195–2202CrossRefGoogle Scholar
  8. 8.
    Pawlak DA, Kolodziejak K, Rozniatowski K, Diduszko R, Kaczkan M, Malinowski M, Piersa M, Kisielewski J, Lukasiewicz T (2008) PrAlO3–PrAl11O18 eutectic: its microstructure and spectroscopic properties. Cryst Growth Des 8:1243–1249CrossRefGoogle Scholar
  9. 9.
    Pawlak DA, Turczynski S, Gajc M, Kolodziejak K, Diduszko R, Rozniatowski K, Smalc J, Vendik I (2010) Metamaterials: How far are we from making metamaterials by self-organization? The microstructure of highly anisotropic particles with an SRR-like geometry. Adv Funct Mater 20:1116–1124CrossRefGoogle Scholar
  10. 10.
    Sadecka K, Gajc M, Orlinski K, Surma HB, Klos A, Joywik-Biala I, Sobczak K, Dluyewski P, Toudert J, Pawlak DA (2015) When eutectics meet plasmonics: nanoplasmonic, volumetric, self-organized, silver-based eutectic. Adv Opt Mater 3:381–389CrossRefGoogle Scholar
  11. 11.
    Kolodziejak K, Gajc M, Rozniatowski K, Diduszko R, Pawlak DA (2016) Synthesis and structural study of a self-organized MnTiO3–TiO2 eutectic. J. Alloys Compd 659:152–158CrossRefGoogle Scholar
  12. 12.
    Merino RI, Acosta MF, Orera VM (2012) New polaritonic materials in the THz range made of directionally solidified halide eutectics. J Eur Ceram Soc 34:2061–2069CrossRefGoogle Scholar
  13. 13.
    Pawlak DA (2008–2012) EU FP7 Collaborative Project ENSEMBLE, CF-FP 213669, Grant Agreement, Annex I.
  14. 14.
    Oliete PB, Peña JI, Larrea A, Orera VM, Lorca JL, Pastor JY, Martín A, Segurado J (2007) Ultra-high-strength nanofibrillar Al2O3–YAG–YSZ eutectics. Adv Mater 19:2313–2318CrossRefGoogle Scholar
  15. 15.
    Orera VM, Merino RI, Pardo JA, Larrea A, de la Fuente G, Contreras L, Peña JI (2000) Oxide eutectics: role of interfaces in the material properties. Acta Phys Slovaca 50:549–557Google Scholar
  16. 16.
    Pastor Y, Martin A, Molina-Aldareguia JM, Llorca J, Oliete PB, Larrea A, Peña JI, Orera VM, Arenal R (2013) Superplastic deformation of directionally solidified nanofibrillar Al2O3–Y3Al5O12–ZrO2 eutectics. J Eur Ceram Soc 33:2579–2586CrossRefGoogle Scholar
  17. 17.
    Fullman RL, Wood DJ (1954) Origin of spiral eutectic structures. Acta Metall 2:188–193CrossRefGoogle Scholar
  18. 18.
    Steinbach I (2009) Phase-field models in materials science. Model Simul Mater Sci Eng 17:073001-1–073001-31CrossRefGoogle Scholar
  19. 19.
    Nestler B, Choudhury A (2011) Phase-field modeling of multi-component systems. Curr Opin Solid State and Mater Sci 15:93–105CrossRefGoogle Scholar
  20. 20.
    Akamatsu S, Plapp M (2016) Eutectic and peritectic solidification patterns. Curr Opin Solid State and Mater Sci 20:46–54CrossRefGoogle Scholar
  21. 21.
    Boettinger WJ (2016) The solidification of multicomponent alloys. J Phase Equilib Diffus 37:4–18CrossRefGoogle Scholar
  22. 22.
    Akamatsu S, Bottin-Rousseau S, Serefoglu M, Faivre G (2012) A theory of thin lamellar eutectic growth with anisotropic interphase boundaries. Acta Mater 60:3199–3205CrossRefGoogle Scholar
  23. 23.
    Akamatsu S, Bottin-Rousseau S, Serefoglu M, Faivre G (2012) Lamellar eutectic growth with anisotropic interphase boundaries experimental study using the rotating directional solidification method. Acta Mater 60:3206–3214CrossRefGoogle Scholar
  24. 24.
    Ghosh S, Choudhury A, Plapp M, Bottin-Rousseau S, Faivre G, Akamatsu S (2015) Interphase anisotropy effects on lamellar eutectics: a numerical study. Phys Rev E 91:022407–1–022407–13Google Scholar
  25. 25.
    Lahiri A, Choudhury A (2015) Effect of surface energy anisotropy on the stability of growth fronts in multiphase alloys. Trans Indian Inst Met 68:1053–1057CrossRefGoogle Scholar
  26. 26.
    Rátkai L, Szállás A, Pusztai T, Mohri T, Gránásy L (2015) Ternary eutectic dendrites: pattern formation and scaling properties. J Chem Phys 142:154501–1–154501–11CrossRefGoogle Scholar
  27. 27.
    Llorca J, Orera V (2006) Directionally solidified eutectic ceramic oxides. Prog Mater Sci 51:711–809CrossRefGoogle Scholar
  28. 28.
    Karma A (2001) Phase-field formulation for quantitative modeling of alloy solidification. Phys Rev Lett 87:115701–1–115701–4Google Scholar
  29. 29.
    Echebarria B, Folch R, Karma A, Plapp M (2004) Quantitative phase-field model of alloy solidification. Phys Rev E 73:061604–1–061604–22Google Scholar
  30. 30.
    Hötzer J, Jainta M, Steinmetz P, Nestler B, Dennstedt A, Genau A, Bauer M, Köstler H, Rüde U (2015) Large scale phase-field simulations of directional ternary eutectic solidification. Acta Mater 93:194–204CrossRefGoogle Scholar
  31. 31.
    Folch R, Plapp M (2005) Quantitative phase-field modeling of two-phase growth. Phys Rev E 72:011602–1–011602–27Google Scholar
  32. 32.
    Kim SG (2007) A phase-field model with antitrapping current for multicomponent alloys with arbitrary thermodynamic properties. Acta Mater 55:4391–4399CrossRefGoogle Scholar
  33. 33.
    Gránásy L, Börzsönyi T, Pusztai T (2002) Nucleation and bulk crystallization in binary phase field theory. Phys Rev Lett 88:206105–1–206105–4CrossRefGoogle Scholar
  34. 34.
    Warren JA, Boettinger WJ (1995) Prediction of dendritc growth and microsegregation patterns in a binary alloy using the phase-field method. Acta Metall Mater 43:689–703CrossRefGoogle Scholar
  35. 35.
    Levitas VI, Warren JA (2016) Phase field approach with anisotropic interface energy and interface stresses: large strain formulation. J Mech Phys Solids 91:94–125CrossRefGoogle Scholar
  36. 36.
    Hoyt JJ, Asta M, Karma A (2003) Atomistic and continuum modeling of dendritic solidification. Mater Sci Eng R 41:121–163CrossRefGoogle Scholar
  37. 37.
    Gránásy L, Pusztai T, Warren JA, Douglas JF, Börzsönyi T, Ferreiro V (2003) Growth of ‘dizzy dendrites’ in a random field of foreign particles. Nat Mater 2:92–96CrossRefGoogle Scholar
  38. 38.
    Gránásy L, Pusztai T, Börzsönyi T, Warren JA, Douglas JF (2004) A general mechanism of polycrystalline growth. Nat Mater 3:645–650CrossRefGoogle Scholar
  39. 39.
    Haxhimali T, Karma A, Gonzales F, Rappaz M (2006) Orientation selection in dendritic evolution. Nat Mater 5:660–664CrossRefGoogle Scholar
  40. 40.
    Davidchack R, Laird BB (2005) Direct calculation of the crystal-melt interfacial free energy via molecular dynamics computer simulation. J Phys Chem B 109:17802–17812CrossRefGoogle Scholar
  41. 41.
    Hoyt JJ, Asta M, Karma A (2001) Method for computing the anisotropy of the solid-liquid interfacial free energy. Phys Rev Lett 86:5530–5533CrossRefGoogle Scholar
  42. 42.
    Majaniemi S, Provatas N (2009) Deriving surface-energy anisotropy for phenomenological phase-field models of solidification. Phys Rev E 79:011607–1–011607–12CrossRefGoogle Scholar
  43. 43.
    Hartel A, Oettel M, Rozas RE, Egelhaff SU, Horbach J, Löwen H (2012) Tension and stiffness of the hard sphere crystal-fluid interface. Phys Rev Lett 108:226101–1–226101–4CrossRefGoogle Scholar
  44. 44.
    Podmaniczky F, Tóth GI, Pusztai T, Gránásy L (2014) Free energy of the bcc-liquid interface and the Wulff shape as predicted by the phase-field crystal model. J Cryst Growth 385:148–153CrossRefGoogle Scholar
  45. 45.
    Tegze G, Bansel G, Tóth GI, Pusztai T, Fan Z, Gránásy L (2009) Advanced operator-splitting-based semi-implicit spectral method to solve the binary phase-field crystal equation with variable coefficients. J Comput Phys 228:1612–1623CrossRefGoogle Scholar
  46. 46.
    Andersson JO, Helander T, Höglund L, Shi PF, Sundman B (2002) Thermo-Calc and DICTRA, computational tools for materials science. Calphad 26:273–312. Thermo-Calc Software,
  47. 47.
    Gránásy L, Tegze M (1991) Crystal-melt interfacial free energy of elements and alloys. Mater Sci Forum 77:243–256CrossRefGoogle Scholar
  48. 48.
    Singman CN (1984) Atomic volume and allotropy of the elements. J Chem Educ 61:137–142CrossRefGoogle Scholar
  49. 49.
    Cline HE, Lee D (1970) Strengthening of lamellar versus equiaxed Ag–Cu eutectic. Acta Metall 18:315–323CrossRefGoogle Scholar
  50. 50.
    Jackson KA, Hunt JD (1966) Lamellar and rod eutectic growth. Trans Metall Soc AIME 236:1129–1142Google Scholar
  51. 51.
    Galenko PK, Herlach DM (2006) Diffusionless crystal growth in rapidly solidifying eutectic systems. Phys Rev Lett 96:150602–1–150602–4CrossRefGoogle Scholar
  52. 52.
    Kobayashi R, Giga Y (2001) On anisotropy and curvature effects for growing crystals. Jpn J Ind Appl Math 18:207–230CrossRefGoogle Scholar
  53. 53.
    Jackson KA, Hunt JD (2011) Eutectic solidification in transparent materials, in Jackson KA Crystal Growth (video). Jackson KA (2011) Personal communicationGoogle Scholar
  54. 54.
    Wang N, Trivedi R (2011) Limit of steady-state lamellar eutectic growth. Scr Mater 64:848–851CrossRefGoogle Scholar
  55. 55.
    Perrut M, Parisi A, Akamatsu S, Botin-Rousseau S, Faivre G, Plapp M (2010) Role of transverse temperature gradients in the generation of lamellar eutectic solidification patterns. Acta Mater 58:1761–1769CrossRefGoogle Scholar
  56. 56.
    Akamatsu S, Perrut M, Bottin-Rousseau S, Faivre G (2010) Spiral two-phase dendrites. Phys Rev Lett 104:056101–1–056101–4CrossRefGoogle Scholar
  57. 57.
    Pusztai T, Rátkai L, Szállás A, Gránásy L (2013) Spiraling eutectic dendrites. Phys Rev E 87:032401–1–032401–4CrossRefGoogle Scholar
  58. 58.
    Akamatsu S, Perrut M, Bottin-Rousseau S, Brener EA (2014) Scaling theory of two-phase dendritic growth in undercooled ternary melts. Phys Rev Lett 112:105502–1–105502–4Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Wigner Research Centre for PhysicsBudapestHungary
  2. 2.Department of Physics and TechnologyUniversity of BergenBergenNorway
  3. 3.Department of Mathematics and Computational SciencesSzéchenyi István UniversityGyőrHungary
  4. 4.Brunel Centre for Advanced Solidification TechnologyBrunel UniversityUxbridge, MiddlesexUK

Personalised recommendations