Journal of Materials Science

, Volume 53, Issue 13, pp 9755–9770 | Cite as

Primary dendrite spacing selection during directional solidification of multicomponent nickel-based superalloy: multiphase-field study

  • Cong Yang
  • Qingyan Xu
  • Baicheng Liu


The primary dendrite spacing selection in a multicomponent Ni-based superalloy during directional solidification was systematically studied using two-dimensional phase-field simulations. The alloy thermodynamic and kinetic data were obtained from Pandat software with PanNickel database and directly coupled into the multiphase-field model. All the simulations were performed on a GPU server, and an optimized computing scheme using GPU shared memory was adopted. First, the morphology of the solidification front was studied, and the segregation pattern was investigated and compared with the experimental results. Then, the dendritic spacing distribution under a wide range of pulling velocities Vp (10–500 μm s−1) and temperature gradients G (2–200 K mm−1) was obtained and analyzed. The simulation results agree well with analytical model that the primary dendrite spacing scales as \( \varLambda \propto V_{\text{p}}^{ - b} G^{ - c} \). The coefficient b is near a constant value of 0.38 and varies slightly between 0.34 and 0.42, while coefficient c increases monotonously from 0.27 to 0.56 with the increasing G. The predicted dendritic spacing agrees well with the experimental data, but exhibits a major difference when under very low cooling rate (R < 0.1 K s−1). The effect of grain inclination angle θ on the final primary dendritic spacing was also studied, and an abnormal decrease in dendritic spacing was found under low grain orientation where θ < 10°. When the grain inclination angle exceeds 20°, the dendritic spacing increases with θ as the power law.



This research was funded by the National Key Research and Development Program of China (2017YFB0701503), National Science and Technology Major Project (No. 2017ZX04014001) and the National Natural Science Foundation of China (No. 51374137).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Versnyder FI, Shank M (1970) The development of columnar grain and single crystal high temperature materials through directional solidification. Mater Sci Eng 6:213CrossRefGoogle Scholar
  2. 2.
    Giamei AF, Tschinkel JG (1976) Liquid metal cooling: a new solidification technique. Metall Trans A 7:1427CrossRefGoogle Scholar
  3. 3.
    Liu L, Huang T, Zhang J, Fu H (2007) Microstructure and stress rupture properties of single crystal superalloy CMSX-2 under high thermal gradient directional solidification. Mater Lett 61:227CrossRefGoogle Scholar
  4. 4.
    Quested P, McLean M (1984) Solidification morphologies in directionally solidified superalloys. Mater Sci Eng 65:171CrossRefGoogle Scholar
  5. 5.
    Whitesell H, Li L, Overfelt R (2000) Influence of solidification variables on the dendrite arm spacings of Ni-based superalloys. Metall Mater Trans B 31:546CrossRefGoogle Scholar
  6. 6.
    Li LC (2002) Microstructural development and segregation effects in directionally solidified nickel-based superalloy PWA 1484. ProQuest Dissertations and Theses, Ph.D. thesis, Auburn University, pp 65–86Google Scholar
  7. 7.
    Konter M, Thumann M (2001) Materials and manufacturing of advanced industrial gas turbine components. J Mater Process Technol 117:386CrossRefGoogle Scholar
  8. 8.
    Reed RC (2008) The superalloys: fundamentals and applications. Cambridge University Press, CambridgeGoogle Scholar
  9. 9.
    Wagner A, Shollock BA, Mclean M (2004) Grain structure development in directional solidification of nickel-base superalloys. Mater Sci Eng A 374:270CrossRefGoogle Scholar
  10. 10.
    Dsouza N, Ardakani M, Wagner A, Shollock B, McLean M (2002) Morphological aspects of competitive grain growth during directional solidification of a nickel-base superalloy, CMSX4. J Mater Sci 37:481. CrossRefGoogle Scholar
  11. 11.
    Liang Z, Xu Q, Li J, Li S, Zhang J, Liu B, Zhong Z (2002) Experimental research on the near net shape casting process of gamma titanium aluminide turbochargers. Rare Met Mater Eng 31:353Google Scholar
  12. 12.
    Hunt J (1979) Solidification and casting of metals. In: Proceedings of conference, Sheffield, England, July 1977Google Scholar
  13. 13.
    Kurz W, Fisher D (1981) Dendrite growth at the limit of stability: tip radius and spacing. Acta Metall 29:11CrossRefGoogle Scholar
  14. 14.
    Gandin CA, Eshelman M, Trivedi R (1996) Orientation dependence of primary dendrite spacing. Metall Mater Trans A 27:2727CrossRefGoogle Scholar
  15. 15.
    Schneider MC, Gu JP, Beckermann C, Boettinger WJ, Kattner UR (1997) Modeling of micro-and macrosegregation and freckle formation in single-crystal nickel-base superalloy directional solidification. Metall Mater Trans A 28:1517CrossRefGoogle Scholar
  16. 16.
    Karma A, Rappel W-J (1996) Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys Rev E 53:R3017CrossRefGoogle Scholar
  17. 17.
    Kim SG, Kim WT, Suzuki T (1999) Phase-field model for binary alloys. Phys Rev E 60:7186CrossRefGoogle Scholar
  18. 18.
    Steinbach I, Pezzolla F (1999) A generalized field method for multiphase transformations using interface fields. Physica D 134:385CrossRefGoogle Scholar
  19. 19.
    Eiken J, Böttger B, Steinbach I (2006) Multiphase-field approach for multicomponent alloys with extrapolation scheme for numerical application. Phys Rev E 73:066122CrossRefGoogle Scholar
  20. 20.
    Tang J, Xue X (2009) Phase-field simulation of directional solidification of a binary alloy under different boundary heat flux conditions. J Mater Sci 44:745. CrossRefGoogle Scholar
  21. 21.
    Warnken N, Ma D, Drevermann A, Reed RC, Fries S, Steinbach I (2009) Phase-field modelling of as-cast microstructure evolution in nickel-based superalloys. Acta Mater 57:5862CrossRefGoogle Scholar
  22. 22.
    Böttger B, Eiken J, Apel M (2015) Multi-ternary extrapolation scheme for efficient coupling of thermodynamic data to a multi-phase-field model. Comput Mater Sci 108:283CrossRefGoogle Scholar
  23. 23.
    Yang C, Xu Q, Liu B (2017) GPU-accelerated three-dimensional phase-field simulation of dendrite growth in a nickel-based superalloy. Comput Mater Sci 136:133CrossRefGoogle Scholar
  24. 24.
    Takaki T, Ohno M, Shimokawabe T, Aoki T (2014) Two-dimensional phase-field simulations of dendrite competitive growth during the directional solidification of a binary alloy bicrystal. Acta Mater 81:272CrossRefGoogle Scholar
  25. 25.
    Takaki T, Sakane S, Ohno M, Shibuta Y, Shimokawabe T, Aoki T (2016) Primary arm array during directional solidification of a single-crystal binary alloy: large-scale phase-field study. Acta Mater 118:230CrossRefGoogle Scholar
  26. 26.
    Kim SG (2007) A phase-field model with antitrapping current for multicomponent alloys with arbitrary thermodynamic properties. Acta Mater 55:4391CrossRefGoogle Scholar
  27. 27.
    Carré A, Böttger B, Apel M (2013) Implementation of an antitrapping current for a multicomponent multiphase-field ansatz. J Cryst Growth 380:5CrossRefGoogle Scholar
  28. 28.
    Karma A (2001) Phase-field formulation for quantitative modeling of alloy solidification. Phys Rev Lett 87:115701CrossRefGoogle Scholar
  29. 29.
    Wang W, Lee PD, Mclean M (2003) A model of solidification microstructures in nickel-based superalloys: predicting primary dendrite spacing selection. Acta Mater 51:2971CrossRefGoogle Scholar
  30. 30.
    Diepers H-J, Ma D, Steinbach I (2002) History effects during the selection of primary dendrite spacing. Comparison of phase-field simulations with experimental observations. J Cryst Growth 237:149CrossRefGoogle Scholar
  31. 31.
    Ganesan M, Dye D, Lee P (2005) A technique for characterizing microsegregation in multicomponent alloys and its application to single-crystal superalloy castings. Metall Mater Trans A 36:2191CrossRefGoogle Scholar
  32. 32.
    Parsa AB, Wollgramm P, Buck H, Somsen C, Kostka A, Povstugar I, Choi PP et al (2015) Advanced scale bridging microstructure analysis of single crystal Ni-base superalloys. Adv Eng Mater 17:216CrossRefGoogle Scholar
  33. 33.
    Seo S, Lee J, Yoo Y, Jo C, Miyahara H, Ogi K (2011) A comparative study of the γ/γ′eutectic evolution during the solidification of Ni-base superalloys. Metall Mater Trans A 42:3150CrossRefGoogle Scholar
  34. 34.
    Eiken J, Apel M, Liang SM, Schmid-Fetzer R (2015) Impact of P and Sr on solidification sequence and morphology of hypoeutectic Al–Si alloys: combined thermodynamic computation and phase-field simulation. Acta Mater 98:152CrossRefGoogle Scholar
  35. 35.
    Higuchi K, Fecht HJ, Wunderlich RK (2010) Surface tension and viscosity of the Ni-based superalloy CMSX-4 measured by the oscillating drop method in parabolic flight experiments. Adv Eng Mater 9:349CrossRefGoogle Scholar
  36. 36.
    Elliott AJ, Pollock TM, Tin S, King WT, Huang SC, Gigliotti MFX (2004) Directional solidification of large superalloy castings with radiation and liquid-metal cooling: a comparative assessment. Metall Mater Trans A 35:3221CrossRefGoogle Scholar
  37. 37.
    Heckl A, Rettig R, Cenanovic S, Göken M, Singer R (2010) Investigation of the final stages of solidification and eutectic phase formation in Re and Ru containing nickel-base superalloys. J Cryst Growth 312:2137CrossRefGoogle Scholar
  38. 38.
    Tien J, Gamble R (1971) The suppression of dendritic growth in nickel-base superalloys during unidirectional solidification. Mater Sci Eng 8:152CrossRefGoogle Scholar
  39. 39.
    Beckermann C, Diepers H-J, Steinbach I, Karma A, Tong X (1999) Modeling melt convection in phase-field simulations of solidification. J Comput Phys 154:468CrossRefGoogle Scholar
  40. 40.
    Clarke AJ, Tourret D, Song Y, Imhoff SD, Gibbs PJ, Gibbs JW, Fezzaa K et al (2017) Microstructure selection in thin-sample directional solidification of an Al–Cu alloy: in situ X-ray imaging and phase-field simulations. Acta Mater 129:203CrossRefGoogle Scholar
  41. 41.
    Takaki T, Rojas R, Sakane S, Ohno M, Shibuta Y, Shimokawabe T, Aoki T (2017) Phase-field-lattice Boltzmann studies for dendritic growth with natural convection. J Cryst Growth 474:146CrossRefGoogle Scholar
  42. 42.
    Zhang H, Xu QY, Sun CB, Qi X, Tang N, Liu BC (2013) Simulation and experimental studies on grain selection behavior of single crystal superalloy II. Spiral part. Acta Metall Sin 49:1521CrossRefGoogle Scholar
  43. 43.
    Zhang H, Xu QY, Tang N, Pan D, Liu BC (2011) Numerical simulation of microstructure evolution during directional solidification process in directional solidified (DS) turbine blades. Sci China Technol Sci 54:3191CrossRefGoogle Scholar
  44. 44.
    Matan N, Cox D, Carter P, Rist M, Rae C, Reed R (1999) Creep of CMSX-4 superalloy single crystals: effects of misorientation and temperature. Acta Mater 47:1549CrossRefGoogle Scholar
  45. 45.
    Wang L, Liu Y, Yu J, Xu Y, Sun X, Guan H, Hu Z (2009) Orientation and temperature dependence of yielding and deformation behavior of a nickel-base single crystal superalloy. Mater Sci Eng A 505:144CrossRefGoogle Scholar
  46. 46.
    Liu B, Xu Q, Jing T, Shen H, Han Z (2011) Advances in multi-scale modeling of solidification and casting processes. JOM 63:19CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory for Advanced Materials Processing Technology, Ministry of Education, School of Materials Science and EngineeringTsinghua UniversityBeijingChina

Personalised recommendations