Journal of Materials Science

, Volume 54, Issue 13, pp 9907–9920 | Cite as

Predicting primary dendrite arm spacing in Al–Si–Mg alloys: effect of Mg alloying

  • Colin D. Ridgeway
  • Cheng Gu
  • Alan A. LuoEmail author


Directional solidification experiments were performed on Al–Si–Mg alloys to examine the effect of compositional variation on dendritic growth and to develop a novel growth model for the prediction of primary dendrite arm spacing in solidification microstructures. Instead of relying on the growth restriction factor or inoculant particle efficacy, this model examines solute effects to describe the final primary dendritic arm spacing. Increased Mg concentration was shown to decrease the dendritic growth velocity by decreasing constitutional undercooling. Solute enrichment at the solid/liquid interface was shown to limit lateral coarsening of primary dendrite arms and create a region of local solute depletion. This phenomenon allowed increased nucleation due to an increase in the resulting local liquidus temperature, and ultimately produced a refined primary dendrite arm spacing. Primary and secondary dendrites were measured and quantitatively analyzed in validation of the model. The model, which shows increased accuracy compared to existing models, was developed with aid of the three-dimensional cellular automaton method and experimentally verified.

List of symbols

\( C_{0} \)

Nominal composition

\( C_{x}^{0} \)

Composition of ‘x

\( C_{x}^{{{\mathrm{L}}*}} \)

Composition of element ‘x’ at the S/L interface

\( C_{i}^{E} \)

Solute concentration of element (i) in state of matter of (E)

\( D \)

Diffusion coefficient

\( d_{\mathrm{gs}} \)

Grain size

\( D_{ij}^{\mathrm{L}} \)

Solute diffusion coefficient in liquid aluminum

\( f \)

Fraction of particles that successfully nucleate a grain

\( \Delta f_{\mathrm{s}} \)

Fraction solid

\( G \)

Thermal gradient

\( \Gamma \)

Gibbs–Thomson coefficient

\( k \)

Partition coefficient

\( \kappa \)

Curvature of S/L interface



\( m \)

Liquidus slope

\( \rho \)

Number density of inoculant particles added to the melt

\( Q \)

Growth restriction factor


Time required to reach final dendritic spacing (completion of competitive growth process)

\( \sigma_{\mathrm{DAS}} \)

Primary and secondary dendrite arm strengthening

\( \sigma_{\mathrm{eutectic}} \)

Eutectic modification strengthening

\( \sigma_{\mathrm{GB}} \)

Grain boundary strengthening

\( \sigma_{i} \)

Intrinsic strength of a material

\( \sigma_{\mathrm{PPT}} \)

Precipitate strengthening

\( \sigma_{\mathrm{SS}} \)

Solid solution strengthening

\( \sigma_{\mathrm{YS}} \)

Yield strength of a material

\( \frac{\partial T}{{\partial C_{x}^{\mathrm{L}} }} \)

Slope of liquidus surface with respect to solute element x

\( T_{0} - T_{\mathrm{local}} \)

Thermal undercooling

\( \Delta T_{n} \)

Undercooling required for nucleation

\( \Delta t \)

Time step for CA

\( \mu_{\mathrm{k}} \)

Dendritic growth kinetic coefficient

\( v \)

Normal dendritic growth velocity

\( \Delta x \)

Mesh size used in CA



The authors would like to acknowledge the National Science Foundation for supporting this work (Award CMMI-1432688). The authors would also like to thank the members of OSU Light Metals and Manufacturing Research Lab for discussions and design assistance.


  1. 1.
    Yildirim M, Özyürek D (2013) The effects of Mg amount on the microstructure and mechanical properties of Al–Si–Mg alloys. Mater Des 51:767–774CrossRefGoogle Scholar
  2. 2.
    Caceres CH, Davidson CJ, Griffiths JR, Wang QG (1999) The effect of Mg on the microstructure and mechanical behavior of Al–Si–Mg casting alloys. Metall Mater Trans A 30:2611–2618CrossRefGoogle Scholar
  3. 3.
    Spear RE, Gardner GR (1963) Dendrite cell size. AFS Trans 71:209–215Google Scholar
  4. 4.
    Mccartney DG, Hunt JD (1981) Measurements of cell and primary dendrite. Acta Met 29:1851–1863CrossRefGoogle Scholar
  5. 5.
    Hunt JD, Lu SZ (1996) Numerical modeling of cellular/dendritic array growth: spacing and structure predictions. Metall Mater Trans A Phys Metall Mater Sci 27:611–623CrossRefGoogle Scholar
  6. 6.
    Greer AL, Bunn AM, Tronche A et al (2000) Modelling of inoculation of metallic melts: application to grain refinement of aluminium by Al–Ti–B. Acta Mater 48:2823–2835. CrossRefGoogle Scholar
  7. 7.
    Spittle JA, Sadli S (1995) Effect of alloy variables on grain refinement of binary aluminium alloys with Al–Ti–B. Mater Sci Technol 11:533–537. CrossRefGoogle Scholar
  8. 8.
    Luo AA (2015) Material design and development: from classical thermodynamics to CALPHAD and ICME approaches. CALPHAD: Comput Coupling Phase Diagr Thermochem 50:6–22. CrossRefGoogle Scholar
  9. 9.
    Twarog D, Apelian D, Luo AA (2016) High Integrity casting of lightweight components. In: NADCAGoogle Scholar
  10. 10.
    Fortini A, Merlin M, Fabbri E et al (2016) On the influence of Mn and Mg additions on tensile properties, microstructure and quality index of the A356 aluminum foundry alloy. Proc Struct Integr 2:2238–2245CrossRefGoogle Scholar
  11. 11.
    Matache G, Stefanescu DM, Puscasu C, Alexandrescu E (2016) Dendritic segregation and arm spacing in directionally solidified CMSX-4 superalloy. Int J Cast Met Res 29:303–316CrossRefGoogle Scholar
  12. 12.
    Stjohn DH, Qian M, Easton MA, Cao P (2011) The interdependence theory: the relationship between grain formation and nucleant selection. Acta Mater 59:4907–4921. CrossRefGoogle Scholar
  13. 13.
    Mitrasinovic AM, Robles Hernandez FC (2012) Determination of the growth restriction factor and grain size for aluminum alloys by a quasi-binary equivalent method. Mater Sci Eng A 540:63–69CrossRefGoogle Scholar
  14. 14.
    Prasad A, Yuan L, Lee PD, Stjohn DH (2013) The Interdependence model of grain nucleation: a numerical analysis of the Nucleation-Free Zone. Acta Mater 61:5914–5927. CrossRefGoogle Scholar
  15. 15.
    Stjohn DH, Prasad A, Easton MA, Qian M (2015) The contribution of constitutional supercooling to nucleation and grain formation. Metall Mater Trans A 46:4868–4885. CrossRefGoogle Scholar
  16. 16.
    Maxwell I, Hellawell A (1975) A simple model for grain refinement during solidification. Acta Metall 23:229–237CrossRefGoogle Scholar
  17. 17.
    Nastac L (1999) Numerical modeling of solidification morphologies and segregation patterns in cast dendritic alloys. Acta Mater 47:4253–4262. CrossRefGoogle Scholar
  18. 18.
    Pan S, Zhu M (2010) A three-dimensional sharp interface model for the quantitative simulation of solutal dendritic growth. Acta Mater 58:340–352. CrossRefGoogle Scholar
  19. 19.
    Luo S, Zhu MY (2013) A two-dimensional model for the quantitative simulation of the dendritic growth with cellular automaton method. Comput Mater Sci 71:10–18. CrossRefGoogle Scholar
  20. 20.
    Gu C, Ridgeway CD, Luo AA (2019) Examination of dendritic growth during solidification of ternary alloys via a novel quantitative 3D cellular automaton model. Metall Mater Trans B 50:123–135. CrossRefGoogle Scholar
  21. 21.
    Gu C, Lu Y, Cinkilic E et al (2019) Predicting grain structure in high pressure die casting of aluminum alloys: a coupled cellular automaton and process model. Comput Mater Sci 161:64–75. CrossRefGoogle Scholar
  22. 22.
    Dobravec T, Mavrič B, Šarler B (2017) A cellular automaton—finite volume method for the simulation of dendritic and eutectic growth in binary alloys using an adaptive mesh refinement. J Comput Phys 349:351–375. CrossRefGoogle Scholar
  23. 23.
    Zhu MF, Stefanescu DM (2007) Virtual front tracking model for the quantitative modeling of dendritic growth in solidification of alloys. Acta Mater 55:1741–1755. CrossRefGoogle Scholar
  24. 24.
    Wei L, Lin X, Wang M, Huang W (2012) A cellular automaton model for a pure substance solidification with interface reconstruction method. Comput Mater Sci 54:66–74. CrossRefGoogle Scholar
  25. 25.
    Gu C, Wei Y, Zhan X, Li Y (2017) A three-dimensional cellular automaton model of dendrite growth with stochastic orientation during the solidification in the molten pool of binary alloy. Sci Technol Weld Join 22:47–58. CrossRefGoogle Scholar
  26. 26.
    Yao X, Dahle AK, Davidson CJ, StJohn DH (2006) Effect of solute on the growth rate and the constitutional undercooling ahead of the advancing interface during solidification of an alloy and the implications for nucleation. J Mater Res 21:2470–2479CrossRefGoogle Scholar
  27. 27.
    Rappaz M, Boettinger WJ (1999) On dendritic solidification of multicomponent alloys with unequal liquid diffusion coefficients. Acta Mater 47:3205–3219. CrossRefGoogle Scholar
  28. 28.
    Chen R, Xu Q, Liu B (2015) Cellular automaton simulation of three-dimensional dendrite growth in Al-7Si-Mg ternary aluminum alloys. Comput Mater Sci 105:90–100. CrossRefGoogle Scholar
  29. 29.
    Chen R, Xu Q, Guo H et al (2017) Correlation of solidification microstructure refining scale, Mg composition and heat treatment conditions with mechanical properties in Al-7Si-Mg cast aluminum alloys. Mater Sci Eng A 685:391–402CrossRefGoogle Scholar
  30. 30.
    Cao X, Campbell J (2006) Morphology of β-Al5FeSi phase in Al–Si cast alloys. Mater Trans 47:1303–1312CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Materials Science and EngineeringThe Ohio State UniversityColumbusUSA
  2. 2.Department of Integrated Systems EngineeringThe Ohio State UniversityColumbusUSA

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