Metallurgical and Materials Transactions A

, Volume 46, Issue 1, pp 377–395 | Cite as

Multigrain and Multiphase Mathematical Model for Equiaxed Solidification

  • Marcelo Aquino MartoranoEmail author
  • Davi Teves Aguiar
  • Juan Marcelo Rojas Arango


A deterministic multigrain and multiphase model of equiaxed solidification of binary alloys is proposed, implemented, and analyzed. An important feature of the present model is the creation of classes of dendritic and globulitic grains according to their instantaneous sizes during solidification. Globulitic and dendritic grain growth, coarsening of secondary dendrite arms, distribution of nucleation undercoolings, and equiaxed eutectic growth are consistently included in the model equations. Important model assumptions are uniform temperature, negligible liquid convection, and negligible grain movement. Calculated cooling curves, solid fraction evolution, average grain sizes, and eutectic fractions agree well with predictions of previous models for dendritic and globulitic equiaxed grains. Predicted grain sizes decrease with an increase in cooling rate for an Al-2.12 pct Cu alloy and with an increase in Si concentration up to 3 pct for Al-Si alloys, agreeing quantitatively with experimental results. Simulations for an Al-7 pct Si alloy predict that an increase in grain size correlates with an increase in the magnitude of the recalescence observed in cooling curves. These calculations agree well with experimental results when the transition from a globulitic to a dendritic morphology occurs in the model before the minimum temperature of recalescence is reached.


Representative Elementary Volume Growth Velocity Interdendritic Liquid Eutectic Cell Eutectic Solidification 
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.



Area of interface (m2) or constant of coarsening law (m sa )


Area of REV boundary (m2)


Constant of eutectic growth law (m s−1 K−2)


Constant of coarsening law


Boundary of REV


Mass fraction of solute (–)

\( \bar{C} \)

Surface average of solute mass fraction (–)


Initial solute mass fraction (–)

\( \left\langle {C_{di} } \right\rangle^{di} \)

Average of solute mass fraction in interdendritic liquid of grains in class i (–)


Mass fraction of solute in the external liquid (–)


Mass fraction of solute in bulk liquid (–)

\( \left\langle {C_{l} } \right\rangle^{l} \)

Average of solute mass fraction in external liquid (–)


Solute mass fraction of the liquidus line (–)


Solute mass fraction of the liquidus line for globulitic grains of class i (–)


Volumetric specific heat (J m−3 K−1)

\( \left\langle {C_{si} } \right\rangle^{si} \)

Average of solute mass fraction in the solid of grains in class i (–)


Interface between eutectic and interdendritic liquid in grains of class i


Coefficient of solute diffusion (m2 s−1)


Complementary error function


Inverse of the Ivantsov function

\( \vec{j}_{k} \)

Diffusive flux of solute (kg m−2 s−1)

k, j

Constituents of the multiphase model


Solute partition coefficient (–)

\( \bar{l} \)

Final average grain size (m)


Volumetric latent heat (J m−3)


Slope of the liquidus line (K pct mass−1)


Total number density of grains of primary solid (m−3)

\( \vec{n}_{k} \)

Normal unit vector at the interfaces of k and pointing out of V k (–)


Number density of eutectic cells (m−3)


Extended number density of eutectic cells (m−3)


Number density of grains in class i (m−3)


Extended number density of grain nuclei in class i (m−3)


Total number density of substrate particles for heterogeneous nucleation (m−3)


Number of existing grain classes


Maximum number of possible grain classes


Peclet number for grains in class i (–)


Average heat flux out of REV (J m−2 s−1)


Radial coordinate (m)

\( \dot{R} \)

Cooling rate of the liquid before solidification (K s−1)


Representative elementary volume


Extended radius of eutectic cells (m)


Radius of spherical unit cell for grains in class i (m)


Radius of grain envelopes in class i (m)


Extended radius of grains in class i (m)


Concentration of interfacial area (m−1)


Time (s)


Time of eutectic nucleation (s)


Time of nucleation of grains in class i (s)


Temperature (K)


Initial temperature (K)


Eutectic temperature (K)


Melting point of the pure metal (K)


Volume of the REV (m3)


Volume of constituent k (m3)

\( \bar{w} \)

Average normal interface velocity (m s−1)

\( \vec{w}_{k} \)

Velocity vector of an interface of constituent k (m s−1)

Greek symbols


Thickness of effective diffusion layer (m)


Gibbs–Thomson coefficient (m K)


Time step of the numerical method (s)


Undercooling of the external liquid (K)


Undercooling for nucleation of eutectic cells (K)


Undercooling for nucleation of the grains in class i (K)


Undercooling range for nucleation of the primary phase (K)

\( \overline{{\Delta T_{N} }} \)

Average nucleation undercooling (K)


Standard deviation of the nucleation undercooling distribution (K)


Volume fraction (–)


Average spacing between secondary dendrite arms in grains of class i (m)


Initial spacing between secondary dendrite arms in grains of class i (m)


Mass density (kg m−3)


Standard deviation of particle size distribution (–)


Geometrical mean diameter of particle size distribution (μm)


General field variable defined in constituent k


Dimensionless undercooling (–)



Interdendritic liquid of grains in class i




External eutectic


Grains of class i


Index of a grain class


External liquid


Interface between external liquid and interdendritic liquid of grains in class i


Interface between external liquid and eutectic


Interface between external liquid and grain envelopes of class i


Interface between external liquid and the solid of grains in class i


Interface between interdendritic liquid and solid of grains in class i


Primary solid


Primary solid of grains in class i



The authors thank the financial support from FAPESP (03/08576-7) and CNPq (475451/04-0).


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Copyright information

© The Minerals, Metals & Materials Society and ASM International 2014

Authors and Affiliations

  • Marcelo Aquino Martorano
    • 1
    Email author
  • Davi Teves Aguiar
    • 2
  • Juan Marcelo Rojas Arango
    • 1
    • 3
  1. 1.Department of Metallurgical and Materials EngineeringUniversity of São PauloSão PauloBrazil
  2. 2.Materials DepartmentPetróleo Brasileiro S.A.Rio De JaneiroBrazil
  3. 3.University of AntioquiaMedellínColombia

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