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A multiphase approach is used to study the macrosegregation phenomena that occur during solidification. Some modeling examples with accompanying animations are presented in this article to increase the understanding of different mechanisms of macrosegregation formation. Examples are presented consecutively with increasing complexity of the mechanisms: macrosegregation in columnar solidification, macrosegregation in globular equiaxed solidification, macrosegregation in the mixed equiaxed-columnar solidification, and Marangoni convection induced macrosegregation.

INTRODUCTION

Knowledge of macrosegregation phenomena in castings has increased significantly in the past few decades, including the development of different analytical and numerical models to study and predict such phenomena in different casting processes.^1–3 Macrosegregation occurs due to the relative motion between different phases during solidification. This motion can arise as a result of thermosolutal convection, solidification-shrinkage-induced flow, flotation and sedimentation of free-moving grains, forced flow by stirring mechanically or electromagnetically, flow caused by pore or gas bubble formation, deformation of the solid framework, capillary (Marangoni) force induced flow, etc.^1–5 As summarized by C. Beckermann: “while some successes have been reported in predicting measured macrosegregation patterns in industrially relevant casting processes, there are still numerous areas where further development is required.”² Obviously, this article cannot cover the entire range of macrosegregation phenomena, thus the current focus is limited to the authors’ recent contributions to this topic. As such, specific examples of macrosegregation in columnar solidification, in globular equiaxed solidification, in mixed equiaxed-columnar solidification, and Marangoni-induced macrosegregation are presented and discussed. Details about the numerical models are presented elsewhere.^6–10

The benchmark of a steel ingot with reduced size is simulated in Figure 1. A two-phase columnar solidification model is used. The purpose is to study the macrosegregation due to the mechanism of thermosolutal convection during columnar solidification. Details about process and thermo-physical parameters can be found in previous publications.^5,8 The important model assumptions are summarized as follows:

The positive macrosegregation in the center is formed gradually during solidification. The interdendritic melt always has a higher concentration than the bulk melt. The interdendritic solute-enriched melt is brought out of the mushy zone by the flow current, causing the c_mix in front of, or slightly behind, the columnar tip front to be enriched gradually. These positively segregated areas of the melt are not stationary; they move with the flow current and finally meet in the casting center, forming a large positive segregation zone.

Figure 2 illustrates the macrosegregation formation mechanism in the corner regions. A local volume taken from the upper corner of the ingot is considered in Figure 2a. The solidified columnar trunks are stationary, while the interdendritic melt flows (e.g., the melt flows in from a horizontal side of the volume and out from a vertical). Due to solidification the interdendritic melt is segregated with solute elements (k < 1), c_l is larger than the bulk melt concentration c₀. Melt with c_lⁱⁿ > c₀ flows out of the volume, being replaced with “fresh” melt with nearly the bulk concentration c_lⁱⁿ ≈ c₀. The consequence is that c_mix in the volume decreases (i.e., a negative macrosegregation occurs). In the lower bottom corners, Figure 2b, melt with c_lⁱⁿ > c₀ flows vertically into the volume. This incoming melt is enriched in solute due to segregation in the solidifying interdendritic region along the mold wall. As the solidification now takes place from that segregated melt, the solid forms with a higher concentration k · c_lⁱⁿ compared to k · c_lⁱⁿ, thus a positive macrosegregation forms.

The benchmark of an Al-4.0wt.%Cu die casting with a two-phase globular equiaxed solidification model is simulated, as shown in Figure 3. The purpose is to study the macrosegregation formed by mechanisms of grain sedimentation, and sedimentation-induced convection during equiaxed solidification. Details about process and thermophysical parameters can be found in previous publications.^6.12 The general model descriptions and assumptions are:

• Mold filling is not simulated; solidification starts with an initial concentration of Al-4.0 wt.%Cu and an initial temperature of 925 K
• Two phases are considered: the melt and equiaxed grains
• The grain morphology is approximated by spheres
• A three-parameter heterogeneous nucleation law is used
• Buoyancy force for the moving grains and thermosolutal convection are accounted for by a Boussinesq approximation

A three-dimensional (3-D) overview of the equiaxed phase distribution at 10 s after the start of cooling is shown in Figure 4. The dynamic evolution sequences of the equiaxed phase, the sedimentation, and the induced macrosegregation are demonstrated in Animation 2. Grains which nucleate in the upper regions and at the sidewalls sink downward. The sinking grains lead to an accumulation of the solid phase in the bottom region of the casting. The grains stop moving and finally settle when the local fraction of solid exceeds the packing limit (0.637). The grain settlement is the main reason for the negative segregation at the bottom of the casting. As solidification proceeds, this negative segregation zone becomes wider and wider. The strongest negative segregations are found in the lower corners near the sidewalls, where the grains tend to accumulate and settle. As the residual positively segregated melt is separated from the sedimentation zone by the settling of grains, a positive segregation zone is formed nearby. It is generally observed that a positive segregation zone exists just near a sedimentation zone. This positively segregated zone is located within the melt, thus it is not stationary and may move with the melt flow. While solidification proceeds, the positively segregated melt areas move toward and accumulate gradually in the last-to-solidify region, forming a large positively segregated zone in the late stage of solidification.

When the same benchmark as Figure 1 was simulated and a 3-D calculation carried out, the benchmark showed a complex case of macrosegregation formation with the mechanisms of coupled thermosolutal convection, grain sedimentation, and sedimentation induced convection. Details about process and thermo-physical parameters can be found in previous publications.⁸ The important model assumptions are:

The simulated solidification sequence including sedimentation of the equiaxed grains, sedimentation-induced melt convection, and thermosolutal melt convection are shown in Figure 5 and Animation 3. The solidification pattern agrees with the classical explanation of steel ingot solidification, summarized by J. Campbell.¹⁴ The columnar dendrites grow from the mold wall and the columnar tip front moves inward. The equiaxed grains nucleate near the mold walls and in the bulk melt. The columnar dendrites are stationary, whereas the equiaxed grains sink and settle in the base region of the ingot. The accumulation of such grains at the base of the ingot has a characteristic cone shape. Two symmetrical melt convection vortices in the ingot are induced by both thermosolutal effects and the drag of sinking grains. The sedimentation of grains and occurring melt convection influence the macroscopic solidification sequence and thus, the final phase distribution: more equiaxed grains will be found at the bottom and in the base region, and larger columnar areas in the upper part of the ingot.

As the columnar tip front is explicitly tracked in the model, the simulation shows that the columnar tip fronts from both sides tend to meet in the center of the casting. However, in the lower part of the casting the large amount of equiaxed grains stops the propagation of the columnar tip front. Its final position indicates the CET. The CET separates areas where only equiaxed grains appear from areas where both columnar dendrites and equiaxed grains might be found.

The final macrosegregation distribution is predicted, as shown in Figure 6. From the simulation results it becomes obvious that the main mechanism for the cone-shaped negative segregation in the base region is grain sedimentation. Since the settling grains are poor in solute elements, their pile-up results in negative segregation in the bottom of the ingot. A further contributing factor to the strength of negative segregation arises from the flow divergence of the residual liquid through this zone at a late solidification stage. The positive segregation at the top region of the ingot is caused by the flow of the segregated melt in the bulk region. This kind of positive segregation coincides with classical experimental results as well.¹³ It should be noted that channel segregations, which are frequently found in steel ingots, are not predicted in such a reduced ingot with the recent model.

The multiphase solidification approach discussed above is applied to model the spatial phase separation and macrosegregation caused by thermo-capillary force-induced convection (i.e., Marangoni) convection in a hypermonotectic alloy. As shown in Figure 7, when a hypermonotectic alloy with initial concentration c0 is cooled down below the binodal, a secondary liquid phase (L₂) is decomposed in the form of droplets from the parent melt (L₁). The surface tension between L₁ and L₂ is a function of the temperature. Therefore, due to the temperature gradient during solidification local convections inside and around the droplet are established. As a consequence, this thermo-capillary force-induced convection leads to the droplet moving from cold toward hot regions.

A 2-D benchmark of Al-10.wt.%Bi alloy with a square cross section is simulated and during solidification, as shown by the phase diagram (Figure 7), at least four phases appear: the parent melt L₁, the secondary liquid phase L₂, the solidified monotectic matrix, and the solidified secondary phase. For simplicity the authors’ approach considers only two phases: L₁ and L₂. During monotectic reaction the monotectic matrix is transformed directly from L₁. Therefore the solidified monotectic matrix is modeled as phase L₁ in such a way that an enlarged viscosity is applied to phase L₁ upon reaching the monotectic temperature. The latent heat of the monotectic reaction is added to phase L₁. L₂ droplets appearing at the monotectic reaction front are modeled to be entrapped in the monotectic matrix by applying a similar enlarged viscosity at or below the monotectic point. Additional assumptions are as follows:

• Filling is not simulated; solidification starts with an initial concentration Al-10 wt.%Bi and an initial temperature of 1,065 K
• Morphology of L₂ is approximated by spherical droplets
• A 3-parameter heterogeneous nucleation law for L₂ droplets is used, no collision and coalescence (coagulation) of the droplets is considered
• Growth of the droplets is controlled by diffusion in L₁
• Marangoni force and buoyancy force are treated by Boussinesq approximation
• Both liquid L₁ and L₂ phases have same viscosity
• Solidification of L₂ is ignored.

Two different simulations are made: one without gravity force and one with gravity force. The results of them are shown in Figure 8 and Animation 4.

For the case of 0-g, Marangoni motion presents the only mechanism for the phase transport. Droplets of L₂ start to nucleate and grow in the casting surface as the local temperature drops below the binodal (1,062.2 K). The Marangoni force causes the L₂ droplets to move from the surface region toward the casting center. The parent melt moves in the reverse direction because the space of the leaving phase L₂ must be replaced by the parent melt L₁. The movement of L₂ results in the depletion of the L₂ phase (f₂) in the corners and surface regions, and enrichment of the L₂ phase in the casting center. As cooling of the casting continues to the monotectic point, the monotectic reaction occurs, the velocity of L₁ vanishes, and the L₂ droplets are entrapped in the monotectic matrix. When solidification is finished, the surface has a lower volume fraction of L₂ phase than the sample center. The spatial separation of the phases is directly responsible for the macrosegregation: c_mix < 7.2%Bi in the corners, c_mix > 12%Bi in the center.


For the case with gravity, both buoyancy force and Marangoni force contribute to the movement of the L₂ droplets. The droplets nucleate in the corners and along the walls, then grow and sink downward along the vertical walls. The parent melt L₁ phase is drawn by the sinking L₂ droplets, forming two symmetric vortices: one clockwise in the right half and one counter-clockwise in the left half of the casting. The convection currents of L₁ are so strong that they in turn influence the movement and the distribution of L₂ droplets. With the monotectic reaction the L₂ droplets are entrapped in the monotectic matrix; the phase and droplet size distribution remain unchanged afterward. The final solidification results show the depletion of phase L₂ in the upper region and an enrichment of L₂ in the central bottom region. Strong spatial separation of the phases leads to a strong macrosegregation: c_mix < 5% Bi in the upper region, c_mix > 25%Bi in the lower bottom region. In this case, although both Marangoni motion and gravity-induced droplet motion are considered in the case with gravity, no influence of Marangoni motion on the solidification result is seen. The function of the gravity overwhelms the Marangoni force under normal terrestrial conditions.

The multiphase model results presented here, with accompanying animations, help to visualize the different flow and sedimentation phenomena and their impact on the formation of macrosegregation. Multiphase modeling provides a deeper insight in several relevant macrosegregation phenomena. The models discussed here may be considered preliminary and qualitative in nature as a result of the model assumptions and simplifications, thus some of the simulation results may not be suitable for quantitative comparisons with industry processes. Future work, however, will include the necessary refinements required to successfully create a multiphase model compatible with industrial castings.

This work was partially sponsored by the Christian-Doppler Society, Austria, in the framework of CD Laboratory-Multiphase Simulation of Metallurgical Processes, and partially supported by the ESA-MONOPHAS project–Advanced Bearing Alloys from Immiscible with Aluminum.

1. M.C. Flemings, “Our Understanding of Macrosegregation: PAST and Present,” ISIJ Int., 40 (9) (2000), pp. 833–841.
2. C. Beckermann, “Modeling of Macrosegregation: Applications and Future Needs,” Inter. Mater. Rev., 47 (5) (2002), pp. 243–261.
3. G. Lesoult, “Macrosegregation in Steel Strands and Ingots: Characterization, Formation and Consequences,” Mater. Sci. Eng. A, 413-414 (2005), pp. 19–29.
4. I. Ohnaka, “Microsegregation and Macrosegregation,” Metals Handbook: Vol. 15 Casting (Materials Park, OH: ASM International, 1998), pp. 136–141.
5. M. Wu et al., “On the Impact of Macroscopic Phase Separation on Solidification and Microstructures,” Adv. Eng. Mater., 7 (9) (2005), pp. 846–851.
6. A. Ludwig and M. Wu, “Modeling of Globular Equiaxed Solidification with a Two-Phase Approach,” Metall. Mater. Trans. A, 33A (2002), pp. 3673–3683.
7. M. Wu et al., “Influence of Convection and Grain Movement on Globular Equiaxed Solidification,” Inter. J. Heat Mass Transfer, 46 (2003), pp. 2819–2832.
8. M. Wu and A. Ludwig, “A Three-Phase Model for Mixed Columnar-Equiaxed Solidification,” Metall. Mater. Trans. A, 37A (2006), pp. 1613–1631.
9. M. Wu, A. Ludwig, and L. Ratke, “Modelling the Solidification of Hypermonotectic Alloys,” Modelling Simul. Mater. Sci. Eng. ., 11 (2003), pp. 755–769.
10. M. Wu, A. Ludwig, and L. Ratke, “Modeling of Marangoni Induced Droplet Motion and Convection during Solidification of Hypermonotectic Alloys,” Metall. Mater. Trans. A, 34A (2003), pp. 3009–3019.
11. R. Mehrabian, M. Keane, and M.C. Flemings, “Interdendritic Fluid Flow and Macrosegregation; Influence of Gravity,” Metall. Trans., 1 (1970), pp. 1209–1220.
12. M. Wu, A. Ludwig, an J. Luo, “Numerical Study of the Thermal-Solutal Convection and Grain Sedimentation during Globular Equiaxed Solidification,” Mater. Sci. Forum, 475-479 (1-5) (2005), pp. 2725–2730.
13. J.D. Hunt, “Steady State Columnar and Equiaxed Growth of Dendrites and Eutectic,” Mater. Sci. Eng., 65 (1984), pp. 75–83.
14. J. Campbell, Castings (Oxford, U.K.: Butterworth Heinemann Ltd, 1991), pp. 151–158.

Menghuai Wu and Andreas Ludwig are with the Department of Metallurgy at the University of Leoben in Austria.

For more information, contact Menghuai Wu, University of Leoben, Department of Metallurgy, A-8700 Leoben, Austria; e-mail menghuai.wu@mu-leoben.at.