Introduction

Al-Si-based cast alloys have been widely used in many industrial applications due to their lightweight, excellent castability, comprehensive mechanical properties, and corrosion resistance1,2. It is well known that the α-Al grain/dendrite microstructure of Al-Si alloys influences the mechanical and corrosion properties of solidification products including castings, welds, and additively manufactured components3. Therefore, extensive experimental and simulation studies4,5 have been reported on the solidification of dendrite microstructure including dendrite morphology, dendrite arm spacing, solute concentration distribution, etc. However, eutectic (α-Al + Si) solidification microstructure in interdendritic regions of Al-Si products can significantly affect their mechanical properties6,7. Additionally, the formation of eutectics which occurs at the final stage of solidification is critical to the castability and the final properties of the alloys8. It is, thus, important to understand the mechanisms of both dendrite and eutectic phase formation during solidification processing, in order to accurately simulate and optimize the solidification microstructure for ICME (integrated computational materials engineering) based design and manufacturing of solidification products9.

The solidification microstructure of aluminum alloys has been extensively investigated using 2-D optical metallography6,10 and 3-D X-ray tomography11,12. Khajeh and Maijer13 applied X-ray microtomography to obtain three-dimensional geometries of eutectic/primary phases. Murphy et al.14 identified the eutectic transformation by X-ray absorption contrast difference. However, it is still difficult to understand the dynamic evolution of dendrite and eutectic formation during solidification processing15. Accurate prediction of the evolution of dendrite and eutectic microstructure is essential to improving the properties and performance of solidification products.

Numerical simulation is an alternative method to study the dynamic process of solidification with the recent advancements in materials science and computer technology16. Numerical models have been developed for predicting solidification dendrites using different techniques such as phase-field (PF)17,18 and cellular automaton (CA)19. Takaki et al.20 performed PF simulations during directional solidification of a binary alloy and conducted a parallel computation to increase the computational speed. Lewis et al.21 established two different PF models to study the morphological evolution of binary eutectics. Guo et al.22 studied eutectic solidification using an atomistic simulation method. However, PF simulations require more computation resources, which limits its industrial applications where large computation domains are needed23.

Compared to the PF method, CA provides strong morphological resolutions but with a reasonable demand for computation resources24,25. Wang et al.26,27 introduced the block-correction technique and the parallel computation approach to improve the computational efficiency of the 3-D CA model for describing the dendritic growth and applied the model to investigate the 3-D dendrite growth of Fe-C alloys. In addition, several groups have published on the simulation of eutectic transformation. Dobravec et al.28 proposed a 2-D CA model for dendritic and eutectic growth. Lei et al.29 developed a quantitative CA model for dendritic and anomalous eutectic growth. However, solidification dendrite and eutectic were considered separately, and the model was only used in 2-D. Zhu and Hong30 proposed a modified CA model for simulating the solidification microstructure evolution of eutectic Al-Si alloys. Chen et al.31 developed a multiphase CA model for the time-dependent Al-Si irregular eutectic growth. Fang et al.32 proposed a 2-D CA model for simulating solidification microstructure and microsegregation based on the phase equilibria method. Although a number of studies have been carried out on dendrite growth simulation, the mechanisms for microstructure evolution during eutectic transformation are still not well understood. There is very limited work on the 3-D simulation of eutectic transformation, especially coupling the simulation of dendritic and eutectic solidifications. It is, therefore, necessary to model and accurately predict the microstructure evolution of complete solidification processing, to provide a critical link in ICME research and applications. Since industrial solidification processes involve various cooling rates and alloys, the effects of cooling rate and solute concentration on the evolution of both dendrite and eutectic microstructure should also be studied.

A multi-component CA model for simulating the microstructure evolution during solidification has been established in our previous work33,34. However, the eutectic transformation was not considered. In this paper, the previously published 3-D CA model was extended to include the nucleation and evolution of eutectic transformation at the end stage of solidification. The complete solidification process of dendrite growth and eutectic transformation of single and multi-dendrite domains were simulated. The average dendrite growth velocities of 3-D simulations were discussed. The simulation results were compared with experimental observations using scanning electron microscopy (SEM) and deep etching techniques.

Results and discussion

Simulation of complete solidification process

The complete solidification process including dendrite growth and eutectic transformation of a single dendrite was simulated. Figure 1 shows the singe dendrite morphology and the solute concentration field at different times. The simulation was performed in a calculation domain of 100 μm × 100 μm and with a cell size of 1.0 μm. The initial undercooling was set as 5 K, and the temperature was decreased with a constant cooling rate of 100 K s−1. As solidification progresses, the single aluminum grain grows to a dendrite morphology. It also shows that secondary dendrites grow at the sides of primary dendrites. Figure 1f displays the Si solute distribution along the red lines A–B and C–D as shown in Fig. 1a. In the solid region, the Si concentrations remain at low values. At the solid/liquid interface, the Si concentration increases to a significantly higher value and then decreases with the distance to the dendrite. This phenomenon agrees well with a previous study by Zhang et al.35. With solidification further proceeding, the primary and secondary dendrites continue to grow and coarsen. When the temperature decreases to the eutectic temperature, the eutectic transformation of aluminum and silicon phases start. In the remaining liquid (the yellow region in Fig. 1c or light blue region in Fig. 1d), nucleation and growth of the α-Al phase and Si phase occur as shown in Fig. 1d. The temperature continues decreasing until reaching an undercooling of 5 K below the eutectic temperature. Then, the simulation continues until solidification is completed with a 100% solid fraction. Figure 1e shows that most of the remaining liquid is transformed into the eutectic microstructure, consisting of α-Al and Si phases. Si phase is shown in yellow color, and α-Al phase is shown in purple color. Therefore, the whole solidification process of a single dendrite including dendrite growth and eutectic transformation can be simulated by using the developed model.

Fig. 1: Singe dendrite morphology.
figure 1

Singe dendrite morphology and the solute concentration field: a At 0.0625 s, b at 0.125 s, c at 0.2167, d at 0.2208 s, e at 0.2705 s, and f solute concentration distribution along A–B and C–D in (a).

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Multi-dendrite growth and subsequent eutectic transformation during the whole solidification process in a 3-D calculation domain (100 μm × 100 μm × 100 μm, and cell size 1.0 μm) was simulated as shown in Fig. 2. The multi-nuclei were randomly placed in the calculation domain. As solidification proceeds, the dendrites start to nucleate and grow until most areas of the calculation domain are transformed into solid. However, with different cooling rates, it is evident that the dendrite morphologies and solute concentration distributions are different. The results in Fig. 2b, e, and h suggest that the dendrite growth occurs in a shorter time with an increasing cooling rate. Meanwhile, solute accumulation occurs around the dendrites due to solutes not having enough time to diffuse away in a shorter time at a higher cooling rate. The higher cooling rate also results in finer dendrite arms and the formation of more secondary dendrites, as shown in Fig. 2d, g, and j. Additionally, Fig. 2a shows that solid fractions under different cooling rates are also different. At a cooling rate of 5 K s−1, the relative slow solute diffusion provides a condition close to equilibrium growth. Therefore, the solid fraction curve of 5 K s−1 fits the lever rule. When the cooling rate is increased to 10 K s−1, a faster increase of solid fraction is observed. However, when the cooling rate is 100 K s−1, the solid fraction becomes lower than at the beginning. This is because the highest cooling rate results in the highest local solute concentration, which leads to the lowest dendrite growth velocity at the beginning. It should be noted that with a cooling rate of 100 K s−1, the temperature decreases considerably faster than in the other two conditions. At higher cooling rates, finer dendrites will form with more secondary dendrites than dendrites formed at slower cooling rates. Meanwhile, the higher cooling rate leads to a higher solid fraction when the temperature is lower than 865 K.

Fig. 2: 3-D multi-dendrite morphology.
figure 2

3-D multi-dendrite morphology with different cooling rates: a Solid fraction as a function of temperature obtained by 3-D multi-dendrite morphology with the cooling rate of 5 K s−1 at b 873.06 K, c 864.73 K, and d 851.31 K, 10 K s−1 at e 871.93 K, f 867.97 K, and g 850.89 K, 100 K s−1 at h 870.06 K, i 865.89 K, and j 850.89 K.

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The average dendrite growth velocities at different cooling rates were recorded as shown in Fig. 3a. Each simulation shows that the average dendrite growth velocity tends to decrease to a steady value. The results also indicate that the average dendrite growth velocity increases with an increasing cooling rate. As shown in Fig. 3b, the solid fraction vs. temperature by the present CA model is compared with calculations by the Scheil model and the lever rule36, respectively. The CA simulations were performed under the condition that diffusions in liquid and solid are controlled by corresponding diffusion coefficients. All the profiles show a similar trend in that the solid fraction increases with decreasing temperature until the eutectic temperature. However, the curve under CA simulations deviates slightly from the prediction of the calculations with the Scheil model and lever rule. It is because, under the Scheil condition, there is no diffusion in solid but complete mixing in liquid; while the lever rule assumes complete diffusion in both solid and liquid. Since diffusion in both solid and liquid states are considered in the CA model in this paper, it is more accurate than the other two curves (Scheil and level rule models) and is closer to industrial solidification processes where diffusion in solid and liquid is neither zero nor infinite.

Fig. 3: Dendrite growth kinetics.
figure 3

a Comparison of the average dendrite growth velocity with different cooling rates, and b solid fraction vs. temperature obtained by 3-D CA simulation, Scheil model, and lever rule.

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Since the eutectic morphology is difficult to be observed clearly in 3-D, the 2-D cross-section as shown in Fig. 2 during eutectic transformation is shown in Fig. 4. During the solidification process, dendrites continue to grow until the temperature reaches eutectic temperature. Although dendrites grow to different morphologies with different cooling rates in Fig. 4a, e, and i, there is no eutectic transformation that occurred in the domains. When the eutectic temperature is reached, nucleation of eutectic phases occurs. Both phases of α-Al and Si nucleate and continue to grow as shown in Fig. 4b, f, and j. When the temperature reaches the end of the simulation, all the remaining liquid has transformed into a solid eutectic mixture, which is composed of α-Al and Si.

Fig. 4: 2-D cross-sections of 3-D simulation results.
figure 4

2-D cross-section of a 3-D simulation result, with the cooling rate of 5 K s−1 at a 851.02 K, b 847.89 K, c 845.07 K, and d 845.01 K, with the cooling rate of 10 K s−1 at e 850.89 K, f 847.14 K, g 845.86 K, and h 845.01 K, with the cooling rate of 100 K s−1 at i 850.48 K, j 847.14 K, k 845.09 K, and l 845.01 K. (α-Al is in blue color, Si is in yellow color, and the remaining liquid is in green color).

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To validate the simulation results, the solidification experiments were conducted for Al-12wt% Si alloy. Figure 5 shows the comparison of the simulated microstructure and experimental microstructure. In the simulation microstructure, the irregular eutectic microstructure distributes in interdendritic regions. It is because the simulation was performed based on the equilibrium thermodynamic parameters. In the experimental microstructure, the eutectic microstructure shows the same distribution, while the needle-like Si morphology is slightly different from the simulation, which is due to the limit of the cell size in the simulation. Meanwhile, with an increasing cooling rate, the dendrites are finer, and the eutectic morphology, such as the length of Si phase, is shorter and finer as well. The simulated results show a well agreement with the experimental results.

Fig. 5: Comparison between simulations and experiments.
figure 5

Comparison between simulated microstructure and experimental microstructure: a–c Cooling rate of 30 K s−1, and d–f cooling rate of 50 K s−1. (α-Al is in blue color and Si is in yellow color).

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The quantitative information including the number, length, and area of the Si phase in the given zone was recorded as shown in Fig. 6. In Fig. 6a, b, most of the length of the Si phase is shorter than 20 μm. When the cooling rate is 30 K s−1 (relatively lower), several Si nuclei can grow longer. However, the length of the Si phase is shorter, and there are significantly more nuclei in the Si phase when the cooling rate is 50 K s−1 (relatively higher). Figure 6c, d shows the area of the Si phase vs. length of the Si phase with a cooling rate of 30 and 50 K s−1, respectively. The area of the Si phase has an obvious relationship with the length of the Si phase, which describes the average thickness of the Si phase. In Fig. 6c, d, the slope of the dashed red line is represented as k. It indicates that the smallest value of k is around 1.0 which means the smallest thickness of the Si phase, in this case, is around 1.0 μm. In addition, the largest thickness of the Si phase is around 3.0 μm. The average thickness increases from 2.58 to 3.77 μm when the cooling rate changes from 30 to 50 K s−1. The length of the Si phase matches well with the above analysis. In addition, average lamellar spacings between the Si phase under different cooling rates were recorded in Fig. 6e. The simulation results agree well with the experimental results in that the average lamellar spacing decreases with an increase in cooling rate.

Fig. 6: Quantitative results of Si phase.
figure 6

Quantitative results compared between simulation results and experiment results: a Number of Si phase vs. length of Si phase with a cooling rate of 30 K s−1, b number of Si phase vs. length of Si phase with a cooling rate of 50 K s−1, c area of Si phase vs. length of Si phase with a cooling rate of 30 K s−1, d area of Si phase vs. length of Si phase with a cooling rate of 50 K s−1, and e average lamellar spacings between Si phase under different cooling rates (The error bars are obtained from the standard deviations of the lamellar spacings in the domain).

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Figure 7 shows the simulated solidification microstructure of Al-Si alloys with different initial Si concentrations of 7.0, 10.0, and 11.5 wt%. With increasing initial Si concentration, it is clear that there is more area of remaining liquid before eutectic temperature. Compared to Fig. 4h, the initial Si concentration increases from 7.0 to 8.5, 10.0, and 11.5 wt%, and thus there is a lower solid fraction before eutectic temperature and more remaining liquid to be transformed to eutectic microstructure. In Fig. 7c, f, i, all the remaining liquid has transformed into an eutectic mixture of α-Al in dark blue and Si in yellow.

Fig. 7: Effect of Si concentration.
figure 7

3-D simulation results of different initial Si concentrations: a Si: 7.0 wt%, b, c cross-section in (a), d Si: 10.0 wt%, e, f cross-section in (c), g Si: 11.5 wt%, and h, i cross-section in (e). (In the cross-sections: α-Al is in blue color, Si is in yellow color, and the remaining liquid is in green color).

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Figure 8 shows the comparison of the simulated Si concentrations in solid by the 3-D CA simulation, Scheil model, and lever rule. The values of Si concentration were calculated by the average Si concentrations in the existing α-Al phase. It shows that with increasing solid fraction, Si concentration in the solid also increases. In the stage of primary dendrite growth, the increasing rate of Si concentration increases due to the decrease in temperature. However, in the stage of eutectic transformation, Si concentration tends to be stable with increasing solid fractions. It is because the formation of Si limits the continuing increase of Si concentration in the liquid. On the other hand, the maximum Si concentration in the α-Al phase confirms the result, which can also be illustrated by Al-Si binary phase diagram. It also can be observed that Si concentrations by 3-D CA simulation is a slightly lower than that of the Scheil model and the lever rule, while the difference between them tends to be smaller with increasing solid fraction. The reason is that the developed CA model includes dendrite nucleation, which is not considered by the Scheil model nor lever rule calculations. The nucleation and growth of primary dendrite keep the Si concentration in the solid phase at a low level, which agrees with the results in Fig. 3b. As solidification proceeds, higher Si concentration in the liquid near the solid Al/liquid (SA/L) interface results in the increase of Si concentration in the solid. It can be indicated that the whole solidification process including primary dendrite growth and eutectic transformation can be predicted by using the 3-D CA model, which is critical to further research and application of ICME in solidification-based product design and manufacturing.

Fig. 8: Model validation.
figure 8

Comparison of the simulated Si concentrations in the α-Al phase vs. the solid fraction by the 3-D CA simulation, Scheil model, and lever rule.

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Based on our previously established 3-D CA model, the eutectic solidification was included, and the 3-D CA model was extended to simulate the complete solidification process including dendrite growth and eutectic transformation. The model was adopted to simulate the complete solidification microstructure of primary and secondary dendrites as well as eutectic transformation in single-dendrite and multi-dendrite domains. The average dendrite growth velocity tends to decrease to a stable value.

With increasing cooling rate, the dendrite arms change from coarse to fine, and with more secondary dendrite formation. The average dendrite growth velocity increases with the increase in cooling rate. The solid fraction profiles vs. temperature by the present CA model and results under the Scheil model and the lever rule were compared. All the profiles show the same trend that the solid fraction increases with decreasing temperature until eutectic temperature, while the curve under CA simulation deviates slightly due to the difference in diffusion conditions. It should be pointed out that the present CA model includes diffusion calculations in both solid and liquid which is closer to industrial solidification processing than the Scheil model (no diffusion in solid) and the lever rule (infinite diffusion in both liquid and solid).

With increasing initial Si concentration, there is more remaining liquid to be transformed to the eutectic microstructure of α-Al and Si at around the eutectic temperature. With increasing solid fraction during solidification processing, Si concentration in solid also increases. The microstructure and quantitative results of the Si phase confirm that the simulation results agree with the experiments and calculations by the Scheil model and the lever rule. The 3-D CA model can accurately predict the complete solidification microstructure including primary/secondary dendrite growth and the eutectic transformation of Al-Si-based alloys, which can be used in ICME-based design and manufacturing of solidification products such as castings, and potentially welded and additively manufactured components.

Methods

A 3-D cellular automaton method is used to simulate the evolution of the whole solidification process including dendrite growth and eutectic transformation. In this paper, Al-Si binary alloy is investigated, and the variable of phase fraction includes solid Al fraction (fs,Al) and solid Si fraction (fs,Si) when considering the growth of α-Al and Si. The variable of the state is consisting of a liquid cell, solid Al cell, solid Al/liquid (SA/L) interface cell, solid Si cell, solid Si/liquid (SS/L) interface cell, solid Al/solid Si (SA/SS) interface cell, and solid Al/solid Si/liquid (SA/SS/L) interface cell. The solidification process starts at the liquidus temperature, and the dendrites nucleate and grow with decreasing temperature until the local temperature is lower than eutectic temperature when a eutectic transformation is activated. It should be noted that the effects of fluid flow and solidification shrinkage are not considered in this model. Figure 9 shows the schematic diagram of solidification dendrites and eutectics in this study.

Fig. 9: Schematic diagram of solidification dendrites and eutectic phases.
figure 9

a Dendrite morphology during solidification in mesoscale, b eutectic phase near dendrite squared in a, and c the variable of the state in this CA model.

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Aluminum dendrite nucleation and growth

Aluminum dendrite nucleation and growth during solidification should be considered to provide accurate dendrite morphology and solute concentration distribution as initial conditions for eutectic transformation. The continuous nucleation distribution37,38 is used to describe dendrite nucleation.

After nucleation, local equilibrium at interface and solute diffusion in liquid and solid should be modeled39, and the nucleated dendrites start to grow with decreasing temperature. The interface thermodynamic equilibrium can be expressed as:

$$\frac{{\partial T}}{{\partial C^L}}\left( {C^0 - C^{L \ast }} \right) = T_0 - T^ \ast - {\Gamma}\kappa$$
(1)

where \(\frac{{\partial T}}{{\partial C^L}}\) is the liquidus slope, \(C^0\) is the initial solute concentration, \(C^{L \ast }\) is the local interface solute concentration in the liquid, T0 is the liquidus temperature, T* is the interface temperature, Γ is the Gibbs–Thomson coefficient, and κ is the curvature of the interface40. Then, based on the lever rule in the interface cell, the solid fraction increment can be calculated39. The increased solute concentration decreases the liquidus temperature, resulting in a decrease in local undercooling41. The local undercooling for dendrite growth ΔTAl should be calculated by:

$${\Delta}T_{Al} = {\Delta}T + \frac{{\partial T}}{{\partial C^L}}\left( {C^L - C^0} \right)$$
(2)

where \(C^L\) is the local solute concentration. More details on solidification dendrite nucleation and growth can be found in authors’ previous publications33,34.

Eutectic solidification

In most previous research, solidification simulation ended at eutectic temperature, ignoring eutectic transformation. In this paper, the temperature continues decreasing to below eutectic temperature, and the eutectic transformation is simulated. The eutectic transformation includes the nucleation and growth of the α-Al and Si. The Rappaz model37,38 is also adopted to predict the heterogeneous eutectic nucleation. For hypoeutectic alloys, it is assumed that a single eutectic grain includes one α-Al nucleus and one adjacent Si nucleus. Both nuclei of α-Al and Si form simultaneously when the eutectic nucleation occurs. And the growth kinetics, such as preferred orientation, of both nuclei are set to be random. After nucleation, α-Al and Si grow simultaneously and independently. The growth of the non-faceted α-Al is simulated by adopting the same model as dendrite growth. The growth velocity of the faceted Si is controlled by the local undercooling at an interface:

$$V_{Si} = \mu _{Si}\left[ {1 + \delta \left( {\cos \left( {2\left( {\varphi _{xy} - \theta _{xy}} \right)} \right) + \cos \left( {2\left( {\varphi _{yz} - \theta _{yz}} \right)} \right) + \cos \left( {2\left( {\varphi _{zx} - \theta _{zx}} \right)} \right)} \right)/3} \right]{\Delta}T_{Si}$$
(3)

where \({\Delta}T_{Si}\) is the local undercooling for Si growth, \(\mu _{Si}\) is the interface kinetic coefficient, \(\delta\) is the kinetic anisotropy degree, θ is the preferential growth orientation in the specific plane x-y, y-z, and z-x, and \(\varphi\) is the angle between the interface moving direction and the axis:

$$\begin{array}{*{20}{c}} {\varphi _{xy} = \arctan \left( {\frac{{\partial f_{Si}/\partial y}}{{\partial f_{Si}/\partial x}}} \right)} \\ {\varphi _{yz} = \arctan \left( {\frac{{\partial f_{Si}/\partial z}}{{\partial f_{Si}/\partial y}}} \right)} \\ {\varphi _{zx} = \arctan \left( {\frac{{\partial f_{Si}/\partial x}}{{\partial f_{Si}/\partial z}}} \right)} \end{array}$$
(4)

where \(f_{Si}\) is the solid fraction of Si in the local cell. Therefore, the increment of the solid fraction of Si can be calculated by:

$${\Delta}f_{Si} = V_{Si}{\Delta}t/{\Delta}x$$
(5)

where \({\Delta}t\) is the time step, and \({\Delta}x\) is the cell size. Assumptions are adopted that the faceted Si always grows in its preferred orientation in the preferred plane.

It is important to point out that the solute diffusion during eutectic solidification is different from that during dendrite growth since the growth of the Si phase absorbs Si atoms which decreases local solute concentration. The governing equation for solute diffusion can be expressed as:

$$\frac{{\partial C}}{{\partial t}} = \nabla D\nabla C + \left( {C^{L \ast } - C^{S \ast }} \right)\frac{{\partial f_{Al}}}{{\partial t}} + \left( {C^{L \ast } - C^{Si}} \right)\frac{{\partial f_{Si}}}{{\partial t}}$$
(6)

where D is the solute diffusion coefficient, \(C^{S \ast }\) is the local interface solute concentration in the solid Al phase, \(C^{Si}\) is the local solute concentration in the Si phase, and \(f_{Al}\) is the solid fraction of Al. Based on Eq. (6), the rejected or absorbed Si atoms as a result of the growth of the Al phase and Si phase can be considered in the model, and solute conservation can be maintained in the whole calculation domain. The parameters are shown in Table 1. To prevent the simulated dendrite and eutectic from always growing parallel to the system coordinate, an optimization algorithm is used33, and thus the dendrite and eutectic can grow in their original preferred orientations. By adopting this model, eutectic transformations of α-Al and Si with different preferred orientations can be simulated.

Table 1 Parameters used in this paper.
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Experimental validation

To validate the new CA model and the simulation results, casting experiments of Al-12wt% Si binary alloys were performed. K-type thermal couples were used to determine the temperature history during casting experiments. Before eutectic temperature, temperature decreases at a high cooling rate of around 30 K s−1. When it reaches eutectic temperature, the cooling rate decreases to 0 due to the nucleation of eutectic phases. Subsequently, the nucleated eutectic phases start to grow, and the temperature decreases again at a cooling rate of around 10 K s−1. Following casting experiments, the solidified sample was cut by electrical discharge machining, and the solidified microstructure was observed using scanning electron microscopy (SEM). Deep etched samples were also prepared by submerging in 30% NaOH solution for 20 min with sonication to reveal 3-D Si morphology under SEM.