J. Mater. Sci. Technol. ›› 2023, Vol. 134: 163-177.DOI: 10.1016/j.jmst.2022.06.042
• Research Article • Previous Articles Next Articles
Pan Wua, Yubing Zhanga, Jiaqi Hua, Shaojie Songa,*(
), Yong Lic, Huiyuan Wangd, Guo Yuanc, Zhaodong Wangc, Shizhong Weie, Feng Liua,b,*()
Received:2022-05-18
Revised:2022-06-27
Accepted:2022-06-29
Published:2023-01-20
Online:2023-01-10
Contact:
Shaojie Song,Feng Liu
About author:liufeng@nwpu.edu.cn (F. Liu).Pan Wu, Yubing Zhang, Jiaqi Hu, Shaojie Song, Yong Li, Huiyuan Wang, Guo Yuan, Zhaodong Wang, Shizhong Wei, Feng Liu. Generalized stability criterion for controlling solidification segregation upon twin-roll casting[J]. J. Mater. Sci. Technol., 2023, 134: 163-177.
| Symbol | Description | Unit |
|---|---|---|
| Thermal diffusion coefficients of casting roller | m2 s−1 | |
| as | Thermal diffusion coefficients of solid | m2 s−1 |
| ali | Thermal diffusion coefficient of mush | m2 s−1 |
| Aαβ | Interfacial area concentration of α/β interface | m−1 |
| Ae | Interfacial area concentration of mush/liquid interface | m−1 |
| As | Interfacial area concentration of solid/mush interface | m−1 |
| Area of α/β interface | m2 | |
| Heat capacity of casting roller | J kg−1 K−1 | |
| Cp | Heat capacity of solid and mush | J kg−1 K−1 |
| Interfacial concentration of kth component in mush | at.% | |
| Interfacial concentration of kth component in solid | at.% | |
| Initial concentration of kth component | at.% | |
| Transient interfacial concentration of kth component in mush | at.% | |
| Transient interfacial concentration of kth component in solid | at.% | |
| Average concentration of kth component in solid | at.% | |
| Average concentration of kth component in mush | at.% | |
| Average concentration of kth component in liquid | at.% | |
| Concentration at the grain envelope | at.% | |
| d | Cooling channel diameter | m |
| Dα | Solute diffusion coefficient in α phase | m2 s−1 |
| Solute diffusion coefficient of kth component at mush | m2 s−1 | |
| Solute interaction diffusion coefficient between kth and jth components in mush | m2 s−1 | |
| Solute interaction diffusion coefficient between kth and jth components in liquid | m2 s−1 | |
| Γ | Gibbs-Thomson coefficient | K m |
| Gs | Temperature gradient of solid at solid/mush interface | K m−1 |
| Gli | Temperature gradient of mush at solid/mush interface | K m−1 |
| ΔG | Thermodynamic driving force for in interface migration | J mol−1 |
| ΔGC | Molar chemical change in Gibbs energy | J mol−1 |
| ΔGD | Gibbs energy dissipated by solute diffusion | J mol−1 |
| ΔGt | Transient thermodynamic driving force for interfacial migration | J mol−1 |
| Initial molar chemical change in Gibbs energy | J mol−1 | |
| Transient Gibbs energy dissipated by solute diffusion | J mol−1 | |
| gα | Volume fraction of α phase | dimensionless |
| gs | Volume fraction of solid | dimensionless |
| Volume fraction of mush | dimensionless | |
| gl | Volume fraction of liquid | dimensionless |
| h0 | Thermal transfer coefficients at water/roller boundary | W m−2 K−1 |
| h1 | Thermal transfer coefficients at roller/solid boundary | W m−2 K−1 |
| ΔHf | Latent heat of solidification | J m−3 |
| Solute diffusion flux of kth component at the interface | mol/s | |
| KR | Thermal conductivity of casting roller | W m−2 K−1 |
| KW | Thermal conductivity of cooling water | W m−2 K−1 |
| Ks | Thermal conductivity of solid | W m−1 K−1 |
| Thermal conductivity of mush | W m−1 K−1 | |
| l | Roller width | m |
| lαβ | Diffusion length of α/β interface | m |
| Solute diffusion length of jth component in liquid at mush/liquid interface | m | |
| Thermal diffusion length in mush at solid/mush interface | m | |
| Thermal diffusion length in liquid at mush/liquid interface | m | |
| Mobility for solute diffusion of kth component | mol mJ−1 s−1 | |
| M0 | Mobility for the interfacial migration | m/s |
| PC | Solute Peclet number | dimensionless |
| q0 | Heat flux at water/roller boundary | W m−2 |
| q1 | Heat flux at roller/solid boundary | W m−2 |
| Effective kinetic energy barrier for interfacial migration | kJ mol−1 | |
| Transient kinetic effective energy barrier for interfacial migration | kJ mol−1 | |
| Q0 | Initial transient kinetic energy barrier for interfacial migration | kJ mol−1 |
| Transient kinetic energy barrier dissipated by solute diffusion | kJ mol−1 | |
| R | Dendritic tip radius | m |
| Rt | Transient dendritic tip radius | m |
| Rg | Gas constant | J mol−1 K−1 |
| Edge segregation index | dimensionless | |
| Center segregation index | dimensionless | |
| t | Solidification time | s |
| TR | Temperature of casting roller | K |
| Ts | Temperature of solid | K |
| Temperature of casting roller at water/roller boundary | K | |
| Temperature of solid at roller/solid boundary | K | |
| Temperature of casting roller at roller/solid boundary | K | |
| Ti | Temperature of mush | K |
| Temperature on mth node at solid/mush interface | K | |
| Temperature on the third node from the mth node | K | |
| ΔT | Bath undercooling | K |
| ΔTR | Curvature undercooling | K |
| ΔTt | Thermal undercooling | K |
| ΔTk | Kinetic undercooling | K |
| Tis | Temperature of solid at solid/mush interface | K |
| Til | Temperature of solid at solid/mush interface | K |
| Average temperature of solid | K | |
| Average temperature of mush | K | |
| Average temperature of liquid | K | |
| μw | Dynamic viscosity of cooling water | N s m−2 |
| Chemical potential of kth component in solid at solid/mush interface | J mol−1 | |
| Chemical potential of kth component in solid at solid/mush interface | J mol−1 | |
| Initial chemical potential of kth component in solid at solid/mush interface | J mol−1 | |
| Initial chemical potential of kth component in mush at solid/mush interface | J mol−1 | |
| Initial jump of chemical potential of the kth component | J mol−1 | |
| v | Velocity at solid/mush interface | m s−1 |
| vw | Flow speed of cooling water | m s−1 |
| Diffusion velocity of kth component at solid/mush interface | m s−1 | |
| Diffusion velocity of kth component at liquid | m s−1 | |
| v0 | Maximum velocity proportional to the sound velocity in liquid | m s−1 |
| vt | Velocity at solid/mush interface at t s | m s−1 |
| Average velocity of α/β interface | m s−1 | |
| Vm | Molar volume | m3 mol−1 |
| Vw | Flow volume per unit time of cooling water | m3 s−1 |
| ω | Casting speed | m min−1 |
| Δx | Space step in solid used by the finite difference method | m |
| ρ | Density of cooling water | kg m−3 |
| Non-equilibrium solute diffusion factor of kth component | dimensionless |
Table 1. Nomenclature.
| Symbol | Description | Unit |
|---|---|---|
| Thermal diffusion coefficients of casting roller | m2 s−1 | |
| as | Thermal diffusion coefficients of solid | m2 s−1 |
| ali | Thermal diffusion coefficient of mush | m2 s−1 |
| Aαβ | Interfacial area concentration of α/β interface | m−1 |
| Ae | Interfacial area concentration of mush/liquid interface | m−1 |
| As | Interfacial area concentration of solid/mush interface | m−1 |
| Area of α/β interface | m2 | |
| Heat capacity of casting roller | J kg−1 K−1 | |
| Cp | Heat capacity of solid and mush | J kg−1 K−1 |
| Interfacial concentration of kth component in mush | at.% | |
| Interfacial concentration of kth component in solid | at.% | |
| Initial concentration of kth component | at.% | |
| Transient interfacial concentration of kth component in mush | at.% | |
| Transient interfacial concentration of kth component in solid | at.% | |
| Average concentration of kth component in solid | at.% | |
| Average concentration of kth component in mush | at.% | |
| Average concentration of kth component in liquid | at.% | |
| Concentration at the grain envelope | at.% | |
| d | Cooling channel diameter | m |
| Dα | Solute diffusion coefficient in α phase | m2 s−1 |
| Solute diffusion coefficient of kth component at mush | m2 s−1 | |
| Solute interaction diffusion coefficient between kth and jth components in mush | m2 s−1 | |
| Solute interaction diffusion coefficient between kth and jth components in liquid | m2 s−1 | |
| Γ | Gibbs-Thomson coefficient | K m |
| Gs | Temperature gradient of solid at solid/mush interface | K m−1 |
| Gli | Temperature gradient of mush at solid/mush interface | K m−1 |
| ΔG | Thermodynamic driving force for in interface migration | J mol−1 |
| ΔGC | Molar chemical change in Gibbs energy | J mol−1 |
| ΔGD | Gibbs energy dissipated by solute diffusion | J mol−1 |
| ΔGt | Transient thermodynamic driving force for interfacial migration | J mol−1 |
| Initial molar chemical change in Gibbs energy | J mol−1 | |
| Transient Gibbs energy dissipated by solute diffusion | J mol−1 | |
| gα | Volume fraction of α phase | dimensionless |
| gs | Volume fraction of solid | dimensionless |
| Volume fraction of mush | dimensionless | |
| gl | Volume fraction of liquid | dimensionless |
| h0 | Thermal transfer coefficients at water/roller boundary | W m−2 K−1 |
| h1 | Thermal transfer coefficients at roller/solid boundary | W m−2 K−1 |
| ΔHf | Latent heat of solidification | J m−3 |
| Solute diffusion flux of kth component at the interface | mol/s | |
| KR | Thermal conductivity of casting roller | W m−2 K−1 |
| KW | Thermal conductivity of cooling water | W m−2 K−1 |
| Ks | Thermal conductivity of solid | W m−1 K−1 |
| Thermal conductivity of mush | W m−1 K−1 | |
| l | Roller width | m |
| lαβ | Diffusion length of α/β interface | m |
| Solute diffusion length of jth component in liquid at mush/liquid interface | m | |
| Thermal diffusion length in mush at solid/mush interface | m | |
| Thermal diffusion length in liquid at mush/liquid interface | m | |
| Mobility for solute diffusion of kth component | mol mJ−1 s−1 | |
| M0 | Mobility for the interfacial migration | m/s |
| PC | Solute Peclet number | dimensionless |
| q0 | Heat flux at water/roller boundary | W m−2 |
| q1 | Heat flux at roller/solid boundary | W m−2 |
| Effective kinetic energy barrier for interfacial migration | kJ mol−1 | |
| Transient kinetic effective energy barrier for interfacial migration | kJ mol−1 | |
| Q0 | Initial transient kinetic energy barrier for interfacial migration | kJ mol−1 |
| Transient kinetic energy barrier dissipated by solute diffusion | kJ mol−1 | |
| R | Dendritic tip radius | m |
| Rt | Transient dendritic tip radius | m |
| Rg | Gas constant | J mol−1 K−1 |
| Edge segregation index | dimensionless | |
| Center segregation index | dimensionless | |
| t | Solidification time | s |
| TR | Temperature of casting roller | K |
| Ts | Temperature of solid | K |
| Temperature of casting roller at water/roller boundary | K | |
| Temperature of solid at roller/solid boundary | K | |
| Temperature of casting roller at roller/solid boundary | K | |
| Ti | Temperature of mush | K |
| Temperature on mth node at solid/mush interface | K | |
| Temperature on the third node from the mth node | K | |
| ΔT | Bath undercooling | K |
| ΔTR | Curvature undercooling | K |
| ΔTt | Thermal undercooling | K |
| ΔTk | Kinetic undercooling | K |
| Tis | Temperature of solid at solid/mush interface | K |
| Til | Temperature of solid at solid/mush interface | K |
| Average temperature of solid | K | |
| Average temperature of mush | K | |
| Average temperature of liquid | K | |
| μw | Dynamic viscosity of cooling water | N s m−2 |
| Chemical potential of kth component in solid at solid/mush interface | J mol−1 | |
| Chemical potential of kth component in solid at solid/mush interface | J mol−1 | |
| Initial chemical potential of kth component in solid at solid/mush interface | J mol−1 | |
| Initial chemical potential of kth component in mush at solid/mush interface | J mol−1 | |
| Initial jump of chemical potential of the kth component | J mol−1 | |
| v | Velocity at solid/mush interface | m s−1 |
| vw | Flow speed of cooling water | m s−1 |
| Diffusion velocity of kth component at solid/mush interface | m s−1 | |
| Diffusion velocity of kth component at liquid | m s−1 | |
| v0 | Maximum velocity proportional to the sound velocity in liquid | m s−1 |
| vt | Velocity at solid/mush interface at t s | m s−1 |
| Average velocity of α/β interface | m s−1 | |
| Vm | Molar volume | m3 mol−1 |
| Vw | Flow volume per unit time of cooling water | m3 s−1 |
| ω | Casting speed | m min−1 |
| Δx | Space step in solid used by the finite difference method | m |
| ρ | Density of cooling water | kg m−3 |
| Non-equilibrium solute diffusion factor of kth component | dimensionless |
| Roller diameter (mm) | Roller width (mm) | Rolling force (kN) | Cooling channel diameter (mm) | Casting speed (m/min) | The pressure of cooling water (MPa) | Pouring temperature (K) |
|---|---|---|---|---|---|---|
| 360 | 100 | 400 | 300 | 0-30 | 0-0.4 | 954-1000 |
Table 2. Equipment parameters of TRC.
| Roller diameter (mm) | Roller width (mm) | Rolling force (kN) | Cooling channel diameter (mm) | Casting speed (m/min) | The pressure of cooling water (MPa) | Pouring temperature (K) |
|---|---|---|---|---|---|---|
| 360 | 100 | 400 | 300 | 0-30 | 0-0.4 | 954-1000 |
Fig. 1. Schematic illustration of centerline segregation (CS) and edge segregation (ES) during the sub-rapid solidification of TRC, where, the melt is poured directly into the gap of two counter-rotating rollers through a tundish. The solidification starts from the two sides in contact with the rollers to the center, accompanied by the sequential growth of fine grains, columnar and equiaxial dendrites. The ES, i.e., the microscale segregation, is formed at the beginning of solidification due to solute redistribution at the dendritic interface, and ultimately retains at the edge of sheets. The CS, i.e., the macroscale segregation, is formed at the end of solidification due to enrichment of solute in the liquid, and finally appears at the center of sheets. Herein, the casting speed (ω) and the flow speed of cooling water (vw) are selected as the greatest influential parameters controlling the CS and the ES.
Fig. 2. Modular framework for the present modeling of solidification upon TRC. Firstly, the alloy composition (e.g. Al-1.72Mg–1.67Si (in at.%)) and the processing parameters (e.g. the casting speed ω and the flow speed of cooling water vw) serve as inputs into the dendritic growth model (Section 2.2), and the microscale parameters for steady-state interface can be solved. Secondly, incorporating these parameters into the overall solidification kinetic model (Section 2.3), the transient parameters for the processing of solidification can be obtained through multiple iterations. Finally, the transient thermodynamic driving force ΔGt, the transient kinetic energy barrier $Q_{\text{eff}}^{\text{t}}$ and the generalized stability can be derived (Section 2.4), and the high ΔGt-high GS criterion is proposed to adjust the processing and/or alloy composition to suppress the segregation.
| Parameters | Units | Values | Refs. |
|---|---|---|---|
| Thermal diffusion coefficients of s and li, as and | m2 s−1 | 1.5 × 10−5 | [ |
| Heat capacity of s and li, Cp | J kg−1 K−1 | variables | Thermal-Calc |
| Latent heat of solidification, ΔHf | J m−3 | variables | Thermal-Calc |
| Thermal transfer coefficients between RT and s, h1 | W m−2 K−1 | 5500 | [ |
| Thermal conductivity of RT, KR | W m−1 K−1 | 398 | [ |
| Heat capacity of RT, | J kg−1 K−1 | 386 | [ |
| Thermal conductivity of WT, KW | W m−1 K−1 | 0.59 | [ |
| Dynamic viscosity of WT, μw | N s m−2 | 56 × 10−6 | [ |
| Density of WT, ρ | Kg m−3 | 103 | [ |
| Upper limit of interface velocity, v0 | m s−1 | 1000 | [ |
| Diffusion speed at interface, | m s−1 | 0.001 | [ |
| Diffusion speed in the bulk liquid, | m s−1 | 1 | [ |
| Mobility for the rapid interface migration, M0 | m s−1 | 290 | [ |
| Primary dendritic arm spacing, λ1 | m | 1.3 × 10−3 | [ |
| Diffusion coefficient of Al, Mg and Si in s and l, Dk | m2 s−1 | variables | Thermal-Calc |
| Thermal transfer coefficients between RT and s, h1 | W m−1 K−1 | 5500 | [ |
| Gibbs-Thomson coefficient, Γ | K m | 1.3 × 10−7 | [ |
Table 3. Physical parameters for model calculations.
| Parameters | Units | Values | Refs. |
|---|---|---|---|
| Thermal diffusion coefficients of s and li, as and | m2 s−1 | 1.5 × 10−5 | [ |
| Heat capacity of s and li, Cp | J kg−1 K−1 | variables | Thermal-Calc |
| Latent heat of solidification, ΔHf | J m−3 | variables | Thermal-Calc |
| Thermal transfer coefficients between RT and s, h1 | W m−2 K−1 | 5500 | [ |
| Thermal conductivity of RT, KR | W m−1 K−1 | 398 | [ |
| Heat capacity of RT, | J kg−1 K−1 | 386 | [ |
| Thermal conductivity of WT, KW | W m−1 K−1 | 0.59 | [ |
| Dynamic viscosity of WT, μw | N s m−2 | 56 × 10−6 | [ |
| Density of WT, ρ | Kg m−3 | 103 | [ |
| Upper limit of interface velocity, v0 | m s−1 | 1000 | [ |
| Diffusion speed at interface, | m s−1 | 0.001 | [ |
| Diffusion speed in the bulk liquid, | m s−1 | 1 | [ |
| Mobility for the rapid interface migration, M0 | m s−1 | 290 | [ |
| Primary dendritic arm spacing, λ1 | m | 1.3 × 10−3 | [ |
| Diffusion coefficient of Al, Mg and Si in s and l, Dk | m2 s−1 | variables | Thermal-Calc |
| Thermal transfer coefficients between RT and s, h1 | W m−1 K−1 | 5500 | [ |
| Gibbs-Thomson coefficient, Γ | K m | 1.3 × 10−7 | [ |
Fig. 3. Evolution of the interfacial parameters at the dendritic tip with the interfacial velocity v during the solidification, in the TRC of Al-1.72Mg–1.67Si (in at.%) alloy. (a) The undercooling contributions: the constitutional undercooling ΔTc, the kinetic undercooling ΔTk and the curvature undercooling ΔTR, vs. v; (b) The liquidus temperature Tm, the interfacial temperature Ti and the dendritic tip radius R, vs. v; (c) The interfacial concentration vs. v, where the parameter of v(Rmax) corresponding to the maximal $C_{k}^{\text{il}}$ is calculated as 0.00542 m/s; (d) The non-equilibrium solute partition coefficient kv of Si and Mg vs. v.
Fig. 4. Evolution of interfacial concentration in the liquid side $C_{k}^{\text{il}}$ and the corresponding edge segregation $S_{k}^{\text{e}}$ for (a) Si and (b) Mg, and average concentration in the liquid $\langle C_{k}^{\text{l}}{{\rangle }^{\text{l}}}$ and the corresponding centerline segregation $S_{k}^{\text{c}}$ for (c) Si and (d) Mg, with the solidification time t in the TRC of Al-1.72Mg-1.67Si (in at.%) alloy subjected to ω = 2, 4, 6 m/min and vw=0.068 m/s. Corresponding evolutions as functions of the distance d (from the sides in contact with the rollers to the center) by processing parameters of ω =6 m/min and vw=0.043 m/s are also shown.
Fig. 5. (a) As for Al-1.72Mg-1.67Si (in at.%) alloy by TRC, parking method was used to prepare as-cast sheets, where, unit boxes of 30 × 30 μm were taken every 170 μm to perform EDS surface images, and contents of Si and Mg were used to represent the averaged concentration of each box. Taking Si as an example, the surface images at the edge and the center of as-cast and rolled specimens by processing parameters of (b) ω = 2 m/min and vw = 0.068 m/s, (e) ω = 6 m/min and vw = 0.068 m/s, and (h) ω = 6 m/min and vw = 0.043 m/s, respectively. The experimental concentration of Si and Mg from the boxes of as-cast specimens (shown in (a)) by processing parameters of (c) ω = 2 m/min and vw = 0.068 m/s, (f) ω = 6 m/min and vw = 0.068 m/s, and (i) ω = 6 m/min and vw =0.043 m/s evolving with the distance d (from the sides in contact with the rollers to the center). The calculated total concentration Ck of Si and Mg during the sub-rapid solidification by processing parameters of (d) ω = 2 and vw = 0.068 m/s, (g) ω = 6 m/min and vw = 0.068 m/s, and (j) ω = 6 m/min and vw= 0.043 m/s, evolving with d.
Fig. 6. Evolution of (a) the thermodynamic driving force ΔGt and (b) the kinetic energy barrier $Q_{\text{eff}}^{\text{t}}$ for the interfacial migration during the sub-rapid solidification, in the TRC of Al-1.72Mg–1.67Si (in at.%) alloy subject to ω = 2, 4 and 6 m/min and vw = 0.068 m/s, with the solidification time t. Corresponding evolution with the distance d (from the sides in contact with the rollers to the center) by processing parameters of ω =6 m/min and vw = 0.043 m/s are also shown.
Fig. 7. Evolution of (a) the initial, the maximal and the final thermodynamic driving force $\text{ }\!\!\Delta\!\!\text{ }{{G}^{\text{t}}}$ and the initial, the minimal and the final kinetic energy barrier $Q_{\text{eff}}^{\text{t}}$ for the interfacial migration, and (b) the centerline segregation $S_{k}^{\text{c}}$ and the edge segregation $S_{k}^{\text{e}}$ during the sub-rapid solidification, in the TRC of Al-1.72Mg–1.67Si (in at.%) alloy, with the casting speed ω and the flow speed of cooling water vw.
Fig. 8. Evolution of (a) the generalized stability during the sub-rapid solidification with the solidification time t, and (b) the total segregation Sk of Si for as-cast specimens with the distance from the center, in the TRC of Al-1.72Mg-1.67Si (in at.%) alloy, by processing parameters of ω = 2, 4 and 6 m/min and vw= 0.068 m/s. Corresponding evolutions by processing parameters of ω = 6 m/min and vw= 0.043 m/s with the distance d (from the sides in contact with the rollers to the center) and with the distance from the center are also shown in (a) and (b), respectively.
Fig. 9. Evolution of (a) the initial thermodynamic driving force $\text{ }\!\!\Delta\!\!\text{ }{{G}^{\text{t}}}$ and the final generalized stability for interfacial migration, and (b) the centerline segregation $S_{k}^{\text{c}}$ and the edge segregation $S_{k}^{\text{e}}$ during the sub-rapid solidification, in the TRC of Al-1.72Mg–1.67Si (in at.%) alloy, with the casting speed ω and the flow speed of cooling water vw. As ω increases, the increased initial $\text{ }\!\!\Delta\!\!\text{ }{{G}^{\text{t}}}$ and the decreased final GS, respectively, suppresses the CS and enriches the ES, and vice versa. Analogously, as vw decreases, the decreased initial $\text{ }\!\!\Delta\!\!\text{ }{{G}^{\text{t}}}$ and the increased final GS, respectively, strengthens the CS and weakens the ES, and vice versa. As such, a trade-off intersecting point for ω and vw can be formed, which guarantees an optimal combination of ω= 6 m/min and vw= 0.043 m/s to satisfy relatively low $S_{k}^{\text{c}}$ and $S_{k}^{\text{e}}$. This philosophy can be clearly evidenced by the present model calculations, following the criterion of high $\text{ }\!\!\Delta\!\!\text{ }{{G}^{\text{t}}}$-high GS.
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