Phase field method study on the directional solidification microstructure of a Fe–C alloy under forced convection
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摘要: 定向凝固技術能夠獲得特定柱狀晶結構，對于優化合金軸向力學性能具有非常顯著的效果。本文采用耦合流場的相場模型模擬了定向凝固過程中枝晶的生長過程，研究了各向異性系數、界面能對定向凝固枝晶生長的影響以及強制對流作用下枝晶的生長行為。數值求解過程中，選用基于均勻網格的有限差分方法對控制方程進行離散，實現了格子中標記點算法（MAC）和相場離散計算方法的聯合求解。處理微觀速度場和壓力場耦合時，采用MAC算法求解Navier-Stokes方程和壓力Poisson方程，采用交錯網格法處理復雜的自由界面。結果表明：隨著各向異性系數的增大，枝晶尖端生長速度增大，曲率半徑減小，枝晶根部溶質濃度逐漸降低；隨著界面能的增大，枝晶尖端曲率半徑增大，當界面能為最大（0.6 J·m?2）時，凝固呈現平界面的凝固方式向前推進；強迫對流對定向凝固枝晶生長方向影響較大，上游方向定向凝固枝晶粗大且生長速度更快，其現象隨流速的增大而愈加明顯。Abstract: A specific columnar crystal structure is obtained using the directional solidification technique, which has a substantial effect on the optimization of the axial mechanical properties of the alloy. Additionally, the convection phenomenon in the melt changes the temperature field and concentration field at the front of the solid–liquid interface, affecting the shape of this interface. Thus, the influence on alloy properties cannot be ignored. Although the phase field method has more research on the microdendrite growth morphology, the results of coupling the flow field into the phase field and exploring the microdendrite morphology of directional solidification are still scarce. In this paper, the phase field model of a coupled flow field is used to simulate dendritic growth during directional solidification. The effects of the anisotropy coefficient and interfacial energy on the growth of directionally solidified dendrites and the growth behavior of dendrites under forced convection were studied. For the numerical solution procedure, a uniform grid of the finite difference method was used to discretize the governing equations. A combined solution of the MAC algorithm and a phase field discrete calculation was realized. When addressing the coupling of the microvelocity and pressure fields, the MAC algorithm was used to solve the Navier–Stokes equation and pressure Poisson equation, and the interlocked grid method was applied to handle the complex free interface. The results show that the growth rate of the dendrite tip increases, and the radius of curvature and the solute concentration at the root of the dendrite decrease with an increasing anisotropy coefficient. When the anisotropy coefficient is a maximum of 0.065, the wall of the dendrite tends to develop toward a secondary dendrite because of the influence of the anisotropy coefficient; with increasing interfacial energy, the radius of curvature of the dendrite tip increases. When the interfacial energy is a maximum of 0.6 J·m?2, the solidification shows a flat interface advancing mode; forced convection has a great influence on the growth direction of directional solidification dendrites. The directional solidification of dendrites in the upstream direction is coarse and grows faster with increasing flow rate. Additionally, the dendrite growth morphology observed using an optical microscope agrees well with the experimental results.
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Key words:
- Fe–C alloys /
- directional solidification /
- phase field method /
- convection /
- columnar dendrite
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圖 2 相場與溶質場模擬結果. (a) 相場; (b) 溶質場; (c) Fe?C合金定向凝固柱狀枝晶形態
Figure 2. Simulation results of the phase and solute fields: (a) phase field; (b) solute field; (c) directional solidification columnar dendritic morphology of the Fe–C alloy
圖 3 各向異性系數對定向凝固柱狀晶組織的影響. (a) 各向異性系數0.04; (b) 各向異性系數0.05; (c) 各向異性系數0.065
Figure 3. Effect of the anisotropy coefficient on the directionally solidified columnar crystal structure: (a) anisotropy coefficient of 0.04; (b) anisotropy coefficient of 0.05; (c)anisotropy coefficient of 0.065
圖 4 實驗觀察Fe?C合金定向凝固柱狀枝晶生長形態SEM圖. (a) 弱各向異性系數; (b) 強各向異性系數（小過冷度）; (c) 強各向異性系數（大過冷度）
Figure 4. Growth morphology of the columnar dendrite of the Fe–C alloy during directional solidification observed using scanning electron microscopy: (a) weak anisotropy coefficient; (b) strong anisotropy coefficient (low undercooling); (c) strong anisotropy coefficient (high undercooling)
圖 5 同一計算步長下不同各向異性系數的平均枝晶尺寸及直徑變化圖
Figure 5. Variation in the average dendrite size and diameter with different anisotropy coefficients at the same calculation step
圖 6 界面能對定向凝固柱狀枝晶組織的影響. (a) 界面能為0.3 J·m?2; (b) 界面能為0.35 J·m?2; (c) 界面能為0.45 J·m?2; (d) 界面能為0.6 J·m?2
Figure 6. Effect of the interfacial energy on the directionally solidified columnar dendrite structure: (a) interfacial energy of 0.3 J·m?2; (b) interfacial energy of 0.35 J·m?2; (c) interfacial energy of 0.45 J·m?2; (d) interfacial energy of 0.6 J·m?2
圖 7 對流作用下枝晶生長的相場與溶質場模擬結果. (a) 相場; (b) 溶質場
Figure 7. Simulation results of the phase and solute fields of dendritic growth under convection: (a) phase field; (b) solute field
圖 8 不同微觀對流速度下的枝晶生長形貌. (a) 對流速度0.02 m·s?1; (b) 對流速度0.08 m·s?1; (c) 對流速度0.1 m·s?1
Figure 8. Dendrite growth morphology under different microconvection velocities: (a) convective velocity of 0.02 m·s?1; (b) convective velocity of 0.08 m·s?1; (c) convective velocity of 0.1 m·s?1
表 1 Fe-C合金熱力學參數
Table 1. Thermophysical data for dilute Fe?C alloy
Solute mass fraction of carbon/% ${\rm{ \sigma }}/({\rm{J}} \cdot {{\rm{m}}^{ - 2} })$ ${T_m}/{\rm{K}}$ ${V_m}/({{\rm{m}}^3} \cdot {\rm{mol}}^{ - 1} )$ ${k^{\rm{e}}}$ ${D_{\rm{L}}}/({{\rm{m}}^2} \cdot {{\rm{s}}^{ - 1} })$ ${D_{\rm{S}}}/({{\rm{m}}^2} \cdot {{\rm{s}}^{ - 1} })$ ${m^{\rm{e} } }$ 0.5 0.204 1810 7.7 × 10?6 0.204 2×10?8 6×10?9 ?1836 
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