Ni-Cu二元单相合金枝晶生长的相场法数值模拟
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摘要
微观组织数值模拟对金属材料的发展和应用有着重要意义,也是计算机应用于材料科学领域的主要发展方向之一。微观组织数值模拟的方法主要有:确定性方法、随机方法及相场法。其中相场法是一种用于描述在非平衡状态中复杂相界面演变强有力的工具,不需要跟踪复杂固液界面,就可实现模拟金属凝固过程中枝晶生长的复杂形貌,是目前凝固组织模拟的国际前沿研究领域。
本文采用相场法对合金凝固过程的枝晶生长进行了数值模拟,深入探讨了合金凝固过程的枝晶生长机制,为最终实现铸件机械性能的预测奠定了良好的基础。采用基于均匀网格的有限差分离散控制方程,在网格剖分时采用了双重网格法。数值计算时,为了避免时间步长的限制,温度控制方程则采用交替方向隐式法(ADI算法);并采用了窄带固液界面法和捕获溶质扩散层边界法两种方法共同优化相场模型的数值求解算法,大大减少了计算量,提高了计算效率。
在等温凝固条件下,模拟了Ni-Cu二元合金凝固过程枝晶的等轴生长的演变过程;再现了等轴枝晶生长过程侧向分支的竞争生长、熟化过程和溶质偏析。采用Neumann温度边界条件进行了合金凝固枝晶生长的非等温模拟,研究了过冷度对枝晶生长模拟结果的影响,随着温度的降低,枝晶的生长速率越快,枝晶的二次晶臂越发达,枝晶中的溶质偏析越严重,但当温度降到临界温度1569K时,微观偏析迅速降低。
多晶粒的模拟结果不同于单晶粒,枝晶相互接触后,枝晶生长收到抑制,发生弯曲,枝晶不再对称,连续形核模型下的多晶粒模拟结果与真实情况更为逼近。定向条件下出现初始平界面失稳,形成胞晶,胞晶间距逐渐调整,最后稳定生长的胞晶间距约为1.8μm。随着初始温度的降低,界面逐渐由柱状晶向胞状晶再到平界面演变。
Phase-field method can be used to describe the complicated morphologies of dendritic growth without explicitly tracking the complex phase boundaries. It is expected as a powerful tool to describe complex phase transitions in non-equilibrium state. It is the frontier domain of the numerical simulation during solidification processes at present.
The dendritic growth in the solidification of alloy is simulated by the phase-field method, the mechanism of the dendritic growth during solidification of the alloy is discussed, and a favorable base for prediction of mechanical property of casting is established. The governing equations are discretized on uniform grids using the Finite Difference method, and a double grid method is used for the mesh slice. The thermal governing equation is numerically solved using an alternating direct implicit (ADI) method, which is unconditionally stable, irrespective of the time step employed. The narrow solid/liquid interface method and the capturing solute diffusing boundary method are put forward to optimize the numerical computation of the phase-field model.
The free growth of dendrite in the solidification of Ni-Cu binary alloy are simulated on the isothermal condition, the competitive growth of the secondary arms, ripen and solute microsegregation in the free dendritic growth are realized. The dendritic growth of alloy solidification is simulated using the non-isothermal model with Neumann boundary conditions, and the effect of undercooling on the dendritic growth is studied. With the decreasing of freezing storage temperature, the dendrite growth rate of the dendrite faster, secondary crystal arm more developed, and solute segregation moreserious, but when the temperature drops to the critical temperature 1569K, micro segregation lowers quickly.
More grain results and single grain are not the same. When the dendrite interaction, dendrite growth is suppressed, bending, dendrite no longer symmetrical. More grain simulation of Continuous nucleation model and real situation is approaching. In the directional conditions, flat interface chip cell instability, cellular crystal spacing gradually adjusted, and finally the steady growth of cellular crystal spacing is about 1.8 um. As the initial temperature decrease, interface gradually from the columnar crystal to the cellular crystal to flat interface evolution.
本文采用相场法对合金凝固过程的枝晶生长进行了数值模拟,深入探讨了合金凝固过程的枝晶生长机制,为最终实现铸件机械性能的预测奠定了良好的基础。采用基于均匀网格的有限差分离散控制方程,在网格剖分时采用了双重网格法。数值计算时,为了避免时间步长的限制,温度控制方程则采用交替方向隐式法(ADI算法);并采用了窄带固液界面法和捕获溶质扩散层边界法两种方法共同优化相场模型的数值求解算法,大大减少了计算量,提高了计算效率。
在等温凝固条件下,模拟了Ni-Cu二元合金凝固过程枝晶的等轴生长的演变过程;再现了等轴枝晶生长过程侧向分支的竞争生长、熟化过程和溶质偏析。采用Neumann温度边界条件进行了合金凝固枝晶生长的非等温模拟,研究了过冷度对枝晶生长模拟结果的影响,随着温度的降低,枝晶的生长速率越快,枝晶的二次晶臂越发达,枝晶中的溶质偏析越严重,但当温度降到临界温度1569K时,微观偏析迅速降低。
多晶粒的模拟结果不同于单晶粒,枝晶相互接触后,枝晶生长收到抑制,发生弯曲,枝晶不再对称,连续形核模型下的多晶粒模拟结果与真实情况更为逼近。定向条件下出现初始平界面失稳,形成胞晶,胞晶间距逐渐调整,最后稳定生长的胞晶间距约为1.8μm。随着初始温度的降低,界面逐渐由柱状晶向胞状晶再到平界面演变。
Phase-field method can be used to describe the complicated morphologies of dendritic growth without explicitly tracking the complex phase boundaries. It is expected as a powerful tool to describe complex phase transitions in non-equilibrium state. It is the frontier domain of the numerical simulation during solidification processes at present.
The dendritic growth in the solidification of alloy is simulated by the phase-field method, the mechanism of the dendritic growth during solidification of the alloy is discussed, and a favorable base for prediction of mechanical property of casting is established. The governing equations are discretized on uniform grids using the Finite Difference method, and a double grid method is used for the mesh slice. The thermal governing equation is numerically solved using an alternating direct implicit (ADI) method, which is unconditionally stable, irrespective of the time step employed. The narrow solid/liquid interface method and the capturing solute diffusing boundary method are put forward to optimize the numerical computation of the phase-field model.
The free growth of dendrite in the solidification of Ni-Cu binary alloy are simulated on the isothermal condition, the competitive growth of the secondary arms, ripen and solute microsegregation in the free dendritic growth are realized. The dendritic growth of alloy solidification is simulated using the non-isothermal model with Neumann boundary conditions, and the effect of undercooling on the dendritic growth is studied. With the decreasing of freezing storage temperature, the dendrite growth rate of the dendrite faster, secondary crystal arm more developed, and solute segregation moreserious, but when the temperature drops to the critical temperature 1569K, micro segregation lowers quickly.
More grain results and single grain are not the same. When the dendrite interaction, dendrite growth is suppressed, bending, dendrite no longer symmetrical. More grain simulation of Continuous nucleation model and real situation is approaching. In the directional conditions, flat interface chip cell instability, cellular crystal spacing gradually adjusted, and finally the steady growth of cellular crystal spacing is about 1.8 um. As the initial temperature decrease, interface gradually from the columnar crystal to the cellular crystal to flat interface evolution.
引文
[1] Spittle A, Brown R. Computer simulation of the effects of alloy variables on the grain structures of casting[J]. Acta Metall, 1989, 37 (7):1803
[2] Stefanescu D.M., Pang H. Modeling of Casting Solidification Stochastic or eterministic[J]. Canadian Metallurgical Quarierly, 1998, 37 (3-4):229-239
[3] Rappaz M. Stochastic modeling of grain structure formation in solidification process[J]. MRS Bulletin, 1994, 20:
[4] Spittle J.A., Brown S.G.R. Computer simulation of the influence of processing conditions on as-cast grain structures[J]. Mater. Sci., 1989, 24 (5):1777-1781
[5] Spittle J.A., Brown S.G.R. Computer simulation of the effects of alloy variables on the grain structures of castings[J]. Acta Metallurgica, 1989, 37 (7):1803-1810
[6] Zhu P.P., Smith R. W. Dynamic simulation of crystal growth by Monte Carlo method[J]. Acta Metall, 1992, 40 (12):3369-3379
[7] Rappaz M, Gandin Ch-A. Probabilostic Modelling of Microstructure Formation in Solidification Processes[J]. Acta Metall, 1993,(41):350-360
[8] Gandin Ch-A, Rappaz M. A Coupled Finite Element-Cellular Automation Model for The Prediction of Dendritic Grain Structures in Solidification Processes[J]. Acta Metall, 1994 (42):2233-2246
[9] Zhu M.F., Hong C.P. A three Dimensional modified cellular automation model for the prediction of solidification microstructures[J]. ISIJ International, 2002, 42 (5):520-526
[10]李殿中,苏仕方等.镍基合金叶片凝固过程微观组织模拟及工艺优化研究[J].铸造, 1997, 8:1-7
[11]李殿中,杜强,胡志勇等.金属成形过程中组织演变的CelIuaIr Automaton模拟技术[J].金属学报, 1999, 35 (1):1201-1205
[12]许庆彦,柳百成等.采用Cellular Autumation法模拟铝合金的微观组织[J].中国机械工程, 2001, 12 (3):328-331
[13]王同敏,金俊泽.金属凝固过程微观模拟的两个辅助算法[J].铸造, 2000, 49 (8):523-527
[14] Boettinger W J, Wheeler A A, Murray B T, McFadden G B. Prediction of solute trapping at high solidification rates using a diffuse interface phase field theory of alloy solidification[J]. Mater. Sci. and Eng., 1994, A178 (10):217-223
[15] Fix G J. Phase field for free boundary problems. In: Fasono A, Primicerio M,Pitman eds. Free Boundary Problems: Theory and Applications[J]. London, 1983:580-589
[16] Kobayashi R. Modeling and numerical simulations of dendritic crystal growth[J]. Phvsica D, 1993, 63 (10):410-422
[17] Wheeler A A, Murray B T,Schaefer R J. Computation of dendrites using a phase field model[J]. Phys. D, 1993, 66 (10):243-262
[18] Wang S L, Sekerka R F,Wheeler A A. Thermodynamically-consistent phase-field models for solidification[J]. Phys. D, 1993, 69 (10):189-200
[19] Karma A, Rappel W J. Phase field method for computationally efficient modeling of solidification with arbitrary interface kinetics[J]. Phys. Rev. E, 1996, 53 (4):3017-3020
[20] Karma A, Rappel W J. Numerical simulation of three-dimensional dendritic growth[J]. Phys. Rev. Lett., 1996, 77 (10):4050-4053
[21] Karma A, Rappel W J. Quantitative phase-field modeling of dendritic growth in two and three dimensions[J]. Phys. Rev. E, 1998, 57 (4):4323-4349
[22] Warren J A, Boettinger W J. Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method[J]. Acta metal.Mater, 1995, 43 (2):689-703
[23] Wheeler A A, Boettinger W J, McPadden G B. Phase-field model for isothermalphase transition in binary alloy[J]. Phys. Rev. E, 1992, 45 (10):7424-7439
[24] Wheeler A A, Boettinger W J, McFadden G B. Phase field model of trapping during solidification[J]. Phys. Rev. E, 1993, 47 (4):1893-1909
[25] Suzuki T, Ode M,Kim S G. Phase-field model of dendritic growth[J]. Crystal Growth, 2002, 237-239:125-131
[26] Loginova I, Amberg G,Agern J. Phase-field simulation of non-isothermal binary alloy solidification[J]. Acta mater., 2001, 49:573-581
[27] Tonhardt R, Amberg G. Phase field simulation of dendritic growth in a shear flow[J]. Crystal Growth, 1998, 194 (7):406-425
[28] Beckermann C, Diepers H J,Steinbach I. Modeling melt convection in phase-field simulations of solidification[J]. Comut. Phys, 1999, 154:468-496
[29] Tong X, Beckermann C. Velocity and shape selection of dendritic crystals in aforced flow[J]. Phvs. Rev. E, 2000, 61 (1):49-52
[30] Diepers H J, Beckermann C,Steinbach I. Simulation of convection and ripening in a binary alloy mush using the phase-field method[J]. Acta. Mater, 1999, 47 (13):3663-3678
[31] Boettinger W J, Warren J A. Simulation of the cell to plane front transition during directional solidification at high velocity[J]. Crystal Growth, 1999, 200-201:583-591
[32] Kim S C, Kim W T. Phase-field modeling of rapid solidification[J]. Mater.Sci.Eng., 2001, A304:281-286
[33] Diepers H J, Ma DSteinbach,D M. History effects during the selection of primary dendrite spacing.Comparison of phase-field simulations with experimental observations[J]. Crystal Growth, 2002, 237:149-153
[34]张玉妥,李殿中,李依依等.用相场法模拟纯物质等轴晶生长[J].金属学报, 2000, 36 (6):589-591
[35]于艳梅,杨根仓,赵达文等.过冷熔体中枝晶生长的相场法数值模拟[J].物理学报, 2001, 50 (12):2423-2428
[36]朱昌盛,菊涛,柳百成等.热噪声促发枝晶侧向分支的相场法数值模拟[J].铸造, 2004, s3 (12):1040-1042
[37]张光跃.相场法模拟铝合金微观组织研究.清华大学博士学位论文. 2002
[38] Guo Jinjie, Li Xinzhong, Su Yanqing. Phase-field simulation of structure evolution at high growth velocities during directional solidification of Ti55Al45 alloy[J]. Intermetallics, 2005, 13:275-279
[39]李梅娥,杨根仓,周尧和等.二元合金高速定向凝固过程的相场法数值模拟[J].物理学报, 2005, 54 (1):454-458
[40]龙文元.铝合金凝固过程枝晶生长的相场法数值模拟.华中科技大学博士学位论文. 2004
[41]刘静.金属凝固过程微观组织形成的相场法模拟研究.西北工业大学硕士学位论文. 2006
[42]肖荣振.二元合金定向凝固的相场法数值模拟.兰州理工大学硕士学位论文, 2006
[43]冯力,王智平,路阳等.二元合金多晶粒的枝晶生长的等温相场模型[J].物理学报, 2007, 57 (2):1084-1090
[44]张宙庆.铝合金凝固过程微观组织模拟技术研究.中北大学硕士学位论文. 2005
[45]李新中,郭景杰,苏彦庆等.金属熔体凝固微观组织模拟的相场方法研究[J].特种铸造及有色合金, 2005, 25(4):222-225
[46]汤进军.基于thermo-calc热力学计算的Al-Cu共晶合金微观组织模拟及其实验验证.哈尔滨工业大学工学博士学位论文, 2009
[47] W.W.Mullins, R.F.Sekerka. The stability of planar interface during solidification[J]. App.Phys., 1964, 35:444-451
[48] W.W.Mullins. Morphological stability of a particle growing by diffusion of heat flow[J]. App.Phys., 1963, 34:323-329
[49] D.Kessler, H.Levine. Theory of the saffman-taylor‘finger’pattern[J]. Phys.Rev.A, 1986, 33:2621-2633
[50] D.Kessler,H.Levine. Dendritic growth in a channel[J]. Phys.Rev.A, 1986, 34:4980-4987
[51] Marrone R.E. Numerical simulation of solidification part I,Low carbon steel casting-L.Shape[J]. AFS Casting Metals Research 1970, 12:184-192
[52]杨全,张真等.金属凝固与铸造过程数值模拟.浙江大学出版社.1996
[53]朱昌盛,王智平,冯力等.二元合金多晶粒枝晶生长相场法模拟[J].系统仿真学报, 2008, 20 (23):6514-6518
[2] Stefanescu D.M., Pang H. Modeling of Casting Solidification Stochastic or eterministic[J]. Canadian Metallurgical Quarierly, 1998, 37 (3-4):229-239
[3] Rappaz M. Stochastic modeling of grain structure formation in solidification process[J]. MRS Bulletin, 1994, 20:
[4] Spittle J.A., Brown S.G.R. Computer simulation of the influence of processing conditions on as-cast grain structures[J]. Mater. Sci., 1989, 24 (5):1777-1781
[5] Spittle J.A., Brown S.G.R. Computer simulation of the effects of alloy variables on the grain structures of castings[J]. Acta Metallurgica, 1989, 37 (7):1803-1810
[6] Zhu P.P., Smith R. W. Dynamic simulation of crystal growth by Monte Carlo method[J]. Acta Metall, 1992, 40 (12):3369-3379
[7] Rappaz M, Gandin Ch-A. Probabilostic Modelling of Microstructure Formation in Solidification Processes[J]. Acta Metall, 1993,(41):350-360
[8] Gandin Ch-A, Rappaz M. A Coupled Finite Element-Cellular Automation Model for The Prediction of Dendritic Grain Structures in Solidification Processes[J]. Acta Metall, 1994 (42):2233-2246
[9] Zhu M.F., Hong C.P. A three Dimensional modified cellular automation model for the prediction of solidification microstructures[J]. ISIJ International, 2002, 42 (5):520-526
[10]李殿中,苏仕方等.镍基合金叶片凝固过程微观组织模拟及工艺优化研究[J].铸造, 1997, 8:1-7
[11]李殿中,杜强,胡志勇等.金属成形过程中组织演变的CelIuaIr Automaton模拟技术[J].金属学报, 1999, 35 (1):1201-1205
[12]许庆彦,柳百成等.采用Cellular Autumation法模拟铝合金的微观组织[J].中国机械工程, 2001, 12 (3):328-331
[13]王同敏,金俊泽.金属凝固过程微观模拟的两个辅助算法[J].铸造, 2000, 49 (8):523-527
[14] Boettinger W J, Wheeler A A, Murray B T, McFadden G B. Prediction of solute trapping at high solidification rates using a diffuse interface phase field theory of alloy solidification[J]. Mater. Sci. and Eng., 1994, A178 (10):217-223
[15] Fix G J. Phase field for free boundary problems. In: Fasono A, Primicerio M,Pitman eds. Free Boundary Problems: Theory and Applications[J]. London, 1983:580-589
[16] Kobayashi R. Modeling and numerical simulations of dendritic crystal growth[J]. Phvsica D, 1993, 63 (10):410-422
[17] Wheeler A A, Murray B T,Schaefer R J. Computation of dendrites using a phase field model[J]. Phys. D, 1993, 66 (10):243-262
[18] Wang S L, Sekerka R F,Wheeler A A. Thermodynamically-consistent phase-field models for solidification[J]. Phys. D, 1993, 69 (10):189-200
[19] Karma A, Rappel W J. Phase field method for computationally efficient modeling of solidification with arbitrary interface kinetics[J]. Phys. Rev. E, 1996, 53 (4):3017-3020
[20] Karma A, Rappel W J. Numerical simulation of three-dimensional dendritic growth[J]. Phys. Rev. Lett., 1996, 77 (10):4050-4053
[21] Karma A, Rappel W J. Quantitative phase-field modeling of dendritic growth in two and three dimensions[J]. Phys. Rev. E, 1998, 57 (4):4323-4349
[22] Warren J A, Boettinger W J. Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method[J]. Acta metal.Mater, 1995, 43 (2):689-703
[23] Wheeler A A, Boettinger W J, McPadden G B. Phase-field model for isothermalphase transition in binary alloy[J]. Phys. Rev. E, 1992, 45 (10):7424-7439
[24] Wheeler A A, Boettinger W J, McFadden G B. Phase field model of trapping during solidification[J]. Phys. Rev. E, 1993, 47 (4):1893-1909
[25] Suzuki T, Ode M,Kim S G. Phase-field model of dendritic growth[J]. Crystal Growth, 2002, 237-239:125-131
[26] Loginova I, Amberg G,Agern J. Phase-field simulation of non-isothermal binary alloy solidification[J]. Acta mater., 2001, 49:573-581
[27] Tonhardt R, Amberg G. Phase field simulation of dendritic growth in a shear flow[J]. Crystal Growth, 1998, 194 (7):406-425
[28] Beckermann C, Diepers H J,Steinbach I. Modeling melt convection in phase-field simulations of solidification[J]. Comut. Phys, 1999, 154:468-496
[29] Tong X, Beckermann C. Velocity and shape selection of dendritic crystals in aforced flow[J]. Phvs. Rev. E, 2000, 61 (1):49-52
[30] Diepers H J, Beckermann C,Steinbach I. Simulation of convection and ripening in a binary alloy mush using the phase-field method[J]. Acta. Mater, 1999, 47 (13):3663-3678
[31] Boettinger W J, Warren J A. Simulation of the cell to plane front transition during directional solidification at high velocity[J]. Crystal Growth, 1999, 200-201:583-591
[32] Kim S C, Kim W T. Phase-field modeling of rapid solidification[J]. Mater.Sci.Eng., 2001, A304:281-286
[33] Diepers H J, Ma DSteinbach,D M. History effects during the selection of primary dendrite spacing.Comparison of phase-field simulations with experimental observations[J]. Crystal Growth, 2002, 237:149-153
[34]张玉妥,李殿中,李依依等.用相场法模拟纯物质等轴晶生长[J].金属学报, 2000, 36 (6):589-591
[35]于艳梅,杨根仓,赵达文等.过冷熔体中枝晶生长的相场法数值模拟[J].物理学报, 2001, 50 (12):2423-2428
[36]朱昌盛,菊涛,柳百成等.热噪声促发枝晶侧向分支的相场法数值模拟[J].铸造, 2004, s3 (12):1040-1042
[37]张光跃.相场法模拟铝合金微观组织研究.清华大学博士学位论文. 2002
[38] Guo Jinjie, Li Xinzhong, Su Yanqing. Phase-field simulation of structure evolution at high growth velocities during directional solidification of Ti55Al45 alloy[J]. Intermetallics, 2005, 13:275-279
[39]李梅娥,杨根仓,周尧和等.二元合金高速定向凝固过程的相场法数值模拟[J].物理学报, 2005, 54 (1):454-458
[40]龙文元.铝合金凝固过程枝晶生长的相场法数值模拟.华中科技大学博士学位论文. 2004
[41]刘静.金属凝固过程微观组织形成的相场法模拟研究.西北工业大学硕士学位论文. 2006
[42]肖荣振.二元合金定向凝固的相场法数值模拟.兰州理工大学硕士学位论文, 2006
[43]冯力,王智平,路阳等.二元合金多晶粒的枝晶生长的等温相场模型[J].物理学报, 2007, 57 (2):1084-1090
[44]张宙庆.铝合金凝固过程微观组织模拟技术研究.中北大学硕士学位论文. 2005
[45]李新中,郭景杰,苏彦庆等.金属熔体凝固微观组织模拟的相场方法研究[J].特种铸造及有色合金, 2005, 25(4):222-225
[46]汤进军.基于thermo-calc热力学计算的Al-Cu共晶合金微观组织模拟及其实验验证.哈尔滨工业大学工学博士学位论文, 2009
[47] W.W.Mullins, R.F.Sekerka. The stability of planar interface during solidification[J]. App.Phys., 1964, 35:444-451
[48] W.W.Mullins. Morphological stability of a particle growing by diffusion of heat flow[J]. App.Phys., 1963, 34:323-329
[49] D.Kessler, H.Levine. Theory of the saffman-taylor‘finger’pattern[J]. Phys.Rev.A, 1986, 33:2621-2633
[50] D.Kessler,H.Levine. Dendritic growth in a channel[J]. Phys.Rev.A, 1986, 34:4980-4987
[51] Marrone R.E. Numerical simulation of solidification part I,Low carbon steel casting-L.Shape[J]. AFS Casting Metals Research 1970, 12:184-192
[52]杨全,张真等.金属凝固与铸造过程数值模拟.浙江大学出版社.1996
[53]朱昌盛,王智平,冯力等.二元合金多晶粒枝晶生长相场法模拟[J].系统仿真学报, 2008, 20 (23):6514-6518