凝固过程微观组织的相场法模拟
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摘要
凝固过程微观组织的模拟经历了由确定性方法、概率方法和相场方法的模拟;其中相场法是最新的方法,它可以模拟枝晶生长的过程。相场法是一种用于描述在非平衡状态中复杂相界面演变行为的强有力的工具。它不需要跟踪复杂液/固界面,就可实现模拟金属凝固过程中枝晶生长的复杂形貌。
本文采用相场法,对Al-2-mole-%-Si合金凝固过程的多个枝晶生长和单个枝晶的定向生长进行了模拟,研究了相场模型参数对枝晶形貌的影响,主要内容如下:
1.模拟了Al-2-mole-%-Si合金凝固过程多个枝晶生长过程,获得了具有多次分枝的枝晶形貌。不同取向的枝晶在熔液中自由生长,最终互相接触。再现了枝晶生长过程枝晶臂之间的竞争生长、枝晶之间的熟化和熔合现象。
2.预测了Al-2-mole-%-Si合金凝固过程多个枝晶生长过程中溶质场分布。当枝晶尖端扩散场与相邻枝晶上生长出来的分枝的扩散场相遇时,枝晶停止生长。
3.研究了相场模型中参数如:各向异性、过冷度、界面厚度和扰动对多个枝晶生长的影响。
4.模拟了Al-2-mole-%-Si合金凝固过程单个枝晶的定向生长,不仅再现了凝固过程的枝晶生长,而且发现三次臂只在二次晶臂的一侧生长的这种特殊现象,这一模拟结果与Clicksman等人的实验吻合。
The simulation of microstructure evolution during solidification has experienced deterministic method, stochastic method and phase field method. Phase field method is the newest method and it can simulate the growth of dendritic crystal. Phase field method is an effective tool to describe the complicate solid-liquid interface in non-equilibrium state. Without explicitly tracking the complex phase boundaries, it is expected to simulate the complex dendritic growth during the solidification processes.
Taking Al-2-mole-%-Si alloy for example, this work aimed at simulating multiple grains growth and oriented growth during solidification by using phase-field method. The effect of model parameters on dendrite morphology was studied. The main contents are as follows:
1. The dendritic growth of multiple grains of Al-2-mole-%-Si alloy during the solidification processes was simulated. Dendrite morphology of multiple branches was obtained. The arbitrarily oriented crystals grow freely in undercooled melt. Eventually, gains impinge against each other. The competitive growth of grains, ripen, coalescence and solute microsegregation in the dendritic growth are realized.
2. The solute distribution of dendritic growth of multiple grains of Al-2-mole-%-Si alloy during the solidification processes was predicted. When the diffusion field of dendrite tip encounter diffusion field of dendritic branch growth from adjacent dendrite, the dendrites then stop growing.
3. The influence of different anisotropy coefficients, disturbance intension, interface thickness and undercooling on dendrite morphology and distribution of solute for grains were studied.
4. Oriented growth of single grain of Al-2-mole-%-Si alloy during the solidification processes was simulated. Not only the process of dendritic growth during solidification is reappearing, but also the asymmetry phenomenon is found in the secondary and tertiary arms. The side branching occurs only at one side of the arms, which are consistent to experiments made by Clicksman.
本文采用相场法,对Al-2-mole-%-Si合金凝固过程的多个枝晶生长和单个枝晶的定向生长进行了模拟,研究了相场模型参数对枝晶形貌的影响,主要内容如下:
1.模拟了Al-2-mole-%-Si合金凝固过程多个枝晶生长过程,获得了具有多次分枝的枝晶形貌。不同取向的枝晶在熔液中自由生长,最终互相接触。再现了枝晶生长过程枝晶臂之间的竞争生长、枝晶之间的熟化和熔合现象。
2.预测了Al-2-mole-%-Si合金凝固过程多个枝晶生长过程中溶质场分布。当枝晶尖端扩散场与相邻枝晶上生长出来的分枝的扩散场相遇时,枝晶停止生长。
3.研究了相场模型中参数如:各向异性、过冷度、界面厚度和扰动对多个枝晶生长的影响。
4.模拟了Al-2-mole-%-Si合金凝固过程单个枝晶的定向生长,不仅再现了凝固过程的枝晶生长,而且发现三次臂只在二次晶臂的一侧生长的这种特殊现象,这一模拟结果与Clicksman等人的实验吻合。
The simulation of microstructure evolution during solidification has experienced deterministic method, stochastic method and phase field method. Phase field method is the newest method and it can simulate the growth of dendritic crystal. Phase field method is an effective tool to describe the complicate solid-liquid interface in non-equilibrium state. Without explicitly tracking the complex phase boundaries, it is expected to simulate the complex dendritic growth during the solidification processes.
Taking Al-2-mole-%-Si alloy for example, this work aimed at simulating multiple grains growth and oriented growth during solidification by using phase-field method. The effect of model parameters on dendrite morphology was studied. The main contents are as follows:
1. The dendritic growth of multiple grains of Al-2-mole-%-Si alloy during the solidification processes was simulated. Dendrite morphology of multiple branches was obtained. The arbitrarily oriented crystals grow freely in undercooled melt. Eventually, gains impinge against each other. The competitive growth of grains, ripen, coalescence and solute microsegregation in the dendritic growth are realized.
2. The solute distribution of dendritic growth of multiple grains of Al-2-mole-%-Si alloy during the solidification processes was predicted. When the diffusion field of dendrite tip encounter diffusion field of dendritic branch growth from adjacent dendrite, the dendrites then stop growing.
3. The influence of different anisotropy coefficients, disturbance intension, interface thickness and undercooling on dendrite morphology and distribution of solute for grains were studied.
4. Oriented growth of single grain of Al-2-mole-%-Si alloy during the solidification processes was simulated. Not only the process of dendritic growth during solidification is reappearing, but also the asymmetry phenomenon is found in the secondary and tertiary arms. The side branching occurs only at one side of the arms, which are consistent to experiments made by Clicksman.
引文
[1] Oldfield W. A qualitative approach to casting solidification [J]. Freezing cast Iron, ASM Trans,1966.59: 945-960
[2] ZHU Pangping, Smith R W. Dynamic simulation of crystal growth by Monte Carlo method [J]. Acta Metall. 1992.40(12): 3369-3379
[3] Stefaneseu D M, Pang H. Modeling of casting solidification stochastic or deterministic [J]. Canadian Metrical Quarterly. 1998.37(3-4): 229-239
[4] Lee H N, Ryoo H S. Monte Carlo simulation of microstructure evolution based on grain boundary character distribution [J]. Mater Sci Enging. 2000.281:176-188
[5] Crespo D, Prade U T. Kinetic theory of microstructural evolution in nucleation and growth process [J]. Mater Sci Enging, 1997.238: 160-165
[6] Wang Liguo, Paulette Clancy. Kinetic Monte Carlo simulation of the growth of polycrystalline Cu films [J]. Surface Science, 2001.473: 25-38
[7]金俊泽,王宗延等.金属凝固组织形成的仿真研究[J].金属学报,1998.34(9):928-932
[8]宋晓艳,刘国权等.一种改进的晶粒长大Monte Carlo模拟方法[J].自然科学进展,1998.8(3):337-341
[9] SONG Xiaoyan, LIU Guoquan, A simple and efficient three-dimensional Monte Carlo simulation of grain growth [J]. Scripta Materirlia, 1998. 38(11): 1691-1696
[10]宋晓艳,刘国权等.一种简便高效的材料三维组织及其演变的Monte Carlo仿真[J].科学通报,1999.44(5):491-495
[11]刘国权,宋晓艳等.一种仿真新算法及其对三维个体晶粒长大速率拓扑依赖性理论方程的验证[J].金属学报,1999.35(3):245-248
[12] Xiaoyan S, Guoquan L, Nanju G. Re-analysis on grain size distribution during normal grain growth based on Monte Carlo simulation [J]. Scripta Mater, 2000.43(4): 355-359
[13] guoquan L, Haibo Y, Xiaoyan S. A new model of three-dimensional grain growth: theory and computer simulation of topology-dependency of individual grain growth rate [J]. Materials and Design, 2001.22(1): 33-38
[14] Zhu P, Smith R W. Dynamic Simulation of Crystal Growth by Monte Carlo Method I. Model Description and Kinetics [J]. Acta Metal.1992.40(4): 683-692
[15] Zhu P, Smith R W. Dynamic Simulation of Crystal Growth by Monte Carlo Method II. Ingot Microstructures [J]. Acta Metal, 1992.40(12): 3369-3379
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[18] Rappaz M, Gandin Ch A, Tintillier R. Three-Dimensional simulation of the grain formation in investment casting [J]. Metallurgical trans, 1994.25(3): 629-635
[19] Gandin Ch A, Rappaz M. A coupled finite element-cellular automaton model for the prediction of dendritic grain structures in solidification process [J]. Acta Metal Mater, 1994.42(7): 2233-2246
[20] Geiger J, Roosz. A simulation of grain coarsening in two dimensions by cellular-automaton [J]. Acta Mater, 2001.49(4): 23
[21]李强.元胞自动机方法模拟枝晶演变过程的分析与研究[J].特种铸造及有色合金,2005.25(12):724-726
[22]李殿中,杜强,胡志勇等.金属成型过程中组织演变的cellular-automaton模拟技术[J].金属学报,1999.35(1):1201-1205
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[29] Braun R J, Murray B T. Adaptive phase-field computations of dendritic crystal growth [J]. Journal of Crystal Growth, 1997.174: 41-53
[30] Karma A, Rappel W J. Phase-field model of dendritic side branching with thermal noise [J]. Physical Rev E, 1999.60 (4): 6614-3625
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[36] Tonhardt R, Amberg G. Phase-field simulation of dendritic growth in a shear flow [J]. Journal of Crystal Growth, 1998.194(7): 406～425
[37] Tonhardt R, Amberg G. Dendritic growth of randomly oriented nuclei in a shear flow [J]. Journal of Crystal Growth, 2000.213: 161～187
[38] Beckermann C, Diepers H J, Steinbach I, et al. Modeling melt convection in phase-field simulations of solidification [J]. Journal of Computational Physics, 1999.154: 468-496
[39] Tong Xinglin. Phase-field simulation of convective effects on dendritic growth in two dimensions [J]. Ph.D. thesis, Iowa USA:The University of Iowa, 2002:237-242
[40] Tong X, Beckermann C, Karma A. Velocity and shape selection of dendritic crystals in a forced flow [J]. Physical Rev E. 2000.61(1): 49-52
[41] Beckermann C, Li Q, Tong X. Microstructure evolution of equiaxed dendritic growth. Science and Technology of Advanced Materials [J]. 2001.(2): 117-126
[42] Wheeler A A, Boettinger W J, McFadden G B. Phase-field model for isothermal phase transition in binary alloys [J]. Physical Rev E. 1992. 45(10): 7424-7439
[43] Boettinger W J, Warren J A. The phase-field method: simulation of alloy dendritic solidification during decalescence [J]. Metall And Materi Trans A, 1996.27A (3): 657-668
[44] Loginova I, Amberg G, Agren J. Phase-field Simulations of Non-isothermal Binary Alloy Solidification. Acta Mater, 2001. 49:573-581
[45]李新中,郭景杰,苏彦庆等.金属熔体凝固微观组织的相场法模拟研究[J].特种铸造及有色金属, 2005.25(4):222-225
[46] Kim S G., Kim W T, Suzuki T. Phase-field model for binary alloys [J]. Physical Rev E, 1999.60 (6): 7186-7197
[47] Ode M, Suzuki T. Numerical simulation of initial microstructure evolution of Fe-C alloys using a phase-field model [J]. ISIJ International, 2002.42(4): 368-374
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[49] Ode M, Kim S G., Kim W T. Numerical prediction of the secondary dendrite arm spacing using a phase-field model [J]. ISIJ International, 2001.41 (4): 345-349
[50] Ode M, Suzuki T. Numerical simulation of initial microstructure evolution of Fe-C alloys using a phase-field model [J]. ISIJ International, 2002.42(4): 368-374
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[73] Kobayashi R. Modeling and numerical simulations of dendritic crystal growth.
[74] Physica D, 1993.63(10): 410-423
[75] Kobayashi R, Warren JA, Carter WC. A continuum model of grain boundaries [J]. Physica D.2000:140:141
[76] Kim S G, Kim W T, Suzuki T. Phase-field model for binary alloys. Phys Rev E, 1999.60(6):7186—7197
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[2] ZHU Pangping, Smith R W. Dynamic simulation of crystal growth by Monte Carlo method [J]. Acta Metall. 1992.40(12): 3369-3379
[3] Stefaneseu D M, Pang H. Modeling of casting solidification stochastic or deterministic [J]. Canadian Metrical Quarterly. 1998.37(3-4): 229-239
[4] Lee H N, Ryoo H S. Monte Carlo simulation of microstructure evolution based on grain boundary character distribution [J]. Mater Sci Enging. 2000.281:176-188
[5] Crespo D, Prade U T. Kinetic theory of microstructural evolution in nucleation and growth process [J]. Mater Sci Enging, 1997.238: 160-165
[6] Wang Liguo, Paulette Clancy. Kinetic Monte Carlo simulation of the growth of polycrystalline Cu films [J]. Surface Science, 2001.473: 25-38
[7]金俊泽,王宗延等.金属凝固组织形成的仿真研究[J].金属学报,1998.34(9):928-932
[8]宋晓艳,刘国权等.一种改进的晶粒长大Monte Carlo模拟方法[J].自然科学进展,1998.8(3):337-341
[9] SONG Xiaoyan, LIU Guoquan, A simple and efficient three-dimensional Monte Carlo simulation of grain growth [J]. Scripta Materirlia, 1998. 38(11): 1691-1696
[10]宋晓艳,刘国权等.一种简便高效的材料三维组织及其演变的Monte Carlo仿真[J].科学通报,1999.44(5):491-495
[11]刘国权,宋晓艳等.一种仿真新算法及其对三维个体晶粒长大速率拓扑依赖性理论方程的验证[J].金属学报,1999.35(3):245-248
[12] Xiaoyan S, Guoquan L, Nanju G. Re-analysis on grain size distribution during normal grain growth based on Monte Carlo simulation [J]. Scripta Mater, 2000.43(4): 355-359
[13] guoquan L, Haibo Y, Xiaoyan S. A new model of three-dimensional grain growth: theory and computer simulation of topology-dependency of individual grain growth rate [J]. Materials and Design, 2001.22(1): 33-38
[14] Zhu P, Smith R W. Dynamic Simulation of Crystal Growth by Monte Carlo Method I. Model Description and Kinetics [J]. Acta Metal.1992.40(4): 683-692
[15] Zhu P, Smith R W. Dynamic Simulation of Crystal Growth by Monte Carlo Method II. Ingot Microstructures [J]. Acta Metal, 1992.40(12): 3369-3379
[16] Hunt J D. Steady State Columnar and Equiaxed Growth of Dendrites and Eutectic [J].Mater. Sci. Eng.1984.65: 75-83
[17] Thevoz Ph, Desbiolies J L, Rappaz M. Modeling of Equiaxed Microstructure Formation in Casting [J]. Metal Trans A, 1989.20A: 311-321
[18] Rappaz M, Gandin Ch A, Tintillier R. Three-Dimensional simulation of the grain formation in investment casting [J]. Metallurgical trans, 1994.25(3): 629-635
[19] Gandin Ch A, Rappaz M. A coupled finite element-cellular automaton model for the prediction of dendritic grain structures in solidification process [J]. Acta Metal Mater, 1994.42(7): 2233-2246
[20] Geiger J, Roosz. A simulation of grain coarsening in two dimensions by cellular-automaton [J]. Acta Mater, 2001.49(4): 23
[21]李强.元胞自动机方法模拟枝晶演变过程的分析与研究[J].特种铸造及有色合金,2005.25(12):724-726
[22]李殿中,杜强,胡志勇等.金属成型过程中组织演变的cellular-automaton模拟技术[J].金属学报,1999.35(1):1201-1205
[23] Caginalp G, Fife P. Phase-field methods for interfacial boundaries [J]. Phys Rev B, 1986.4 (11): 7792–7794
[24] Kobayashi R. Modeling and numerical simulations of dendritic crystal growth [J]. Physica D, 1993.63(10): 410-423
[25] Wheeler A A, Murrary B T, Schaefer R.J. Computation of dendrites using a phase field model [J]. Physica D, 1993.66(10): 243-262
[26] Wang S L, Sekerka R F, Wheeler A.A. et al. Thermodynamically consistentphase-field models for solidification [J]. Physica D, 1993. 69: 189-200
[27] Penrose O, Fife P C. Thermodynamically consistent models of phase field type for the kinetics of phase transitions [J]. Physica D, 1990. 43: 44-62
[28] Murrary B T, Wheeler A A, Glicksman M.E. Simulation of experimentally observed dendritic growth behavior using a phase-field model [J]. Journal of Crystal Growth, 1995.154: 386-400
[29] Braun R J, Murray B T. Adaptive phase-field computations of dendritic crystal growth [J]. Journal of Crystal Growth, 1997.174: 41-53
[30] Karma A, Rappel W J. Phase-field model of dendritic side branching with thermal noise [J]. Physical Rev E, 1999.60 (4): 6614-3625
[31] Kim Y T, Provatas N, Goldenfeld N. Dantzig J. Universal dynamics of phase-field models for dendritic growth [J]. Physical Rev E, 1999. 59 (9): 2546-2549
[32] Karma A, Rappel W J. Phase field method for computationally efficient modeling of solidification with arbitrary interface kinetics [J]. Physical Rev E, 1996.53 (4): 3017-3020
[33] Karma A, Rappel W J. Phase-field simulation of three-dimensional dendrite: is microscopic solvability theory correct [J]. Crystal Growth, 1997.174: 54-64
[34] Karma A, Rappel W J. Numerical simulation of three-dimensional dendritic growth [J]. Physical Review Letter, 1996.77(10), 4050-4053
[35] Karma A, Lee Y H, Plapp M. Three-dimensional dendrite-tip morphology at low undercooling [J]. Physical Rev E, 2000.61: 3996-4006
[36] Tonhardt R, Amberg G. Phase-field simulation of dendritic growth in a shear flow [J]. Journal of Crystal Growth, 1998.194(7): 406～425
[37] Tonhardt R, Amberg G. Dendritic growth of randomly oriented nuclei in a shear flow [J]. Journal of Crystal Growth, 2000.213: 161～187
[38] Beckermann C, Diepers H J, Steinbach I, et al. Modeling melt convection in phase-field simulations of solidification [J]. Journal of Computational Physics, 1999.154: 468-496
[39] Tong Xinglin. Phase-field simulation of convective effects on dendritic growth in two dimensions [J]. Ph.D. thesis, Iowa USA:The University of Iowa, 2002:237-242
[40] Tong X, Beckermann C, Karma A. Velocity and shape selection of dendritic crystals in a forced flow [J]. Physical Rev E. 2000.61(1): 49-52
[41] Beckermann C, Li Q, Tong X. Microstructure evolution of equiaxed dendritic growth. Science and Technology of Advanced Materials [J]. 2001.(2): 117-126
[42] Wheeler A A, Boettinger W J, McFadden G B. Phase-field model for isothermal phase transition in binary alloys [J]. Physical Rev E. 1992. 45(10): 7424-7439
[43] Boettinger W J, Warren J A. The phase-field method: simulation of alloy dendritic solidification during decalescence [J]. Metall And Materi Trans A, 1996.27A (3): 657-668
[44] Loginova I, Amberg G, Agren J. Phase-field Simulations of Non-isothermal Binary Alloy Solidification. Acta Mater, 2001. 49:573-581
[45]李新中,郭景杰,苏彦庆等.金属熔体凝固微观组织的相场法模拟研究[J].特种铸造及有色金属, 2005.25(4):222-225
[46] Kim S G., Kim W T, Suzuki T. Phase-field model for binary alloys [J]. Physical Rev E, 1999.60 (6): 7186-7197
[47] Ode M, Suzuki T. Numerical simulation of initial microstructure evolution of Fe-C alloys using a phase-field model [J]. ISIJ International, 2002.42(4): 368-374
[48] Ode M, Suzuki T, Kim S G. et al. Phase-field model for solidification of Fe-C alloys [J]. Science and Technology of Advanced Materials, 2000. (1): 43-49
[49] Ode M, Kim S G., Kim W T. Numerical prediction of the secondary dendrite arm spacing using a phase-field model [J]. ISIJ International, 2001.41 (4): 345-349
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