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。
本文基于金兹堡－朗道相变理论，分别建立了一个以熵增原理的二元合金相场模型和自由能减小原理的纯物质相场模型，并在薄界面限制条件下，建立了相场参数与材料热物性参数的关系；采用基于均匀网格的有限差分离散控制方程，网格剖分采用了双重网格法。数值计算时，为了避免时间步长的限制，提高计算效率，相场和溶质场控制方程采用了用显式算法，即向前Euler法；温度控制方程则采用交替方向隐式法（ADI算法）。
分析了各向异性对平衡形貌的影响，发现在界面各向异性系数变化中，这之间有一临界值，二维的临界值为0.067；建立了引入各向异性的二维相场模型，并对二维各向异性项进行了离散；分别对二维界面能各向异性进行了数值模拟，得到了模拟结果与理论分析结果基本相符；在大于临界值时，相场模型模拟出的结果出现了失真。
在相场模型的基础上推导了噪声的数学模型，给出了相场噪声和温度场噪声的数学方程；从理论上分析了相场模型中噪声的引入，提出了三种噪声引入的方法；对噪声产生侧向分支进行了模拟，发现热噪声能够引发枝晶侧向不稳定,是侧向分支形成的主要起因。
模拟了不同过冷度下纯镍凝固过程枝晶生长，发现温度场可解释过冷度对二次枝晶生长的影响的原因；低过冷度下枝晶半径随过冷度的增加而减少，与边缘稳定性理论和微观可解性理论一致。
Numerical simulation of microstructure plays an important role on the development and application of metallic materials, and it is also one of the main development directions which a computer used in the field of materials science. There are some microstructure numerical simulation methods: deterministic methods, stochastic methods and phase field method. Phase-field methods 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.
Based on the Ginzburg - Landau theory of phase transitions, respectively, it established a principle of entropy of binary alloy phase field model and the principle of freedom to reduce the phase-field model of pure substances , and in the thin interface limit condition, it established a relationship between phase-field parameters and thermal physical parameters. Using uniform grid based on finite difference equations to discrete control equation, mesh generation used a dual grid. Numerical calculations, in order to avoid time step constraints, and improving computational efficiency, phase field and solute field equations used an explicit algorithm, that is forward Euler method; temperature control equation is adopted alternating direction implicit method (ADI method) .
We analyzed the impact of anisotropy on the equilibrium shape and found a critical value. The critical value is 0.067,also we established a two-dimensional anisotropic phase field model, and two-dimensional anisotropic discrete items; respectively, two-dimensional interface energy anisotropy of the numerical simulation results obtained the result was consistent with the theoretical analysis; In greater than the critical value, the phase field model to simulate came out with the distortion.
In the phase field model based on the noise derived mathematical model of phase field and temperature field disturbance disturbance mathematical equation; from the theoretical analysis of the phase field model, the introduction of noise, making noise, the introduction of three ways; noise generated on the lateral branches were simulated and found that thermal noise can lead to dendritic lateral instability is the main cause of the formation of lateral branches.
Simulation under different undercooling solidification process of pure nickel dendrite growth, found that could explain the undercooling temperature on the secondary branch growth of the reasons; low undercooling dendrite radius increases with the undercooling decrease and the result is consistent with the edge of stability theory and the microscopic solvability of the theory.
引文
[1] H. Assadi, M. Oghabi .Influence of Ordering Kinetics on Dendritic Growth Morphology. Acta Materialia .2009,57 (2):1639–1647.
[2] M.Asta,C.Beckermann. Solidification Microstructures and Solid-State Parallels: Recent Developments, Future Directions. Acta Materialia. 2009, 57 (2):941–971.
[3]侯卫周,侯华.铸件凝固过程微观组织模拟技术的研究.华北工学院学报.2002,06(06):0416-0419.
[4] M.Rappaz,C.A.Gandin. Probabilistc Modeling of Microstructure Formation in Solidification Processes. Aeta Metall Mater.1993,41(2):345-360.
[5] C.A.Gandin,M.RaPPaz,R.Tintillier.3-Dimensional Simulation of the Grain Formation in Investrnent Castings.Metal MaterTransA.1994,25(3):629-635.
[6] W.DeLimasilva,L.C.Wrobel. A Front-Tracking BEM Formulation for One Phase Solidification Melting Problems.Eng Anal Bound Elem.1995,16(2):171-182.
[7] D.Jurie,GTryggvason. A Front-Tracking Method for Dendritie Solidification. J Comp Phys.1996,123(l):127-148.
[8] J.F.MeCarthy,N.W.Blake. A Front-Tracking Model for the Rapid Solidification of Dendritie Alloys.Aeta Mater.1996,44(5):2093-2100.
[9] Y.Saito,G.Goldbeck-wood. Numerical Simulation of Dendritie Growth.Phys RevA.1988,38:2148-2153.
[10] A.Schmidt,Computation of Three Dimensional Dendrite with Finite Elements.J Comp Phys.1996,125:293-312.
[11] H.L.TSai,B.Rubinsky,A Numerical Study Using Front Tracking Finite-Elements on the Morphological Stability of a Planar Interface during Transient Solidification Processes. J Cryst Growth.1984,69(l):29-46.
[12] H.L.TSai,B.Rubinsky. A Front Tracking Finite-Element Study on Change Of Phase Interface Stability during Solidification Processes in Solutions. J Cryst Growth.1984,70(l-2):56-63.
[13] N. Zabaras,Y.Ruan,O.Richmond. Front Tracking Thermomechanical Model for Hypoelastic Viscoplastic Behavior in a Solidifying Body. Comp Meth Appl Mech Eng.1990,81(3):333-364.
[14] JoséLuiz Boldrini , Bianca Morelli Calsavara Caretta. Analysis of a Two-Phase Field Model for the Solidification of an Alloy. J. Math. Anal. Appl. 2009, 357(2):25–44.
[15] R.S. Qin, H.K.D.H. Bhadeshia. Phase-Field Model Study of the Crystal Morphological Evolution of Hcp Metals. Acta Materialia . 2009,57(2):3382–3390.
[16] Yutuo ZHANG , Chengzhi WANG . Phase-Field Modeling of Dendrite Growth. Acta Metall. 2009,22(3):197-201.
[17]丁雨田,袁训锋,胡勇.过冷金属熔体中枝晶生长的相场法模拟.材料热处理. 2007,22(36):0040-0047.
[18] A.Karma , W.Rappel. Phase-Field Method for Computationally Efficient Modelling of Solidification with Arbitrary Interface Kineties. Phys Rev Lett.1996,53(3017-3020).
[19] S.Wang,R.F.Sekerka. Computation of the Dendritie Operating State at Large Supercoolings by the Phase Field Model.Phys Rev E.1996,53:3760-3766.
[20] S.L.Wang , R.F.Sekerka. Algorithms for Phase-Field Computation of the Dendritie Operation State at Large Supercoolings.J Comp Phys.1996 ,127:110-117.
[21] A.Karrna,W.J.Rappel. Quantitative Phase-Field Modeling of Dendritie Growth in Two and Three Dimensions. Phys Rev E.1998,57(4):4323-4349.
[22] H.Zhang,L.L.Zheng,Y.Prasad,et al. A Curvilinear Level Set Formulation For Highly Deformable Free Surface Problems with Application to Solidification.Num Heat Trans B.1998,34(l):1-20.
[23] YT.Kim,N.Provatas,N.Goldenfeld,et al. Universal Dynamics of Phase-Field Models for Dendritie Growth.Phys RevE.2000,59:2546-2549.
[24] F.Gibou,R.Fedkiw,R.Cafliseh,et al. A Level Set Approach for the Numerical Simulation of Dendritie Growth.JSeiComp.2003,19(l-3):183-199.
[25] L.J.Tan,N.Zabaras. A Level Set Simulation of Dendritie Solidification with Combined Features of Front-Tracking and Fixed-Domain Methods.J Comput Phys.2006,211(1):36-63.
[26]王狂飞,孙瑞霞,米国发,傅恒志.凝固显微组织数值模拟研究新进展.河南理工大学报. 2008,(03):0348-0351.
[27] Doru M.Stefanseu. Methodologies for Modeling of Solidification Micro-structure and their Capabilities. ISU International,135(6):637.
[28]Doru M.Stefanscu. Modeling of solidification stochastic or Deterministic. Canadian Metallurgical Quarterly.1998,37(3-4):229.
[29] D Raabe.计算材料学.北京:化学工业出版社,2002,9,第1版.
[30] WANG Kuang-fei, LI Bang-sheng .Modeling of Cell/Dendrite Transition During Directional Solidification of Ti-A1 Alloy Using Cellular Automaton Method. Journal of Iron and Steel Research, International. 2008, 15(3) 82-86.
[31]傅廷亮尹雪涛张扬. Voronoi算法模型及其程序实现.计算机模拟技术[M].合肥:中国科学技术大学出版社,2006.
[32] J B Collins, H Levin. Diffuse Interface model of Diffusion-Limited Crystal Growth. Phys. Rev. B, 1985, 31(9): 6119-6122.
[33] J S Langer. in: Directions in Condenesd Physicics. Grinstein G and Maxenko G (World Science, 1986):164-168.
[34] G Caginalp, P C Fife. Phase-Field Methods for Interfacial Boundaries. Phys. Rev. B, 1986, 33(11): 7792-7794.
[35] G Caginalp, P C Fife. High-Order Phase-Field Models and Detailed Anisotropy. Phys. Rev. B,1986, 34(7): 4940-4943.
[36] G Caginalp, P C Fife. Dynamics of Layered Interfaces Arising from Phase Boundaries. SIAM J. of Appl.Math., 1988, 48(3): 506-510.
[37] G Caginalp. Stefan and Hele-Shaw Type Models as Asymptotic Limits of the Phase-Field Equations. Phys. Rev. A, 1989, 39(11): 5887-5890.
[38] P C Fife, G S Gill. The phase-Field Description of Mushy Zones [J]. Phys. D, 1989, 35: 267-275.
[39] P C Fife, G S Gill. Phase-Transition Mechanisms for the Phase-Field Model under Internal Heating. Phys. Rev. A, 1991, 43(2): 843-851.
[40] G J Fix, in: Free Boundary Problems: Theory and Applications, Vol.Ⅱ, edited by A Fasano, M Primicerio. 580.
[41] B. Nestler, D. Danilov , P. Galenko. Crystal Growth of Pure Substances: Phase-Field Simulations in Comparison with Analytical and Experimental Results. Journal of Computational Physics .2005,207 :221-239.
[42]张玉妥,李殿中.用相场方法模拟纯物质等轴枝晶生长.金属学报,2000, 36(6).
[43]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.物理学报,2001, 50(12): 2423-2428.
[44]赵代平,荆涛.用捕获液态改进的相场方法模拟三维枝晶生长.金属学报,2002,38 (12): 1238-1240.
[45]赵达文,杨根仓,过冷熔体凝固行为的自适应有限元相场法模拟,材料导报,2010.
[46] Tonhardt R., Amberg G. Phase-Field Simulation of Dendritic Growth in a Shear Flow.Journal of Crystal Growth, 1998,194(7): 406-425.
[47] Tonhardt R., Amberg G. Dendritic Growth of Randomly Oriented Nuclei in a shear flow. J. Crystal Growth, 2000, 213: 161-187.
[48] D.M. Anderson, G.B. McFadden , A.A. Wheeler .A Phase-Field Model of Solidification with Convection . Physica D .2000,135:175-194.
[49] Beckermann C, Diepers H.J, Steinbach I. et al. Modeling Melt Convection in Phase-Field Simulations of Solidification. Journal of Computational. Physics,1999,154: 468-496.
[50] Tong Xinglin. Phase-Field Simulation of Convective Effects on Dendritic Growth in two Dimensions: Ph.D. thesis, Iowa USA: The University of Iowa, 2002.
[51] Tong X , Beckermann C , Karma A. Velocity and Shape Selection of Dendritic Crystals in a Forced Flow. Physical Rev. E. 2000, 61(1): 49-52.
[52] Beckermann C., Li Q., Tong X. Microstructure Evolution of Equiaxed Dendritic Growth. Science and Technology of Advanced Materials. 2001, (2): 117-126.
[53] Jeong J.H ,Goldenfeld N,Dantzig J.A. Phase-Field Model for Three-Dimensional Dendritic Growth with Fluid Flow. Physical Rev. E.2001, 64: 041602-1- 041602-14.
[54]王颖硕,陈长乐.对流作用下纯物质枝晶生长的相场法模拟.铸造. 2008,(03):0249-0254
[55] Wheeler A.A, Boettinger W.J., McFadden G.B. Phase-Field Model for Isothermal Phase Transition in Binary Alloys. Physical Rev. E. 1992, 45(10): 7424-7439.
[56] Wheeler A.A., Murrary B.T., Schaefer R.J. Computation of Dendrites Using a Phase-Field Model. Physica D, 1993, 66: 243-262.
[57] Loginova I., Amberg G., Agren J. Phase-Field Simulation of Non-isothermal Binary Alloy Solidification. Acta mater., 2001, 49: 573-581.
[58] Lan C.W., Chang Y.C., Shih C.J. Adaptive phase field simulation of non-isothermal free dendritic growth of a binary alloy. Acta Materialia, 2003, 51: 1857-1869.
[59] Peter Galenko. Phase-Field Fodel with Relaxation of the Diffusion Flux in Nonequilibrium Solidification of a Binary System. Physics Letters A ,2001,287 :190-197.
[60] Losert W., Stillman D.A., Cummins H.Z. et al. Selection of Doublet Cellular Patterns in Directional Solidification Through Spatially Periodic Perturbations. Physical Rev. E, 1998, 58 (6): 7492-7506.
[61] Kim S. G., Kim W. T., Suzuki T. Phase-Field Model for Binary Alloys. Physical Rev.E, 1999, 60 (6): 7186-7197.
[62]张光跃,荆涛,柳百成,赵代平.铝合金枝晶生长形貌数值模拟研究.铸造,2002, 51 (12): 764-766.
[63]龙文元,蔡启舟,魏伯康,陈立亮.相场法模拟多元合金过冷熔体中的枝晶生长.物理学报. 2006 ,03.
[64]朱昌盛,王智平.二元合金多晶粒枝晶生长相场法模拟.系统仿真学报.2008,20(23):6514-6518.
[65] Kobayashi R., Warren J.A., Carter W.C. Vector-Valued Phase-Field Model for Crystallization and Grain Boundary Formation. Physica D. 1998, 119: 415-423.
[66] Kobayashi R., Warren J.A., Carter W.C. A Continuum Model of Grain Boundaries. Physica D.2000, 140: 141-150.
[67] Lobkovsky A.E., Warren J.A. Phase-Field Model of Crystal Grains. J. Crystal Growth, 2001, 225: 282-288.
[68] Krill C.E., Chen L.Q. Computer Simulation of 3-D Grain Growth Using a Phase-Field Model. Acta Materialia.2002, 50: 3057-3073.
[69] Sekerka R.F MorPhology:from Sharp Interface to Phase Field models. Journal of Crystal Growth.2004,264:530-540.
[70]刘儒勋,王志峰数值模拟方法和运动界面追踪.中国科学技术人学出版社.合肥.2001.
[71] Karma A., Rappel W.J. Numerical Simulation of Three-Dimensional Dendritic Growth. Physical Review Letter.1996, 77(10), 4050-4053.
[72] G Caginalp. Stefan and Hele-Shaw Type Models as Asymptotic Limits of the Phase-Field Equations. Phys. Rev. A.1989, 39(11): 5887-5890.
[73] P C Fife, G S Gill. Phase-Transition Mechanisms for the Phase-Field Model under Internal Heating. Phys. Rev. A.1991, 43(2): 843-851.
[74] A.Karma, W.J.Rappal. Quantitative Phase-Field Modeling of Dendritic Growth in Two and Three Dimension. Phys Rev E.1998,57(4):4323-4349.
[75]A.Karma,W.J.Rappal. Phase-Field Theory for Computationally Efficient Modeling of Solidification with Arbitrary Interface Kinetics. Phys Rev E.1996,53(4):R3017-R3020.
[76] R.F.Almgren,Second-Order Phase-Field Asymtotics for Unequal Conductivities. Siam J Appl Math.1999,59(6):2086-2107.
[77] A.Karma,W.J.Rappal. Numercial Simulation of Three-Dimentional Dendritic Growth. Phys Rev Lett.1996,77(19):4050-4053.
[78] Allen S M, Cahn D W. Acta Metall.1979, 27:1085
[79] Eggleston J J. A Phase Model for Highly Ansiotropic Interfacial Energy[J] . Physica D. 2001,150:91~103.
[80]许泉,王锦程,李俊杰,杨根仓.各向异性条件下晶粒长大过程的相场法模拟.材料导报. 2007, 21(09):21-25.
[81]王智平,杨世银.各向异性影响非等温凝固过程的相场法模拟.兰州理工大学学报. 2007,33(06):0001-0005.
[82]张国伟,侯华,赵宇辉,程军.晶粒生长中高界面能各向异性的相场模拟.铸造技术. 2008(02):0239-0243.
[83] R.S. Qin, H.K.D.H. Bhadeshia. Phase-Field Model Study of the Effect of Interface Anisotropy on the Crystal Morphological Evolution of Cubic Metals. Acta Materialia. 2009,57(2):2210–2216.
[84]贾伟建,王帆.相场参数对枝晶生长形貌的影响.铸造.2006,55(1):51-54.
[85]侯卫周,向兵,侯华. Ni-Cu二元单相合金枝晶生长相场法模拟研究.铸造. 2008, 1l (2):l170-1176.
[86]徐宏,侯华. Al - Si合金枝晶生长相场法模拟影响因素.北京科技大学学报. 2009.
[87]丁雨田,袁训锋,胡勇.噪声影响凝固微观组织的相场法模拟.铸造技术. 2008,29(l1):55-59.
[88]张国伟,侯华.噪声影响枝晶生长的相场法模拟.铸造技术. 2008,29(06):0777-0781.
[89] Julie Greensmith , Uwe Aickelin, Gianni Tedesco. Information fusion for anomaly detection with the dendritic cell algorithm. Information Fusion. 2010, 57(11):21–34.
[90] Stanislav G. Pavlik_, Robert F. Sekerka. Forces Due to Fluctuations in the Anisotropic Phase Field Model of Solidification. Physica A .1999 , 268:283~290
[91] Stanislav G. Pavlik, Robert F. Sekerka. Fluctuations in the Phase-field Model of Solidification. Physica A ,2000,277:415~431
[92]陈长乐.相场法模拟Ni-Cu合金枝晶生长中过冷度的影响.铸造技术. 2007,28(4):0538-0548.