We developed a model in PROCAST base on Cellular Automaton (CA) and Finite Element (FE) methods in order to simulate micro structure of ultra-fast and slow solidification processes. Using this approach, new hopes can be achieved in order to enhance and improve material properties made during solidification processes. The coupled CA-FE model is applied to two important directional solidification processes: silicon ingot casting and laser surface melting which can be considered as ultra-slow and fast solidifications, respectively.
Here first, finite element (FE) and cellular automaton (CA) models are used in order to predict the microstructure of Poly-Crystalline Silicon (PC-Si) ingots during casting in a directional solidification furnace. During solidification, the 3D-FE model is used to simulate thermal field inside the furnace, while the nucleation and growth of PC-Si are simulated by the modified version of 2D-CA model. In this model both nucleation at silicon-crucible interface and nucleation of equiaxed grains on impurities are modeled. The origin of impurity is considered SiC particles when carbon segregates during the solidification and precipitates as SiC particles if carbon solubility limit is reached. The model is evaluated with experimental data for different parameters, such as: the temperature profile at the top and bottom of the PC-Si, grain size, and the height of solidified silicon during solidification. The effects of the cooling rate and competitive growth on the final microstructure of PC-Si are also investigated. A simulated model is proposed which shows that by changing the density and location of nucleation sites it is possible to achieve more effective control on final PC-Si micro structure.
We know during the solidification of silicon ingot, the carbon concentration may play a critical role with regard to the final silicon microstructure. If the carbon concentration passes the solubility limit, SiC particles precipitate and it is possible the grain structure changes from columnar to equiaxed grains. Furthermore, carbon atoms are confined inside these equiaxed grains during solidification, which may decrease the carbon concentration inside the liquid silicon, and the equiaxed grains then become columnar again. Therefore, both solidification and carbon concentration will impact each other. A 3D multiscale model is also developed here to simulate the microstructural evolution of Multi-Crystalline Silicon (mc-Si) ingots coupled with the carbon concentration during the casting process. Our microstructural model is based on the simulation of nucleation and crystal growth. Besides modelling the nucleation due to silicon-crucible interaction, our model can successfully represent the occurrence of equiaxed grains observed ahead of a planar faceted interface due to carbon segregation during solidification. In addition, by coupling concentration and crystallization, the transition from columnar to equiaxed growth and equiaxed to columnar growth can be modelled. Several comparisons are made between the mc-Si ingot’s microstructure obtained by our model and the experimental data in order to evaluate our model.
There are serious questions about the grain structure of metals after laser melting and the ways that it can be controlled. In this regard, in the second approach, the current study explains the grain structure of metals after laser melting using a new model based on combination of 3D finite element (FE) and cellular automaton (CA) models validated by experimental observation. Competitive grain growth, relation between heat flows and grain orientation and the effect of laser scanning speed on final micro structure are discussed with details. Moreover, using the secondary laser heat source (SLHS) as a new approach to control the grain structure during the laser melting is presented.

1. Sergueeva, A., et al., Advanced mechanical properties of pure titanium with ultrafine grained structure. Scripta Materialia, 2001. 45(7): p. 747-752.
2. Armstrong, R.W. Hall-Petch Analysis of yield, flow and fracturing. in MRS Proceedings. 1994. Cambridge Univ Press.
3. Hall, E., Variation of hardness of metals with grain size. 1954.
4. Armstrong, R., The influence of polycrystal grain size on several mechanical properties of materials. Metallurgical and Materials Transactions, 1970. 1(5): p. 1169-1176.
5. Garofalo, F., Fundamentals of creep and creep-rupture in metals. 1965: Macmillan.
6. Feltham, P. and J.D. Meakin, Creep in face-centred cubic metals with special reference to copper. Acta metallurgica, 1959. 7(9): p. 614-627.
7. Matsui, T., et al., Correlation between microstructure and photovoltaic performance of polycrystalline silicon thin film solar cells. Japanese journal of applied physics, 2002. 41(1R): p. 20.
8. Wang, Z.-J., et al., Relationship between electrical activity and grain boundary structural configuration in polycrystalline silicon. Interface Science, 1999. 7(2): p. 197-205.
9. Zhu, M. and C. Hong, A Three Dimensional Modified Cellular Automaton Model for the Prediction of Solidification Microstructures. ISIJ international, 2002. 42(5): p. 520-526.
10. Gandin, C.A., et al., Grain texture evolution during the columnar growth of dendritic alloys. Metallurgical and Materials Transactions A, 1995. 26(6): p. 1543-1551.
11. Dantzig, J.A. and M. Rappaz, Solidification. 2009: EPFL press.
12. Flemings, M.C., Solidification processing. Metallurgical transactions, 1974. 5(10): p. 2121-2134.
13. Kremeyer, K., Cellular automata investigations of binary solidification. Journal of Computational Physics, 1998. 142(1): p. 243-263.
14. Pruppacher, H.R., J.D. Klett, and P.K. Wang, Microphysics of clouds and precipitation. 1998.
15. Lee, K.-Y. and C.P. Hong, Stochastic Modeling of Solidification Grain Structures of Al-Cu Crystalline Ribbons in Planar Flow Casting. ISIJ international, 1997. 37(1): p. 38-46.
16. Stefanescu, D.M., Methodologies for Modeling of Solidification Microstructure and Their Capabilities. Isij International, 1995. 35(6): p. 637-650.
17. Flood, S. and J. Hunt, Columnar and equiaxed growth: II. Equiaxed growth ahead of a columnar front. Journal of Crystal Growth, 1987. 82(3): p. 552-560.
18. Möller, H.J., et al., Multicrystalline silicon for solar cells. Thin Solid Films, 2005. 487(1): p. 179-187.
19. Surek, T., Crystal growth and materials research in photovoltaics: progress and challenges. Journal of Crystal Growth, 2005. 275(1): p. 292-304.
20. Zhang, H., et al., Nucleation and bulk growth control for high efficiency silicon ingot casting. Journal of Crystal Growth, 2011. 318(1): p. 283-287.
21. Wang, Y.-C., et al., Grain-size-related transient terahertz mobility of femtosecond-laser-annealed polycrystalline silicon. Applied Physics B, 2009. 97(1): p. 181-185.
22. Bo, X.-Z., et al., Large-grain polycrystalline silicon films with low intragranular defect density by low-temperature solid-phase crystallization without underlying oxide. Journal of applied physics, 2002. 91(5): p. 2910-2915.
23. Yamamoto, K., et al., A high efficiency thin film silicon solar cell and module. Solar Energy, 2004. 77(6): p. 939-949.
24. Roy, A., et al., Growth of large diameter silicon tube by EFG technique:: modeling and experiment. Journal of Crystal Growth, 2001. 230(1): p. 224-231.
25. Barvinschi, F., et al., Modeling the multi-crystalline silicon ingots solidification process in a vertical square furnace. Journal of Optoelectronics and Advanced Materials, 2003. 5(1): p. 293-300.
26. Vizman, D., J. Friedrich, and G. Mueller, 3D time-dependent numerical study of the influence of the melt flow on the interface shape in a silicon ingot casting process. Journal of crystal growth, 2007. 303(1): p. 231-235.
27. Wu, B., et al., Bulk multicrystalline silicon growth for photovoltaic (PV) application. Journal of Crystal Growth, 2008. 310(7): p. 2178-2184.
28. Wei, J., et al., Modeling and improvement of silicon ingot directional solidification for industrial production systems. Solar Energy Materials and Solar Cells, 2009. 93(9): p. 1531-1539.
29. Black, A., et al., Optimizing seeded casting of mono-like silicon crystals through numerical simulation. Journal of Crystal Growth, 2012. 353(1): p. 12-16.
30. Ouadjaout, D., et al., Growth by the Heat Exchanger Method and Characterization of Multi-crystalline Silicon ingots for PV. Rev. Energ. Ren, 2005. 8: p. 49-54.
31. Fujiwara, K., et al., Growth of structure-controlled polycrystalline silicon ingots for solar cells by casting. Acta Materialia, 2006. 54(12): p. 3191-3197.
32. Huali, Z., et al., Growth of multicrystalline silicon ingot with both enhanced quality and yield through quartz seeded method. Journal of Crystal Growth, 2015.
33. Lan, C., et al., Grain control in directional solidification of photovoltaic silicon. Journal of Crystal Growth, 2012. 360: p. 68-75.
34. Kühn, G., W. Kurz, DJ Fisher, Fundamentals of Solidification. Trans Tech Publications, Switzerland‐Germany‐UK‐USA, 1986 (Erstauflage 1984), 242 Seiten, zahlreiche Abbildungen und Tabellen, Sachwortindex, SFr 54.00, ISBN 0‐87849‐523‐3. Crystal Research and Technology, 1986. 21(9): p. 1176-1176.
35. Tsoutsouva, M., et al., Undercooling measurement and nucleation study of silicon droplet solidification. Crystal Research and Technology, 2015. 50(1): p. 55-61.
36. Brynjulfsen, I. and L. Arnberg, Nucleation of silicon on Si 3 N 4 coated SiO 2. Journal of Crystal Growth, 2011. 331(1): p. 64-67.
37. Beaudhuin, M., et al., Impurities influence on multicrystalline photovoltaic Silicon. Transactions of the Indian Institute of Metals, 2009. 62(4-5): p. 505-509.
38. Liu, L., S. Nakano, and K. Kakimoto, Carbon concentration and particle precipitation during directional solidification of multicrystalline silicon for solar cells. Journal of Crystal Growth, 2008. 310(7): p. 2192-2197.
39. Beaudhuin, M., et al., One-dimensional model of the equiaxed grain formation in multi-crystalline silicon. Journal of crystal growth, 2011. 319(1): p. 106-113.
40. Delannoy, Y., F. Barvinschi, and T. Duffar, 3D dynamic mesh numerical model for multi-crystalline silicon furnaces. Journal of Crystal Growth, 2007. 303(1): p. 170-174.
41. Watanabe, K., K. Nagayama, and K. Kuribayashi. Morphological transition in crystallization of Si from undercooled melt. in Journal of Physics: Conference Series. 2011. IOP Publishing.
42. Nagashio, K. and K. Kuribayashi, Growth mechanism of twin-related and twin-free facet Si dendrites. Acta Materialia, 2005. 53(10): p. 3021-3029.
43. Fujiwara, K., et al., In situ observation of Si faceted dendrite growth from low-degree-of-undercooling melts. Acta Materialia, 2008. 56(11): p. 2663-2668.
44. Fujiwara, K., et al., In-situ observations of melt growth behavior of polycrystalline silicon. Journal of Crystal Growth, 2004. 262(1): p. 124-129.
45. Glicksman, M. and A. Lupulescu, Dendritic crystal growth in pure materials. Journal of crystal growth, 2004. 264(4): p. 541-549.
46. Atwater, H.A., C.V. Thompson, and H.I. Smith, Mechanisms for crystallographic orientation in the crystallization of thin silicon films from the melt. Journal of Materials Research, 1988. 3(06): p. 1232-1237.
47. Aoyama, T., Y. Takamura, and K. Kuribayashi, Dendrite growth processes of silicon and germanium from highly undercooled melts. Metallurgical and Materials Transactions A, 1999. 30(5): p. 1333-1339.
48. Fujiwara, K., et al., In situ observations of crystal growth behavior of silicon melt. Journal of crystal growth, 2002. 243(2): p. 275-282.
49. Fujiwara, K., et al., Directional growth method to obtain high quality polycrystalline silicon from its melt. Journal of Crystal Growth, 2006. 292(2): p. 282-285.
50. Ruhl, R.C., Cooling rates in splat cooling. Materials Science and Engineering, 1967. 1(6): p. 313-320.
51. Majumdar, J.D., et al., Effect of laser surface melting on corrosion and wear resistance of a commercial magnesium alloy. Materials Science and Engineering: A, 2003. 361(1): p. 119-129.
52. Flanagan, A., Laser polishing of medical devices. 2002, Google Patents.
53. Abbas, G., Z. Liu, and P. Skeldon, Corrosion behaviour of laser-melted magnesium alloys. Applied Surface Science, 2005. 247(1): p. 347-353.
54. Conde, A., et al., Corrosion behaviour of steels after laser surface melting. Materials & Design, 2000. 21(5): p. 441-445.
55. Vorobyev, A. and C. Guo, Femtosecond laser structuring of titanium implants. Applied surface science, 2007. 253(17): p. 7272-7280.
56. Yilbas, B., et al., Laser melting of plasma nitrided Ti 6A1 4V alloy. Wear, 1997. 212(1): p. 140-149.
57. Gasser, A., et al., Laser additive manufacturing. Laser Technik Journal, 2010. 7(2): p. 58-63.
58. Emmelmann, C., et al., Laser additive manufacturing and bionics: redefining lightweight design. Physics Procedia, 2011. 12: p. 364-368.
59. Gu, D., et al., Laser additive manufacturing of metallic components: materials, processes and mechanisms. International materials reviews, 2012. 57(3): p. 133-164.
60. Murr, L.E., et al., Metal fabrication by additive manufacturing using laser and electron beam melting technologies. Journal of Materials Science & Technology, 2012. 28(1): p. 1-14.
61. Roberts, I., et al., A three-dimensional finite element analysis of the temperature field during laser melting of metal powders in additive layer manufacturing. International Journal of Machine Tools and Manufacture, 2009. 49(12): p. 916-923.
62. Rickey, K.M., et al., Welding of Semiconductor Nanowires by Coupling Laser-Induced Peening and Localized Heating. Scientific reports, 2015. 5.
63. Hofmann, D.C., et al., Developing gradient metal alloys through radial deposition additive manufacturing. Scientific reports, 2014. 4.
64. Wei, H., J. Mazumder, and T. DebRoy, Evolution of solidification texture during additive manufacturing. Scientific reports, 2015. 5.
65. Zenou, M., A. Sa’ar, and Z. Kotler, Laser jetting of femto-liter metal droplets for high resolution 3D printed structures. Scientific reports, 2015. 5.
66. Fogagnolo, J.B., et al., Surface stiffness gradient in Ti parts obtained by laser surface alloying with Cu and Nb. Surface and Coatings Technology, 2016. 297: p. 34-42.
67. Tian, Y., et al., Research progress on laser surface modification of titanium alloys. Applied Surface Science, 2005. 242(1): p. 177-184.
68. Cui, Z., H. Man, and X. Yang, The corrosion and nickel release behavior of laser surface-melted NiTi shape memory alloy in Hanks' solution. Surface and Coatings Technology, 2005. 192(2): p. 347-353.
69. Yang, J., et al., Role of molten pool mode on formability, microstructure and mechanical properties of selective laser melted Ti-6Al-4V alloy. Materials & Design, 2016. 110: p. 558-570.
70. Guo, W., et al., Microstructure and mechanical characteristics of a laser welded joint in SA508 nuclear pressure vessel steel. Materials Science and Engineering: A, 2015. 625: p. 65-80.
71. Sundqvist, J., et al., Numerical optimization approaches of single-pulse conduction laser welding by beam shape tailoring. Optics and Lasers in Engineering, 2016. 79: p. 48-54.
72. Tian, J., et al., An analysis of the heat conduction problem for plates with the functionally graded material using the hybrid numerical method. Computers, Materials & Continua (CMC), 2009. 10(3): p. 229.
73. Goncalves Assuncao, E., Investigation of conduction to keyhole mode transition. 2012.
74. Watkins, K., M. McMahon, and W. Steen, Microstructure and corrosion properties of laser surface processed aluminium alloys: a review. Materials Science and Engineering: A, 1997. 231(1): p. 55-61.
75. Kulka, M. and A. Pertek, Microstructure and properties of borided 41Cr4 steel after laser surface modification with re-melting. Applied Surface Science, 2003. 214(1): p. 278-288.
76. Jiang, P., et al., Wear resistance of a laser surface alloyed Ti–6Al–4V alloy. Surface and Coatings Technology, 2000. 130(1): p. 24-28.
77. Xianqing, Y., et al., Microstructure evolution of WC/steel composite by laser surface re-melting. Applied Surface Science, 2007. 253(9): p. 4409-4414.
78. Hussein, A., et al., Finite element simulation of the temperature and stress fields in single layers built without-support in selective laser melting. Materials & Design, 2013. 52: p. 638-647.
79. Zeng, K., D. Pal, and B. Stucker. A review of thermal analysis methods in laser sintering and selective laser melting. in Proceedings of Solid Freeform Fabrication Symposium Austin, TX. 2012.
80. Dezfoli, A.A. and Z. Adabavazeh, Nanoscale modeling of conduction heat transfer in metals using the two-temperature model. Canadian Journal of Physics, 2015. 93(11): p. 1402-1406.
81. Jabbareh, M.A. and H. Assadi, Modeling of grain structure and heat-affected zone in laser surface melting process. Metallurgical and Materials Transactions B, 2013. 44(4): p. 1041-1048.
82. Thiessen, R., I. Richardson, and J. Sietsma, Physically based modelling of phase transformations during welding of low-carbon steel. Materials Science and Engineering: A, 2006. 427(1): p. 223-231.
83. Lopez-Botello, O., et al., Two-dimensional simulation of grain structure growth within selective laser melted AA-2024. Materials & Design, 2017. 113: p. 369-376.
84. Zinoviev, A., et al., Evolution of grain structure during laser additive manufacturing. Simulation by a cellular automata method. Materials & Design, 2016. 106: p. 321-329.
85. Matassi, F., et al., Porous metal for orthopedics implants. Clinical Cases in Mineral and Bone Metabolism, 2013. 10(2): p. 111.
86. Vandenbroucke, B. and J.-P. Kruth, Selective laser melting of biocompatible metals for rapid manufacturing of medical parts. Rapid Prototyping Journal, 2007. 13(4): p. 196-203.
87. Childs, T., C. Hauser, and M. Badrossamay, Selective laser sintering (melting) of stainless and tool steel powders: experiments and modelling. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2005. 219(4): p. 339-357.
88. Deligianni, D., et al., Effect of surface roughness of the titanium alloy Ti–6Al–4V on human bone marrow cell response and on protein adsorption. Biomaterials, 2001. 22(11): p. 1241-1251.
89. Gandin, C.-A. and M. Rappaz, A coupled finite element-cellular automaton model for the prediction of dendritic grain structures in solidification processes. Acta Metallurgica et Materialia, 1994. 42(7): p. 2233-2246.
90. Zhu, M. and C. Hong, A Modified Cellular Automaton Model for the Simulation of Dendritic Growth in Solidification of Alloys. Isij International, 2001. 41(5): p. 436-445.
91. Guillemot, G., C.-A. Gandin, and M. Bellet, Interaction between single grain solidification and macrosegregation: Application of a cellular automaton—Finite element model. Journal of Crystal Growth, 2007. 303(1): p. 58-68.
92. Dubois, F. and P. Le Floch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, in Nonlinear Hyperbolic Equations—Theory, Computation Methods, and Applications. 1989, Springer. p. 96-104.
93. Li, Z., et al., Effects of argon flow on heat transfer in a directional solidification process for silicon solar cells. Journal of Crystal Growth, 2011. 318(1): p. 298-303.
94. Modest, M.F., Radiative heat transfer. 2013: Academic press.
95. Hewitt, G.F., G.L. Shires, and T.R. Bott, Process heat transfer. Vol. 113. 1994: CRC press Boca Raton, FL.
96. Raithby, G. and E. Chui, A finite-volume method for predicting a radiant heat transfer in enclosures with participating media. ASME, Transactions, Journal of Heat Transfer, 1990. 112: p. 415-423.
97. Howell, J.R., M.P. Menguc, and R. Siegel, Thermal radiation heat transfer. 2010: CRC press.
98. Özışık, M.N., Radiative transfer and interactions with conduction and convection. 1973: Werbel & Peck.
99. Chorin, A.J., The numerical solution of the Navier-Stokes equations for an incompressible fluid. Bulletin of the American Mathematical Society, 1967. 73(6): p. 928-931.
100. Donea, J., A Taylor–Galerkin method for convective transport problems. International Journal for Numerical Methods in Engineering, 1984. 20(1): p. 101-119.
101. Huebner, K.H., et al., The finite element method for engineers. 2008: John Wiley & Sons.
102. Li, S., Y. Cheng, and Y.-F. Wu, Numerical manifold method based on the method of weighted residuals. Computational Mechanics, 2005. 35(6): p. 470-480.
103. Oldfield, W., A quantitative approach to casting solidification: freezing of cast iron. 1966.
104. Maxwell, I. and A. Hellawell, A simple model for grain refinement during solidification. Acta Metallurgica, 1975. 23(2): p. 229-237.
105. Goettsch, D.D. and J.A. Dantzig, Modeling microstructure development in gray cast irons. Metallurgical and Materials Transactions A, 1994. 25(5): p. 1063-1079.
106. Thevoz, P., J. Desbiolles, and M. Rappaz, Modeling of equiaxed microstructure formation in casting. Metallurgical Transactions A, 1989. 20(2): p. 311-322.
107. Stefanescu, D.M., G. Upadhya, and D. Bandyopadhyay, Heat transfer-solidification kinetics modeling of solidification of castings. Metallurgical Transactions A, 1990. 21(3): p. 997-1005.
108. Hunt, J. and K. Jackson, Nucleation of solid in an undercooled liquid by cavitation. Journal of Applied Physics, 1966. 37(1): p. 254-257.
109. Popovici, A. and D. Popovici. Cellular automata in image processing. in Fifteenth International Symposium on Mathematical Theory of Networks and Systems. 2002.
110. Dezfoli, A.R.A., et al., Modeling of poly-crystalline silicon ingot crystallization during casting and theoretical suggestion for ingot quality improvement. Materials Science in Semiconductor Processing, 2016. 53: p. 36-46.
111. Gandin, C.-A., et al., A three-dimensional cellular automation-finite element model for the prediction of solidification grain structures. Metallurgical and Materials Transactions A, 1999. 30(12): p. 3153-3165.
112. Mangelinck-Noël, N. and T. Duffar, Modelling of the transition from a planar faceted front to equiaxed growth: application to photovoltaic polycrystalline silicon. Journal of crystal growth, 2008. 311(1): p. 20-25.
113. Mangelinck-Noël, N., D. Ballutaud, N. Quang, G. Goaer, in 20th European Photovoltaic Solar Energy Conference and Exhibition. 2005: Barcelona,Spain. p. 6-10.
114. Teng, Y.-Y., et al., The carbon distribution in multicrystalline silicon ingots grown using the directional solidification process. Journal of crystal growth, 2010. 312(8): p. 1282-1290.
115. Fredriksson, H. and U. Åkerlind, Faceted and Dendritic Solidification Structures. Solidification and Crystallization Processing in Metals and Alloys: p. 475-586.
116. Wu, H., et al., Quantifying and analyzing neighborhood configuration characteristics to cellular automata for land use simulation considering data source error. Earth Science Informatics, 2012. 5(2): p. 77-86.
117. Anderson, M., et al., Computer simulation of grain growth—I. Kinetics. Acta metallurgica, 1984. 32(5): p. 783-791.
118. N. Mangelinck-Noe, T.D., D. Ballutaud, N. Le Quang, G. Goae, in 20th European Photovoltaic Solar Energy Conference and Exhibition,
. 2005: Barcelona,Spain. p. 6-10.
119. Durand, F. and J. Duby, Carbon solubility in solid and liquid silicon—A review with reference to eutectic equilibrium. Journal of phase equilibria, 1999. 20(1): p. 61-63.
120. Fujiwara, K., et al., Grain growth behaviors of polycrystalline silicon during melt growth processes. Journal of crystal growth, 2004. 266(4): p. 441-448.
121. Ujihara, T., et al., Effects of growth temperature on the surface morphology of silicon thin films on (111) silicon monocrystalline substrate by liquid phase epitaxy. Journal of crystal growth, 2004. 266(4): p. 467-474.
122. Obreten, W., D. Kashchiev, and V. Bostanov, Unified description of the rate of nucleation-mediated crystal growth. Journal of crystal growth, 1989. 96(4): p. 843-848.
123. Xi, Z., et al., Texturization of cast multicrystalline silicon for solar cells. Semiconductor science and technology, 2004. 19(3): p. 485.
124. Tsai, D.-C. and W.-S. Hwang, A Three Dimensional Cellular Automaton Model for the Prediction of Solidification Morphologies of Brass Alloy by Horizontal Continuous Casting and Its Experimental Verification. Materials transactions, 2011. 52(4): p. 787-794.
125. He, X., P. Fuerschbach, and T. DebRoy, Heat transfer and fluid flow during laser spot welding of 304 stainless steel. Journal of Physics D: Applied Physics, 2003. 36(12): p. 1388.
126. Ping, W.S., et al., Numerical simulation of microstructure evolution of Ti-6Al-4V alloy in vertical centrifugal casting. Materials Science and Engineering: A, 2006. 426(1): p. 240-249.
127. Lipton, J., M. Glicksman, and W. Kurz, Dendritic growth into undercooled alloy metals. Materials Science and Engineering, 1984. 65(1): p. 57-63.
128. Ivantsov, G., Temperature field around a spherical, cylindrical, and needle-shaped crystal, growing in a pre-cooled melt. Temperature field around a spherical, cylindrical, and needle-shaped crystal, growing in a pre-cooled melt Transl. into ENGLISH of, 1985. 1: p. 567-569.
129. Trivedi, R., Morphological stability of a solid particle growing from a binary alloy melt. Journal of Crystal Growth, 1980. 48(1): p. 93-99.
130. Kurz, W. and D. Fisher, Dendrite growth at the limit of stability: tip radius and spacing. Acta Metallurgica, 1981. 29(1): p. 11-20.
131. Wang, Z.-j., et al., Simulation of Microstructure during Laser Rapid Forming Solidification Based on Cellular Automaton. Mathematical Problems in Engineering, 2014. 2014.
132. Xu, Z.-m., G.-x. Geng, and J.-g. Li, Numerical Analysis for Position and Shape of Solid-Liquid Interface during Continuous Casting of Single Crystal Cu. JOURNAL-SHANGHAI JIAOTONG UNIVERSITY-CHINESE EDITION-, 2001. 35(3): p. 406-410.
133. Mills, K.C., Recommended values of thermophysical properties for selected commercial alloys. 2002: Woodhead Publishing.
134. Yang, J., et al., Experimental investigation and 3D finite element prediction of the heat affected zone during laser assisted machining of Ti6Al4V alloy. Journal of Materials Processing Technology, 2010. 210(15): p. 2215-2222.
135. Mapelli, C., R. Venturini, and C. Tagliabue, Extrusion simulation of TI-6AL-4V for the production of special shaped cross sections. Metallurgical Science and Tecnology, 2013. 22(2).
136. Mishra, S. and T. DebRoy, Measurements and Monte Carlo simulation of grain growth in the heat-affected zone of Ti–6Al–4V welds. Acta Materialia, 2004. 52(5): p. 1183-1192.
137. Wu, B. and R. Clark, Influence of inclusion on nucleation of silicon casting for photovoltaic (PV) application. Journal of Crystal Growth, 2011. 318(1): p. 200-207.
138. Bhattacharya, P., R. Fornari, and H. Kamimura, Comprehensive Semiconductor Science and Technology, Six-Volume Set. Vol. 1. 2011: Newnes.
139. Huang, W.-C., et al., 3D Printing Optical Engine for Controlling Material Microstructure. Physics Procedia, 2016. 83: p. 847-853.
140. Li, S. and G. Cui, Dependence of strength, elongation, and toughness on grain size in metallic structural materials. Journal of applied physics, 2007. 101(8): p. 083525.
141. Vrancken, B., et al., Heat treatment of Ti6Al4V produced by selective laser melting: microstructure and mechanical properties. Journal of Alloys and Compounds, 2012. 541: p. 177-185.
142. Gu, D., et al., Densification behavior, microstructure evolution, and wear performance of selective laser melting processed commercially pure titanium. Acta Materialia, 2012. 60(9): p. 3849-3860.
143. Simchi, A. and H. Pohl, Direct laser sintering of iron–graphite powder mixture. Materials Science and Engineering: A, 2004. 383(2): p. 191-200.