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A two-dimensional axisymmetric immersed boundary thermal lattice Boltzmann (IB-TLB) model is presented to study the phase transition in Czochralski silicon crystal growth for improving the morphology of the melt-crystal interface and the crystal quality. Specifically, the Euler grid and the Lagrange grid are established, respectively. The melt-crystal interface is considered as an immersed boundary, and it is described by a series of Lagrange nodes. In this paper, the melt-crystal interface is tracked by the immersed boundary method, and the melt flow and heat transfer are simulated by the lattice Boltzmann method. The D2Q9 model is adopted to solve the axial velocity, radial velocity, swirling velocity and temperature of the melt. The finite difference method is used to solve the temperature distribution of the crystal. Then the solid-liquid transition in crystal growth with moving boundary is solved by the proposed IB-TLB model. The proposed model is validated by the solid-liquid phase transition benchmark. In addition, the flatness of the melt-crystal interface is evaluated by the mean value of the absolute value of the interface deviation and the standard deviation of the interface deviation. The effects of the process parameters on the morphology of melt-crystal interface, melt flow structure and temperature distribution are analyzed. The results show that the morphology of the melt-crystal interface is relevant to the interaction of the crystal pulling rate, the crystal rotation parameter, and the crucible rotation parameter. When the crystal and crucible rotate together, the deviation and fluctuation of the melt-crystal interface can be effectively adjusted, whether they rotate in the same direction or rotate in the opposite directions. And a flat melt-crystal interface can be obtained by appropriately configurating the ratio of crystal rotation parameter to crucible rotation parameter. Finally, according to a series of computations, it is found that when the crucible and crystal rotate in the opposite directions, the crystal rotation parameter and the crucible rotation parameter satisfy a functional relation, with a flat interface maintained. The obtained relationship has a certain reference for adjusting and improving the crystal growth parameters in practice.
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Keywords:
- crystal growth /
- solid-liquid phase change /
- immersed boundary method /
- lattice Boltzmann method
[1] Liu D, Zhao X G, Zhao Y 2017 Control Theor. Appl. 34 1 (in Chinese)[刘丁, 赵小国, 赵跃 2017 控制理论与应用 34 1]
[2] Nikitin N, Polezhaev V 2001 J. Cryst. Growth 230 30
[3] Zhang N, Liu D 2018 Results Phys. 10 882
[4] Jana S, Dost S, Kumar V, Durst F 2006 Int. J. Eng. Sci. 44 554
[5] Liu L J, Kakimoto K 2008 J. Cryst. Growth 310 306
[6] Chen J C, Chiang P Y, Chang C H, et al. 2014 J. Cryst. Growth 401 813
[7] Peng Y, Shu C, Chew Y T 2003 J. Comput. Phys. 186 295
[8] Mencinger J 2004 J. Comput. Phys. 198 243
[9] Miller W, Rasin I, Succi S 2006 Physica A 362 78
[10] Miller W 2001 J. Cryst. Growth 230 263
[11] Zhao X, Dong B, Li W Z, Dou B L 2017 Appl. Therm. Eng. 111 1477
[12] Huang R Z, Wu H Y 2016 J. Comput. Phys. 315 65
[13] Huang R Z, Wu H Y 2014 J. Comput. Phys. 277 305
[14] Wu X D, Liu H P, Chen F 2017 Acta Phys. Sin. 66 224702 (in Chinese)[吴晓笛, 刘华坪, 陈浮 2017 物理学报 66 224702]
[15] Qian Y H, d'Humières D, Lallemand P 1992 Europhys. Lett. 17 479
[16] Guo Z L, Zheng C G, Shi B C 2002 Chin. Phys. 11 366
[17] Liu K T, Tang A P 2014 J. Sichuan Univ. (Eng. Sci.) 46 73 (in Chinese)[刘克同, 汤爱平 2014 四川大学学报(工程科学版) 46 73]
[18] Guo Z L, Zheng C G, Shi B C 2002 Phys. Fluids 14 2007
[19] Huang R, Wu H, Cheng P 2013 Int. J. Heat. Mass. Tran. 59 295
[20] Jiang L, Liu D, Zhao Y, Liu Z S 2012 J. Synth. Cryst. 41 1762 (in Chinese)[姜雷, 刘丁, 赵跃, 刘志尚 2012 人工晶体学报 41 1762]
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[1] Liu D, Zhao X G, Zhao Y 2017 Control Theor. Appl. 34 1 (in Chinese)[刘丁, 赵小国, 赵跃 2017 控制理论与应用 34 1]
[2] Nikitin N, Polezhaev V 2001 J. Cryst. Growth 230 30
[3] Zhang N, Liu D 2018 Results Phys. 10 882
[4] Jana S, Dost S, Kumar V, Durst F 2006 Int. J. Eng. Sci. 44 554
[5] Liu L J, Kakimoto K 2008 J. Cryst. Growth 310 306
[6] Chen J C, Chiang P Y, Chang C H, et al. 2014 J. Cryst. Growth 401 813
[7] Peng Y, Shu C, Chew Y T 2003 J. Comput. Phys. 186 295
[8] Mencinger J 2004 J. Comput. Phys. 198 243
[9] Miller W, Rasin I, Succi S 2006 Physica A 362 78
[10] Miller W 2001 J. Cryst. Growth 230 263
[11] Zhao X, Dong B, Li W Z, Dou B L 2017 Appl. Therm. Eng. 111 1477
[12] Huang R Z, Wu H Y 2016 J. Comput. Phys. 315 65
[13] Huang R Z, Wu H Y 2014 J. Comput. Phys. 277 305
[14] Wu X D, Liu H P, Chen F 2017 Acta Phys. Sin. 66 224702 (in Chinese)[吴晓笛, 刘华坪, 陈浮 2017 物理学报 66 224702]
[15] Qian Y H, d'Humières D, Lallemand P 1992 Europhys. Lett. 17 479
[16] Guo Z L, Zheng C G, Shi B C 2002 Chin. Phys. 11 366
[17] Liu K T, Tang A P 2014 J. Sichuan Univ. (Eng. Sci.) 46 73 (in Chinese)[刘克同, 汤爱平 2014 四川大学学报(工程科学版) 46 73]
[18] Guo Z L, Zheng C G, Shi B C 2002 Phys. Fluids 14 2007
[19] Huang R, Wu H, Cheng P 2013 Int. J. Heat. Mass. Tran. 59 295
[20] Jiang L, Liu D, Zhao Y, Liu Z S 2012 J. Synth. Cryst. 41 1762 (in Chinese)[姜雷, 刘丁, 赵跃, 刘志尚 2012 人工晶体学报 41 1762]
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