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为改善晶体相变界面形态，提高晶体品质，提出了一种融合浸入边界法（immersed boundary method，IBM）和格子Boltzmann法（lattice Boltzmann method，LBM）的二维轴对称浸入边界热格子Boltzmann模型来研究直拉法硅单晶生长中的相变问题.将相变界面视为浸没边界，用拉格朗日节点显式追踪相变界面；用LBM求解熔体中的流场和温度分布；用有限差分法求解晶体中的温度分布.实现了基于IB-LBM的动边界晶体生长过程研究.得到了不同晶体生长工艺参数作用下的相变界面，并用相变界面位置偏差绝对值的均值和偏差的标准差来衡量界面的平坦度，得到平坦相变界面对应工艺参数的调整方法.研究表明，相变过程与晶体提拉速度、晶体旋转参数和坩埚旋转参数的相互作用有关，合理地配置晶体旋转参数和坩埚旋转参数的比值，能够得到平坦的相变界面.
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关键词:
- 晶体生长 /
- 固-液相变 /
- 浸入边界法 /
- 格子Boltzmann法
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.-
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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