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A capillary model is developed for calculating the wetting angle of molten silicon on different walls by using the microfluidic two-phase flow level set method and studying the characteristics of the rising process. A mathematical model formulation rigorously accounts for the mass and momentum conservation by using the improved Navier-Stokes equation and considering the Marangoni effect. Compared with the experimental data, the change of the wetting angle on the chemical vapor deposition (CVD) diamond wall indicates the grids independence and the validity of the numerical algorithm. We also discuss the influence of surface tension, and Marangoni stress induced by the gradient of surface tension coefficient, and wall adhesion to the change of wetting angle for three different walls, which include SiC wall, graphite wall, and CVD diamond wall, at different temperatures (1683-1873 K). Result shows that at the same temperature, the thermal-capillary effects that induce the molten silicon to undulation are raised. The wetting angle is reduced after first being increased and finally stabilized. At the initial stage, the fluctuation of the liquid-air interface is volatile due to the large changes of the liquid-air and the wall-air surface tensions, and subsequently, the fluctuation tends to be stable while the wetting angle is close to a fixed value. It is also found that with the graphite wall, these changes are more likely to be stable. This research provides a theoretical guide to obtain a stable growth environment for silicon belt fabricated from the molten silicon.
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Keywords:
- wetting angle /
- molten silicon /
- stability /
- capillary effect
[1] Qian J, Zerda T W, He D, Daemen L, Zhao Y 2003 Journal of Materials Research 18 1173
[2] Ekimov E A, Gavriliuk A G, Palosz B, GierlotkaS, Dluzewski P, Tatianin E, Kluev Y, Naletov A M, Presz A 2000 Applied Physics Letters 77 954
[3] Xu S H, Zhou H W, Wang C X, Wang L W, Sun Z W 2013 Acta Phys. Sin. 62 134702 (in Chinese) [徐升华, 周宏伟, 王彩霞, 王林伟, 孙祉伟 2013 62 134702]
[4] Yamamoto Y, Tokieda K, Wakimoto T, Ito T, Katoh K 2014 International Journal of Multiphase Flow 59 106
[5] Li Y Q, Liu L, Zhang C H, Duan L, Kang Q 2013 Acta Phys. Sin. 62 024701 (in Chinese) [李永强, 刘玲, 张晨辉, 段俐, 康琦 2013 62 024701]
[6] Messmer B, Lemee T, Ikebukuro K, Ueno I, Narayanan R 2014 International Journal of Heat and Mass Transfer 78 1060
[7] Daggolu P, Yeckel A, Bleil C E, Derby J J 2012 Journal of Crystal Growth 355 129
[8] Sasaki H, Tokizaki E, Huang X M 1995 Japanese journal of applied physics 34 3432
[9] Sedlmeier F, Janecek J, Sendner C, Bocquet L, Netz R R, Horinek D 2008 Biointerphases 3 FC23
[10] Landau L D, Lifshitz E M (translated by Li Z) 2013 Fluid Mechanics (Beijing:Higher Education Press) pp267-269 (in Chinese) [朗道, 栗夫席兹著(李植译) 2013 流体动力学(第五版)(北京:高等教育出版社)第267-269页]
[11] Peng L, Li Y R, Zeng D L 2004 JournaI of Chongging University 27 60 (in Chinese) [彭岚, 李友荣, 曾丹苓 2004 重庆大学学报:自然科学版 27 60]
[12] Peng L, Zhang W, Li X R, Meng H Y 2011 Journal of Synthetic Crystals 40 556 (in Chinese) [彭岚, 张伟, 李友荣, 孟海泳 2011 人工晶体学报 40 556]
[13] Wang C X, Xu S H, Sun Z W, Hu W R 2010 International Journal of Heat and Mass Transfer 53 1801
[14] Mlungwane K, Sigalas I, Herrmann M, Rodríguez M 2009 Ceramics International 35 2435
[15] Li Y Q, Liu L 2014 Acta Phys. Sin. 63 214704 (in Chinese) [李永强, 刘玲 2014 63 214704]
[16] Hitoshi S, Eiji T, Kazutaka T, Shigeyuki K 1994 Jpn. J. Appl. Phys. 33 6078
[17] Huang X M 1997 Physics 26 37 (in Chinese) [黄新明1997 物理 26 37]
[18] Rowlinson J, Widom B 1982 Molecular Theory of Capillarity (Oxford:Oxford University Press) p86
[19] Li Y R, Deng N B, Wu S Y, Peng L, Li M W 2005 Chinese Journal of Materials Research 19 395 (in Chinese) [李友荣, 邓努波, 吴双应, 彭岚, 李明伟 2005 材料研究学报 19 395]
[20] Son G H 2014 International Communications in Heat and Mass Transfer 58 156
[21] Daggolu P 2013 Ph. D. Dissertation (Minnesota:University of Minnesota)
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[1] Qian J, Zerda T W, He D, Daemen L, Zhao Y 2003 Journal of Materials Research 18 1173
[2] Ekimov E A, Gavriliuk A G, Palosz B, GierlotkaS, Dluzewski P, Tatianin E, Kluev Y, Naletov A M, Presz A 2000 Applied Physics Letters 77 954
[3] Xu S H, Zhou H W, Wang C X, Wang L W, Sun Z W 2013 Acta Phys. Sin. 62 134702 (in Chinese) [徐升华, 周宏伟, 王彩霞, 王林伟, 孙祉伟 2013 62 134702]
[4] Yamamoto Y, Tokieda K, Wakimoto T, Ito T, Katoh K 2014 International Journal of Multiphase Flow 59 106
[5] Li Y Q, Liu L, Zhang C H, Duan L, Kang Q 2013 Acta Phys. Sin. 62 024701 (in Chinese) [李永强, 刘玲, 张晨辉, 段俐, 康琦 2013 62 024701]
[6] Messmer B, Lemee T, Ikebukuro K, Ueno I, Narayanan R 2014 International Journal of Heat and Mass Transfer 78 1060
[7] Daggolu P, Yeckel A, Bleil C E, Derby J J 2012 Journal of Crystal Growth 355 129
[8] Sasaki H, Tokizaki E, Huang X M 1995 Japanese journal of applied physics 34 3432
[9] Sedlmeier F, Janecek J, Sendner C, Bocquet L, Netz R R, Horinek D 2008 Biointerphases 3 FC23
[10] Landau L D, Lifshitz E M (translated by Li Z) 2013 Fluid Mechanics (Beijing:Higher Education Press) pp267-269 (in Chinese) [朗道, 栗夫席兹著(李植译) 2013 流体动力学(第五版)(北京:高等教育出版社)第267-269页]
[11] Peng L, Li Y R, Zeng D L 2004 JournaI of Chongging University 27 60 (in Chinese) [彭岚, 李友荣, 曾丹苓 2004 重庆大学学报:自然科学版 27 60]
[12] Peng L, Zhang W, Li X R, Meng H Y 2011 Journal of Synthetic Crystals 40 556 (in Chinese) [彭岚, 张伟, 李友荣, 孟海泳 2011 人工晶体学报 40 556]
[13] Wang C X, Xu S H, Sun Z W, Hu W R 2010 International Journal of Heat and Mass Transfer 53 1801
[14] Mlungwane K, Sigalas I, Herrmann M, Rodríguez M 2009 Ceramics International 35 2435
[15] Li Y Q, Liu L 2014 Acta Phys. Sin. 63 214704 (in Chinese) [李永强, 刘玲 2014 63 214704]
[16] Hitoshi S, Eiji T, Kazutaka T, Shigeyuki K 1994 Jpn. J. Appl. Phys. 33 6078
[17] Huang X M 1997 Physics 26 37 (in Chinese) [黄新明1997 物理 26 37]
[18] Rowlinson J, Widom B 1982 Molecular Theory of Capillarity (Oxford:Oxford University Press) p86
[19] Li Y R, Deng N B, Wu S Y, Peng L, Li M W 2005 Chinese Journal of Materials Research 19 395 (in Chinese) [李友荣, 邓努波, 吴双应, 彭岚, 李明伟 2005 材料研究学报 19 395]
[20] Son G H 2014 International Communications in Heat and Mass Transfer 58 156
[21] Daggolu P 2013 Ph. D. Dissertation (Minnesota:University of Minnesota)
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