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A quantitative phase-field (PF) model with an anti-trapping current (ATC) is developed to simulate the dendritic growth with two-sided diffusion. The asymptotic analysis is performed at the second-order for the PF equations coupled with nonlinear thermodynamic properties and an ATC term under the equal chemical potential condition. The PF mobility and ATC are derived based on the asymptotic analysis in the thin interface limit, and the solute drag model. Then the model is reduced to the dilute solution limit for dendrite solidification of binary alloys. The test of convergence with respect to the interface width exhibits an excellent convergent behavior of the proposed model. The performance of the model is then validated by comparing PF simulations with the predictions of the Gibbs-Thomson relation, the linearized solvability theory, and the modified-Lipton-Glicksman-Kurz (M-LGK) analytical model, for the isothermal dendritic growth of an Fe-0.15 mol%C alloy. The results demonstrate quantitative capabilities of the model that effectively suppresses the abnormal solute trapping effect when the interface is taken artificially to be wide. It is also found that the present model can quantitatively describe dendrite growth with various solid diffusivities, ranging from the case with one-sided diffusion to the symmetrical model.
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Keywords:
- phase-field modelling /
- solidification /
- dendritic growth /
- two-sided diffusion
[1] Yao W J, Yang C, Han X J, Chen M, Wei B B, Guo Z Yun 2003 Acta Phys. Sin. 52 448 (in Chinese) [姚文静, 杨春, 韩秀君, 陈民, 魏炳波, 过增元 2003 52 448]
[2] Zang D Y, Wang H P, Wei B B 2007 Acta Phys. Sin. 56 4804 (in Chinese) [臧渡洋, 王海鹏, 魏炳波 2007 56 4804]
[3] Wang J Y, Chen C L, Zhai W, Jin K X 2009 Acta Phys. Sin. 58 6554 (in Chinese) [王建元, 陈长乐, 翟薇, 金克新 2009 58 6554]
[4] Zhao D P, JingT, Liu B C 2003 Acta Phys. Sin. 52 1737 (in Chinese) [赵代平, 荆涛, 柳百成 2003 52 1737]
[5] Zong Y P, Wang M T, Guo W 2009 Acta Phys. Sin. 58 S161 (in Chinese) [宗亚平, 王明涛, 郭巍 2009 58 S161]
[6] Shan B W, Lin X, Wei L, Huang W D 2009 Acta Phys. Sin. 58 1132 (in Chinese) [单博炜, 林鑫, 魏雷, 黄卫东 2009 58 1132]
[7] Li Q, Li D Z, Qian B N 2004 Acta Phys. Sin. 53 3477 (in Chinese) [李强, 李殿中, 钱百年2004 53 3477]
[8] Pan S Y, Zhu M F 2009 Acta Phys. Sin. 58, S278 (in Chinese) [潘诗琰, 朱鸣芳 2009 58 S278]
[9] Chen Y, Kang X H, Li D Z 2009 Acta Phys. Sin. 58 390 (in Chinese) [陈云, 康秀红, 李殿中 2009 58 390]
[10] Long W Y, Cai Q Z, Chen L L, Wei B K 2005 Acta Phys. Sin. 54 256 (in Chinese) [龙文元, 蔡启舟, 陈立亮, 魏伯康 2005 54 256]
[11] Zhu C S, Wang Z P, Jing T, Xiao R Z 2006 Acta Phys. Sin. 55 1502 (in Chinese) [朱昌盛, 王智平, 荆涛, 肖荣振2006 55 1502]
[12] Li J J, Wang J C, Xu Q, Yang G C 2007 Acta Phys. Sin. 56 1514 (in Chinese) [李俊杰, 王锦程, 许泉, 杨根仓 2007 56 1514]
[13] Feng L, Wang Z P, Lu Y, Zhu C S 2008 Acta Phys. Sin. 57 1084 (in Chinese) [冯力, 王智平, 路阳, 朱昌盛 2008 57 1084]
[14] Karma A, Rappel W J 1998 Phys. Rev. E 57 4323
[15] Almgren R F 1999 SIAM J. Appl. Math. 59 2086
[16] Karma A 2001 Phys. Rev. Lett. 87 115701
[17] Echebarria B, Folch R, Karma A, Plapp M 2004 Phys. Rev. E 70 061604
[18] Gopinath A, Armstrong R C, Brown R A 2006 J. Cryst. Growth 291 272
[19] Steinbach I 2009 Modelling Simul. Mater. Sci. Eng. 17 073001
[20] Ohno M, Matsuura K 2009 Phys. Rev. E 79 031603
[21] Svoboda J, Fischer F D, Gamsjäger E 2002 Acta Mater. 50 967
[22] Hillert M 1999 Acta Mater. 47 4481
[23] Kim S G, Kim WT, Suzuki T 1999 Phys. Rev. E 60 7186
[24] Ode M, Suzuki T, Kim S G, Kim W T 2000 Sci. Tech. Adv. Mater. 1 43
[25] Ohno M, Matsuura K 2010 Acta Mater. 58 5749
[26] Ramirez J C, Beckermann C, Karma A, Diepers H J 2004 Phys. Rev. E 69 051607
[27] Ramirez J C, Beckermann C 2005 Acta Mater. 53 1721
[28] Barbieri A, Langer J S 1989 Phys. Rev. A 39 5314
[29] Lipton J, Clicksman M E, Kurz W 1984 Mater. Sci. Eng. 65 57
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[1] Yao W J, Yang C, Han X J, Chen M, Wei B B, Guo Z Yun 2003 Acta Phys. Sin. 52 448 (in Chinese) [姚文静, 杨春, 韩秀君, 陈民, 魏炳波, 过增元 2003 52 448]
[2] Zang D Y, Wang H P, Wei B B 2007 Acta Phys. Sin. 56 4804 (in Chinese) [臧渡洋, 王海鹏, 魏炳波 2007 56 4804]
[3] Wang J Y, Chen C L, Zhai W, Jin K X 2009 Acta Phys. Sin. 58 6554 (in Chinese) [王建元, 陈长乐, 翟薇, 金克新 2009 58 6554]
[4] Zhao D P, JingT, Liu B C 2003 Acta Phys. Sin. 52 1737 (in Chinese) [赵代平, 荆涛, 柳百成 2003 52 1737]
[5] Zong Y P, Wang M T, Guo W 2009 Acta Phys. Sin. 58 S161 (in Chinese) [宗亚平, 王明涛, 郭巍 2009 58 S161]
[6] Shan B W, Lin X, Wei L, Huang W D 2009 Acta Phys. Sin. 58 1132 (in Chinese) [单博炜, 林鑫, 魏雷, 黄卫东 2009 58 1132]
[7] Li Q, Li D Z, Qian B N 2004 Acta Phys. Sin. 53 3477 (in Chinese) [李强, 李殿中, 钱百年2004 53 3477]
[8] Pan S Y, Zhu M F 2009 Acta Phys. Sin. 58, S278 (in Chinese) [潘诗琰, 朱鸣芳 2009 58 S278]
[9] Chen Y, Kang X H, Li D Z 2009 Acta Phys. Sin. 58 390 (in Chinese) [陈云, 康秀红, 李殿中 2009 58 390]
[10] Long W Y, Cai Q Z, Chen L L, Wei B K 2005 Acta Phys. Sin. 54 256 (in Chinese) [龙文元, 蔡启舟, 陈立亮, 魏伯康 2005 54 256]
[11] Zhu C S, Wang Z P, Jing T, Xiao R Z 2006 Acta Phys. Sin. 55 1502 (in Chinese) [朱昌盛, 王智平, 荆涛, 肖荣振2006 55 1502]
[12] Li J J, Wang J C, Xu Q, Yang G C 2007 Acta Phys. Sin. 56 1514 (in Chinese) [李俊杰, 王锦程, 许泉, 杨根仓 2007 56 1514]
[13] Feng L, Wang Z P, Lu Y, Zhu C S 2008 Acta Phys. Sin. 57 1084 (in Chinese) [冯力, 王智平, 路阳, 朱昌盛 2008 57 1084]
[14] Karma A, Rappel W J 1998 Phys. Rev. E 57 4323
[15] Almgren R F 1999 SIAM J. Appl. Math. 59 2086
[16] Karma A 2001 Phys. Rev. Lett. 87 115701
[17] Echebarria B, Folch R, Karma A, Plapp M 2004 Phys. Rev. E 70 061604
[18] Gopinath A, Armstrong R C, Brown R A 2006 J. Cryst. Growth 291 272
[19] Steinbach I 2009 Modelling Simul. Mater. Sci. Eng. 17 073001
[20] Ohno M, Matsuura K 2009 Phys. Rev. E 79 031603
[21] Svoboda J, Fischer F D, Gamsjäger E 2002 Acta Mater. 50 967
[22] Hillert M 1999 Acta Mater. 47 4481
[23] Kim S G, Kim WT, Suzuki T 1999 Phys. Rev. E 60 7186
[24] Ode M, Suzuki T, Kim S G, Kim W T 2000 Sci. Tech. Adv. Mater. 1 43
[25] Ohno M, Matsuura K 2010 Acta Mater. 58 5749
[26] Ramirez J C, Beckermann C, Karma A, Diepers H J 2004 Phys. Rev. E 69 051607
[27] Ramirez J C, Beckermann C 2005 Acta Mater. 53 1721
[28] Barbieri A, Langer J S 1989 Phys. Rev. A 39 5314
[29] Lipton J, Clicksman M E, Kurz W 1984 Mater. Sci. Eng. 65 57
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