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The effect of interface kinetics on the stability of deep cell morphology in directional solidification is studied. By using the multiple variable method and the matching asymptotic methodandby finding the mode solution of the system,the dispersion relation satisfied by the change rate of the disturbance amplitude of the cell-crystal interface is derived , and the quantization condition of the interface morphology is obtained. The results show that there are two global instability mechanisms in the directional solidification system with considering the growth of deep cell crystal with interfacial dynamic parameters : global oscillation instability mechanism and low-frequency instability. The stability analysis shows that the interface stability parameter
$\mathrm{\varepsilon }$ is related to the cell relative parameter$ {\mathrm{\lambda }}_{0} $ , and thatthe larger the interface dynamic parameter$ {M}_{\text{*}} $ , the larger the stable region of the overall fluctuation instability of the dendrite structure in the overall oscillation modeis.-
Keywords:
- directional solidification /
- deep cellular crystal growth /
- interface kinetics /
- morphological stability
[1] Peng P, Li S Y, Zheng W C, Lu L, Zhou S D 2021 Trans. Nonferrous Met. Soc. China 31 3096
Google Scholar
[2] Mullins W W, Sekerka R F 1963 Appl. Phys. 34 323
Google Scholar
[3] Pelcé P 1988 Dynamics of Curved Fronts (New York: Academic Press) pp345–352
[4] Nash G E, Glicksman M E 1974 ScriptaMetal. 8 xxix
Google Scholar
[5] KruskalMD, Segur H 1991 Stud. Appl. Math. 85 129
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[6] Xu J J 1991 Phys. Rev. A. 43 930
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[7] Xu J J 1991 Eur. J. Appl. Math. 2 105
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[8] Pocheau A, Georgelin M 1999 J. Cryst. Growth 206 215
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[9] Ding G, Huang W, Xin L, Zhou Y 1997 J. Cryst. Growth 177 281
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[10] Coriell S R, Sekerka R F 1976 J. Cryst. Growth 34 157
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[12] Li J F, Zhou Y H 2005 中国科学: E辑 35 10
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李金富, 周尧和 2005 Science in China(Series E) 35 10
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[13] Tan Y, Wang H 2012 J. Mater. Sci. 47 5308
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[14] 蒋晗, 陈明文, 王涛, 王自东 2017 物理学报 66 10
Google Scholar
Jiang H, Chen M W, Wang T, Wang Z D 2017 Acta Phys. Sin. 66 10
Google Scholar
[15] Chen M W, Wang Z D, Xu J J 2008 Sci. China Ser. E 51 225
Google Scholar
[16] 陈明文, 倪锋, 王艳林, 王自东, 谢建新 2011 物理学报 60 068103
Google Scholar
Chen M W, Ni F, Wang Y L, Wang Z D, Xie J X 2011 Acta Phys. Sin. 60 068103
Google Scholar
[17] 郭洪民, 杨湘杰 2008 中国有色金属学报 18 651
Google Scholar
Guo H M, Yang X J 2008 Chin. J. Nonferrous Met. 18 651
Google Scholar
[18] 统雷雷, 林鑫, 赵力宁, 黄卫东 2009 金属学报 45 737
Google Scholar
Tong L L, Lin X, Zhao L N, Huang W D 2009 ActaMetall. Sin. 45 737
Google Scholar
[19] Pelcé P 1988 Dynamics of Curved Fronts(New York: Academic. Press) pp155–174
[20] Chen Y Q, Xu J J 2011 Phys. Rev. E 83 041601
Google Scholar
[21] Xu J J, Chen Y Q 2011 Phys. Rev. E 83 061605
Google Scholar
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图 1 基于Saffmen-Taylor解构造的曲线坐标系
$ \left(\xi , \eta \right) $ [20]Figure 1. Curve coordinate system
$ \left(\xi , \eta \right) $ based on Saffmen-Taylor solution.
图 2 首级近似与一级近似的GTW-S中性模式曲线. 参数分别为
$ n=0 $ ,$ {\mathrm{\lambda }}_{G}=0.3991 $ ,$ \kappa =0.29 $ ,$ {G}_{\mathrm{c}}=0.14485\times {10}^{-4} $ ,${\varepsilon }_{\mathrm{c}}=0.5388\times {10}^{-2}$ ,$ M=0.09552 $ ,$ E=0.25 $ ,$ {m}_{\text{*}}=1 $ Figure 2. The neutral curves of GTW-S-modes with zero-th-order approximation and first-order approximation for the case
$ n=0 $ ,$ {\lambda }_{G}=0.3991 $ ,$ \kappa =0.29 $ ,$ {G}_{\mathrm{c}}=0.14485\times {10}^{-4} $ ,${\varepsilon }_{\mathrm{c}}=0.5388\times {10}^{-2}$ ,$ M=0.09552 $ ,$ E=0.25 $ ,${m}_{\text{*}}=1$ 
图 3 一级近似下的GTW-S中性模式曲线. 参数分别为
$ n=0, 1, 2 $ ,$ {\mathrm{\lambda }}_{G}=0.3991 $ ,$ \kappa =0.29 $ ,$ {G}_{\mathrm{c}}=0.14485\times {10}^{-4} $ ,${\varepsilon }_{\mathrm{c}}=0.5388\times {10}^{-2}$ ,$ M=0.09552 $ ,$ E=0.25 $ ,$ {m}_{\text{*}}=1 $ Figure 3. The neutral curves of GTW-S-modes with first-order approximation for the case
$ n=\mathrm{0, 1}, 2 $ ,$ {\mathrm{\lambda }}_{G}=0.3991 $ ,$ \kappa =0.29 $ ,$ {G}_{\mathrm{c}}=0.14485\times {10}^{-4} $ ,${\varepsilon }_{\mathrm{c}}=0.5388\times {10}^{-2}$ ,$ M= $ $ 0.09552 $ ,$ E=0.25 $ ,$ {m}_{\text{*}}=1 $ .
图 4 一级近似下的GTW-S中性模式曲线. 参数分别为
$ E = 0.1, 0.25 $ ,$ n = 0 $ ,$ {\mathrm{\lambda }}_{G} = 0.3991 $ ,$ \kappa = 0.29 $ ,$ {G}_{\mathrm{c}} = 0.14485\times $ $ {10}^{-4} $ ,${\varepsilon }_{\mathrm{c}}=0.5388\times {10}^{-2}$ ,$ M=0.09552 $ ,$ {m}_{\text{*}}=1 $ Figure 4. The neutral curves of GTW-S-modes with first-order approximation for the case of
$ E=0.1, 0.25 $ ,$ n=0 $ ,$ {\mathrm{\lambda }}_{G}=0.3991 $ ,$ \kappa =0.29 $ ,$ {G}_{\mathrm{c}}=0.14485\times {10}^{-4} $ ,${{\varepsilon }}_{\mathrm{c}}=0.5388\times $ $ {10}^{-2}$ ,$ M=0.09552 $ ,${m}_{\text{*}}=1$ 
图 5 一级近似下的GTW-S中性模式曲线. 参数分别为
$ {m}_{\text{*}}=1, 5, 10 $ ,$ E=0.25 $ ,$ {\mathrm{\lambda }}_{G}=0.3991 $ ,$ \kappa =0.29 $ ,$ {G}_{\mathrm{c}}= $ $ 0.14485\times {10}^{-4} $ ,${\varepsilon }_{\mathrm{c}}=0.5388\times {10}^{-2}$ ,$ M=0.09552 $ Figure 5. The neutral curves of GTW-S-modes with first-order approximation for the case of
$ {m}_{\text{*}}=1, 5, 10 $ ,$ E=0.25 $ ,$ n=0 $ ,$ {\lambda }_{G}=0.3991 $ ,$ \kappa =0.29 $ ,$ {G}_{\mathrm{c}}=0.14485\times {10}^{-4} $ ,${\varepsilon }_{\mathrm{c}}= $ $ 0.5388\times {10}^{-2}$ ,$M=0.09552$ -
[1] Peng P, Li S Y, Zheng W C, Lu L, Zhou S D 2021 Trans. Nonferrous Met. Soc. China 31 3096
Google Scholar
[2] Mullins W W, Sekerka R F 1963 Appl. Phys. 34 323
Google Scholar
[3] Pelcé P 1988 Dynamics of Curved Fronts (New York: Academic Press) pp345–352
[4] Nash G E, Glicksman M E 1974 ScriptaMetal. 8 xxix
Google Scholar
[5] KruskalMD, Segur H 1991 Stud. Appl. Math. 85 129
Google Scholar
[6] Xu J J 1991 Phys. Rev. A. 43 930
Google Scholar
[7] Xu J J 1991 Eur. J. Appl. Math. 2 105
Google Scholar
[8] Pocheau A, Georgelin M 1999 J. Cryst. Growth 206 215
Google Scholar
[9] Ding G, Huang W, Xin L, Zhou Y 1997 J. Cryst. Growth 177 281
Google Scholar
[10] Coriell S R, Sekerka R F 1976 J. Cryst. Growth 34 157
Google Scholar
[11] Trivedi R, Seetharaman V, Eshelman M A 1991 Metall. Mater. Trans. A. 22 585
[12] Li J F, Zhou Y H 2005 中国科学: E辑 35 10
Google Scholar
李金富, 周尧和 2005 Science in China(Series E) 35 10
Google Scholar
[13] Tan Y, Wang H 2012 J. Mater. Sci. 47 5308
Google Scholar
[14] 蒋晗, 陈明文, 王涛, 王自东 2017 物理学报 66 10
Google Scholar
Jiang H, Chen M W, Wang T, Wang Z D 2017 Acta Phys. Sin. 66 10
Google Scholar
[15] Chen M W, Wang Z D, Xu J J 2008 Sci. China Ser. E 51 225
Google Scholar
[16] 陈明文, 倪锋, 王艳林, 王自东, 谢建新 2011 物理学报 60 068103
Google Scholar
Chen M W, Ni F, Wang Y L, Wang Z D, Xie J X 2011 Acta Phys. Sin. 60 068103
Google Scholar
[17] 郭洪民, 杨湘杰 2008 中国有色金属学报 18 651
Google Scholar
Guo H M, Yang X J 2008 Chin. J. Nonferrous Met. 18 651
Google Scholar
[18] 统雷雷, 林鑫, 赵力宁, 黄卫东 2009 金属学报 45 737
Google Scholar
Tong L L, Lin X, Zhao L N, Huang W D 2009 ActaMetall. Sin. 45 737
Google Scholar
[19] Pelcé P 1988 Dynamics of Curved Fronts(New York: Academic. Press) pp155–174
[20] Chen Y Q, Xu J J 2011 Phys. Rev. E 83 041601
Google Scholar
[21] Xu J J, Chen Y Q 2011 Phys. Rev. E 83 061605
Google Scholar
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