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Lamellar eutectic solidification is very important in the development of new materials in which the periodic multiphase structures each may have a remarkable or enhanced functionality. The morphological instability during solidification may lead to various eutectic microstructures and greatly affect the physical and mechanical properties of final solidification products. In this paper, the morphological stability of lamellar eutectic growth with the anisotropic surface tension is studied by using the matched asymptotic expansion method and multiple variable expansion method. We assume that the process of solidification is viewed as a two-dimensional problem, The anisotropic surface tension is a four-fold symmetry function. The solute diffusion in the solid phase is negligible, and there is no convection in the system. On the basis of the basic state solution for the lamellar eutectic in directional solidification, the asymptotic solution for the perturbed interface shape of lamellar eutectic growth under the anisotropic surface tension is obtained in the case where the Peclet number is small, and then the quantization conditions of interfacial morphology for lamellar eutectic crystal is obtained. A dispersion relation between the wave number and the perturbation amplification rate, and the stability criterion of lamellar eutectic growth under the anisotropic surface tension are also obtained. The result shows that the anisotropic surface tension has a significant effect on lamellar eutectic growth in directional solidification. It shows that comparing the directional solidification system of isotropic surface tension, the interface morphological stability of anisotropic surface tension also involves two types of global instability mechanisms: the exchange of stability induced by the non-oscillatory, unstable modes and the global wave instability caused by four types of oscillatory unstable modes, namely antisymmetric antisymmetric (AA)-, symmetric antisymmetric (SA)-, antisymmetric symmetric (AS)-, and symmetric symmetric (SS)- modes. The linear stability analysis reveals that the stability of lamellar eutectic growth depends on stability critical number *. When *, the eutectic growth system is unstable; When q *, the eutectic growth system is stable. The anisotropic surface tension, by reducing the corresponding stability critical number *, stabilizes both the exchange of stability mechanism and the global instability mechanism for the AA-, SA- and SS-modes. It implies that the anisotropic surface tension parameter tends to reduce the stability zone. However, by increasing the corresponding stability critical number *, the anisotropic surface tension destabilizes the global instability mechanism for the AS-mode. It implies that the anisotropic surface tension parameter tends to increase the stability zone for AS-mode.
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Keywords:
- directional solidification /
- eutectic growth /
- anisotropic surface tension /
- morphological stability
[1] Jackson K A, Hunt J D 1966 Trans. Metall. Soc. AIME 236 1129
[2] Trivedi R, Mangnin P, Kurz W 1987 Acta Metall. 35 971
[3] Liu J M, Zhou Y H, Shang B L 1990 Acta Metall. Mater. 38 1625
[4] Kassner K, Misbah C 1991 Phys. Rev. A 44 6533
[5] Li X, Ren Z M, Fautrelle Y, Zhang Y D, Esling C 2010 Acta Mater. 58 1403
[6] Meng G H, Lin X, Huang W D 2007 J. Mater. Sci. Technol. 23 851
[7] Karma A, Sarkissian A 1996 Metall. Mater. Trans. A 27 635
[8] Parisi A, Plapp M 2008 Acta Mater. 56 1348
[9] Ginibre M, Akamatsu S, Faivre G 1997 Phys. Rev. E 56 780
[10] Datye V, Langer J S 1981 Phys. Rev. B 24 4155
[11] Xu J J, Li X M, Chen Y Q 2014 J. Cryst. Growth 401 93
[12] Xu J J, Chen Y Q, Li X M 2014 J. Cryst. Growth 401 99
[13] Xu J J 2017 Interfacial Wave Theory of Pattern Formation in Solidification: Dendrites, Fingers, Cells and Free Boundaries (2nd Ed.) (New York: Springer) pp503-572
[14] Xu J J 2006 Introduction to kinetics of solidification and stability theory of the interface (Beijing: Science Press) pp33-44 (in Chinese) [徐鉴君 2006 凝固过程动力学与交界面稳定性理论导引(北京: 科学出版社)第3344页]
[15] Wang Z J, Wang J C, Yang G C 2008 Acta Phys. Sin. 57 1246(in Chinese) [王志军, 王锦程, 杨根仓 2008 57 1246]
[16] Wang Z J, Wang J C, Yang G C 2010 Chin. Phys. B 19 017305
[17] Chen M W, Wang Z D, Xu J J 2014 J. Cryst. Growth 385 115
[18] Xu J J 1991 Physica D 51 579
[19] Chen M W, Chen Y C, Zhang W L, Liu X M, Wang Z D 2014 Acta Phys. Sin. 63 038101(in Chinese) [陈明文, 陈奕臣, 张文龙, 刘秀敏, 王自东 2014 63 038101]
[20] Seetharaman V, Trivedi R 1988 Metall. Trans. A 19 2955
[21] Mergy J, Faivre G, Guthmann C, Mellet R 1993 J. Cryst. Growth 134 353
[22] Mullins W W, Sekerka R F 1964 J. Appl. Phys. 35 444
[23] Xu J J 2002 J. Cryst. Growth 245 134
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[1] Jackson K A, Hunt J D 1966 Trans. Metall. Soc. AIME 236 1129
[2] Trivedi R, Mangnin P, Kurz W 1987 Acta Metall. 35 971
[3] Liu J M, Zhou Y H, Shang B L 1990 Acta Metall. Mater. 38 1625
[4] Kassner K, Misbah C 1991 Phys. Rev. A 44 6533
[5] Li X, Ren Z M, Fautrelle Y, Zhang Y D, Esling C 2010 Acta Mater. 58 1403
[6] Meng G H, Lin X, Huang W D 2007 J. Mater. Sci. Technol. 23 851
[7] Karma A, Sarkissian A 1996 Metall. Mater. Trans. A 27 635
[8] Parisi A, Plapp M 2008 Acta Mater. 56 1348
[9] Ginibre M, Akamatsu S, Faivre G 1997 Phys. Rev. E 56 780
[10] Datye V, Langer J S 1981 Phys. Rev. B 24 4155
[11] Xu J J, Li X M, Chen Y Q 2014 J. Cryst. Growth 401 93
[12] Xu J J, Chen Y Q, Li X M 2014 J. Cryst. Growth 401 99
[13] Xu J J 2017 Interfacial Wave Theory of Pattern Formation in Solidification: Dendrites, Fingers, Cells and Free Boundaries (2nd Ed.) (New York: Springer) pp503-572
[14] Xu J J 2006 Introduction to kinetics of solidification and stability theory of the interface (Beijing: Science Press) pp33-44 (in Chinese) [徐鉴君 2006 凝固过程动力学与交界面稳定性理论导引(北京: 科学出版社)第3344页]
[15] Wang Z J, Wang J C, Yang G C 2008 Acta Phys. Sin. 57 1246(in Chinese) [王志军, 王锦程, 杨根仓 2008 57 1246]
[16] Wang Z J, Wang J C, Yang G C 2010 Chin. Phys. B 19 017305
[17] Chen M W, Wang Z D, Xu J J 2014 J. Cryst. Growth 385 115
[18] Xu J J 1991 Physica D 51 579
[19] Chen M W, Chen Y C, Zhang W L, Liu X M, Wang Z D 2014 Acta Phys. Sin. 63 038101(in Chinese) [陈明文, 陈奕臣, 张文龙, 刘秀敏, 王自东 2014 63 038101]
[20] Seetharaman V, Trivedi R 1988 Metall. Trans. A 19 2955
[21] Mergy J, Faivre G, Guthmann C, Mellet R 1993 J. Cryst. Growth 134 353
[22] Mullins W W, Sekerka R F 1964 J. Appl. Phys. 35 444
[23] Xu J J 2002 J. Cryst. Growth 245 134
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