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In this paper, we study the effects of anisotropic interface kinetics and surface tension on deep cellular crystal growth in directional solidification. The following assumptions are made: the process of solidification is viewed as a two-dimensional problem; the minor species in this binary mixture system is considered as an impurity; the solute diffusion in the solid phase is negligible; the thermodynamic properties other than the diffusivities are the same for both solid and liquid phases; there is no convection in the system; the anisotropic interface kinetics and the anisotropic surface tension are a four-fold symmetry function each; neither the preferred directions of the anisotropic interface kinetics nor the anisotropic surface tensions are necessarily the same as their counterparts for the solid and liquid phases respectively; the angle between the preferred directions of the two anisotropies is 0. By using the matched asymptotic expansion method and the multiple variable expansion method, we obtain the diagram of interface morphology for a deep cellular crystal in directional solidification. The results show that there exists a discrete set of the steady-state solutions subject to the quantization condition (35). The quantization condition yields the eigenvalue ???106801-20170033???* as a function of parameter and other parameters of the system, which determines the interface morphology of the cell. The results also show the variation of the minimum eigenvalue ???106801-20170033???*(0) with parameter . It is seen that when the preferred directions of the two anisotropies are the same, i.e., 0 = 0, the minimum eigenvalue ???106801-20170033???*(0) reduces with the increase of anisotropic surface-tension coefficient 4 , increases with the augment of parameter , and is unrelated to anisotropic interface kinetic coefficient 4 in the low order; when the angle 0 0 /4, as the 0 increases, the minimum eigenvalue ???106801-20170033???*(0) increases; when the angle /4 0 /2, as the 0 increases, the minimum eigenvalue ???106801-20170033???*(0) decreases. In addition, the results show the composite solution for the interface shape function B described on (X, Y) plane. It is seen that both of the anisotropy and the angle 0 have a significant effect on the total length and the root of deep cellular crystal, however, have little influence on the other solid-liquid interface, such as the top of deep cellular crystal. When the angle 0 is 0, as anisotropic coefficient increases, the total length of the finger increases, the curvature of the interface near the root increases or the curvature radius decreases. It is found that the influence of the anisotropic surface-tension coefficient on interface morphology is more remarkable than that of the anisotropic interface kinetics coefficient. when the angle 0 0 /4, as the 0 increases, the total length of the finger decreases, the curvature of the interface near the root decreases or the curvature radius increases; when the angle /4 0 /2, as 0 increases, the total length of the finger increases, the curvature of the interface near the root increases or the curvature radius decreases.
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Keywords:
- directional solidification /
- anisotropic surface tension /
- anisotropic interface kinetics /
- deep cellular crystal growth
[1] Ding G L, Huang W D, Lin X, Zhou Y 1997 J. Cryst. Growth 177 281
[2] Ding G L, Lin H, Huang W D, Zhou Y H 1995 Acta Metall. Sin. 31 469
[3] Georgelin M, Pocheau A 2006 Phys. Rev. E 73 011604
[4] Georgelin M, Pocheau A 2009 Phys. Rev. E 81 031601
[5] Chen Y Q, Xu J J 2011 Phys. Rev. E 83 041601
[6] Xu J J, Chen Y Q 2015 Eur. J. Appl. Math. 26 1
[7] Mullins W W, Sekerka R F 1963 J. Appl. Phys. 34 323
[8] Mullins W W, Sekerka R F 1964 J. Appl. Phys. 35 444
[9] Saffman P G, Taylor G I 1958 Proc. R. Soc. London A 245 312
[10] Coriell S R, Sekerka R F 1976 J. Cryst. Growth 34 157
[11] 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 物理学报 57 1246]
[12] Wang Z J, Wang J C, Yang G C 2010 Chin. Phys. B 19 017305
[13] Trivedi R, Seetharaman V, Eshelman M A 1991 Mater. Trans. A 22 585
[14] 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 物理学报 63 038101]
[15] Jiang H, Chen M W, Shi G D, Wang T, Wang Z D 2016 Mod. Phys. Lett. B 30 1650205
[16] Ihle T 2000 Eur. Phys. J. B 16 337
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[1] Ding G L, Huang W D, Lin X, Zhou Y 1997 J. Cryst. Growth 177 281
[2] Ding G L, Lin H, Huang W D, Zhou Y H 1995 Acta Metall. Sin. 31 469
[3] Georgelin M, Pocheau A 2006 Phys. Rev. E 73 011604
[4] Georgelin M, Pocheau A 2009 Phys. Rev. E 81 031601
[5] Chen Y Q, Xu J J 2011 Phys. Rev. E 83 041601
[6] Xu J J, Chen Y Q 2015 Eur. J. Appl. Math. 26 1
[7] Mullins W W, Sekerka R F 1963 J. Appl. Phys. 34 323
[8] Mullins W W, Sekerka R F 1964 J. Appl. Phys. 35 444
[9] Saffman P G, Taylor G I 1958 Proc. R. Soc. London A 245 312
[10] Coriell S R, Sekerka R F 1976 J. Cryst. Growth 34 157
[11] 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 物理学报 57 1246]
[12] Wang Z J, Wang J C, Yang G C 2010 Chin. Phys. B 19 017305
[13] Trivedi R, Seetharaman V, Eshelman M A 1991 Mater. Trans. A 22 585
[14] 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 物理学报 63 038101]
[15] Jiang H, Chen M W, Shi G D, Wang T, Wang Z D 2016 Mod. Phys. Lett. B 30 1650205
[16] Ihle T 2000 Eur. Phys. J. B 16 337
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