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Solute diffusion is an important process that determines the dendrite growth during solidification. The theoretical model generally simplifies the solute diffusion coefficient in liquid phase into a constant. Nevertheless, the composition of the boundary layer changes greatly in the solidification process, the diffusion coefficient will no longer be a constant and is dependent on concentration. In this paper, the quantitative phase field model is used to simulate the effect of concentration-dependent diffusion coefficient on dendrite growth in directional solidification. In the model, the concentration-dependent diffusion process is investigated by coupling the concentration-dependent diffusion coefficient in the liquid solute diffusion equation. A series of simulation results confirms that the concentration-dependent diffusion process has a significant effect on the dendrite growth. The results show that the increase of the coupling intensity of solute concentration will enhance the diffusion of solute in the mushy zone between primary dendrites to the dendrite tip, resulting in the increase of solute enrichment at the dendrite tip, thereby increasing the tip undercooling. The variation of diffusion coefficient in liquid phase has little effect on the tip radius of dendrite, and the simulation results are in good agreement with those from the theoretical model. Moreover, the amplitude of dendritic side branches decreases with the increase of solute diffusion coefficient. In the study of dendrite arrays, it is found that the concentration-dependent diffusion coefficient increases the primary spacing and reduce the tip position. The results of this study indicate that for a system with a concentration-dependent coefficient significantly, the effect of concentration-dependent diffusion on tip undercooling and side branches should be considered in the quantitative and experimental verification of the existing model.
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Keywords:
- phase field method /
- directional solidification /
- dendrite /
- diffusion coefficient
[1] Bower T F, Brody H D, Flemings M C 1966 Trans. AIME 236 624
[2] Burden M H, Hunt J D 1974 J. Cryst. Growth 22 109
Google Scholar
[3] Laxmanan V 1985 Acta Metall. 33 1023
Google Scholar
[4] Kurz W, Fisher D J 1981 Acta Metall. 29 11
Google Scholar
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Google Scholar
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Wang Z J, Wang J C, Yang G C 2008 Acta Phys. Sin. 57 1246
Google Scholar
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[15] Eechebarria B, Folch R, Karma A, Plapp M 2004 Phys. Rev. E 70 061604
Google Scholar
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Google Scholar
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Kurz W, Fisher D J (translated by Li J G, Hu Q D) 2010
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Google Scholar
[19] Spencer B J, Huppert H E 1999 J. Cryst. Growth 200 287
Google Scholar
[20] 张云鹏, 林鑫, 魏雷, 彭东剑, 王猛, 黄卫东 2013 物理学报 62 178105
Google Scholar
Zhang Y P, Lin X, Wei L, Peng D J, Wang M, Huang W D 2013 Acta Phys. Sin. 62 178105
Google Scholar
[21] Hunt J D 1979 Solidification of Casting of Metals (London: The Metals Society) p3
[22] Hunt J D, Lu S Z 1992 J. Cryst. Growth. 123 17
Google Scholar
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图 1 耦合强度对枝晶尖端生长形貌的影响 (a) vp = 20 μm/s; (b) vp = 32 μm/s; (c) vp = 50 μm/s
Figure 1. Effect of coupling intensities on the morphology of dendrite tip growth: (a) vp = 20 μm/s; (b) vp = 32 μm/s; (c) vp = 50 μm/s.
图 2 vp = 32 μm/s,
$\eta _0^2 = 80$ 时枝晶尖端前沿(实线)与边界(虚线)处液相的溶质场特征 (a)溶质场; (b)溶质浓度; (c)与浓度相关的扩散系数Figure 2. The solute field along the full line and the hidden line of the channel when vp = 32 μm/s and
$\eta _0^2 = 80$ : (a) Solute field; (b) solute concentration; (c) concentration-dependent diffusion coefficient.
图 3 不同模拟条件下枝晶的尖端特征 (a)尖端半径; (b)尖端位置; (c)尖端过冷度
Figure 3. The characteristics of dendrite tip with different simulated conditions: (a) Tip radius; (b) tip position; (c) tip undercooling.
图 4 vp = 32 μm/s时不同耦合强度下枝晶尖端附近的浓度特征 (a)边界浓度; (b)尖端浓度与尖端扩散系数
Figure 4. Concentration characteristics near the dendrite tip under different coupling intensities with vp = 32 μm/s: (a) Boundary concentration; (b) tip concentration and tip diffusion coefficient.
图 5 (a1)−(a3)
$\eta _0^2 = 0$ 时枝晶列的演化过程; (b1)−(b3)$\eta _0^2 = 80$ 时枝晶列的演化过程Figure 5. (a1)−(a3) The evolution of dendrite columns when
$\eta _0^2 = 0$ ; (b1)−(b3) the evolution of dendrite columns when$\eta _0^2 = 80$ .表 1 材料物性参数
Table 1. Material physical parameters.
物性参数 取值 熔点温度/K 500 初始浓度/at.% 0.01 平衡分配系数 0.3 液相线斜率/K·at.%–1 –200 液相中的扩散系数/m2·s–1 0.45 × 10–9 Gibbs-Thomson系数/K·m 6.48 × 10–8 
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[1] Bower T F, Brody H D, Flemings M C 1966 Trans. AIME 236 624
[2] Burden M H, Hunt J D 1974 J. Cryst. Growth 22 109
Google Scholar
[3] Laxmanan V 1985 Acta Metall. 33 1023
Google Scholar
[4] Kurz W, Fisher D J 1981 Acta Metall. 29 11
Google Scholar
[5] Trivedi R 1980 J. Cryst. Growth 49 219
Google Scholar
[6] Shampine L F 1973 Quart. Appl. Math. 30 441
Google Scholar
[7] Lee J H, Liu S, Miyahara H, Trivedi R 2004 Metall. Mater. Trans. B 35B 909
Google Scholar
[8] Dahlborg U, Besser M, Calvo-Dahlborg M, Janssen S, Juranyi F, Kramer M J, Morris J R, Sordelet D J 2007 J. Non-Cryst. Solids 353 3295
Google Scholar
[9] Li J J, Wang Z J, Wang Y Q, Wang J C 2012 Acta Mater. 60 1478
Google Scholar
[10] Wang Z J, Li J J, Wang J C, Zhou Y H 2012 Acta Mater. 60 1957
Google Scholar
[11] Diepers H J, Ma D, Steinbach D M 2002 J. Cryst. Growth 237 149
Google Scholar
[12] 王志军, 王锦程, 杨根仓 2008 物理学报 57 1246
Google Scholar
Wang Z J, Wang J C, Yang G C 2008 Acta Phys. Sin. 57 1246
Google Scholar
[13] Boettinger W J, Warren C, Beckermann C, Karma A 2002 Ann. Rev. Mater. Res. 32 163
Google Scholar
[14] Karma A 2001 Phys. Rev. Lett. 87 115701
Google Scholar
[15] Eechebarria B, Folch R, Karma A, Plapp M 2004 Phys. Rev. E 70 061604
Google Scholar
[16] Losert W, Shi B Q, Cummins H Z 1998 Proc. Acad. Sci. USA 95 431
Google Scholar
[17] 凝固原理 (Beijing: Higher Education Press) p60
Kurz W, Fisher D J (translated by Li J G, Hu Q D) 2010
[18] Tiller W A, Jackson K A, Rutter J W, Chalmers B 1953 Acta Metall. 1 428
Google Scholar
[19] Spencer B J, Huppert H E 1999 J. Cryst. Growth 200 287
Google Scholar
[20] 张云鹏, 林鑫, 魏雷, 彭东剑, 王猛, 黄卫东 2013 物理学报 62 178105
Google Scholar
Zhang Y P, Lin X, Wei L, Peng D J, Wang M, Huang W D 2013 Acta Phys. Sin. 62 178105
Google Scholar
[21] Hunt J D 1979 Solidification of Casting of Metals (London: The Metals Society) p3
[22] Hunt J D, Lu S Z 1992 J. Cryst. Growth. 123 17
Google Scholar
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