於凝固的過程中，溫度與濃度場之變化會直接影響到凝固材料之顯微結構，而此顯微結構與凝固材料之品質與物理特性有著很密切的關係。所以本文研究目的是希望藉用模擬網狀晶，來瞭解凝固時顯微結構之變化。首先建立一簡化模式，此模式是以含向前擴散之謝荷方程式為基礎，來建立包含表面能效應之濃度場模式，接者進一步建立包含溫度與濃度場、表面能效應及原子附著效應之完整成長模式。在液固界面上，使用直接法來調整液固界之位置，以疊代方式來求得自我滿足之網狀晶形狀。最後以此模式來探討在不同初始濃度、溫度梯度、成長速率及網狀晶間距對網狀晶的影響。期望本文之分析結果可作為進一步研究之參考。

In the process of solidification, the variations of the temperature and concentration fields will directly affect the microstructures of materials, which have very close relations with the qualities of materials. This paper is to build two models to study the cellular shape of constrained growth. The first model, based on the Scheil equation with forward diffusion, was set up. This model includes the solute diffusion and the capillarity effect. The second model is a complete one, which consists of the concentration and temperature fields, and the effects of capillarity effect and atom attachment. In the second model, the shape of the solid/liquid interface of cellular growth is not known a priori, rather it is calculated as part of solution to the field problem. The cellular shape predicted by the first model will be used as the initial shape of the second model. The direct iteration method was used to compute the self-consistent cellular shape. With these models, the effects of different control parameters, which are growth rate, temperature gradient and the initial concentration, were investigated. It is hoped that the results of this study can be referred to for the further study.

1.Flemings, M.C., Solidification Processing, McGraw-Hill Book Company, New York, USA, 1974.
2.Kurz, W., and D.J. Fisher, Fundamentals of Solidification, 4th ed., Trans Tech Publication, Aedermannsdrof, Switzerl and, 1998.
3.Gruzleski, J.E., B.M. Closset, The Treatment of Liquid Aluminum- Silicon Alloys, The American Foundarymen’s Society, 1990.
4.Bates, C.E.,“Alloy Elements Effects on Gray Iron Properties: Part Ⅱ,”AFS Transactions, Vol. 94,pp. 889-912, 1986.
5.Sachar, H. and J.F. Wallace, “Effect of Microstructure and Testing Mode on the Fatigure Properties of Gray Iron,”AFS Transactions, Vol. 90,pp. 777-793, 1982.
6.Kasap, S.,“Principles of Electrical Engineering Material and Devices,”revised Edition, McGraw-Hill Book company, New York, USA, 2000.
7.Desbiolles, J.D., Droux, J.J., Rappaz, M.,“Simulation of Solidification of Alloys by the Finite-Element Method,”Computer Physics Reports, Vol. 6, pp.371-383, 1987.
8.Gandin, Ch.-A., Rappaz, M., and Tinillier, R.,“Three-Dimension Probabilistic Simulation of Solidification Grain Structure Application to Superalloy Precision Casting,”Metallurgical Transactions A, Vol. 24A, pp.467-479, 1993.
9.Kurz, W.,“Microsegregation in Rapidly Solidified Ag-15wt-percent- Cu,”Journal of Crystal Growth, Vol. 91, pp.123-125, 1988.
10.Rappaz, M., and Gandin, Ch-A. ,“Probabilistic Modeling of Micro- structure Formation In Solidification Processes.,”Acta Metall., Vol. 59, p.945, 1966.
11.Chao Long-Sun and Du Wu-Chang,“Macro-Micro Modeling of Solidification,”Pro. Natl. Sci. Counc. ROC(A), Vol. 23, NO. 5, pp. 622-629, 1999.
12.Fisher, J.C., B.Chalmers in Principles of Solidification, Wiley, New York, USA, p. 105,1966.
13.Ivantsov, G.P., “ The temperature field around a spherical,
cylindrical, or point crystal growing in a cooling solution ”,Doklady Akademii Nauk SSSR, Vol 58, p.567,1947.
14.Glicksman, M.E., R.J. Schaefer, J.D. Ayers, Dendritic Growth---A Test of Theory, Metallurgical Transactions, Vol. 7A, p.1747, 1976.
15.Langer, J.S., and H. Müller-Krumbahaar, Stability Effect in Dendritic Crystal Growth, J. of Crystal Growth, Vol. 42,p. 11,1977.
16.Mullins, W.W., R.F. Sekerka, Stability of a Planar Interface During Solidification of a Dilute Binary Alloy, J. of Applied Physics, Vol. 35,p. 444, 1964.
17.Dantzig, J.A. and Chao L.S., Low-Gravity Fluid Dynamics and Tranport Phenomenon, edited by J.N. Koster and R.L Sani, Vol. 150 of Progress in AIAA, Washington, DC, USA, pp. 477-436,1990.
18.Han Q, and Hunt JD, Numerical modeling of the growth of a cellular /dendritic array in multi-component alloys, Metall. and Materials Trans. A, Vol. 238(1), pp.192-195, 1997.
19.Brown SGR, Simulation of diffusional composite growth using the cellular automaton finite difference (CAFD), Journal of Materials Science, Vol. 33(19), pp. 4769-4776, 1998.
20.Yu Ym M, Yang GC, Zhao DW, and Lu YL, Numerical simulation of dendritic growth in undercooled melt using phase-field approach, ACTA PHYSICA SINICA, Vol. 50(12), pp. 2423-2428, 2001
21.Bower, T.F., H.D. Brody, and M.C. Fleming, MET. Trans., p. 624, Vol. 236, 1966.
22.Kreysig, E., Introduction to Differential Geometry and Riemanian Geometry, University of Toronto Press, 1968.
23.Engelman, M.S., FIDAP Rev. 8.0 Theoretical Manual, Fluent Inc., Lebanon, New Hampshire, USA, 1998.
24.Richard L. Burden, J. Douglas Faiers , and Albert C. Reynolds, Numerical Analysis, 2th ed. , The Southeast Book Company, USA, pp. 200-243 , 1981.
25.Anderson, D. A., Tannehill, J. C. and Pletcher, R. H., "Computational Fluid Mechanics and Heat Trans.," Hemisphere 1990.
26.D. G. MCCARTNEY and J. D. HUNT, A Numerical FiniteDifference Model of Steady State Cellular and Dendritic Growth, Metallurgical transactions A, Vol. 15A , pp. 983-994, 1984.
27.J. D. HUNT and D. G. MCCARTNEY , Numerical FiniteDifference Model for Steady State Cellular Array Growth, Metallurgical transactions A, Vol. 35 , pp. 89-99, 1987.