相場模式法(phase field method)是模擬自由介面問題的眾多方式之一。本實驗室利用二維適應性相場模式法（以下簡稱為此法）處理固化問題已有四年時間，且有許多豐碩的成果。然而因數值上的本性，使此法在合金固化模擬上存著難以將結果定量化之難題。近年來這些影響模擬定量化結果的關鍵因素皆已被研究者標明出來（A. Karma, Phys. Rev. Lett. 87, 115701, 2001），且所提出之解決方法-薄介面分析(thin-interface analysis)-亦陸續在近幾年的研究經由與理論解的比較而證實其可行性(J. C. Ramirez, C. Beckermann, A. Karma, and H-J. Diepers, Phys. Rev. E 69, 051607, 2004)。
在本論文中，吾人將上述提及之薄介面分析應用在此法上面，以期達到量化模擬的目標。經由一些嚴謹的一維與二維測試，我們發現利用此法所計算之結果不僅可以逼近理論解析解，其模擬出之二維結果跟他人文獻比較，也有高度的一致性。接著，吾人對Shih(C. J. Shih, M.S. Diss, National Taiwan University, 2004)在處理溶質包覆效應時所提出的簡單界面模式（Simple-Interface-Model, SIM）做出一連串完整的一維與二維之比較研究，除了與sharp-interface model所預測之解析解比較，亦與既有文獻之ATC 模式作比較。其結果顯示簡單界面模式在抑制溶質包覆效應上，與ATC模式一樣，能成為一可行、可依賴的方法。此外吾人亦賦予了簡單界面模式更完整之物理詮釋。
本論文最後段落則試圖在自然對流對垂直固化過程中型態變化的影響作探討。初部結果顯示在強自然對流的效應之下，因為側向的濃度梯度被改變，因此溶質的輕重與否會成為改變其起初的固液界面的重要因素。這是第一個用相場模式法模擬此現象的研究。

The phase field method has been regarded as one of the many methods used to simulate free-boundary problems. It has been four years that our laboratory used the two-dimensional adaptive phase field model to deal with the solidification process, and many fruitful results have been reported. However, due to the numerical nature of this method, it does not support accurate quantitative measurements when simulating the alloy solidification process. The key factors affecting the quantitative modeling, have been clearly identified in recent works (A. Karma, Phys. Rev. Lett. 87, 115701, 2001). The so-called ‘thin-interface analysis’ methodology has consequently been developed to tackle these difficulties, and the feasibility of which has been justified very recently (J. C. Ramirez, C. Beckermann, A. Karma, and H-J. Diepers, Phys. Rev. E 69, 051607, 2004).

In this thesis, we successfully developed a quantitative phase field model by adopting the ‘thin-interface analysis’ methodology. Through careful examinations, we find that results from the present model not only match closely with analytic solutions but are highly identical to other researchers'' results. Moreover, we tested the practicability of the Simple-Interface-Model (SIM), proposed by Shih (C. J. Shih, M.S. Diss., National Taiwan University, 2004), on the solute trapping effect. In addition to comparison with sharp-interface model, a comparative study of SIM and ATC model is also reported. The simulated results indicate that SIM is indeed a viable alternative method when suppressing the solute trapping effect. What''s more, a full physical explanation of SIM is given for the first time in this thesis.

The last part of this thesis considers the impact that thermalsolutal convection has on the morphological change of the directional solidification process. Preliminary simulation results show that the buoyancy does significantly change the melt/solid interface morphology, due to the lateral change of the concentration profile. This is the first research that uses the phase field simulation method to investigate this phenomenon.

Acknowledgements…………………………………………..………..Ⅰ
Abstract …………………………………………………......………...Ⅱ
Table of Contents ………………………………………………..…... Ⅳ
Nomenclature ………………………...…………………………….…Ⅵ
List of Tables ……………………………………………………….... Ⅹ
List of Figures ………………………………………………………...Ⅺ

Chapter 1 Introduction ……………………………….….…………1
1.1 A Review of Numerical Methods in Simulation of the Solidification ……..2
1.2 A Review of Theory of Morphological Instabilities in the Solidification ....7
1.3 Motivations ……………………………………………………………….10
Chapter 2 Models and Numerical Method ………………........….12
2.1 Sharp Interface Model ………………………….…………………………12
2.2 Phase Field Model …………………………………………………..….... 13
2.2.1 Diffusive System …………………………………………………..13
2.2.2 Flow System ……………………………………………………….19
2.3 Toward Quantitative Phase Field Modeling -Thin-Interface Analysis …...21
2.3.1 Thin-interface Analysis Applied to Phase Field Equation ...………23
2.3.2 Thin-interface Analysis Applied to Species Transport Equation…..24
2.3.3 Anti-Trapping Current Model (ATC) ……………………………...27
2.3.4 Simple-Interface model (SIM) ………………………………….....29
2.4 Dimensionless Equations ………………………………………………....31
2.5 Numerical Methods ……………………………………………………..…33
2.4.1 Adaptive Mesh Refinement (AMR) …………………………….….33
2.4.2 Finite Volume Method (FVM) ……………………………………..36
Chapter 3 Results and Discussions …………………………...……40
3.1 Validation of Quantitative Phase Field Model ………………….………....40
3.1.1 One-dimensional Solidification of an Alloy………………….….…40
3.1.2 Curvature Effect in Phase Field Equation …………………………49
3.1.3 Isothermal Alloy Dendrite Growth …………………….…………..54
3.2 Comparative Study of SIM and ATC Model in Alloy Solidification…...…56
3.2.1 One-dimensional Directional Solidification……………………….57
3.2.2 Isothermal Alloy Dendrite Growth ………………………………..65
3.2.3 Alloy Directional Solidification …………………………………...69
3.3 Effect of Thermosolutal Convection during Directional Solidification...…74
Chapter 4 Conclusions …………………………….…………….…80
Reference ……………………………………………..…………….…83

[1] J. F. Thompson, Z. U. A. Warsi and C. W. Mastin, J. Comput. Phys. 47, 1, 1982
[2] J. F. Thompson, Z. U. A. Warsi and C. W. Mastin, Numerical Grid Generation, North-Holland, Amsterdam, 1985
[3] L. H. Ungar and R. A. Brown, Phys. Rev. B 29, 1367, 1984
[4] L. H. Ungar and R. A. Brown, Phys. Rev. B 30, 3993, 1984
[5] L. H. Ungar, M. J. Bennett and R. A. Brown, Phys. Rev. B 31, 5923, 1985
[6] L. H. Ungar and R. A. Brown, Phys. Rev. B 31, 5931, 1985
[7] N. Ramprasad, M. J. Bennett and R. A. Brown, J. Cryst. Growth 121, 536, 1993
[8] K. Tsiveriotis and R. A. Brown, Phys. Rev. B 48, 13495, 1993
[9] K. Tsiveriotis and R. A. Brown, Int. J. Num. Met. Fluid 14, 981, 1992
[10] K. Tsiveriotis and R. A. Brown, Int. J. Num. Met. Fluid 16, 827, 1993
[11] L.H. Ungar, N. Ramprassd and R.A. Brown, J. Sci. Comput. 3, 77, 1988
[12] C. W. Lan, Chemical Engineer. Science 59, 1437, 2004
[13] M. Sussmann, P. Smereka and S. Osher, J. Comput. Phys. 114, 146, 1994
[14] M. Sussman, A. S. Almgren, J. B. Bell, L. H. Howell and M . L. Welcome, J. Compt. Phys. 148, 81, 1999
[15] G. Liao, F. Liu, G. C. de la Pena, D. Peng, and S. Osher, J. Comp. Phys. 159, 103, 2000
[16] V. Sochnikov and S. Efrima, Int. J. Numer. Meth. Engng 56, 1913, 2003
[17] Y. T. Kim, N. Goldenfeld and J. Dantzig, Phys. Rev. E. 62, 2471, 2000
Y. T. Kim, Ph. D. dissertation, University of Illinois, 2003
[18] W.J. Boettinger, J.A. Warren, C. Beckermann and A. Karma, Annu. Rev. Mater. Res. 32, 164, 2002
[19] L. Q. Chen, Annu. Rev. Mater. Res. 32, 113, 2002
[20] B. I. Halperin, P. C. Hohenberg, and S-K. Ma, Phys. Rev. B 10, 139, 1974
[21] J. S. Langer, Directions in Condensed Matter Physics, p.165
[22] J.B. Collins and H. Levin, Phy. Rev. B 31, 6119, 1985
[23] J. A. Warren, W. J. Boettinger, and G. B. McFadden, Phys. Rev. A 45, 689, 1992
[24] C, Beckermann, H.-J. Diepers, I. Steinbach, A. Karma and X. Tong, J. Comput. Phys. 154, 468, 1999
[25] G. Caginalp and E. A. Socolovsky, J. Comp. Phys. 95, 85, 1991
[26] R. Kobayashi, Exp. Math. 3, 60, 1994
[27] A. A. Wheeler, B. T. Murray, and R. J. Schaefer, Physica D 66, 243, 1993
[28] B. T. Murray, W. J. Boettinger, G. B. McFadden, and A. A. Wheeler, in Heat Transfer in Melting, Solidification and Crystal Growth, HTD Vol. 234, edited by I. S. Habib and S. Thynell (ASME, New York, 1993), p. 67.
[29] S.-L.Wang, R. F. Sekerka, A. A. Wheeler, B. T. Murray, S. R. Coriell, R. J. Braun, and G. B. McFadden, Physica D 69, 189 (1993).
[30] A. A. Wheeler, B. T. Murray and R. Schaefer, Physica D 66, 243, 1993
[31] S-L. Wang and R. F. Sekerka, Phys. Rev. E 53, 3760, 1996
[32] A. Karma and W-J. Rappel, Phys. Rev. E 53, R3017, 1996
[33] A. Karma and W-J. Rappel, Phys. Rev. E 57, 4323, 1998
[34] N. Provatas, N. Goldenfeld, and J. Dantzig, Phys. Rev. Lett. 80, 3308, 1999
[35] C.W. Lan, C.M. Hsu, C.C. Liu, and Y.C. Chang, Phys. Rev. E 65, 061601, 2002
[36] A. Karma, Phys. Rev. Lett. 87, 115701, 2001
[37] G. Amberg, Phys. Rev. Lett. 91, 265505, 2003
[38] J.C. Ramirez, C. Beckermann, A. Karma, and H-J. Diepers, Phys. Rev. E 69, 051607, 2004
[39] J. C. Ramirez and C. Beckermann, Acta Materialia 53, 1721, 2005
[40] K. Glasner, J. Comput. Phys. 174, 695, 2001
[41] J. H. Jeong, N. Goldenfeld, and J. A. Dantzig, Phys. Rev. E 64, 041602, 2001
[42] C. C. Liu, M.S. Diss., National Taiwan University, 2000
[43] C. W. Lan, and C. M. Hsu, J. Crystat Growth 241, 379, 2002
[44] C. W. Lan, and C. M. Hsu, Phys. Rev. E. 65, 061601, 2002
[45] C.W. Lan, Y.C. Chang, and C.J. Shih, Acta Mater. 51, 1857, 2003
[46] C. W. Lan, and C. J. Shih, Phys. Rev. E. 69, 031601, 2004
[47] C. W. Lan, and Y. C. Chang, J. Crystal Growth, 250, 525, 2003
[48] C. J. Shih, M.S. Diss., National Taiwan University, 2004
[49] M.J. Aziz and T. Kaplan, Acta. Metall. 36, 2335, 1988
[50] N.A. Ahmad, A.A. Wheeler, W.J. Boettinger and G.B. McFadden, Phys. Rev. E 58, 3436, 1998
[51] C.W. Lan, C. J. Shih, and M. H. Lee, Acta Materialia 53, 2285, 2005
[52] W. A. Tiller, K. A. Jackson, J. W. Rutter and B. Chalmers, Acta Metall 1, 428, 1953
[53] W. W. Mullins and R. F. Sekerka, J. Appl. Phys. 35, 444, 1964
[54] J. A. Warren and J. S. Langer, Phys. Rev. E 47, 2072, 1993
[55] S. R. Coriell, M. R. Cordes, W. J. Boettinger and R. F. Sekerka, J. Crystal Growth 49, 13, 1980.
[56] S. H. Davis. Theory of Solidification. Cambridge University Press, 2001
[57] R. Tonhardt , and G. Amberg , J Crystal Growth 194, 406, 1998
[58] R. Tonhardt , and G. Amberg , J Crystal Growth 213, 161, 2000
[59] R. Tonhardt , and G. Amberg, Phys. Rev E 62, 828, 2000
[60] B. Echebarria, R. Folch, A. Karma, and M. Plapp, Phys. Rev. E 70, 061604, 2004
[61] W. Kurz and D. J. Fisher, Fundamentals of Solidification, Trans Tech, Addermannsdorf, Switzerland, 1992
[62] R. F. Almgren, SIAM J. Appl. Math 59, 2086, 1999
[63] S. R. Murray and J. Y. Murthy, Numer. Hear Trans. B 31, 195, 1996
[64] S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, D. C. 1980
[65] C. M. Rhie and W. L. Chow, AIAA J. 21, 1527, 1983
[66] Trivedi, R; Liu, S; Echebarria, B; Karma, A Solidification Processes and Microstructures: A Symposium in Honor of Wilfried Kurz as held at the 2004 TMS Annual Meeting; Charlotte, NC; USA; 14-18 Mar. 2004. pp. 307-318. 2004
[67] P. G. Saffman and G. I. Taylor, Proc. R. Soc. London A 254, 312, 1958
[68] A. Karma and P. Pelce, Phys. Rev. A 41, 6741, 1990
[69] A. La Magna, P. Alippi and V. Privitera, J. Appl. Phys. 95, 4806, 2004
[70] P. Zhao, M. Venere, J.C. Heinrich, and D.R. Poirier, J. Compt. Phys. 188, 434, 2003
[71] P. Zhao, J.C. Heinrich, and D.R. Poirier, Int. J. Numer. Meth. Engng. 61, 928, 2004