The Diffusion and Solid-Liquid Phase Transformation in Directional Solidification of Alloy: A Quantitative Phase Field Characterization and Real-Time Observation

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Directional solidification is a paradigm process to gain the desired microstructure via certain applied solidification parameters. A thorough understanding of the diffusion-limited solid-liquid interface morphology evolution from initial transient to steady state is of uppermost importance to optimize the solidification processes. The rapid development of quantitative phase-field model provides a feasible computational tool to explore the underlying physics of the morphological transition at different stages. On basis of the diffusion-limited quantitative phase-field simulations using adaptive finite element method, the directional solidification of Al-4wt.%Cu alloy is characterized and both the solid interface propagation speed and solute profile are analyzed. The simulations are then compared with the in situ and real-time observation by means of synchrotron radiation x-ray radiography image. Good agreements are obtained between simulations and experimental data. Detailed mechanism that controls the morphological instability and transition are then addressed.

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Diffusion Foundations (Volume 15)

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97-127

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February 2018

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[1] J. Lipton, W. Kurz, R. Trivedi. Rapid dendrite growth in undercooled alloys, Acta Metall. 35 (1987) 957-964.

DOI: 10.1016/0001-6160(87)90174-x

Google Scholar

[2] J. Lipton, M.E. Glicksman, W. Kurz. Dendritic growth into undercooled alloy metals, Mater. Sci. Eng. 65 (1984) 57-63.

DOI: 10.1016/0025-5416(84)90199-x

Google Scholar

[3] R. Trivedi, W. Kurz. Solidification microstructures: A conceptual approach, Acta Metall. Mater. 42 (1994) 15-23.

Google Scholar

[4] B. Billia, R. Trivedi. Pattern formation in crystal growth. In: D.T.J. Hurle, editor. Handbook of crystal growth. Amsterdam: Elsevier, (1993).

Google Scholar

[5] W.J. Boettinger, S.R. Coriell, R. Trivedi. Rapid solidification processing: Principles and technologies. In: R. Mehrabian, P.A. Parrish, editors. Baton Rouge (LA): Claitor's Publishing Division, 1988. pp.13-25.

Google Scholar

[6] R. Trivedi, W. Kurz. Morphological stability of a planar interface under rapid solidification conditions, Acta Metall. 34 (1986) 1663-1670.

DOI: 10.1016/0001-6160(86)90112-4

Google Scholar

[7] R. Trivedi. Interdendritic spacing: Part ii. A comparison of theory and experiment, Metall. Trans. A 15 (1986) 977-982.

DOI: 10.1007/bf02644689

Google Scholar

[8] M. Glicksman, R. Schaefer, J. Ayers. Dendritic growth-a test of theory, Metall. Mater. Trans. A 7 (1976) 1747-1759.

Google Scholar

[9] W. Losert, D.A. Stillman, H.Z. Cummins, P. Kopczynski, W.J. Rappel, A. Karma. Selection of doublet cellular patterns in directional solidification through spatially periodic perturbations, Phys. Rev. E 58 (1998) 7492-7506.

DOI: 10.1103/physreve.58.7492

Google Scholar

[10] W. Kurz, D.J. Fisher, editors. Fundamentals of solidification. Switzerland: Trans Tech Publications Ltd, (1998).

Google Scholar

[11] W.W. Mullins, R.F. Sekerka. Stability of a planar interface during solidification of a dilute binary alloy, J. Appl. Phys. 35 (1964) 444-451.

DOI: 10.1063/1.1713333

Google Scholar

[12] R. Kobayashi. Modeling and numerical simulations of dendritic crystal growth, Physica D 63 (1993) 410-423.

DOI: 10.1016/0167-2789(93)90120-p

Google Scholar

[13] A.A. Wheeler, W.J. Boettinger, G.B. McFadden. Phase-field model for isothermal phase transitions in binary alloys, Phys. Rev. A 45 (1992) 7424-7439.

DOI: 10.1103/physreva.45.7424

Google Scholar

[14] J.A. Warren, W.J. Boettinger. Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method, Acta Metall. Mater. 43 (1995) 689-703.

DOI: 10.1016/0956-7151(94)00285-p

Google Scholar

[15] A. Karma, W.J. Rappel. Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys. Rev. E 57 (1998) 4323-4349.

DOI: 10.1103/physreve.57.4323

Google Scholar

[16] A. Karma, W.-J. Rappel. Phase-field model of dendritic sidebranching with thermal noise, Phys. Rev. E 60 (1999) 3614-3625.

DOI: 10.1103/physreve.60.3614

Google Scholar

[17] A. Karma. Phase-field formulation for quantitative modeling of alloy solidification, Phys. Rev. Lett. 87 (2001) 115701.

DOI: 10.1103/physrevlett.87.115701

Google Scholar

[18] I. Loginova, G. Amberg, J. Ågren. Phase-field simulations of non-isothermal binary alloy solidification, Acta Mater. 49 (2001) 573-581.

DOI: 10.1016/s1359-6454(00)00360-8

Google Scholar

[19] B. Echebarria, R. Folch, A. Karma, M. Plapp. Quantitative phase-field model of alloy solidification, Phys. Rev. E 70 (2004) 061604.

DOI: 10.1103/physreve.70.061604

Google Scholar

[20] W.J. Boettinger, J.A. Warren, C. Beckermann, A. Karma. Phase-field simulation of solidifcation, Annu. Rev. Mater. Res. 32 (2002) 163-194.

DOI: 10.1146/annurev.matsci.32.101901.155803

Google Scholar

[21] D. Tourret, Y. Song, A.J. Clarke, A. Karma. Grain growth competition during thin-sample directional solidification of dendritic microstructures: A phase-field study, Acta Mater. 122 (2017) 220-235.

DOI: 10.1016/j.actamat.2016.09.055

Google Scholar

[22] Y. Chen, A.-A. Bogno, N.M. Xiao, B. Billia, X.H. Kang, H. Nguyen-Thi, X.H. Luo, D.Z. Li. Quantitatively comparing phase-field modeling with direct real time observation by synchrotron X-ray radiography of the initial transient during directional solidification of an Al–Cu alloy, Acta Mater. 60 (2012).

DOI: 10.1016/j.actamat.2011.09.028

Google Scholar

[23] Y. Chen, A.-A. Bogno, B. Billia, X.H. Kang, H. Nguyen-Thi, D.Z. Li, X.H. Luo, J.-M. Debierre. Phase-field modeling of the initial transient in directional solidification of Al-4wt%Cu alloy, ISIJ Int. 50 (2010) 1895-(1900).

DOI: 10.2355/isijinternational.50.1895

Google Scholar

[24] J.A. Warren, R. Kobayashi, A.E. Lobkovsky, W. Craig Carter. Extending phase field models of solidification to polycrystalline materials, Acta Mater. 51 (2003) 6035-6058.

DOI: 10.1016/s1359-6454(03)00388-4

Google Scholar

[25] L. Granasy, T. Pusztai, T. Borzsonyi, J.A. Warren, J.F. Douglas. A general mechanism of polycrystalline growth, Nat. Mater. 3 (2004) 645-650.

Google Scholar

[26] L. Granasy, T. Pusztai, J.A. Warren. Modelling polycrystalline solidification using phase field theory, J. Phys.: Condens. Mat. 16 (2004) R1205-R1235.

DOI: 10.1088/0953-8984/16/41/r01

Google Scholar

[27] P. Tamás, T. György, I.T. Gyula, K. László, B. Gurvinder, F. Zhungyun, G. László. Phase-field approach to polycrystalline solidification including heterogeneous and homogeneous nucleation, J. Phys.: Condens. Mat. 20 (2008) 404205.

DOI: 10.1088/0953-8984/20/40/404205

Google Scholar

[28] N. Moelans. A quantitative and thermodynamically consistent phase-field interpolation function for multi-phase systems, Acta Mater. 59 (2011) 1077-1086.

DOI: 10.1016/j.actamat.2010.10.038

Google Scholar

[29] A. Karma. Phase-field model of eutectic growth, Phys. Rev. E 49 (1994) 2245-2250.

DOI: 10.1103/physreve.49.2245

Google Scholar

[30] B. Nestler, A.A. Wheeler. A multi-phase-field model of eutectic and peritectic alloys: Numerical simulation of growth structures, Physica D 138 (2000) 114-133.

DOI: 10.1016/s0167-2789(99)00184-0

Google Scholar

[31] A. Karma, W.J. Rappel. Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics, Phys. Rev. E 53 (1996) R3017-R3020.

DOI: 10.1103/physreve.53.r3017

Google Scholar

[32] R. Li. On multi-mesh h-adaptive methods, J. Sci. Comput. 24 (2005) 321-341.

Google Scholar

[33] A. Schmidt. Computation of three dimensional dendrites with finite elements, J. Comput. Phys. 125 (1996) 293-312.

Google Scholar

[34] R.J. Braun, B.T. Murray. Adaptive phase-field computations of dendritic crystal growth, J. Cryst. Growth 174 (1997) 41-53.

DOI: 10.1016/s0022-0248(96)01059-7

Google Scholar

[35] N. Provatas, N. Goldenfeld, J. Dantzig. Efficient computation of dendritic microstructures using adaptive mesh refinement, Phys. Rev. Lett. 80 (1998) 3308-3311.

DOI: 10.1103/physrevlett.80.3308

Google Scholar

[36] N. Provatas, N. Goldenfeld, J. Dantzig. Adaptive mesh refinement computation of solidification microstructures using dynamic data structures, J. Comput. Phys. 148 (1999) 265-290.

DOI: 10.1006/jcph.1998.6122

Google Scholar

[37] A. Karma, W.-J. Rappel. Numerical simulation of three-dimensional dendritic growth, Phys. Rev. Lett. 77 (1996) 4050-4053.

DOI: 10.1103/physrevlett.77.4050

Google Scholar

[38] J.-H. Jeong, N. Goldenfeld, J.A. Dantzig. Phase field model for three-dimensional dendritic growth with fluid flow, Phys. Rev. E 64 (2001) 041602.

DOI: 10.1103/physreve.64.041602

Google Scholar

[39] C.W. Lan, C.J. Shih. Phase field simulation of non-isothermal free dendritic growth of a binary alloy in a forced flow, J. Cryst. Growth. 264 (2004) 472-482.

DOI: 10.1016/j.jcrysgro.2004.01.016

Google Scholar

[40] C.W. Lan, Y.C. Chang, C.J. Shih. Adaptive phase field simulation of non-isothermal free dendritic growth of a binary alloy, Acta Mater. 51 (2003) 1857-1869.

DOI: 10.1016/s1359-6454(02)00582-7

Google Scholar

[41] S.C. Huang, M.E. Glicksman. Overview 12: Fundamentals of dendritic solidification--i. Steady-state tip growth, Acta Metall. 29 (1981) 701-715.

DOI: 10.1016/0001-6160(81)90115-2

Google Scholar

[42] S.C. Huang, M.E. Glicksman. Overview 12: Fundamentals of dendritic solidification--ii development of sidebranch structure, Acta Metall. 29 (1981) 717-734.

DOI: 10.1016/0001-6160(81)90116-4

Google Scholar

[43] A. Bogno, H. Nguyen-Thi, G. Reinhart, B. Billia, J. Baruchel. Growth and interaction of dendritic equiaxed grains: In situ characterization by synchrotron X-ray radiography, Acta Mater. 61 (2013) 1303-1315.

DOI: 10.1016/j.actamat.2012.11.008

Google Scholar

[44] Y. Chen, D.Z. Li, B. Billia, H. Nguyen-Thi, X.B. Qi, N.M. Xiao. Quantitative phase-field simulation of dendritic equiaxed growth and comparison with in situ observation on Al – 4 wt.% Cu alloy by means of synchrotron X-ray radiography, ISIJ Int. 54 (2014).

DOI: 10.2355/isijinternational.54.445

Google Scholar

[45] A. Prasad, E. Liotti, S.D. McDonald, K. Nogita, H. Yasuda, P.S. Grant, D.H. StJohn. Real-time synchrotron X-ray observations of equiaxed solidification of aluminium alloys and implications for modelling, IOP Conf. Ser.: Mater. Sci. Eng. 84 (2015).

DOI: 10.1088/1757-899x/84/1/012014

Google Scholar

[46] E. Liotti, A. Lui, R. Vincent, S. Kumar, Z. Guo, T. Connolley, I.P. Dolbnya, M. Hart, L. Arnberg, R.H. Mathiesen, P.S. Grant. A synchrotron X-ray radiography study of dendrite fragmentation induced by a pulsed electromagnetic field in an Al–15Cu alloy, Acta Mater. 70 (2014).

DOI: 10.1016/j.actamat.2014.02.024

Google Scholar

[47] G. Reinhart, A.G. Ch, N. Mangelinck-Noël, H. Nguyen-Thi, B. Billia, J. Baruchel. Direct simulation of a directional solidification experiment observed in situ and real-time using X-ray imaging, IOP Conf. Ser.: Mater. Sci. Eng. 33 (2012) 012077.

DOI: 10.1088/1757-899x/33/1/012077

Google Scholar

[48] A. Bogno, H. Nguyen-Thi, B. Billia, G. Reinhart, N. Mangelinck-Noël, N. Bergeon, T. Schenk, J. Baruchel. In situ and real-time analysis of the growth and interaction of equiaxed grains by synchrotron x- ray radiography, IOP Conf. Ser.: Mater. Sci. Eng. 27 (2012).

DOI: 10.1088/1757-899x/27/1/012089

Google Scholar

[49] A.B. Phillion, R.W. Hamilton, D. Fuloria, A.C.L. Leung, P. Rockett, T. Connolley, P.D. Lee. In situ X-ray observation of semi-solid deformation and failure in Al–Cu alloys, Acta Mater. 59 (2011) 1436-1444.

DOI: 10.1016/j.actamat.2010.11.005

Google Scholar

[50] A. Bogno, G. Reinhart, A. Buffet, H. Nguyen Thi, B. Billia, T. Schenk, N. Mangelinck-Noël, N. Bergeon, J. Baruchel. In situ analysis of the influence of convection during the initial transient of planar solidification, J. Cryst. Growth. 318 (2011).

DOI: 10.1016/j.jcrysgro.2010.10.184

Google Scholar

[51] A. Bogno, H. Nguyen-Thi, A. Buffet, G. Reinhart, B. Billia, N. Mangelinck-Noël, N. Bergeon, J. Baruchel, T. Schenk. Analysis by synchrotron X-ray radiography of convection effects on the dynamic evolution of the solid-liquid interface and on solute distribution during the initial transient of solidification, Acta Mater. 59 (2011).

DOI: 10.1016/j.actamat.2011.03.059

Google Scholar

[52] G. Reinhart, N. Mangelinck-Noël, H. Nguyen-Thi, T. Schenk, J. Gastaldi, B. Billia, P. Pino, J. Härtwig, J. Baruchel. Investigation of columnar–equiaxed transition and equiaxed growth of aluminium based alloys by X-ray radiography, Mater. Sci. Eng. A 413–414 (2005).

DOI: 10.1016/j.msea.2005.08.197

Google Scholar

[53] R.H. Mathiesen, L. Arnberg. X-ray radiography observations of columnar dendritic growth and constitutional undercooling in an Al–30wt%Cu alloy, Acta Mater. 53 (2005) 947-956.

DOI: 10.1016/j.actamat.2004.10.050

Google Scholar

[54] C.Y. Wang, C. Beckermann. Equiaxed dendritic solidification with convection ii: Numerical simulations for an Al-4wt% Cu alloy Metall. Mater. Trans. A 27 (1996) 2765-2783.

DOI: 10.1007/bf02652370

Google Scholar

[55] D.J. Browne, M. Rebow. On the dendritic tip stability parameter for aluminium alloy solidification, Scr. Mater. 56 (2007) 481-484.

DOI: 10.1016/j.scriptamat.2006.11.025

Google Scholar

[56] S. Liu, R.E. Napolitano, R. Trivedi. Measurement of anisotropy of crystal-melt interfacial energy for a binary Al-Cu alloy, Acta Mater. 49 (2001) 4271-4276.

DOI: 10.1016/s1359-6454(01)00306-8

Google Scholar

[57] M. Amoorezaei, S. Gurevich, N. Provatas. Spacing characterization in Al–Cu alloys directionally solidified under transient growth conditions, Acta Mater. 58 (2010) 6115-6124.

DOI: 10.1016/j.actamat.2010.07.029

Google Scholar

[58] R. Li, W.B. Liu. http://dsec.pku.edu.cn/~rli, last accessing date March 01st, (2015).

Google Scholar

[59] B. Billia, H. Nguyen-Thi, N. Mangelinck-Noel, N. Bergeon, H. Jung, G. Reinhart, A. Bogno, A. Buffet, J. Hartwig, J. Baruchel, T. Schenk. In situ synchrotron X-ray characterization of microstructure formation in solidification processing of al-based metallic alloys, ISIJ Int. 50 (2010).

DOI: 10.2355/isijinternational.50.1929

Google Scholar

[60] A. Buffet, H. Nguyen-Thi, A. Bogno, T. Schenk, N. Mangelinck-Noël, G. Reinhart, N. Bergeon, B. Billia, J. Baruchel. Measurement of solute profiles by means of synchrotron X-ray radiography during directional solidification of Al-4 wt% Cu alloys, Mater. Sci. Forum 649 (2010).

DOI: 10.4028/www.scientific.net/msf.649.331

Google Scholar

[61] J.A. Warren, J.S. Langer. Prediction of dendritic spacings in a directional-solidification experiment, Phys. Rev. E 47 (1993) 2702-2712.

DOI: 10.1103/physreve.47.2702

Google Scholar

[62] W.A. Tiller, K.A. Jackson, J.W. Rutter, B. Chalmers. The redistribution of solute atoms during the solidification of metals, Acta Metall. 1 (1953) 428-437.

DOI: 10.1016/0001-6160(53)90126-6

Google Scholar

[63] B. Caroli, C. Caroli, B. Roulet. On the emergence of one-dimensional front instabilities in directional solidification and fusion of binary mixtures, J. Phys. France 43 (1982) 1767-1780.

DOI: 10.1051/jphys:0198200430120176700

Google Scholar

[64] Y. Chen, H. Nguyen-Thi, D.Z. Li, A.-A. Bogno, B. Billia, N.M. Xiao. Influence of natural convection on microstructure evolution during the initial solidification transient: Comparison of phase-field modeling with in situ synchrotron X-ray monitoring data, IOP Conf. Ser.: Mater. Sci. Eng. 33 (2012).

DOI: 10.1088/1757-899x/33/1/012102

Google Scholar

[65] R. Trivedi, H. Miyahara, P. Mazumder, E. Simsek, S.N. Tewari. Directional solidification microstructures in diffusive and convective regimes, J. Cryst. Growth. 222 (2001) 365-379.

DOI: 10.1016/s0022-0248(00)00761-2

Google Scholar

[66] N. Provatas, Q. Wang, M. Haataja, M. Grant. Seaweed to dendrite transition in directional solidification, Phys. Rev. Lett. 91 (2003) 155502.

DOI: 10.1103/physrevlett.91.155502

Google Scholar

[67] M. Qu, L. Liu, F.-t. Tang, W.-g. Zhang, T.-w. Huang, H.-z. Fu. Dendrite tip splitting and primary spacing adjustment mechanism of Al-Cu alloy during directional solidification, Chin. J. Nonferr. Metals 18 (2008) 1813-1818.

Google Scholar

[68] E. Brener, H. Müller-Krumbhaar, D. Temkin, T. Abel. Morphology diagram of possible structures in diffusional growth, Physica A 249 (1998) 73-81.

DOI: 10.1016/s0378-4371(97)00433-0

Google Scholar

[69] E. Brener, H. Muller-Krumbhaar, D. Temkin. Structure formation and the morphology diagram of possible structures in two-dimensional diffusional growth, Phys. Rev. E 54 (1996) 2714-2722.

DOI: 10.1103/physreve.54.2714

Google Scholar

[70] S. Akamatsu, G. Faivre, T. Ihle. Symmetry-broken double fingers and seaweed patterns in thin-film directional solidification of a nonfaceted cubic crystal, Phys. Rev. E 51 (1995) 4751-4773.

DOI: 10.1103/physreve.51.4751

Google Scholar

[71] H.M. Singer, I. Singer-Loginova, J.H. Bilgram, G. Amberg. Morphology diagram of thermal dendritic solidification by means of phase-field models in two and three dimensions, J. Cryst. Growth. 296 (2006) 58-68.

DOI: 10.1016/j.jcrysgro.2006.07.033

Google Scholar

[72] E. Brener, H. Mullerkrumbhaar, D. Temkin. Kinetic phase diagram and scaling relations for stationary diffusional growth, Europhys. Lett. 17 (1992) 535-540.

DOI: 10.1209/0295-5075/17/6/010

Google Scholar

[73] E. Brener, H. Müller-Krumbhaar, D. Temkin, T. Abel. Morphology diagram of possible structures in diffusional growth, Physica A249 (1998) 73-81.

DOI: 10.1016/s0378-4371(97)00433-0

Google Scholar

[74] T. Ihle, H. Müller-Krumbhaar. Diffusion-limited fractal growth morphology in thermodynamical two-phase systems, Phys. Rev. Lett. 70 (1993) 3083-3086.

DOI: 10.1103/physrevlett.70.3083

Google Scholar

[75] T. Ihle, H. Müller-Krumbhaar. Fractal and compact growth morphologies in phase transitions with diffusion transport, Phys. Rev. E 49 (1994) 2972-2991.

DOI: 10.1103/physreve.49.2972

Google Scholar

[76] B. Utter, R. Ragnarsson, E. Bodenschatz. Alternating tip splitting in directional solidification, Phys. Rev. Lett. 86 (2001) 4604-4607.

DOI: 10.1103/physrevlett.86.4604

Google Scholar

[77] B. Utter, E. Bodenschatz. Dynamics of low anisotropy morphologies in directional solidification, Phys. Rev. E 66 (2002) 051604.

DOI: 10.1103/physreve.66.051604

Google Scholar

[78] A. Pocheau, J. Deschamps, M. Georgelin. Dendrite growth directions and morphology in the directional solidification of anisotropic materials, JOM 59 (2007) 71-76.

DOI: 10.1007/s11837-007-0093-5

Google Scholar

[79] A. Buffet, H.N. Thi, A. Bogno, T. Schenk, N. Mangelinck-Noel, G. Reinhart, N. Bergeon, B. Billia, J. Baruchel. Measurement of solute profiles by means of synchrotron X-ray radiography during directional solidification of Al-4 wt% Cu alloys, Mater. Sci. Forum 649 (2010).

DOI: 10.4028/www.scientific.net/msf.649.331

Google Scholar