Tilted Dendritic Growth Dynamics and Dendrite to Degenerate Seaweed Transition in Directional Solidification: Insights from Phase-Field Simulations

Abstract:

In this paper, we review our results from phase field simulations of tilted dendritic growth dynamics and dendrite to seaweed transition in directional solidification of a dilute alloy. We focus on growth direction selection, stability range and primary spacing selection, and degenerate seaweed-to-tilted dendrite transition in directional solidification of non-axially orientated crystals. For growth direction selection, the DGP law (Phys. Rev. E, 78 (2008) 011605) was modified through take the anisotropic strength and pulling velocity into account. We confirm that the DGP law is only validated in lower pulling velocity. For the stability range and primary spacing selection, we found that the lower limit of primary spacing is irrelative to the misorientation angle but the upper limit is nonlinear with respect to the misorientation angle. Moreover, predicted results confirm that the power law relationship with the orientation correction by Gandin et al. (Metall. Mater. Trans. A. 27A (1996) 2727-2739) should be a universal scaling law for primary spacing selection. For the seaweed-to-dendrite transition, we found that the tip-splitting instability in degenerate seaweed growth dynamics is related to the M-S instability dynamics, and this transition originates from the compromise in competition between two dominant mechanisms, i.e., the macroscopic thermal field and the microscopic interfacial energy anisotropy.

• Periodical: Diffusion Foundations (Volume 15)
• Pages: 128-153
• DOI: https://doi.org/10.4028/www.scientific.net/DF.15.128
• Online since: February 2018
• Authors: Hui Xing*, Xiang Lei Dong, Jian Yuan Wang, Ke Xin Jin
• Keywords: Crystal Growth, Directional Solidification, Phase Field Simulation

References

[1] J. S. Langer, Instabilities and pattern formation in crystal growth, Rev. of Mod. Phys. 52 (1980) 1-28.

[2] J. S. Langer, Dendrites, viscous fingers, and the theory of pattern formation, Sci. 243 (1989) 1150.

DOI: 10.1126/science.243.4895.1150

[3] P. Pelcé, Dynamics of Curved Fronts, Academic Press, Boston,(1988).

[4] E. Ben-Jacob, and P. Garik, The formation of patterns in non-equilibrium growth, Nat. 343 (1990) 523-530.

DOI: 10.1038/343523a0

[5] W. Kurz, D. Fisher, Fundamentals of Solidification, Trans Tech Publications, (1998).

[6] J. Dantzig, M. Rappaz, Solidification, EPFL Press, Lausanne, Swizerland, (2009).

[7] V. Fleury, J. F. Gouyet, and M. Léonetti, Branching in Nature. Springer, Berlin Heidelberg, (2001).

[8] A. Karma, D. Tourret, Atomistic to continuum modeling of solidification microstructures, Curr. Opin. Solid State Mater. Sci., 20 (2016) 25-36.

DOI: 10.1016/j.cossms.2015.09.001

[9] Y. Couder, J. Maurer, R. González-Cinca, et al. Side-branch growth in two-dimensional dendrites. I. Experiments. Phys. Rev. E, 71 (2005) 031602.

DOI: 10.1103/physreve.71.031602

[10] A. J. Melendez, C. Beckermann, Measurements of dendrite tip growth and sidebranching in succinonitrile–acetone alloys, J Cryst. Growth, 340 (2012) 175-189.

DOI: 10.1016/j.jcrysgro.2011.12.010

[10] K.G. Libbrecht, Physical Dynamics of Ice Crystal Growth, Annu. Rev. Mater. Res. 47 (2017) 271-295.

DOI: 10.1146/annurev-matsci-070616-124135

[11] G. Demange, H. Zapolsky, R. Patte, et al., A phase field model for snow crystal growth in three dimensions, Npj Comp. Mater. 3 (2017) 15.

DOI: 10.1038/s41524-017-0015-1

[12] Information on http://www.snowcrystals.com.

[13] T. Yang, Y. Han, J. Li. Manipulating silver dendritic structures via diffusion and reaction, Chem. Engin. Sci. 138 (2015) 457-464.

DOI: 10.1016/j.ces.2015.08.017

[14] H. Wang, Y. Han, J. Li. Dominant Role of Compromise between Diffusion and Reaction in the Formation of Snow-Shaped Vaterite. Cryst. Growth Des., 13 (2013) 1820-1825.

DOI: 10.1021/cg301241s

[15] H. Thomé, M. Rabaud, V. Hakim, et al. The Saffman–Taylor instability: From the linear to the circular geometry, Phys. Fluids A, 1 (1989) 224-240.

DOI: 10.1063/1.857493

[16] G.P. Ivantsov. Temperature field around spheroidal, cylindrical, and acicular crystal growing in a supercooled melt, Dokl. Akad.Nauk. SSSR 58 (1947)567.

DOI: 10.1016/b978-0-08-092523-3.50026-5

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

[18] Y. Saito, Statistical Physics of Crystal Growth, World Scientific, New Jersey, (1996).

[19] E. Brener, H. Müllerkrumbhaar, 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

[20] H.M. Singer, I. Singer-Loginova, J.H. Bilgram, et al. 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

[21] T. Ihle. Competition between kinetic and surface tension anisotropy in dendritic growth. Eur. Phys. J. B 16 (2000) 337–344.

DOI: 10.1007/pl00011060

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

[23] M. Asta, C. Beckermann, A. Karma, et al. Solidification microstructures and solid-state parallels: Recent developments, future directions. Acta Mater., 57 (2009) 941-971.

DOI: 10.1016/j.actamat.2008.10.020

[24] A. Barbieri and J. Langer. Predictions of dendritic growth rates in the linearized solvability theory. Phys. Rev. A 39 (1989) 5314–5325.

DOI: 10.1103/physreva.39.5314

[25] A. M. Ben and E. Brener. Theory of pattern selection in three-dimensional nonaxisymmetric dendritic growth. Phys. Rev. Lett. 71 (1993) 589.

DOI: 10.1103/physrevlett.71.589

[26] M.H. Burden, J.D. Hunt. Cellular and dendritic growth. I. J. Cryst. Growth 22 (1974) 99-108.

[27] W. Kurz, D.J. Fisher. Dendrite growth at the limit of stability: tip radius and spacing. Acta Metall. 29 (1981) 11-20.

DOI: 10.1016/0001-6160(81)90082-1

[28] M. Greenwood, M. Haataja, and N. Provatas. Crossover Scaling of Wavelength Selection in Directional Solidification of Binary Alloys. Phys. Rev. Lett. 93 (2004) 246101.

DOI: 10.1103/physrevlett.93.246101

[29] J. Warren and J. Langer. Stability of dendritic arrays. Phys. Rev. A 42 (1990) 3518–3525.

[30] W. Huang, X. Geng, Y. Zhou. Primary spacing selection of constrained dendritic growth. J Cryst. Growth 134 (1993) 105–115.

[31] W. Losert, B. Shi, H. Cummins, et al. Spatial Period-Doubling Instability of Dendritic Arrays in Directional Solidification. Phys. Rev. Lett. 77 (1996) 889.

DOI: 10.1103/physrevlett.77.889

[32] W. Losert, B. Q. Shi, H. Z. Cummins. Evolution of dendritic patterns during alloy solidification: onset of the initial instability. Proc. Natl. Acad. Sci. USA, 95 (1998) 431.

DOI: 10.1073/pnas.95.2.431

[33] A. Borisov, S. Fedorov, V. Maslov. Growth of succinonitrile dendrites in different crystallographicdirections. J Cryst. Growth, 112 (1991) 463-466.

DOI: 10.1016/0022-0248(91)90324-x

[34] R. Trivedi, V. Seetharaman, M. Eshelman. The effects of interface kinetic anisotropy on the growth direction ofcellular microstructures. Metall. Trans. A. 22A (1991) 585-593.

DOI: 10.1007/bf02656826

[35] T. Okada, Y. Saito. Simulation of unidirectional solidification with a tilted crystalline axis. Phys.Rev.R, 54 (1996) 650-655.

DOI: 10.1103/physreve.54.650

[36] S. Akamatsu, T. Ihle. Similarity law for the tilt angle of dendrites in directional solidification of non-axially-oriented crystals. Phys. Rev. E, 56 (1997) 4479-4485.

DOI: 10.1103/physreve.56.4479

[37] S. Akamatsu, G. Faivre. Anisotropy-driven dynamics of cellular fronts in directional solidification in thin samples. Phys. Rev. E, 58 (1998) 3302-3315.

DOI: 10.1103/physreve.58.3302

[38] G. L. Ding, S. N. Tewari. Dendritic morphologies of directionally solidified single crystals along different crystallographic orientations. J. Cryst. Growth, 236 (2002) 420-428.

DOI: 10.1016/s0022-0248(01)01787-0

[39] D. Walton, B. Chalmers. The origin of the preferred orientation in the columnar zone of ingots. Trans. Metall. Soc. AIME, 215 (1959) 447–457.

[40] N. D'Souza, M. G. Ardakani, A. Wagner, et al. Morphological aspects of competitive grain growth during directional solidification of a nickel-base superalloy, CMSX4. J. Mater. Sci., 37 (2002) 481-487.

[41] Y.Z. Zhou, A. Volek, N.R. Green NR. Mechanism of competitive grain growth in directional solidification of a nickel-base superalloy. Acta Mater. 56 (2008) 2631-2637.

DOI: 10.1016/j.actamat.2008.02.022

[42] J. Li, Z. Wang, Y. Wang, J. Wang. Phase-field study of competitive dendritic growth of converging grains duringdirectional solidification. Acta Mater., 60 (2012) 1478-1493.

DOI: 10.1016/j.actamat.2011.11.037

[43] T. Takaki, M. Ohno, T. Shimokawabe, et al. Two-dimensional phase-field simulations of dendrite competitive growth during the directional solidification of a binary alloy bicrystal. Acta Mater. 81 (2014) 272–283.

DOI: 10.1016/j.actamat.2014.08.035

[44] D. Tourret, Y. Song, A. 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

[45] J. Deschamps, M. Georgelin, A. Pocheau. Growth directions of microstructures in directional solidification of crystalline materials. Phys. Rev. E, 78 (2008) 011605.

DOI: 10.1103/physreve.78.011605

[46] J. Ghmadh, J.-M. Debierre, J. Deschamps, M. Georgelin, R. Guérin, A. Pocheau. Directional solidification of inclined structures in thin samples. Acta Mater. 74 (2014) 255-267.

DOI: 10.1016/j.actamat.2014.04.023

[47] Z. Wang, J. Li, J. Wang. Predicting growth direction of tilted dendritic arrays during directional solidification. J. Crys. Growth 328 (2011) 108-113.

DOI: 10.1016/j.jcrysgro.2011.06.021

[48] C. Rios, R. Caram. Primary dendrite spacing as a function of directional solidification parameters in an Al-Si-Cu alloy. J. Crys. Growth 174 (1997) 65-69.

DOI: 10.1016/s0022-0248(96)01061-5

[49] D. Ma, P. Sahm. Primary spacing in directional solidification. Metall. Mater. Trans. A, 29 (1998) 1113-1119.

DOI: 10.1007/s11661-998-1020-2

[50] D. Ma. Modeling of primary spacing selection in dendrite arrays during directional solidification. Metall. Mater. Trans. B 33 (2002) 223-233.

DOI: 10.1007/s11663-002-0007-4

[51] D. Ma. Response of primary dendrite spacing to varying temperature gradient during directional solidification. Metall. Mater. Trans. B 35 (2004) 735-742.

DOI: 10.1007/s11663-004-0013-9

[52] S. Tewari, Y. Weng, G. Ding, et al. Cellular array morphology during directional solidification. Metall. Mater. Trans. A, 33 (2002) 1229-1243.

DOI: 10.1007/s11661-002-0224-0

[53] H. Chen, Y. Chen, X. Wu, et al. History dependence of primary dendrite spacing during directional solidification of binary metallic alloys and interdendritic convection. J. Crys. Growth, 253 (2003) 413-423.

DOI: 10.1016/s0022-0248(03)00945-x

[54] J. Spinelli, K. Cruz, M. Canté, et al. Primary dendrite arm spacing during transient directional solidification of Al alloys with low redistribution coefficients. Philos. Mag. Lett. 89(2009) 779-786.

DOI: 10.1080/09500830903292338

[55] Z. Min, J. Shen, Z. Feng, et al. Effects of melt flow on the primary dendrite spacing of Pb–Sn binary alloy during directional solidification. J. Crys. Growth 320 ( 2011) 41-45.

DOI: 10.1016/j.jcrysgro.2011.01.013

[56] Z. Wang, J. Wang, J. Li, et al. Quantitative investigation of cellular growth in directional solidification by phase-field simulation. Phys. Rev. E 84 (2011) 041604.

DOI: 10.1103/physreve.84.041604

[57] Z. Wang, J. Li, J. Wang, Y. Zhou. Phase field modeling the selection mechanism of primary dendritic spacing in directional solidification. Acta Mater. 60 (2012)1957-(1964).

DOI: 10.1016/j.actamat.2011.12.029

[58] Y. Shibuta, M. Ohno, T. Takaki. Solidification in a Supercomputer: From Crystal Nuclei to Dendrite Assemblages. JOM 67 (2015) 1793-1804.

DOI: 10.1007/s11837-015-1452-2

[59] T. Takaki, S. Sakane, M. Ohno M, et al. Primary arm array during directional solidification of a single-crystal binary alloy: Large-scale phase-field study. Acta Mater. 118 (2016) 230-243.

DOI: 10.1016/j.actamat.2016.07.049

[60] A. Clarke, D. Tourret, Y. Song, et al. Microstructure selection in thin-sample directional solidification of an Al-Cu alloy: in situ X-ray imaging and phase-field simulations. Acta Mater. 129 (2017) 203-216.

DOI: 10.1016/j.actamat.2017.02.047

[61] J.D. Hunt, Solidification and Casting of Metals, The Metals Society, London, (1979).

[62] W. Kurz, D. J. Fisher. Dendrite growth at the limit of stability: tip radius and spacing. Acta Metall. 29 (1981)11-20.

DOI: 10.1016/0001-6160(81)90082-1

[63] R. N.Grugel, Y. Zhou. Primary dendritie spacing and the effect of off-axis heat flow, Metall. Trans. A. 20A (1989) 969-973.

DOI: 10.1007/bf02651662

[64] Ch.-A. Gandin, M. Eshelman and R. Trivedi. Orientation dependence of primary spacing. Metall. Mater. Trans. A. 27A (1996) 2727-2739.

DOI: 10.1007/bf02652367

[65] D. Tourret, A. Karma. Growth competition of columnar dendritic grains: A phase-field study. Acta Mater. 82 (2015) 64-83.

DOI: 10.1016/j.actamat.2014.08.049

[66] M. A. Chopra, M.E. Glicksman, N.B. Singh. Measurement of the diffusion coefficient of acetone in succinonitrile at its melting point. J Cryst. Growth. 92 (1988) 543-546.

DOI: 10.1016/0022-0248(88)90039-5

[67] U. Bisang, J.H. Bilgram. Shape of the tip and the formation of sidebranches of xenon dendrites. Phys. Rev. E. 75 (1996) 5309-5326.

DOI: 10.1103/physreve.54.5309

[68] Q. Li, C. Beckermann. Scaling behavior of three-dimensional dendrites. Phys. Rev. E 57 (1998) 3176-3188.

DOI: 10.1103/physreve.57.3176

[69] D. P. Corrigan, M.B. Koss, J. C. Lacombe, et al. Experimental measurements of sidebranching in thermal dendrites under terrestrial-gravity and microgravity conditions. Phys. Rev. E 60 (1999) 7217.

DOI: 10.1103/physreve.60.7217

[70] A. J. Melendez, C. Beckermann. Measurements of dendrite tip growth and sidebranching in succinonitrile–acetone alloys. J Cryst. Growth 340 (2012) 175-189.

DOI: 10.1016/j.jcrysgro.2011.12.010

[71] E. Brener and D. Temkin. Noise-induced sidebranching in the three-dimensional nonaxisymmetric dendritic growth. Phys. Rev. E 51(1995) 351.

DOI: 10.1103/physreve.51.351

[72] Y. Couder, J. Maurer, R. González-Cinca, et al. Side-branch growth in two-dimensional dendrites. I. Experiments. Phys. Rev. E 71 (2005) 031602.

DOI: 10.1103/physreve.71.031602

[73] B. Echebarria, A. Karma, S. Gurevich. Onset of sidebranching in directional solidification. Phys. Rev. E 81(2010):275-282.

DOI: 10.1103/physreve.81.021608

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

DOI: 10.1103/physreve.66.051604

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

DOI: 10.1103/physrevlett.86.4604

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

DOI: 10.1103/physrevlett.91.155502

[77] R.H. Mathiesen, L. Arnberg, F. Mo, T. Weitkamp, A. Snigirev. Time Resolved X-Ray Imaging of Dendritic Growth in Binary Alloys. Phys. Rev. Lett. 83 (1999) 5062–5065.

DOI: 10.1103/physrevlett.83.5062

[78] T. Schenk, H.N. Thi, J. Gastaldi, G. Reinhart, V. Cristiglio, N. Mangelinck-Noël, H. Klein, J. Härtwig, B. Grushko, B. Billia. Application of synchrotron X-ray imaging to the study of directional solidification of aluminium-based alloys. J. Cryst. Growth 275 (2005).

DOI: 10.1016/j.jcrysgro.2004.10.081

[79] B. Billia, N.T. Henri, M.N. Nathalie, N. Bergeon, H. Jung, G. Reinhart, A. Bogno, A. Buffet, J. Hartwig, J. Baruchel. In Situ Synchrotron X-ray Characterization of Microstructure Formation in Solidification Processing of Al-based Metallic Alloys. ISIJ International. 50 (2010).

DOI: 10.2355/isijinternational.50.1929

[80] K. Nogita, H. Yasuda, A. Prasad, S.D. Mcdonald, T. Nagira, N. Nakatsuka, K. Uesugi, D.H. Stjohn. Real time synchrotron X-ray observations of solidification in hypoeutectic Al–Si alloys. Mater. Charact. 85 (2013) 134-140.

DOI: 10.1016/j.matchar.2013.08.015

[81] E. Liotti, A. Lui, R. Vincent, S. Kumar, Z. Guo, T. Connolley, I.P. Dolbnya, M. Hart, L. Arnberg, R.H. Mathiesen, 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

[82] B. Cai, J. Wang, A. Kao, K. Pericleous, A.B. Phillion, R.C. Atwood, P.D. Lee. 4D synchrotron X-ray tomographic quantification of the transition from cellular to dendrite growth during directional solidification. Acta Mater. 117 (2016) 160-169.

DOI: 10.1016/j.actamat.2016.07.002

[83] C. Kenel, D. Grolimund, J.L. Fife, V.A. Samson, S.V. Petegem, H.V. Swygenhoven, C. Leinenbach. Combined in situ synchrotron micro X-ray diffraction and high-speed imaging on rapidly heated and solidified Ti–48Al under additive manufacturing conditions. Scri. Mater. 2016 (2016).

DOI: 10.1016/j.scriptamat.2015.12.009

[84] K. Chen, X. Sha, X. Zhang, Y. Li. Rapid solidification of Cu25at.% Ni alloy: molecular dynamics simulations using embedded atom method. Mater. Sci. Engin. A 214 (1996) 139-145.

DOI: 10.1016/0921-5093(96)10207-0

[85] H.J. Chang, L.F. Chen, X.F. Zhu. First principle molecular dynamic simulation of the rapid solidification process of Ca50Mg20Cu30 alloy. J. Appl. Phys. 112 (2012) 81-870.

[86] G.S. Fishman, Monte Carlo, Springer, New York, (1996).

[87] A. Das, S. Ji, Z. Fan. Morphological development of solidification structures under forced fluid flow: a Monte-Carlo simulation. Acta Mater. 50 (2002) 4571-4585.

DOI: 10.1016/s1359-6454(02)00305-1

[88] Ch.-A. Gandin, M. Rappaz. Coupled finite element-cellular automation model for the prediction of dendritic grain structures in solidification processes. Acta Metall. 42 (1994) 2233-2246.

DOI: 10.1016/0956-7151(94)90302-6

[89] A. Rai, M. Markl, C. Körner. A coupled Cellular Automaton–Lattice Boltzmann model for grain structure simulation during additive manufacturing. Comp. Mater. Sci. 127 (2016) 37-48.

DOI: 10.1016/j.commatsci.2016.07.005

[90] X. Zhang, J. Zhao, H. Jiang, et al. A three-dimensional cellular automaton model for dendritic growth in multi-component alloy. Acta Mater. 60 (2012) 2249-2257.

DOI: 10.1016/j.actamat.2011.12.045

[91] A.A. Wheeler, B.T. Murray, R.J. Schaefer. Computation of dendrites using a phase field model. Phys. D. 66 (1993) 243-262.

DOI: 10.1016/0167-2789(93)90242-s

[92] I. Steinbach. Why Solidification? Why Phase-Field? JOM. 65 (2013) 1096-1102.

DOI: 10.1007/s11837-013-0681-5

[93] L. Q. Chen. Phase-Field Models for Microstructure Evolution, Annu. Rev. Mater. Res. 2002, 32(32):113-140.

DOI: 10.1146/annurev.matsci.32.112001.132041

[94] 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

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

DOI: 10.1103/physrevlett.77.4050

[96] A. Karma, W.-J. Rappel. Phase-field simulation of three-dimensional dendrites: is microscopic solvability theory correct? J. Cryst. Growth 174 (1997) 54-64.

DOI: 10.1016/s0022-0248(96)01060-3

[97] 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

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

DOI: 10.1103/physreve.60.3614

[100] A. Karma. Phase-Field Formulation for Quantitative Modeling of Alloy Solidification. Phys. Rev. Lett. 87 (2001) 115701-1-4.

DOI: 10.1103/physrevlett.87.115701

[101] B. Echebarria, R. Folch, A. Karma, M. Plapp. Quantitative phase-field model of alloy solidification. Phys. Rev. E 70 (2004) 061604-1-22.

DOI: 10.1103/physreve.70.061604

[102] H. Xing, X. Zhao, P. Duan, et al. Stability range of tilted dendritic arrays during directional solidification. Sci. China Tech. Sci. 57 (2014) 2530-2535.

DOI: 10.1007/s11431-014-5676-0

[103] H. Xing, X.L. Dong, C.L. Chen, et al. Phase-field simulation of tilted growth of dendritic arrays during directional solidification. Int. J Heat and Mass Transfer 90 (2015) 911-921.

DOI: 10.1016/j.ijheatmasstransfer.2015.07.029

[104] H. Xing, X. Dong, H. Wu, et al. Degenerate seaweed to tilted dendrite transition and their growth dynamics in directional solidification of non-axially oriented crystals: a phase-field study. Sci. Rep. 6 (2016) 26625.

DOI: 10.1038/srep26625

[105] H. Xing, L. M. Zhang, K. Song. Effect of interface anisotropy on growth direction of tilted dendritic arrays in directional solidification of alloys: insights from phase-field simulations. Int J Heat Mass Transfer. 104 (2017) 607-614.

DOI: 10.1016/j.ijheatmasstransfer.2016.08.096

[106] H. Xing, K.Anikt, X.L. Dong, et al. Growth direction selection of tilted dendritic arrays in directional solidification over a wide range of pulling velocity: a phase-field study. Int J Heat Mass Transfer. 117 (2018) 1107-1114.

DOI: 10.1016/j.ijheatmasstransfer.2017.10.086

[107] H. Xing, X.L. Dong, J.Y. Wang, K.X. Jin. Orientation dependence of columnar dendritic growth with sidebranching behaviors in directional solidification: insights from phase-field simulations. Unpublished.

DOI: 10.1007/s11663-018-1265-0

[108] S.-C. Huang, M. 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

[109] 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

[110] S. Gurevich, M. Amoorezaei, N. Provatas. Phase-field study of spacing evolution during transient growth. Phys. Rev. E 82 (2010) 051606.

DOI: 10.1103/physreve.82.051606

[111] J. Li, J. Zhang, W. Ge, et al. Multi-scale methodology for complex systems. Chem. Eng. Sci. 59 (2004) 1687−1700.

[112] J. Li, W. Ge, W. Wang, N. Yang. Focusing on the meso-scales of multi-scale phenomena−In search for a new paradigm in chemical engineering. Particuology 8 (2010) 634−639.

DOI: 10.1016/j.partic.2010.09.007

[113] T. Jiang, M. Georgelin, A. Pocheau. Wave dynamics on directional solidification interfaces swept by a flow in a thin sample. J Cryst. Growth 417 (2015) 37-43.

DOI: 10.1016/j.jcrysgro.2014.09.026