Home Technology Numerische Simulation der Erstarrung Teil 4: Unterkühlungseffekte
Article
Licensed
Unlicensed Requires Authentication

Numerische Simulation der Erstarrung Teil 4: Unterkühlungseffekte

  • Torsten Kraft and Hans Eckart Exner
Published/Copyright: December 2, 2021

Abstract

In diesem abschließenden Teil der Übersicht über die in den letzten drei Dekaden entwickelten numerischen Verfahren für die Simulation von Erstarrungsvorgängen werden Unterkühlungseffekte (Unterkühlung der Dendritenspitze und der eutektischen Reaktion) und ihr Einfluß auf die Vorhersage von Gefüge und Mikroseigerungen in metallischen Legierungen diskutiert. Die kinetischen Effekte, die bei hohen und sehr hohen Abkühlraten die Phasengleichgewichte beeinflussen, werden beschrieben. Die verschiedenen Ansätze werden im Hinblick auf ihre Verwendung in Simulationsprogrammen kritisch bewertet und offene Fragen definiert.

Abstract

In this final part of the overview on the numerical techniques for the simulation of solidification processes developed during the last three decades, the effects of undercooling (undercooling of the dendrite tip and the eutectic reaction) and their influence on the prediction of microstructures and microsegregation in metallic alloys are discussed. The kinetic effects, which affect the phase equilibria at high and very high cooling rates, are described. The various models are critically assessed with respect to their utilization in simulation programs, and open questions are defined.


T. Kraft und H. E. Exner Technische Hochschule Darmstadt Fachgebiet Physikalische Metallkunde Fachbereich Materialwissenschaft Petersenstr. 23, D-64287 Darmstadt

Literatur

1 Kraft, T.; Exner, H.E.: Numerische Simulation der Erstarrung. Teil 1: Mikroseigerungen in binären Legierungen, Z. Metallkd. 87 (1996) 598–611.Search in Google Scholar

2 Kraft, T.; Exner, H.E.: Numerische Simulation der Erstarrung. Teil 2: Mikroseigerungen in ternären und höherkomponentigen Legierungen, Z. Metallkd. 87 (1996) 652–660.Search in Google Scholar

3 Kraft, T.; Exner, H.E.: Numerische Simulation der Erstarrung. Teil 3: Vorhersage der Gefügestruktur, Z. Metallkd. 88 (1997) 278–290Search in Google Scholar

4 Herlach, D.M.: Non-Equilibrium Solidification of Undercooled Metallic Melts, Mater. Sci. Eng. R 12 (1994) 177–272.Search in Google Scholar

5 Trivedi, R.; Kurz,W.: Dendritic Growth, Int. Mater. Rev. 39 (1994) 49–74.10.1179/imr.1994.39.2.49Search in Google Scholar

6 Boettinger, W.J.; Coriell, S.R.: Microstructure Formation in Rapidly Solidified Alloys, in: P.R. Sahm, H. Jones, C.M. Adams (eds.), Science and Technology of the Undercooled Melt, Martinus Nijhoff Pub., Dordrecht, The Netherlands (1986) 81–108.10.1007/978-94-009-4456-5_5Search in Google Scholar

7 Turnbull, D.: On the Relation Between Crystallisation Rate and Liquid Structure, J. Phys. Chem. 66 (1962) 609–613.Search in Google Scholar

8 Turnbull, D.; Bagley, B.G.: Transitions in Viscous Liquids and Glasses, in: N.B. Hannay (ed.), Treatise on Solid State Chemistry, Vol. 5, Plenum Press, New York, NY (1975) 513–554.Search in Google Scholar

9 Turnbull, D.: Metastable Structures in Metallurgy, Metall. Trans. A 12 (1981) 695–708.10.1007/BF02648333Search in Google Scholar

10 Coriell, S.R.; Turnbull, D.: Relative Roles of Heat Transport and Interface Rearrangement Rates in the Rapid Growth of Crystals in Undercooled Melts, Acta Metall. 39 (1982) 2135–2139.Search in Google Scholar

11 Baker, J. C.; Cahn, J.W.: Solute Trapping by Rapid Solidification, Acta Metall. 17 (1969) 575–578.Search in Google Scholar

12 Baker, J.C.; Cahn, J.W.: Thermodynamics of Solidification, in: Solidification, American Society for Metals, Metals Park, OH (1970) 23–58.Search in Google Scholar

13 Jackson, K.A.; Gilmer, G.H.; Leamy, H.J.: Solute Trapping, in: C.W. White, P.S. Peercy (eds.), Laser and Electron Beam Processing of Materials, Academic Press, New York, NY (1980) 104–110.10.1016/B978-0-12-746850-1.50017-7Search in Google Scholar

14 Wood, R.F.: Model for Nonequilibrium Segregation during Pulsed Laser Annealing, Appl. Phys. Lett. 37 (1980) 302–304.Search in Google Scholar

15 Aziz, M.J.: Model for Solute Redistribution during Rapid Solidification, J. Appl. Phys. 53 (1982) 1158–1168.Search in Google Scholar

16 Kar, A.; Mazumder, J.: Model für Nonequilibrium Segregation during Rapid Solidification, in: M. Rappaz, M.R. Özgü, K.W. Mahin (eds.), Modeling of Casting, Welding and Advanced Solidification Processes V, TMS, Warrendale, PA (1991) 467–472.Search in Google Scholar

17 Kar, A.; Mazumder, J.: Model for Nonequilibrium Partioning during Rapid Solidification of Binary Concentrated Solutions, Acta Metall. Mater. 40 (1992) 1873–1881.Search in Google Scholar

18 Aziz, M.J.; Kaplan, T.: Continous Growth Model for Interface Motion during Alloy Solidification, Acta Metall. 36 (1988) 2335–2347.Search in Google Scholar

19 Sobolev, S.L.: Local- Nonequilibrium Model for Rapid Solidification of Undercooled Melts, Phys. Lett. A. 199 (1995) 383–386.Search in Google Scholar

20 Sobolev, S.L.: Effects of Local Non-Equilibrium Solute Diffusion on Rapid Solidification of Alloys, Phys. Stat. Sol. A. 156 (1996) 293–303.Search in Google Scholar

21 Jackson, K.A.; Gilmer, G.H.; Temkin, D.E.: Monte Carlo Simulation of the Rapid Crystallization of Bismuth-Doped Silicon, Phys. Rev. Lett. 75 (1995) 2530–2533.Search in Google Scholar

22 Aziz, M.J.: Interface Attachment Kinetics in Alloy Solidification, Metall. Mater. Trans. A 27 (1996) 671–686.10.1007/BF02648954Search in Google Scholar

23 Smith, P.M.; Aziz, M.J.: Solute Trapping in Aluminium Alloys, Acta Metall. Mater. 42 (1994) 3515–3525.10.1016/0956-7151(94)90483-9Search in Google Scholar

24 Cook, S.J.; Clancy, P.: Impurity Segregation in Lennard-Jones A/AB Heterostructures: I. The Effect of Lattice Strain, J. Chem. Phys. 99 (1993) 2175–2191.Search in Google Scholar

25 Yu, Q.; Thompson, M.O.; Clancy, P.: Solidification Kinetics in SiGe Alloys, Phys. Rev. B 53 (1996) 8386–8397.Search in Google Scholar

26 Aziz, M.J.: private Mitteilung, zitiert in Ref. 24.Search in Google Scholar

27 Aziz, M.J.: Non-Equilibrium Interface Kinetics during Rapid Solidification, Mater. Sci. Eng. A 178 (1994) 167–170.Search in Google Scholar

28 Goldman, L.M.; Aziz, M.J.: Aperiodic Stepwise Growth Model for the Velocity and Orientation Dependence of Solute Trapping, J. Mater. Res. 2 (1987) 524–527.Search in Google Scholar

29 Kittl, J.A.; Aziz, M.J.; Brunco, D.P.; Thompson, M. O.: Nonequilibrium Partitioning During Rapid Solidification of Si-As Alloys, J. Cryst. Growth 148 (1995) 172–182.Search in Google Scholar

30 Aziz, M.J.; Boettinger, W.J.: On the Transition from Short-Range Diffusion-Limited to Collision-Limited Growth in Alloy Solidification, Acta Metall. Mater. 42 (1994) 527–537.Search in Google Scholar

31 Eckler, K.; Herlach, D.M.; Aziz, M.J.: Search for a Solute-Drag Effect in Dendritic Solidification, Acta Metall. Mater. 42 (1994) 975–979.Search in Google Scholar

32 Kittl, J.A.; Aziz, M.J.; Brunco, D.P.; Thompson, M.O.: Absence of Solute Drag in Solidification, Appl. Phys. Lett. 64 (1994) 2359–2361.10.1063/1.111614Search in Google Scholar

33 Hillert, M.; Sundman, B.: ATreatment of the Solute Drag on Moving Grain Boundaries and Phase Interfaces in Binary Alloys, Acta Metall. 24 (1976) 731–743.Search in Google Scholar

34 Wheeler, A.A.; Boettinger, W.J.; McFadden, G.B.: Phase-Field Model of Solute Trapping during Solidification, Phys. Rev. E 47 (1993) 1893–1909.Search in Google Scholar

35 Boettinger, W.J.; Wheeler, A.A.; Murray, B.T.; McFadden, G.B.: Prediction of Solute Trapping at High Solidification Rates using a Diffuse Interface Phase-Field Theory of Alloys Solidification, Mater. Sci. Eng. A 178 (1994) 217–223.Search in Google Scholar

36 Conti, M.: Planar Isothermal Solidification from an Undercooled Melt: Unsteady Solute Segregation Studied with the Phase-Field Model, Phys. Rev. E 55 (1997) 701–707.Search in Google Scholar

37 Conti, M.: Solidification of Binary Alloys: Thermal Effects Studied with the Phase-Field Model, Phys. Rev. E 55 (1997) 765–771.Search in Google Scholar

38 Carrard, M.; Gremaud, M.; Zimmermann, M.; Kurz, W.: About the Banded Structure in Rapidly Solidified Dendritic and Eutectic Alloys, Acta Metall. Mater. 40 (1992) 983–996.Search in Google Scholar

39 Kraft, T.; Rettenmayr, M.; Exner, H.E.: An Extended Numerical Procedure for Predicting Microstructures and Microsegregation in Multicomponent Alloys, Model. Simul. Mater. Sci. Eng. 4 (1996) 161–177.Search in Google Scholar

40 Mullins, W.W.; Sekerka, R.F.: Stability of a Planar Interface during Solidification of a Dilute Binary Alloy, J. Appl. Phys. 35 (1964) 444–451.Search in Google Scholar

41 Coriell, S.R.; Sekerka, R.F.: Oscillatory Morphological Instabilities due to Non-Equilibrium Segregation, J. Cryst. Growth 61 (1983) 499–508.Search in Google Scholar

42 Sekerka, R.F.: Morphological and Hydrodynamic Instabilities during Phase Transformations, Physica D 12 (1984) 212–214.Search in Google Scholar

43 Coriell, S.R.; McFadden, G.B.; Sekerka, R.F.: Cellular Growth during Directional Solidification, Ann. Rev. Mater. Sci. 15 (1985) 119–145.10.1146/annurev.ms.15.080185.001003Search in Google Scholar

44 Coriell, S.R.; McFadden, G.B.: Morphological Stability, in: D.T.J. Hurle (ed.), Handbook of Crystal Growth, Vol. 1B, Elsevier Science B.V., Amsterdam, The Netherlands (1993) 785–857.Search in Google Scholar

45 Flesslles, J.M.; Simon, A.J.; Libchaber, A.J.: Dynamics of One-Dimensional Interfaces: An Experimentalist’s View, Adv. Phys. 40 (1991) 1–51.Search in Google Scholar

46 Davis, S.H.; Schulze, T.P.: Effects of Flow on Morphological Stability During Directional Solidification, Metall. Mater. Trans. A 27 (1996) 583–593.Search in Google Scholar

47 Faivre, G.: Morphological Instabilities of Lamellar Eutectic Growth Fronts: A Survey of Recent Experimental and Numerical Results, J. Cryst. Growth 166 (1996) 29–39.Search in Google Scholar

48 Jones, H.: The Critical Concentration for Formation of Segregation-Free Solid by Solute Trapping during Rapid Solidification from the Melt, Mater. Lett. 6 (1988) 181–182.Search in Google Scholar

49 Boettinger, W.J.; Coriell, S.R.; Sekerka, R.F.: Mechanisms of Microsegregation-Free Solidification, Mater. Sci. Eng. 65 (1984) 27–36.Search in Google Scholar

50 Trivedi, R.; Kurz, W.: Morphological Stability of a Planar Interface under Rapid Solidification Conditions, Acta Metall. 34 (1986) 1663–1670.Search in Google Scholar

51 Laxmanan, V.: Analysis of Stability of a Planar Solid/Liquid Interface in a Dilute Binary Alloy, J. Mater. Res. 5 (1990) 223–228.Search in Google Scholar

52 Merchant, G.J.; Davis, S.H.: Morphological Instability in Rapid Directional Solidification, Acta Metall. Mater. 38 (1990) 2683–2693.Search in Google Scholar

53 Huntley, D.A.; Davis, S.H.: Thermal Effects in Rapid Solidification: Linear Theory, Acta Metall. Mater. 41 (1993) 2025–2043.Search in Google Scholar

54 Ludwig, A.: Limit of Absolute Stability for Crystal Growth into Undercooled Alloy Melts, Acta Metall. Mater. 39 (1991) 2795–2798.Search in Google Scholar

55 Willnecker, R.: Messungen zur Unterkühlung, Keimbildung und schnellen Erstarrung metallischer Systeme, Forschungsbericht 88-39, DFVLR, Institut für Raumsimulation, Köln, Germany (1988).Search in Google Scholar

56 Ivantsov, G.P.: Doklady Akademii Nauk SSSR 58 (1947) 567, zitiert in Ref. 61.Search in Google Scholar

57 Papapetrou, A.: Z. Kristallograph. 92 (1935) 89, zitiert in Ref. 61.Search in Google Scholar

58 La Combe, J.C.; Koss, M.B.; Fradkov, V.E.; Glicksman, M.E.: Three-Dimensional Dendrite Tip Morphology, Phys. Rev. E 52 (1995) 2778–2786.Search in Google Scholar

59 Laxmanan, V.: The Gibbs-Thomson Effect during Cellular and Dendritic Solidification, Scripta Mater. (eingereicht).Search in Google Scholar

60 Spencer, B.J.; Huppert, H.E.: Steady State Solutions for an Array of Strongly Interacting needle Crystals in the Limit of Small Undercoolings, J. Cryst Growth 148 (1995) 305–323.Search in Google Scholar

61 Glicksman, M.E.; Marsh, S.P.: The Dendrite, in: D.T.J. Hurle (ed.), Handbook of Crystal Growth, Vol. 1B, Elsevier Science B.V., Amsterdam, The Netherlands (1993) 1075–1122.Search in Google Scholar

62 Temkin, D.E.: Doklady Akademii Nauk SSSR 132 (1960) 1307, zitiert in Ref. 61.Search in Google Scholar

63 Trivedi, R.: Growth of Dendritic Needles from a Supercooled Melt, Acta Metall. 18 (1970) 287–294.Search in Google Scholar

64 Langer, J.S.: Recent Developments in the Theory of Dendritic Solidification, Crystal Properties and Preparation, Vol. 22–25, Pt. 1, Trans Tech Pub., Aedermannsdorf, Switzerland (1989) 1–9.Search in Google Scholar

65 Bower, T.F.; Brody, H.D.; Flemings, M. C.: Measurements of Solute Redistribution in Dendritic Solidification, Trans. AIME 236 (1966) 624–634.Search in Google Scholar

66 Hunt, J.D.; McCartney, D. G.: Numerical Finite Difference Model for Steady State Cellular Array Growth, Acta Metall. 35 (1987) 89–99.Search in Google Scholar

67 Ungar, L.H.; Brown, R.A.: Cellular Interface Morphologies in Directional Solidification. IV. Deep Cells, Phys. Rev. B 31 (1985) 5931–5940.Search in Google Scholar

68 Lu, S.Z.; Hunt, J.D.: A Numerical Analysis of Dendritic and Cellular Array Growth: The Spacing Adjustment Mechanisms, J. Cryst. Growth 123 (1992) 17–34.Search in Google Scholar

69 Hunt, J.D.; Lu, S.Z.: Numerical Modelling of Cellular and Dendritic Array Growth: Spacing and Structure Predictions, Mater. Sci. Eng. A 173 (1993) 79–83.Search in Google Scholar

70 Lu, S.Z.; Hunt, J.D.; Gilgien, P.; Kurz, W.: Cellular and Dendritic Growth in Rapidly Solidified Al-Fe and Al-Cu Alloys, Acta Metal. Mater. 42 (1994) 1653–1660.Search in Google Scholar

71 Bolling, G.F.; Tiller, W.A.: Growth from the Melt: III. Dendritic Growth, J. Appl. Phys. 32 (1961) 2587–2605.Search in Google Scholar

72 Billia, B.; Trivedi, R.: Pattern Formation in Crystal Growth, in: D.T.J. Hurle (ed.), Handbook of Crystal Growth, Vol. 1B, Elsevier Science B.V., Amsterdam, The Netherlands (1993) 899–1073.Search in Google Scholar

73 Burden, M.H.; Hunt, J.D.: Cellular and Dendritic Growth I/II, J. Cryst. Growth 22 (1974) 99–108, 109–116.10.1016/0022-0248(74)90126-2Search in Google Scholar

74 Laxmanan, V.: Dendritic Solidification – I. Analysis of Current Theories and Models. II. A Model for Dendritic Growth under an Imposed Thermal Gradient, Acta Metall. 33 (1985) 1023–1035, 1037–1049.Search in Google Scholar

75 Laxmanan, V.: Comments on Undercooling Effects in Microsegregation Modelling, Scripta Mater. 36 (1997) 687–692.Search in Google Scholar

76 Laxmanan, V.: Dendritic Solidification – III. Some Further Refinements to the Model for Dendritic Growth under an Imposed Thermal Gradient, Acta Metall. 33 (1985) 1475–1480.Search in Google Scholar

77 Langer, J.S.; Müller-Krumbhaar, H.: Stability Effects in Dendritic Crystal Growth, J. Cryst. Growth 42 (1977) 11–14.Search in Google Scholar

78 Langer, J.S.; Müller-Krumbhaar, H.: Theory of Dendritic Growth – I. Elements of a Stability Analysis. II. Instabilities in the Limit of Vanishing Surface Tension. III. Effects of the Surface Tension, Acta Metall. 26 (1978) 1681–1687, 1689–1695, 1697–1708.Search in Google Scholar

79 Glicksman, M.E.; Schäfer, R.J.; Ayers, J.D.: Dendritic Growth – A Test of Theory, Metall. Trans. A 7 (1976) 1747–1759.10.1007/BF03186673Search in Google Scholar

80 Kurz, W.; Fisher, D.J.: Dendrite Growth at the Limit of Stability: Tip Radius and Spacing, Acta Metall. 29 (1981) 11–20.Search in Google Scholar

81 Esaka, H.; Kurz, W.: Columnar Dendrite Growth: A Comparison of Theory, J. Cryst. Growth 69 (1984) 362–366.10.1016/0022-0248(84)90343-9Search in Google Scholar

82 Esaka, H.; Kurz, W.: Columnar Dendrite Growth: Experiments on Tip Growth, J. Cryst. Growth 72 (1985) 578–584.10.1016/0022-0248(85)90208-8Search in Google Scholar

83 Esaka, H.; Kurz, W.: Modelling of Columnar Dendritic Growth, Z. Metallkd. 76 (1985) 127–133.Search in Google Scholar

84 Kurz, W.; Giovanola, B.; Trivedi, R.: Theory of Microstructural Development during Rapid Solidification, Acta Metall. 34 (1986) 823–830.Search in Google Scholar

85 Kurz, W.; Giovanola, B.; Trivedi, R.: Microsegregation in Rapidly Solidified Ag-15 wt.% Cu, J. Cryst. Growth 91 (1988) 123–125.Search in Google Scholar

86 Ludwig, A.; Kurz, W.: Direct Observation of Solidification Microstructures Around Absolute Stability, Acta Mater. 44 (1996) 3643–3654.Search in Google Scholar

87 Trivedi, R.: Theory of Dendritic Growth during the Directional Solidification of Binary Alloys, J. Cryst. Growth 49 (1980) 219–232.Search in Google Scholar

88 Tewari, S.N.; Laxmanan, V.: A Critical Examination of the Dendrite Growth Models: Comparison of Theory with Experimental Data, Acta Metall. 35 (1987) 175–183.Search in Google Scholar

89 Lipton, J.; Glicksman, M.E.; Kurz,. W.: Dendritic Growth into Undercooled Melts, Mater. Sci. Eng. 65 (1984) 57–63.Search in Google Scholar

90 Lipton, J.; Glicksman, M.E.; Kurz, W.: Equiaxed Dendrite Growth in Alloys at Small Supercooling, Metall. Trans. A 18 (1987) 341–345.Search in Google Scholar

91 Laxmanan, V.: Cellular and Dendritic Growth in a Binary Alloy Melt: A Marginal Stability Approach, J. Cryst. Growth 75 (1986) 573–590.Search in Google Scholar

92 Rettenmayr, M.; Pompe, O.: Instability of Coarse Microstructures during Equiaxed Growth, in: A. Roósz, M. Rettenmayr (eds.), Solidification and Gravity, Materials Science Forum Vols. 215–216, Transtec Pub., Zürich, Switzerland (1996) 201–208; Interface Instabilities on Solidifying Globulitic Particles, J. Cryst. Growth (im Druck).10.4028/www.scientific.net/MSF.215-216.201Search in Google Scholar

93 Lipton, J.; Kurz,W.; Trivedi, R.: Rapid Dendrite Growth in Undercooled Alloys, Acta Metall. 35 (1987) 957–964.Search in Google Scholar

94 Trivedi, R.; Lipton, J.; Kurz, W.: Effect of Growth Rate Dependent Partition Coefficient on the Dendritic Growth in Undercooled Melts, Acta Metall. 35 (1987) 965–970.Search in Google Scholar

95 Wu, Y.; Piccone, T.J.; Shiohara, Y.; Flemings, M.C.: Dendritic Growth of Undercooled Nickel-Tin: Part I; Part II, Metall. Trans. A 18 (1987) 915–924, 925–932.Search in Google Scholar

96 Boettinger, W.J.; Coriell, S.R.; Trivedi, R.: Application of Dendritic Growth Theory to the Interpretation of Rapid Solidification Microstructures, in: R. Mehrabian, P.A. Parrish (eds.), Rapid Solidification Processing: Principles and Technologies IV, Claitor’s Publishing Division, Baton Rouge, LA (1988) 13–25.Search in Google Scholar

97 Suzuki, M.; Piccone, T.J.; Flemings, M.C.; Brody, H.D.: Solidification in Highly Undercooled Fe-P Alloys, Metall. Trans. A 22 (1991) 2761–2768.Search in Google Scholar

98 Ananth, R.; Gill, W.N.: Self-Consistent Theory of Dendritic Growth with Convection, J. Cryst. Growth 108 (1991) 173–189.Search in Google Scholar

99 Beckermann, C.: Modeling Segregation and Grain Structure Development in Equiaxed Solidification with Convection, JOM 49 (Nr. 3) (1997) 13–17.Search in Google Scholar

100 Ramani, A.; Beckermann, C.: Dendrite Tip Growth Velocities of Settling NH4CI Equiaxed Crystals, Scripta Mater. 36 (1997) 633–638.Search in Google Scholar

101 Langer, J.S.: Existence of Needle Crystals in Local Models Solidification, Phys. Rev. A 33 (1986) 435–441.Search in Google Scholar

102 Pelcé, P.: Dynamics of Curved Fronts, Academic Press, New York, NY (1988).Search in Google Scholar

103 Brener, E.A.; Melnikow, V.I.: Pattern Selection in Two-Dimensional Dendritic Growth, Adv. Phys. 40 (1991) 53–97.Search in Google Scholar

104 Caroli, B.; Müller-Krumbhaar, H.: Recent Advances in the Theory of Free Dendritic Growth, ISIJ Int. 35 (1995) 1541–1550.Search in Google Scholar

105 Langer, J.S.: Dendrite Sidebranching in the Three-Dimensional Symmetric Model in the Presence of Noise, Phys. Rev. A 36 (1987) 3350–3358.Search in Google Scholar

106 Kessler, D.A.; Levine, H.: Pattern Selection in Three Dimensional Dendritic Growth, Acta Metall. 36 (1988) 2693–2706.Search in Google Scholar

107 Brener, E.: Nonaxisymmetric Patterns in the Saffman-Taylor Problem and in Three-Dimensional Directional Solidification at Low Velocity, Phys. Rev. E 48 (1993) 4437–4443.Search in Google Scholar

108 Brener, E.: Needle-Crystal Solution in Three-Dimensional Dendritic Growth, Phys. Rev. Lett. 71 (1993) 3653–3656.Search in Google Scholar

109 Ben Amar, M.; Brener, E.: Theory of Pattern Selection in Three-Dimensional Nonaxisymmetric Dendritic Growth, Phys. Rev. Lett. 71 (1993) 589–592.Search in Google Scholar

110 Ben Amar, M.; Brener, E.: Theory of Dendritic Growth in Three Dimensions, Mater. Sci. Eng. A 178 (1994) 147–152.Search in Google Scholar

111 Brener, E.A.: Three-Dimensional Dendrite Growth, J. Cryst. Growth 166 (1996) 339–346.10.1016/0022-0248(96)00046-2Search in Google Scholar

112 Ben Amar, M.; Pomeau, Y.: Theory of Dendritic Growth in a Weakly Undercooled Melt, Europhys. Lett. 2 (1986) 307–314.Search in Google Scholar

113 Müller-Krumbhaar, H.; Kurz, W.: Solidification, in: P. Haasen (ed.), Materials Science and Technology: Vol. 5 Phase-Transformations in Materials, VCH-Verlag, Weinheim, Germany (1991) 553–632.Search in Google Scholar

114 Classen A.; Misbah, C.; Müller-Krumbhaar, H.; Saito, Y.: Direction Solidification with Interphase Dissipation, Phys. Rev. A 43 (1991) 6920–6933.Search in Google Scholar

115 Glicksman, M.E.; Koss, M.B.; Winsa, E.A.: Dendritic Growth Velocities in Microgravity, Phys. Rev. Lett. 73 (1994) 573–576: The Chronology of a Microgravity Spaceflight Experiment: IDGE, JOM 47 (Nr. 8) (1995) 49–54.Search in Google Scholar

116 Glicksman, M.E.; Koss, M.B.; Bushnell, L.T.; LaCombe, J.C.: The Isothermal Dendritic Growth Experiment: Implications for Theory, in: M. Cross, J. Campbell (eds.), Modelling of Casting, Welding and Advanced Solidification Processes VII, TMS, Warrendale PA (1995) 664-670.Search in Google Scholar

117 Glicksman, M.E.; Koss, M.B.; Bushnell, L.T. LaCombe, J.C.; Winsa, E.A.: Dendritic Growth Measurements in Microgravity, in: L. Ratke, H. Walter, B. Feuerbacher (eds.), Materials and Fluids Under Low Gravity, Lecture Notes in Physics, Vol. 464, Springer Verlag, Berlin, Germany (1996) 63–75.10.1007/BFb0102513Search in Google Scholar

118 Sekerka, R.F.; Coriell, S.R.; McFadden, G.B.: Stagnant Film Model of the Effect of Natural Convection on the Dendrite Operating State, J. Cryst. Growth 154 (1995) 370–376.Search in Google Scholar

119 Jin, I.; Purdy, G.R. Controlled Solidification of a Dilute Binary Alloy: I. Theory, II. Experiment, J. Cryst. Growth 23 (1974) 29–36, 37–44.10.1016/0022-0248(74)90038-4Search in Google Scholar

120 Billia, B.; Capella, L.: Étude en Thermodynamique Irréversible de la Croissance Cellulaire d’un Mono Cristal d’Alliage Binaire Dilue, J. Cryst. Growth 44 (1978) 235–240.Search in Google Scholar

121 Kirkaldy, J.S.: A Zener-Hillert Model for Growth of Binary Alloy Cells, Scripta Metall. 14 (1980) 739–744.Search in Google Scholar

122 Venugopalan, D.; Kirkaldy, J.S.: Theory of Cellular Solidification of Binary Alloys with Applications to Succinonitrile-Salol, Acta Metall. 32 (1984) 893–906.Search in Google Scholar

123 Brown, S.G.R.; Williams, T.; Spittle, J.A.: A Cellular Automaton Model of the Steady-State ,Free‘ Growth of a Non-Isothermal Dendrite, Acta Metall. Mater. 42 (1994) 2893–2898.Search in Google Scholar

124 Wheeler, A.A.; Murray, B.T.; Schaeffer, R.J.: Computation of Dendrites Using a Phase Field Model, Physica D 66 (1993) 243–262.Search in Google Scholar

125 Bobadilla, M.; Lacaze, J.; Lesoult, G.: Influence des conditions de solidification sur le déroulement de la solidification des aciers inoxydables austénitiques, J. Cryst. Growth 89 (1988) 531–544.Search in Google Scholar

126 Siredey, N.; Lacaze, J.: Growth Conditions at the Solidification Front of Multicomponent Alloys, Scripta Metall. Mater. 29 (1993) 759–764.Search in Google Scholar

127 Rappaz, M.; David, S.A.; Vitek, J.M.; Boatner, L.A.: Analysis of Solidification Microstructures in Fe-Ni-Cr Single-Crystals Welds, Metall. Trans. A 21 (1990) 1767–1782.Search in Google Scholar

128 Löser, W.; Herlach, D.M.: Theoretical Treatment of the Solidification of Undercooled Fe-Cr-Ni Melts, Metall. Trans. A 23 (1992) 1585–1591.Search in Google Scholar

129 Kurz, W.; Sahm, P.R.: Gerichtet erstarrte eutektische Werkstoffe, Springer Verlag, Berlin, Germany (1975).10.1007/978-3-642-65993-5Search in Google Scholar

130 Biloni, H.; Boettinger, W.J.: Solidification, in: R.W. Cahn, P. Haasen (eds.), Physical Metallurgy, North-Holland, Amsterdam, The Netherlands (1996) 669–842.10.1016/B978-044489875-3/50013-2Search in Google Scholar

131 Jones, H.: Development in Aluminium Alloys by Solidification at Higher Cooling Rates, Aluminium 54 (1978) 274–281.Search in Google Scholar

132 Jackson, K.A.; Hunt, J.D.: Lamellar and Rod Eutectic Growth, Trans. AIME 236 (1966) 1129–1142.Search in Google Scholar

133 McCartney, D.G.; Hunt, J.D.; Jordan, R.M.: The Structures in a Simple Ternary Eutectic System: Part I. Theory, Metall. Trans. A 11 (1980) 1243–1249.Search in Google Scholar

134 Delamore, G.W.; Hill, J.M.: Solute Concentration Profiles in Ternary Lamellar Eutectic Growth, Scripta Metall. 14 (1980) 809–813.Search in Google Scholar

135 Donaghey, L.F.; Tiller, W.A.: On the Diffusion of Solute during the Eutectoid and Eutectic Transformations, Mater. Sci. Eng. 3 (1968) 231–239.Search in Google Scholar

136 Series, R.W.; Hunt, J.D.; Jackson, K.A.: The Use of an Electric Analogue to Solve the Lamellar Eutectic Diffusion Problem, J. Cryst. Growth 40 (1977) 221–233.Search in Google Scholar

137 Exner, H.E.; Paul, J.: Simplified Determination of Cooling Conditions of Aluminium-Silicon Alloys, Metal. Sci. 17 (1983) 141–143.Search in Google Scholar

138 Pompe, O.: Geometrische Charakterisierung von Erstarrungsgefügen, Naturwissenschaftliche Reihe, Bd. 7, Dissertations Druck Darmstadt, Darmstadt, Germany (1996).Search in Google Scholar

139 Shingu, P.H.: The Extremum Condition for the Rate of Cellular Phase Separation, J. Appl. Phys. 50 (1979) 5743–5746.Search in Google Scholar

140 Flemings, M. C.: Solidification Processing, McGraw-Hill, New York, NY (1974).10.1007/BF02643923Search in Google Scholar

141 Trivedi, R.; Mason, J.T.; Verhoeven, J.D.; Kurz,W.: Eutectic Spacing Selection in Lead-Based Alloy Systems, Metall. Trans. A 22 (1991) 2523–2533.Search in Google Scholar

142 Qurdjini, A.; Litu, J.; Elliott, R.: Eutectic Spacing Selection in Al-Cu System, Mater. Sci. Tech. 10 (1994) 312–318.Search in Google Scholar

143 Liu, J.; Elliott, R.: Eutectic Spacing Selection in the Lead-Tin Eutectic System, Metall. Mater. Trans. A 26 (1995) 471–476.Search in Google Scholar

144 Brattkus, K.; Caroli, B.; Caroli, C.; Roulet, B.: Lamellar Eutectic Growth at Large Thermal Gradient: I. Stationary Patterns, J. Phys. France 51 (1990) 1847–1864; Caroli, B.; Caroli, C.; Roulet, B.: Lamellar Eutectic Growth at Large Thermal Gradient: II. Linear Stability, J. Phys. France 51 (1990) 1865–1876.Search in Google Scholar

145 Liu, J.; Zhou, Y.; Shang, B.: Lamellar Eutectic Stable Growth – I. Modelling; II. Experiment on Al-Si Eutectic, Acta Metall. 38 (1990) 1625–1630, 1631–1634.Search in Google Scholar

146 Magnin, P.; Trivedi, R.: Eutectic Growth: A Modification of the Jackson and Hunt Theory, Acta Metall. Mater. 39 (1991) 453–467.Search in Google Scholar

147 Jones, H.; Kurz, W.: Relation of Interphase Spacing and Growth Temperature to Growth Velocity in Fe-C and Fe-Fe3C Eutectic Alloys, Z. Metallkd. 72 (1981) 792–797.Search in Google Scholar

148 Grugel, R.; Kurz, W.: Growth of Interdendritic Eutectic in Directionally Solidified Al-Si Alloys, Metall. Trans. A 18 (1987) 1137–1142.Search in Google Scholar

149 Magnin, P.; Mason, J.T.; Trivedi, R.: Growth of Irregular Eutectics and the Al-Si System, Acta Metall. Mater. 39 (1991) 469–480.10.1016/0956-7151(91)90115-HSearch in Google Scholar

150 Catalina, A.V.; Stefanescu, D.M.: Lamellar Growth of Eutectic Equiaxed Grains, Metall. Mater. Trans. A 27 (1996) 4205–4210.10.1007/BF02595668Search in Google Scholar

151 Östlund, Å.; Lundbäck, E.; Fredriksson, H.: Rapidly Solidified Eutectic Fe-C-Si Alloys, in: G. Ohiro, T. Kusakawa, E. Niyama (eds.), Physical Metallurgy of Cast Iron IV, MRS Symp. Proc., Materials Research Society, Pittsburgh, PA (1989) 423–431.Search in Google Scholar

152 Wei, B.; Herlach, D.M.; Feuerbacher, B.; Sommer, F.: Dendrite and Eutectic Solidification of Undercooled Co-Sb Alloys, Acta Metall. Mater. 41 (1993) 1801–1809.10.1016/0956-7151(93)90200-CSearch in Google Scholar

153 Trivedi, R.; Magnin, P.; Kurz, W.: Theory of Eutectic Growth under Rapid Solidification Conditions, Acta Metall. 35 (1987) 971–980.Search in Google Scholar

154 Zimmermann, M.; Karma, A.; Carrard, M.: Oscillatory Lamellar Microstructure in Off-Eutectic Al-Cu Alloys, Phys. Rev. B 42 (1990) 833–837.Search in Google Scholar

155 Zimmermann, M.; Carrard, M.; Kurz, W.: Rapid Solidification of Al-Cu Eutectic Alloy by Laser Remelting, Acta Metall. 37 (1989) 3305–3313.Search in Google Scholar

156 Trivedi, R.; Kurz, W.: Microstructure Selection in Eutectic Alloy Systems, in: D.M. Stefanescu, G.J. Abbaschian, R.J. Bayuzick (eds.), Solidification Processing of Eutectic Alloys, TMS, Warrendale, PA (1988) 3–34.Search in Google Scholar

157 Kurz, W.; Trivedi, R.: Eutectic Growth under Rapid Solidification Conditions, Metall. Trans. A 22 (1991) 3051–3057.Search in Google Scholar

158 Rappaz, M.; Carrupt, B.; Zimmermann, M.; Kurz, W.: Numerical Simulation of Eutectic Solidification in the Laser Treatment of Materials, Helv. Phys. Acta 60 (1987) 924–936.Search in Google Scholar

159 Zryd, A.; Gremaud, M.; Kurz, W.: Lamellar Eutectics: A Comparison of Three Models Valid at High Growth Rates, Mater. Sci. Eng. A 181/182 (1994) 1392–1396.Search in Google Scholar

160 Strässler, S.; Schneider, W.R.: Stability of Lamellar Eutectics, Phys. Condensed Matter 17 (1974) 153–178.10.1007/BF01475921Search in Google Scholar

161 Langer, J.S.: Eutectic Solidification and Marginal Stability, Phys. Rev. Lett. 44 (1980) 1023–1026.10.1103/PhysRevLett.44.1023Search in Google Scholar

162 Datye, V.; Langer, J.S.: Stability of Thin Lamellar Eutectic Growth, Phys. Rev. B 24 (1981) 4155–4168.Search in Google Scholar

163 Fisher, D.J.; Kurz,W.: ATheory of Branching Limited Growth off Irregular Eutectics, Acta Metall. 28 (1980) 777–794.Search in Google Scholar

164 Liu, J.; Elliott, R.: Selfconsistent Solutions for Lamellar Eutectic Growth, Acta Metall. Mater. 43 (1995) 3301–3311.10.1016/0956-7151(95)00051-VSearch in Google Scholar

165 Kim, K.B.; Liu, J.; Marasli, N.; Hunt, J.D.: The Effect of Different Atomic Volumes in the Three Phases during Lamellar Eutectic Growth. A Comparison of Experiment and Theory in the Al-Al2Cu System, Acta Metall. Mater. 43 (1995) 2143–2147.Search in Google Scholar

166 Karma, A.: Beyond Steady-State Lamellar Eutectic Growth, Phys. Rev. Lett. 59 (1987) 71–74.Search in Google Scholar

167 Xiao, R.F.; Alexander, J.I.D.; Rosenberger, F.: Microscopic-Growth Morphologies in Binary Systems, Phys. Rev. A 45 (1992) 571–574.Search in Google Scholar

168 Xiao, R.F.; Alexander, J.I.D.; Rosenberger, F.: Eutectic and Off-Eutectic Growth Patterns, Mater. Sci. Eng. A 178 (1994) 233–238.Search in Google Scholar

169 Kassner, K.; Misbah, C.: Growth of Lamellar Eutectic Structures: The Axisymetric State, Phys. Rev. A 44 (1991) 6513–6532.Search in Google Scholar

170 Kassner, K.; Misbah, C.: Similarity Laws in Eutectic Growth, Phys. Rev. Lett. 66 (1991) 445–448.10.1103/PhysRevLett.66.445Search in Google Scholar PubMed

171 Kassner, K.; Misbah, C.: Parity Breaking in Eutectic Growth, Phys. Rev. Lett. 65 (1990) 1458–1461.10.1103/PhysRevLett.65.1458Search in Google Scholar PubMed

172 Kassner, K.; Misbah, C.: Spontaneous Parity-Breaking Transition in Directional Growth of Lamellar Eutectic Structures, Phys. Rev. A 44 (1991) 6533–6543.Search in Google Scholar

173 Kassner, K.; Misbah, C.: Coupling Between Crystalline Anisotropy and Spontaneous Parity-Breaking in Lamellar Eutectic Growth, Phys. Rev. A 45 (1992) 7372–7384.Search in Google Scholar

174 Karma, A.; Sarkissian, A.: Morphological Instabilities of Lamellar Eutectics, Metall. Mater. Trans. A 27 (1996) 635–656.10.1007/BF02648952Search in Google Scholar

175 Elder, K.R.; Drolet, F.; Kosterlitz, J.M.; Grant, M.: Stochastic Eutectic Growth, Phys. Rev. Lett. 72 (1994) 677–680.10.1103/PhysRevLett.72.677Search in Google Scholar PubMed

176 Karma, A.: Phase-Field Model of Eutectic Growth, Phys. Rev. E 49 (1994) 2245–2250.Search in Google Scholar

177 Wheeler, A.A.; McFadden, G.B.; Boettinger, W.J.: Phase-Field Model for Solidification of a Eutectic Alloys, Proc. Royal Soc. London, Series A 452 (1996) 495–525.Search in Google Scholar

178 Alexander, J.I.D.; Xiao, R.F.; Rosenberger, F.: Modelling Growth Morphologies on Different Length Scales, in: J.E. Mark, M.E. Glicksman, S.P. Marsh (eds.), Computational Methods in Materials Science, MRS Symp. Proc. Vol. 278, Materials Research Society, Pittsburgh, PA (1992) 269–280.Search in Google Scholar

179 Spittle, J.A.; Brown, S.G.R.: A 3D Cellular Automaton Model of Coupled Growth in Two Component Systems, Acta Metall. Mater. 42 (1994) 1811–1815.Search in Google Scholar

180 Gill, S. C.; Kurz, W.: Rapidly Solidified Al-Cu Alloys – II. Calculation of the Microstructure Selection Map, Acta Metall. Mater. 43 (1995) 139–151.Search in Google Scholar

181 Bhadeshia, H.K.D.H.; David, S.A.; Vitek, J.M.: Solidification Sequences in Stainless Steel Dissimilar Alloy Welds, Mater. Sci. Tech. 7 (1991) 50–61.Search in Google Scholar

182 Beer, S.Z.: Liquid Metals: Chemistry and Physics, Marcel Decker, New York (1972).Search in Google Scholar

183 Frohberg, G.; Kraatz, K.H.;Wever, H.: Investigations on Self- and Interdiffusion in Liquid Metals, in: C. Abromeit, H. Wollenberger (eds.), Vacancies and Interstitials in Metals and Alloys, Materials Science Forum Vols. 15–18, Transtec Pub., Aedermannsdorf, Switzerland (1987) 529–534; Frohberg, G.: Wenn es kritisch wird . . ., VDI Nachrichten Magazin 6 (1994) 18–19.Search in Google Scholar

184 Trivedi, R.; Kurz, W.: Modeling of Solidification Microstructures in Concentrated Solutions and Intermetallic Systems, Metall. Trans. A 21 (1990) 1311–1318.Search in Google Scholar

185 Kraft, T.; Roósz, A.; Rettenmayr, M.: Undercooling Effects in Microsegregation Modelling, Scripta Mater. 35 (1996) 77–82.Search in Google Scholar

186 Gácsi, Z.; Roó sz, A.: Stereometric Characterisation of Unidirectionally Solidified Dendritic Structure, Acta Stereol. 13 (1994) 335–341.Search in Google Scholar

Received: 1996-08-12
Published Online: 2021-12-02

© 1997 Carl Hanser Verlag, München

Downloaded on 10.3.2026 from https://www.degruyterbrill.com/document/doi/10.3139/ijmr-1997-0084/html
Scroll to top button