Numerische Simulation der Erstarrung Teil 4: Unterkühlungseffekte
-
Torsten Kraft
and Hans Eckart Exner
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.
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
© 1997 Carl Hanser Verlag, München
Articles in the same Issue
- Frontmatter
- Aufsätze
- Phase Formation During Dissolution of Nickel in Liquid Aluminium
- Comment on the Article: Phase Transformations in CuAu and Cu3Au
- Numerische Simulation der Erstarrung Teil 4: Unterkühlungseffekte
- Integral Enthalpy of Mixing in Ternary (In, Pb, Zn) Monophase Liquid – Calorimetric Measurements. Modeling Outside and Inside the Miscibility Gap
- Lanthanum-containing Ternary Solid Solutions with the NaZn13-, ThMn12- and Th2Zn17-Type Crystal Structures
- Modelling of the Thermodynamic Properties of Bi–As2Te3 and Tl–As2Te3 Ternary Alloys
- Solubility of Metals in Low-melting Melts
- Effect of Aging on the Structure of Spark Erosion Treated Surface of Cobalt
- Fracture Toughness of Supported Ni–P Films Prepared by Autocatalytic Chemical Deposition
- Tribological Behaviour of Electroless Ni –P–MoS2 Composite Coatings
- Effect of Various Heat Treatments on Microstructure and Mechanical Properties of 34CrNiMo6 Steel
- Simulation des Deformationsverhaltens von Metallen mit Formgedächtniseffekt unter Berücksichtigung des Versetzungsgleitens
- Mitteilungen der Deutschen Gesellschaft für Materialkunde e.V.
- Personen
- Veranstaltungen
- Buchbesprechungen
- Terminkalender
Articles in the same Issue
- Frontmatter
- Aufsätze
- Phase Formation During Dissolution of Nickel in Liquid Aluminium
- Comment on the Article: Phase Transformations in CuAu and Cu3Au
- Numerische Simulation der Erstarrung Teil 4: Unterkühlungseffekte
- Integral Enthalpy of Mixing in Ternary (In, Pb, Zn) Monophase Liquid – Calorimetric Measurements. Modeling Outside and Inside the Miscibility Gap
- Lanthanum-containing Ternary Solid Solutions with the NaZn13-, ThMn12- and Th2Zn17-Type Crystal Structures
- Modelling of the Thermodynamic Properties of Bi–As2Te3 and Tl–As2Te3 Ternary Alloys
- Solubility of Metals in Low-melting Melts
- Effect of Aging on the Structure of Spark Erosion Treated Surface of Cobalt
- Fracture Toughness of Supported Ni–P Films Prepared by Autocatalytic Chemical Deposition
- Tribological Behaviour of Electroless Ni –P–MoS2 Composite Coatings
- Effect of Various Heat Treatments on Microstructure and Mechanical Properties of 34CrNiMo6 Steel
- Simulation des Deformationsverhaltens von Metallen mit Formgedächtniseffekt unter Berücksichtigung des Versetzungsgleitens
- Mitteilungen der Deutschen Gesellschaft für Materialkunde e.V.
- Personen
- Veranstaltungen
- Buchbesprechungen
- Terminkalender