In this work, we develop one- and two-dimensional phase-field simulations to approximate dendritic growth of a binary Al–2 wt% Si alloy. Simulations are performed for both isothermal as well as directional solidification. Anisotropic interface energies are included with fourfold symmetries, and the dilute alloy assumption is imposed. The isothermal results confirm the decrease in the maximum concentration for larger interface velocities as well as reveal the presence of parabolic, dendrite tips evolving along directions of maximum interface energy. The directional solidification results further confirm the formation of distinctive secondary dendritic arm structures that evolve at regular intervals along the unstable solid/liquid interface.
Issue Section:
Research Papers
References
1.
Maxwell
, I.
, and Hellawell
, A.
, 1975
, “A Simple Model for Grain Refinement During Solidification
,” Acta Metall.
, 23
(2
), pp. 229
–237
. 2.
Wolfram
, S.
, 1984
, “Cellular Automata as Models of Complexity
,” Nature
, 311
(4
), pp. 601
–644
. 3.
Saito
, Y.
, Goldbeck-Wood
, G.
, and Muller-Krumbhaar
, H.
, 1988
, “Numerical Simulation of Dendritic Growth
,” Phys. Rev. A
, 38
(4
), pp. 2148
–2157
. 4.
Spittle
, J. A.
, and Brown
, S. G. R.
, 1989
, “Computer Simulation of the Effects of Alloy Variables on the Grain Structures of Castings
,” Acta Metall.
, 37
(7
), pp. 1803
–1810
. 5.
Kobayashi
, R.
, 1993
, “Modeling and Numerical Simulations of Dendritic Crystal Growth
,” Phys. D
, 63
(3–4
), pp. 410
–423
. 6.
Wheeler
, A. A.
, Boettinger
, W. J.
, and McFadden
, G. B.
, 1992
, “Phase-Field Model for Isothermal Phase Transitions in Binary Alloy
,” Phys. Rev. E
, 45
(10
), pp. 7424
–7439
. 7.
Wheeler
, A. A.
, Boettinger
, A. A.
, and McFadden
, G. B.
, 1993
, “Phase Field Model of Trapping During Solidification
,” Phys. Rev. E
, 47
(4
), pp. 1893
–1909
. 8.
Kim
, S. G.
, Kim
, W. T.
, and Suzuki
, T.
, 1998
, “Interfacial Compositions of Solid and Liquid in a Phase-Field Model With Finite Interface Thickness for Isothermal Solidification in Binary Alloys
,” Phys. Rev. E
, 58
(3
), pp. 3316
–3323
. 9.
Kim
, S. G.
, Kim
, W. T.
, and Suzuki
, T.
, 1999
, “Phase-Field Model for Binary Alloys
,” Phys. Rev. E
, 60
(6
), pp. 7186
–7197
. 10.
Conti
, M.
, 1997
, “Solidification of Binary Alloys: Thermal Effects Studied With the Phase-Field Model
,” Phys. Rev. E
, 55
(1
), pp. 765
–771
. 11.
Loginova
, I.
, Amberg
, G.
, and Agern
, J.
, 2001
, “Phase-Field Simulation of Non-Isothermal Binary Alloy Solidification
,” Acta Mater.
, 49
(4
), pp. 573
–581
. 12.
Ghosh
, S.
, Ma
, L.
, Ofori-Opoku
, N.
, and Guyer
, J. E.
, 2017
, “On the Primary Spacing and Microsegregation of Cellular Dendrites in Laser Deposited Ni-Nb Alloys
,” Modell. Simul. Mater. Sci. Eng.
, 25
(6
), pp. 1
–25
. 13.
Ode
, M.
, Kim
, S. G.
, Kim
, W. T.
, and Suzuki
, T.
, 2001
, “Numerical Prediction of the Secondary Dendrite Arm Spacing Using a Phase-Field Model
,” ISIJ Int.
, 41
(4
), pp. 345
–349
. 14.
Caginalp
, G.
, and Xie
, W.
, 1993
, “Phase-Field and Sharp Interface Alloy Models
,” Phys. Rev. E
, 48
(3
), pp. 1897
–1909
. 15.
Hohenberg
, P. C.
, and Krekhov
, A. P.
, 2015
, “An Introduction to the Ginzburg-Landau Theory of Phase Transitions and Nonequilibrium Patterns
,” Phys. Rep.
, 572
(4
), pp. 1
–42
. 16.
Ferreira
, A. F.
, Ferreira
, I. L.
, Pereira da Cunha
, J.
, and Salvino
, I. M.
, 2015
, “Simulation of the Microstructural Evolution of Pure Material and Alloys in an Undercooled Melts via Phase-Field Method and Adaptive Computational Domain
,” Mater. Res.
, 18
(3
), pp. 644
–653
. 17.
Hu
, S.
, Baskes
, M.
, Stan
, M.
, and Mitchell
, J.
, 2007
, “Phase-Field Modeling of Coring Structure Evolution in PuGa Alloys
,” Acta Mater.
, 55
(11
), pp. 3641
–3648
. 18.
Euler
, H.
, Institutiones calculi integralis. Volumen Primum, Opera Omnia
, Vol. XI B G Teubneri Lipsiae et Berolini MCMXIII, 1768.19.
Mullis
, A. M.
, 2006
, “Quantification of Mesh Induced Anisotropy Effects in the Phase-Field Method
,” Comput. Mater. Sci.
, 36
(3
), pp. 345
–353
. 20.
Murray
, J. L.
, and McAllister
, A. J.
, 1984
, “The Al-Si (Aluminum-Silicon) System
,” Bull. Alloy Phase Diagrams
, 5
(1
), pp. 74
–84
. 21.
Ohno
, M.
, and Matsuura
, K.
, 2009
, “Quantitative Phase-Field Modeling for Dilute Alloy Solididication Involving Diffusion in the Solid
,” Phys. Rev. E
, 79
(3
), pp. 31603
. 22.
Sakane
, S.
, Takaki
, T.
, Ohno
, M.
, Shimokawabe
, T.
, and Aoki
, T.
, 2015
, “GPU-Accelerated 3D Phase-Field Simulations of Dendrite Competitive Growth During Directional Solidification of Binary Alloy
,” IOP Conv. Ser. Mater. Sci. Eng
, 84
(1
), pp. 1
–7
. 23.
Echebarria
, B.
, Folch
, R.
, Karma
, A.
, and Plapp
, M.
, 2004
, “Quantitative Phase-Field Model of Alloy Solidification
,” Phys. Rev. E
, 70
(6
), p. 61604
. 24.
Takaki
, T.
, Sakane
, S.
, Ohno
, M.
, Shibuta
, Y.
, Shimokawabe
, T.
, and Aoki
, T.
, 2016
, “Primary Arm Array During Directional Solidification of a Single-Crystal Binary Alloy: Large-Scale Phase-Field Study
,” Acta Mater.
, 118
(1
), pp. 230
–243
. 25.
Danilov
, D.
, and Nestler
, B.
, 2006
, “Phase-Field Modelling of Solute Trapping During Rapid Solidification of a Si-As Alloy
,” Acta Mater.
, 54
(18
), pp. 4659
–4664
. 26.
Pieters
, R.
, and Langer
, J. S.
, 1986
, “Noise-Driven Sidebranching in the Boundary-Layer Model of Dendritic Solidification
,” Phys. Rev. Lett.
, 56
(18
), pp. 1948
–1951
. 27.
Brener
, E.
, and Temkin
, D.
, 1995
, “Noise-Induced Sidebranching in the Three-Dimensional Nonaxisymmetric Dendritic Growth
,” Phys. Rev. E
, 51
(1
), pp. 351
–359
. 28.
Mullins
, W. W.
, and Sekerka
, R. F.
, 1964
, “Stability of a Planar Interface During Solidification of a Dilute Binary Alloy
,” J. Appl. Phys.
, 35
(2
), pp. 444
–451
. 29.
Badillo
, A.
, and Beckermann
, C.
, 2006
, “Phase-Field Simulation of the Columnar-to-Equiaxed Transition in Alloy Solidification
,” Acta Mater.
, 54
(8
), pp. 2015
–2026
. 30.
Kattamis
, T. Z.
, and Flemmings
, M. C.
, 1965
, “Dendrite Morphology, Microsegregation and Homogenization of 4340 Low Alloy Steel
,” Trans. TMS-AIME
, 233
(1
), pp. 992
–999
.31.
Zhu
, J. Z.
, Wang
, T.
, Zhou
, S. H.
, Liu
, Z. K.
, and Chen
, L. Q.
, 2004
, “Quantitative Interface Models for Simulating Microstructure Evolution
,” Acta Mater.
, 52
(4
), pp. 833
–840
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