Numerical Modeling of Dendrite Growth During Directional Solidification: A Review

doi:10.11900/0412.1961.2025.00301

Acta Metall Sin

2026, Vol. 62

Issue (5): 756-769 DOI: 10.11900/0412.1961.2025.00301

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Numerical Modeling of Dendrite Growth During Directional Solidification: A Review

DONG Hongbiao¹(

), HAO Wenshuo²

1 School of Metallurgy and Materials, University of Birmingham, Birmingham, B15 2TT, UK
2 School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China

Cite this article:

DONG Hongbiao, HAO Wenshuo. Numerical Modeling of Dendrite Growth During Directional Solidification: A Review. Acta Metall Sin, 2026, 62(5): 756-769.

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Abstract

Solidification is a fundamental process in advanced manufacturing, playing a pivotal role in determining the microstructure and performance of components. Directional solidification is a critical technique for producing single-crystal superalloy components; however, precise control of dendritic structure remains challenging. This article aims to systematically review the current status and future trends of dendrite growth simulations in directional solidification. In addition, it seeks to elucidate the key role of multiscale modeling in predicting dendritic morphology, dendrite competition, and defect formation. This article focuses on dendritic growth and summarizes the constitutional undercooling theory, dendrite growth models, and typical directional solidification processes. Multiscale numerical simulation methods, including finite element, finite difference, cellular automaton, and phase-field approaches, are emphasized, with an analysis of their applicability and a discussion of their representative achievements from macroscopic physical-field simulation to microscopic dendrite growth modeling. Finally, the key challenges are outlined, and prospective advancements in the numerical simulation of dendrite growth in directional solidification are delineated.

Key words: solidification dendrite growth directional solidification numerical simulation

Received: 30 September 2025

ZTFLH:

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Corresponding Authors: DONG Hongbiao, professor, Tel: 0044(116)2522528, E-mail: h.dong.1@bham.ac.uk

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https://www.ams.org.cn/EN/10.11900/0412.1961.2025.00301 OR https://www.ams.org.cn/EN/Y2026/V62/I5/756

Fig.1 Solute concentration at the solidification front (solid/liquid interface) (a) and constitutional undercooling zone indicated by the shadow region (b) (C—compositon; C_S, C_L, and C_∞—compositions of solid, liquid, and alloys, respectively. Z—distance from solid/liquid interface. T—temperature, T(C_L)—equilibrium liquidus temperature at the solidification front, T(C_∞)—equilibrium liquidus temperature corresponding to C_∞. G—applied thermal gradient, G_T_L—thermal gradient of equilibrium liquidus temperature)

Fig.2 Schematic of dendrite growth illustrating the mechanism of competitive growth during directional solidification^[19] (θ—misorientation of the <100> cry-stallographic direction relative to the heat-flow direction. v—velocity, v_L—velocity of the liquidus isotherm, v_θ —growth velocity of dendrites with orientation θ. z—coordinate along the heat-flow direction. Δz₀, Δz_N, and Δz_θ —distances from the dendrite (or equiaxed grain) tip to the liquidus isotherm, for dendrite parallel to the heat-flow direction, for an equiaxed grain, and for dendrite at θ, respectively. ΔT—undercooling)

Fig.3 Schematics of directional solidification process^[17]
(a) high rate solidification (b) liquid metal cooling

Table 1 Multiscale numerical modeling methods for directional solidification and dendritic growth

Fig.4 Temperature fields and mushy zone evolutions during single-crystal blade fabrication by high rate solidification at a constant withdrawal rate of 3 mm/min (a1-a5) and liquid metal cooling at a constant withdrawal rate of 8 mm/min (b1-b5)^[18]
(a1) t = 241 s, f_s = 30% (t—solidification time, f_s—solid volume fraction)
(a2) t = 2028 s, f_s = 40% (a3) t = 2411 s, f_s = 50%
(a4) t = 2632 s, f_s = 60% (a5) t = 2987 s, f_s = 80%
(b1) t = 205 s, f_s = 30% (b2) t = 689 s, f_s = 40%
(b3) t = 910 s, f_s = 50% (b4) t = 1031 s, f_s = 60%
(b5) t = 1193 s, f_s = 80%

Fig.5 Numerical simulation results of solidification sequence and flow pattern (a1-a3), solute distributions on the convex (b1-b3), and concave (c1-c3) sides of the aerofoil^[14] (SI—segregation index)
(a1-c1) t = 100 s
(a2-c2) t = 300 s s
(a3-c3) t = 500

Fig.6 Cellular automaton finite element (CAFE) predicted results of 3D grain structures in the starter block of grain selector (a), the cross-section grain structures at different heights from the base (b1-e1) and corresponding <100> pole figures (b2-e2) at the height from the base of b—2 mm, c—5 mm, d—10 mm, e—29 mm shown in Fig.6a^[64]

Fig.7 CAFE predicted results of 3D grain structures in a spiral selector (a), the cross-section grain structures (b1-f1) and corresponding <001> pole figures along the direction of heat flux (b2-f2), and grain deviation distributions (b3-f3) for the cross sections at the height of b—0.1 mm, c—1.1 mm, d—6.25 mm, e—11.5 mm, and f—16.5 mm from the bottom shown in Fig.7a^[9]

Fig.8 Cellular automaton finite difference (CAFD) simulation results of columnar-to-equiaxed transition showing the typical predicted microstructures for an Al-3%Cu alloy at time of 5 s (a), 14 s (b), 17 s (c), 19 s (d), 20 s (e), and 21 s (f)^[88]

Fig.9 Abnormal elimination of dendrites on the converging side during directional solidification simulated by phase-field method^[40] (θ_UO—unfavorably oriented (UO) grain inclination angle)
(a) 0.05 × 10⁶th step (1.3 s) (b) 0.25 × 10⁶th step (6.7 s) (c) 10⁶th step (26.8 s)
(d) 3 × 10⁶th step (80.4 s) (e) 7 × 10⁶th step (187.5 s)

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