Quantitative phase-field simulation of precipitation kinetics of γ′ phase in multicomponent Co-based superalloys

doi:10.1016/j.metadv.2026.02.021

Abstract

Abstract:

Co-Al-W-based alloys have been recognized as promising next-generation structural materials for high-temperature applications due to the formation of the γ′ strengthened phase. However, optimizing these superalloys through experimental studies is a significant challenge because of complex element interactions. This paper investigates the temporal processing and mechanisms of γ′ precipitates in novel Co-based superalloys using the phase-field method. By coupling the calculation of phase diagrams (CALPHAD) approach for the phase free energy and chemical mobility involving element interactions, and by calculating thermophysical parameters of alloys using first-principles, we developed a quantitative phase-field model for multicomponent systems. Using this model, the element diffusion path and γ′ evolution path are simulated in ternary diffusion couples and aging alloys, and the results align well with experimental observations. Furthermore, the effects of W content in Co-Al-W alloys on the coarsening kinetics and mechanisms of γ′ precipitates during long-term aging are systematically studied. It is found that increasing W content results in higher γ′ volume fraction and increased γ′ coarsening rate. The accelerated coarsening is primarily attributed to the shortened inter-particle spacing between γ′ precipitates, which is comprehensively analyzed through element diffusion distance and flux. Additionally, the model is extended to quaternary systems and successfully applied to Co-Ni-Al-W alloy. This study provides a novel method for the quantitative prediction of γ/γ′ microstructures and contributes to the alloy design and processing optimization of novel superalloys.

Key words: Co-based superalloys, γ′ precipitate, Phase-field model, Calculation of phase diagrams method, Multicomponent alloy

Jia Chen, Min Guo, Tingting Cui, Xingbing Zhang, Fusheng Tan, Min Yang, Wenchao Yang, Taiwen Huang, Haijun Su, Lin Liu. Quantitative phase-field simulation of precipitation kinetics of γ′ phase in multicomponent Co-based superalloys[J]. Metals Advances, 2026, 45: 52-66.

Figures/Tables 18

Table 1. Thermodynamic parameters of γ and γ′ phases in Co-Al-W ternary alloy (Unit: J/mol) [25].

Fcc — γ phase
GAlfcc= −11278.378 + 188.684153T − 31.748192TlnT − 1.231 × 10²⁸T⁻⁹ GCofcc= 310.241 + 133.36601T − 25.0861TlnT − 2.654739 × 10⁻³T² − 1.7348 × 10⁻⁷T³ + 72526.9T⁻¹ GWfcc= −7646.311 + 130.4T − 24.1TlnT − 1.936 × 10⁻³T² + 2.07 × 10⁻⁷T³ − 5.33 × 10⁻¹¹T⁴ + 44500T⁻¹
L0Al,Cofcc= −124200 + 17.24T, L1Al,Cofcc= 0, L2Al,Cofcc= −28740; L0Al,Wfcc= −45000 + 14.6469T L0Co,Wfcc= −16432.7 + 11.3688T, L1Co,Wfcc= −22554.7 + 3.3885T L0Al,Co,Wfcc= −70808.4, L2Al,Co,Wfcc= 3454298.5
L1₂ — γ′ phase
UAlCo= −3088.7, UAlW= 17463.8, UCoW= −4996.2 LAl,Co1= 10116.5 − 3.7079T, LCo,W1= 1440.1 − 10.0T

Fig. 1. Gibbs free energy of γ and γ′ phases used in phase-field simulation. (a) Energy surfaces as a function of cW and cAl, in which black solid lines represent contour lines, (b) energy curves as a function of cW for specific cAl. From the side of high W concentration, the curve with high free energy belongs to γ phase, and the other is γ′ phase.

Fig. 2. Schematic of three typical supercells of FCC structure (a) Co24W8, (b) Co24Al4W4, (c) Co24Co2Al3W3; and the elastic constants Cij in the pseudo-ternary Co-Co3Al-Co3W system calculated by Materials Studio software: (d) C11, (e) C12, (f) C44.

Table 2. Elastic constants for studied alloys.

Alloy	Elastic constant of γ phase (GPa)			Elastic constant of γ′ phase (GPa)
Alloy	$C_{11}^{\mathrm{m}}$	$C_{12}^{\mathrm{m}}$	$C_{44}^{\mathrm{m}}$	$C_{11}^{\mathrm{p}}$	$C_{12}^{\mathrm{p}}$	$C_{44}^{\mathrm{p}}$
Co-9Al-8W	315	-	-	361	-	-
Co-9Al-9W	316	209	160	363	190	212
Co-9Al-10W	317	-	-	364	-	-

Table 3. Diffusion activation energy parameters of Co-Al-W ternary alloy [43].

Empty Cell	$Q_{i}^{\mathrm{Co}}$ (J/mol)	$Q_{i}^{\mathrm{Al}}$ (J/mol)	$Q_{i}^{\mathrm{W}}$ (J/mol)
i = Co	−286175 − 75.98T	−211490 − 61.48T	−359115 − 8.98T
i = Al	−247067 − 103.79T	−126719 − 95.09T	−253494 − 101.01T
i = W	−222104 − 134.55T	−347963 − 52.88T	−311423 − 70.05T

Fig. 3. (a) Diffusion couple composition and (b) corresponding penetration curves predicted using established phase-field model in γ single phase region at 1250 °C for Co-Al-W ternary alloy. Simulated and experimental results of composition profile of the diffusion couples of (a) Co/Co-3Al-10W at 1573 K for 7200 s, (b) Co-8Al/Co-4W at 1173 K for 86400 s λ in Fig. 3(c) represents two times of diffusion distance of element W. Experimental data Fig. 3(c) and (d) are obtained from Ref. [47].

Fig. 4. Evolution of single γ′ precipitate. Element distribution of (a) Al, (b) W and (c) Co; element profiles of (d) Al, (e) W and (f) Co along the white horizontal dotted lines; temporal evolution of (g) γ′ volume fraction, (h) γ′ equivalent radius, and (i) γ′ shape factor.

Fig. 5. Evolution of multiple γ′ precipitates. (a) W concentration field, (b) order parameter fields, (c) γ′ volume fraction, (d) γ′ equivalent radius, and (e) γ′ shape factor. The two coarsening mechanisms are highlighted by dashed ellipses: Ostwald ripening, characterized by the growth of large precipitates at the expense of small ones, is marked by red dashed ellipses; Coalescence, characterized by the merging of adjacent particles, is indicated by white dashed ellipses.

Fig. 6. Distribution of elastic strain energy density at 2.5 × 105 s: (a) single γ′ precipitate system, (b) multiple γ′ precipitates system.

Fig. 7. γ/γ′ two-phase aging microstructures at 900 °C from (a-c) simulation and (d-f) experiment [54] for Co-Al-W ternary alloys with different W content. (a, d) 8 W, (b, e) 9 W, (c) 10 W, and (f) 11 W (Co-9Al-11W).

Fig. 8. Microstructure characteristics for alloys with different W content. (a) Shape factor at 10.0 × 105 s, (b) evolution of γ′ volume fraction obtained from phase-field simulation, (c) comparison of γ′ volume fraction obtained from simulation, calculation and experiment [10], [52], [54].

Fig. 9. Temporal evolutions of γ′ microstructure characteristics during aging at 900 °C. (a1-a4) γ′ mean equivalent radius, <R>, (b1-b4) particle number density, NV. Scatters are simulation results, and solid lines are fitting results.

Fig. 10. γ′ particle size distribution (PSD) at (a1-c1) 2.5 × 105 s, (a2-c2) 10.0 × 105 s for alloys (a1, a2) 8 W, (b1, b2) 9 W, (c1, c2) 10 W. The gray dashed line denotes the peak position, and the black dashed line and red dash dot represent the theoretical PSD of MSLW and LSEM model [55], respectively.

Fig. 11. Plots of <R(t)>3 −<R(t0)>3 vs. t for Co-9Al-xW (x = 8, 9, 10 at.%). The slope of linear fitting gives KR of γ′ precipitate.

Fig. 12. Temporal evolution of average inter-particle spacing of precipitates, <λ>, during coarsening regime for Co-9Al-xW (x = 8, 9, 10 at.%) alloy.

Fig. 13. Evolutions of mean chemical mobility in γ matrix during coarsening regime: (a) <MAlAl>, (b) <MAlW>, (c) <MWW>.

Fig. 14. Evolutions of (a, b) diffusion potential and (c, d) γ/γ′ two-phase diffusion potential difference for the 8 W, 9 W, and 10 W alloys. (a) δF/δcAl, (b) δF/δcW, (c) $\Delta g_{\mathrm{Al}}^{\gamma^{\prime} / \gamma}$, (d) $\Delta g_{\mathrm{W}}^{\gamma^{\prime} / \gamma}$.

Fig. 15. Temporal evolution of γ/γ′ two-phase microstructures of Co-10Ni-10Al-7.5W (at.%) aging at 900 °C. Element distribution of (a) Co, (b) Ni, (c) Al, and (d) W, and right column shows corresponding compositional profile at γ/γ′ interface along white lines.

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