Precipitation Modeling via the Synergy of Thermodynamics and Kinetics

doi:10.11900/0412.1961.2020.00413

Acta Metall Sin

2021, Vol. 57

Issue (1): 55-70 DOI: 10.11900/0412.1961.2020.00413

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Precipitation Modeling via the Synergy of Thermodynamics and Kinetics

LIU Feng¹^,²(

), WANG Tianle¹

^1.State Key Laboratory of Solidification Processing, Northwestern Polytechnical University ;Xi'an 710072, China
^2.Analytical & Testing Center, Northwestern Polytechnical University ;Xi'an 710072, China

Cite this article:

LIU Feng, WANG Tianle. Precipitation Modeling via the Synergy of Thermodynamics and Kinetics. Acta Metall Sin, 2021, 57(1): 55-70.

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Abstract

Owing to the critical role precipitation hardening plays in the improved mechanical performance of metals, understanding the formation mechanisms of precipitates is significant for the rational control of the corresponding and correlated effects. From the perspective of the synergetic variation of thermodynamics and kinetics, the current work briefly reviews the mesoscale methods for precipitation modeling based on the computational thermodynamics of CALPHAD (including the DICTRA simulation, Kampmann-Wagner numerical model, Svoboda-Fischer-Fratzl-Kozeschnik model, and diffusion field cell model) and the multiscale methods based on first-principles calculations (including the phase field model and multiscale structural modeling using the Fokker-Planck equation). On this basis, the research and development of precipitation modeling for heat-treated metals is discussed in detail.

Key words: precipitate thermodynamics kinetics structural modeling

Received: 16 October 2020

ZTFLH:

TG113.12

Fund: National Key Research and Development Program of China(2017YFB0703001);National Natural Science Foundation of China(51134011);China Postdoctoral Science Foundation(2018M643729);Natural Science Basic Research Plan in Shaanxi Province(2019JQ-091);Research Fund of the State Key Laboratory of Solidification Processing(2019-TZ-01)

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https://www.ams.org.cn/EN/10.11900/0412.1961.2020.00413 OR https://www.ams.org.cn/EN/Y2021/V57/I1/55

Fig.1 Schematic representation of the Gibbs energy changes associated with precipitate formation as a function of their radius R in the classical nucleation theory (ΔG^* is the nucleation barrier, R^* is the critical radius for stable precipitates, $R k B T *$ is the radius at which stable precipitates nucleate, Z is the Zeldovich factor, k_B is the Boltzmann constant and T is the temperature)^[11]

Fig.2 Work flow for DICTRA simulation^[37]

Fig.3 Three-cell simulation of competitive growth of stable M₂₃C₆ and of metastable M₇C₃ and M₃C (M represents the substitutional alloying elements) in Fe-12Cr-0.1C at 1053 K (μ_i—chemical potential of component i)^[39]

Fig.4 Log-normal particle size distribution with a mean particle radius $R ˉ$ (Particles with radius R<R^* are continuously dissolving in the matrix. In the Langer and Schwartz approach, only the hatched area with particle radius R>R^* contributes to the mean particle radius $R ˉ L S$ . $R ˉ$ is the mean radius for the full log-normal distribution, $R ˉ L S$ is the mean radius of the stable particles (R>R^*), f(R) is the particle size distribution and f_a(R^*) is the density of the particles with the critical nucleation size)^[46]

Fig.5 Growth of precipitates in the “Euler-like approach” (At each time step, fluxes between each neighboring classes are calculated. N is the number of particles in a specific size class, and $d R d t$ is the growth rate)^[11]

Fig.6 The obtained precipitation sequences in a Fe-Mn-Si-Cr-Mo-Ti-C system at 600℃ via SFFK model using different interface energies^[29]

Fig.7 Schematic of the multiscale model, showing different constituent models and their links along with their associated length scales (F_vibrational—free energy contributed from lattice vibration, F_{configurational}—configurational free energy)^[93]

Fig.8 Phase-field simulation using thermodynamic parameters from first principles, showing θ' morphologies obtained with different anisotropic contributions^[94]

Fig.9 Schematic of an integrable deep neural network (X_k is the k-th input parameter, Y is the output of deep neural network, $? Y ? X k$ is the output of integrable deep neural network, α and β are adjustable parameters of the hidden layers)^[98]

Fig.10 The size evolution of θ' at 473 K (a) and the precipitation sequence, i.e. GP zone→θ"→θ' (b) in Al-2%Cu (atomic fraction) alloy (The abbreviations FP, MD, FPE, and PFM represent first-principles, molecular dynamics, Fokker-Planck equation, and phase field method, respectively)^[30]

Fig.11 Evolution for the energy of system and the number of primary building units (PBUs) in carbides during precipitations of ε-Fe₂C, η-Fe₂C, and θ-Fe₃C in Fe-2%C (atomic fraction) alloy precipitated at 473 K (τ₁ represents the moment for ε-Fe₂C→η-Fe₂C transition corresponding to the point o, and τ₂ represents the time needed for η-Fe₂C to grow to the same size as ε-Fe₂C corresponding to the points F and Z)^[58]

Fig.12 Variations of the driving force (thermodynamics) and energy barrier (kinetics) with respect to the number of PBUs in precipitates and temperature for precipitations of ε-Fe₂C and η-Fe₂C^[58]

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