Precipitation Modeling via the Synergy of Thermodynamics and Kinetics
Received date: 2020-10-16
Revised date: 2020-11-10
Online published: 2020-11-16
Supported by
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)
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
Feng LIU , Tianle WANG . Precipitation Modeling via the Synergy of Thermodynamics and Kinetics[J]. Acta Metall Sin, 2021 , 57(1) : 55 -70 . DOI: 10.11900/0412.1961.2020.00413
| 1 | Liu F, Wang K. Discussions on the correlation between thermodynamics and kinetics during the phase transformations in the TMCP of low-alloy steels [J]. Acta Metall. Sin., 2016, 52: 1326 |
| 1 | 刘 峰, 王 慷. 低合金钢TMCP中相变热力学/动力学相关性探讨 [J]. 金属学报, 2016, 52: 1326 |
| 2 | Wang K, Shang S L, Wang Y, et al. Martensitic transition in Fe via Bain path at finite temperatures: A comprehensive first-principles study [J]. Acta Mater., 2018, 147: 261 |
| 3 | Liu Z K. Ocean of data: Integrating first-principles calculations and CALPHAD modeling with machine learning [J]. J. Phase Equilib. Diffus., 2018, 39: 635 |
| 4 | Luo Q, Chen H C, Chen W, et al. Thermodynamic prediction of martensitic transformation temperature in Fe-Ni-C system [J]. Scr. Mater., 2020, 187: 413 |
| 5 | Hohenberg P, Kohn W. Inhomogeneous electron gas [J]. Phys. Rev., 1964, 136: B864 |
| 6 | Kohn W, Sham L J. Self-consistent equations including exchange and correlation effects [J]. Phys. Rev., 1965, 140: A1133 |
| 7 | de Pablo J J, Jones B, Kovacs C L, et al. The materials genome initiative, the interplay of experiment, theory and computation [J]. Curr. Opin. Solid State Mater. Sci., 2014, 18: 99 |
| 8 | de Pablo J J, Jackson N E, Webb M A, et al. New frontiers for the materials genome initiative [J]. npj Comput. Mater., 2019, 5: 41 |
| 9 | Wagner R, Kampmann R, Voorhees P W. Homogeneous second-phase precipitation [A]. Phase Transformations in Materials [M]. New York: Wiley-VCH, 2001 |
| 10 | Svoboda J, Turek I, Fischer F D. Application of the thermodynamic extremal principle to modeling of thermodynamic processes in material sciences [J]. Philos. Mag., 2005, 85: 3699 |
| 11 | Perez M, Dumont M, Acevedo-Reyes D. Implementation of classical nucleation and growth theories for precipitation [J]. Acta Mater., 2008, 56: 2119 |
| 12 | Volmer M, Weber A. Keimbildung in übers?ttigten gebilden [J]. Z. Phys. Chem., 1926, 119: 277 |
| 13 | Becker R, D?ring W. Kinetische behandlung der keimbildung in übers?ttigten d?mpfen [J]. Ann. Phys., 1935, 416: 719 |
| 14 | Zeldovich Y B. On the theory of new phase formation: Cavitation [J]. Acta Physicochim., 1943, 18: 1 |
| 15 | Russell K C. Phase Transformations [M]. Ohio: American Society for Metals, 1968: 219 |
| 16 | Kampmann R, Wagner R. Kinetics of precipitation in metastable binary alloys—Theory and application to Cu-1.9 at % Ti and Ni-14 at % Al [A]. Decomposition of Alloys: The Early Stages [M]. Oxford, U.K.: Pergamon Press, 1984: 91 |
| 17 | Zener C. Theory of growth of spherical precipitates from solid solution [J]. J. Appl. Phys., 1949, 20: 950 |
| 18 | Perez M. Gibbs-Thomson effects in phase transformations [J]. Scr. Mater., 2005, 52: 709 |
| 19 | Voorhees P W. The theory of Ostwald ripening [J]. J. Stat. Phys., 1985, 38: 231 |
| 20 | Lifchitz I M, Slyosov V V. The kinetics of precipitation from supersaturated solid solutions [J]. J. Phys. Chem. Solids, 1961, 19: 35 |
| 21 | Wagner C. Theorie der Alterung von Niederschl?gen durch Uml?sen (Ostwald-Reifung) [J]. Z. Elektrochem., 1961, 65: 581 |
| 22 | Onsager L. Reciprocal relations in irreversible processes. I [J]. Phys. Rev., 1931, 37: 405 |
| 23 | Ziegler H. Some extremum principles in irreversible thermodynamics with applications to continuum mechanics [A]. Progress in Solid Mechanics [M]. Amsterdam: North-Holland, 1963: 1 |
| 24 | Ziegler H. An Introduction to Thermomechanics [M]. Amsterdam: North-Holland, 1977: 1 |
| 25 | Fischer F D, Svoboda J, Petryk H. Thermodynamic extremal principles for irreversible processes in materials science [J]. Acta Mater., 2014, 67: 1 |
| 26 | Hackl K, Fischer F D. On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials [J]. Proc. Roy. Soc., 2008, 464A: 117 |
| 27 | Fischer F D, Svoboda J. Thermodynamic treatment of diffusive phase transformation (reactive diffusion) [A]. Handbook of Solid State Diffusion [M]. Amsterdam: Elsevier, 2017: 391 |
| 28 | Svoboda J, Fischer F D, Fratzl P, et al. Modelling of kinetics in multi-component multi-phase systems with spherical precipitates: I: Theory [J]. Mater. Sci. Eng., 2004, A385: 166 |
| 29 | Kozeschnik E, Svoboda J, Fratzl P, et al. Modelling of kinetics in multi-component multi-phase systems with spherical precipitates: II: Numerical solution and application [J]. Mater. Sci. Eng., 2004, A385: 157 |
| 30 | Wang K, Zhang L, Liu F. Multi-scale modeling of the complex microstructural evolution in structural phase transformations [J]. Acta Mater., 2019, 162: 78 |
| 31 | Liu Z K. First-principles calculations and CALPHAD modeling of thermodynamics [J]. J. Phase Equilib. Diffus., 2009, 30: 517 |
| 32 | Saunders N, Miodownik A P. CALPHAD: Calculation of Phase Diagrams: A Comprehensive Guide [M]. Berlin: Pergamon, 1998: 1 |
| 33 | Kaufman L, ?gren J. CALPHAD, first and second generation—Birth of the materials genome [J]. Scr. Mater., 2014, 70: 3 |
| 34 | Olson G B, Kuehmann C J. Materials genomics: From CALPHAD to flight [J]. Scr. Mater., 2014, 70: 25 |
| 35 | Andersson J O, ?gren J. Models for numerical treatment of multicomponent diffusion in simple phases [J]. J. App. Phys., 1992, 72: 1350 |
| 36 | Spencer P J. A brief history of CALPHAD [J]. Calphad, 2008, 32: 1 |
| 37 | Zhang L J, Chen Q. CALPHAD—Type modeling of diffusion kinetics in multicomponent alloys [A]. Handbook of Solid State Diffusion [M]. Amsterdam: Elsevier, 2017: 321 |
| 38 | Thomson R C. Characterization of carbides in steels using atom probe field-ion microscopy [J]. Mater. Charact., 2000, 44: 219 |
| 39 | Schneider A, Inden G. Simulation of the kinetics of precipitation reactions in ferritic steels [J]. Acta Mater., 2005, 53: 519 |
| 40 | Hu X B, Zhang M, Wu X C, et al. Simulations of coarsening behavior for M23C6 carbides in AISI H13 steel [J]. J. Mater. Sci. Technol., 2006, 22: 153 |
| 41 | Bj?rbo A, H?ttestrand M. Complex carbide growth, dissolution, and coarsening in a modified 12 pct chromium steel—An experimental and theoretical study [J]. Metall. Mater. Trans., 2001, 32A: 19 |
| 42 | Bratberg J, ?gren J, Frisk K. Diffusion simulations of MC and M7C3 carbide coarsening in bcc and fcc matrix utilising new thermodynamic and kinetic description [J]. Mater. Sci. Technol., 2008, 24: 695 |
| 43 | Sanhueza J P, Rojas D, Prat O, et al. Precipitation kinetics in a 10.5%Cr heat resistant steel: Experimental results and simulation by TC-PRISMA/DICTRA [J]. Mater. Chem. Phys., 2017, 200: 342 |
| 44 | Li Y, Holmedal B, Li H X, et al. Precipitation and strengthening modeling for disk-shaped particles in aluminum alloys: Size distribution considered [J]. Materialia, 2018, 4: 431 |
| 45 | Langer J S, Schwartz A J. Kinetics of nucleation in near-critical fluids [J]. Phys. Rev., 1980, 21A: 948 |
| 46 | Zhao D D, Xu Y J, Gouttebroze S, et al. Modelling the age-hardening precipitation by a revised Langer and Schwartz approach with log-normal size distribution [J]. Metall. Mater. Trans., 2020, 51A: 4838 |
| 47 | Kampmann R, Eckerlebe H, Wagner R. Precipitation kinetics in metastable solid solutions—Theoretical considerations and application to Cu-Ti alloys [J]. Mater. Res. Soc. Symp. Proc., 1987, 57: 525 |
| 48 | Robson J D. Modelling the evolution of particle size distribution during nucleation, growth and coarsening [J]. Mater. Sci. Technol., 2004, 20: 441 |
| 49 | Myhr O R, ? Grong. Modelling of non-isothermal transformations in alloys containing a particle distribution [J]. Acta Mater., 2000, 48: 1605 |
| 50 | Nicolas M, Deschamps A. Characterisation and modelling of precipitate evolution in an Al-Zn-Mg alloy during non-isothermal heat treatments [J]. Acta Mater., 2003, 51: 6077 |
| 51 | Du Q, Poole W J, Wells M A. A mathematical model coupled to CALPHAD to predict precipitation kinetics for multicomponent aluminum alloys [J]. Acta Mater., 2012, 60: 3830 |
| 52 | Hasting H S, Fr?seth A G, Andersen S J, et al. Composition of β″ precipitates in Al-Mg-Si alloys by atom probe tomography and first principles calculations [J]. J. Appl. Phys., 2009, 106: 123527 |
| 53 | Biswas A, Siegel D J, Wolverton C, et al. Precipitates in Al-Cu alloys revisited: Atom-probe tomographic experiments and first-principles calculations of compositional evolution and interfacial segregation [J]. Acta Mater., 2011, 59: 6187 |
| 54 | Ohmori Y, Tamura I. Epsilon carbide precipitation during tempering of plain carbon martensite [J]. Metall. Trans., 1992, 23A: 2737 |
| 55 | Holmedal B, Osmundsen E, Du Q. Precipitation of non-spherical particles in aluminum alloys part I: Generalization of the Kampmann-Wagner numerical model [J]. Metall. Mater. Trans., 2016, 47A: 581 |
| 56 | Chen Q, Wu K S, Sterner G, et al. Modeling precipitation kinetics during heat treatment with calphad-based tools [J]. J. Mater. Eng. Perform., 2014, 23: 4193 |
| 57 | Rojhirunsakool T, Meher S, Hwang J Y, et al. Influence of composition on monomodal versus multimodal γ′ precipitation in Ni-Al-Cr alloys [J]. J. Mater. Sci., 2013, 48: 825 |
| 58 | Wang T L, Du J L, Liu F. Modeling competitive precipitations among iron carbides during low-temperature tempering of martensitic carbon steel [J]. Materialia, 2020, 12: 100800 |
| 59 | Robson J D. Modelling the overlap of nucleation, growth and coarsening during precipitation [J]. Acta Mater., 2004, 52: 4669 |
| 60 | Popov V V, Gorbachev I I, Pasynkov A Y. Simulation of precipitates evolution in multiphase multicomponent systems with consideration of nucleation [J]. Philos. Mag., 2016, 96: 3632 |
| 61 | Popov V V, Gorbachev I. Simulation of the evolution of precipitates in multicomponent alloys [J]. Phys. Met. Metallogr., 2003, 95: 417 |
| 62 | Popov V V, Gorbachev I I, Alyabieva J A. Simulation of VC precipitate evolution in steels with consideration for the formation of new nuclei [J]. Philos. Mag., 2005, 85: 2449 |
| 63 | Kozeschnik E, Svoboda J, Fischer F D. Modified evolution equations for the precipitation kinetics of complex phases in multi-component systems [J]. Calphad, 2004, 28: 379 |
| 64 | Cerjak H. Mathematical Modelling of Weld Phenomena 5 [M]. London: CRC Press, 2001: 349 |
| 65 | Leitner H, Bischof M, Clemens H, et al. Precipitation behaviour of a complex steel [J]. Adv. Eng. Mater., 2006, 8: 1066 |
| 66 | Shim J H, Povoden-Karadeniz E, Kozeschnik E, et al. Modeling precipitation thermodynamics and kinetics in type 316 austenitic stainless steels with varying composition as an initial step toward predicting phase stability during irradiation [J]. J. Nucl. Mater., 2015, 462: 250 |
| 67 | Kim M Y, Chu D J, Lee Y S, et al. Mechanical property change and precipitate evolution during long-term aging of 1.25Cr-0.5Mo steel [J]. Mater. Sci. Eng., 2020, A789: 139663 |
| 68 | McDowell D L. Microstructure-sensitive computational structure-property relations in materials design [A]. Computational Materials System Design [M]. Cham: Springer International Publishing, 2018: 1 |
| 69 | Chen W. High-throughput computing for accelerated materials discovery [A]. Computational Materials System Design [M]. Cham: Springer International Publishing, 2018: 169 |
| 70 | Hafner J. Ab-initio simulations of materials using VASP: Density-functional theory and beyond [J]. J. Comput. Chem., 2008, 29: 2044 |
| 71 | Giannozzi P, Baroni S, Bonini N, et al. QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials [J]. J. Phys.: Condens. Matter, 2009, 21: 395502 |
| 72 | Gonze X, Amadon B, Anglade P M, et al. ABINIT: First-principles approach to material and nanosystem properties [J]. Comput. Phys. Commun., 2009, 180: 2582 |
| 73 | Fang C M, Van Huis M A, Zandbergen H W. Structure and stability of Fe2C phases from density-functional theory calculations [J]. Scr. Mater., 2010, 63: 418 |
| 74 | Barrow A T W, Kang J H, Rivera-Díaz-del-Castillo P E J. The ε→η→θ transition in 100Cr6 and its effect on mechanical properties [J]. Acta Mater., 2012, 60: 2805 |
| 75 | Medvedeva N I, Van Aken D C, Medvedeva J E. Stability of binary and ternary M23C6 carbides from first principles [J]. Comput. Mater. Sci., 2015, 96: 159 |
| 76 | Konyaeva M A, Medvedeva N I. Electronic structure, magnetic properties, and stability of the binary and ternary carbides (Fe, Cr)3C and (Fe, Cr)7C3 [J]. Phys. Solid State, 2009, 51: 2084 |
| 77 | Ande C K, Sluiter M H F. First-principles prediction of partitioning of alloying elements between cementite and ferrite [J]. Acta Mater., 2010, 58: 6276 |
| 78 | Liu Y Z, Jiang Y H, Xing J D, et al. Mechanical properties and electronic structures of M23C6 (M=Fe, Cr, Mn)-type multicomponent carbides [J]. J. Alloys Compd., 2015, 648: 874 |
| 79 | Sanchez J M. Cluster expansions and the configurational energy of alloys [J]. Phys. Rev., 1993, 48B: 14013 |
| 80 | van de Walle A, Asta M. Self-driven lattice-model Monte Carlo simulations of alloy thermodynamic properties and phase diagrams [J]. Modell. Simul. Mater. Sci. Eng., 2002, 10: 521 |
| 81 | Wang Y, Hector L G, Zhang H, et al. Thermodynamics of the Ce γ-α transition: Density-functional study [J]. Phys. Rev., 2008, 78B: 104113 |
| 82 | Wang Y, Hector L G, Zhang H, et al. A thermodynamic framework for a system with itinerant-electron magnetism [J]. J. Phys.: Condens. Matter, 2009, 21: 326003 |
| 83 | Fang C M, Sluiter M H F, van Huis M A, et al. Origin of predominance of cementite among iron carbides in steel at elevated temperature [J]. Phys. Rev. Lett., 2010, 105: 055503 |
| 84 | Kaplan B, Blomqvist A, ?rhammar C, et al. Structural determination of (Cr, Co)7C3[A]. 18th Plansee Seminar [C]. Reutte, Austria, 2013: 1 |
| 85 | Chen L Q. Phase-field models for microstructure evolution [J]. Annu. Rev. Mater. Res., 2002, 32: 113 |
| 86 | Cahn J W. On spinodal decomposition [J]. Acta Metall., 1961, 9: 795 |
| 87 | Allen S M, Cahn J W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening [J]. Acta Metall., 1979, 27: 1085 |
| 88 | Wang Y Z, Li J. Phase field modeling of defects and deformation [J]. Acta Mater., 2010, 58: 1212 |
| 89 | Moelans N, Blanpain B, Wollants P. An introduction to phase-field modeling of microstructure evolution [J]. Calphad, 2008, 32: 268 |
| 90 | Steinbach I. Phase-field models in materials science [J]. Modell. Simul. Mater. Sci. Eng., 2009, 17: 073001 |
| 91 | Xiong H, Huang Z H, Wu Z Y, et al. A generalized computational interface for combined thermodynamic and kinetic modeling [J]. Calphad, 2011, 35: 391 |
| 92 | Philippe T, Erdeniz D, Dunand D C, et al. A phase-field study of the aluminizing of nickel [J]. Philos. Mag., 2015, 95: 935 |
| 93 | Vaithyanathan V, Wolverton C, Chen L Q. Multiscale modeling of θ′ precipitation in Al-Cu binary alloys [J]. Acta Mater., 2004, 52: 2973 |
| 94 | Vaithyanathan V, Wolverton C, Chen L Q. Multiscale modeling of precipitate microstructure evolution [J]. Phys. Rev. Lett., 2002, 88: 125503 |
| 95 | Khachaturyan A G. Theory of Structural Transformations in Solids [M]. New York: Wiley, 1983: 1 |
| 96 | Kim K, Roy A, Gururajan M P, et al. First-principles/phase-field modeling of θ′ precipitation in Al-Cu alloys [J]. Acta Mater., 2017, 140: 344 |
| 97 | Teichert G H, Natarajan A R, Van der Ven A, et al. Machine learning materials physics: Integrable deep neural networks enable scale bridging by learning free energy functions [J]. Comput. Methods Appl. Mech. Eng., 2019, 353: 201 |
| 98 | Teichert G H, Natarajan A R, Van der Ven A, et al. Scale bridging materials physics: Active learning workflows and integrable deep neural networks for free energy function representations in alloys [J]. Comput. Methods Appl. Mech. Eng., 2020, 371: 113281 |
| 99 | Zhang L, Chen L Q, Du Q. Diffuse-interface description of strain-dominated morphology of critical nuclei in phase transformations [J]. Acta Mater., 2008, 56: 3568 |
| 100 | Boussinot G, Finel A, Le Bouar Y. Phase-field modeling of bimodal microstructures in nickel-based superalloys [J]. Acta Mater., 2009, 57: 921 |
| 101 | Simmons J P, Shen C, Wang Y. Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events [J]. Scr. Mater., 2000, 43: 935 |
| 102 | Vaithyanathan V, Chen L Q. Coarsening kinetics of δ′-Al3Li precipitates: Phase-field simulation in 2D and 3D [J]. Scr. Mater., 2000, 42: 967 |
| 103 | Li Y S, Yu Y Z, Cheng X L, et al. Phase field simulation of precipitates morphology with dislocations under applied stress [J]. Mater. Sci. Eng., 2011, A528: 8628 |
| 104 | Du J L, Zhang A, Zhang Y B, et al. Atomistic determination on stability, cluster and microstructures in terms of crystallographic and thermo-kinetic integration of Al-Mg-Si alloys [J]. Mater. Today Commun., 2020, 24: 101220 |
| 105 | Huang L K, Lin W T, Zhang Y B, et al. Generalized stability criterion for exploiting optimized mechanical properties by a general correlation between phase transformations and plastic deformations [J]. Acta Mater., 2020, 201: 167 |
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