Abstract
A tomistic kinetic Monte Carlo (AKMC) is a powerful technique to study the microstructural and microchemical evolution of alloys controlled by diffusion processes. AKMC simulations are thus ideal tools to study precipitation, under irradiation and during thermal aging. In this chapter, we briefly present the method, underlining the different hypotheses usually made in the studies which have been done so far and the increasing contribution of density functional theory (DFT) calculations. We then proceed to present several simulations of the first stages of precipitation that can be quantitatively compared with experimental studies, in order to show the complexity introduced by the irradiation. We move to the mesoscale and introduce event kinetic Monte Carlo (EKMC) and object kinetic Monte Carlo (OKMC) methods which until now have mostly dealt with point defect cluster distributions in pure metals or “gray alloys” and were thus not really appropriate to study precipitation. However, they can be coupled with AKMC to speed up the calculations and recent developments take into account solute atoms more explicitly. We expose then recent advances that relieve some of the simplifying assumptions of standard AKMC models and conclude with a few challenging issues that we feel need to be addressed to predict correctly the behavior of alloys under irradiation but have been barely introduced in the models.
Abbreviations
- ABC:
-
Autonomous basin climbing
- AKMC:
-
Atomic kinetic Monte Carlo
- ANN:
-
Artificial neural network
- bac-MRM:
-
Basin auto-constructing mean rate method
- CRP:
-
Copper-rich precipitates
- DFT:
-
Density functional theory
- EKMC:
-
Event kinetic Monte Carlo
- HSLA:
-
High-strength low-alloy
- KMC:
-
Kinetic Monte Carlo
- NEB:
-
Nudged elastic band
- ODS:
-
Oxide dispersion strengthened
- OKMC:
-
Object kinetic Monte Carlo
- PD:
-
Protective domain
- RIP:
-
Radiation-induced precipitation
- RIS:
-
Radiation-induced segregation
- RPV:
-
Reactor pressure vessel
- SIA:
-
Self-interstitial atom
References
Russell KC. Phase stability under irradiation. Prog Mater Sci. 1984;28:229–434.
Was GS. Fundamentals of radiation materials science: metals and alloys. Berlin/Heidelberg: Springer Berlin Heidelberg; 2007. p. 433–90.
Martin G, Bellon P. Driven alloys. In: Spaepen F, Ehrenreich H, editors. Solid state physics. New York: Academic Press; 1996. p. 189–331.
Averback RS, de la Rubia TD. Displacement damage in irradiated metals and semiconductors. In: Spaepen F, Ehrenreich H, editors. Solid state physics. New York: Academic Press; 1998. p. 281–402.
Soisson F, Becquart CS, Castin N, Domain C, Malerba L, Vincent E. Atomistic Kinetic Monte Carlo studies of microchemical evolutions driven by diffusion processes under irradiation. J Nucl Mater. 2010;406:55–67.
Becquart CS, Domain C. Introducing chemistry in atomistic kinetic Monte Carlo simulations of Fe alloys under irradiation. Phys Status Solidi B. 2010;247:9–22. https://doi.org/10.1002/pssb.200945251.
Becquart CS, Wirth BD. Kinetic Monte Carlo simulations of irradiation effects. Compr Nucl Mater Elsevier. 2012:393–410.
Landau DP, Binder K. A guide to Monte Carlo simulations in statistical physics. Cambridge: Cambridge University Press; 2000.
Young WM, Elcock EW. Monte Carlo studies of vacancy migration in binary ordered alloys: I. Proc Phys Soc. 1966;89:735–46.
Bortz AB, Kalos MH, Lebowitz JL. A new algorithm for Monte Carlo simulation of Ising spin systems. J Comput Phys. 1975;17:10–8.
Müller S, Wang L-W, Zunger A. First-principles kinetic theory of precipitate evolution in Al-Zn alloys. Model Simul Mater Sci Eng. 2002;10:131–45.
Varvenne C, Finel A, Le Bouar Y, Fèvre M. Alloy microstructures with atomic size effects: a Monte Carlo study under the lattice statics formalism. Phys Rev B. 2012;86:184203.
Vineyard GH. Frequency factors and isotope effects in solid state rate processes. J Phys Chem Solids. 1957;3:121–7.
Voter AF. Introduction to the kinetic Monte Carlo method. In: Radiation effects in solids. Dordrecht: Springer; 2007. p. 1–23.
Henkelman G, Uberuaga BP, Jónsson H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J Chem Phys. 2000;113:9901.
Henkelman G, Jónsson H. A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J Chem Phys. 1999;111:7010.
Fan Y, Kushima A, Yildiz B. Unfaulting mechanism of trapped self-interstitial atom clusters in bcc Fe: a kinetic study based on the potential energy landscape. Phys Rev B. 2010;81:104102.
Barkema GT, Mousseau N. Event-based relaxation of continuous disordered systems. Phys Rev Lett. 1996;77:4358–61.
El-Mellouhi F, Mousseau N, Lewis LJ. Kinetic activation-relaxation technique: an off-lattice self-learning kinetic Monte Carlo algorithm. Phys Rev B. 2008;78:153202.
Mantina M, Wang Y, Arroyave R, Chen LQ, Liu ZK, Wolverton C. First-principles calculation of self-diffusion coefficients. Phys Rev Lett. 2008;100:215901.
Huang S, Worthington DL, Asta M, Ozolins V, Ghosh G, Liaw PK. Calculation of impurity diffusivities in α-Fe using first-principles methods. Acta Mater. 2010;58:1982–93.
Ganeshan S, Hector LG Jr, Liu Z-K. First-principles calculations of impurity diffusion coefficients in dilute Mg alloys using the 8-frequency model. Acta Mater. 2011;59:3214–28.
Messina L, Nastar M, Garnier T, Domain C, Olsson P. Exact ab initio transport coefficients in bcc Fe-X (X=Cr,Cu,Mn,Ni,P,Si) dilute alloys. Phys Rev B. 2014;90:104203.
Bocquet JL. On-the-fly evaluation of diffusional parameters during a Monte Carlo simulation of diffusion in alloys: a challenge. In: Defect and diffusion forum. Zürich: Trans Tech Publications; 2002. p. 81–112.
Senninger O, Martínez E, Soisson F, Nastar M, Bréchet Y. Atomistic simulations of the decomposition kinetics in Fe-Cr alloys: influence of magnetism. Acta Mater. 2014;73:97–106.
Huang C-H, Marian J. A generalized Ising model for studying alloy evolution under irradiation and its use in kinetic Monte Carlo simulations. J Phys Condens Matter. 2016;28:425201.
Martínez E, Senninger O, Fu C-C, Soisson F. Decomposition kinetics of Fe-Cr solid solutions during thermal aging. Phys Rev B. 2012;86:224109.
Kang HC, Weinberg WH. Dynamic Monte Carlo with a proper energy barrier: surface diffusion and two-dimensional domain ordering. J Chem Phys. 1989;90:2824.
Vincent E, Becquart CS, Pareige C, Pareige P, Domain C. Precipitation of the FeCu system: a critical review of atomic kinetic Monte Carlo simulations. J Nucl Mater. 2008;373:387–401.
Costa D, Adjanor G, Becquart CS, Olsson P, Domain C. Vacancy migration energy dependence on local chemical environment in Fe-Cr alloys: a density functional theory study. J Nucl Mater. 2014;452:425–33.
Bouar YL, Soisson F. Kinetic pathways from embedded-atom-method potentials: influence of the activation barriers. Phys Rev B. 2002;65:94103.
Nastar M, Soisson F. Atomistic modeling of phase transformations: point-defect concentrations and the time-scale problem. Phys Rev B. 2012;86:220102.
Allnatt AR, Lidiard AB. Atomic transport in solids. Cambridge: Cambridge University Press; 2003.
Clouet E, Laé L, Épicier T, Lefebvre W, Nastar M, Deschamps A. Complex precipitation pathways in multicomponent alloys. Nat Mater. 2006;5:482–8.
Dhua SK, Ray A, Sarma DS. Effect of tempering temperatures on the mechanical properties and microstructures of HSLA-100 type copper-bearing steels. Mater Sci Eng A. 2001;318:197–210.
Othen PJ, Jenkins ML, GDW S. High-resolution electron microscopy studies of the structure of Cu precipitates in α-Fe. Philos Mag A. 1994;70:1–24.
Soisson F, Barbu A, Martin G. Monte Carlo simulations of copper precipitation in dilute iron-copper alloys during thermal ageing and under electron irradiation. Acta Mater. 1996;44:3789–800.
Schmauder S, Binkele P. Atomistic computer simulation of the formation of Cu-precipitates in steels. Comput Mater Sci. 2002;24:42–53.
Vincent E, Becquart CS, Domain C. Solute interaction with point defects in α Fe during thermal ageing: a combined ab initio and atomic kinetic Monte Carlo approach. J Nucl Mater. 2006;351:88–99.
Soisson F, Fu C-C. Cu-precipitation kinetics in α- Fe from atomistic simulations: vacancy-trapping effects and Cu-cluster mobility. Phys Rev B. 2007;76:214102.
Liu CL, Odette GR, Wirth BD, Lucas GE. A lattice Monte Carlo simulation of nanophase compositions and structures in irradiated pressure vessel Fe-Cu-Ni-Mn-Si steels. Mater Sci Eng A. 1997;238:202–9.
Bonny G, Pasianot RC, Castin N, Malerba L. Ternary Fe–Cu–Ni many-body potential to model reactor pressure vessel steels: first validation by simulated thermal annealing. Philos Mag. 2009;89:3531–46.
Clouet E, Barbu A, Laé L, Martin G. Precipitation kinetics of Al3Zr and Al3Sc in aluminum alloys modeled with cluster dynamics. Acta Mater. 2005;53:2313–25.
Bonny G, Terentyev D, Malerba L, Van Neck D. Early stages of α-α’ phase separation in Fe-Cr alloys: an atomistic study. Phys Rev B. 2009;79:104207.
Pareige C, Domain C, Olsson P. Short- and long-range orders in Fe-Cr: a Monte Carlo study. J Appl Phys. 2009;106:104906.
Pareige C, Roussel M, Novy S, Kuksenko V, Olsson P, Domain C, Pareige P. Kinetic study of phase transformation in a highly concentrated Fe-Cr alloy: Monte Carlo simulation versus experiments. Acta Mater. 2011;59:2404–11.
Pareige C, Soisson F, Martin G, Blavette D. Ordering and phase separation in Ni–Cr–Al: Monte Carlo simulations vs three-dimensional atom probe. Acta Mater. 1999;47:1889–99.
Mao Z, Sudbrack CK, Yoon KE, Martin G, Seidman DN. The mechanism of morphogenesis in a phase-separating concentrated multicomponent alloy. Nat Mater. 2007;6:210–6.
Domain C, Becquart CS, van Duysen JC. Kinetic Monte Carlo simulations of cascades in Fe alloys. MRS Online Proc Libr Arch. 2000. https://doi.org/10.1557/PROC-650-R3.25.
Sizmann R. The effect of radiation upon diffusion in metals. J Nucl Mater. 1978;69:386–412.
Arakawa K, Ono K, Isshiki M, Mimura K, Uchikoshi M, Mori H. Observation of the one-dimensional diffusion of nanometer-sized dislocation loops. Science. 2007;318:956–9.
Soisson F, Jourdan T. Radiation-accelerated precipitation in Fe-Cr alloys. Acta Mater. 2016;103:870–81.
Meslin E, Soisson F, Tissot O. In preparation. 2017.
Enrique RA, Bellon P. Compositional patterning in immiscible alloys driven by irradiation. Phys Rev B. 2001;63:134111.
Soisson F. Kinetic Monte Carlo simulations of radiation induced segregation and precipitation. J Nucl Mater. 2006;349:235–50.
Senninger O, Soisson F, Martínez E, Nastar M, Fu C-C, Bréchet Y. Modeling radiation induced segregation in iron–chromium alloys. Acta Mater. 2016;103:1–11.
Hardie CD, Williams CA, Xu S, Roberts SG. Effects of irradiation temperature and dose rate on the mechanical properties of self-ion implanted Fe and Fe-Cr alloys. J Nucl Mater. 2013;439:33–40.
Bhattacharya A, Meslin E, Henry J, Pareige C, Décamps B, Genevois C, Brimbal D, Barbu A. Chromium enrichment on the habit plane of dislocation loops in ion-irradiated high-purity Fe-Cr alloys. Acta Mater. 2014;78:394–403.
Huang C-H, Gharaee L, Zhao Y, Erhart P, Marian J. Mechanism of nucleation and incipient growth of Re clusters in irradiated W-Re alloys from kinetic Monte Carlo simulations. Phys. Rev. B 2017; 96, 094108.
Vincent E, Becquart CS, Domain C. Atomic kinetic Monte Carlo model based on ab initio data: simulation of microstructural evolution under irradiation of dilute Fe–CuNiMnSi alloys. Nucl Instrum Methods Phys Res Sect B Beam Interact Mater At. 2007;255:78–84.
Potter DI, McCormick AW. Irradiation-enhanced coarsening in Ni-12.8 at.% Al. Acta Metall. 1979;27:933–41.
Odette GR, Nanstad RK. Predictive reactor pressure vessel steel irradiation embrittlement models: issues and opportunities. JOM. 2009;61:17–23.
Ngayam-Happy R, Becquart CS, Domain C, Malerba L. Formation and evolution of MnNi clusters in neutron irradiated dilute Fe alloys modelled by a first principle-based AKMC method. J Nucl Mater. 2012;426:198–207.
Bonny G, Terentyev D, Bakaev A, Zhurkin EE, Hou M, Van Neck D, Malerba L. On the thermal stability of late blooming phases in reactor pressure vessel steels: an atomistic study. J Nucl Mater. 2013;442:282–91.
Odette GR, Alinger MJ, Wirth BD. Recent developments in irradiation-resistant steels. Annu Rev Mater Res. 2008;38:471–503.
Wharry JP, Swenson MJ, Yano KH. A review of the irradiation evolution of dispersed oxide nanoparticles in the b.c.c. Fe-Cr system: current understanding and future directions. J Nucl Mater. 2017;486:11–20.
Hin C, Wirth BD. Formation of oxide nanoclusters in nanostructured ferritic alloys during anisothermal heat treatment: a kinetic Monte Carlo study. Mater Sci Eng A. 2011;528:2056–61.
Chiapetto M, Malerba L, Becquart CS. Nanostructure evolution under irradiation in FeMnNi alloys: a “grey alloy” object kinetic Monte Carlo model. J Nucl Mater. 2015;462:91–9.
Chiapetto M, Malerba L, Becquart CS. Effect of Cr content on the nanostructural evolution of irradiated ferritic/martensitic alloys: an object kinetic Monte Carlo model. J Nucl Mater. 2015;465:326–36.
Castin N, Chiapetto M, Messina L, Malerba L. In preparation. 2017.
Castin N, Pascuet MI, Malerba L. Modeling the first stages of Cu precipitation in α-Fe using a hybrid atomistic kinetic Monte Carlo approach. J Chem Phys. 2011;135:64502.
Pannier B. Towards the prediction of microstructure evolution under irradiation of model ferritic alloys with an hybrid AKMC-OKMC approach. Lille: Université Lille; 2017.
Molnar D, Mukherjee R, Choudhury A, Mora A, Binkele P, Selzer M, Nestler B, Schmauder S. Multiscale simulations on the coarsening of Cu-rich precipitates in α-Fe using kinetic Monte Carlo, molecular dynamics and phase-field simulations. Acta Mater. 2012;60:6961–71.
Mason DR, Rudd RE, Sutton AP. Atomistic modelling of diffusional phase transformations with elastic strain. J Phys Condens Matter. 2004;16:S2679.
Trochet M, Béland LK, Joly J-F, Brommer P, Mousseau N. Diffusion of point defects in crystalline silicon using the kinetic activation-relaxation technique method. Phys Rev B. 2015;91:224106.
Henkelman G, Jónsson H. Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table. J Chem Phys. 2001;115:9657.
Castin N, Fernández JR, Pasianot RC. Predicting vacancy migration energies in lattice-free environments using artificial neural networks. Comput Mater Sci. 2014;84:217–25.
Messina L, Nastar M, Sandberg N, Olsson P. Systematic electronic-structure investigation of substitutional impurity diffusion and flux coupling in bcc iron. Phys Rev B. 2016;93:184302.
Ding H, Razumovskiy VI, Asta M. Self diffusion anomaly in ferromagnetic metals: a density-functional-theory investigation of magnetically ordered and disordered Fe and Co. Acta Mater. 2014;70:130–6.
Lazauskas T, Kenny SD, Smith R. Influence of the prefactor to defect motion in α -Iron during long time scale simulations. J Phys Condens Matter. 2014;26:395007.
Maydet SI, Russell KC. Precipitate stability under irradiation: point defect effects. J Nucl Mater. 1977;64:101–14.
Mason DR, Rudd RE, Sutton AP. Atomistic modelling of diffusional phase transformations with elastic strain. J Phys Condens Matter. 2004;16:S2679.
Joshi K, Chaudhuri S. Empirical force field-based kinetic Monte Carlo simulation of precipitate evolution and growth in Al–Cu alloys. Model Simul Mater Sci Eng. 2016;24:75012.
Djurabekova FG, Domingos R, Cerchiara G, Castin N, Vincent E, Malerba L. Artificial intelligence applied to atomistic kinetic Monte Carlo simulations in Fe-Cu alloys. Nucl Instrum Methods Phys Res Sect B Beam Interact Mater At. 2007;255:8–12.
Castin N, Malerba L, Bonny G, Pascuet MI, Hou M. Modelling radiation-induced phase changes in binary FeCu and ternary FeCuNi alloys using an artificial intelligence-based atomistic kinetic Monte Carlo approach. Nucl Instrum Methods Phys Res Sect B Beam Interact Mater At. 2009;267:3002–8.
Opplestrup T, Bulatov VV, Gilmer GH, Kalos MH, Sadigh B. First-passage Monte Carlo algorithm: diffusion without all the hops. Phys Rev Lett. 2006;97:230602.
Béland LK, Brommer P, El-Mellouhi F, Joly J-F, Mousseau N. Kinetic activation-relaxation technique. Phys Rev E. 2011;84:46704.
Athènes M, Bulatov VV. Path factorization approach to stochastic simulations. Phys Rev Lett. 2014;113:230601.
Shim Y, Amar JG. Hybrid asynchronous algorithm for parallel kinetic Monte Carlo simulations of thin film growth. J Comput Phys. 2006;212:305–17.
Sadigh B, Erhart P, Stukowski A, Caro A, Martinez E, Zepeda-Ruiz L. Scalable parallel Monte Carlo algorithm for atomistic simulations of precipitation in alloys. Phys Rev B. 2012;85:184203.
Arampatzis G, Katsoulakis MA, Plecháč P, Taufer M, Xu L. Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms. J Comput Phys. 2012;231:7795–814.
Esteves A, Moura A. Distributed memory implementation strategies for the kinetic Monte Carlo algorithm. New York: ACM Press; 2016. p. 130–9.
Martínez E, Marian J, Kalos MH, Perlado JM. Synchronous parallel kinetic Monte Carlo for continuum diffusion-reaction systems. J Comput Phys. 2008;227:3804–23.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this entry
Cite this entry
Becquart, C.S., Soisson, F. (2018). Monte Carlo Simulations of Precipitation Under Irradiation. In: Schmauder, S., Chen, CS., Chawla, K., Chawla, N., Chen, W., Kagawa, Y. (eds) Handbook of Mechanics of Materials. Springer, Singapore. https://doi.org/10.1007/978-981-10-6855-3_24-1
Download citation
DOI: https://doi.org/10.1007/978-981-10-6855-3_24-1
Received:
Accepted:
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-6855-3
Online ISBN: 978-981-10-6855-3
eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering