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Atomistic Kinetic Monte Carlo and Solute Effects

  • Charlotte S. BecquartEmail author
  • Normand Mousseau
  • Christophe Domain
Living reference work entry
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Abstract

The atomistic approach of the kinetic Monte Carlo methods allows one to explicitly take into account solute atoms. In this chapter, we present and discuss the different pathways available at this point to go behind nearest neighbor pair interaction for binary alloys on rigid lattices as well as their perspectives. Different strategies to treat complex alloys with several solutes with improved cohesive models are exposed and illustrated as well as the modeling of self-interstitial diffusion under irradiation and its complexity compared to vacancy diffusion.

Abbreviations

AKMC

Atomic kinetic Monte Carlo

BKL

Bortz, Kalos, and Lebowitz

CE

Cluster expansion

DFT

Density functional theory

FIA

Foreign interstitial atom

FISE

Final initial system energy

GAP

Gaussian approximation potential

KMC

Kinetic Monte Carlo

KRA

Kinetically resolved activation

LAE

Local atomic environment

NEB

Nudged elastic band

PD

Point defect

RPV

Reactor pressure vessel

RTA

Residence time algorithm

SFT

Stacking fault tetrahedra

SIA

Self interstitial atom

SNAP

Spectral neighbor analysis potential

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Notes

Acknowledgments

This work is part of the EM2VM laboratory. It has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euroatom research and training program 2014–2018 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission and the Commission is not responsible for any use that may be made of the information it contains. Further funding from the Euratom research and training program 2014–2018 under Grant Agreement No 661913 (Soteria) is acknowledged. This work contributes also to the Joint Programme on Nuclear Materials (JPNM) of the European Energy Research Alliance (EERA).

References

  1. Aidhy DS, Lu C, Jin K, Bei H, Zhang Y, Wang L, Weber WJ (2016) Formation and growth of stacking fault tetrahedra in Ni via vacancy aggregation mechanism. Scr Mater 114:137–141CrossRefGoogle Scholar
  2. Alloy Theoretic Automated Toolkit (ATAT) Home Page. https://www.brown.edu/Departments/Engineering/Labs/avdw/atat/. Accessed 7 Nov 2017
  3. Athènes M, Bulatov VV (2014) Path factorization approach to stochastic simulations. Phys Rev Lett. https://doi.org/10.1103/PhysRevLett.113.230601
  4. Barkema GT, Mousseau N (1996) Event-based relaxation of continuous disordered systems. Phys Rev Lett 77:4358–4361ADSCrossRefGoogle Scholar
  5. Barnard L, Young GA, Swoboda B, Choudhury S, Van der Ven A, Morgan D, Tucker JD (2014) Atomistic modeling of the order–disorder phase transformation in the Ni2Cr model alloy. Acta Mater 81:258–271CrossRefGoogle Scholar
  6. Bartók AP, Csányi G (2015) Gaussian approximation potentials: a brief tutorial introduction. Int J Quantum Chem 115:1051–1057CrossRefGoogle Scholar
  7. Becquart CS, Soisson F (2018) Monte carlo simulations of precipitation under irradiation. In: Hsueh CH et al. (eds) Handbook of mechanics of materials. Springer, SingaporeGoogle Scholar
  8. Behler J (2016) Perspective: machine learning potentials for atomistic simulations. J Chem Phys 145:170901ADSCrossRefGoogle Scholar
  9. Bonny G, Castin N, Domain C, Olsson P, Verreyken B, Pascuet MI, Terentyev D (2017) Density functional theory-based cluster expansion to simulate thermal annealing in FeCrW alloys. Philos Mag 97:299–317ADSCrossRefGoogle Scholar
  10. Bortz AB, Kalos MH, Lebowitz JL (1975) A new algorithm for Monte Carlo simulation of Ising spin systems. J Comput Phys 17:10–18ADSCrossRefGoogle Scholar
  11. Bouar YL, Soisson F (2002) Kinetic pathways from embedded-atom-method potentials: influence of the activation barriers. Phys Rev B. https://doi.org/10.1103/PhysRevB.65.094103
  12. Brommer P, Kiselev A, Schopf D, Beck P, Roth J, Trebin H-R (2015) Classical interaction potentials for diverse materials from ab initio data: a review of potfit. Model Simul Mater Sci Eng 23:074002ADSCrossRefGoogle Scholar
  13. CASM Developers (2016) Casmcode: V0.2.0. https://doi.org/10.5281/zenodo.60142
  14. CASMcode (2017) First-principles statistical mechanical software for the study of multi-component crystalline solids. PRISMS CenterGoogle Scholar
  15. Castin N, Messina L, Domain C, Pasianot RC, Olsson P (2017) Improved atomistic Monte Carlo models based on ab-initio -trained neural networks: application to FeCu and FeCr alloys. Phys Rev B. https://doi.org/10.1103/PhysRevB.95.214117
  16. Cerezo A, Hirosawa S, Rozdilsky I, Smith GDW (2003) Combined atomic-scale modelling and experimental studies of nucleation in the solid state. Philos Trans R Soc Math Phys Eng Sci 361:463–477ADSCrossRefGoogle Scholar
  17. Clouet E, Hin C, Gendt D, Nastar M, Soisson F (2006) Kinetic Monte Carlo simulations of precipitation. Adv Eng Mater 8:1210–1214CrossRefGoogle Scholar
  18. Costa D (2012) Modelling the thermal ageing evolution of Fe-Cr alloys using a lattice based kinetic Monte Carlo approach based on DFT calculations. PhD dissertation, Université LilleGoogle Scholar
  19. Costa D, Adjanor G, Becquart CS, Olsson P, Domain C (2014) Vacancy migration energy dependence on local chemical environment in Fe–Cr alloys: a density functional theory study. J Nucl Mater 452:425–433ADSCrossRefGoogle Scholar
  20. Danielson T, Sutton JE, Hin C, Savara A (2017) SQERTSS: dynamic rank based throttling of transition probabilities in kinetic Monte Carlo simulations. Comput Phys Commun 219:149–163ADSCrossRefGoogle Scholar
  21. Daw MS, Baskes MI (1984) Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals. Phys Rev B 29:6443–6453ADSCrossRefGoogle Scholar
  22. Djurabekova FG, Domingos R, Cerchiara G, Castin N, Vincent E, Malerba L (2007) Artificial intelligence applied to atomistic kinetic Monte Carlo simulations in Fe–Cu alloys. Nucl Instrum Methods Phys Res Sect B Beam Interact Mater At 255:8–12ADSCrossRefGoogle Scholar
  23. El-Mellouhi F, Mousseau N, Lewis LJ (2008) Kinetic activation-relaxation technique: an off-lattice self-learning kinetic Monte Carlo algorithm. Phys Rev B 78:153202ADSCrossRefGoogle Scholar
  24. Fan Y, Kushima A, Yildiz B (2010) Unfaulting mechanism of trapped self-interstitial atom clusters in bcc Fe: a kinetic study based on the potential energy landscape. Phys Rev B 81:104102ADSCrossRefGoogle Scholar
  25. Fichthorn KA, Weinberg WH (1991) Theoretical foundations of dynamical Monte Carlo simulations. J Chem Phys 95:1090ADSCrossRefGoogle Scholar
  26. Handley CM, Behler J (2014) Next generation interatomic potentials for condensed systems. Eur Phys J B. https://doi.org/10.1140/epjb/e2014-50070-0
  27. Henkelman G, Jónsson H (1999) A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J Chem Phys 111:7010ADSCrossRefGoogle Scholar
  28. Henkelman G, Uberuaga BP, Jónsson H (2000) A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J Chem Phys 113:9901ADSCrossRefGoogle Scholar
  29. Herder LM, Bray JM, Schneider WF (2015) Comparison of cluster expansion fitting algorithms for interactions at surfaces. Surf Sci 640:104–111ADSCrossRefGoogle Scholar
  30. Hin C, Bréchet Y, Maugis P, Soisson F (2008) Kinetics of heterogeneous grain boundary precipitation of NbC in α-iron: a Monte Carlo study. Acta Mater 56:5653–5667CrossRefGoogle Scholar
  31. Hocker S, Binkele P, Schmauder S (2014) Precipitation in α $\alpha$ -Fe based Fe-cu-Ni-Mn-alloys: behaviour of Ni and Mn modelled by ab initio and kinetic Monte Carlo simulations. Appl Phys A Mater Sci Process 115:679–687ADSCrossRefGoogle Scholar
  32. Johnson RA (1964) Interstitials and vacancies in α Iron. Phys Rev 134:A1329–A1336ADSCrossRefGoogle Scholar
  33. Kang HC, Weinberg WH (1989) Dynamic Monte Carlo with a proper energy barrier: surface diffusion and two-dimensional domain ordering. J Chem Phys 90:2824ADSCrossRefGoogle Scholar
  34. Kushima A, Yildiz B (2010) Oxygen ion diffusivity in strained yttria stabilized zirconia: where is the fastest strain? J Mater Chem 20:4809CrossRefGoogle Scholar
  35. Lazauskas T, Kenny SD, Smith R (2014) Influence of the prefactor to defect motion in α -Iron during long time scale simulations. J Phys Condens Matter 26:395007CrossRefGoogle Scholar
  36. Lear CR, Bellon P, Averback RS (2017) Novel mechanism for order patterning in alloys driven by irradiation. Phys Rev B 96:104108ADSCrossRefGoogle Scholar
  37. Liu CL, Odette GR, Wirth BD, Lucas GE (1997) A lattice Monte Carlo simulation of nanophase compositions and structures in irradiated pressure vessel Fe-Cu-Ni-Mn-Si steels. Mater Sci Eng A 238:202–209CrossRefGoogle Scholar
  38. Mahmoud S, Trochet M, Restrepo OA, Mousseau N (2018) Study of point defects diffusion in nickel using kinetic activation-relaxation technique. Acta Mater. https://doi.org/10.1016/j.actamat.2017.11.021CrossRefGoogle Scholar
  39. Mantina M, Wang Y, Arroyave R, Chen LQ, Liu ZK, Wolverton C (2008) First-principles calculation of self-diffusion coefficients. Phys Rev Lett 100:215901ADSCrossRefGoogle Scholar
  40. Marinica MC, Willaime F (2007) Orientation of interstitials in clusters in α-Fe: a comparison between empirical potentials. Solid State Phenom 129:67–74CrossRefGoogle Scholar
  41. Martínez E, Senninger O, Fu C-C, Soisson F (2012) Decomposition kinetics of Fe-Cr solid solutions during thermal aging. Phys Rev B. https://doi.org/10.1103/PhysRevB.86.224109
  42. McKay BD, Piperno A (2014) Practical graph isomorphism, II. J Symb Comput 60:94–112MathSciNetCrossRefGoogle Scholar
  43. Messina L, Nastar M, Garnier T, Domain C, Olsson P (2014) Exact ab initio transport coefficients in bcc Fe-X (X=Cr,Cu,Mn,Ni,P,Si) dilute alloys. Phys Rev B 90:104203ADSCrossRefGoogle Scholar
  44. Messina L, Malerba L, Olsson P (2015) Stability and mobility of small vacancy–solute complexes in Fe–MnNi and dilute Fe–X alloys: a kinetic Monte Carlo study. Nucl Instrum Methods Phys Res Sect B Beam Interact Mater At 352:61–66ADSCrossRefGoogle Scholar
  45. Messina L, Castin N, Domain C, Olsson P (2017) Introducing ab initio based neural networks for transition-rate prediction in kinetic Monte Carlo simulations. Phys Rev B. https://doi.org/10.1103/PhysRevB.95.064112
  46. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087–1092. https://doi.org/10.1063/1.1699114ADSCrossRefGoogle Scholar
  47. Murali D, Posselt M, Schiwarth M (2015) First-principles calculation of defect free energies: general aspects illustrated in the case of bcc Fe. Phys Rev B. https://doi.org/10.1103/PhysRevB.92.064103
  48. Ngayam-Happy R, Olsson P, Becquart CS, Domain C (2010) Isochronal annealing of electron-irradiated dilute Fe alloys modelled by an ab initio based AKMC method: influence of solute–interstitial cluster properties. J Nucl Mater 407:16–28ADSCrossRefGoogle Scholar
  49. Nguyen-Manh D, Lavrentiev MY, Dudarev SL (2008) The Fe–Cr system: atomistic modelling of thermodynamics and kinetics of phase transformations. Comptes Rendus Phys 9:379–388ADSCrossRefGoogle Scholar
  50. Olsson P, Klaver TPC, Domain C (2010) Ab initio study of solute transition-metal interactions with point defects in bcc Fe. Phys Rev B. https://doi.org/10.1103/PhysRevB.81.054102
  51. Pannier B (2017) Towards the prediction of microstructure evolution under irradiation of model ferritic alloys with an hybrid AKMC-OKMC approach. PhD dissertation, Université LilleGoogle Scholar
  52. Pareige C, Domain C, Olsson P (2009) Short- and long-range orders in Fe–Cr: a Monte Carlo study. J Appl Phys 106:104906ADSCrossRefGoogle Scholar
  53. Pareige C, Roussel M, Novy S, Kuksenko V, Olsson P, Domain C, Pareige P (2011) Kinetic study of phase transformation in a highly concentrated Fe–Cr alloy: Monte Carlo simulation versus experiments. Acta Mater 59:2404–2411CrossRefGoogle Scholar
  54. Peters B, Heyden A, Bell AT, Chakraborty A (2004) A growing string method for determining transition states: comparison to the nudged elastic band and string methods. J Chem Phys 120:7877–7886ADSCrossRefGoogle Scholar
  55. Piochaud JB (2013) Modelling of radiation induced segregation in austenitic Fe alloys at the atomistic level. PhD dissertation, Université LilleGoogle Scholar
  56. Posselt M, Murali D, Schiwarth M (2017) Influence of phonon and electron excitations on the free energy of defect clusters in solids: a first-principles study. Comput Mater Sci 127:284–294CrossRefGoogle Scholar
  57. Rehman T, Jaipal M, Chatterjee A (2013) A cluster expansion model for predicting activation barrier of atomic processes. J Comput Phys 243:244–259ADSMathSciNetCrossRefGoogle Scholar
  58. Sanchez JM, Ducastelle F, Gratias D (1984) Generalized cluster description of multicomponent systems. Phys Stat Mech Its Appl 128:334–350MathSciNetCrossRefGoogle Scholar
  59. Soisson F (2006) Kinetic Monte Carlo simulations of radiation induced segregation and precipitation. J Nucl Mater 349:235–250ADSCrossRefGoogle Scholar
  60. Soisson F, Jourdan T (2016) Radiation-accelerated precipitation in Fe–Cr alloys. Acta Mater 103:870–881CrossRefGoogle Scholar
  61. Soisson F, Barbu A, Martin G (1996) Monte Carlo simulations of copper precipitation in dilute iron-copper alloys during thermal ageing and under electron irradiation. Acta Mater 44:3789–3800CrossRefGoogle Scholar
  62. Tchitchekova DS, Morthomas J, Ribeiro F, Ducher R, Perez M (2014) A novel method for calculating the energy barriers for carbon diffusion in ferrite under heterogeneous stress. J Chem Phys 141:034118ADSCrossRefGoogle Scholar
  63. Thompson AP, Swiler LP, Trott CR, Foiles SM, Tucker GJ (2015) Spectral neighbor analysis method for automated generation of quantum-accurate interatomic potentials. J Comput Phys 285:316–330ADSMathSciNetCrossRefGoogle Scholar
  64. Tucker JD, Najafabadi R, Allen TR, Morgan D (2010) Ab initio-based diffusion theory and tracer diffusion in Ni–Cr and Ni–Fe alloys. J Nucl Mater 405:216–234ADSCrossRefGoogle Scholar
  65. Van der Ven A, Ceder G, Asta M, Tepesch PD (2001) First-principles theory of ionic diffusion with nondilute carriers. Phys Rev B. https://doi.org/10.1103/PhysRevB.64.184307
  66. Van der Ven A, Thomas JC, Xu Q, Bhattacharya J (2010) Linking the electronic structure of solids to their thermodynamic and kinetic properties. Math Comput Simul 80:1393–1410MathSciNetCrossRefGoogle Scholar
  67. Vincent E, Becquart CS, Domain C (2006) Solute interaction with point defects in α Fe during thermal ageing: a combined ab initio and atomic kinetic Monte Carlo approach. J Nucl Mater 351:88–99ADSCrossRefGoogle Scholar
  68. Vincent E, Becquart CS, Pareige C, Pareige P, Domain C (2008a) Precipitation of the FeCu system: a critical review of atomic kinetic Monte Carlo simulations. J Nucl Mater 373:387–401ADSCrossRefGoogle Scholar
  69. Vincent E, Becquart CS, Domain C (2008b) Microstructural evolution under high flux irradiation of dilute Fe–CuNiMnSi alloys studied by an atomic kinetic Monte Carlo model accounting for both vacancies and self interstitials. J Nucl Mater 382:154–159ADSCrossRefGoogle Scholar
  70. Vineyard GH (1957) Frequency factors and isotope effects in solid state rate processes. J Phys Chem Solids 3:121–127ADSCrossRefGoogle Scholar
  71. Voter AF (2007) Introduction to the kinetic Monte Carlo method. In: Radiat. Eff. Solids. Springer, Dordrecht, pp 1–23Google Scholar
  72. Xu H, Stoller RE, Béland LK, Osetsky YN (2015) Self-evolving atomistic kinetic Monte Carlo simulations of defects in materials. Comput Mater Sci 100, Part B:135–143CrossRefGoogle Scholar
  73. Young WM, Elcock EW (1966) Monte Carlo studies of vacancy migration in binary ordered alloys: I. Proc Phys Soc 89:735ADSCrossRefGoogle Scholar
  74. Yuge K (2012) Modeling configurational energetics on multiple lattices through extended cluster expansion. Phys Rev B. https://doi.org/10.1103/PhysRevB.85.144105
  75. Yuge K (2017) Graph representation for configurational properties of crystalline solids. J Phys Soc Jpn 86:024802ADSCrossRefGoogle Scholar
  76. Yuge K, Okawa R (2014) Cluster expansion approach for modeling strain effects on alloy phase stability. Intermetallics 44:60–63CrossRefGoogle Scholar
  77. Zhang Y, Jiang C, Bai X (2017) Anisotropic hydrogen diffusion in α-Zr and Zircaloy predicted by accelerated kinetic Monte Carlo simulations. Sci Rep 7:41033ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Charlotte S. Becquart
    • 1
    Email author
  • Normand Mousseau
    • 2
  • Christophe Domain
    • 3
  1. 1.Univ.Lille, CNRS, INRA, ENSCL, UMR 8207UMET, Unité Matériaux et TransformationsLilleFrance
  2. 2.Département de physique and Regroupement québécois sur les matériaux de pointeUniversité de MontréalMontréalCanada
  3. 3.Département MMCLes RenardièresMoret sur LoingFrance

Section editors and affiliations

  • Michael P. Short
    • 1
  • Kai Nordlund
    • 2
  1. 1.Department of Nuclear Science and EngineeringMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Computational Materials PhysicsUniversity of HelsinkiHelsinkiFinland

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