Abstract
Exact modeling of the dynamics of chemical and material systems over experimentally relevant time scales still eludes us even with modern computational resources. Fortunately, many systems can be described as rare event systems where atoms vibrate around equilibrium positions for a long time before a transition is made to a new atomic state. For those systems, the kinetic Monte Carlo (KMC) algorithm provides a powerful solution. In traditional KMC, mechanism and rates are computed beforehand, limiting moves to discretized positions and largely ignoring strain. Many systems of interest, however, are not well-represented by such lattice-based models. Moreover, materials often evolve with complex and concerted mechanisms that cannot be anticipated before the start of a simulation. In this chapter, we describe a class of algorithms, called off-lattice or adaptive KMC, which relaxes both limitations of traditional KMC, with atomic configurations represented in the full configuration space and reaction events are calculated on-the-fly, with the possible use of catalogs to speed up calculations. We discuss a number of implementations of off-lattice KMC developed by different research groups, emphasizing the similarities between the approaches that open modeling to new classes of problems.
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References
Alexander KC, Schuh CA (2016) Towards the reliable calculation of residence time for off-lattice kinetic Monte Carlo simulations. Model Simul Mater Sci Eng 24(6):65014. http://stacks.iop.org/0965-0393/24/i=6/a=065014?key=crossref.38e788234d74209ed2f8ad8b6b21fa51, https://doi.org/10.1088/0965-0393/24/6/065014
Althorpe S, Angulo G, Astumian RD, Beniwal V, Bolhuis PG, Brandão J, Ellis J, Fang W, Glowacki DR, Hammes-Schiffer S et al (2016) Application to large systems: general discussion. Faraday Discuss 195:671–698
Barkema GT, Mousseau N (1996) Event-based relaxation of continuous disordered systems. Phys Rev Lett 77(21):4358–4361
Béland LK, Mousseau N (2013) Long-time relaxation of ion-bombarded silicon studied with the kinetic activation-relaxation technique: microscopic description of slow aging in a disordered system. Phys Rev B 88(21):214201
Béland LK, Brommer P, El-Mellouhi F, Joly JF, Mousseau N (2011) Kinetic activation-relaxation technique. Phys Rev E 84(4):046704. https://doi.org/10.1103/PhysRevE.84.046704
Béland LK, Anahory Y, Smeets D, Guihard M, Brommer P, Joly JFF, Pothier JcC, Lewis LJ, Mousseau N, Schiettekatte F, Postale C, Centre-ville S (2013) Replenish and relax: explaining logarithmic annealing in ion-implanted c-Si. Phys Rev Lett 111(10):105502–105506. https://doi.org/10.1103/PhysRevLett.111.105502, http://arxiv.org/abs/1304.2991
Béland LK, Osetsky YN, Stoller RE, Xu H (2015a) Interstitial loop transformations in FeCr. J Alloys Compd 640:219–225
Béland LK, Osetsky YN, Stoller RE, Xu H (2015b) Slow relaxation of cascade-induced defects in Fe. Phys Rev B 91(5):054108
Béland LK, Samolyuk GD, Stoller RE (2016) Differences in the accumulation of ion-beam damage in Ni and NiFe explained by atomistic simulations. J Alloys Compd 662:415–420
Boulougouris GC, Frenkel D (2005) Monte Carlo sampling of a Markov web. J Chem Theory Comput 1:389–393
Boulougouris GC, Theodorou DN (2007) Dynamical integration of a Markovian web: a first passage time approach. J Chem Phys 127:084903
Brommer P, Béland LK, Joly JF, Mousseau N (2014) Understanding long-time vacancy aggregation in iron: a kinetic activation-relaxation technique study. Phys Rev B 90(13):134109–134117. https://doi.org/10.1103/PhysRevB.90.134109
Chill ST, Henkelman G (2014) Molecular dynamics saddle search adaptive kinetic Monte Carlo. J Chem Phys 140:214110
Chill ST, Stevenson J, Ruhle V, Shang C, Xiao P, Farrell J, Wales D, Henkelman G (2014a) Benchmarks for characterization of minima, transition states and pathways in atomic systems. J Chem Theory Comput 10:5476–5482
Chill ST, Welborn M, Terrell R, Zhang L, Berthet JC, Pedersen A, Jónsson H, Henkelman G (2014b) Eon: software for long time scale simulations of atomic scale systems. Model Simul Mater Sci Eng 22:055002
Duncan J, Harjunmaa A, Terrell R, Drautz R, Henkelman G, Rogal J (2016) Collective atomic displacements during complex phase boundary migration in solid-solid phase transformations. Phys Rev Lett 116(3):035701
El-Mellouhi F, Mousseau N, Lewis L (2008) Kinetic activation-relaxation technique: an off-lattice self-learning kinetic Monte Carlo algorithm. Phys Rev B 78(15):153202. https://doi.org/10.1103/PhysRevB.78.153202
Faken D, Jónsson H (1994) Systematic analysis of local atomic structure combined with 3D computer graphics. Comput Mater Sci 2:279–286
Fichthorn KA, Lin Y (2013) A local superbasin kinetic Monte Carlo method. J Chem Phys 138(16):164104
Guteŕrez M, Argaéz C, Jónsson H (2016) Improved minimum mode following method for finding first order saddle points. J Chem Theory Comput 13:125–134
Henkelman G, Jónsson H (1999) A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. J Chem Phys 111:7010–7022
Henkelman G, Jónsson H (2000) Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. J Chem Phys 113:9978–9985
Henkelman G, Jónsson H (2001) Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table. J Chem Phys 115(21):9657–9666. https://doi.org/10.1063/1.1415500
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 113:9901–9904
Jay A, Raine M, Richard N, Mousseau N, Goiffon V, Hemeryck A, Magnan P (2017) Simulation of single particle displacement damage in silicon part II: generation and long time relaxation of damage structure. IEEE Trans Nucl Sci 64(1):141–148. https://doi.org/10.1109/TNS.2016.2628089, http://ieeexplore.ieee.org/document/7742370/
Joly JF, Béland LK, Brommer P, El-Mellouhi F, Mousseau N (2012) Optimization of the kinetic activation-relaxation technique, an off-lattice and self-learning kinetic Monte-Carlo method. J Phys Conf Ser 341:012007. https://doi.org/10.1088/1742-6596/341/1/012007, http://stacks.iop.org/1742-6596/341/i=1/a=012007?key=crossref.dfb01ebf3ff94111aa93a5794b3384f8
Joly JF, Béland LK, Brommer P, Mousseau N (2013) Contribution of vacancies to relaxation in amorphous materials: a kinetic activation-relaxation technique study. Phys Rev B 87(14):144204. http://link.aps.org/doi/10.1103/PhysRevB.87.144204
Jónsson H, Mills G, Jacobsen KW (1998) Nudged elastic band method for finding minimum energy paths of transitions. In: Berne BJ, Ciccotti G, Coker DF (eds) Classical and quantum dynamics in condensed phase simulations. World Scientific, Singapore, pp 385–404
Kim WK, Tadmor EB (2014) Entropically stabilized dislocations. Phys Rev Lett 112(10):105501
Koziatek P, Barrat JL, Derlet P, Rodney D (2013) Inverse Meyer-Neldel behavior for activated processes in model glasses. Phys Rev B 87:224105. https://doi.org/10.1103/PhysRevB.87.224105
Lu C, Jin K, Béland LK, Zhang F, Yang T, Qiao L, Zhang Y, Bei H, Christen HM, Stoller RE et al (2016) Direct observation of defect range and evolution in ion-irradiated single crystalline Ni and Ni binary alloys. Sci Rep 6:19994
Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117:1–19
Machado-Charry E, Béland LK, Caliste D, Genovese L, Deutsch T, Mousseau N, Pochet P (2011) Optimized energy landscape exploration using the ab initio based activation-relaxation technique. J Chem Phys 135(3):034102. https://doi.org/10.1063/1.3609924, http://www.ncbi.nlm.nih.gov/pubmed/21786982
Mahmoud S, Trochet M, Restrepo OA, Mousseau N (2018) Study of point defects diffusion in nickel using kinetic activation-relaxation technique. Acta Mater 144:679–690. https://doi.org/10.1016/j.actamat.2017.11.021, http://www.sciencedirect.com/science/article/pii/S1359645417309643
Malek R, Mousseau N (2000) Dynamics of Lennard-Jones clusters: a characterization of the activation-relaxation technique. Phys Rev E 62(6):7723–7728. https://doi.org/10.1103/PhysRevE.62.7723
Marinica MC, Willaime F, Mousseau N (2011) Energy landscape of small clusters of self-interstitial dumbbells in iron. Phys Rev B 83(9):094119. https://doi.org/10.1103/PhysRevB.83.094119
Martínez E, Marian J, Kalos MH, Perlado JM (2008) Synchronous parallel kinetic Monte Carlo for continuum diffusion-reaction systems. J Comput Phys 227(8):3804–3823
McKay BD, Piperno A (2014) Practical graph isomorphism, II. J Symb Comput 60:94–112. https://doi.org/10.1016/j.jsc.2013.09.003, http://www.sciencedirect.com/science/article/pii/S0747717113001193
McKay BD et al (1981) Practical graph isomorphism. Congr Numer 30:45–87
Mousseau N, Barkema GT (1998b) Traveling through potential energy landscapes of disordered materials: the activation-relaxation technique. Phys Rev E 57:2419–2424. https://doi.org/10.1103/PhysRevE.57.2419
Munro LJ, Wales DJ (1999) Defect migration in crystalline silicon. Phys Rev B 59:3969
Nocedal J (1980) Updating quasi-Newton matrices with limited storage. Math Comput 35:773–782
Novotny MA (1995) Monte Carlo algorithms with absorbing Markov chains: fast local algorithms for slow dynamics. Phys Rev Lett 74:1–5
Novotny MA (2001) A tutorial on advanced dynamic monte carlo methods for systems with discrete state spaces. In: Stauffer D (ed) Annual reviews of computational physics IX. World Scientific, Singapore, pp 153–210
Ojifinni RA, Froemming NS, Gong J, Pan M, Kim TS, White J, Henkelman G, Mullins CB (2008) Water-enhanced low-temperature CO oxidation and isotope effects on atomic oxygen-covered Au (111). J Am Chem Soc 130(21):6801–6812
Osetsky YN, Béland LK, Stoller RE (2016) Specific features of defect and mass transport in concentrated FCC alloys. Acta Mater 115:364–371
Pedersen A, Jónsson H (2009) Simulations of hydrogen diffusion at grain boundaries in aluminum. Acta Mater 57:4036–4045
Pedersen A, Luiser M (2014) Bowl breakout: escaping the positive region when searching for saddle points. J Chem Phys 141(2):024109
Pedersen A, Henkelman G, Schiøtz J, Jónsson H (2009) Long time scale simulation of a grain boundary in copper. New J Phys 11:073034
Pedersen A, Berthet JC, Jónsson H (2012) Simulated annealing with coarse graining and distributed computing. Lect Notes Comput Sci 7134:34–44
Perez D, Luo SN, Voter AF, Germann TC (2013) Entropic stabilization of nanoscale voids in materials under tension. Phys Rev Lett 110(20):206001
Puchala B, Falk ML, Garikipati K (2010) An energy basin finding algorithm for kinetic Monte Carlo acceleration. J Chem Phys 132(13):134104. https://doi.org/10.1063/1.3369627, http://www.ncbi.nlm.nih.gov/pubmed/20387918
Raine M, Jay A, Richard N, Goiffon V, Girard S, Member S, Gaillardin M, Paillet P, Member S (2017) Simulation of single particle displacement damage in silicon part I: global approach and primary interaction simulation. IEEE Trans Nucl Sci 64(1):133–140. https://doi.org/10.1109/TNS.2016.2615133, http://ieeexplore.ieee.org/document/7582531/
Restrepo OA, Mousseau N, El-Mellouhi F, Bouhali O, Trochet M, Becquart CS (2016) Diffusion properties of Fe-C systems studied by using kinetic activation-relaxation technique. Comput Mater Sci 112:96–106. https://doi.org/10.1016/j.commatsci.2015.10.017, http://www.sciencedirect.com/science/article/pii/S0927025615006643, http://linkinghub.elsevier.com/retrieve/pii/S0927025615006643
Restrepo OA, Becquart CS, El-Mellouhi F, Bouhali O, Mousseau N (2017) Diffusion mechanisms of C in 100, 110 and 111 Fe surfaces studied using kinetic activation-relaxation technique. Acta Mater 136:303–314. https://doi.org/10.1016/j.actamat.2017.07.009, http://www.sciencedirect.com/science/article/pii/S135964541730558X
Shim Y, Amar JG (2005) Semirigorous synchronous sublattice algorithm for parallel kinetic Monte Carlo simulations of thin film growth. Phys Rev B 71:125432
Shim Y, Amar JG (2006) Hybrid asynchronous algorithm for parallel kinetic Monte Carlo simulations of thin film growth. J Comput Phys 212(1):305–317
Sinha AK (1972) Topologically close-packed structures of transition metal alloys. Prog Mat Sci 15:81
Sørensen MR, Voter AF (2000) Temperature-accelerated dynamics for simulation of infrequent events. J Chem Phys 112:9599–9606
Sørensen MR, Jacobsen KW, Jónsson H (1996) Thermal diffusion processes in metal-tip-surface interactions: contact formation and adatom mobility. Phys Rev Lett 77:5067–5070
Terentyev D, Malerba L, Klaver P, Olsson P (2008) Formation of stable sessile interstitial complexes in reactions between glissile dislocation loops in BCC Fe. J Nucl Mater 382(2):126–133
Terrell R, Welborn M, Chill ST, Henkelman G (2012) Database of atomistic reaction mechanisms with application to kinetic Monte Carlo. J Chem Phys 137:014105
Trochet M, Mousseau N (2017) Energy landscape and diffusion kinetics of lithiated silicon: a kinetic activation-relaxation technique study. Phys Rev B 96(13):134118. https://doi.org/10.1103/PhysRevB.96.134118
Trochet M, Béland LK, Joly JF, Brommer P, Mousseau N (2015) Diffusion of point defects in crystalline silicon using the kinetic activation-relaxation technique method. Phys Rev B 91(22):224106. https://doi.org/10.1103/PhysRevB.91.224106
Trochet M, Sauvé-Lacoursière A, Mousseau N (2017) Algorithmic developments of the kinetic activation-relaxation technique: accessing long-time kinetics of larger and more complex systems. J Chem Phys 147(15):152712. https://doi.org/10.1063/1.4995426
Trushin O, Karim A, Kara A, Rahman TS (2005) Self-learning kinetic Monte Carlo method: application to Cu(111). Phys Rev B 72(11):115401. https://doi.org/10.1103/PhysRevB.72.115401
Valiquette F, Mousseau N (2003) Energy landscape of relaxed amorphous silicon. Phys Rev B 68:125209. https://doi.org/10.1103/PhysRevB.68.125209
Vernon LJ (2010) Modelling the growth of TiO2. Ph.D. thesis, Loughborough University
Vernon LJ (2012) PESTO: potential energy surface tools. https://github.com/louisvernon/pesto
Vernon L, Kenny SD, Smith R, Sanville E (2011) Growth mechanisms for TiO2 at its rutile (110) surface. Phys Rev B 83(7):75412. https://doi.org/10.1103/PhysRevB.83.075412
Wales DJ (2002) Discrete path sampling. Mol Phys 100:3285–3305
Xiao P, Wu Q, Henkelman G (2014) Basin constrained κ-dimer method for saddle point finding. J Chem Phys 141:164111
Xu H, Osetsky YN, Stoller RE (2011) Simulating complex atomistic processes: on-the-fly kinetic Monte Carlo scheme with selective active volumes. Phys Rev B 84(13):132103. https://doi.org/10.1103/PhysRevB.84.132103
Xu H, Stoller RE, Osetsky YN, Terentyev D et al (2013) Solving the puzzle of <100> interstitial loop formation in BCC iron. Phys Rev Lett 110(26):265503
Xu H, Stoller RE, Béland LK, Osetsky YN (2015) Self-evolving atomistic kinetic Monte Carlo simulations of defects in materials. Comput Mater Sci 100:135–143
Xu L, Mei DH, Henkelman G (2009) Adaptive kinetic Monte Carlo simulation of methanol decomposition on Cu(100). J Chem Phys 131:244520
Zeng Y, Xiao P, Henkelman G (2014) Unification of algorithms for minimum mode optimization. J Chem Phys 140:044115
Zhou XW, Wadley HNG, Johnson RA, Larson DJ, Tabat N, Cerezo A, Petford-Long AK, Smith GDW, Clifton PH, Martens RL, Kelly TF (2001) Atomic scale structure of sputtered metal multilayers. Acta Mater 49:4005–4015
Acknowledgments
This work was supported in part by a grant from the Natural Science and Engineering Research Council of Canada. MT and NM are grateful to Calcul Québec and Compute Canada for providing extensive computer time and computer access. The work in Austin was supported by the National Science Foundation (CHE-0645497, CHE-1152342, and CHE-1534177) and the Welch Foundation (F-1841). Sustained computational resources have been provided by the Texas Advanced Computing Center.
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Trochet, M., Mousseau, N., Béland, L.K., Henkelman, G. (2020). Off-Lattice Kinetic Monte Carlo Methods. In: Andreoni, W., Yip, S. (eds) Handbook of Materials Modeling. Springer, Cham. https://doi.org/10.1007/978-3-319-44677-6_29
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