(Luận văn thạc sĩ) atomistically kinetic simulations of carbon diffusion in ɑ iron with point defects001

VIETNAM NATIONAL UNIVERSITY, HANOI VIETNAM JAPAN UNIVERSITY HO NGOC NAM ATOMISTICALLY KINEMATIC SIMULATIONS OF CARBON DIFFUSION IN α-IRON WITH POINT DEFECTS MASTER’S THESIS Hanoi, 2019 VIETNAM NATIONAL UNIVERSITY, HANOI VIETNAM JAPAN UNIVERSITY HO NGOC NAM ATOMISTICALLY KINEMATIC SIMULATIONS OF CARBON DIFFUSION IN α-IRON WITH POINT DEFECTS MAJOR: NANOTECHNOLOGY CODE: PILOT RESEARCH SUPERVISORS: Prof Dr YOJI SHIBUTANI Dr NGUYEN TIEN QUANG Hanoi, 2019 ACKNOWLEDGMENT To accomplish this thesis, I have received great support, helpful advice, and guidance from respectful professors, lecturers, researchers, and staff in Vietnam Japan University and Osaka University I would like to express my gratefulness to my supervisors, Prof Dr Yoji Shibutani and Dr Nguyen Tien Quang for supplying great researching environments in laboratories, and for giving helpful instructions, guidance, advice, and inspirations during my master course Finally, I am thankful to my family for the support, companion, and mobilization, which is an essential element for me to finish the thesis Hanoi, 10 June 2019 Student HO NGOC NAM TABLE OF CONTENTS ACKNOWLEDGMENT i LIST OF FIGURES i LIST OF TABLES iii LIST OF ABBREVIATIONS iv CHAPTER INTRODUCTION .1 CHAPTER THEORETICAL BASICS CHAPTER RESULTS AND DISCUSSION 21 CONCLUSION 54 FUTURE PLAN 55 LIST OF PUBLISCATIONS 56 REFFRENCES 57 APPENDIX .63 LIST OF FIGURES Figure 1.1 The relation between elongation (ductility) and tensile strength in low carbon steel for general applications [4] .2 Figure 1.2 Phase diagram of iron-carbon alloy by different carbon content [19] Figure 1.3 Simulation picture of typical defects in iron-carbon alloy Figure 2.1 Reaction energy diagram as a function of reaction coordinate q  for an isomerization reaction [37] Figure 2.2 Illustration of finding the minimum energy path by NEB Each image on the chain of the system is connected by spring forces which located along the minimum energy line between two minimum energy points [44] .10 Figure 2.3 Decomposition of force on an image [38] 11 Figure 2.4 Contour plot of the potential energy surface for an energy-barrierlimited infrequent-event system After many vibrational periods, the trajectory finds a way out of the initial basin, passing a ridgetop into a new state The dots indicate saddle points [45] 13 Figure 2.5 the transition of atom when diffusing from the state (i) to the state (j) by crossing the energy barrier E m [44] .15 Figure 2.6 The K-th transition is chosen because its assigned value of s(K) intercepts r2 i [44] .15 Figure 3.1 Positions 1, of carbon correspond to O site, and corresponds to T site 22 Figure 3.2: Positions carbon is adopted in iron system 23 Figure 3.3 Configurations of BCC iron structure in case of two carbons 24 Figure 3.4: Configurations of BCC iron structure in case of three carbons 25 Figure 3.5: Configurations of BCC iron structure in case four carbons 26 Figure 3.6 Energy landscape (a, b) and energy contour line (c, d) of iron-carbon system in case of vacancy/without vacancy is created by [010] and [001] directions .30 Figure 3.7: The change position and angle of iron atoms around carbon atom, which is doped between two iron atoms lead to relaxing configuration 31 Figure 3.8 Binding energy of carbon-vacancy is calculated by DFT calculation and MD method in case 1V-1C 33 Figure 3.9: Configuration after optimized in case carbon atoms .34 Figure 3.10 Configuration after optimized 36 Figure 3.11 The most stable configuration in case carbon atoms in iron 37 i Figure 3.12 Trapping energy is calculated in two ways: “sequential” way (blue line) and “simultaneous” way (red line) 38 Figure 3.13 Minimum energy paths of carbon in case 1C by two possible ways 39 Figure 3.14 Eight diffusion paths of 2nd carbon around vacancy in case of carbon atoms 41 Figure 3.15 Minimum energy paths of carbon in case 2C 42 Figure 3.16 Seven diffusion paths of the 3rd carbon atom around the vacancy in case of carbon atoms 45 Figure 3.17 Minimum energy paths of carbon in case 3C 46 Figure 3.18 Jumping rate in case of 1, and carbon atoms as an inverse function of temperature 50 Figure 3.19 Diffusion coefficient vs temperature in cases: perfect case and vacancy case 53 ii LIST OF TABLES Table 1.1 Different phases of steel based on carbon content [23] .3 Table 3.1 Configuration of system when carbon is adopted in position 1, 2, 29 Table 3.2 The binding energy between vacancy-carbon at position P1 to P7 for both size 3x3x3/8x8x8 was calculated with the consideration of the distance 32 Table 3.3 Binding energy at P1 by DFT method from some authors is collected for system size 3x3x3 and 4x4x4 .32 Table 3.4: Binding energy from MD and DFT method (for size 3x3x3) is computed for seven configurations .34 Table 3.5 Position of carbon atoms before and after optimized 35 Table 3.6 The binding energy of seven configurations in case of carbons 35 Table 3.7: Position of carbon atoms before and after optimized in case 3C 36 Table 3.8: Binding energy of configurations in case carbon atoms 37 Table 3.9: The change position of carbon in configuration after optimized 37 Table 3.10 Relaxation configurations in two carbon case .43 Table 3.11 Comparison between two durable configurations of two carbon atoms in perfect and vacancy case 44 Table 3.12 Relaxation configurations in three carbon case 47 Table 3.13 Comparison between two stable configurations of three carbon atoms in perfect and vacancy case 48 Table 3.14 Mean square displacement of carbon atom vs time of some temperatures in both case: perfect case and vacancy case by kMC .52 iii LIST OF ABBREVIATIONS Abbreviation Description ABOP Analytic Bond-order Potential BCC Body-Centered Cubic DFT Density Functional Theory MD Molecular Dynamics MEP Minimum energy path kMC Kinetic Monte Carlo FDM Finite Different Method CI-NEB Climbing image – Nudged Elastic Band O-site Octahedral site T-site Tetragonal site TST Transition State Theory LAMMPS Large-scale Atomic/Molecular Massively Parallel Simulator iv CHAPTER INTRODUCTION Nowadays, along with the steady development of science and technology, the achievements in scientific research are increasingly contributing to society, especially in the field of nanotechnology Research, development, and application of potential and unique properties from nanoscale materials have brought many improvements and breakthroughs compared to previous traditional materials [15] The field of computational materials science is considered as one of the areas of top concern in material science today [9] Calculations are implemented based on the theoretical foundations, which apply to specific subjects under the simulation process supported by modern computer systems, acting as useful tools in describing, verifying, predicting the rules, physical phenomena occurring inside objects and between objects The development process of computational science is an essential and inseparable part of the practical application in industry In particular, the calculation related to iron-carbon alloys is a good example and plays a crucial role in the development of the steel industry Until now, the steel industry has an extraordinary development, which can be divided into three main generations The first generation - Conventional low carbon steels can be mentioned as high strength low-alloy products (HSLA) steels, advanced high strength steels (AHSS), IF (Interstitial Free), DP (Dual Phase) or so-called TRIP / TWIP (Transformation or Twinning Induced Plasticity), etc is incredibly famous and widely used steel generation today [4] The second generation - Austenitic-Based Steels has been developed, and the third generation is still being researched and developed For different generations, superiorities and disadvantages still exist not only on mechanical properties but also on product costs Therefore, the main goal of this third-generation material system is to continue to improve the desired mechanical MSD and simulation time, combined with Einstein relation (equation 3.8) The diffusion coefficient of carbon atom is determined as a function of temperature in Figure 3.19 Figure 3.19 Diffusion coefficient vs temperature in cases: perfect case and vacancy case Figure 3.19 illustrates the relationship between the diffusion coefficient and temperature as a linear function By steadily increasing the simulated temperature range from 300K to over 1000K, the diffusion coefficient also increases gradually In the low-temperature range, we found that the diffusion coefficient of carbon impurity in the perfect case is higher when compared to the case of vacancy, but the difference is negligible However, at some high-temperature points that are close to the phase transition temperature, the diffusion coefficient of carbon at these states in the case of vacancy is higher than the perfect case An assumption to explain this result may come from the increase in temperature When the temperature is high enough, and the system provides enough energy for the carbon atom, it can escape from the confinement of the vacancy (the energy barriers at vacancy site for carbon atom to escape in 1C case are 0.359 eV and 0.488 eV) However, in order to verify this result, we need to conduct additional investigations to give a specific conclusion 53 CONCLUSION In this work, we used the newly developed potential for MD simulation to investigate the stability of carbon for cases 1, 2, 3, and carbon atoms with the system size of 3x3x3 and 8x8x8 In particular, the binding energy for the most stable configuration is in cases 1, 2, and carbon are -0.375 eV, -1.944 eV, -2.658 eV and -2.857 eV, respectively Based on the binding energy, we found that the effect of vacancy on carbon atoms is considered to be the strongest within Å Besides, the trend of clustering formation of carbon atoms is also investigated In 2C case, the binding energy of the carbon pair is -1.588 eV with the bond length of about 1.5 Å, which means that the interaction between them becomes stronger than the nonvacancy case (binding energy: -0.140 eV, bond length: 1.7 Å) to approach the bond length of C-C pair in graphene (1.42 Å) In 3C case, the 3rd carbon atom tends to be localized close to the carbon pair to form carbon clusters Therefore, we can conclude that under the influence of vacancy, the bonding between carbon atoms becomes more stable; the clustering formation is also observed in the case of 3C; Finally, the jumping rate and diffusion coefficient of carbon impurity in the system also has the significant changes in the presence of vacancy 54 FUTURE PLAN In the framework of the thesis, we have studied the effect of point defects on the diffusion of carbon atoms in  -iron system In the future, the topic may expand and continue relevant studies as follows: • Using DFT method to make a comparison with the results from MD method; • Conducted kMC simulation for multiple carbon atoms; • Expanding research for other types of defects: grain boundary, dislocation, etc • Investigate the diffusion of multi-carbon in the system 55 LIST OF PUBLICATIONS Ngoc Nam Ho, Tien Quang Nguyen, Yoji Shibutani, “Atomistically kinematic properties of carbon diffusion in alpha iron with point defects”, Vietnam-Japan Science and Technology Symposium (VJST 2019), paper 114, 2019.05.04 Tien Quang Nguyen, Ngoc Nam Ho, Thi Thu Dinh Ngo, Kazunori Sato, Yoji Shibutani, “Diffusion properties of carbon in Fe-C alloy using new Tersoff potential”, 日 本 機 械 学 会 第 31 回 計 算 力 学 講 演 会 / Proceedings of the Computational Mechanics Conference https://doi.org/10.1299/jsmecmd.2018.31.237 56 31 (2018) 237, REFERENCES [1] Atkins, P., & De Paula, J (1989a) Atkin’s Physical Chemistry In Journal of Chemical Information and Modeling (Vol 53) https://doi.org/10.1017/CBO9781107415324.004 [2] Bakaev, A., Terentyev, D., He, X., Zhurkin, E E., & Van Neck, D (2014b) Interaction of carbon–vacancy complex with minor alloying elements of ferritic steels Journal of Nuclear Materials, 451(1–3), 82–87 https://doi.org/10.1016/J.JNUCMAT.2014.03.031 [3] Band, Y B., Avishai, Y., Band, Y B., & Avishai, Y (2013c) Density Functional Theory Quantum Mechanics with Applications to Nanotechnology and Information Science, 871–889 https://doi.org/10.1016/B978-0-444-537867.00015-0 [4] Barella, S., Bondi, E., Di Cecca, C., Ciuffini, A F., Gruttadauria, A., Mapelli, C., & Mombelli, D (2014d) New perspective in steelmaking activity to increase competitiveness and reduce environmental impact Metallurgia Italiana, 106(11–12), 31–39 [5] Bhadeshia, H K D H (Harshad K D H., & Honeycombe, R W K (Robert W K (n.d.-e) Steels : microstructure and properties Retrieved from https://books.google.com.vn/books?id=4Rt5CgAAQBAJ&printsec=frontcover &dq=the+properties+of+steel&hl=vi&sa=X&ved=0ahUKEwjjoffl_N_iAhUQ_ GEKHat6DXAQ6AEINjAC#v=onepage&q=the properties of steel&f=false [6] Bortz, A B., Kalos, M H., & Lebowitz, J L (1975f) A new algorithm for Monte Carlo simulation of Ising spin systems Journal of Computational Physics, 17(1), 10–18 https://doi.org/10.1016/0021-9991(75)90060-1 [7] Cai, W., Bulatov, V V, Justo, J F., Argon, A S., & Yip, S (2002g) Kinetic Monte Carlo approach to modeling dislocation mobility Computational Materials Science, 23(1–4), 124–130 https://doi.org/10.1016/S0927- 0256(01)00223-3 [8] Carlo, M., Carlo, K M., & Kratzer, P (2009h) Multiscale Simulation Methods 57 in Molecular Sciences In Forschungszentrum J ă ulich (Vol 42) Retrieved from http://www.fz-juelich.de/nic-series/volume42 [9] de Borst, R (2008i) Challenges in computational materials science: Multiple scales, multi-physics and evolving discontinuities Computational Materials Science, 43(1), 1–15 https://doi.org/10.1016/j.commatsci.2007.07.022 [10] Dever, D J (1972j) Temperature dependence of the elastic constants in α‐iron single crystals: relationship to spin order and diffusion anomalies Journal of Applied Physics, 43(8), 3293–3301 https://doi.org/10.1063/1.1661710 [11] Dinh, N T T (2018k) Computational Kinetics on Diffusion Process of Carbon in Iron Bulk Vietnam Japan University [12] Domain, C., Becquart, C S., & Foct, J (2004l) Ab initio study of foreign interstitial atom (C, N) interactions with intrinsic point defects in α-Fe Physical Review B - Condensed Matter and Materials Physics, 69(14), 144112 https://doi.org/10.1103/PhysRevB.69.144112 [13] Elton, D C (2013m) Energy Barriers and Rates -Transition State Theory for Physicists Retrieved from http://moreisdifferent.com/wp- content/uploads/2015/07/transition_state_theory_dan_elton1.pdf [14] Först, C J., Slycke, J., Van Vliet, K J., & Yip, S (2006n) Point Defect Concentrations in Metastable Fe-C Alloys Physical Review Letters, 96(17), 175501 https://doi.org/10.1103/PhysRevLett.96.175501 [15] Foster, L E., & E., L (2006o) Nanotechnology : science, innovation and opportunity Retrieved from https://dl.acm.org/citation.cfm?id=1121649 [16] Frenkel, D., & Smit, B (n.d.-p) Understanding molecular simulation : from algorithms to applications [17] Fu, C.-C., Torre, J D., Willaime, F., Bocquet, J.-L., & Barbu, A (2004q) Multiscale modelling of defect kinetics in irradiated iron Nature Materials, 4(1), 68–74 https://doi.org/10.1038/nmat1286 [18] Fu, C C., Meslin, E., Barbu, A., Willaime, F., & Oison, V (2008r) Effect of C on Vacancy Migration in α-Iron Solid State Phenomena, 139, 157–164 https://doi.org/10.4028/www.scientific.net/SSP.139.157 58 [19] H Föll (n.d.-s) The Iron Carbon Phase Diagram Retrieved June 6, 2019, from https://www.tf.uni-kiel.de/matwis/amat/iss/kap_6/illustr/s6_1_2.html [20] Henkelman, G., & Jónsson, H (1999t) A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives The Journal of Chemical Physics, 111(15), 7010–7022 https://doi.org/10.1063/1.480097 [21] Henkelman, G., & Jónsson, H (2000u) Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points The Journal of Chemical Physics, 113(22), 9978–9985 https://doi.org/10.1063/1.1323224 [22] Henkelman, G., Uberuaga, B P., & Jónsson, H (2000v) Climbing image nudged elastic band method for finding saddle points and minimum energy paths Journal of Chemical Physics, 113(22), 9901–9904 https://doi.org/10.1063/1.1329672 [23] Hosford, W (2013w) Tempering and Surface Hardening In Physical Metallurgy, Second Edition https://doi.org/10.1201/b15858-19 [24] How Steel is Made (2015x) Scientific American, 10(2), 20–21 https://doi.org/10.1038/scientificamerican01091864-20a [25] Jiang, D E., & Carter, E A (2003y) Carbon dissolution and diffusion in ferrite and austenite from first principles Physical Review B - Condensed Matter and Materials Physics, 67(21) https://doi.org/10.1103/PhysRevB.67.214103 [26] Liu, Y.-L., Zhou, H.-B., Zhang, Y., & Duan, C (2012z) Point defect concentrations of impurity carbon in tungsten Computational Materials Science, 62, 282–284 https://doi.org/10.1016/j.commatsci.2012.05.012 [27] Liu, Y.-L., Zhou, H.-B., Zhang, Y., Lu, G.-H., & Luo, G.-N (2011aa) Interaction of C with vacancy in W: A first-principles study Computational Materials Science, 50(11), 3213–3217 https://doi.org/10.1016/J.COMMATSCI.2011.06.003 [28] McCool, M A., & Woolf, L A (1972ab) Pressure and temperature dependence of the self-diffusion of carbon tetrachloride Journal of the Chemical Society, Faraday Transactions 1: Physical Chemistry in Condensed Phases, 68(0), 1971 59 https://doi.org/10.1039/f19726801971 [29] Müller, M., Erhart, P., & Albe, K (2007ac) Analytic bond-order potential for bcc and fcc iron - Comparison with established embedded-atom method potentials Journal of Physics Condensed Matter, 19(32), 23 https://doi.org/10.1088/0953-8984/19/32/326220 [30] Munro, L J., & Wales, D J (1999ad) Defect migration in crystalline silicon Physical Review B, 59(6), 3969–3980 https://doi.org/10.1103/PhysRevB.59.3969 [31] Nguyen, T Q., Sato, K., & Shibutani, Y (2018ae) Development of Fe-C interatomic potential for carbon impurities in α-iron Computational Materials Science, 150, 510–516 https://doi.org/10.1016/J.COMMATSCI.2018.04.047 [32] Ohnuma, T., Soneda, N., & Iwasawa, M (2009af) First-principles calculations of vacancy–solute element interactions in body-centered cubic iron Acta Materialia, 57(20), 5947–5955 https://doi.org/10.1016/J.ACTAMAT.2009.08.020 [33] Ohtsuka, H., Dinh, V A., Ohno, T., Tsuzaki, K., Tsuchiya, K., Sahara, R., … Nakamura, T (2014ag) First-Principles Calculation of the Effects of Carbon on Tetragonality and Magnetic Moment of BCC-Fe Tetsu-to-Hagane, 100(10), 1329–1338 https://doi.org/10.2355/tetsutohagane.100.1329 [34] Plimpton, S (1995ah) Fast Parallel Algorithms for Short-Range Molecular Dynamics In Journal of Computational Physics (Vol 117) Retrieved from http://www.cs.sandia.gov/∼sjplimp/main.html [35] Ridnyi, Y M., Mirzoev, A A., Schastlivtsev, V M., & Mirzaev, D A (2018ai) Ab initio Computer Simulation of Carbon–Carbon Interactions for Various Spacings in BCC and BCT Lattices of Ferrite and Martensite Physics of Metals and Metallography, 119(6), 576–581 https://doi.org/10.1134/s0031918x18060121 [36] Rilwan Okunnu (2015aj) High Strength Solution-Strengthened Ferritic Ductile Cast Iron (Aalto University - School of Engineering) Retrieved from www.aalto.fi 60 [37] Santen, R A., & Niemantsverdriet, J W (1995ak) Chemical Kinetics and Catalysis Springer US [38] Sheppard, D., Terrell, R., & Henkelman, G (2008al) Optimization methods for finding minimum energy paths The Journal of Chemical Physics, 128(13), 134106 https://doi.org/10.1063/1.2841941 [39] Sholl, D S., & Steckel, J A (2009am) Density functional theory [electronic book] : a practical introduction In Online access with subscription: Ebrary Retrieved from https://www.wiley.com/en- us/Density+Functional+Theory%3A+A+Practical+Introduction-p9780470373170 [40] Simonovic, D., Ande, C K., Duff, A I., Syahputra, F., & Sluiter, M H F (2010an) Diffusion of carbon in bcc Fe in the presence of Si Physical Review B - Condensed Matter and Materials Physics, 81(5), 054116 https://doi.org/10.1103/PhysRevB.81.054116 [41] Steadman, R (1970ao) Materials science Physics Education, 5(2), 70–71 https://doi.org/10.1088/0031-9120/5/2/304 [42] Terentyev, D., Bonny, G., Bakaev, A., & Van Neck, D (2012ap) On the thermal stability of vacancy–carbon complexes in alpha iron Journal of Physics: Condensed Matter, 24(38), 385401 https://doi.org/10.1088/0953- 8984/24/38/385401 [43] Trygubenko, S A., & Wales, D J (2004aq) A doubly nudged elastic band method for finding transition states The Journal of Chemical Physics, 120(5), 2082–2094 https://doi.org/10.1063/1.1636455 [44] Veiga, A (2011ar) Computational insights into the strain aging phenomenon in bcc iron at the atomic scale (L’Institut National des Sciences Appliquees de Lyon) Retrieved from http://theses.insa- lyon.fr/publication/2011ISAL0084/these.pdf [45] Voter, A F (2005as) Introduction to kinetic Monte Carlo https://doi.org/10.1007/978-1-4020-5295-8_1 [46] Wilson, E B., Decius, J C., Cross, P C., & Sundheim, B R (2007at) Molecular 61 Vibrations: The Theory of Infrared and Raman Vibrational Spectra Journal of The Electrochemical Society, 102(9), 235C https://doi.org/10.1149/1.2430134 [47] Winn, E B (1950au) The Temperature Dependence of the Self-Diffusion Coefficients of Argon, Neon, Nitrogen, Oxygen, Carbon Dioxide, and Methane Physical Review, 80(6), 1024–1027 https://doi.org/10.1103/PhysRev.80.1024 [48] Wunderlich, W., & Pilkey, W (2010av) The Finite Difference Method In Mechanics of Structures (pp 495–531) https://doi.org/10.1201/9781420041835.ch8 [49] Yamamoto, M., Matsunaka, D., & Shibutani, Y (2008aw) Modeling of heteroepitaxial thin film growth by Kinetic Monte Carlo Japanese Journal of Applied Physics, 47(10 https://doi.org/10.1143/JJAP.47.7986 62 PART 1), 7986–7992 APPENDIX 63 CopyrightⒸ2018 一般社団法人 日本機械学会 237 元 Tersoff ポテンシャルを用いた Fe-C 合金における炭素の拡散特性 Diffusion Properties of Carbon in Fe-C Alloy using New Tersoff Potential ○NGUYEN Tien Quang*1,2, HO Ngoc Nam*2, NGO Thi Thu Dinh*2, 佐藤 和則*1，渋谷 陽二*1,2 Tien Quang NGUYEN*1,2, Ngoc Nam HO*2, Thi Thu Dinh NGO*2, Kazunori SATO*1 and Yoji SHIBUTANI*1,2 大阪大学 Osaka University 日越大学 Vietnam-Japan University *1 *2 Diffusion mechanisms for one to three carbon interstitials in BCC iron are studied using new Fe-C interatomic potential Firstly, carbon diffusion and stability are investigated by using nudged-elastic band method The diffusion paths are also verified and analyzed by constructing potential energy landscapes Finally, diffusion rates are calculated based on the transition state theory at different temperatures to study the precipitation of carbon in iron bulk For the case of 1C, the diffusion barrier from O-site to 1NN O-site is about 1.106×10-19J In the case of 2C, two carbon atoms tend to form a C-C pair, where both are shifted off the O-sites The C-C binding energy is about -0.224×10-19J In 3C case, the third carbon prefers to bind to pre-located C atoms in the neighbor cells with a binding energy of -1.057×10-19J In addition, from diffusion rate calculations, it shows that carbon atoms tend to combine to stable pairs instead of larger clusters Key Words : Tersoff Potential, Iron-Carbon Alloy, Carbon Diffusion, Diffusion Rate, Nudged-Elastic Band INTRODUCTION Designing materials which could simultaneously demonstrate multiple promising properties like high strength and good ductility has been a long-standing demand for steel industry Since the mechanical properties of materials depend on their microstructure(1), to obtain both properties, one considers to use numerous innovative methods to control the microstructure of steel by adequate processing While the excellent mechanical properties have been observed experimentally, the physics behind its operation are still unclear In carbon steel, the distribution and diffusion of carbon in iron can control the formation and kinetics of many important processes such as carbide precipitation, martensite ageing, and ferrite transformation(2)(3) While the level of carbon is essential for optimizing microstructure, the location of carbon determines whether the steel is in martensitic or ferritic forms Diffusion of carbon strongly linked to processes of production of steels To better choose thermal treatments for desired phase composition it is necessary to predict the precipitation kinetics Hence, the study of diffusion properties of carbon can be a good hint to manipulate the microstructure composition, from that the mechanical properties can be controlled METHODOLOGY 2・1 Interacting potential The method of classical atomistic simulation is used in this work The interactions between atoms in Fe-C system are described by a newly developed Tersoff potential, in which the potential parameters are fit to the density functional theory (DFT) data using force-matching method The details of potential construction and parameters are described in our previous work(4) We note that, in BCC iron, this potential predicts octahedral site (O-site) energetically more stable over tetrahedral site (T-site) for carbon occupation, which is in good agreement with previous density functional theory (DFT) calculations(5) In [No.18-8] 第 31 回計算力学講演会（CMD2018） 講演論文集 〔2018.11.23-25，（徳島）〕 CopyrightⒸ2018 一般社団法人 日本機械学会 addition, it reproduces very well the energy barrier and the minimum energy path of single carbon diffusion in BCC iron Hence, this potential is suitable for investigating the diffusion properties of carbon in Fe-C alloys 2・2 System and simulation An 8×8×8 supercell model with 1024 iron atoms was chosen for atomistic simulations The optimal lattice constant for BCC iron crystal using the new Tersoff potential is about 2.889×10-10m (2.889Å) This supercell is large enough to avoid carboncarbon interaction in neighbor cells while the computation cost is reasonable for modern computer resource Interstitial carbon atoms, one-by-one (up to three), are introduced into the supercell according to the details of each simulation Thus, the atomic percentage of carbon is in the range of 0.021÷0.063 mass% Total energy calculations and structural optimization are carried out using the public domain code LAMMPS(6) The diffusion properties of carbon in Fe-C alloy can be interpreted through the jumping rates of diffusion events of carbon, which in turn is defined by the following equation from the Transition State Theory (TST)(7): Γ ( j ) = Γ (0j ) exp(−Ed( j ) / kBT ) ($) where, Γ" （1） ($) is the attempt frequency of the j-th event, 𝐸' is the diffusion barrier, which is the difference in potential energies between an initial state and a saddle point along the diffusion path, 𝑘) is the Boltzmann constant, and 𝑇 is the temperature In order to determine diffusion barrier, the climbing-image nudged-elastic band method (CI-NEB)(8) is used Firstly, several possible diffusion paths of carbon are assumed For each diffusion path, the initial and final positions of carbon are pre-located and fully optimized After that, in the search for the transition state and the minimum energy path, the initial and final states are kept fixed while other intermediate structures (so called “images” or “replica”) are allowed to relax using force-based optimizers For the effective attempt frequency, it can be evaluated from the TST using the following equation: 3N 3N−1 i=1 i=1 Γ (0j ) = ∏ν i / ∏ ν i# （2） where, ν" is the normal mode frequency at the initial position and ν#" is that at the saddle point, where one and only one of the vibration frequencies should be negative Within the harmonic approximation, the normal mode frequencies are obtained by diagonalization of dynamical matrices, which are constructed using the finite-difference method for evaluation of the second derivatives of energy with respect to the atomic positions RESULTS AND DISCUSSIONS 3・1 Single carbon diffusion In BCC iron, even though T-site has more space, due to the asymmetry arrangement of surrounding iron atoms, carbon atoms primary occupies O-sites The total energy calculations using new potential confirm that the dissolution energy of carbon at Osite is about 1.354×10-19J, that is much lower than the value of 2.634×10-19J of T-site(4) This is similar to the first-principles results(5)(9) As a consequence, the diffusion of carbon atom from O-site to others neighbour O-sites is investigated In Fig 1a, the diffusion paths of carbon to the first (1NN), second (2NN), and third (3NN) nearest neighbours O-sites are shown The diffusion barriers are found to be 1.086×10-19J (0.687eV), 1.059×10-19J (0.661eV), and 1.942×10-19J (1.212eV) for 1NN, 2NN, and 3NN diffusion paths, respectively For the 1NN case, the transition state of O-O diffusion is located exactly at T-site To verify this, the energy landscape of (001) plane which contains the three O and T points above is scanned and shown in Fig 1b The diffusion mechanism simply follows an O-T-O route, which is consistent with previous DFT results(4) For the 2NN path, we found two transition states located at two T-sites which has similar barrier with that of 1NN case This initially implies an O-T-T-O diffusion route for carbon However, we note that between these saddle points, there is a straight line in the range of 1.5ữ2.5ì10-10m (1.5ữ2.5) To check this, the energy landscape of (200) plane which contains the four mentioned O and T [No.18-8] 第 31 回計算力学講演会（CMD2018） 講演論文集 〔2018.11.23-25，（徳島）〕 CopyrightⒸ2018 一般社団法人 日本機械学会 points is scanned and plotted in Fig 1c The energy landscape suggests that the natural diffusion better follows an O-T-O-T-O route because the potential energy along T-T line is higher than that of T-O-T line This is due to the fact that the CI-NEB is original constructed for finding single saddle point For complicated diffusion path like the 2NN case, the CI-NEB method might not predict the path correctly Finally, for the case of 3NN, the barrier is much higher than those of 1NN and 2NN paths This leads to a conclusion that the mechanism for single carbon diffusion will highly follows the O-T-O route 1.4 - 1NN - 2NN - 3NN Energy related to initial state (eV) (a) 1.212eV 1.2 (b) (c) 0-1NN 0-2NN (d) T 0-3NN 1.0 T 0.8 0.687eV 0.661eV 0.6 O T O 0.4 0.2 0.0 Reaction coordinate (Å) Fig (a) Minimum energy paths of carbon diffusion from O-sites to neighbour O-sites in BCC iron Iron atoms are shown in yellow colour, while only diffusion trajectories are shown for carbon (b, c, d) Potential energy landscapes of single carbon in BCC iron within primitive cell for the 1NN, 2NN, and 3NN cases T-sites and O-sites are marked with red stars and blue circles, respectively Iron atoms at four corners and centre are note shown 3・2 Diffusion in two-carbon and three-carbon cases 0.8 0.6 2C stable 0.6 1st C 1st C 2nd C 0.438eV 0.4 2nd C 0.2 C-C distance: 1.737Å 0.0 C-C binding energy: -0.140eV Diffusion along [100], from O-site to 1NN O-site -0.2 0.0 0.5 1.0 1.5 2.0 Reaction coordinate (Å) 2.5 3.0 (b) Energy related to initial state (eV) Energy related to initial state (eV) (a) 3C stable 1st C 1st C 0.4 2nd C 2nd C 0.256eV 0.2 3rd C trajectory 3rd C 0.0 -0.2 0.0 C-C distance: C1-C2: 1.544Å C1-C3: 4.066Å C2-C3: 3.800Å Diffusion along [001], from O-site to 1NN O-site 0.5 1.0 1.5 2.0 2.5 3.0 C-C-C binding energy: -0.663eV Reaction coordinate (Å) Fig (a) Minimum energy paths of carbon diffusion from O-sites to neighbour O-sites with the presence of one neighbour carbon and the most stable structure of C-C pair (b) Minimum energy paths of carbon diffusion from O-sites to neighbour O-sites with the presence of two carbon atoms and the most stable configuration Now, we investigate the diffusion of system with two carbon atoms To simplify the problem, one carbon is assumed to be pre-located at O-site The second carbon then will be placed at other O-sites surrounding the first carbon and is allowed to diffuse in all directions along three cartesian axes By including symmetry, a numerous diffusion paths are proposed Among them, the path which has smallest diffusion barrier is shown in Fig 2a In this path, the 2nd carbon occupies initially at the 1NN O-site, on the (001) plane and along the [010] direction From here, it diffuses along the [100] direction with an energy barrier of 0.702×10-19J (0.438eV) By examining the binding energy between carbon atoms for all cases, we find that the two atoms form a stable pair with a binding energy of -0.224×10-19J The C-C bonding distance is about 1.737×10-10m For the case of three-carbon, in a similar manner, we assume that the first two carbon atoms are pre-located in the form of stable pair as described above The 3rd carbon is set up in the vicinity of the pair and placed at O-sites As seen in Fig 2b, the 3rd carbon does not prefer to stay near to the two pre-located carbon atoms to form a bigger cluster It tends to be away from the pair and resides in the neighbor cells The binding energy of this carbon with the pair is about -1.057×10-19J This is actually the final state of the diffusion process with lowest barrier of 0.410×10-19J between two O-sites along the [00-1] direction [No.18-8] 第 31 回計算力学講演会（CMD2018） 講演論文集 〔2018.11.23-25，（徳島）〕 CopyrightⒸ2018 一般社団法人 日本機械学会 3・3 Comparison of diffusion rates In the diffusion process of carbon interstitials, the factors that govern diffusion rate are temperature, carbon concentration, and so on To understand about the dependence of diffusion rate on these factors, the temperature is varied for three cases above In Fig 3, the diffusion rates calculated from equations (1) and (2) are compared We can see that, single carbon case has the lowest diffusion rate, whereas three-carbon case has the highest one These results support the conclusion in previous sections that the carbon couple binding is more dominated in lieu of the formation of large carbon cluster 10 Diffusion rate log10 (k )/s-1 -10 -20 -30 -40 -50 -60 0.000 Single carbon Two-carbon Three-carbon 0.005 0.010 0.015 0.020 Reciprocal temprature (K-1 ) Fig Comparison of jumping rate at different temperatures for one-, two-, and three-carbon cases CONCLUSION By using newly-developed Tersoff potential, we have investigated the diffusion properties of carbon in BCC iron for one to three carbon atoms For single carbon, the fundamental mechanism for diffusion should be from O-site to the 1NN O-site In case of two carbons, these two atoms tend to form C-C pairs, where one is shifted off the O-site and another is located closely to T-site, with binding energy of -0.224×10-19J The smallest energy barrier of carbon when diffusing to other position is about 0.702×10-19J In three-carbon case, the third carbon prefers to bind to the two pre-located carbon atoms in the neighbor cells with a binding energy about -1.057×10-19J The smallest values of activation energies are about -0.417×10-19J Moreover, the results of diffusion rates showed the binding ability of carbon atoms: when carbon atoms were inserted into Fe bulk, the formation of large clusters was not a priority because the carbon atoms tend to combine to form a stable C-C pair REFERENCES (1) Takaki, S., Kawasaki, K., and Kimura, Y., “Mechanical Properties of Ultra Fine Grained Steels”, Journal of Materials Processing Technology, Vol 117 (2001), pp 359-363 (2) Taylor, K.A., and Cohen, M., “Aging of Ferrous Martensites”, Progress in Materials Science, Vol 36 (1992), pp 225-272 (3) Hillert, M., “Diffusion and Interface Control of Reactions in Alloys”, Metallurgical Transactions A, Vol (1975), pp 5-19 (4) Nguyen, T.Q., Sato, K., and Shibutani, Y., “Development of Fe-C Interatomic Potential for Carbon Impurities in ɑ-Iron”, Computational Materials Science, Vol 150 (2018), pp 510-516 (5) Domain, C., Becquart, C.S., and Foct, J., “Ab initio Study of Foreign Interstitial Atom (C, N) Interactions with Intrinsic Point Defects in α-Fe”, Physical Review B, Vol 69 (2004), pp 144112(16) and references therein (6) Plimpton, S., “Fast Parallel Algorithms for Short-Range Molecular Dynamics”, Journal of Computational Physics, Vol 117 (1995), pp 1-19 Available at: http://lammps.sandia.gov/ (7) Vineyard, G.H., “Frequency Factors and Isotope Effects in Solid State Rate Processes”, Journal of Physics and Chemistry of Solids, Vol (1957), pp 121-127 (8) Henkelman, G.B., Uberuga, P., and Jonsson, H., “A Climbing Image Nudged Elastic Band Method for Finding Saddle Points and Minimum Energy Paths”, The Journal of Chemical Physics, Vol 113 (2000), pp 9901-9904 (9) Nguyen, T.Q., Sato, K., and Shibutani, Y., “First-Principles Study of BCC/FCC Phase Transition Promoted by Interstitial Carbon in Iron”, Materials Transactions, Vol 59 (2018), pp 870-875 [No.18-8] 第 31 回計算力学講演会（CMD2018） 講演論文集 〔2018.11.23-25，（徳島）〕 ... binding energy of seven configurations in case of carbons 35 Table 3.7: Position of carbon atoms before and after optimized in case 3C 36 Table 3.8: Binding energy of configurations in case carbon. .. results in describing minimum energy path (MEP) of carbon with T site found as a transition point [25] This topic is intended to provide a clearer and more objective view of the point defect in the... Illustration of finding the minimum energy path by NEB Each image on the chain of the system is connected by spring forces which located along the minimum energy line between two minimum energy points