Computational kinetics on diffusion process of carbons in iron bulk

VIETNAM NATIONAL UNIVERSITY, HANOI VIETNAM JAPAN UNIVERSITY - NGO THI THU DINH COMPUTATIONAL KINETICS ON DIFFUSION PROCESS OF CARBONS IN IRON BULK MASTER'S THESIS Hanoi, 2018 VIETNAM NATIONAL UNIVERSITY, HANOI VIETNAM JAPAN UNIVERSITY NGO THI THU DINH COMPUTATIONAL KINETICS ON DIFFUSION PROCESS OF CARBONS IN IRON BULK MAJOR: NANO TECHNOLOGY SUPERVISORS: Prof Dr YOJI SHIBUTANI Dr NGUYEN TIEN QUANG Hanoi, 2018 Acknowledgment First and foremost, I am so grateful to my supervisor, Professor Yoji Shibutani, Division of Mechanical Engineering, Graduate School of Engineering of Osaka University, for his progressive outlook and meticulous guidance to conduct my thesis Without his constant reminder and valuable advice from time-to-time, I would probably not manage to complete the work in an appropriate manner My special gratitude also to the second supervisor - Dr Nguyen Tien Quang, Osaka University, who spent a lot of time for me to develop and give basic instructions His patient guidance, encouragement and suggestions provided me the necessaries to struggle the research problems and set the stage for the meaningful ending of the work in a short duration Finally, I would wish to express my deepest gratitude to Vietnam-Japan University and Osaka University as well as my classmates for their strong support and great help during the time I did my thesis i TABLE OF CONTENTS Acknowledgment i LIST OF FIGURES, SCHEMES iii LIST OF TABLES iv LIST OF ABBREVIATIONS .v INTRODUCTION .1 CHAPTER LITERATURE REVIEW 1.1 Introduction to iron and steel 1.2 Potential 1.3 The purpose of study CHAPTER THEORETICAL BASICS 2.1 The transition state theory 2.2 Analytic Bond-Oder Potential 2.3 Nudged Elastic Band method 14 2.4 Molecule vibrations 17 2.4.1 Small Vibrations in Classical Mechanics 17 2.4.2 Normal Modes of Vibration and Normal Coordinates 20 CHAPTER METHODOLOGY AND RESULTS 21 3.1 Implementation methods 21 3.1.1 Lattice constant 21 3.1.2 Optimization structure 22 3.1.3 Diffusion of carbon in BCC iron 26 3.1.4 Attempt frequency 27 3.1.5 Diffusion rate 28 3.2 Results and discussion 28 3.2.1 Diffusion paths .28 3.2.2 Diffusion rate 38 CONCLUSION 43 REFERENCES 44 ii LIST OF FIGURES, SCHEMES Figure 1.1 The ―banana curve‖ of steels shows the dependence of elongation and tensile strength Figure 2.1 The dependence of reaction energy diagram on reaction coordinate for an isomerization reaction Figure 2.2 Diffusion of carbon in BCC iron 14 Figure 2.3 Decomposition of force on an image 15 Figure 3.1 Octahedral and tetrahedral interstitial sites in the (a) Body-centered cubic (BCC) and (b) Face-centered cubic (FCC) lattice 22 Figure 3.2.The optimized structure of one carbon in BCC iron at different o-sites: 23 Figure 3.3 The optimized structures of two carbon atoms in BCC iron 24 Figure 3.4 Some of the optimized structures in BCC iron with adopted 25 Figure 3.5 Diffusion paths in the case of one carbon 29 Figure 3.6 Schematic of diffusion directions in the case of one carbon 30 Figure 3.7 Diffusion paths in the case of two carbon atoms 32 Figure 3.8 Diffusion path of carbon from C01 (initial state) to the ―down‖ direction 35 Figure 3.9 Diffusion paths in the case of three carbons 37 Figure 3.10 Dependence of the diffusion rate on temperature .41 Figure 3.11 Comparison diffusion rate with the concentration of carbon .42 iii LIST OF TABLES Table 1.1 Comparison of different phases of steel Table 2.1 The basic assumptions of the transition-state theory Table 2.2 Optimal iron-carbon potential parameters .13 Table 3.1 Comparison of lattice constant of BCC iron with different potentials and methods 22 Table 3.2 Diffusion energy barrier in the case of one carbon 29 Table 3.3 Diffusion energy barriers (eV) in the case of two carbon atoms 33 Table 3.4 Binding energies (eV) of carbon pair in the case of two carbon atoms 34 Table 3.5 Stability of carbon pair in the case of 2C 35 Table 3.6 Relaxation configurations in three carbon case .38 Table 3.7 Activation energies and attempt frequencies for carbon in BCC iron 39 iv LIST OF ABBREVIATIONS 1C case One carbon case 2C case Two carbon case 3C case Three carbon case AHSS Advanced high strength steels BCC Body-centered cubic BCT Body-centered tetragonal EOM Newton’s equation of motion FCC Face-centered cubic HSLA High-strength low-alloy steels LAMMPS Large-scale Atomic/ Molecular Massively Parallel Simulator MEP Minimum energy path NEB Nudged elastic band method NN Nearest neighbor O-site Octahedral site T-site Tetrahedral site TST Transition state theory v INTRODUCTION Nowadays, computing has become an important tool in scientific study, numerical simulation is used to incorporate between science and technology Computation technology plays a valuable role in advancing scientific knowledge, especially in the understanding of materials at atomic and molecular levels By using computational simulation, the scientists can select favorable materials for specific purposes, even improve advanced materials for applications In the moment, carbon steel is one of the most popular materials in engineering structure Basing on the carbon content, carbon steel has phase such as ferrite, austenite, or martensite with carbon occurring as interstitial atoms All these phases usually comprise of cementite and ferrite at room temperature It is required to enhance the carbon steel's strength and ductility for the enable lightweight construction as well as struggling environmental issues Then, a study to design novel steel is needed to figure out not only mechanical but also thermal properties There are many methods to get both properties, such as precipitation control, grain refinement and so on The diffusion of impurities is affected by several phenomena For instance, for the carbon diffusion in iron, because of its relatively high diffusivity, the kinetics of phase transformations in steel and the resulting microstructure is influenced by the interstitial diffusion of carbon However, interstitial diffusion might result in problems related to strain aging, embrittlement, and steel erosion In this research, in order to understand deeply the behavior of steel depending on different environments, we concentrated in study to explore the interstitial diffusion process in iron CHAPTER LITERATURE REVIEW 1.1 Introduction to iron and steel Ferrous alloys, remarkably steels, are very popular in today's society owing to the availability of cheap iron and excellent flexibility of completed products All of the understanding about processing changes of steel is described in Figure 1.1 with tensile range for low, medium, high and ultra-high strength steels, named the ―banana curve‖ It guides the continuity of characteristics as steel increases from the lowest to the highest level Figure 1.1 The ―banana curve‖ of steels shows the dependence of elongation and tensile strength [1] In Figure 1.1, the ellipses show the range of feature levels available in each steel There are three separate generations, included in: (i) Generation with advanced high strength steels (AHSS) and high-strength low-alloy (HSLA) steels that indicates the types of steel produced at the moment, (ii) Generation represents austenitic-based steels (Cr-Ni (3XX) stainless steel and twinning-induced plasticity (TWIP) steel), and (iii) Generation is being developed These types of steel will be designed for parts requiring less elongation than second generation steel, at less cost and better connecting power As we known, pure iron and steel are mostly the same material The main difference between pure iron and steel is the amount of carbon they contain Generally, in the iron-carbon alloy, a higher carbon content leads to a harder and less ductile material Alloys of iron with a carbon content above 2.1wt% are considered as cast iron Cast iron has a very high carbon content which makes it very hard, but also very brittle Wrought iron containing less than 0.08wt% carbon makes it a much better material for the application such as structural steel Because of low carbon content, it is a lot softer than cast iron This allows wrought iron to be bent under loads without breaking Steel is in between the two with a carbon content between 0.2wt% and 2wt%, giving it an ideal balance between hardness and ductility History of iron is determined by our capability to control carbon content Table 1.1 Comparison of different phases of steel [ [2], p 25] Phase Term Structure  - Fe Ferrite BCC  - Fe  - Ferrite FCC Fe3C Cementite Orthorhombic Fe-C solid Martensite solution Temperature Conditions T < 922.50C 911.50C < T < 13960C Notes Hard to Solubility C is an "Austenite stabilizer": add C,  field widens Hard ceramic, lower nucleation barrier than for graphite Metastable, generated by quenching BCT Table 1.1 shows several phases of steel with the presence of carbon at the different temperature Also depending on the carbon content, the microstructure may be ferrite, austenite, or martensite with carbon occurring as interstitial atoms In these phases, ferrite has slight contents of carbon, so it gets some properties as soft, easy to shape Once it has much harder, the phase is martensite Besides, with In table 3.5, information of atomic distances between atoms are shown, this indicates that the C01_down_min configuration is formed by one carbon at O-site while another occupies T-site The binding energy of C-C is found to be about -0.14 eV Note that, there is another position played the role in an intermediate state called C02_down_min which has some similar properties as the one mentioned above But it is not important to consider because two diffusion processes are just reversible Hence, it is clear that the second carbon has the tendency to bind to the first carbon at T-site This is an important conclusion as it allows us to limit the case of carbon diffusion when the number of carbon atoms increases up to three carbons In this case, we considered the configuration that fixes the first carbon at O-site and the second carbon at T-site as shown in the C01_down_min configuration 3.2.1.3 Three carbon case Likewise, the resulting paths from forty possible diffusion paths can be found in Figure 3.9 After optimizing configurations in three-carbon case, we obtained the total energy , So as to investigate the formation of carbon cluster, the binding energy between carbon pair with the third carbon is estimated (see Appendix A) There are many configurations which have negative binding energies such as C6_down, C6.3_left, C6.3, C6.2, C3.2, C2.3 configuration with and are -0.663 and - 0.802; -0.638 and -0.777; -0.595 and -0.734; -0.461 and -0.600; -0.460 and -0.599; 0.338 and -0.477 in eV, respectively In other words, these configurations are favorable Moreover, we also observed the optimized structures of three smallest energy configurations (Table 3.6) It was found that the third carbon does not prefer to stay near to two pre-located carbon atoms to form a bigger cluster It has a tendency to be away pre-located carbon atoms and reside in the neighbor cells 36 0.8 Energy related to initial state (eV) C3.2 C3.2_in C3.2_right 0.6 0.4 0.2 0.0 -0.2 -0.4 0.0 0.5 1.0 1.5 2.0 2.5 C4 Energy related to initial state (eV) 3.5 Reaction coordinate (Angstrom) 1.0 C4.2_left C4.2_out C4.3_out C4.3_right 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 0.0 0.6 Energy related to initial state (eV) 3.0 0.5 1.0 1.5 2.0 2.5 3.0 Reaction coordinate (Angstrom) C6 C6.2_up C6.3_down C6.3_left 0.4 0.2 0.0 -0.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Reaction coordinate (Angstrom) Figure 3.9 Diffusion paths in the case of three carbons 37 Table 3.6 Relaxation configurations in three carbon case C6.3 Binding energy: -0.595 eV Structural information: C1-C2:1.54293 Å C1-C3: 3.74237 Å C2-C3: 3.74235 Å Fe63-C3: 5.05635 Å Fe65-C3: 5.05635 Å C6.3_down Binding energy: -0.663 eV Structural information: C1-C2: 1.54165 Å C1-C3: 4.06619 Å C2-C3:3.79954 Å Fe63-C3: 5.23208 Å Fe65-C3: 5.23208 Å C6.3_left Binding energy: -0.638 eV Structural information: C1-C2: 1.54403 Å C1-C3: 3.78062 Å C2-C3: 4.05833 Å Fe63-C3: 5.22448 Å Fe65-C3: 5.22448 Å 3.2.2 Diffusion rate To make certain the minimal interaction of carbon atoms in the neighboring cells, the effect of the cell size on the vibration frequencies have been checked, specifically for the 1C case We found that most of the states have similar frequencies However, in the cases of 6x6x6 and 8x8x8 supercells, the real frequencies of the 0-1, 0-2 and 0-3diffusion paths are (24.2, 24.2, 32.5) THz for the initial state, (24.3, 25.5) THz, (2.6, 60.2) THz and 81.7 THz for saddle points with, respectively While for the 3x3x3 supercell, the differences are about (0.02, 0.02, 0.44) THz with the initial state; (0.01, 0.22) THz and 1.95 THz at saddle points with diffusion paths of 0-1 and 0-2, respectively (see at Appendix B) Therefore, bulk with 6x6x6 supercell size is enough to describe all properties of our system However, the reason why the 8x8x8 supercell is chosen is that it is essential to 38 investigate the system large enough to ensure the negligible change of volume when interstitial atoms present Table 3.7 Activation energies and attempt frequencies for carbon in BCC iron Case Process Activation energy Attempt frequency (eV) (THz) 1C 0-1 0.687 30.870 1C 0-2 0.661 121.399 2C C03_right 0.558 30.282 2C C04_right 0.584 28.167 2C C04_out 0.438 27.815 2C C06_down 0.438 24.416 2C C06_right 0.501 24.632 3C C3.2_in 0.714 25.026 3C C3.2_right 0.450 56.385 3C C4.2_left 0.880 21.324 3C C4.2_out 0.593 23.029 3C C4.3_out 0.496 33.535 3C C4.3_right 0.575 16.218 3C C6.2_up 0.510 52.330 3C C6.3_down 0.256 16.726 3C C6.3_left 0.258 16.482 As we mentioned, there are three vibration frequencies at the initial state and just two real frequencies for the saddle point, another is imaginary In the case of 1C, for the diffusion path of carbon from O-site to the 3NN O-site, two frequencies are found to be imaginary and one frequency is real This implies that the saddle point, in this case, corresponds to a second-order saddle point [49] The minimum energy 39 paths, in this case, was not found Therefore, it is not necessary to consider the diffusion path as well as the diffusion rate of the 0-3 path in the 1C case From formula (3.1), the attempt frequency was estimated Table 3.7 represents activation energies and attempt frequencies for all cases of diffusion models The activation energy is considered independent of the temperature in a small range of temperature where kinetic processes take place Likewise, in a wide range of practical conditions, it is reasonable to neglect the weak temperature dependence of the pre-exponential factor which is compared to the temperature dependence In the case of "barrierless" or diffusion-limited reactions, the preexponential factor is dominant and is observed directly Apply the values of above table in formula (3.2), the diffusion rate can be obtained depend on temperature Figure 3.10 shows the diffusion rates as a function of reciprocal temperature, when the system consists of one, two, and three carbon atoms From these plots, it may be concluded that when the temperature increases, the rate of diffusion also increases In all practical cases, the diffusion rate k increases rapidly with T As we know, diffusion process is defined by transferring of atoms from one material to another material through random atomic motion The factors that influent diffusion rate are temperature, concentration difference, and so on In which temperature is the greatest effect and the easiest to change When the energy is added to each particle, the atoms more frequently collide each other and spread evenly throughout the material volume Therefore, the temperature raises, the diffusion rate also increases In contract, with reducing of the temperature, the diffusion rate decreases Mathematically, at very high temperatures so that, diffusion rate levels off and approaches attempt frequency as a limit, but this case does not occur under practical conditions Obviously, there is not a significant difference between diffusion rates at high temperature 40 Diffusion rate log10 ()/s-1 -20 1C -40 0-1 -60 0-2 -28 -56 -84 2C C03_right C04_right C04_out C06_down C06_right -28 -56 -84 0.000 C3.2_in C3.2_right C4.2_left C4.2_out C4.3_out C4.3_right C6.2_up C6.3_down C6.3_left 0.005 3C 0.010 0.015 0.020 Reciprocal temperature T/K-1 Figure 3.10 Dependence of the diffusion rate on temperature Furthermore, the low diffusion rate indicates that interstitial atoms could not easily pass through in supercell Hence, it is important to consider the dependence of carbon concentration with highest diffusion rate in the cases of 1C, 2C and 3C Figure 3.11 illustrates the concentration difference of carbon on diffusion rate From these plots, we could see that 1C case has the lowest diffusion rate, whereas 3C case has the highest one It is reasonable to accept this conclusion due to the dependence 41 on the difference between concentrations across the host material With higher concentration, the difference resulting in higher diffusion rates Diffusion rate 10 Diffusion rate log10 ()/s-1 -10 -20 -30 -40 -50 -60 0.000 1C 0-1 2C C04_out 3C C6.3_down 0.005 0.010 0.015 0.020 Reciprocal temprature, T/(K-1) Figure 3.11 Comparison diffusion rate with the concentration of carbon As shown in above figure, the diffusion rate of the 3C case is higher than of 2C In addition, comparable values of activation energy for carbon diffusion for three cases at room temperature (0.687 eV, 0.438 eV and 0.256 eV, respectively) can give support for the binding ability of carbon atoms: When more than two carbon atoms are inserted into Fe system, third carbon tends to go far away from a couple one This means the former ability of carbon cluster is not a priority as well as the carbon couple binding is dominated 42 CONCLUSION By using the analytic bond-order potential [12], we have investigated the diffusion and the stability 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