Atomistically kinetic simulations of carbon diffusion in ɑ-iron with point defects

Seven diffusion paths of the 3rd carbon atom around the vacancy in case of 3 carbon atoms.. Nudged Elastic Band Method In energy surface analysis, it is known that finding the minimum e

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 1 INTRODUCTION 1

CHAPTER 2 THEORETICAL BASICS 6

CHAPTER 3 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] 3

Figure 1.3 Simulation picture of typical defects in iron-carbon alloy 5

Figure 2.1 Reaction energy diagram as a function of reaction coordinate q for an isomerization reaction [37] 7

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-barrier-limited 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 m E [44] 15

Figure 2.6 The K-th transition is chosen because its assigned value of s(K) intercepts r 2 i [44] 15

Figure 3.1 Positions 1, 2 of carbon correspond to O site, and 3 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 3 after optimized in case 2 carbon atoms 34

Figure 3.10 Configuration 3 after optimized 36

Figure 3.11 The most stable configuration in case 4 carbon atoms in iron 37

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 2 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 3 carbon atoms 45

Figure 3.17 Minimum energy paths of carbon in case 3C 46 Figure 3.18 Jumping rate in case of 1, 2 and 3 carbon atoms as an inverse function

of temperature 50

Figure 3.19 Diffusion coefficient vs temperature in 2 cases: perfect case and

vacancy case 53

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, 3 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 4 carbons 35

Table 3.7: Position of carbon atoms before and after optimized in case 3C 36

Table 3.8: Binding energy of 7 configurations in case 4 carbon atoms 37

Table 3.9: The change position of 4 carbon in configuration 4 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

CI-NEB Climbing image – Nudged Elastic Band

LAMMPS Large-scale Atomic/Molecular Massively Parallel

Simulator

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

properties while cutting the costs and enhancing the connectivity of materials compared to previous generations

Figure 1.1 The relation between elongation (ductility) and tensile strength in low

carbon steel for general applications [4]

Overview of iron-carbon alloy

With its long history of development, steel is still one of the most widely used materials in our modern world [24], and it can be seen that steel is present in most buildings from small houses to skyscrapers, roads, and bridges The reason for this material becoming popular and preferable comes from its characteristics The versatility, durability, and strength of steel can meet requirements as well as applications for a variety of purposes, and it is also an affordable and environmentally friendly option [5] Research on steel is still an exciting field that scientists, especially

in material science, are interested in improving the properties of this traditional material

Figure 1.2 Phase diagram of iron-carbon alloy by different carbon content [19]

Table 1.1 Different phases of steel based on carbon content [23]

Phase Term Structure Temperature

δ - Fe δ – Ferrite BCC 13920 C< T

<15360 C

Dissolve as much as 0.08% of carbon

Fe3C Cementite Orthorhombic

Hard ceramic, lower nucleation barrier than for graphite Fe-C solid

solution Martensite BCT

Metastable, formed

by quenching

Based on the structure of pure iron and steel, it is easy to see that these are similar structural materials The most significant and vital difference comes from the occurrence of carbon impurity concentrations in the system More specifically, when the carbon concentration in the alloy of iron exceeds the 2.1wt% threshold, the alloy

is considered as cast iron, which is very hard and also very brittle In the case of carbon concentration less than 0.08wt%, it becomes softer when compared to cast iron, but its ability as incurvation or distortion was better without breaking, which is necessary to play a role as a structural steel in the building When carbon concentration is between 0.2wt% and 2wt%, the properties of steel become special thanks to the balance between hardness and ductility [36] However, how to control both level and location of carbon in iron is the most challenging problem we faced

So, there is no denying that the history of the steel industry is defined based on carbon concentration control techniques

The appearance of carbon atoms in the iron system even in small quantities is still thought to have a significant effect based on the energy and kinetic properties of the system It can be seen that carbide formation comes from exceeding the limit of carbon solubility, which contributes significantly to improving the durability and hardness of metals as in steel On the opposite side, when the carbon concentration

in the system is below the solubility limit, the thermal and mechanical properties of the system can change significantly only by a minimal amount of carbon atoms (several tens of ppm) in interstitial sites or when they interact strongly with defects

in steel [27]

The purpose and objectives of research

The real lattice is not perfect but contains many types of defects, which can be referred to as vacancy, dislocation, or grain boundary [41] While vacancy is well known as a typical case of point defect and also a simple case which we can consider Study about the vacancy case in BCC structure of iron will help us understand clearly about the role and the effects of vacancy to the diffusion and clustering of carbon in iron matrix

Figure 1.3 Simulation picture of typical defects in iron-carbon alloy

The cause of the interaction between carbon and metals has a tremendous scientific and technological interest which has essential effects on the yield stress and the sub-consequent mechanical properties and also a broad range of implications in the scope

of material science [26] Research on atomic carbon concentration dissolved in iron

as well as its distribution and diffusion in iron plays a vital role in making a view insight of phenomena such as carbide precipitation, martensite aging, and ferrite transformation [31] The restriction of system size when calculating using First principle method causes Molecular Dynamic (MD) to be a reasonable substitute for large systems [39] However, the accuracy of MD simulations largely depends on the

choice of interatomic potential Recently, Nguyen et al [31] developed a new

interatomic potential to describe the interaction of Fe-C system based on the analytic bond-order potential (ABOP) formalism [29], which gives good 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 iron-carbon alloy as well as its effect on the diffusion process of carbon impurities in the alpha iron system through the use of atomistically kinematic simulations

CHAPTER 2 THEORETICAL BASICS

The transition state theory

Transition state theory is a theoretical method used to predict the rate of chemical reactions The theory was proposed by Erying and Polanyi in 1935 to explain the bipolar reactions based on the relationship between kinetics and thermodynamics This theory is based on the initial assumption that the reaction speed can be calculated completely, so it is also called the theory of absolute reaction speed [1] In particular,

if there is only one barrier between the reactant and the product, the transition state theory specifies how to calculate the reaction rate constant The transitional state theory assumes the validity of only one condition, but a significant condition, namely

on one side of the barrier, the states of the system in equilibrium If there is only one barrier between reactants and products, then the reactant should be kept in equilibrium The simplicity of the transition state theory is lost if the reactants are selected according to the state The assumption of statistics given by the transitional state theory is not about the dynamics of the reaction; instead, it is about the balanced nature of the reactants placed on one side of the barrier One fundamental meaning

of this assumption is that it allows theory to be cast in anatomical terms The statistical assumption given by transitional state theory is a specification of reactants that theory can be applied

The basic foundation for this theory can be understood as being based on the ability

to activate the internal bonds of a molecule In other words, the reaction only occurs when the activation energy is high enough for it to overcome the activation barrier When the activation energy required for the reaction is higher than that of the thermal energy supplied, k T B then the probability of activating a molecule is very low In order to provide more energy for the reaction, more collisions occur as a significant event for the reaction When a molecule is activated, the probability for it to cross the energy barrier becomes more accessible and faster In which the constant representing the speed of the reaction is determined as ( ) #

/ h

B

k T K , with (k T B / h) being the rate

of a molecule when it passes through the barrier and K is the equilibrium constant

of activated complexes

The transition state or activated complex can be assumed to have all the attributes of

a typical molecule except that one of the vibrational degrees of freedom is converted into a translation degree of freedom along with the reaction coordinates The reaction

is thought to proceed through an activated complex, the transition state, located at an energy barrier separating reactants and products It can be visualized by the travel over a potential energy surface, such as a mountain landscape where the barrier lies

at the saddle point, the mountain pass or col The event is described by one degree of freedom (e.g., a vibration in case of a dissociation reaction) called the reaction coordinate, q#

For an isomerization reaction, a representation of the potential energy dependence of the reaction on reaction coordinate qis given in Table 2.1 The difference in energy

is defined by

' 0

U is the energy of reactant The overall reaction energy E reactiondepends on the energy difference of product and reactant

Figure 2.1 Reaction energy diagram as a function of reaction coordinate q for an

isomerization reaction [37]

As mentioned by the TST, the activation complex or transition state is considered to

be in equilibrium with reactant molecules; the rate of reaction is equal to the number

of activation complexes which pass over the product side per unit time The transition-state expression for the rate of reaction is described as below:

#

B TST

The constant equilibrium #

K is related to the free energy of activation, #

G

 , through the relation

 is the enthalpy of activation and #

T S is the entropy of activation The goal of transitional state theory is to predict the rate of a reaction on a known potential energy surface The potential energy surface is, in general, a high dimensional surface, but usually, a large number of degrees of freedom (such as the

orientation of molecules) can be neglected Ideally, one would like to project the potential energy surface onto a single dimension, which is called the reaction coordinate The reaction coordinate can be as simple as the distance between two molecules One of the accomplishments of transition state theory is the theoretical justification of the Arrhenius law, which was proposed by Svante August Arrhenius

in 1889 that the effective attempt frequency k0( )j of the j-th event can be evaluated

N i

N i i

From that, the rate of each event will be defined as the following:

( )

0

j d

Where E d( )j is the activation energy, k B is the Boltzmann factor, T is the temperature

2.2 Nudged Elastic Band Method

In energy surface analysis, it is known that finding the minimum energy path between two minimum energy points becomes an important problem for determining the diffuse properties of atoms in the matrix Until now, the Nudged Elastic Band method (NEB) is still the most advanced method for determining minimum energy paths [22] The basic idea of this method comes from creating a series of replicas (or images) between two minimum energy points, from which the replicas are linked together to form a chain bonded by fictitious spring force Finally, the actual minimum energy

path will be revealed when the total energy of the string of replicas is minimized by

a suitable algorithm NEB method can be used for:

➢ Chemical reactions on surface;

➢ Diffusion processes;

➢ Phase transformations, etc

A modified version based on the NEB method is Climbing Image- Nudged Elastic Band (CI-NEB) After minimizing the energy of all the replicas together based on the virtual elastic force, the appropriate algorithm will be used to push the highest energy image from another up to the saddle point by maximizing its energy along the direction defined by the band [44] In this way, the CI-NEB method not only helps to determine a saddle point more accurately but also provides an overview of the minimum energy line shape, which also allows us to identify more than a saddle point along with the atomic movement of the atom Therefore, it has helped to give us a more precise view as well as providing the necessary parameters for calculating the diffusion properties of atoms

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]

Figure 2.3 Decomposition of force on an image [38]

By a force projection scheme in which potential forces act perpendicular to the band and spring forces act along the band, the images along the NEB are relaxed to the MEP The process is carried out through the following formulas [38]:

Spring force on each image is given by:

of the total force:

no spring forces and climbs to the saddle via a reflection in the force along the tangent [20], [21], [30]

The force at the max-energy image without spring forces:

Kinetic Monte Carlo method

The kinetic Monte Carlo (kMC) is a simulation method intended to simulate the time evolution of some processes occurring in nature [8] Typically, these are processes that occur with known transition rates among states It is important to understand that these rates are inputs to the kMC algorithm, the method itself cannot predict them [7] The kinetic Monte Carlo method provides a simple yet powerful and flexible tool for exercising the concerted action of fundamental, stochastic, physical mechanisms to create a model of the phenomena that they produce By using kMC method, we can

easily do the simulation related to the events which take a long time as the diffusion

of impurities in the material, surface growth, coarsening of domain evolution, defect mobility and clustering in ion or neutron irradiated solids, vacancy diffusion in alloys, etc [49]

Figure 2.4 Contour plot of the potential energy surface for an

energy-barrier-limited 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]

kMC method with an appealing property, in principle, give the exact dynamical evolution of a system As we know, molecular dynamics (MD) simulation is a popular tool in the class of atomistic simulation methods, in which one propagates the classical equations of motion forward in time However, a serious drawback of MD method is that accurate integration requires time steps short enough ( 15

~ 10− s) to resolve the atomic vibrations Therefore, the total simulation time is usually limited

to less than a microsecond, while processes such as diffusion or surface growth often occur at much longer time scales [45] Because our focus is on simulating the dynamical evolution of systems of atoms, kinetic Monte Carlo attempts to overcome this limitation by exploiting the fact that the long-time dynamics of this kind of system typically consists of diffusive jumps from state to state Rather than following

the trajectory through every vibrational period, these state-to-state transitions are treated directly As a result, kMC can reach vastly longer time scales, typically seconds and often well beyond

A simple example of a brief description of kMC is shown below, the central quantity

in kMC, calculated at each iteration, is the residence time τ, which determines how long the system remains in a given state before jumping to one of the adjacent states Therefore, it is important to know the diffusion rate of all states, the diffusion rate

from the event (i) to adjacent (j) event, i→j will be determined by:

E and min

i

E are the total energies of the system at the saddle point and the

state (i), respectively The residence time will be computed:

It should be noted that the value of  depends on all diffusion rates, not only on the

rate to escape to state (j); r1 is a random number in the range of (0, 1] Besides, the total probability rates of all possible Z transitions which the system can encounter

from the initial state (i) is

Figure 2.5 the transition of atom when diffusing from the state (i) to the state (j) by

crossing the energy barrier m

=

= is defined and its value is assigned to the K-th transition, with K ≤ Z (Z is the total number of transitions that are allowed from the current state

(i)) Then, a random number r2 in the range of [0, 1] is generated and the transition

corresponding to the smallest s(k) that is greater than r 2 i will be chosen (see Figure 2.6) At this point, a single kMC iteration is finished The program will proceed until the maximum number ofkMC steps (or any other stopping condition) is reached

Molecular Dynamic method

Among methods used in material simulation field, molecular dynamics (MD) is the standard method used to simulate the temporal evolution of an atomic system by integrating Newton’s equations of motion for all its particles In particular, time in

this simulation is discretized, and the time step t must be small enough to resolve the fastest atomic oscillations ( 15

10−

 seconds) Because the time selected is very short, so the force applied to the particle during this period is considered constant Different finite methods are used for numerical integration

The total potential energy E of the system at any instant of time is determined from

the relative positions of the particles, which interact with each other through an interatomic potential, and is evaluated at each MD step The use of molecular dynamics for computational material science can be supported by computational software packages such as LAMMPS, NAMD, Amber or Gromacs In general, the common points of these programs still use the same basic logic through a minimalist outline of a typical MD program as follows:

• A simulation protocol defines the general conditions under which the system will be simulated: initial temperature, initial pressure, target temperature, target pressure, boundary conditions, number of MD steps, time step (t), etc

• System initialization: initial positions and velocities are assigned to the particles, and all necessary information about the system, such as the interatomic potential, is provided

• The simulation itself is launched MD is an iterative method, where each iteration corresponds to a time step t At each iteration:

1) The total potential energy of the system as a function of particle positions is calculated

2) The forces acting on all particles are computed

3) The new positions and velocities of the particles are computed by

integrating the equations of motion with a suitable algorithm

4) The total simulated time is incremented by ∆t

5) Optionally, the quantities of interest (e.g., the positions and velocities of the particles) are stored into appropriate data files for postprocessing

6) If the total number of MD steps is reached, the simulation stops; otherwise, the next iteration starts

When performing MD simulation, the simulated trajectories need to be long enough

to provide representative samples of the system configurations Depending on the characteristics of the system in specific conditions, in order to perform statistical analyzes where the number of steps required for simulation, simulation temperature

is significant to be considered In particular, the MD simulations for fluid systems and soft systems may be shorter and more comfortable to perform at low temperatures while simulations for solid systems are often carried out at high temperatures, this is because the mobility of atoms in different problems is different [16]

Molecular Vibrations

Small Vibrations in Classical Mechanics

To consider small vibrations in a purely classical system, starting by understanding the motion of an atom “α” as it moves away from its equilibrium value Then, the position change can be described in three axes of atom as x, y and z corresponding to the change value of x, y and z, respectively The kinetic energy is given by [46]:

1

12

1 1 1

q = M x, q2 = M1y1, q3 = M1z1, q4= M2x2 , q5= M2y2, … (2.19) Then the kinetic energy operator will be changed to

3 2 1

1 2

N i i

( V / q i 0 = 0) and then V0 may be eliminated (V =0 0) Besides, when all the q s'

are zero, the atoms are all in their equilibrium positions so that the energy must be a minimum for q = i 0,i =1, 2,3, Therefore

0

0

i i

V

f q

In which, T is a function of the velocities only (in this coordinate system) and V is a

function of the coordinates only Substitution of the expressions for T and V given above yields the equations

0

N

i i

1, 2, ,3

i= N , corresponding to no vibration Based on the eigenvalue equation

F a=a in matrix type, which has a solution only if  has special values obtainable from the secular determinant

Normal Modes of Vibration and Normal Coordinates

The matrix eigenvalue equation is equivalent to matrix diagonalization, which is

equivalent to solving the secular determinant for each λ (N of them) From the value

ofk, we can get the corresponding a k,

ik ik k k

q =a  t+  (2.31) Where a k are the normal modes of vibration; It is important to know that for each normal mode, all atoms will be activated and they will always oscillate at the same frequency and phase, but the amplitude of each atom varies

Based on normal modes, new set of coordinates can be defined as:

3

1

N

k ik i i

=

= with k= 1, 2, , 3N (2.32) When the eigenvectors of a real and Symmetric matrix (F ) are orthogonal, T and V

will be diagonal (no cross terms):

3 2 1

12

N k k

1

12

N

k k k

=

CHAPTER 3 RESULTS AND DISCUSSION

Methodology

For this work, our calculations were based on the framework of MD simulation with simulation package from The Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [34] All calculations of LAMMPS simulation use new

Tersoff potential developed by Nguyen et al [31] for Fe-C system constructed based

on the analytic bond-order potential (ABOP) formalism Besides, Density Functional Theory (DFT) method [3] is also used as implemented in the Vienna Ab initio Simulation Package (VASP) to get the comparison with the results from MD simulation in some cases By using kinetic Monte Carlo (kMC) method, the effect of vacancy on diffusion coefficient in case of one carbon atom has been investigated All the binding energy calculations have used the supercell model of BCC iron with the presence of one vacancy as point defect by using two sizes of 3x3x3 (53 iron atoms) and 8x8x8 (1023 iron atoms) Note that this work is investigated primarily based on the system size of 8x8x8 The reason for this choice is to ensure minimal interaction of carbon atoms in neighboring cells Also, we know that when interstitial carbon atoms appear in iron, it will lead to a change in the volume of the system [33] Therefore, the selection of the 8x8x8 system size to be considered is large enough to ensure this change is negligible However, data on calculations for large size systems are limited, especially from highly reliable sources such as DFT calculations Therefore, system sizes such as 3x3x3 and 4x4x4 have been used in some calculations

of LAMMPS to make an objective comparison with some computational data of the DFT that has the same system size

Thereby, the main points in this research are related to:

(i) Study about the stability of carbon atoms around the vacancy by an investigation based on new Tersoff potential;

(ii) The trend of carbon-trapping of vacancy

(iii) The effect of vacancy on diffusion properties of carbon in iron system

Stable configurations

3.1.1.1 One carbon atom

In previous research [12], [35], [40], O-site is found as a stable position where carbon atom prefers to occupy in BCC perfect lattice The most likely diffusion path for carbon is the linear movement from O-site to O-site by crossing a transition point, T site So, in this study, we also started by studying the stable position of carbon atoms

in the case of point defect: one vacancy in the iron system Previous stable sites, such

as O-site and T-site, were surveyed to find the most stable configuration when there was carbon in the system

Figure 3.1 Positions 1, 2 of carbon correspond to O site, and 3 corresponds to T site

The simple model of the system can be observed in Figure 3.1 Here, the red balls represent iron atoms, the small orange balls represent carbon atoms, and the middle gray box illustrates one vacancy in the center of the box Note that we just described the image to illustrate for positions where carbon lays on in supercell of 3x3x3 (including 53 iron atoms) and 8x8x8 (including 1023 iron atoms) Then, a carbon atom will be adopted into the system in terms of positions 1, 2, 3 By calculating the energy of the system in these different positions, we can determine the suitable location for carbon atoms to occupied, thereby finding the most stable configuration

of the system in the case of one carbon It is important to determine the stable position for one carbon case Through this case, we can find out more stable configurations of

23

the system in the case of multiple carbon atoms by adding atoms and determining the system energy for each configuration Therefore, the investigation will become more accurate and logical

Also, considering the interactive limit between vacancy and carbon, some reasonable positions of carbon around vacancy site were considered In Figure 3.2, we compute the interaction by dropping carbon at positions such as P1, P2, P3, P4, P5, P6, P7 The positions on Figure 3.2 are arranged in ascending order, starting from the nearest position P1 to the far position P7 Then, the binding energy between carbon and vacancy is calculated based on the positions considered by the formula:

E V C = E +E − − E +E (3.1) This work aims to identify the positions where carbon atoms are strongly influenced

by the interaction of vacancy From there, we can delineate what is the distance (Å) where the interaction between vacancy and carbon atom is considered strong Out of this limit, the interaction can be ignored (the zero point)

Figure 3.2: Positions carbon is adopted in iron system

3.1.1.2 Two carbon atoms

In this case, two carbons are doped into iron bulk with the 1st carbon position is determined in the previous part, and then the 2nd carbon is adopted in some possible positions as Figure 3.3 showed below

Figure 3.3 Configurations of BCC iron structure in case of two carbons

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3.1.1.3 Three carbon atoms

In this part, the 1st carbon and the 2nd carbon is fixed in stable positions, which is reached in two carbons case, then stable position for the 3rd carbon is considered as the schematic structure of Figure 3.4

Figure 3.4: Configurations of BCC iron structure in case of three carbons

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3.1.1.4 Four carbon atoms

In four carbon atoms case, the positions of the 1st, the 2nd, the 3rd carbon are fixed in stable positions reached in three carbons case and then, stable position for the 4thcarbon is considered by some configurations below

Figure 3.5: Configurations of BCC iron structure in case four carbons

Trapping energy

Binding energy helps us determine which configuration of the system is the most stable configuration However, if we want to determine the tendency of the interaction between carbon and vacancy, then we need to calculate the energy which describes the ability of vacancy to trap carbon atom, or trapping energy Based on the most stable configuration in all case, trapping energy is calculated for four configurations (V-C, V-2C, V-3C, V-4C)

There are two common ways to calculate the trapping energy of vacancy for carbon atoms The first is trapping energy in a “sequential” way, and the other way is

"simultaneous” way In which, the following formula defines the "sequential" way:

To investigate the diffuse properties of carbon atoms in the system and also the effect

of vacancy on their diffusion, we need to learn about the MEPs The determination

of MEPs in this work is done by the help of the CI-NEB method with the number of images considered is 32; Quickmin is minimized algorithm used in finding MEPs

It will be clear and more specific if we can consider all possible diffusion paths of carbon atoms around the vacancy site However, because of the complexity of the potential energy landscape, finding the correct shape of MEP also faces some difficulties (lack of support algorithms, etc) Therefore, after determining the MEPs

from the NEB method, we also examined these paths by checking the position of local minima and saddle points on the MEP through the Hessian matrix based on the Finite Difference method [48]

In the case of multiple carbon atoms, the bond between carbon pairs will also be considered based on the bonding length and the binding energy Thereby, it helps us

to have a more unobstructed view of the role of vacancy by comparing with the case

of non-vacancy as in the previous study [11]

Besides, based on TST theory, the diffusion rate of carbon atoms is also calculated These input data with other parameters such as temperature, the energy barrier is used

to calculate the diffusion coefficient of carbon using the kMC method From there,

we can elucidate the effect of vacancy on the diffusion of carbon atoms when compared with the case of non-vacancy

Results and Discussion

Stable configurations

3.2.1.1 One carbon atom

By adopting one carbon in three possible positions which are well known as O-site (position 1, 2) and T-site (position 3) We tried to identify what is the most stable position for one carbon occupied by considering data from dissolution energy, volume change The results are shown in Table 3.1

Table 3.1 Configuration of system when carbon is adopted in position 1, 2, 3

(unit lattice, a = 2.8886 Å)

Dissolution energy (eV)

Volume change (%)

Note that dissolution energy is defined by the formula:

E V C = E +E − − E +E (3.4) From the data above, we can see that O-site with configuration 1 gives the smallest value of dissolution energy and the change of volume with both system size 3x3x3 and 8x8x8, it means that the system size does not affect much of the dissolution energy Besides, when carbon is adopted at T-site in 8x8x8 system, after optimized structure of system we can see the change position of carbon move to near position 1

of O-site to reach the minimum value of energy, it means that is the most stable position of carbon, and we can use it to consider for other cases

The appearance of vacancy in the center of the cubic distorted the potential surface

of the system Specifically, in Figure 3.6, an illustration is created by scanning the potential energy landscape of carbon atom on the surface formed by two directions, [010] and [001] In the absence of vacancy, the energy landscape gives the agreement with previous calculations that the O-site is found to be the most stable position for carbon occupied; T-site is the transition point between two minimum energy points