Atomistically kinetic simulations of carbon diffusion in ɑ iron with point defects002

Figure 2.5 the transition of atom when diffusing from the state i to the state j by crossing the energy barrier Figure 2.6 The K-th transition is chosen because its assigned value

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

To accomplish this thesis, I have received great support, helpful advice, andguidance from respectful professors, lecturers, researchers, and staff in VietnamJapan University and Osaka University

I would like to express my gratefulness to my supervisors, Prof Dr Yoji Shibutaniand Dr Nguyen Tien Quang for supplying great researching environments inlaboratories, and for giving helpful instructions, guidance, advice, and inspirationsduring 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 2019Student

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]

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 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]

Figure 2.3 Decomposition of force on an image [38]

Figure 2.4 Contour plot of the potential energy surface for an energy-barrier-limited infrequent-event system After many vibrational periods, the trajectory findsa way out of the initial basin, passing a ridgetop into a new state The dots indicate saddle points [45]

Figure 2.5 the transition of atom when diffusing from the state ( i ) to the state ( j ) by crossing the energy barrier Figure 2.6 The K-th transition is chosen because its assigned value of s(K) intercepts r 2 i [44] .

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

Figure 3.2: Positions carbon is adopted in iron system

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

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

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

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

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

Figure 3.8 Binding energy of carbon-vacancy is calculated by DFT calculation and MD method in case 1V-1C

Figure 3.9 : Configuration 3 after optimized in case 2 carbon atoms

Figure 3.10 Configuration 3 after optimized

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

i

Figure 3.12 Trapping energy is calculated in two ways: “sequential” way (blue

line) and “simultaneous” way (red line) 38

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

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, 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

iii

Tetragonal siteTransition State TheoryLarge-scale Atomic/Molecular Massively ParallelSimulator

iv

CHAPTER 1 INTRODUCTION

Nowadays, along with the steady development of science and technology, theachievements in scientific research are increasingly contributing to society,especially in the field of nanotechnology Research, development, and application ofpotential and unique properties from nanoscale materials have brought manyimprovements and breakthroughs compared to previous traditional materials [15].The field of computational materials science is considered as one of the areas of topconcern in material science today [9] Calculations are implemented based on thetheoretical foundations, which apply to specific subjects under the simulationprocess supported by modern computer systems, acting as useful tools in describing,verifying, predicting the rules, physical phenomena occurring inside objects andbetween objects The development process of computational science is an essentialand inseparable part of the practical application in industry In particular, thecalculation related to iron-carbon alloys is a good example and plays a crucial role inthe development of the steel industry

Until now, the steel industry has an extraordinary development, which can be dividedinto three main generations The first generation - Conventional low carbon steels can

be mentioned as high strength low-alloy products (HSLA) steels, advanced highstrength steels (AHSS), IF (Interstitial Free), DP (Dual Phase) or so-called TRIP /TWIP (Transformation or Twinning Induced Plasticity), etc is incredibly famous andwidely used steel generation today [4] The second generation - Austenitic-Based Steelshas been developed, and the third generation is still being researched and developed.For different generations, superiorities and disadvantages still exist not only onmechanical properties but also on product costs Therefore, the main goal of this third-generation material system is to continue to improve the desired mechanical

1

properties while cutting the costs and enhancing the connectivity of materialscompared 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 usedmaterials in our modern world [24], and it can be seen that steel is present in mostbuildings from small houses to skyscrapers, roads, and bridges The reason for thismaterial becoming popular and preferable comes from its characteristics Theversatility, durability, and strength of steel can meet requirements as well asapplications for a variety of purposes, and it is also an affordable andenvironmentally friendly option [5] Research on steel is still an exciting field thatscientists, especially in material science, are interested in improving the properties

of this traditional material

2

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].

3

Based on the structure of pure iron and steel, it is easy to see that these are similarstructural materials The most significant and vital difference comes from theoccurrence 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 thecase 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 carbonconcentration is between 0.2wt% and 2wt%, the properties of steel become specialthanks to the balance between hardness and ductility [36] However, how to controlboth 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 oncarbon concentration control techniques

The appearance of carbon atoms in the iron system even in small quantities is stillthought to have a significant effect based on the energy and kinetic properties of thesystem It can be seen that carbide formation comes from exceeding the limit ofcarbon solubility, which contributes significantly to improving the durability andhardness 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 ofthe 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 bereferred to as vacancy, dislocation, or grain boundary [41] While vacancy is wellknown as a typical case of point defect and also a simple case which we canconsider Study about the vacancy case in BCC structure of iron will help usunderstand clearly about the role and the effects of vacancy to the diffusion andclustering of carbon in iron matrix

4

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 andtechnological 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 ofmaterial science [26] Research on atomic carbon concentration dissolved in iron aswell 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 ferritetransformation [31] The restriction of system size when calculating using Firstprinciple method causes Molecular Dynamic (MD) to be a reasonable substitute forlarge 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 analyticbond-order potential (ABOP) formalism [29], which gives good results in describingminimum 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 pointdefect in the iron-carbon alloy as well as its effect on the diffusion process of carbonimpurities in the alpha iron system through the use of atomistically kinematicsimulations

5

CHAPTER 2 THEORETICAL BASICS

The transition state theory

Transition state theory is a theoretical method used to predict the rate of chemicalreactions The theory was proposed by Erying and Polanyi in 1935 to explain thebipolar reactions based on the relationship between kinetics and thermodynamics.This theory is based on the initial assumption that the reaction speed can becalculated 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, thetransition state theory specifies how to calculate the reaction rate constant Thetransitional state theory assumes the validity of only one condition, but a significantcondition, 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 thereactants are selected according to the state The assumption of statistics given bythe transitional state theory is not about the dynamics of the reaction; instead, it isabout the balanced nature of the reactants placed on one side of the barrier Onefundamental meaning of this assumption is that it allows theory to be cast inanatomical terms The statistical assumption given by transitional state theory is aspecification of reactants that theory can be applied

The basic foundation for this theory can be understood as being based on the ability toactivate the internal bonds of a molecule In other words, the reaction only occurs whenthe activation energy is high enough for it to overcome the activation barrier When theactivation energy required for the reaction is higher than that of the thermal

energy supplied, k T then the probability of activating a molecule is very low In B

order to provide more energy for the reaction, more collisions occur as a significantevent for the reaction When a molecule is activated, the probability for it to cross theenergy barrier becomes more accessible and faster In which the constant representingthe speed of the reaction is determined as (kB T/ h)K# , with (kB T/ h) being the rate

6

of a molecule when it passes through the barrier and

is the equilibrium constant

The transition state oractivated complex can

be assumed to haveall the attributes of atypical moleculeexcept that one of thevibrational degrees offreedom is convertedinto a translationdegree of freedomalong with thereaction coordinates.The reaction isthought to proceedthrough an activatedcomplex, thetransition state,located at an energybarrier separatingreactants andproducts It can bevisualized by thetravel over a potentialenergy surface, such

as a mountainlandscape where thebarrier lies at thesaddle point, the

mountain pass or col The event is described by onedegree of

Where E b is activation energy (the energy at q b

overall reaction energy

reactant

Figure 2.1 Reaction energy diagram as a function

of reaction coordinate q for an isomerization

reaction [37]

7

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:

k TST

The goal of transitional state theory is to predict the rate of a reaction on a knownpotential energy surface The potential energy surface is, in general, a highdimensional surface, but usually, a large number of degrees of freedom (such as the

8

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

in 1889 that the effective attempt frequency

is the activation energy, k 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 betweentwo minimum energy points becomes an important problem for determining the diffuseproperties of atoms in the matrix Until now, the Nudged Elastic Band method (NEB) isstill the most advanced method for determining minimum energy paths [22] The basicidea of this method comes from creating a series of replicas (or images) between twominimum energy points, from which the replicas are linked together to form a chainbonded by fictitious spring force Finally, the actual minimum energy

9

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:

➢ Diffusion processes;

➢ Phase transformations, etc.

A modified version based on the NEB method is Climbing Image- Nudged ElasticBand (CI-NEB) After minimizing the energy of all the replicas together based onthe virtual elastic force, the appropriate algorithm will be used to push the highestenergy image from another up to the saddle point by maximizing its energy alongthe direction defined by the band [44] In this way, the CI-NEB method not onlyhelps to determine a saddle point more accurately but also provides an overview ofthe minimum energy line shape, which also allows us to identify more than a saddlepoint 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 calculatingthe 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]

10

Figure 2.3 Decomposition of force on an image [38].

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

Spring force on each image is given by:

=− V −i

Removes perpendicular component of spring force:

(2.12)

=− V −i

The saddle point is exceptionally crucial for characterizing the

transition state within harmonic transition state theory (HTST)

The difference between the saddle point energy and that of the

initial state determines the exponential term in the Arrhenius

rate, and the MEP can be obtained by minimizing from the

saddle point(s) An efficient approach for seeking a saddle

between known states is to use the Climbing Image - NEB

(CI-NEB) method [43] In this method, the highest energy image

feels 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:

F MAX NEB = − V MAX

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

12

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, defectmobility and clustering in ion or neutron irradiated solids, vacancy diffusion inalloys, 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 dynamicalevolution of a system As we know, molecular dynamics (MD) simulation is a populartool in the class of atomistic simulation methods, in which one propagates the classicalequations of motion forward in time However, a serious drawback of MD

method is that accurate integration requires time steps short enough ( ~ 10

resolve the atomic vibrations Therefore, the total simulation time is usually limited toless than a microsecond, while processes such as diffusion or surface growth oftenoccur at much longer time scales [45] Because our focus is on simulating thedynamical evolution of systems of atoms, kinetic Monte Carlo attempts to overcomethis limitation by exploiting the fact that the long-time dynamics of this kind of systemtypically consists of diffusive jumps from state to state Rather than following

13

the trajectory through every vibrational period, these state-to-state transitions aretreated directly As a result, kMC can reach vastly longer time scales, typicallyseconds 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 howlong the system remains in a given state before jumping to one of the adjacentstates 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:

0 is the attempt frequency,kis the Boltzmann constant, T is temperature andthe

energy barrier for atom to escape from the state (i) to the adjacent (j) by :

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, thetotal probability rates of all possible Z transitions which the system can encounter

from the initial state (i) is i = i →k

k =1

14

Figure 2.5 the transition of

crossing the energy barrier

atom when diffusing from the state (i) to the state (j) by

K

[6] A quantity s(k)= R i→jis 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 r2 i will be chosen (see Figure

2.6) At this point, a singlekMC iteration is finished.The program will proceeduntil the maximum numberofkMC steps (or any otherstopping condition) isreached.

15

Molecular Dynamic method

Among methods used in material simulation field, molecular dynamics (MD) is thestandard method used to simulate the temporal evolution of an atomic system byintegrating Newton’s equations of motion for all its particles In particular, time inthis simulation is discretized, and the time step t must be small enough to resolvethe fastest atomic oscillations ( 10

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

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

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

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

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

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

2) The forces acting on all particles are computed

16

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 theparticles) 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 toprovide representative samples of the system configurations Depending on thecharacteristics of the system in specific conditions, in order to perform statisticalanalyzes where the number of steps required for simulation, simulation temperature issignificant to be considered In particular, the MD simulations for fluid systems andsoft systems may be shorter and more comfortable to perform at low temperatureswhile simulations for solid systems are often carried out at high temperatures, this isbecause 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 themotion of an atom “α” as it moves away from its equilibrium value Then, the positionchange can be described in three axes of atom as x, y and z corresponding to

the change value of

[46]:

T =

By converting the coordinates fromx1 , , z n

displacement coordinates q1 , ,q3N with

y wil

l

be somefunction

of thedisplacementsan

d therefore

of the

q ' s For small values of the displacements, we can express the potential energy

0

may be eliminated

(V

0 =1) Besides, when all

the

q ' s

are zero, the atoms are all in their equilibrium positions so that the energy must be a

d T + V

dt q j q j

= 0 with j=1, 2, ,3N (2.25)

In which,function ofthe

coordinatesonly

Substitution

expressionsfor T and V

above yieldsthe equations

V

isagiven

i=

=0 j,= (2.2

6)

This is a set of 3N

simultaneous second-order linear

differential equations with one

possible solution is q i = a i

in harmonic oscillator)

It is easy to replace the final expression intothe upper differential

equation to obtain the

following equation

(j

= 1, 2,.

, 3

i =1, 2, , 3N , corresponding to no vibration Based

F a = a in matrix type, which has a solution only if

from the secular determinant

Using the Born - Oppenheimer

x y

(2.29)

To convert to mass-weighted coordinates, theformula below can be used:

(2.30)

By replacingthe result into(2.29), we can

a molecule aresquare roots ofthe eigenvaluesderived from

equation

Normal Modes of Vibration and Normal Coordinates

The matrix eigenvalue equation is equivalent to matrix diagonalization, which isequivalent to solving the

secular determinant for each λ (N of them) From the

a 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:

i=

k =1, 2, ,3

N

(2.32)

When the eigenvectors of a real and

Symmetric matrix ( will be diagonal

(no cross terms):

T =

V =

F) are orthogonal,

T and V

CHAPTER 3 RESULTS AND DISCUSSION

Methodology

For this work, our calculations were based on the framework of MD simulation withsimulation package from The Large-scale Atomic/Molecular Massively ParallelSimulator (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 SimulationPackage (VASP) to get the comparison with the results from MD simulation in somecases By using kinetic Monte Carlo (kMC) method, the effect of vacancy on diffusioncoefficient in case of one carbon atom has been investigated

All the binding energy calculations have used the supercell model of BCC iron withthe presence of one vacancy as point defect by using two sizes of 3x3x3 (53 ironatoms) and 8x8x8 (1023 iron atoms) Note that this work is investigated primarilybased on the system size of 8x8x8 The reason for this choice is to ensure minimalinteraction of carbon atoms in neighboring cells Also, we know that wheninterstitial carbon atoms appear in iron, it will lead to a change in the volume of thesystem [33] Therefore, the selection of the 8x8x8 system size to be considered islarge enough to ensure this change is negligible However, data on calculations forlarge size systems are limited, especially from highly reliable sources such as DFTcalculations Therefore, system sizes such as 3x3x3 and 4x4x4 have been used insome calculations of LAMMPS to make an objective comparison with somecomputational 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

21