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 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]
-Fe Ferrite BCC T < 922.50C Solubility is very low
C is an "Austenite stabilizer": add C, field widens δ - Fe δ – Ferrite BCC 13920 C< T
Dissolve as much as 0.08% of carbon
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
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.
THEORETICAL BASICS
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 ( k T / h ) K # , with ( k T / 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
Where E b is activation energy (the energy at q b # ); U 0 ' is the energy of reactant The overall reaction energy E reaction depends 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:
Where k B is the Boltzmann constant; h is the Plank constant; T is the temperature, and K # is an equilibrium constant
The constant equilibrium K # is related to the free energy of activation, G # , through the relation
(2.2) where G # equal to the change in the enthalpy of the system minus the change in the product of the temperature times the entropy of the system in their standard states
= − (2.3) Substituting the value of G # in (2.3):
Then, the expression for the rate constant of a reaction on the basic of TST is shown below
Where H # 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 k 0 ( ) j of the j-th event can be evaluated as [13]:
(2.7) where i is the i th− normal mode frequency at the initial position and i # is that at the saddle point, where one of the vibration frequencies should be imaginary frequency
From that, the rate of each event will be defined as the following:
Where E d ( ) j is the activation energy, k B is the Boltzmann factor, T is the temperature
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:
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:
To find MEP, we need to minimize the following objective function:
After that, the total force on the image is considered as:
Projecting the component of the total force perpendicular to the reaction pathway out of the total force:
Removes perpendicular component of spring force:
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
The force at the max-energy image without spring forces:
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]
RESULTS AND DISCUSSION
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 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:
( , ) Fe Fe Fe Fe b perfect V C V C
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
In this case, two carbons are doped into iron bulk with the 1 st carbon position is determined in the previous part, and then the 2 nd 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
In this part, the 1 st carbon and the 2 nd carbon is fixed in stable positions, which is reached in two carbons case, then stable position for the 3 rd carbon is considered as the schematic structure of Figure 3.4
Figure 3.4: Configurations of BCC iron structure in case of three carbons
In four carbon atoms case, the positions of the 1 st , the 2 nd , the 3 rd carbon are fixed in stable positions reached in three carbons case and then, stable position for the 4 th carbon is considered by some configurations below
Figure 3.5: Configurations of BCC iron structure in case four carbons
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:
( n ) ( n 1 ) trap FeVC Fe FeVC FeC
While the “simultaneous” way is defined by the formula:
1 1 trap FeVC n Fe FeV FeC
Where E FeVC is the total energy of system, including one vacancy and n carbon atoms;
E Fe , E FeV are total energies of the system without/with vacancy; E FeC is the total energy of system, including one carbon atom
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]