2021 Volume 61 Issue 11 Pages 2677-2686
Carbon is the most important alloying element in steels and Fe–C alloys have been studied intensively. Carbon has various functions and has a strong influence on the transformed structures and mechanical properties. There are plenty of experimental data of Fe–C alloys but we still have many unsolved problems. The first-principles calculation and molecular dynamics (MD) method can evaluate effects of the slight change of the position and distribution of carbon atoms on energy and physical properties. So they are very useful for clarifying the nature and state of carbon in steels and for solving the problems which cannot be made clear by experiments. The electronic structure of carbon in iron, diffusion of carbon, site occupation of carbon atom in martensite, tetragonality, C–C interaction, carbon cluster and spinodal decomposition of martensite are the topics of this review paper. The studies using first-principles calculation and MD method are mainly reviewed. How much of the unsolved issues are clarified and what kind of problems remain are shown.
Carbon is the most important atomic element in steels and the Fe–C system has been intensively studied for a long time, but we still have many unsolved questions. In this paper, relatively new computational studies of “the nature and state of carbon atoms in iron” are mainly reviewed, and what were the unsolved questions, how they have been tackled, and what is still unsolved, are summarized. The “nature and state of carbon atoms in iron” means the electron state, energy, position, and distribution of carbon atoms in iron when (1) a single carbon atom exists as solute atom in iron, (2) more than two carbon atoms are distributed in iron and (3) carbon atom clusters are formed. The topics of this paper are, the electron state and energy of carbon atoms in iron, diffusion of carbon atom in iron, site occupancy of carbon atoms in Fe–C martensite and tetragonality, carbon-carbon interaction, ordering of carbon atoms and phase decomposition. In many of these topics, the position and distribution of carbon atoms are very important factors and are difficult to clarify experimentally. The first principles calculation and molecular dynamics (MD) are usually effective for these topics because the position and distribution of atoms can be precisely treated in these simulation methods. The recent researches are picked up and the main targets of the first principles calculation or MD method are the carbon atoms in solution of bcc iron or in the form of cluster, and the precipitation of carbon in iron are excluded. The old papers are also picked up if necessary.
The first principles calculation has become easy due to the progress of software and computational resource, and many fundamental information of electron state of carbon in iron are reported recently. There are many types of first principles calculations and most of the calculation introduced in this paper are based on the density functional theory (DFT: The total energy of entire electron system is expressed as a function of electron density distribution and when the electron density distribution is correct, the total energy is minimum.1)). The solution energy, carbon-M (M: substitutional solution atom) interaction energy, density of states, electron charge density and magnetization are introduced in the followings.
2.1. Solution EnergyThe solution energy is important for the evaluation of phase stability. The solution energy for Fe–C system has been evaluated by first principles calculation in many papers.2,3,4,5,6,7,8,9) The solution energy, ΔE, is also dubbed as solution enthalpy or formation enthalpy and ΔE for Fe–C system which consists of n iron atoms and 1 carbon atom is expressed as
| (1) |
There are some methods to evaluate the energy of graphite and diamond. Fang et al. calculated the energy of diamond and added a correction term (−17 meV/C) to obtain the energy of graphite, E[C], and calculated the formation enthalpy of carbides. This is because the ground state of carbon is graphite and experiments have determined that graphite is about 17 meV/C more stable than diamond.10) Graphite has a layered structure and the interlayer bonding in graphite is largely determined by van der Waals interaction, which is not correctly described by conventional DFT method. On the other hand, the DFT works well for diamond. Therefore, they performed calculations for diamond and subtracted the correction term to obtain the energy of graphite.10) Jiang et al. calculated the energy of graphite using the experimental data of lattice parameter of graphite. They did not optimize the structure during calculation.2) Ridnyi et al. also calculated the energy of graphite using the experimentally measured lattice parameter.8) Domain et al. calculated the energy of graphite and diamond by optimizing the structure.3) For the graphite, they also used the experimentally determined lattice parameter and calculated the energy of graphite without structural optimization. The energy of graphite obtained with structural optimization is minimum, and the interlayer distance is different from the experimental data by about 10%. The difference of the energy of graphite calculated by the above methods is only 10 mV and small compared to the energy difference between diamond and graphite. This indicates that the most significant contribution to the energy comes from the interactions within the graphite layers rather than the weak interactions between layers. Souissi et al. computed the solution energy of graphite with an improved description of the van der Waals interaction.7) Nguyen et al. used the energy of diamond9) as the energy of carbon atom. Other methods to evaluate the energy of a carbon atom are, calculating the energy of a carbon atom in a fairly wide vacuum space or calculating the energy of a carbon atom from the energy of bcc-iron including a carbon atom. Ande and Sluiter showed that it is appropriate to calculate the energy of an alloying element from the energy of that in solution of bcc iron or cementite instead of the standard state.11) Sawada reported the solution energy of carbon and boron in various standard states.12)
2.2. Interaction between Carbon Atom and Substitutional AtomThe interaction between atoms can be estimated from the solution energy. For example, the interaction energy between substitutional atom and interstitial atom, ΔE, can be calculated from the following equation.13)
| (2) |
The solution energy and interaction energy can be divided into strain energy term and chemical energy term. The strain energy is produced by the strain from substitutional and interstitial atoms. When the strain energy is subtracted from the solution energy or the interaction energy, the remaining part is the chemical energy. All of these three energies can be evaluated by the first principles calculation. By the calculation of strain energy and chemical energy, it can be clarified whether the attractive or repulsive interaction between two atoms in iron mainly comes from the strain or chemical interaction. The example for this will be shown at the end of section 5.1.
2.3. Density of States and Electron Density DistributionThe density of states (DOS: The number of electron states in some energy interval.) gives some information of energy, bonding state and magnetization. Domain et al.,3) Fors and Wahnström,4) Baik et al.,5) Nguyen et al.9) and Ohtsuka et al.6) calculated DOS of Fe–C system. In all of the above calculations, it is shown that a strong hybridization between carbon p orbital and iron d orbital is observed at around −6 eV. In the first three papers, the differences of the electron structure between Fe–C, Fe–N and Fe–B systems, and the differences of DOS of Fe–C system with carbon atom at O site, T site and substitutional site are shown. Ohtsuka et al. showed that the magnetization of iron atom at the first nearest neighbor of carbon atom is reduced and the magnetization of iron increases, decreases and increases again with increasing distance between carbon and iron atom.6)
Domain et al.3) calculated the charge density distribution for Fe–C and Fe–N systems and found that the area of charge transfer is narrower around the carbon atom. Baik et al.5) calculated the Partial Density of States (PDOS: DOS decomposed into contributions of fragment molecular orbitals.) and charge distribution for Fe–X (X = B, C, N) systems with X at O site or substitutional site. They discussed the atomic bonding based on the changes produced by the different interstitial atoms and different occupation sites. They found that the hybridization between 2p orbital of interstitial atoms and iron 3d orbital is stronger for interstitial site than substitutional site and the bonding between interstitial atoms and iron atoms is also stronger for interstitial site rather than the substitutional site. They reported that the hybridized orbital is formed in the lower energy region, the total energy is decreased and the distance between X and iron atoms becomes shorter in the order of boron, carbon, and nitrogen. They showed that the charge density between X and iron atoms increases and the Fe–X bonding is intensified in the order of boron, carbon, and nitrogen. It is the common analysis of DOS to conclude that covalent or strong bonding is formed because the strong hybridization between two atoms are observed in DOS. However, this is not enough. In order to confirm that covalent bonding is formed, Mulliken population analysis,16) Crystal Orbital Hamilton Population (COHP) analysis17,18,19) or both DOS and charge density distribution should be investigated in detail. In general, if two atomic orbitals interact with each other, bonding orbital with low energy and antibonding orbital with high energy are formed. The degree of bonding and antibonding can be quantitatively determined by COHP analysis. Although it appears from DOS analysis that the hybridization is formed and covalent bonding exists, it has to be confirmed by the above mentioned methods because it is also possible that the hybridization is formed but they are from the antibonding and it is not the covalent bonding. Nguyen et al.20) have performed the COHP analysis for Fe–C system, and found that the bonding is formed between iron and carbon atoms and the hybridization is due to the covalent bonding. Ohtsuka et al. showed that the bonding is formed between iron and carbon atoms and antibonding increases in the order of boron, carbon, nitrogen, and oxygen*1. In this way, the stronger bonding is formed between Fe–C atoms than Fe–Fe atoms. In the case of a molecule with two different atoms, the energy of atomic orbital is different and both covalent bonding and ionic bonding exist.21) The Fe–C supersaturated solution also has both covalent and ionic bonding between iron and carbon atoms*1. One of the examples of Fe–C chemical bonding analysis can be found in the paper of Jiang et al. for the case of cementite and the bonding is a complex mixture of metallic, ionic and covalent bonding from the analysis of DOS and charge distribution.22) The chemical bonding between Fe atoms or between Fe and C atoms should be analyzed in detail. It is very important to find how the bonding strength is affected by the change of electron density formed by the hybridization between Fe and C atoms.
*1 H. Ohtsuka, Z. Hou and K. Tsuzaki: Oral presentation at the 177th ISIJ Meeting (March, 2019, Tokyo).
The diffusion of carbon is an important factor in structural control of steels and experimental data is already reported.23) Because the movement and route of individual atom in the process of diffusion are hard to follow in the experiments, the elementary process of carbon diffusion in bcc iron has been investigated by the first principles calculation recently.2,3,4,7,9,24,25) (There are many studies such as the effect of vacancy on diffusion, interaction between vacancy and dislocation, and diffusion under stress, but these are excluded in this paper. Only the primary process of diffusion is reported here.). Many researchers calculated the activation energy of diffusion by the Nudged Elastic Band (NEB) method.26,27) The ‘nudge’ means ‘push gently’ or ‘fine tune’ and ‘elastic band’ means the ‘rubber band’. As shown in Fig. 1, NEB method determines the Minimum Energy Path (MEP: The solid line which connects point 0 and 7 in Fig. 1) that links two energy minimum states. In this method, appropriate number of points are aligned with equal distance along the line (point 0 to 7 in Fig. 1) and it is assumed that one elastic band is expanded between the two energy minimum states and the neighboring points are connected by a spring and the energy is calculated with the spring constant. When this energy becomes minimum, the real MEP can be determined.28) Some of the calculation results of migration energy are summarized by Souissi et al.7) They applied correction for spurious elastic interaction of the solute atom with its images in the periodic supercells. As a result, the calculated migration energy is virtually unchanged by this correction. Chen et al. performed the full first principles calculation of the diffusion coefficient. The atomic diffusion coefficient can be calculated with the distance of an atomic free movement and the atomic trial frequency. They performed a bunch of calculations such as electron state calculation at ground state, activation energy calculation during atomic movement, phonon vibration spectrum calculation, and vibration entropy calculation. They calculated the self-diffusion coefficient of oxygen in ZrO2*2, diffusion coefficient of carbon and nitrogen in iron as a function of temperature. The pre-exponential factor for carbon agrees well with the experimental data. On the other hand, the experimental data of the pre-exponential factor for nitrogen have a large scattering and further consideration of the experimental data is necessary.
Schematic illustration of a two-dimensional energy surface with two local minima separated by a transition state. The MEP is indicated with a dark line. Filled circles show the location of images used in an elastic band calculation.28) Figure is redrawn.
*2 Y. Chen, H. Luo and T. Mohri: Oral presentation at the 2017 Autumn Meeting of JIM (September, 2017, Sapporo). Collected Abstracts of 2017 Autumn Meeting of JIM, 136.
The Fe–C martensite has been widely used for structural materials. The basic and practical studies have been made for a long time by many researchers. The research covers a broad range of topics and the following three topics have been intensively studied, (1) the occupation site of carbon atom in martensite, (2) tetragonality (c/a) of martensite and (3) martensitic transformation mechanism. The history of these researches is written in textbooks and review papers.29,30,31) However, in spite of these studies, there are still many unsolved questions. The occupancy of carbon atom in martensite and tetragonality are reviewed in this section.
4.1. Occupation Site of Carbon Atom in MartensiteOne of the unsolved questions is, ‘Where is the occupation site of carbon atoms in Fe–C martensite ?’ The tetragonality is affected by the occupation site of carbon atoms and the martensitic transformation mechanism can be deduced by comparing the occupation site of carbon atoms before and after the martensitic transformation. Therefore, the occupation site of carbon atoms is an important topic which is involved in all of the above three topics of (1) to (3). It is commonly accepted that the carbon atoms exist in the O site of Fe–C martensite. However, for the relatively high carbon steels (more than 1.2 mass% C) or some alloyed steels, it was confirmed by the experiments that both O site and T site are the occupation sites of carbon atoms. After these reports, no major progress has been made. Now the history of the research of carbon atom occupancy in martensite is briefly summarized.
Carbon atoms sit in the interstitial sites in martensite. There are two kinds of interstitial sites, (a) O site and (b) T site in bcc iron. The rigid body model indicates that the radius of maximum sphere which occupies O site and T site is, 0.15 and 0.29, respectively if the radius of iron atom is 1.32) Thus, T site has a larger void than O site, but many textbooks read ‘The carbon atom prefers the smaller O site because the smaller strain is induced.’33,34,35,36) In these textbooks, only the above description is found and no references are shown. Why O site is now commonly accepted as the occupation site of carbon atoms in martensite, is shown as follows.29,30,31,33) Which site do carbon atoms prefer, had been discussed for some years. Nishiyama29) considered O site as the occupation site of carbon atoms because carbon atoms occupy the O site in austenite and all the atoms still occupy the O site in martensite after transformation as indicated by the Bain correspondence. He further considered X-ray diffraction and Mässbauer data as the evidence of the O site occupancy of carbon atoms since the data indicate the dipolar strain, which cannot be induced by the carbon atom at T site. This idea was accepted as the common model. This history was introduced in detail by Fujita and Ino. Fujita described Nishiyama’s idea as ‘The widely accepted theory of the site occupancy of carbon atoms in martensite.’ and wrote that ‘It is a common knowledge for many researchers and engineers.’.30) Ino introduced Nishiyama’s idea as ‘the common knowledge and its evidence’.31) In addition, the textbook written by Enami (he translated Kurdjumov’s textbook into Japanese) reads, ‘The site occupancy of carbon atoms in martensite was directly confirmed for the first time in 1972 by neutron diffraction experiment using the isotope of Fe and Ni. The hypothesis of occupation of O site by carbon atoms was proved by this experiment.37)’ and ‘Carbon atoms preferentially occupy single variant of O site but about 20% of them occupy other two variants of O site. It was clarified by some special analysis that carbon atoms do not occupy T site.38)’ This result is also found in Kurdjumov and Khachaturyan’s paper.40)
On the other hand, some experimental data which show carbon atoms occupy both O site and T site in the specimens with more than 1.2mass%C or some alloying elements like Mn, were reported.30,41,42) Thus, the occupation site of carbon atoms in Fe–C martensite is classified into the following three models.31) There are three equivalent variants for O site and each variant extends a, b or c axis. Carbon atoms occupy (1) only one variant of O site (ordering of carbon atoms known as Zener ordering),43,44) (2) two variants of O site40) and (3) both O site and T site. The fraction of O site and T site occupancy for model (3) is varied in papers.30,31,41,42) Yoshimoto et al. recently applied the CK-edge X-ray absorption spectroscopy (XAS) and full potential real-space multiple-scattering calculations to Fe-0.79 mass%C and found that the water quenched specimen consists of a single martensite phase and carbon atoms exist in solution at O site.45)
Recently many papers report that energy of Fe–C system with carbon atom at O site and T site is evaluated by the first principles calculation, and it is found that O site has a lower energy and is a stable site for carbon atoms in the case of relatively low carbon content.2,3,4,5,6,7,8,9,25) (Ruda et al.46) showed that T site has a lower energy than O site with EAM potential, but made a correction later,47) which will be explained in detail at section 6 later.). However some issues are not solved yet. Which is more stable for carbon atoms, O or T site, when carbon content is relatively high or some alloying elements are added, and is it possible that carbon atoms occupy both O site and T site at the same time, and if this is possible, why is it, and what is the fraction of O and T site occupation. These problems should be solved by the calculation.
4.2. Tetragonality of MartensiteThe value of tetragonality of Fe–C system with carbon atom at O site is an important topic. More researchers evaluate the tetragonality value by the first principles calculation recently and experimental research to re-examine the experimental values of tetragonality is increasing. Figure 2(a) is the experimental data of tetragonality which appears in Nishiyama’s textbook and can be expressed by the following linear equation except relatively low carbon content region.48)
| (3) |
Sherby et al. showed that tetragonality value is 1 (bcc iron) when carbon content is less than 0.6 mass%C as shown in Fig. 2(b), and named this composition H point.49) On the other hand, Cadeville et al. measured the lattice constant of Fe–C alloys made by splat quenching in a wide range of carbon content (0–5 at%C) and found that the tetragonality value is larger than 1 in the range of 1–5 at%C.50) Ohtsuka et al. evaluated the tetragonality of Fe–C by the first principles calculation (Fig. 2(c)) and showed that the calculated data (solid marks) agree very well with the experimental data represented by the solid line, which is the same line with that in Fig. 2(a).6) This agreement between calculation and experimental data is reported in some papers.2,6,8,51) Considering the calculation data by Ohtsuka et al., if all carbon atoms exist at single variant O site and carbon is ordered, the tetragonality value should be on the line shown in Fig. 2(c). It is highly expected that when the tetragonality value obtained by experiment is on the line shown in Fig. 2(c), all the carbon atoms are supersaturated in iron and occupy one variant of O site and ordered*3. This is explained as follows. Ohtsuka et al. calculated the total energy and tetragonality for all the configurations of carbon atoms in the system consisting of 54 iron and 2 carbon atoms at O site.6) The total energy is relatively low and the calculated tetragonality value agrees with experimental value when two Fe–C–Fe pairs are parallel, that is, carbon atoms are ordered at the single variant O site. Here, Fe–C–Fe pair consists of a carbon atom at O site and two iron atoms at the first nearest neighbor site of carbon. On the other hand, the total energy is relatively high and the calculated tetragonality value does not agree with experimental value when two Fe–C–Fe pairs are perpendicular to each other, that is, carbon atoms occupy the two variants of O site. The tetragonality value is clearly different between the single variant occupancy and two variants occupancy of O site even if the total energy is almost the same. We consider that whether the calculated tetragonality value agree with that of experiment or not clearly indicates whether carbon atoms are ordered at the single variant O site or occupy two variants of O site.
Ohtsuka et al. performed the first principles calculation and clarified the reason why the tetragonality values of Fe–C and Fe–N systems lie on one straight line as shown in Fig. 2(a) and showed that the tetragonality value of Fe–N is slightly smaller than that of Fe–C*4. Recently Maruyama et al. reported that they have measured the tetragonality value of Fe–Mn–C and Fe–Mn–Si–C alloys and found that the tetragonality value is more than 1 with carbon content of less than 0.6 mass%C as shown in Fig. 2(d).52) The abscissa shows the carbon content of the specimen. Mn affects the tetragonality value when more than 4 mass% Mn is added, which will be discussed in the next paragraph. The Mn content of the specimen is about 1 mass% and the effects of Mn on tetragonality can be neglected in Maruyama’s experiment. They determined the actual amount of carbon in solid solution from the atom probe tomography data and plotted the tetragonality as a function of the actual amount of carbon in Fig. 2(e). The measured tetragonality value linearly increases with increasing carbon content in solid solution and agrees well with the data by Honda-Nishiyama (dashed line) and Ohtsuka’s calculation (three cross marks in the figure). The tetragonality can be detected at room temperature in spite of the auto-tempering during quenching. This is because the strain introduced in martensite by the tetragonality is hard to be relaxed due to the restraint by the surrounding grains and the kinetics of the relaxation of strain is much more sluggish than the kinetics of carbon diffusion.52) Tanaka and Wilkinson measured the tetragonality of local area by EBSD.53) Now the precision of measurement of tetragonality has improved, so it is expected that the experimental data of tetragonality is increased and the old experimental data is re-examined.
The tetragonality value is affected by the alloying elements. The experimental data show Al and Ni increase the tetragonality, but Mn and Re decrease the tetragonality compared with the tetragonality value of Fe–C system.54) Some calculation show the effects of alloying elements on the tetragonality. Al-Zoubi et al.55) reported that both Al and Ni increase the tetragonality but Co and Cr have very little effect, which can be explained by the effects of alloying elements on the elastic constant. That is, the addition of Al or Ni reduce the elastic constant and the specimen becomes easier to be deformed and the tetragonality increases. Chentouf et al. showed that the addition of Ni increases the tetragonality because Ni reduces the elastic constant.56) Kurdjumov and Khachaturyan considered that the tetragonality is increased by the addition of Ni because carbon atoms are completely ordered at the single variant of O site in Ni-containing alloys though they are not completely ordered and some of them occupy other variants of O site in Fe–C alloys.40) On the other hand, Chentouf et al. considered that the increase of tetragonality by the addition of Ni can be explained by the effect of Ni on the elastic constant.56) This is because the tetragonality is increased by the addition of Ni though the calculation was performed with carbon atoms completely ordered at single variant of O site for both Fe–C and Fe–Ni–C systems. In this case, the increase of the tetragonality by the addition of Ni is ascribed to the reduction of the elastic constant. This is one of the reasons for the increase of tetragonality by the addition of Ni. However, other reasons, for example, formation of C–Fe–C or C–Ni–C dumbbells are proposed54) but the above mentioned calculations do not clarify these points. Whether these dumbbells exist or not, and do they contribute to the increase of tetragonality, should be clarified by the first principles calculation.
*3 H. Ohtsuka, Z. Hou and K. Tsuzaki: Oral presentation at the International Conference on Martensitic Transformation 2017 (ICOMAT2017) (July, 2017, U.S.A.).
*4 H. Ohtsuka, Z. Hou and K. Tsuzaki: Oral presentation at the 2016 Autumn Meeting of JIM (September, 2016, Osaka).
The C–C interaction influences the configuration of carbon atoms in iron, so it is a very important factor that affects the ordering of carbon atoms, formation of tetragonality and clusters. Sato57) estimated the ordering temperature of carbon atoms, which will be described in detail at section 5.2. He considered the C–C interaction and mentioned that ‘The large strain energy is produced since a carbon atom occupies the small void at interstitial sites of iron, so it is obvious the carbon atoms stay away from each other as much as possible to minimize the interaction energy. Therefore carbon atoms apply repulsive force to each other.’ and that ‘we cannot precisely estimate the C–C interaction purely theoretically nor we can precisely estimate it experimentally since the solubility of carbon in α iron is too small.’ Ino31) mentioned that the C–C interaction is reported to be negative, that is, attractive from the calculation based on elastic theory or electron calculation. Mou and Aaronson obtained the nearest neighbor carbon interaction by analyzing the experimental data of activity of carbon in ferrite with statistical thermodynamics and found that it is attractive at 1000 and 1070 K.58) Mou and Aaronson pointed out that the necessary condition for analysis is not satisfied and the correct values are not obtained in the researches which report the repulsive C–C interaction.58) They further reported that the origin of C–C interaction is static electron interaction rather than the elastic interaction. Bhadeshia considered that C–C interaction is repulsive both in γ and α based on thermodynamic calculation but concluded that it is difficult to evaluate the C–C interaction energy in α precisely by experiment because the solubility of carbon in α is very small.59) Now the evaluation of C–C interaction by the first principles calculation is introduced in chronological order. First, Jiang and Carter performed the first principles calculation for Fe–C system. The supercell was optimized in structure, internal stress was reduced enough and the solution enthalpy was compared for Fe16C1 and Fe54C1 systems.2) It is lower by 0.14 eV in the former case and they considered this is because the C–C interaction is negative in bcc iron. They considered that the precipitation of cementite is also the evidence of negative C–C interaction. In their calculation, one carbon atom exists in one supercell and the interaction between one carbon atom and another carbon atom in the neighboring supercell is included in the solution energy. The C–C distance before the structural optimization is 2ao and 3ao (ao is the lattice constant of pure iron) for Fe16C1 and Fe54C1, respectively and is not close enough. Ohtsuka et al. have done the first principles calculation for the minimum C–C distance of 0.5ao,6) which will be shown later. Domain et al. calculated the C–C interaction for various carbon configurations when the C–C distance is changed from 0.5ao to about 2ao.3) It was shown C–C interaction is repulsive for 54 iron atoms. Ruban has done the calculation with further C–C distances and showed that C–C interaction is repulsive in the short distance but is nearly zero in the long distance and the interaction produced from the strain is attractive.60) Yan and Ruban have done the calculation based on DFT or EAM and found that the C–C interaction is a strong repulsion in a short distance but the repulsion becomes weak with more than 1.6ao atomic distance and rather attractive in some configurations.61) They mentioned that electron structure and local chemical environment can be made clear by DFT, but the calculation results is affected by the size of a supercell and the interaction between one carbon atom and another one in the next neighbor supercell and the results are different from the C–C interaction which is obtained in the infinitely large supercell. On the other hand, they pointed out that the chemical interaction in the short distance evaluated by EAM is not reliable. Ohtsuka et al. performed the first principles calculation for all the configuration of carbon atoms in the Fe–C system with 54 iron atoms and two carbon atoms at O site.6) When two carbon atoms are at the first nearest neighbor site with each other, the solution energy is positive and maximum. The solution energy is classified into strain energy (mechanical energy in the paper6) and chemical energy and both of these energies are higher than those for other configurations. The high chemical energy is due to the antibonding interaction between two carbon atoms*5. On the other hand, when two carbon atoms are farthest away with each other, the solution energy is minimum, the chemical energy is nearly zero with very little C–C interaction and the solution energy is almost the same with the strain energy. In this way, the solution energy is classified into the strain energy and chemical energy, and one of the origin of chemical energy is the bonding condition between atoms. It is important to investigate the chemical bonding between atoms and the chemical bonding is helpful to clarify the origin of atomic interaction. It is expected that more and more researchers perform the calculation of the chemical bonding between atoms.
*5 H. Ohtsuka, Z. Hou and K. Tsuzaki: Oral presentation at the 2018 Autumn Meeting of JIM (September, 2018, Sendai). Collected Abstracts of 2018 Autumn Meeting of JIM, J38.
As already shown in the previous section, Zener proposed the idea of ordering of carbon atoms in 1946.62,63) In this ordering, carbon atoms occupy only one variant of O site and the tetragonality appears. He considered that the elastic strain in bcc iron becomes minimum by the ordering of carbon atoms. Sato derived the ordering temperature of carbon as a function of carbon content and the C–C interaction using the Bragg-Williams theory for order-disorder transformation.57) As shown in Fig. 3, the order-disorder transformation temperature increases linearly with carbon content. It is obvious that order-disorder transformation occurs at around room temperature if the carbon content is about 0.2 mass%C (the cross point of the solid line and the dot-dash line in Fig. 3), and this content is referred to as critical content. Various theoretical approaches have been applied to estimate the critical content, but fairly large scattering of values between 0.18 to 0.64 mass%C is reported.64) As shown in Fig. 4(a),64) Naraghi et al. calculated the Gibbs energy at room temperature for disordered bcc (carbon atoms occupy three variants of O site), Zener ordered (carbon atoms occupy the single variant of O site) and α″ ordered (Fe16C2 structure shown in Fig. 5) structures using CALPHAD method. In Fig. 4(a), Xs1 and Xs2 represent the spinodal decomposition points, and Xs1 also represents the critical content. The energy of ordered phase and disordered phase is equal at less than Xs1 content. Xe1 and Xe2 represent the miscibility gap. They considered that ordering and spinodal decomposition coexist, the ordered martensite is unstable and spinodal decomposition occurs during aging. Figure 4(b) shows the Zener ordering temperature as a function of carbon content. The solid line indicates the calculated data by Naraghi, the dotted line show the past research data and black point shows the cubic-tetragonal transition content at room temperature obtained by experiments. The order-disorder temperature is different from the previous data and seems interesting but is still have to be investigated in detail.
Calculated order/disorder transformation temperature as a function of carbon content.57) Figure is redrawn.
(a) Gibbs energy curves for disordered, Zener ordered, and α″ ordered Fe–C solutions with bcc iron and graphite as the reference at 300 K. (b) Calculated Zener-order/disorder transformation curves from the present evaluation and previous estimations in the literature.64) (Online version in color.)
α″-Fe16C2 structure. (Online version in color.)
The α″ Fe16C2 is an important structure for considering the carbon ordering. Taylor et al. measured the aging behavior of martensite at room temperature in Fe–Ni–C alloys and found that the as-quenched martensite is unstable, spinodal decomposition occurs by aging of martensite and high carbon phase is Fe16C2 (11.1 at%C).65) Zhu et al. showed that carbon enriched region of about 10 at%C is formed before ε carbide precipitates at the room temperature aging of martensite in Fe–Ni–C alloys.66) Sinclair et al. have done the MD calculation and found that Fe16C2 is more stable at 0 K than either α′-Fe8C or the structure with three variants of O site occupied randomly by carbon atoms.67) Kandaskalov and Maugis performed the first principles calculation for Fe16C2 and Fe16N2, and evaluated the solution energy and elastic constants and concluded that the bonding of Fe–C and Fe–N is highly ionic and Fe16C2 is metastable.68) Yan and Ruban performed the Monte Carlo simulation and found that the disordered α phase and Fe16C1 or Fe16C2 coexist at low temperatures.61) However, Van Genderen et al. reported that Fe16C2 is not formed by the aging of martensite in Fe–C and Fe–Ni–C alloys at room temperature or above room temperature.69) The reason for this is, C–C interaction is attractive and the carbon atoms tend to form clusters. This effect is greater than that of the decrease of elastic energy produced by the long range ordering through the formation of Fe16C2. In this way, Fe16C2 is experimentally recognized in Fe–Ni–C alloys but not in Fe–C alloys. The reason for this is not clarified yet and is very important.
In the case of relatively small number of carbon atoms, the first principles calculation is possible. However, in the case of ordering or cluster formation of carbon atoms, relatively large number of carbon atoms are included and it takes rather long time for the first principles calculation. Then MD calculation is performed by making the interatomic potential. How many atoms can be used for the first principles calculation depends on the ability of computer and software, computational resource and cost. In this review paper, maximum number of iron atom is 25060,61) and maximum number of carbon atom is 16 with 128 iron atoms.56) There are many reports of the interatomic potential for Fe–Fe, Fe–C and C–C. The traditional interatomic potentials are sometimes re-examined and revised or new potential is produced. It is notable that some new potentials focus on the chemical bonding between atoms.
The new potentials are produced so that the parameters well reproduce the results obtained by the first principles calculation. The Embedded Atom Method (EAM) is one of the representative methods to make a new potential. In this method, the total energy is expressed by the sum of the repulsive potential between the atomic cores and the energy required to place an atom into the electron gas with homogeneous electron density.70) The MD simulation is performed using the potential made by EAM. Becquart et al. developed the new potential by EAM and calculated the activation energy for the diffusion of carbon atoms.24) Hepburn and Ackland developed the new potential in the framework of EAM considering the energy required for embedding the carbon atoms and the energy required for the covalent bonding.71) It was proved that activation energy for diffusion and the interaction energy between carbon atoms and vacancies agree with those obtained by the first principles calculation better than the other reported potentials. As was already mentioned at the end of the section 4.1, Ruda et al. developed the new potential and showed that T site is a more stable site for carbon occupation than O site.46) However, they later made a correction and reported that O site is a more stable site.47) The reason for the correction is that their former potential was less accurate and the reference data of the first principles calculation was not enough much. They made the new potential based on the more accurate potential.47) Nguyen et al. considered that Ruda et al. reported the more stable T site because the potential developed by EAM is spherically symmetric and does not include the bonding angle dependence which is necessary for covalent bonding which has angular dependence of bonding.9) So they did not use EAM and adopted the analytic bond-order potential, which includes the three body interaction and bond-order factor with angular dependence.9) They compared the calculation results obtained by the new potential and by the ordinary DFT, and found that in the former case the formation enthalpy is about +6% and the activation energy for diffusion is about −14%~−19%. It is important to know how much difference is brought about for the calculations of C–C or C–M interaction with the traditional potential and with the new potential which takes account of the chemical bonding. The potential is empirically developed and it is hard to decide which potential is the best one. So the validity of the potential should be discussed in detail by scrutinizing how the potential was developed, what data was brought about with the potential and how much difference is produced for the new calculation results compared with other traditional calculation results and experimental data.
In this review paper, the topics are limited to the nature and state of solute carbon atoms in iron. It is recognized that carbon atoms play an important, various and profound roles in steels. The carbon atoms diffuse swiftly and easily form clusters and various kinds of precipitates and therefore the experiment and analysis are hard. The first principles calculation and MD simulation reflect the slight change of the position and configuration of carbon atoms and give us useful information of nature, state and behavior of carbon atoms in iron. These calculations have recently made a remarkable progress. The new analysis which is based on the investigation of chemical bonding is expected in the future. Such analysis will give us more fundamental information for the atomic interactions. It is highly expected that the essence of old and new issues, such as the nature, state and function of carbon atoms in iron, is made clear from a new point of view.
Kaneaki Tsuzaki gratefully recognizes the financial support provided by Japan Society for the Promotion of Science (JSPS) KAKENHI (JP16H06365) and the Japan Science and Technology Agency (JST) (grant: 20100113) under the Industry-Academia Collaborative R&D Program “Heterogeneous Structure Control: Towards Innovative Development of Metallic Structural Materials”.
