Segregation of Carbon in α-Fe Symmetrical Tilt Grain Boundaries Studied by First-Principles Based Interatomic Potential

Abstract

Carbon segregation is known to have an extreme influence on the cohesive energies and mechanical properties of grain boundaries (GBs) in steel. In this paper, the stability of a series of α-Fe⟨110⟩ symmetrical tilt grain boundaries (STGBs) with C was systematically investigated based on first-principles calculations. We used the newly-developed Tersoff/ZBL potential of Fe–C interaction constructed from the forces and disolution energies of various defect complexes with C in Fe calculated from first-principles. This potential shows a great effectiveness in considering large complex systems of STGB and determining the segregation sites of various STGB configurations. The stable location of C was analyzed from the view point of free volume formed by the GB systems. We found that the compact GBs were less attractive to C than the open ones. The GBs exhibited a strong attractive interaction with C compared to vacancies, therefore, a higher solubility of C can be expected in GB systems. The present simulation results are consistent with the experimental observations by TEM and APT method, and qualitatively explains the behaviour of C in Fe.

Fig. 4 Scanning planes (colored in pink on the left-hand side figures) and dissolution energy landscapes (right-hand side figures) of Fe–C systems of (a) Σ3(112) STGB and (b) Σ9(221) STGB. The location of GB is indicated by green dashed line.

1. Introduction

Iron–carbon alloy or steel is one of the most widely used structural materials in various industries, infrastructures and our society. The versatility, durability and strength of steel can meet requirements for a variety of purposes, and it is also an affordable and environmentally friendly.¹⁾ Improving their mechanical properties by modifying polycrystal structures of Fe matrix or doping impurity atoms is still an attractive research field.^2,3) Among the impurities for such purpose, carbon is the most important foreign interstitial atom and frequently used in determining the strength and hardness of steels even though its concentration is quite low as 0.022 mass% in body centered cubic (BCC)-Fe.⁴⁾ It is known that the formation energy of octahedral C interstitial is very high,^5,6) therefore many experimental and theoretical studies have been devoted to identify the location of C in Fe. In the fabrication process of steel-related material, various types of defects, such as vacancies, dislocations or grain boundaries, are naturally expected to exist. Among them, grain boundaries (GBs) significantly affect the physical and mechanical properties of polycrystalline material.^7,8) Experimentally, C is well known to enhance the stability of GBs in steels, in contrast to H, P and S, thereby improving its crack-resistant properties.^9–11) Hence, to control the strength of steel, it is important to investigate the interaction between GBs and C.¹²⁾ In addition, the segregation of carbon causes the formation of carbon clusters and the precipitation of carbides in aging process, and it affects the mechanical properties of carbon steel.¹³⁾ However, the experimental assignment of the location of carbon is still difficult.

So far, the GBs have been visualized by using transmission electron microscopy (TEM) and atom probe tomography (APT).^14,15) The excess of carbon at the GBs was detected by combination of mass spectroscopy and ion projection microscopy in grain boundary space and the linear relationship was observed between solubility of C and misorientation angle ω for ω < 0.44 rad (25°) in α-Fe GBs.¹⁶⁾ However, such kind of correlation has not been found for large ω-GBs. Since the local atomic structure depends strongly on ω, it is desirable to investigate the correlation between local atomic structure of GBs and stability of carbon at the GBs. For this purpose, numerical simulations might be helpful to complement the experimental observations. As for the computational approaches, the stability of various pure α-Fe ⟨110⟩ symmetric tilt grain boundaries (STGBs) with common ⟨110⟩ tilt axis was investigated by performing the molecular dynamics simulations with classical interatomic potentials of Fe such as pair potential proposed by Johnson^17,18) or Embedded Atom Method (EAM),¹⁹⁾ however the behaviour of impurity atoms has been studied only in a few typical α-Fe ⟨110⟩ STGB configurations^20–23) due to the lack of the universal and reliable interatomic potentials for the systems with impurities.

In this paper, we try to predict the location of C in α-Fe ⟨110⟩ STGB and clarify the relation between the stability of C and its local structures. First-principles calculations based on the density-functional theory (DFT) have been widely applied to give highly accurate prediction and insight into chemical behavior of impurity segregation. However, it is computationally too expensive to apply the DFT to complex STGB systems. Therefore, alternative methods in conjunction with classical force-field have been employed, namely the classical interatomic potentials for Fe–C systems are constructed within the framework of Tersoff/ZBL potential by fitting its parameters to reproduce the results of first-principles calculations of various α-Fe systems with C and Fe vacancies.^24,25) Tersoff potential is a bond-order potential which was found as a suitable potential model for metallic/non-metallic compound.^26–28) The Tersoff/ZBL interatomic potential of Fe–C was constructed by fitting to forces and energies of DFT calculations of C in bulk BCC Fe with and without Fe vacancy. This potential was shown to be effective in reproducing C diffusion paths and BCC/FCC transformation by introducing C. In the followings, by using the Tersoff/ZBL potential determined in Refs. 24) and 25), the carbon segregation in a set of α-Fe⟨110⟩STGBs is systematically investigated. Firstly, the reliability of the new Tersoff/ZBL potential is demonstrated by comparing the grain boundary energies with those obtained by DFT. The results are also compared with those calculated by other interatomic potentials. Secondly, the effect of carbon on the stability of grain boundary is studied by calculating the formation energy of GBs with and without C. By calculating energy landscape of the GB-C systems, the segregation sites are clearly obtained. Finally, the dependence of segregation energy of C in various α-Fe⟨110⟩STGBs as a function of distance between C and the GB is analyzed. Based on the calculated results, the relation between local atomic structure and stability of C is discussed by considering free volume around C based on the Voronoi construction.

2. Model and Calculation Method

In the present simulations, we assume periodic boundary condition, namely, atomic structure of STGBs are simulated by using supercell. For each STGB, corresponding supercell is generated by GBstudio software.²⁹⁾ In this study, 9 structures of α-Fe ⟨110⟩ STGBs with the range of misorientation angle of 0.68–2.68 rad (38.9°–153.5°) are considered. The unit cell sizes of STGB configurations before relaxation are provided in Table 1. The translation vectors a₀, b₀ and c₀ define the supercell and are used for the DFT calculations. a₀, b₀ are the minimum two-dimensional CSL (coincidence site lattice) sizes which define the GB plane and c₀ defines the width of twin. The rotation axis of the STGBs, namely ⟨110⟩ direction of BCC structure, is parallel to b₀ and its length is $\sqrt{2} a_{Fe}$ where a_Fe is the lattice constant of BCC Fe lattice. The directions of two perpendicular axes a₀ and c₀ and their lengths are also indicated in Table 1. The distance of two adjacent STGBs is |c₀|/2.

Table 1 Unit cell used for the DFT calculations on grain boundary energies of typical STGBs. Here, ω is misorientation angle, m is number of Fe in the unit cell. γ_GB^* values are obtained from previous DFT studies.

The information of the supercells of the STGBs used for the simulations with classical interatomic potentials is shown in Table 2. The GB plane is constructed by two vectors N₁a₀ and N₂b₀ (where, N₁ and N₂ are integers) by using a₀, b₀ defined in Table 1 for each case. The width of twin is c and also indicated in Table 2. Among the considered STGBs, the simplest GB is Σ3(112) with misorientation angle of 1.91 rad (109.5°) which is well known as a twin GB configuration with [112] Miller index of grains. In other configurations with higher CSL Σ value, the structure of GB becomes more complex. For example, in Σ9(221) GB some Fe pairs come too close with each other. In such cases, to simplify the calculations, some Fe atoms are removed as shown in Fig. 1(a). The resulting STGB structure becomes asymmetric but by relaxing the structure, the symmetric arrangement is recovered for most cases, as shown for the case of Σ9(221) GB in Fig. 1(b).

Table 2 Supercells used for the calculations using Tersoff/ZBL potential on various ⟨110⟩STGBs. Here, ω is misorientation angle and grain boundary plane is perpendicular to c-axis. The supercell size is indicated by using vectors a₀ and b₀ defined in Table 1. m is number of Fe in the supercell. Grain boundary energy, segregation energy of C and Voronoi volume of C are also summarized.

Fig. 1

Grain boundary structures of (a) modified Σ9(221) 38.9° STGB and (b) its optimized structure. The grain boundary is perpendicular to c-axis. Orange and gray spheres represent Fe atoms lying on two adjacent atomic layers along rotation axis ⟨110⟩ (b-axis).

All the calculations with classical interatomic potentials are carried out by using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS).³⁰⁾ In addition to the new Tersoff/ZBL interatomic potential,^24,25) the calculations are also performed in conjunction with the EAM potential³¹⁾ for comparison. To avoid strong interaction of two adjacent GBs, calculations using classical potentials are performed with large cells of STGBs. For some typical cells of STGBs with small number of atoms (the sizes of STGBs are listed in Table 1), we performed DFT calculations, in order to confirm the reliability of presently constructed Tersoff/ZBL potential. Our DFT calculations were performed by using the spin-polarized version of the Vienna Ab initio Simulation Package (VASP).³²⁾ The Perdew-Burke-Ernzerhof (PBE) functional based on the generalized gradient approximation (GGA)^33,34) was employed for the electronic exchange-correlation interaction. The projector-augmented-wave (PAW) pseudopotential was taken from PAW database where the 3d, 4s states of Fe are treated as valence states. An energy cut-off of 720.981 × 10⁻¹⁹ J (450 eV) is used for all calculated systems, and all relaxation calculations are performed until a residual force of 0.016 × 10⁻¹⁰ N (10⁻² eV/Å) is achieved. The calculated lattice constant a_Fe of BCC Fe is 2.83 Å with the k-point mesh of 15 × 15 × 15.

3. Results and Discussion

3.1 Grain boundary energy

In order to examine the stability of GBs, grain boundary energies (GBEs) with C were calculated by the following equation:   

\begin{equation} \gamma_{\textit{GB}} = \frac{E_{\textit{GB}}^{\textit{mFe} + \textit{nC}} - m\mu_{\textit{Fe}}- n\mu_{\textit{C}}}{2A}, \end{equation} (1)

where $E_{\textit{GB}}^{\textit{mFe} + \textit{nC}}$ is the total energy of the supercell of the STGBs systems containing m Fe atoms and n C atoms, and μ_Fe, μ_C are the chemical potentials of Fe atom in perfect BCC structure and C in diamond, respectively. A is the area of calculated STGB. For GBE calculations, periodic boundary conditions along three directions are applied. Hence, 2A is used in eq. (1) for indicating two symmetric GB interfaces for each calculated supercell. Firstly, we assume there is no C in the system, namely n = 0, and calculated values of γ_GB by the DFT and the classical interatomic potentials are compared. The obtained values of γ_GB are compared and illustrated in Fig. 2. For both cases, the STGB structures are fully optimized. The information of DFT calculations and the supercell sizes used for the classical force-field calculations are summarized in Table 1 and 2, respectively. For both cases, the STGB structures are fully optimized. Both of the present DFT and classical force-field calculations confirm that the most stable STGB is Σ3(112), which is in good agreement with previous studies.^20,35–38) The Σ9(221) is predicted as the most unstable STGB among the presently calculated ones. The results of classical force-field calculations using the new Tersoff/ZBL potential reproduce DFT results reasonably as compared with those obtained by the EAM and Johnson potentials. Therefore, the present Tersoff/ZBL interatomic potential has shown to be a good force-field for calculating complex GB system, and we use the potential for all of the following simulations.

Fig. 2

Grain boundary energy as a function of misorientation angle. The GBE values calculated by Johnson^* potential are taken from previous study (Ref. 18)).

Next, to determine the influence of C on the stability of STGBs, the GBEs ($\gamma _{\textit{GB}}^{\textit{C}}$ and $\gamma _{\textit{GB}}^{\textit{ref}}$) of STGBs including one C (i.e., n = 1) are calculated and compared by using eq. (1). Herein, the GBEs $\gamma _{\textit{GB}}^{\textit{C}}$ are obtained by introducing C at the most stable position in GB. The areal concentration of C in GBs is shown in Table 2. The reference of GBEs ($\gamma _{\textit{GB}}^{\textit{ref}}$) were calculated for the configuration in which the C atom is inserted at the most stable position in bulk BCC Fe, which is about 2.2–4.4 nm away from the GBs. At these distances, the interaction of C with GBs is negligible, this means that C atom exhibits as in bulk BCC Fe while the GBEs in these cases are considered to be the same as in the case of no C (γ_GB). As shown in Fig. 3, generally the energy of STGBs is reduced by introducing C at GB comparing with C at a distance away from GB. In the Σ3(112) GB, carbon at GB is shown to have very small influence on the GBE. This is reasonable, because this GB is considerably stable and its local structure is similar to bulk Fe. On the other hand, C shows strong effect on other STGBs such as Σ11(332), Σ9(114), and Σ19(116). They are basically unstable GBs and the local structures are very much different from the bulk Fe. Quantitative discussion between the stability of C and the local structure will be discussed in sec. 3.3.

Fig. 3

Grain boundary energy of STGBs with C calculated by using classical interatomic potential. $\gamma _{\textit{GB}}^{\textit{C}}$ and $\gamma _{\textit{GB}}^{\textit{ref}}$ values are indicated by red circles and blue triangles, respectively.

3.2 Segregation sites

The segregation sites of C are determined by examining the landscape of the dissolution energy (E_dis) of C in the GB systems. E_dis is calculated as follows:   

\begin{equation} E_{\textit{dis}} = E_{X}^{\textit{mFe} + \textit{C}} - E_{X}^{\textit{mFe}} - \mu_{C}, \end{equation} (2)

where X distinguishes the kinds of defects such as GB or vacancy.

In this calculation, a single C is inserted on a middle plane of two Fe layers corresponding to [022] plane in the bulk BCC structure. This plane includes set of octahedral sites (O-site), which is the most stable site of C in BCC Fe, therefore assumed as the most preferred location of C. On this plane, a 100 × 100 mesh along two lattice vectors is set and the dissolution energy of C for each mesh point is obtained by optimizing the structure with C position kept fixed. In Fig. 4, the scanning planes and dissolution energy landscapes of two typical STGBs, namely, the most stable Σ3(112) STGB (Fig. 4(a)) and the most unstable ones Σ9(221) STGB (Fig. 4(b)) are illustrated. Here, the darker color reflects the higher energy area in which C is less preferred, whereas the lighter area illustrates the favourable location of C. As can be seen that the unstable area of C is the area close to Fe atoms, indicating the strong repulsive interaction between C and Fe atoms. The lighter area of GB planes indicates an attractive source of C compared to the one in the distant GB. Besides, possible positions of C are found at O-sites in the distant GB area. Similar calculations are also performed for the other STGBs. All obtained results confirm that C is preferably located at the GB plane considered in the present study.

Fig. 4

Scanning planes (colored in pink on the left-hand side figures) and dissolution energy landscapes (right-hand side figures) of Fe–C systems of (a) Σ3(112) STGB and (b) Σ9(221) STGB. The location of GB is indicated by green dashed line.

3.3 Segregation energy

The difference between dissolution energy of C in the GB configuration and the one in pure bulk α-Fe is indicated as segregation energy (E_seg), which is calculated as,   

\begin{align} E_{\textit{seg}} &= E_{\textit{dis}}^{\textit{GB}} - E_{\textit{dis}}^{\textit{bulk}} \\ &= (E_{\textit{GB}}^{\textit{Fe} + \textit{C}} - E_{\textit{GB}}^{\textit{Fe}} - \mu_{\textit{C}}) - (E_{\textit{bulk}}^{\textit{Fe}+\textit{C}} - E_{\textit{bulk}}^{\textit{Fe}} - \mu_{\textit{C}}), \end{align} (3)

where $E_{\textit{GB}}^{\textit{Fe} + \textit{C}}$ and $E_{\textit{bulk}}^{\textit{Fe} + \textit{C}}$ represent the total energy of the supercell of the grain boundary containing C and the total energy of bulk Fe with C located at the O-site, respectively. $E_{\textit{GB}}^{\textit{Fe}}$, $E_{\textit{bulk}}^{\textit{Fe}}$ are the total energy of the supercell of the GBs and the total energy of bulk Fe without C, respectively. E_dis is the dissolution energy of single C defined as eq. (2) and written down for respective structures in the latter half of the eq. (3). The $E_{\textit{bulk}}^{\textit{Fe} + \textit{C}}$, $E_{\textit{bulk}}^{\textit{Fe}}$ are calculated by using 8 × 8 × 8 supercells of perfect bulk BCC Fe with and without one C atom.

In order to figure out the behavior of C, the E_seg is calculated for various STGBs and the results are summarized in Table 2. For all of the STGBs considered in this work, the segregation energies E_seg are found to be negative, which indicates a strong segregation tendency of C atoms to the GBs from the bulk region. Among the considered STGBs, the Σ3(112) GB has the highest segregation energy of −0.848 × 10⁻¹⁹ J (−0.53 eV), and the Σ9(221) GB has lowest E_seg of −2.848 × 10⁻¹⁹ J (−1.78 eV). In order to discuss the origin of the difference in segregation energy from the view point of the local atomic structure, we focus on the free volume defined by neighboring Fe atoms around C. For this purpose, we define Voronoi cell around C as free volume. In Fig. 5, the correlation between the Voronoi volume of C and its E_seg is shown. In the figure, C is assumed to be located not only on the GB but also off the GB for each GB. It is clearly observed that there is a negative correlation between them, namely the larger free volume STGBs is, the more negative E_seg is predicted. The negative correlation was not clear in the previous DFT calculation²⁰⁾ due to the limited number of calculated GBs, but the trend of the most stable position of C at GBs is consistent with the present results.

Fig. 5

The relationship between segregation energy and Voronoi volume of C in different STGBs. In this plot not only the C on the GB plane but also the C off the GB plane are included.

In order to extend the above discussion, we calculate E_seg as a function of distance between C and GB plane for STGBs, namely Σ3(112), Σ3(111), Σ9(221), Σ9(114), and Σ11(332). At the same time, for each position of C, the Voronoi volume is calculated. For this procedure, a single C atom is inserted in the stable positions which are predicted based on the energy landscapes obtained in Section 3.2. The results are shown in Fig. 6. E_seg takes the lowest value when C is placed at the GB plane (d_GB-C = 0) then gradually increases and approaches zero as C is moved far away from the GB plane. The distance dependence of E_seg corresponds very well with the change of the Voronoi volume of C, namely the Voronoi volume reaches to the highest value when C located at the GB plane and from there the lowest E_seg is obtained. A gradual decrease in Voronoi volume is found when increasing the distance from C to the GB, d_GB-C. The minimum of Voronoi volume of around 7.0 × 10⁻³ nm³ is obtained when C is located in distant of GB. This is almost the same to the value of C in bulk Fe (7.05 × 10⁻³ nm³). By considering the effect of the Voronoi volume to the behaviour of C in various STGBs, it can be concluded that the GB creates open space for C occupation, resulting in C being strongly trapped by GB.

Fig. 6

The segregation energy of C (top-panel) and Voronoi volume (bottom-panel) as a function of the distance between GB and C. The horizontal dashed line in the bottom panel indicates the Voronoi volume of C located at O-site in BCC bulk iron.

In the most stable GB configuration Σ3(112), E_seg is small but we can still find the anti-correlation between E_seg and the Voronoi volume of C. As already shown in Fig. 6, this behavior is also clear for the other GBs. It is suggested that the anti-correlation between Voronoi volume around C and the segregation energy is general for Fe–C systems. Experimental confirmation of this finding is desirable. For Σ3(112) GB, a small difference in E_seg between the position in the GB plane and distant GB is shown, it means that such GB behaves as a weak attraction to C. It was observed that a significantly lower excess of C is detected in the experimental study for special stable Σ3 and Σ5 GBs.¹⁶⁾ On the contrary, in the case of Σ9(221) configuration, C is strongly trapped at the GB plane when compared to other locations. The strong interaction of Fe–C in the GB plane prevents the difusion of C to the bulk. Therefore, a high solubility of C can be expected in Fe matrix by controlling the inclination of GB planes.

In order to compare the strength of interactions between C and GB with those of C and the other defects, we estimate the dissolution energy of single C in different structures by eq. (2). The dissolution energies of C in Fe with different defects are listed in Table 3. In perfect BCC Fe, a single C is located at O-site in the supercell (Bulk Fe in Table 3). For calculating the interaction with vacancy in bulk Fe, the location of C was assumed to be the O-site nearest to the single Fe vacancy (DefectedBulk_C@O-site in Table 3) and at vacancy (DefectedBulk_C@Vac in Table 3) in BCC. The interaction energy is calculated for the cases of GB and single vacancy. It is confirmed that the present results obtained by the new Tersoff/ZBL potential are very close to those of the previous DFT studies.^5,39) Compared to C location in single vacancy cases, the E_dis of C at the Σ3(112) GB is smaller. However, even for the smallest case, it is still about 0.288 × 10⁻¹⁹ J (0.18 eV) and remains a positive value. It means that C is attracted by Σ3(112) GB rather than single Fe vacancy cases, but C still tends to move out from the BCC Fe matrix. The strongest interaction of C with GB is found for the unstable GB Σ9(221), and the dissolution energy is strongly reduced and becomes a negative value of −1.234 × 10⁻¹⁹ J (−0.77 eV), which is much lower than the one in bulk and vacancy cases. Therefore, if such unstable GBs exist, C is strongly trapped during metallurgical process. Moreover, the Voronoi volume values of perfect, single vacancy, and grain boundary cases are compared in Table 3. We can see that the Voronoi volumes of a C located in single vacancy cases (about 9.99 ų and 9.86 ų for the “inside” and “near to” vacancy cases, respectively) are much higher than the ones in perfect and grain boundary cases (7.06 ų, 7.50 ų and 8.39 ų, respectively) even though the interactions of C in the former cases are weaker than in grain boundary cases. It can be explained that for the single vacancy cases, the distances of Fe and C are slightly longer than the equilibrium Fe–C bond length²⁴⁾ which lead to the weak interaction of Fe and C. In conclusion, GBs exhibit stronger attraction to C than other point defects in the BCC Fe matrix, and the lowering of the GB energy due to the existence of C might partly contribute to the strengthening effect of C.²⁰⁾

Table 3 Dissolution energy and concentration of C in bulk Fe, and Fe with different defected structures.

4. Conclusion

In this work, the segregation behavior of C is explored in different GBs. For the present simulations, we use the Tersoff/ZBL potential newly constructed based on the first-principles total energy calculations. It is confirmed that the potential shows a good performance in reproducing the DFT results of GB energy for α-Fe⟨110⟩STGBs with a range of misorientation angle of 0.68–2.68 rad. A decrease in GB energies due to the introduction of C has been found for all of the considered GBs. Thus, C is strongly trapped by the unstable STGBs, resulting in a high solubility of C surrounding those GBs. These results are partially confirmed by the recent experimental observations.¹⁶⁾ By performing the Voronoi analysis, the anti-correlation between Voronoi volume around C and the segregation energy is found, i.e., for the site with larger Voronoi volume, the smaller segregation energy of C is predicted. Grain boundary with open free space represents as strong attraction for C. Moreover, grain boundary exhibits a stronger interaction with C compared to bulk Fe or vacancy cases. The most stable STGB slightly attracts C atoms because of compact atomic structure. This finding can be used as a guideline to analyze the local concentration of C in Fe and the local atomic structure around C. Experimental verification is strongly desirable.

Acknowledgement

This work is partly supported by the Building of Consortia for the Development of Human Resources in Science and Technology project, implemented by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.

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