1. Introduction
Many mechanical properties of materials strongly depend on defects in the structure of the material such as vacancies, dislocations and grain boundaries, which present space to accommodate impurity elements.1–4) For iron/steel materials, the location of foreign atoms such as carbon, which is one of the essential interstitial solute atoms, in defect structure strongly affects their strength and ductility.5,6) The concentration of carbon atoms in the iron system is considered to have a significant effect on the mechanical properties of steel. For instance, the formation of carbides occurs by exceeding the limit of carbon solubility, which contributes significantly to the improvement of the durability and hardness of steel.7) Besides, when the carbon concentration in the system is below the solubility limit, the thermal and mechanical properties of the system can be changed significantly only by a minimal amount of carbon atoms (several tens of ppm) at interstitial sites.8) The control of C solubility strongly depends on the types of defects occurring in the manufacturing process. Therefore, knowledge of location of C and its segregation energy in iron/steel under the existence of several kinds of defects is essential for material design.
In simulations of defects in iron/steel, large scale calculations are usually required to minimize the spurious interactions between defects. For such calculations, classical force-field method in conjunction with interatomic potentials becomes a suitable choice to simulate the defect systems. In our previous work, Fe–C interatomic potential was 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 involving C and Fe vacancies.9,10) Tersoff/ZBL potential formalism originates from the concept of bond-order potential which was found as a suitable potential model for metallic/non-metallic systems.10,11) The reliability of the new Tersoff/ZBL potential is demonstrated by reproducing the grain boundary energy and segregation of C in some typical symmetrical tilt grain boundaries (STGBs) as compared to those obtained by density functional theory (DFT) calculations.11) It was found that the segregation energy of carbon in STGBs and the Voronoi volume of C exhibit the correlation, namely, for the STGB with larger Voronoi volume, the segregation of C is predicted to be stronger. To investigate the behaviour of C in different defect structures systematically, in this work, several dislocation configurations and their interaction with C are studied by using the same Fe–C Tersoff/ZBL interatomic potential.
Dislocations can be observed by using X-ray diffraction, transmission electron microscopy (TEM) or scanning transmission electron microscope (STEM).12–14) It is well known that the creation and motion of dislocations are the fundamental mechanisms of plastic deformation. The influence of impurity atoms and vacancies on the mobility of dislocation core leads to the significant change in mechanical properties of materials. In BCC metal, the dislocations with Burgers vector 1/2⟨111⟩ are usually observed in experiments.15,16) Therefore, many previous studies focused on 1/2⟨111⟩ dislocation by using classical interatomic potentials.17–20) In steel, the interaction of dislocation with C impurity has attracted a large attention from scientific community. The most popular edge dislocation model 1/2⟨111⟩$\{ 1\bar{1}0\} $ which is also known as the most stable dislocation in BCC Fe has been studied with C segregation. The carbon–dislocation interaction was simulated for the 1/2⟨111⟩ dislocation using the pairwise interatomic potentials for Fe–C pair proposed by Johnson21) and the Embedded Atom Method (EAM) potential.22,23) The Johnson and EAM interatomic potentials24,25) show reasonable results for pure metallic systems while the covalent bond in metal-nonmetal system is essential for Fe–C interaction. In addition, based on DFT calculations, the formation of strong covalent-like bonds between C and adjacent Fe was investigated for 1/2⟨111⟩ and ⟨100⟩ edge dislocations, thereby leading to formation of carbon–dislocation complexes.26) Therefore, the interatomic potential considering the characteristic of covalent bond is expected to be effective for the Fe–C system. The theoretical study of carbon–dislocation interactions might complement experimental studies where the detail of segregation sites of C in dislocation system has not been clearly obtained yet.
In order to confirm the universality of the relation between open space around C and its segregation energy, in this paper, the location of C in BCC Fe with some typical edge dislocation structures with Burgers vectors 1/2⟨111⟩, ⟨100⟩ and ⟨110⟩ is investigated by considering the relation between the stability of C and its local structures. Firstly, the dislocation energies are explored to figure out the stability of several configurations of edge dislocations. Secondly, the segregation energy of C is explored as a function of distance from dislocation core. After that, the dependence of segregation energy of C and its Voronoi volume formed by surrounding Fe atoms is analyzed. Finally, within the calculation framework using the same classical potential as in previous work,11) the segregation energy of C in different defect structures are compared to figure out the dominant location of C in α-Fe with the existence of different extended defects.
3. Results and Discussions
3.1 Stability of dislocation
Before determining the stability of edge dislocations the performance of Tersoff/ZBL potential is examined by considering the core structure calculated by DFT and various interatomic potentials calculations. In a previous paper,29) a cylindrical slab geometries of edges dislocation was constructed with dislocation core at the center of cylinder. All input files of DFT and classical force-field calculations were provided in the previous paper. In order to check the reliability of Tersoff/ZBL potential used in this work, the geometries of ⟨100⟩(010) dislocation are explored. After relaxation, a cluster of Fe atoms containing dislocation core is extracted and the distances of Fe atoms in the vicinity of dislocation core to the core are calculated by the following equation:
| \begin{equation}
\bar{d}_{\textit{Fe–core}} = \frac{d_{\textit{Fe–core}}}{\boldsymbol{a}_{\boldsymbol{Fe}}}
\end{equation}
| (1) |
The color map of $\bar{d}_{\textit{Fe–core}}$ in the edge dislocations are shown in Fig. 2. As we can see, the relative positions of Fe atoms around the dislocation core are almost unchanged by DFT and other interatomic potential calculations.30–32) Thus, our Tersoff/ZBL potential constructed for Fe–C system is reliable for reproducing the core structure at the level of DFT.
The supercell size of quadrupole dislocations is chosen by considering the change of the formation energy of the dislocation ΔEdislocation when increasing the size of the supercell. ΔEdislocation is estimated by following equation:
| \begin{equation}
\Delta E_{\textit{dislocation}} = \frac{E_{\textit{dislocation}}^{\textit{mFe}} - m\mu_{\textit{Fe}}}{4L},
\end{equation}
| (2) |
where
$E_{\textit{dislocation}}^{\textit{mFe}}$ is the total energy of the supercell containing 4 dislocations with
m Fe atoms and μ
Fe is the chemical potential of Fe atom in the perfect BCC structure. It is obtained after optimizing the supercell volume and atomic positions.
L is the length of dislocation line in the supercell. Since the quadrupole model has 4 dislocations in one supercell, we have a factor 1/4 in
eq. (2). The definition in
eq. (2) implies that, the smaller Δ
Edislocation, the more stable dislocation is. In this work, the stability of edge dislocations is figured out by comparing the dislocation energy Δ
Edislocation with supercell size dependence being taken into account. The size of supercell is gradually increased in both x and y directions which corresponds to increasing the distance of two adjacent dislocation cores (D). The dependence of Δ
Edislocation and D is illustrated in
Fig. 3. The dislocation energies of edge dislocations rise continuously with the increasing of the distance D. The size dependence of the dislocation energy shows logarithmic-like behavior which is consistent with the prediction of elastic theory, where the energy of an edge dislocation increases logarithmically as a function of distance from the dislocation core.
33) Among considered edge dislocations, it is found that 1/2⟨111⟩{110} edge dislocation has the smallest Δ
Edislocation and 1/2⟨111⟩{112} dislocation is the second smallest one. This is to say, the most stable dislocation is 1/2⟨111⟩{110} and the second stable dislocation is 1/2⟨111⟩{112}, which are consistent with the conclusion from experimental observation.
34,35) The dislocations with Burgers vector ⟨100⟩ and glide planes {010} and {011} are shown to be less stable than the ones with Burgers vector 1/2⟨111⟩, with a small energy difference. This might explain the transformation of 1/2⟨111⟩ dislocation loop to ⟨100⟩ dislocation in previous theoretical studies and experiment observation at high temperature.
36–38) In present dislocations, the Δ
Edislocation value of ⟨110⟩ dislocation is the highest one which is significantly different as compared to other dislocations, namely, it is the most unstable one from computational investigation. This result is consistent with the fact that this dislocation is not observed in experiments and known as unstable dislocation in contrast to the case of FCC metals. The order of stability obtained by the present calculations can be reasonably explained by the elastic theory which predicts that the dislocation energy is proportional to |
b|
2 (
Table 1).
3.2 Segregation of carbon
The reliability of our newly-developed potential was examined in cases of carbon segregation at vacancy and grain boundary in BCC iron.9–11) For the calculations of edge dislocation, this procedure is also considered by comparing the results calculated by our new potential to the results from previous DFT calculations. In a previous study, the interaction of C with ⟨100⟩(001) edge dislocation core was investigated using DFT calculation.39) Because the limitation of number of atoms in DFT a cluster of Fe containing dislocation core was extracted from the optimized configuration of ⟨100⟩(001) edge studied by using Finnis-Sinclair potential. Here, the structure of Fe cluster, as shown in Fig. 4, is divided into two regions: a compression region (CR) and an expansion region (ER). The black and white balls represent Fe atoms in two adjacent planes along ⟨001⟩ direction (or dislocation line). To determine the interaction energy between C and dislocation core, C is singly inserted into the center of plane A (CA, above dislocation core) and plane B (CB, under dislocation core), octahedral site in compression region (CR-O) and expansion region (ER-O), tetrahedral site in compression and expansion region (CR-T and ER-T, respectively). Similar to previous work, the periodic boundary condition is applied to the ⟨001⟩ direction while kept fixed along ⟨100⟩ and ⟨010⟩ directions. In addition, for the Fe–C system, some empirical interatomic potentials have already been implemented in LAMMPS. Therefore, to find out the advantages of our newly-developed Tersoff/ZBL potential, the calculations are also carried out by using other potentials, namely EAM40) and MEAM.41) In the previous study,39) the interaction of C and edge dislocation was discussed by using segregation energies. However, the definition of those segregation energies is different to our present study. To avoid confusion hereafter the segregation energies presented in Ref. 39) is denoted as dissolution energy Ediss and expressed as follows:39)
| \begin{equation}
E_{\textit{diss}} = \frac{E_{b}^{\textit{dop}} - E_{b}^{\textit{clean}}}{N}
\end{equation}
| (3) |
where
N is number of C atoms,
$E_{b}^{\textit{dop}}$ and
$E_{b}^{\textit{clean}}$ are binding energies of the C-doped system and clean system, respectively. The calculated dissolution energies
Ediss for different C sites are listed in
Table 2. By using Tersoff/ZBL potential, we can see that the most stable position of C around dislocation core is located in CB. This result is in good agreement with those obtained by other interatomic potentials and DFT. For the next stable site of C, our Tersoff/ZBL potential predicts ER-T site, which is also consistent with DFT calculation, while for EAM and MEAM potentials, the second most stable is found to be CA site. Overall, the values of
Ediss calculated from Tersoff/ZBL potential are higher than DFT results by at most 1.68 eV, which is much lower than those obtained by MEAM (2.87 eV) and EAM (6.02 eV). The small discrepancy implies a better performance of our new potential for this specific Fe–C system as compared to other interatomic potentials. This is also complemented by the fact that the trend of dissolution energy reproduced by our potential agrees very well with the DFT calculations as compared to others (see
Table 2).
In our paper, the difference between formation energy of C in defect configuration (X) and the one in perfect BCC Fe is indicated as segregation energy (Eseg),11) and it is calculated as,
| \begin{align}
E_{\textit{seg}} & = (E_{X}^{\textit{Fe}+C} - E_{X}^{\textit{Fe}} - \mu_{C}) - (E_{\textit{BCC}}^{\textit{Fe}+C} - E_{\textit{BCC}}^{\textit{Fe}} - \mu_{C})\\
& = (E_{X}^{\textit{Fe}+C} - E_{X}^{\textit{Fe}}) - (E_{\textit{BCC}}^{\textit{Fe}+C} - E_{\textit{BCC}}^{\textit{Fe}}),
\end{align}
| (4) |
where μ
C is the chemical potential of C, and
$E_{X}^{\textit{Fe} + C}$ and
$E_{X}^{\textit{Fe}}$ represent the total energies of the supercells containing defect X together with and without C atom, respectively.
$E_{\textit{BCC}}^{\textit{Fe} + C}$ and
$E_{\textit{BCC}}^{\textit{Fe}}$ are the total energies of perfect bulk Fe with and without C, respectively. For perfect BCC Fe, the size of supercell is set to 8 × 8 × 8 of the conventional BCC unit cell with the optimized lattice constant of
aFe = 2.8886 Å (obtained from the current Fe–C Tersoff/ZBL interatomic potential). It is noted that the
ab initio calculations have shown that the octahedral site (O-site) is the most preferred position for C interstitial in α-Fe.
9,10) Therefore, only O-site is considered as the interstitial site of C in the following calculations with dislocations. In carbon segregation calculations, C is inserted in dislocation configurations whose supercell sizes and number of Fe atoms are listed in
Table 3. Here, the supercell containing dislocations is constructed by multiplying the basic unit cell with size of
Lx ×
Ly ×
Lz to
N1Lx ×
N2Ly ×
N3Lz (
N1,
N2,
N3 are integers). The length of the dislocation line is about 1.7–2.1 nm (
N3Lz in
Table 3), which is enough to avoid C–C interaction when considering the periodic boundary conditions. In the following sub-sections, the segregation of C to the investigated dislocations will be discussed in more detail.
Table 3 Size of simulated supercell containing quadruple edge dislocations, segregation energy (Eseg) of C, Voronoi volume (V) around C and shortest Fe–C bond distance (dFe–C). m is number of Fe atoms in a simulation cell. Here, the supercell size is indicated by using size of basic unit cell of Lx, Ly, Lz.
Here, we consider the 1/2⟨111⟩{110} edge dislocation with Burgers vector 1/2⟨111⟩ and glide plane {110}. For this dislocation, the glide plane is normal to $\langle \bar{1}10\rangle $ direction and the dislocation line lies along $\langle 11\bar{2}\rangle $ direction. The dislocation model is created based on a basic unit cell of Lx × Ly × Lz (Fig. 5(a)). The dislocation system contains 151200 atoms.
The O-sites in α-Fe are located at the middle of edges or the centre of faces of BCC lattice which could be in ⟨100⟩, ⟨010⟩, or ⟨001⟩ directions corresponding to TDA(100), TDA(010), and TDA(001) sites (Fig. 5(a)). Here, TDA is an abbreviation for “tetragonal distortion axis” which indicates the largest distortion of lattice due to the insertion of C.17,22) The position of C in different layers along with $\langle 11\bar{\mathbf{2}}\rangle $ direction is shown in Fig. 5(b). In Fig. 5(b), the Voronoi volume of Fe atoms around dislocation core is illustrated using a colour map and positions of C under consideration at different layers from the glide plane are shown. The value of Voronoi volume increases as the colour changes from red to blue. As we can see, the area above the dislocation core is more compressed while the area below the dislocation core is more expanded. The compression and expansion are reduced by moving away from the dislocation core. The C in the TDA(010) and TDA(100) located along 0.5 and 1.5 layers from the glide plane while the ones in the TDA(100) are right on the glide plane or along 1.0 and 2.0 layers.
In Fig. 6, the segregation energies of C along TDA(001) and TDA(010) as functions of distance from C to the dislocation core (in unit of aFe) are shown. Results for the C segregation in TDA(001) are presented in Fig. 6(a) and those in TDA(100) and TDA(010) are presented in Fig. 6(b). It can be seen that the C segregation in TDA(001) is highly symmetric for both sides of the dislocation core (Fig. 6(a)). While, the asymmetry for C segregation is shown in Fig. 6(b) for the left- and right-side of the dislocation core, which can be explained by considering the local atomic structure of iron. As shown in Fig. 5(a) the octahedral geometry of the TDA(001) sites is symmetric along Burgers vector ⟨111⟩, in contrast to those of the TDA(010) and the TDA(100) sites. This mainly contributes to the asymmetry of the local structures on two sides of the plane which is normal to the glide plane and contains the dislocation line. In both cases, the interaction between carbon and dislocation is weaker when C is located at a distance layer along $\langle 11\bar{\mathbf{2}}\rangle $ direction from the glide plane. Based on the calculated segregation energy, the strong interaction range between C and dislocation is within 5aFe (about 1.44 nm) for TDA(001) and 10aFe (about 2.89 nm) for TDA(010). When C atom is located far from the dislocation core, the Eseg values approach to that of C in bulk BCC Fe. Considering the Eseg of C in different layers from the glide plane, interaction of C and dislocation is weaker in the layer which is distant from the glide plane. As shown in Fig. 6, C lying along the TDA(010) shows the lowest segregation energy of −1.11 eV at the dislocation core, which is lower than those obtained from Johnson and EAM potentials.17,22) This is because of the different dislocation model and interatomic potential used in the present study.
The 1/2⟨111⟩{112} dislocation is characterized by Burgers vector 1/2⟨111⟩ and glide plane {112} containing 180000 Fe atoms in calculation of C segregation. The positions of C at the O-sites are also considered along the TDA(100), TDA(010) and TDA(001) as in previous case. The calculated position-dependence of C segregation energy is plotted in Fig. 7. The figure shows the interaction between dislocation and C located along the TDA(001) (Fig. 7(a)). The interaction is repulsive when C is on the right side of the dislocation line (⟨110⟩ direction) and attractive if C is on the left side of the dislocation line. In contrast, the C lying along the TDA(010) and TDA(100) (Fig. 7(b)) does not prefer to locate on the left side of the dislocation line which is qualitatively in agreement with previous study.17) The strong interaction range of C and 1/2⟨111⟩{112} dislocation is within 10aFe (about 2.89 nm) for both TDA(001) and TDA(010). As shown in Fig. 7(b), C belonging to the TDA(100) shows the lowest segregation energy of −1.35 eV at the dislocation core. This energy is lower than the case of the 1/2⟨111⟩{110} dislocation, which is also observed in previous calculations.17,22)
The supercell containing ⟨100⟩{010} dislocation is constructed from the conventional BCC unit cell of Fe with Burgers vector ⟨100⟩ and glide plane {010}, including 115200 Fe atoms. For this edge dislocation, the positions of C at O-sites are symmetric along the Burgers vector ⟨100⟩ and equivalent along TDA(010), TDA(001) and TDA(100), leading to the symmetrical dependence of segregation energy of C with respect to the distance between C and the dislocation core (Fig. 8(a)). In this dislocation, C is localized in dislocation core area with an interaction range about of 4aFe (1.15 nm). The lowest segregation energy of C, which is −1.77 eV, is obtained when C locates at the dislocation core. The maximum absolute value of Eseg is larger than that in the cases of 1/2⟨111⟩ dislocations.
The ⟨100⟩{011} dislocation with Burgers vector ⟨100⟩ and glide plane (011) which contains 160000 atoms is considered. The segregation energy of C in this dislocation is shown in Fig. 8(b). It is found that C atom is localized more strongly in the ⟨100⟩{011} dislocation core as compared to the one in ⟨100⟩{010} dislocation, with a segregation energy of 2.00 eV. The interaction range between C and dislocation in the cases of ⟨100⟩ dislocations is around 5aFe (1.44 nm) which is shorter than that in the cases of 1/2⟨111⟩ dislocations.
The supercell of ⟨110⟩{110} dislocation, characterized by ⟨110⟩ Burgers vector {110} glide plane, including 226800 Fe atoms. Here, the most unstable dislocation structure is very complex. Thus, the interaction of C with the most unstable ⟨110⟩{110} dislocation is studied with C located at dislocation core which is known as the favorable location of C. The lowest segregation energy of −2.57 eV is obtained and shown much lower than in others edge dislocations (Table 3).
3.3 Local atomic structure and carbon segregation in extended defects
To figure out the relation between the local atomic structure around C in the dislocation structure and the segregation tendency, the Eseg of C for the cases of 1/2⟨111⟩, ⟨100⟩ and ⟨110⟩ dislocations is plotted as a function of Voronoi volume formed by Fe atoms around C atom in Fig. 9. It is shown that there is an anti-correlation relation between segregation energy of C and its Voronoi volume. A similar tendency was discovered for the case of grain boundaries.11) In Table 3, the Eseg of C at the most stable positions in different dislocation configurations and two indicators for the geometry of local atomic structure, namely the Voronoi volume (V) and the shortest Fe–C bond length (dFe–C). It is found that, in most cases, the most stable segregation site has the largest Voronoi volume and the largest dFe–C which was also figured out in the case of grain boundaries. For ⟨100⟩{010} dislocation, although the dFe–C are much longer than the one in other dislocation cases, the interaction of C is weaker than in ⟨100⟩{011} and ⟨110⟩{011} edge dislocations. It can be explained that the distances of Fe and C are slightly longer than the equilibrium Fe–C bond length which lead to the weak interaction of Fe and C, similar to the case of grain boundaries.9,11) Here, equilibrium Fe–C bond length is the shortest bond length of Fe and C in O-site in BCC Fe (1.790 Å). Besides, in addition to the Voronoi volume and the shortest Fe–C bond length, the shape of the Voronoi volume formed by neighbour Fe atoms might contribute to the strength of Fe–C interaction.
In order to compare the strength of interactions between C and defects, the segregation energy of C with different kinds of defect structures is summarized in Fig. 10. In the figure, not only the present results of the Eseg of the dislocations systems but also the previous results11) on the Eseg of grain boundaries and single Fe vacancy are also summarized. As compared to the single vacancy case, the Eseg of C in the cases of STGBs and dislocations are more negative, namely, the later defects are more attractive to C than the vacancy. This order is partly consistent with previous DFT calculation of segregation energy of C in bulk Fe with a single vacancy (−0.44 eV), stable ∑3(112) STGB (−0.67 eV).42,43) The strongest interaction of C with GB is found to be −1.78 eV for the unstable GB ∑9(221), which is lower than the one in 1/2⟨111⟩ edge dislocations. The interaction of C with the stable dislocations 1/2⟨111⟩ is weaker than the unstable ⟨100⟩ and ⟨110⟩ dislocations. The interaction of C with the unstable dislocations is stronger than the most unstable ∑9(221) STGBs, in contrast to the stable 1/2⟨111⟩ dislocation. The interaction range of C and grain boundaries (0.8 nm) is much shorter than in dislocations (2.89 nm).11) It is known that the segregation energy is related to the solubility of C in Fe matrix. Therefore, it can be concluded that the higher solubility can be obtained by BCC Fe matrix which contains unstable STGBs and dislocations structures.44) The obtained results might provide useful instruction for experimental studies for controlling the solubility of C through manipulating different defects.
