Potentials for Describing Interatomic Interactions in γFe-Mn-C-N System

Abstract

Potentials for describing interatomic interactions in a γFe-Mn-C-N multicomponent system, modified Hadfield steel, where face-centered cubic (f.c.c.) iron is the main component, are proposed. To describe the Fe-Fe interactions in austenite, it is proposed to use Lau EAM potential. For all other interactions, Morse potentials are proposed, the parameters of which were found from various experimental characteristics: in particular, the energy of dissolution and migration of an impurity in an f.c.c. iron crystal, the radius of atoms, their electronegativity, mutual binding energy, etc. The found potentials are intended for modeling the atomic structures and processes occurring at the atomic level in Hadfield steel using relatively large computational cells by the molecular dynamics method.

1. Introduction

Despite the interest in Hadfield steel and a long history of research on its unique properties, mainly associated with its excellent work hardening ability [1,2,3], there are very few works devoted to the modeling of its atomic structures and the processes occurring in it under deformation conditions at the atomic level. At the same time, computer modeling can help answer a number of questions regarding the mechanisms and regularities of the deformation behavior of this steel, which relates to the peculiarities of the interaction of dislocations with twins and grain boundaries.

The present work is devoted to the search for potentials for describing interatomic interactions in Hadfield steel for their subsequent application in molecular dynamics simulation. Hadfield steel is a multicomponent system and, in addition to classical iron, manganese, and carbon, it may contain some other alloying elements [1,3]. To describe interatomic interactions even in the three-component system, it is necessary, at least, to specify six potentials (or pair components for many-body potentials) to describe the interactions of different pairs of atoms in a given system: Fe-Fe, Fe-Mn, Mn-Mn, Fe-C, Mn-C, C-C. Unfortunately, there are very few works, within the framework of one approach, where potentials are proposed simultaneously for several of these bonds.

Lee et al., developed modified EAM potentials that take into account two coordination spheres and anisotropy of electron clouds (2N MEAM potentials). Despite the physical validity, the Lee potentials [4,5,6,7] have two disadvantages. Firstly, these potentials are relatively very cumbersome. They include at least 11 parameters and additional calculations of the mutual orientation of atoms, which is used to calculate the screening functions of electron clouds. This significantly slows down the calculation of a computer experiment and is critical for models that include a large number of atoms. Secondly, while strictly taking into account many factors that affect interatomic interactions, the 2N MEAM Lee potentials take into account the interaction of atoms no further than only the 2nd coordination sphere. This is not enough according to our previous work [8]. There are only six atoms in the second coordination sphere of an f.c.c. lattice, at a distance of $\sqrt{3/2}a$, where a is the lattice parameter. There are 24 atoms in the third sphere at a distance of $\sqrt{2}a$, i.e., only 0.19a beyond the 2nd coordination sphere. According to the calculations previously made [8], the third coordination sphere contributes about 20% to the calculation of atomic energy depending on the metals, and the second even less, about 15%. Xie et al. [9] proposed a set of Morse pair potentials for describing all the desired interactions, except for the Fe-Mn bond. However, they are not suitable for modeling interactions in Hadfield steel, since they were selected not for steels, but for Mn₇C₃ and Fe₇C₃ carbides. For example, when using one potential for the Fe-Fe bond [9], it was not possible to obtain satisfactory results for the basic characteristics of f.c.c. iron. In the present paper, we propose a set of Morse potentials to describe all interactions in the four-component γFe-Mn-C-N system.

2. Choice of Potential for Describing Interatomic Interactions in F.C.C. Iron

In view of the foregoing, we decided to independently select the potentials for the γFe-Mn-C system, which must satisfy two important criteria: (1) not be mathematically cumbersome, so that relatively large computational cells can be simulated (including 10⁵–10⁶ atoms); (2) satisfactorily describe the known experimental characteristics of the γFe-Mn-C system. The potential for describing the Fe-Fe bond is a main potential in the γFe-Mn-C system. As a basis, it was decided to take one of the known and tested EAM potentials that well describe the properties of austenite. Most of the known potentials for iron were created to describe its body-centered cubic (b.c.c.) modification, and in the majority of cases, the f.c.c. phase is considered as secondary. As a result, not all known potentials describe austenite well. For this reason, we abandoned the well-known EAM potential of Ackland and Mendelev [10,11,12].

Of all the potentials that we have examined, the potentials for Fe-Fe and Fe-C bonds in austenite proposed by Lau et al. [13] are best suited for our purposes. These are classic EAM potentials, where the energy of the i-th atom is calculated as the sum of the pair and multiparticle components:

$$\begin{array}{c}{E}_{\alpha ,i}=-A_{\alpha }\sqrt{{\displaystyle \sum }_{j\ne i}{\rho }_{\beta \alpha }\left(r_{ij}\right)}+\frac{1}{2}{\displaystyle {\displaystyle \sum }_{j\ne i}}{\varphi }_{\beta \alpha }\left(r_{ij}\right)\\ {\rho }_{\beta \alpha }\left(r_{ij}\right)=t_1{\left(r-r_{c,\rho }\right)}^2+t_2{\left(r-r_{c,\rho }\right)}^3,\hspace{1em}\hspace{1em}r \le r_{c,\rho }\\ {\varphi }_{\beta \alpha }\left(r_{ij}\right)={\left(r-r_{c,\varphi }\right)}^2\left(k_1+k_{2}r+k_3{r}^2\right),\hspace{1em}\hspace{1em}r \le r_{c,\varphi }\end{array}$$

(1)

The potential for the Fe-Fe bond in the article of Lau et al. [13] is a modified potential from the work of Rosato [14]. We drew attention to this potential primarily because it was tested to describe the structural, energy, and elastic characteristics of austenite [15]. The obtained values are in satisfactory agreement with the experiment: the lattice parameter at 0 K is 3.573 Å, the sublimation energy (per atom) is 4.228 eV, and the bulk modulus is 156.7 GPa. In addition, good agreement with the reference value of thermal expansion coefficient of 16∙10⁻⁶ K⁻¹ was obtained.

3. Fe-C, C-C, Fe-Mn, Mn-Mn, and Mn-C Bonds in the Crystal Lattice of F.C.C. Iron

An adjustment was necessary for the Fe-C bond, since such values as the energy of carbon dissolution in f.c.c. iron; the difference between energies of carbon impurity in octahedral and tetrahedral voids of the lattice; the migration energy of a carbon atom in f.c.c. iron; and the binding energy with a vacancy were unsatisfactorily described by potentials proposed by Lau et al. [13]. It was decided for all the other five bonds to build pair Morse potentials that would satisfactorily describe the known characteristics. The Morse potential is a pair potential, but it is often used in molecular dynamics calculations, including the description of interatomic interactions in metals. Pair potentials are often used by various researchers to describe interatomic interactions in metal-impurity systems [9,16,17,18,19,20,21,22,23]. The procedure for calculating forces using many-body and pair potentials in a computer program is fundamentally different, and therefore, the latter (with the same methods of optimizing the computer code) always wins in the speed of calculating forces in the model. The Morse potential determines the interaction energy of a pair of atoms located at a distance r from each other:

$$\phi \left(r\right)=D\beta e^{-\alpha r}\left(\beta e^{-\alpha r}-2\right)$$

(2)

where α, β, D are the potential parameters.

When determining the parameters of the Morse potential for the Fe-C bond, we relied on the already known potentials from [13,24], only slightly correcting them to obtain good agreement with the experimental data and first-principle calculations on the energy of carbon dissolution in f.c.c. iron (E_sol); the difference between the energies of carbon impurity in octahedral and tetrahedral voids of the lattice (ΔE_OT); the migration energy of a carbon atom in f.c.c. iron (E_m); and the binding energy with a vacancy (E_bv). These energy characteristics are listed in

Table 1. Energy characteristics of carbon impurity in f.c.c. iron.

Energy Characteristics	Our Model	Experiments or FP-Calc.	Potentials from [24]	Potentials from [13]
Esol (eV/atom)	0.38	0.25–0.48 [25,26] (experiment, FP-calc.)	1.01	0.78
ΔEOT (eV)	1.36	1.48 [27] (FP-calc.)	1.03	1.12
Em (eV)	1.20	1.40–1.53 [4,28,29] (experiment)	0.33	0.86
Ebv (eV)	0.41	0.37–0.41 [30] (experiment)	0.50	0.54

. In addition, the radius of the Fe and C atoms was also taken into account. As can be seen, the potentials proposed here simultaneously well describe the considered energy characteristics of the carbon impurity in austenite, even better than the known EAM potentials for this system [13,24].

To describe the interaction between carbon atoms in an austenite lattice, we transformed the potentials [16,24] into Morse potentials. The justification given in these works seemed quite reasonable. When searching for parameters for the Mn-Mn bond, we focused on the radius of Mn atoms, the sublimation energy, and the bulk modulus of metallic Mn [31]. The equilibrium distance for the Fe-Mn bond was calculated as the sum of the radii of Fe and Mn atoms [31], the elastic modulus, as the mean of the modules for Fe and Mn. The depth of the potential, i.e., the parameter D in Formula (2), was determined by the value of the mixing energy of Mn in f.c.c. Fe, where it was calculated by first principles (ab initio) method [32]. For a concentration of 13 wt.% Mn, the mixing energy of Mn in Fe was −0.0182 eV/atom [32]. A minus sign indicates that mixing is energetically favorable, and a relatively small value refers that the tendency to order the atomic structure of the alloy is very weak. The parameter D was selected so that this value was exactly obtained.

The last of the six necessary potentials, for the Mn-C bond, was chosen, taking into account the potentials proposed in [5,7,9], the radii of the Mn and C atoms [31], as well as the binding energy between Mn and C atoms in f.c.c. Fe. The potential parameters were selected in such a way as to accurately reproduce the binding energy of Mn and C atoms experimentally obtained in a f.c.c. Fe lattice, 0.35 eV [33]. This energy seems very large and comparable with the binding energy of a carbon atom with a vacancy in iron. However, if we pay attention to the difference in electronegativities of Mn and C atoms compared to Fe and C, 1.00 and 0.72, respectively [31], such a strong bond becomes clear, which is comparable in strength, probably, to an Al-C bond, for which the difference in electronegativities is close to the Mn-C bond and amounts to 0.94 [31].

When calculating the binding energy of a C atom with an Mn atom in a f.c.c. lattice of Fe, the carbon atom was introduced into the octahedral interstitial closest to the Mn atom, as shown in Figure 1. It is known that impurity atoms of light elements (such as C, N, O, etc.) occupy octahedral voids in f.c.c., h.c.p., and b.c.c. lattices of metals [20,34].

4. Interatomic Potentials for Describing Nitrogen Impurity in Hadfield Steel

Nitrogen impurity is not a classic component of Hadfield steel. At the same time, it is sometimes considered as an additional alloying element that stabilizes austenite and increases strength [1]. The radius of N atom is slightly less than that of C atom: 0.71 and 0.77 Å, respectively [34]. The difference between the electronegativities of iron and nitrogen is 1.21, while for iron and carbon, it is 0.72 [31]. The high difference between the electronegativity means a strong interatomic bond, and its character is close to the covalent type.

The dissolution energy of nitrogen in f.c.c. iron is an important characteristic by which the depth of the interatomic potential can be determined. Unfortunately, there is a very large run-off of this value in the literature: −0.13 eV [35], −0.27 eV [36], and −0.53 eV [37]. Another important energy characteristic is the activation energy of migration of an impurity atom in the metal lattice. Based on the analysis of data on the migration energy of C and N impurity atoms in α-Fe and γ-Fe [6,28,29,38,39], we took a benchmark value of migration energy of nitrogen impurity in the f.c.c. iron lattice of 1.29 eV. Therefore, we were guided by the radius of iron and nitrogen atoms, the energy of dissolution, the energy of migration, and the assumed depth of the potential found from the empirical dependencies [20].

Table 2. Some energy characteristics of nitrogen impurity in f.c.c. iron.

Energy Characteristics	Our Model	Experiment or FP-Calc.
Esol (eV/atom)	−0.13	−0.13 [35] (experiment) −0.27 [36] (ab initio calculations) −0.53 [37] (thermodynamic evaluation)
Em (eV)	1.29	1.29 [6,28,29,38,39]
Ebv (eV)	0.45	−

shows the data by which we were guided when selecting the potentials and the values obtained in our work.

Mn atoms have a high binding energy with N atoms and a large difference in electronegativities compared to iron atoms, which causes a high effect of Mn impurity on the solubility of nitrogen in iron [40]. When searching for the potential parameters for the Mn-N bond in f.c.c. iron, we took into account the radius of Mn and N atoms, empirical regularities [20], and the data on the solubility of Mn and N in iron [40]. To describe the interactions of nitrogen atoms with each other in metals, the considerations and pair potentials proposed by Vashishta et al. [17] and San et al. [19] were taken as a basis. According to these potentials, impurity atoms, such as nitrogen and oxygen, repel each other in the crystal lattice of metals. To describe the N-N and O-O bonds in metals, the parameters of the Morse potentials were calculated [20]. The radius of atoms and their electronegativity were also taken into account to determine the parameters of the C-N potential. The binding energy of C and N atoms in the metal lattice was calculated using Lorentz–Berthelot mixing rules [41], taking into account the ratio of the C-C bond energies in the metal and graphene, as well as the C-N bond in the organic molecule.

5. Potential Parameters

Table 3. Parameters of Morse potentials for the considered interactions.

Bond	α (Å−1)	β	D (эB)
Fe-Fe	1.285	35.878	0.433
Fe-C	1.82	41	0.41
C-C	1.97	50	0.65
Mn-Mn	1.321	39.792	0.373
Mn-Fe	1.306	38.030	0.413
Mn-C	1.87	43	0.777
Fe-N	1.788	34.046	0.579
Mn-N	1.812	36.482	0.940
C-N	2.140	58.323	0.230
N-N	1.556	700	0.001

shows the obtained parameters of Morse potentials. Figure 2 illustrates graphs of the found potentials. Table 3 also contains the parameters for the Fe-Fe bond, found from the values of the γ-Fe lattice parameter, sublimation energy, and bulk modulus obtained using the potential from [13]. The Morse potential for the Fe-Fe bond is a simpler analogue of the EAM potential [13] and similarly well describes the above characteristics, including thermal expansion. It can be used in models with a large number of atoms and in lengthy computer experiments requiring large resources. All Morse potentials were found for a cutoff radius of 4.7 Å, i.e., taking into account the three coordination spheres in f.c.c. Fe.

For the Fe-Fe and Mn-Mn bonds, the classical method by Girifalco and Weiser [42] was used to determine the potential parameters from the crystal lattice parameter, bulk modulus, and sublimation energy, so these characteristics are reproduced quite accurately. For other bonds describing the interaction of C, Mn, and N impurities with atoms of the γ-Fe lattice and with each other, the potential parameters were chosen in such a way as to reproduce, as best as possible, above-mentioned experimental characteristics: in particular, the energy of dissolution and the energy of migration of an impurity in an f.c.c. iron crystal, the radius of atoms, their electronegativity, mutual binding energy, etc.

It should be borne in mind that in complex multicomponent systems such as γFe-Mn-C-N, the bonds of impurity atoms depend on the mutual arrangement of neighboring atoms of different types in the austenite lattice space. However, strict consideration of this in our case, when the system consists of four components, will lead to a significant complication in calculations. It will be not allowed to perform molecular dynamics simulations using relatively large computational cells.

6. Conclusions

Potentials for describing interatomic interactions in a γFe-Mn-C-N multicomponent system, modified Hadfield steel, where f.c.c. iron is the main component, are proposed. To describe the Fe-Fe interactions in austenite, it is proposed to use Lau EAM potential, which reproduces well the structural, energy, and elastic characteristics of austenite. For all other nine interactions, Morse potentials are proposed, the parameters of which were found from various experimental characteristics: in particular, the energy of dissolution and the energy of migration of an impurity in a f.c.c. iron crystal, the radius of atoms, their electronegativity, mutual binding energy, etc. The found potentials are intended for modeling the atomic structure and processes occurring at the atomic level in Hadfield steel using relatively large computational cells by the molecular dynamics method.