Atomistic simulations in the Fe C system


(1)Computational Materials Science 45 (2009) 550–560. Contents lists available at ScienceDirect. Computational Materials Science journal homepage: Atomistic simulations in the Fe–C system Margarita Ruda a,*, Diana Farkas b, Griselda Garcia c a b c. Centro Atómico Bariloche and Universidad N. Comahue, 8400 Bariloche, Argentina Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, United States Facultad de Física, Universidad Católica de Chile, Casilla 306, Santiago 1, Chile. a r t i c l e. i n f o. Article history: Received 6 September 2008 Received in revised form 20 November 2008 Accepted 28 November 2008 Available online 20 January 2009 PACS: 61.72.y Keywords: Simulation EAM method Fe–C Defects Ferrite Cementite. a b s t r a c t We improved an embedded atom (EAM) interatomic potential for the description of defective body-centered cubic a-Fe containing C interstitials. Guided by first principles calculations results the potential was developed to reproduce experimental information such as the dilute heat of solution of carbon, the vacancy-carbon binding energy and its configuration, the location of interstitial carbon atoms and the migration energy of carbon atoms in body-centered cubic (bcc) Fe. The potential reproduces the known physical properties of carbon as an interstitial solute element both in ferrite and austenite. It also successfully calculates energetically favored defect configurations and predicts the formation energy and configurations of multicarbon-vacancy defect clusters. The potential performs well for the atomistic study of the interactions between carbon interstitial solute atoms and extended defects in ferrite, such as free surfaces, dislocations and grain boundaries. Since C–C interactions are taken into account, this potential is applicable to the study of cementite and we present results for a ferrite/cementite interface. Ó 2008 Elsevier B.V. All rights reserved.. 1. Introduction Carbon is one of the most important alloying elements in steel and many different properties originate from the strong interactions between interstitial carbon atoms and various defects such as vacancies, grain boundaries and dislocations. The fundamental physics that govern interactions among interstitial atoms and other defects in the bcc Fe lattice play a critical role in the possibility of designing steels with improved properties. These fundamental phenomena at the atomic level can in some cases be understood using first principles simulations based on density functional theory (DFT). These calculations provide accurate descriptions of binding energies and structure among the atoms in defect clusters (see, for example, Refs. [1–8]), but the technique is limited to a relatively small number of atoms. Empirical potentials are required to study mechanical behavior of steels in digital samples of sizes large enough to extract meaningful macroscopic behavior properties and therefore be useful in materials design. Several efforts in the development of such potentials have been reported in the literature. Johnson et al. [9] con* Corresponding author. Tel.: +54 2944 445278; fax: +54 2944 445299. E-mail addresses: (M. Ruda), (D. Farkas), (G. Garcia). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.11.020. structed a pair potential for this system. Rosato [10] reported potentials for this system using a Finnis–Sinclair type potential for Fe and a Fe–C pair potential. These two potentials neglect C–C interactions. Ruda et al. [11] developed a many-body embedded atom method potential (EAM) that includes C–C interactions, using a description of the C–C bonds that was based on the Tersoff’s C potential [12,13]. While this potential could be used for somewhat increased carbon concentrations [14,15], it wrongly predicts that C occupies tetrahedral sites of a bcc Fe lattice. A modified embedded atom method (MEAM) interatomic potential for the Fe–C binary system has been developed using previous MEAM potentials of Fe and C [16]. More recently Lau et al. [17] have developed a many-body potential of the Finnis–Sinclair formalism using Rosato’s potential for the Fe–Fe interaction [10]. All of these efforts have been based on Fe–Fe potentials that present limitations in their description of two important defects: the core structure of the ½h1 1 1i screw dislocation, and the configuration of the self interstitial. Simultaneously with the efforts in the development of mixed Fe–C potentials, significant progress has been made in the improvement of the description of pure Fe using empirical potentials. In particular the Fe potentials developed by Mendelev et al. [18] and slightly modified in Ackland et al. [19] represent a significant improvement over previous potentials in that they correctly.

(2) 551. M. Ruda et al. / Computational Materials Science 45 (2009) 550–560. predict the configuration of self interstitials as well as the correct dislocation core structure for the ½h1 1 1i screw dislocation. These potentials predict a dislocation core structure that is in agreement with first principles methods. This is of great importance in studies of deformation behavior for many practical applications. For applications under irradiation conditions it is also important to be able to use Fe–C potentials that correctly predict the self interstitial configuration in bcc Fe. For many other applications it is critical to be able to describe the interaction of C with extended defects. For example, Tapasa et al. [20–22] have recently performed simulations of C in Fe using Ackland et al. [19] EAM potential to describe the Fe–Fe interaction while retaining Johnson’s pair potential [9] for the Fe–C interaction. The third important area where significant progress has been made is in first principles calculations of various configurations in the Fe–C system. These results enable the development of more accurate potentials and the validation of the constructed functions for situations that are far from equilibrium. First principles simulations have addressed the interaction of C with vacancies and divacancies [1–5]. These simulations provide reliable relative energies of various configurations of C interstitial and vacancies. More accurate interatomic potentials can be developed by requiring that the most stable configurations of C and vacancy complexes be reproduced. Furthermore, first principles calculations can be used to develop intuition and basic understanding of the physics of the C–Fe bonding in the various structures. This understanding can be used as guidance in the development of more accurate potentials. Becquart et al. [23] recently developed a set of EAM potentials for the Fe system with small C concentration by using the Ackland et al. [19] EAM potential to describe the Fe–Fe interaction while fitting the Fe–C pair potential, the C density and the C embedding function to an ab initio data base. They considered no C–C pair interactions. In our own research group, some time ago we developed a first EAM C–Fe interaction potential [11], using an Fe potential developed by Simonelli et al. [24]. The mixed potential was developed as a linear combination of the pair interaction potential for pure Fe that Simonelli et al. developed and a C–C interaction that was derived from Tersoff’s [12,13] C–C potential without the angular term. It has been shown [25] that if the angular dependence is removed from Tersoff’s formalism, a Tersoff’s potential could be considered equivalent to an EAM potential. In our previous work, we fitted the coefficients of the linear combination to the dilute heat of solution of C in bcc Fe and to the lattice parameter and cohesive energy of unstable FeC NaCl-type obtained by first principles calculations available in the literature at that time. That potential was developed with no other input or guidance from first principles calculations and even though it provided a general description of the Fe–C bond, and it correctly predicted the formation of tetragonal martensite, it failed to predict the correct interstitial site in the bcc lattice. The basic issue in that work is that the tetrahedral site in bcc was predicted stable, largely due to the presence of two Fe atoms very close to the C in the octahedral configuration. The octahedral site is experimentally observed to be the lowest energy and in order to develop an improved potential, in this paper we seek basic guidance from first principles calculations as to the reasons. for the stability of the octahedral site despite the short bond lengths present. We present here a new Fe–C potential based on the combination of the more accurate Fe potential II developed by Mendelev et al. [18] and the C potential from Tersoff [12,13]. The potential is constructed using the guidance of first principles calculations and it is here applied to a variety of defect configurations. The paper is organized as follows: Section 2 reports first principles calculations that bring insight on the nature of the C–Fe bonding both in O and T sites of bcc Fe. Section 3 describes the new EAM potential and in the following sections we present and discuss the results of using the new potential in simulations. Section 4 deals with different point defect configurations involving C interstitials and vacancies in ferrite and C interstitials in austenite. Section 5 shows the application of the potential to extended defects in bcc Fe. In Section 6 the results of the new potential applied to cementite and its free surfaces are discussed and in Section 7 we present results for the structure and energetics of a simulated ferrite/cementite interface. 2. First principles calculations In this section, we present first principles calculations of C interstitial in bcc Fe in octahedral and tetrahedral sites that are used as guidance in the development of our potentials. Initial supercells with C in the octahedral and tetrahedral sites of the bcc lattice were constructed with 16 Fe atoms and 1 C atom in the center of each cell. The unit cell parameter was 5.74 Å, which is twice the value of the experimental lattice constant of pure airon. Relaxed positions of the atoms were determined from an EAM calculation performed with our previous Fe–C potential [11]. The C atom was either at the octahedral (O) or the tetrahedral (T) site, so we could compare both sites in terms of electronic density distribution and total energies. Table 1 shows the calculated distances between the C interstitial and all the neighboring atoms in the relaxed cell for both the octahedral and tetrahedral sites. The calculations were performed using full-potential linearized augmented plane waves method (LAPW), as implemented in the Wien2K program [26], within density functional theory. The electron exchange-correlation energy is described in the generalized gradient approximation in the parametrization of Perdew et al. [27,28]. The LAPW wave functions within the muffin-tin spheres were expanded in spherical harmonics with angular momenta up to l = 10 and the potential and charge density were expanded up to l = 4. The number of the augmented plane plane waves included was about 2000 per atom. A plane wave cut-off parameter of about 44 Ry and a set of 75 k-points in the irreducible wedge of the Brillouin zone (BZ) were chosen throughout the calculations. We chose a muffin-tin radius of 2.2 and 1.2 a.u. for iron and carbon atoms, respectively. Fe 1s, 2s, 2p were treated as core states, the shallow core states 3s and 3p were treated as band states by using the so-called local orbitals, and the 3d, 4s and 4p were treated as band states. For C, just the 1s was treated as a core state; the 2s was treated with a local orbital, and the 2p as a band state. The convergence was assumed when the average root-mean-square difference between the input and output charge density was less than 1  104 e/(a.u.)3.. Table 1 Distances between the C interstitial and the Fe neighbors in both octahedral and tetrahedral sites, as given by the potentials of Ref. [11]. Octahedral site. Fe Fe Fe Fe. (1) (2) (3) (4). Tetrahedral site. Distance (nm). Number of neighbors. Distance (nm). Number of neighbors. 0.174 0.206 0.320 0.356. 2 4 8 2. 0.183 0.250 0.330. 4 4 8.

(3) 552. M. Ruda et al. / Computational Materials Science 45 (2009) 550–560. [010]. [001]. [100]. [100]. Fig. 1. Electronic density maps of bcc Fe with C in T site projected onto two different planes of the unit cell. The contours are equally spaced between 0 and 1 electrons per Å3. C is in the center of the figure.. C in the octahedral site was found to be the configuration with lowest energy and our calculated energy difference from an O site to a T position was 0.978 eV. This is slightly larger than the experimental value [29] for migration of 0.87 eV. Other first principles calculations of the same energy barrier DE [(T)  (O)] were performed by Jiang and Carter [2] and by Domain et al. [1]; they obtained values of 0.86 eV and 0.902 eV, respectively, using supercells of 128 Fe atoms and 1 C atom. Domain et al. [1] also calculated the migration energy to be 0.923 eV with a smaller supercell of 54 Fe atoms and 1 C atom. The electronic densities in real space projected onto two planes of the unit cell for the T site and for the O site are shown in Figs. 1 and 2, respectively. From a close inspection of the electronic densities around the C atom in both T and O sites we found that the electronic density is higher between C and the two closer Fe neighbors in the O site. The T site is regular with four Fe atoms surrounding the C atom at the same distance, 1.87 Å while the O site is not regular and there are two Fe atoms that are closer to the C than the other four Fe atoms, 1.76 Å and 2.06 Å, respectively. The electron density map in Fig. 2 shows that despite these short distances, most of the bonding is directional between the carbon and the two atoms that are at the shortest distances. Therefore, the length of the C–Fe bond in these configurations should be around 1.8 Å. The bond is more attractive at this distance that at the C–Fe distance for the other neighbors in the O site and also for the C–Fe distance in the T site. These results provide guidance for developing an empirical potential and indicate that the minimum of the Fe–C pair function. [001]. [100]. should be located around a distance of 1.8 Å. The location of the minimum at these shorter distances should result in the O site as more energetically favorable than the T site.. 3. Development of the EAM potentials The embedded atom method (EAM) [30] represents the total energy of a block of atoms by the following expression:. X 1X  iÞ Vðr ij Þ þ Fðq 2 i;j i X q i ¼ qðri;j Þ E¼. ð1Þ. i. where V(rij) is the interatomic potential, rij is the distance between atoms i and j, Fðqi Þ is the embedding energy as a function of the host electronic density induced at site i by all the other atoms in the system. In order to ensure compatibility in ternary systems, EAM functions for monoatomic systems may be transformed to a normalized form in which the embedding function has null first derivative at the equilibrium electron density of the perfect lattice, q0. This is known as the effective pair approach, and the effective interatomic potential Veff controls the energy change of a block of atoms when the electronic density at an atom site is not significantly altered [31]. The transformation of the potentials to effective pairs is as follows:. [010]. [100]. Fig. 2. Electronic density maps of bcc Fe with C in O site projected onto two different planes of the unit cell. The contours are equally spaced between 0 and 1.2 electrons per Å3. C is in the center of the figure..

(4) 553. M. Ruda et al. / Computational Materials Science 45 (2009) 550–560. V eff ðr ij Þ ¼ Vðr ij Þ þ 2  qi  F 0 ðq0 Þ. 2.0. ð2Þ. When the monoatomic potentials are in effective pairs, the mixed-atoms pair interactive function can be modeled as a linear combination of the corresponding equal-atoms pair interactive functions. We will follow this procedure using both Fe–Fe and C– C pair functions in the effective pair formulation, as described below. 3.1. EAM potential for carbon: Tersoff’s potential with no angular term Brenner [25] showed that Tersoff’s potentials are equivalent to EAMs potentials when a proper set of functions and parameters is selected and no angular term is taken into account. In our case, after some algebraic manipulation [11] the C–C pair interaction in Tersoff’s potential without the angular dependence was approximated as follows:. Vðr ij Þ ¼ 1396 expð3:4879r ij Þ  346:74 expð2:2119r ij þ 1Þ. ð4Þ. These functions were transformed to effective pairs and the electronic density for the perfect diamond structure was normalized to a value of q0 = 0.417 Å3. The resulting functions are described by the following equations:. V eff ðr ij Þ ¼ 709:1 expð3:161r ij Þ  191:6 expð2:305r ij Þ 1:0050 6 rij < 6:0061 Å pffiffiffiffi 3 F ðqÞ ¼ 12:75 q þ 10:98q 0 6 q 62:73438 Å eff. 1.5. 1.0 Mendelev et al [18] Fe II 0.5. 0.0 Simonelli et al [24] -0.5. 1.5. 2.0. 2.5. 3.0. 3.5. 4.0. r (0.1 nm) Fig. 3. Comparison of Fe–Fe pair part of the potentials developed by Simonelli et al. [24] and Mendelev et al. [18], transformed to the effective pair scheme.. ð3Þ. where energies are in eV and distances in Å. We extrapolated this expression beyond the r < 1.80 Å range of the Tersoff’s potential. The electronic density function q(rij) was fitted to an exponential form similar to the one used by Daw and Baskes [26] for hydrogen. Finally, the embedding function FðqÞ was defined by adjusting to the cohesive energy of diamond. We take the functional form of the embedding function, as in Finnis–Sinclair potentials [28], to be a square root function. pffiffiffiffi FðqÞ ¼ 83:33 q qðrij Þ ¼ 8:5888 expð3:7807rij Þ. Fe-Fe pair interaction energy (eV). F eff ðqi Þ ¼ Fðqi Þ  qi  F 0 ðq0 Þ. ð5Þ. qðrij Þ ¼ 31:35 expð3:7807rij Þ 0 < rij < 8:02826 Å For the diamond structure calculated with these effective potentials, the cohesive energy was E0 = 7.18 eV/atom, the equilibrium lattice constant a0 = 0.361 nm, the vacancy formation energy Evac = 3.588 eV and the bulk modulus B = 462 GPa. All these properties coincide with those calculated using the original Tersoff’s potential [12,13]. They also agree quite well with the corresponding experimental values. Since no angular dependence was taken into account, the individual elastic constants C11, C12, C44 for diamond were not reproduced by the C EAM potentials.. core structures, even for the same material. For bcc Fe, simulations using many pair and EAM potentials predicted a core structure spread asymmetrically on the three {1 1 0} planes of the [1 1 1] zone. However, recent first principles calculations based on the density functional theory (DFT) yielded a core [32] that is spread symmetrically into all sides of the three {1 1 0} planes. This structure is indeed the one predicted by Mendelev’s potentials [18,19]. Furthermore, Mendelev’s potentials predict glide at finite temperatures on a {1 1 0} plane in agreement with experimental data. In contrast, simulations performed with other Fe potentials, such as the Simonelli’s potential [24] predict {1 1 2} average glide planes. We selected Mendelev’s iron potential II extended as shown in Ref. [19] to model Fe. Using this potential the perfect lattice constant for a bcc lattice is a0 = 2.8553 Å and its cohesive energy is 4.011 eV. In the present work we have transformed the extended Mendelev’s potential II to effective pairs as shown on Eq. (2) and re-scaled the perfect lattice electronic density to the value of 0.34. This latter value is the same value of the electronic density of the perfect bcc lattice given by Simonelli’s potential [24]. Fig. 3 shows a comparison of the two Fe potentials in the effective pair form and with the same value of the electronic density of the perfect lattice. The two are remarkably similar, except in the region of very short interatomic distances. These differences are responsible for the different behavior of the self interstitials and dislocation cores. 3.3. Development of the mixed interaction potential The mixed Fe–C pair interaction was obtained by a linear combination of Tersoff’s C–C potentials [12,13] and Mendelev’s Fe–Fe potential II extended [18,19] as follows:. 3.2. EAM potential for iron: Mendelev’s iron potentials. eff eff V eff Fe—C ða þ bxÞ ¼ A½V C—C ðc þ dxÞ þ B½V Fe—Fe ðe þ fxÞ. One of the main limitations of the Fe potential developed by Simonelli et al. [24] is that it does not reproduce the correct configuration of the self interstitial. Mendelev et al. [18] and Ackland et al. [19] specifically developed EAM iron potentials that were by construction fit to obtain the correct self interstitial configuration. This is a significant improvement and is critical for applications where self interstitials may be produced, such as simulations of materials behavior under irradiation. In addition, Mendelev’s potentials predict the correct picture for dislocation core structure, in agreement with first principles calculations [32]. In this respect, we note that different interatomic potentials may predict different. where A = 0.80, B = 0.20, a, c and e = 0.75 Å, b = 5.3 Å, d = 5.76 Å, f = 5.67 Å, and 0 6 x 6 1. Energy values are in eV. The values for the adjustable parameters were obtained in a trial and error procedure to give an overall fit to the same properties used in our previous potential [11] as well as our new first principle calculations. The quantities used for the fitting include the dilute heat of solution of C in bcc Fe, lattice stability, cohesive energy and bulk modulus of metastable B1 FeC. The values of A = 0.80, B = 0.20 where chosen in order to obtain a minimum for the pair potential at short distances, as suggested by the first principles calculations reported in the present work.. ð6Þ.

(5) 554. M. Ruda et al. / Computational Materials Science 45 (2009) 550–560. Pair interaction energy (eV). 1.0. 0.5. Fe-Fe. 0.0 Fe-C -0.5 C-C -1.0 1.0. 1.5. 2.0. 2.5. 3.0. 3.5. 4.0. r (0.1 nm) Fig. 4. Pair interaction potentials used in this work.. Table 2 Location of the effective pair potential minima for Fe–Fe, Fe–C and C–C pairs. Pair. C–C Fe–Fe C–Fe. Ref. [11]. This work. r (nm). V (eV). r (nm). V (eV). 0.200 0.261 0.208. 0.65 0.29 0.52. 0.200 0.268 0.190. 0.65 0.26 0.45. Fig. 4 shows the resulting pair potentials used in this work. At r = 1.71 Å both potentials, C–C and Fe–C have Veff = 0.35 eV. At shorter distances, between 1.58 and 1.71 Å, the Fe–C potential is more attractive than the C–C potential. The positions of the minima for the Fe–Fe, C–C and Fe–C pairs are summarized in Table 2. We note that the minimum for the mixed pair interaction is below 0.2 nm and is 20% smaller than the minimum in our previous Fe–C mixed interaction, developed without the guidance of the first principles calculations. 4. C in point defects in Fe 4.1. Interstitials in ferrite As shown in Table 3, C interstitial in bcc Fe is predicted to be more stable in the octahedral site than in the tetrahedral site. The difference in energy between the two sites predicted by our potential is smaller than that obtained from first principles. This means that our potential will underestimate the migration energy. of C in bcc Fe. We also compared the magnitudes of the atomic relaxation of the Fe atoms around the C with the results of first principles calculations. For the more stable octahedral site the relaxation predicted for the two Fe atoms closest to the C is 24.1% Dd/d0 outwards, which is in good agreement with the 24.3% Dd/d0 outward predicted by first principles calculations [1]. For the other four Fe atoms that constitute the octahedral site the present potential predicts a relaxation inwards of 2.5%, close to the 1.8% Dd/d0 inwards predicted by first principles calculations. Experimental data from [34] give values of 38% Dd/d0 and 2.6% Dd/d0 for first and second neighbors, respectively. For C occupying the tetrahedral sites we predict an outward relaxation of all the Fe first neighbors of 12.0% Dd/d0, and a contraction of the second neighbors with an inwards relaxation of 3.6% Dd/d0; first principles values are 13.9% Dd/d0 and 2.9% Dd/d0, respectively. Our figures compare well with other simulations results such as [23] except at the second neighbors of C in the T position (+0.06% Dd/d0). Because the octahedral site is not regular, the interstitial in bcc Fe causes a tetragonal distortion with relaxation being larger for the two Fe neighbors that are at the shortest distances from the C. We performed calculations of the stability of bct Fe when a C interstitial occupies such octahedral sites. We used a supercell of 16 Fe atoms and 1 C atom and studied the equilibrium lattice parameters. This supercell was chosen to be the same as used in first principles calculations [2,33], so that the results can be compared directly. As mentioned above, because the octahedral site is not regular the equilibrium lattice parameters are different in the direction in which the C has the closest two Fe neighbors. Our results give a c/a relationship of 1.045, compared to first principles results for a similar supercell that give c/a = 1.07 [2,33]. For a carbon concentration of 5.8 at% carbon (1.3 wt%) the experimental tetragonality of martensite is c/a = 3.03/2.85 = 1.06 [34]. With our previous potential [11] and using a large equilibrated block we obtained c/a = 2.890/2.857 = 1.01. This lower value given by our previous potential was due to the fact that with that potential most of the C goes in the tetrahedral sites. The present potential predicts a tetragonality ratio for martensite that is in good agreement with experimental and first principles calculations. Briefly, the potentials predict the correct interstitial location for the C in ferrite, and are in good quantitative agreement with first principles for the tetragonality effect of the C as well as the relaxation behavior in both types of interstitial sites. 4.2. Carbon-vacancy complexes in bcc Fe The potential that we have developed here was also evaluated by conducting a study of the defect energies of various point defect complexes in the bcc structure. The interaction of two C. Table 3 Energetic properties of C interstitials in bcc Fe. EAM Fe–C (this work) Dilute DH of solution, C at octahedral (O) site (eV) Dilute DH of solution, C at tetrahedral (T) site (eV) DE[(O)  (T)] (eV). Relaxation around the C atom in O site Dd/d0 % 1: First neighbors 2: Second neighbors Relaxation around the C atom in T site Dd/d0 % 1: First neighbors 2: Second neighbors Tetragonal distortion around the O site c/a (Fe16C1 with C in O site). 1.81 2.08 0.27. First principles calculations. Experimental values 1.1 [34]. 0.978 (this work, Fe16C1) 0.902(Fe128C1) [1] 0.86 (Fe128C1)[2]. 0.87 [29]. 1: +24.1% 2: 2.5%. 1: +24.3% [2] 2: 1.8%. 1: +38%, [34] 2: 2.6%. 1: +12.0% 2: 3.3% 1.05. 1: +13.9% [2] 2: 2.9% 1.07 [2]. 1.06 [34]. Dilute heats of solution from this work are referred to pure Fe and experimental graphite (Ec = 7.37 eV)..

(6) 555. M. Ruda et al. / Computational Materials Science 45 (2009) 550–560 Table 4 Energetic properties of C containing defect complexes in bcc Fe.. C–C binding energy (eV) C–C distance (a0 units) (a0 = 2.8553 Å). Vacancy-carbon binding energy (eV) Distance C-vacancy (a0 units) Vacancy-two C binding energy (eV) h1 1 0i dumbbell-C binding energy (eV) Configuration 1 of Ref. [2] h1 1 0i dumbbell-C binding energy (eV) Configuration 3 of Ref. [2]. EAM Fe–C (this work). First principles calculations. Experimental values. 0.18 h1 2 0i 0.65 h1 0 0i 1.0 h1 2 0i 1.19 h1 2 0i (relaxed) p 5/2 h1 0 0i 1.16 h1 0 0i (relaxed) 0.78. 0.09 h1 2 0i (Fe128C1) [1] 1.67 h1 0 0i (Fe128C1) [1] 1.0 h1 2 0i (Fe128C1) [1] p 5/2 h1 0 0i (Fe128C1) [1]. 0.08 [35]. 0.47 (Fe128C1) [1]. 0.41 [36], 0.85 [37], 1.1 [38]. 0.303. 0.41 (Fe128C1) [1]. 0.365 [37], 0.41–0.48 Å from center vacancy [33]. 2.123 h1 0 0i C–C align h1 1 0i on the nn sites of the vacancy +0.53. 1.07 h1 0 0i (Fe128C1) [1] C–C align h1 1 0i on the nn sites of the vacancy 0.19 (Fe128C1) [1]. +0.33. +0.56 [9] 0.09 (Fe128C1) [1]. interstitials is important for use of the potential in situations with slightly larger C concentration. C-vacancy complexes in various configurations are also important for the use of the potential in a Fe–C alloy at finite temperatures. First principles DFT calculations are available to compare with the predictions of the present interatomic potential. In what follows we detail such a survey and we compare the results with those of DFT calculations, and with experimental data when available. The results are summarized in Table 4. Complexes of two carbon interstitials are shown to be more stable when they are aligned along the h1 2 0i direction, in agreement with first principles calculations [1]. The binding energy is predicted to be 0.18 eV, which is closer to the experimental value of 0.08 eV [35] than to the value of 0.09 eV obtained by first principles [1]. C–C distances agree with calculated ones [1] when unrelaxed, but are longer when the system relaxes. The C interstitial is shown to bond with an Fe vacancy with a binding energy of 0.78 eV, larger than the 0.47 eV given by first principles but within the range of experimental data (0.41 eV [36], 0.85 eV [37], 1.1 eV [38]. A similar agreement is obtained for distances predicted between the C and the center of the vacancy. Our potential also predicts the correct configuration of a C-two vacancy complex, where the C–C bond is aligned along h1 1 0i in sites that are nearest neighbor to the vacancy. 4.3. C in austenite We have tested the interatomic potentials developed here for the description of C interstitials in fcc Fe. The results are shown in Table 5. Although the predicted enthalpy of solution has a higher endothermic value than expected, the potentials predict the octahedral site to be stable in the fcc structure, in agreement with first principles calculations and experiments. The potentials underestimate the migration energy for the C interstitial in the fcc structure,. +0.11 [38]. but reproduce well the binding energy of the C interstitial with a vacancy. We note that the description of austenite is mostly determined by the selection of the potential that models the pure Fe. A comparative discussion of various Fe potentials is given by Müller et al., including predictions for the fcc phase [40]. The elastic constants of fcc Fe predicted by Mendelev’s potential are too low, and the fcc phase is predicted to be unstable at higher temperatures [40]. Overall our potential predicts quite well the behavior of various point defects involving C in ferrite and austenite. Beyond point defects, the predictions of the interatomic potential for the interaction of C with extended defects in the bcc structure are of great interest for the simulation of mechanical behavior and are described next. 5. Extended defects 5.1. C in free surfaces and grain boundaries in bcc Fe The energetics of carbon in free surfaces and grain boundaries allows predictions of the effects of segregated C on the grain boundary fracture behavior. Based on the thermodynamic theory developed by Rice and Wang [41], a segregating solute will reduce the ‘‘Griffith work” of a brittle boundary separation according to the differences in segregation energies for that solute at the grain boundary, DEb, and at the free surface DEbs. If the segregation energy DEbs, that is the energy difference DEb  DEs, is positive, the solute will be an embrittler, while a negative difference will enhance cohesion. Freeman and co-workers [6] calculated the segreP gation energy for C in a bcc Fe 3 (1 1 1) grain boundary using density functional theory and the LAPW approximation. They showed that C enhances the cohesion of the boundary with an energy of DEbs = 0.61 eV/adatom for the relaxed configuration and. Table 5 Energetic properties of C interstitials in fcc Fe.. Dilute enthalpy of solution (eV) DE[(O)  (T)] (eV) Migration energy barrier (eV) h1 1 0i direction C-vacancy binding energy (eV). EAM Fe–C (this work). First principles calculations. Experimental values. 1.01 1.03 0.33 0.50. 0.12 [5] 1.48 (Fe32C1) [2]. 0.36 [39] 1.4–1.53 (Refs. in [16]) 0.37–0.41 [5]. Dilute heats of solution from this work are referred to pure Fe and experimental graphite (Ec = 7.37 eV). Reference states of all bindings are the separated, non-interacting individual defects..

(7) 556. M. Ruda et al. / Computational Materials Science 45 (2009) 550–560. a. b. [111]. [112 ] Fig. 5. Relaxed configuration of C in (a) a bcc Fe (1 1 1) surface and in (b) a bcc Fe. Table 6 Energetic properties of C interstitials in a grain boundary in bcc Fe.. P = 3 (1 1 1) GB DEbs = DEb  DEs (eV/adatom) Fe GB layers d12 (Å) d23 (Å) d34 (Å) d45 (Å) (see text) Idem previous but with carbon. EAM Fe–C (this work). First principles calculations. 0.74 (relaxed). 0.81 unrelaxed [6] 0.61 relaxed. 1.1617 0.59637 0.8905 0.88742 1.1660 0.65723 0.82796 0.8893. 1.1004 [6] 0.66548 0.786 0.79124 1.1266 [6] 0.7336 0.7336 0.8174. P 3 (1 1 1) grain boundary.. the dislocation core structure in ferrite. In particular, mechanical response of bcc materials at low temperatures is known to be strongly controlled by the core structure of the ½h1 1 1i screw dislocation. The potentials developed here are based on the Fe interatomic potential by Mendelev et al. [18,19] which predicts the correct core structure and glide planes for this particular dislocation [18]. The relaxed configuration of the calculated dislocation core is shown in Fig. 6, with and without C. The C was placed in several different positions within the core region and the. 0.81 eV/adatom for the unrelaxed structure. As a test of our new potentials, we performed simulations for the same grain boundary in its relaxed structure, with and without carbon. We also calculated the energies of the (1 1 1) surface with our potential, with and without carbon. Fig. 5 shows the obtained relaxed configurations for the minimum energy positions of the carbon interstitial in the free surface and grain boundary. Our potentials indeed predict that carbon is a cohesion enhancer in this boundary with an energy of DEbs = 0.74 eV/adatom for the relaxed configuration. The energetics of these results are summarized in Table 6. For this particular grain boundary we also compared the details of the atomistic relaxation around the C atom in the boundary region as given by our potentials and the first principles simulation. Table 6 shows the relaxation in Angstroms of the various interlayer distances within the boundary region, with layer 1 being the location of the grain boundary plane. For the clean boundary our predicted inter-layer relaxations agree with first principles calculations [6] within 10%. We predict that the addition of C to the boundary has almost no effect on the spacing between layers 1 and 2, increases the spacing between layers 2 and 3 by about 0.05 Å and decreases the spacing between layers 3 and 4 by a similar amount. This behavior is exactly what is seen in the first principle calculations. These results suggest that these potentials are adequate for the simulation of the effects of C interstitials segregated to surfaces and grain boundaries in the bcc phase, as well as for the simulation of the effects of C on the fracture behavior of ferrite. 5.2. C in dislocation cores in bcc Fe For studies of deformation behavior it is important to understand the structure of dislocation cores and the effects of C on. Fig. 6. C in the dislocation core of the ½h1 1 1i dislocation in bcc Fe. Visualization using the Y1 invariant of the strain tensor as in Ref. [43]..

(8) M. Ruda et al. / Computational Materials Science 45 (2009) 550–560. configuration shown in Fig. 7b corresponds to the minimum energy position. The visualization used in this figure is based on the invariants of the local strain tensor, as described in previous work [42,43]. The figure shows the Y1 invariant for both cases. A similar visualization technique has also been recently used by Hartley and Mishin [44]. Using this visualization technique one can easily see the distortion of the dislocation core structure caused by the presence of the C interstitial. More importantly, the potentials predict that the C interstitial does find sites with significantly lower energy within the dislocation core as compared to the bulk. The binding energy of C with the dislocation core was found to be 0.64 eV. This value is very similar to that obtained by Tapasa et al. [21] of 0.68 eV as well as Becquart et al. [23] of 0.41 eV and compares well with the experimental range of 0.5–0.75 eV (see Refs. in [23]). The core in pure Fe is non degenerate and Chaussidon et al. [45] have shown that with the potential used here for Fe this corresponds to a {1 1 0} type glide plane. In the presence of C the core is more extended in one of the possible {1 1 0} type glide planes, further stabilizing the {1 1 0} glide.. a. [010] [100]. b. 557. 6. Description of cementite Table 7 shows the results of testing the potentials developed here in the experimentally observed Fe3C cementite structure. The structure of cementite was tested for stability by annealing it using molecular dynamics at constant pressure at temperatures up to 900 K for 600 ps. No transformations to any other structure were observed. The relaxed lattice parameters obtained are similar to the experimental ones and to the calculated by first principles [46]. The heat of formation referred to pure bcc Fe and experimental graphite is 0.18 eV, greater than the experimental value of 0.06 eV [39]; the cohesive energy is 4.66 eV compared to the experimental value of 5.05 eV [47] and first principles calculation result of 5.24 eV [46]. In agreement with known experimental data, cementite was predicted to be unstable with respect to ferrite and graphite. The reference state taken for this calculation for graphite is the experimental value since we used Tersoff’s potential for C without the corresponding angular term, and therefore cannot describe properly the graphite structure. As for the elastic properties of cementite, the calculated bulk modulus is 173 GPa, very similar to the experimental value of 174 GPa at 300 K [48]. We also calculated all nine individual elastic constants to compare with values reported recently in the literature [49,50]. All nine elastic constants are clearly positive and the agreement is reasonably good for the non-shear constants. As expected, the EAM potentials cannot reproduce the surprisingly low value of C44 reported in Refs. [50,51]. The potential does predict that C44 is the lowest of all the shear constants, reproducing the correct trend in the anisotropy, even though not the magnitude. We also computed three different surface energies for this carbide and the results are shown on Table 7. They compare very well to the values and the energetic ordering of the surfaces that were obtained from first principles simulations [46]. Fig. 7 shows the relaxed cementite surface structures for (a) 1 0 0 (b) 0 1 0 (c) 0 0 1 surfaces. We conclude that the potentials developed here are adequate for the simulation of the properties of cementite.. 7. Structure and energetics of a ferrite/cementite interface. [100] [010]. c. [100] [001] Fig. 7. Views of the relaxed surface structure of cementite predicted by the present potentials, in three different orientations corresponding to surfaces perpendicular to the three lattice vectors. (a) 1 0 0, (b) 0 1 0 and (c) 0 0 1.. Different ferrite/cementite orientation relationships in near eutectoid steel have been proposed and observed experimentally, showing a variety of possible habit planes [51,52]. Each of these possesses specific morphological features. For the present simulations we chose the conventional Bagaryatsky orientation relationship [51]. We simulated the interface with a habit plane corresponding to a {1 1 2} plane in ferrite and a {0 0 1} in cementite. The simulation technique is similar to that used in precious studies of interfaces [53,54]. The computational block was designed so that the interface would be in the center of the block. The two directions contained in the interface were treated as periodic. The first of these directions corresponds to the [1 0 0] direction in cement 0 direction in ferrite. The block included four latite and the ½1 1 tice periodicities of cementite (2.06 nm), corresponding to five lattice periodicities in ferrite (2.02 nm). We used a block periodicity of 2.06 nm in this direction. The perpendicular direction contained in the interface corresponds to [0 1 0] in cementite and [1 1 1] in ferrite. The block in this direction contained three lattice periodicities of cementite (1.95 nm) and four fcc lattice periodicities of ferrite (1.98 nm). The periodicity used in the direction for the interface simulation was 2 nm. Fixed boundary conditions were used in the direction perpendicular to the interface. The fixed regions were located at least 5 nm from the.

(9) 558. M. Ruda et al. / Computational Materials Science 45 (2009) 550–560. Table 7 Energetic properties of cementite. EAM Fe–C (this work). First principles calculations. Experimental values. Lattice constants a, b, c (Å). 5.14, 6.52, 4.35. 5.06, 6.74, 4.5 [46] 5.036, 6.724, 4.480 [49] 5.04, 6.72, 4.48 [50]. 5.09, 6.74, 4.52 [34]. Heat of formation (eV) Cohesive energy (eV) Bulk modulus (GPa). +0.18 4.66 173. Elastic constants (GPa) c11 c22 c33 c12 c13 c23 c44 c55 c66 (1 0 0) surface energy, relaxed (J/m2) (0 1 0) surface energy, relaxed (J/m2) (0 0 1) surface energy, relaxed (J/m2). 263 219 247 176 146 143 77 95 123 2.34 2.00 1.96. +0.06 [39] 5.05 [47] 174 (at 300 K) [48]. 5.24 [46] 142 [46] 276–272 [49] 224 [50] 385 [49] 388 [50] 341[49] 345[50] 316 [49] 322 [50] 157 [49] 156 [50] 162 [49] 162 [50] 167 [49] 164 [50] 13 [49] 15 [50] 131 [49] 134 [50] 131 [49] 134 [50] 2.47 [46] 2.26 [46] 2.05 [46]. Heat of formation from this work is referred to pure Fe and experimental graphite (Ec = 7.37 eV) Reference states of all bindings are the separated, non-interacting individual defects.. 0.0. Energy (eV/atom). interface. The simulation block contained 4442 atoms, not including the fixed regions. The atoms were placed initially at the sites corresponding to the bulk crystals and then they were allowed to relax until the simulation block reached a configuration of minimum energy. The initial configuration was constructed so that interatomic distances between Fe atoms were at least 0.15 nm. The block energy was minimized with respect to all local atomic displacements as well as a rigid body displacement perpendicular to the interface. Fig. 8 shows the relaxed structure of the ferrite/cementite block. The area of the interface is 4.12 nm2. The average energy of the atoms in the simulation block is presented in Fig. 9 as a function of the distance from the interface. We can see a steady value of energy for the ferrite side while there are oscillations in the cementite side, reflecting the location of C atoms in the structure. There is a smooth transition at the interface, with no particular increase in the energy per atom in the interface region.. -2.0. -4.0. -6.0 -6. -4. -2. 2. 4. 6. Fig. 9. Average energy per atom (eV/atom) as a function of the distance of the ferrite/cementite interface.. [111] f //[010] c [ 112] f. 0. Distance from the interface (nm). [ 001 ] c. Fig. 8. Relaxed structure of the ferrite (left)/cementite (right) interface..

(10) M. Ruda et al. / Computational Materials Science 45 (2009) 550–560. 200. the core structure of the dislocation becomes more planar in the vicinity of C. Because C–C interactions are taken into account, the potential is adequate to simulate cementite and reproduces the experimental cohesive energy and bulk modulus as well as surface energies for this carbide as calculated by first principles methods. All nine elastic constants are positive and with the exception of C44 agree well with first principle calculations. The stability of the cementite phase obtained enabled us to actually calculate the interface energy and simulate the structure of the interface in the ferrite/ cementite system, in a common orientation relationship.. 160. Excess energy (eV). 559. 120. 80. 40. Acknowledgements. 0 0. 1000. 2000. 3000. 4000. 5000. Block size (number of atoms) Fig. 10. Excess energy of the interface as a function of the simulation block size. The interface area is in the center of the simulation block.. To calculate the energy of the interface we determined first the excess energy for different sizes of computational blocks. This was done by subtracting the cohesive energy of the ferrite and cementite sections from the total energy of each block. The excess energy is composed of two terms, the elastic strain and the interfacial energy. The interfacial energy term does not change with the size of the block as long as the area of the interface stays constant. The strain energy is proportional to the volume of the crystal, consequently the excess energy increases linearly with the thickness of the block for a constant interface area. In Fig. 10 we plotted the excess energy as a function of the total number of atoms in the simulation block, considering that the interface is in the center of each block. We obtained a linear relationship with a slope that represents the misfit elastic energy [53]. From the extrapolation of this line to an infinitely thin simulation block, the value of the interfacial energy is 615 mJ/m2. This value can be compared to 211 mJ/m2 for the ferrite/austenite interface as calculated by Chen et al. [53]. We also analyzed the elastic misfit in the simulation block and found that the ferrite side of the interface is under tension, whereas the cementite side is under compression.. D. Farkas would like to thank R. Asaro for many fruitful discussions. The research was partly supported by Argentina’s ANPCyT. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]. 8. Conclusions The EAM formalism is capable of a reasonable description of the behavior of C interstitials in metals and alloys. We have presented interatomic potentials for Fe–C mixtures that use the standard EAM formalism and are based on a recent Fe potential that correctly reproduces the configuration of self interstitials and dislocation cores in Fe; C–C interactions are taken into account and the mixed pair interaction was fitted to first principles calculations results. The mixed potential developed here predicts the correct type of interstitial site and the magnitude of the tetragonality introduced by C additions in bcc Fe. The potentials also predict the energetics and structure of various C–C and C-vacancy complexes. Furthermore, the potentials also reproduce basic properties of C interstitials in the fcc phase quite well. We have used these potentials to study the effects of C interaction with various extended defects, including surfaces, grain boundaries and dislocation cores in bcc iron. 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