Atomistic investigation into the atomic structure and energetics of the ferrite-cementite interface: The Bagaryatskii orientation

Acta Materialia 119 (2016) 184e192

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Atomistic investigation into the atomic structure and energetics of the ferrite-cementite interface: The Bagaryatskii orientation Matthew Guziewski a, *, Shawn P. Coleman b, Christopher R. Weinberger a a b

Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, USA U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 June 2016 Received in revised form 6 August 2016 Accepted 7 August 2016

Atomistic modeling was used to investigate the energetics and structure of the Bagaryatskii orientation relationship between ferrite and cementite within pearlite. The atomic level results show that the interface structure consists of a rectangular array of dislocations that lie along the high symmetry directions of the interface. The interface can be constructed using three different atomic terminating planes in the cementite structure, which dictates the chemistry and registry of the interface and controls the interfacial energy. The FeC-Fe terminating plane is always the lowest energy because the interfacial dislocations are most easily able to spread on these planes, thus reducing the interfacial energy. These atomistic results compare favorably with results from a continuum model based on O-lattice theory and anisotropic continuum theory. The dislocation spacing and interfacial energies predicted from the continuum level agree well with the atomistic simulation results. © 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Dislocations Nanostructure Plastic deformation Fe-C alloys Atomistic simulations

1. Introduction Steel is one of the most prevalent materials found in modern society, primarily due to its wide availability, high strength and low cost [1]. The wide uses of steel can be partially attributed to the wide range of physical properties that it exhibits. This diversity of behavior arrises from the fact that steel is less a single material, but rather a composite of materials which vary in both composition and structure. Different processing routes and alloy content can dramatically impact the microstructure and properties of steel, including its mechanical properties such as strength, ductility, and toughness [2]. Carbon steels, which may be the most prevalent by volume, can form a wide range of microstructures including pearlite, banite, and martensite that alter the mechanical properties [3]. Despite the enormous amount of research conducted on carbon-based steels, there are still a great number of outstanding questions regarding the microstructure and properties of these unique steel microstructures including pearlite, which is perhaps the most studied. Pearlite is derived from the eutectic transformation of austenite, which forms colonies of ferrite and cementite lamellae that are

* Corresponding author. E-mail address: [email protected] (M. Guziewski). http://dx.doi.org/10.1016/j.actamat.2016.08.017 1359-6454/© 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

aligned in the same orientation. Both the arrangement and orientation of the pearlite lamellae colonies alter its mechanical properties. Specifically, its interlammaler spacing is shown to be linearly related to the strength; whereas the size of the pearlite colony and the size of the preceding austenite colony affects the ductility [4,5]. These strong structure-property relationships highlight the importance of obtaining a deeper understanding of the underlying microstructure within pearlitic steels. While there has been significant effort in understanding the connection between the pearlite structure and its macroscopic properties, there are a number of open questions regarding the atomic-level physics that surrounding the formation of pearlite. While there is consensus that a single pearlite colony will exhibit a single consistent orientation relationship amongst the lamellae, numerous orientation relationships (ORs) have been observed for pearlite colonies [6e8] resulting in three being consistently reported in literature. Using the convention a
b
c for the unit cell of cementite, these are the: Bagaryatskii OR:

½100q ½010q ð001Þq

jj jj jj

 110 a ½111a   112 a


Isaichev OR:

M. Guziewski et al. / Acta Materialia 119 (2016) 184e192

½010q ð103Þq

jj jj

½111a   011 a

Pitsch-Petch OR:

½010q 2:6 from ½131a ð001Þq

jj



 215 a

with q denoting cementite and a for ferrite. However, there is no consensus as to which of the three ORs is the most favorable or prevalent. Some research has shown the Bagaryatskii OR to be the most common [9], while others have suggested the Pitsch-Petch OR [10,11]. Other work found the Pitsch-Petch and Bagaryatskii ORs [12] to occur with similar frequency, while a different study found the Pitsch-Petch and Isachev ORs [13] to be the most prevalent and occur with approximately the same frequency. This is further complicated by the possibility of surface steps occurring on the interface, which could slightly change the macroscopic OR but would not be representative of the true habit plane of the interface. This is especially relevant in the case of the Isaichev OR, which is only 3.8+ from the Bagaryatskii OR. Early work used transmission electron microscopy (TEM) to determine the ORs of pearlite. More recent work has used electron backscatter diffraction (EBSD) to determine the ORs [14,15]. The use of EBSD has validated many of the ORs found through TEM investigations, including the three mentioned above, but has done little to clarify which is the most favorable. In addition, these experimental techniques have primarily focused on the macroscopic ORs and have not specifically investigated the atomic structure. Atomistic modeling, however, can both determine interfacial energy as well as in investigate the local atomic structure, making it ideal for providing additional insight into the atomic level structure of pearlite. Atomistic modeling has been used extensively to investigate the structure, energetics and mobility of metal interfaces. Grain boundaries have been studied in face-centered cubic (FCC) metals including their structure [16,17], energetics [18e20], mobility and migration [21e24], mechanical response [25e28], and damage tolerance [29]. Similar work is ongoing in body-centered-cubic (BCC) metals [30e35] There has also been substantial work on modeling BCC-FCC interfaces, including radiation damage tolerance [36], interfacial structure [37e39], and mechanical properties [40,41] and phase transformations [42]. There have been fewer atomic level studies of interfaces of ceramics or metal-ceramic interfaces. However, work has been conducted on grain boundaries in oxides such as alumina [43,44], ZnO [45], MgO [46] as well as silicon [47], diamond [48], and silicon carbide [49]. While this is not an exhaustive list, it does point to the utility of using atomistic simulation methods to gain insight into the energetics and atomic level structure of interfaces. In this work, we use atomistic simulations to study the structure and energetics of the Bagaryatskii OR. The Bagaryatskii OR, due to its frequent reports in literature and high degree of registry between ferrite and cementite, appears to be the most likely interface to form between cementite and ferrite. The results of this in-depth study will provide insight, combined with other studies, into which interface is most favorable. There have been some preliminary approximations of the interfacial energy of the Bagaryatskii OR by Ruda et al. [50] in their development of an EAM potential for the iron-carbon system as well as by Zhang et al. [51] using density functional theory (DFT) to characterize the transition between austenite, ferrite and cementite. However, the small size of these previous simulations likely suppressed the formation of defects at the interface, such as dislocations, leading to potentially inaccurate

185

calculations of the interfacial energy. To the authors' knowledge, no prior study of pearlite interfaces has fully taken into account of the effect of the chemistry of the terminating planes on the formation of interfacial dislocations, which have previously been observed within pearlite [62,63]. As such, the goal of this work is to determine the local atomic structure of the Bagaryatskii OR using classical atomistic simulations. The reduced strain state enabled by the large simulations allows for a more complete relaxation of the interface and enables the direct observation of defect structures. This work will also investigate the role of local chemistry on interfacial structure and energetics. In order to test for the robustness of the simulation, multiple styles of interatomic potentials were employed, each yielding very similar results. We identified several unique interfacial planes within the Bagaryatskii OR and found that the chemistry of these planes is important in determining the interfacial energy. Upon inspection of the interface structure that formed, it was found that it can be described using a classical dislocation network with spacings that are determined through lattice mismatch. To further reinforce these results, the line directions of dislocations found by atomistic simulations are used in conjunction with O-lattice theory to determine the spacing of the dislocations and the resulting Burgers vectors. The values from Olattice theory match well with those from the atomistic simulations and given Burgers vectors yielded a continuum model prediction of interfacial energy very much in line with the atomistic values. 2. Bagaryatskii orientation relationship Pearlite is comprised alternating lamellae of BCC ferrite (a ¼ 2.87 Å) [52] and orthorhombic cementite (a ¼ 4.52 Å, b ¼ 5.09 Å, c ¼ 6.74 Å) [6], whose unit cell contains 16 atoms (12 Fe, 4 C) [53]. The lattices in the Bagaryatskii OR are related as follows (Fig. 1):

½100q

jj

½010q

jj

ð001Þq

jj

h

110

i a

½111a 112

a

As previously mentioned, the Bagaryatskii OR has a high degree of registry between ferrite and cementite at the interface. Fig. 2 shows the positions of the atoms of cementite overlaid with those of ferrite. The iron atoms for both lattices are very close in position, which should facilitate the formation of a low energy semi-coherent interface. The misfit strain can be approximated as:      2  ε ¼ 2aa11 a þa2  yielding strains of 10.7% 10:7% in the ½100q ½110a direction and 2.4% in the ½010q jj½111a direction, corroborating the hypothesis of a semi-coherent interface. While these relations fully describe the macroscopic degrees of

Fig. 1. The ferrite and cementite structures in the Bagaryatskii OR showing the (a)   ð010Þq jjð111Þa projection (b) ð100Þq ð110Þa projection.

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 a and class b in Fig. 3. In the ½100q ½110a direction, these a and b planes are identical. In the ½010q jj½111a though, the direction of the atoms in cementite relative to that of the ferrite is different between for the a and b planes i.e. nearby atoms to the terminating plane trend in the positive or negative y directions. This requires that each be analyzed separately to detect if this variation affects the energetics or structure of the interface. Our results demonstrate that the class of planes, a or b, has no discernible effect on the the energetics or structure but that chemistry of the terminating plane plays a substantial role.

3. Atomistic simulations

Fig. 2. The relative positions of atoms  of a 3  3 unit cell of cementite overlaid with the  matching ferrite layer; the ð001Þq ð112Þa projection. Blue atoms represent aFe, red represent Fe in Fe3C, brown C in Fe3C (color online). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

freedom of the interface, there are additional microscopic degrees of freedom that need to be accounted for in any atomic level description of an interface. One is the relative in-plane positions of the crystals, while another is the terminating planes within each unit cell that comprises the interface itself. The former can be addressed by running multiple simulations as the two lattices are shifted relative to each other and will be discussed in more detail later. As for the latter, due to the high symmetry of the BCC lattice, the structure and chemistry of the terminating planes of the BCC structure is the same, varying only by an in-plane shift. Within the cementite crystal, however, there are six distinct terminating planes that could form the interface with the ferrite. These planes can be classified by the atoms in the first two layers starting from the interface (Fig. 3): Fe-FeC, FeC-Fe, and Fe-Fe. By this classification, each type of terminating planes occurs twice, denoted as class

Atomistic methods allows one to probe the exact interface structure as well as determine interfacial energies. Of course, the results of such modeling is dependent on the accuracy of the interatomic potential used, and it is well known that ceramics can be difficult to model using empirical potentials. To mitigate this problem, we use multiple interatomic potentials for the ironcarbon interactions. To assure that the potentials are appropriate for modeling ferrite and cementite, the lattice constants, elastic constants and formation energies for known iron-carbon structures are compared with experimental and ab-initio values. This includes a-Fe, g-Fe, Fe3C, Fe5C2, Fe7C3, graphite, diamond and many others. Six potentials were initially considered, two Embedded Atom Method (EAM) potentials, two Modified Embedded Atom Method (MEAM) potentials, and two Tersoff style potentials. In addition to the known structures, an evolutionary algorithm, USPEX [54], was used to determine if the potentials predicted erroneous stable iron carbides, which would suggest problems associated with the quality of the potential. This potential testing process led to the elimination of three candidate potentials, for reasons including negative elastic constants and the prediction of low energy, stable states that have not been experimentally observed. The process of choosing potentials is discussed in greater detail in the supplemental material. Three potentials were eventually chosen, an EAM by Hepburn [55], a Tersoff by Henriksson [56], and a MEAM by Liyanage [57]. Table 1 shows the lattice constants and elastic constants computed using these potentials for cementite, while the values for ferrite are given in Table 2. The accuracy of the lattice constants and elastic constants are of particular importance as they will dictate the structure and elastic energy stored at the interface. In the case of interfacial dislocations, which will be shown to exist here, the dislocation spacing will be controlled by the lattice mismatch while the energy will be controlled by both the lattice mismatch as well as the elastic constants. It is worth noting that the EAM potential predicts a value for the b lattice constant that differs from

Table 1 The lattice constants and elastic constants of cementite predicted by the interatomic potential used in this work.

Fig. 3. Possible terminating planes within the cementite unit cell for the Bagaryatskii OR. The dotted lines represent where the cementite structures is cut to form and interface with ferrite. The atoms below the dashed line represent the structure of cementite at the interface. Three pairs of terminating plane can be described by the atomic content of their first two layers (from dotted line down): Fe-FeC, FeC-Fe, Fe-Fe. Additionally, due to differences in the relative direction of the atoms in the   ð100Þq ð110Þa projection plane, each pair of planes is further differentiated into two classes: a and b.

a (\) b (sÅ) c (Å) C11 (GPa) C22 (GPa) C33 (GPa) C12 (GPa) C23 (GPa) C13 (GPa) C44 (GPa) C55 (GPa) C66 (Gpa)

EAM

MEAM

Tersoff

DFT [53] (Experimental [64])

4.84 4.41 6.66 398 417 368 220 196 213 93 107 94

4.47 5.09 6.67 329 236 336 142 123 175 17.9 106 69.4

4.47 5.07 6.45 332 356 373 183 133 174 68.8 118 128

4.51 (4.52) 5.08 (5.09) 6.73 (6.74) 388 345 322 156 162 164 15 134 134

M. Guziewski et al. / Acta Materialia 119 (2016) 184e192 Table 2 The lattice constants and elastic constants of ferrite predicted by the interatomic potential used in this work.

a (Å) C11 (GPa) C12 (GPa) C44 (GPa)

EAM

MEAM

Tersoff

Experimental [61]

2.87 243 138 122

2.85 213 143 119

2.86 225 142 128

2.87 242 147 112

experiments appreciably. Its elastic constants and atomic positions for cementite are otherwise accurate however, and since its computational cost is significantly lower than that of the MEAM and Tersoff, it proves useful as a benchmark, so its results are included. The simulations domains are constructed to be sufficiently large as to allow for extended defects to form and minimize elastic strain required for in-plane periodicity. All simulations are made to be periodic in the directions of the interface and have free surfaces in the direction normal to the interface. The interface plane dimensions are made to be a multiple of the cementite lattice in the relevant directions and the ferrite lattice was then strained to fit these dimension, followed by subsequent relaxation of the total energy and stress. Dimensions are deemed sufficient when the imposed elastic strain on the ferrite, prior to relaxation, was less than 0.15%. Six repeats of each lattice in the normal direction are found to be sufficient to eliminate the effects of free surfaces on computed interfacial energies. Table 3 shows the dimensions of the simulation, along with in-plane strains. In order to simulate the difference between class a and b planes seen in Fig. 3, the BCC portion of the a terminating plane is rotated 180+. Due to the symmetry of the ½111a direction, this has the same relative positioning as the b planes. All simulations are performed using the LAMMPS molecular dynamics code [58].

3.1. Interfacial energy The interfacial energy is computed using the energy of the structure with the interface as well as the bulk energies of the two phases and the appropriate free surfaces as:

Eint ¼

Etot  EBCC  EFe3 C  gBCC  gFe3 C Lx Ly

(1)

where Lx and Ly are the dimensions of the interface, and gBCC and gFe3C are the free surface energies per unit area in the normal directions. It is also necessary to test the influence of the microscopic

Table 3 The details of the simulation domains: geometry, strain state, and the number of atoms. ½110a

½010q

Length(Å)

Strain (%)

Length(Å)

Strain (%)

EAM

24.19

0.14

213.06

0.02

MEAM

205.60

0.01

167.90

0.01

Tersoff

84.93

0.01

202.73

0.02

½100q

a b c

Fe-FeC. FeC-Fe. Fe-Fe.

jj

jj

½111a

Height(Å)

80.96a 82.91b 84.86c 81.92a 83.92b 85.92c 80.71a 82.64b 84.57c

187

degrees of freedom. For the in-plane DOF, this is done by shifting the origin of the BCC portion of the simulation by increments of 20% of the unit cell. Multiple other random shifts of the origin are also implemented to ensure the robustness of the results. This in-plane shifting has the added effect of modeling the various interface planes of ferrite. However, the different terminating planes of cementite need to be accounted for directly. All six possible terminating planes are modeled, and there is noticeable variation between the three different plane chemistries. However, there is no distinguishable difference in the energies between the a and b planes and thus these differences are not reported. Table 4 shows the results of these simulations. While the magnitudes of the interfacial energies predicted by each potential are different, there is a clear trend regarding the role of the surface chemistry (or terminating plane). The interface energies for the various terminating planes are ordered: FeC-Fe < FeFeC < Fe-Fe. Clearly the terminating plane plays a major role in determining the interfacial energy. It is important to note that at no point do the values for the terminating planes within a potential overlap with each other, even after consider variations associated with in-plane translations. Previous work done on the Bagaryatskii OR found interfacial energies on the same order of magnitude as our results, with a value of 0.45 J/m2 predicted by Zhang for the FeC-Fe interface using DFT. While this is in excellent agreement with the results of the Tersoff potential, the similarities are likely an artifact of the psuedopotential and integration techniques used. Atomistic simulations with the Tersoff potential using similar dimensions used in the DFT study predict interfacial energies to be several times larger than the values seen in Table 4, with a value of 2.5 J/m2 for the lowest energy FeC-Fe plane. Ruda predicted an interfacial energy 0.615 J/m2 using and EAM potential, although there is no mention of which terminating plane this represents; thus, making direct comparisons with our results difficult. Prior atomistic simulations by both Zhang and Ruda were conducted on significantly smaller scale models than those performed in this work. For reasons of computational cost, DFT simulations generally consist of only a few unit cells and several hundred atoms and the simulation dimensions used in the EAM study by Ruda et al. were approximately 20 Å  20 Å and consisted of 4442 atoms. This means that even the smallest simulations performed in this work were larger by an order of magnitude or more than previous work. The smaller simulation domains have larger elastic strains, and therefore generally higher interfacial energies. More importantly, however, the smaller simulation dimensions suppress the relaxation of the interface, and thus the correct interface structure is unlikely to be observed especially if interfacial dislocations form. The importance of the formation of dislocations in reducing the interfacial energy is highlighted by the discrepancy in values between DFT sized atomistic simulations and those discussed in this work. 3.2. Interfacial structure

Atoms

39,792a 41,172b 42,092c 270,576a 279,684b 285,756c 141,624a 146,412b 149,604c

The structure of the relaxed Bagaryatskii OR interface, as elucidated by the atomistic simulations, is a set of orthogonal dislocations along the ½110a and ½111a directions as shown in Fig. 4 and Fig. 5. Fig. 4 (a) plots displacement vectors of the atoms in the interface, which clearly shows the formation of two sets of parallel regions with a higher density of atoms. This strongly suggests the formation of interfacial dislocations. The atomic rearrangement is also shown through an analysis of the change of the energy of the interfacial atoms relative to their bulk values, as shown in Fig. 4b. There are clearly two sets of dislocations that form, one in the ½110a direction and one in the ½111a direction. Similar interfacial structures are formed for all three potentials, each terminating

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Table 4 The interfacial energy of the Bagaryatskii OR predicted by the interatomic potentials for the three terminating planes (J/m2).

Tersoff

EAM

MEAM

FeC-Fe Fe-FeC Fe-Fe FeC-Fe Fe-FeC Fe-Fe FeC-Fe Fe-FeC Fe-Fe

Range

Mean

Standard deviation

0.45e0.58 0.93e1.14 2.26e2.32 1.10e1.32 1.45e1.56 2.48e2.75 0.83e0.92 1.04e1.19 2.21e2.40

0.52 1.03 2.29 1.23 1.47 2.65 0.88 1.12 2.31

0.05 0.06 0.02 0.06 0.03 0.06 0.04 0.03 0.07

Fig. 4. The local interfacial structure of the FeC-Fe interface modeled by the Tersoff potential (a) Displacement map of the atoms between their bulk and interfacial positions (b) Energy map showing the difference of atomic energy in the interface relative to the bulk. These maps highlight the formation of dislocations in the interface.

Fig. 5. Visualization of atoms in ferrite displaced by the formation of interfacial dislocations for the Tersoff potential using cluster analysis. Green represents ferrite atoms with displacement greater than 0.03 Å from BCC lattice positions. (a) FeC-Fe terminating plane (b) Fe-FeC terminating plane (c) Fe-Fe terminating plane. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

plane, as well as all in-plane shifts. Additionally, the dislocation spacing remains constant as the simulation dimensions are increased. This suggests the general features of the interfacial structure is independent of the local chemistry and is tied to the misfit strain. The formation of interfacial dislocations is a common occurrence, although one that has not previously discussed in literature for pearlite. It should be noted that while all potentials exhibited the same behavior, the Tersoff potential was chosen for in-depth structural analysis as it provided the necessary combination of accuracy and a lower computational cost. From the analysis of the interfacial energy, it is noted that there are differences between the terminating planes that suggests there should be structural differences as well. These differences must lie in the details of the dislocation structures since the dislocation structures itself exists for all the analyzed simulations. One approach to quantify and visualize these differences is through the use of cluster analysis which groups atoms by their proximity to their neighbors and therefore should pick up variations in local strain. Fig. 5 shows the results of the cluster analysis applied to ferrite for the various terminating planes modeled using the Tersoff potential and a cutoff radius 0.03 Å less than that of the nearest neighbor in the BCC lattice of ferrite. The resultant regions allows for the determination of the spacing of the dislocations, as well as relative widths and heights of the dislocation for the given cutoff radius. It is noted that the spacings observed using this method are consistent with those observed in the displacement and energy maps (Fig. 4a,b). Fig. 6 shows these values for spacing, height, and width of the dislocations for each terminating plane for the Tersoff potential obtained from the cluster analysis. This reveals three interesting relations: (i) the dislocation spacing remains constant for each dislocation set regardless of terminating plane, (ii) the height and width of the dislocation sets within each terminating plane are notably different, and (iii) the height and width of the dislocation sets vary modestly with terminating plane. A dislocation spacing that is invariant with respect to the terminating plane is a consequence of the lattice mismatch between the two crystals. If we compare the value from atomistic simulations to those predicted from simple lattice mismatch arguments, a2 d ¼ aa11a , we find that the spacing in the ½110a direction matches 2 very well (40.5 Å vs. 42.5 Å) but there is a significant discrepancy in the ½111a direction (106 Å vs. 211.57 Å). However, from closer inspection of the unit cells (Fig. 1b), it can be seen that the effective atomic repeat in these two directions are half of these values, yielding spacing of 105.8 Å, which is in excellent agreement with atomistic simulation. These findings are verified using O-lattice theory later in the paper, with the results also matching atomistic simulations. Figs. 4 and 5 demonstrate that two orthogonal sets of dislocations within any single interface are structurally quite different. The  ½100q ½110a dislocation spreads out significantly (z37Å), while

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189

Fig. 6. The spacing, relative height, and relative width (Å) of the displaced atoms obtained from cluster analysis (see Fig. 5) for the Tersoff Potential. Three trends can observed here: constant dislocation spacing, increasing dislocation height, and decreasing dislocation width.

the ½010q jj½111a dislocations are more compact (z15Å). Kar'kina et al. [59] computed the generalized stacking fault energies of cementite on the ð001Þq plane and found that the ½100q direction has a stacking fault energy significantly higher than that of the ½010q direction. These stacking fault energies are a function of many variables though, including geometry, elastic moduli and terminating plane chemistry. Therefore it is necessary to compute these energies with the potentials used in our simulations. Our own generalized stacking fault surfaces found similar results to Kar'kina, as seen in Fig. 7. The ½100q direction is perpendicular to the more compact dislocation, the ½010q jj½111a , and the higher stacking fault energy would provide a larger resistive force to spreading. Following the arguments of a classical Peierls-Nabarro model for dislocations, the splitting width of the dislocations is inversely related to the stacking fault energy. Thus, we expect that dislocation in h the i ½010q jj½111a would be more compact than the ½100q  110 , as observed in our simulations. a The other notable result from the cluster analysis is that the heights and widths of the dislocations vary with the terminating planes of the cementite structure. By comparing Fig. 6 and Table 4, we note that the width of the dislocations decrease and the heights increase as the interfacial energy increases. This suggests that higher interfacial energy is related, at least in part, to the inability of the dislocations to spread; which again can be related to the lattice resistance in cementite. Kar'kina [59] also considered two planes within the cementite structure, one between an FeC and Fe layer, analogous to the FeC-Fe interface plane, and another between two layers of Fe, which would be similar to the Fe-FeC and Fe-Fe interface. They showed that the stacking fault energy was lower between the FeC and Fe layers, which is consistent with our findings. Thus, the FeC-Fe interface has the lowest energy because the

dislocations are easier to spread due to the low lattice resistance between the FeC and Fe layers. However, these results do not fully explain why the Fe-FeC and Fe-Fe interfaces have different energies. These differences are related to how the interface appropriately continues the cementite lattice, and thus is related to higher order interactions between the metal atoms. The terminating planes of cementite occur in a pattern of Fe-FeC-Fe-Fe-FeC-Fe within the unit cell. For the FeC-Fe and Fe-FeC terminating planes, the next plane would be Fe. Thus, when either of these two terminating sets of planes are matched to ferrite, ferrite acts as a continuation of the cementite lattice. These two terminating planes are found to have significantly lower energy than that of the Fe-Fe plane in this study. Conversely, for the Fe-Fe interface, the next layer would be FeC. However, when placed in an interface with ferrite, the layer it mates only has Fe, which breaks the cementite pattern earlier and increases interfacial energy. This suggests that the placement of the atoms in FeC-Fe interface is more optimal as it has a high registry with the ferrite crystal. Thus, removing atoms from this interface should raise the interfacial energy. To test this hypothesis, as well as search for alternate, lower energy interface structures, simulations are run that remove the highest energy atom from the relaxed simulation domain. The simulations are then relaxed again, and the process repeated. This was done for each terminating plane, with a total of 5000 atoms being removed from each. This corresponds to the number of atoms in the first layer immediately adjacent to the interface in both the cementite and ferrite in the FeC-Fe plane, 4998 atoms, and is significantly more than the amount in the Fe-FeC and Fe-Fe terminating planes, 3402 atoms, for the Tersoff potential. Interfacial energy values are then calculated by reducing the bulk energy of

Fig. 7. Generalized stacking fault surfaces of cementite generated using the Tersoff potential between (a) Fe-Fe layers (b) Fe-FeC layers. These surfaces predict higher stacking fault energy in the ½100q direction for both, and lower overall stacking fault energy between the Fe and FeC layers.

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Interfacial Energy (J/m2)

2.5

2

1.5

1 Fe−Fe FeC−Fe Fe−FeC

0.5

0 0

500

1000

1500

2000 2500 3000 # of Removed Atoms

3500

4000

4500

5000

Fig. 8. Variation in interfacial energy for each terminating planes as the highest energy atoms are sequentially removed. This shows the stability of the FeC-Fe and Fe-FeC interfaces, as well as the lack thereof in the Fe-Fe.

Table 5 Comparison of the interfacial dislocation spacing of the Bagaryatskii OR predicted by atomistics, O-lattice theory and lattice mismatch theories (Å).

½010c

jj

½111a

½100c

jj

½110a

Atomistic simulations

O-lattice

Classic

40.5 106

42.5 105.8

42.5 105.8

the system by the chemical potential of the removed atoms. Fig. 8 shows the interfacial energy plotted against number of atoms removed. For both the FeC-Fe and Fe-FeC interfaces, the interfacial energy increases after each high energy atom is removed. This suggests a high stability of the original interface, and as these are the two lowest energy interface, this is not surprising. Conversely, for the higher energy Fe-Fe terminating plane, the interfacial energy is reduced initially by the removal of these high energy atoms. This supports the idea that the the FeC-Fe interface is indeed optimal as it has a low resistance to dislocation splitting between the FeC and Fe layers that form at the interface and provides and optimal continuation of the cementite lattice.

material interfaces [16] and suggest that continuum models can be used to describe some aspects of the interface. One such methods is O-Lattice theory, a variant of the classic Frank-Bilby equation for characterizing dislocations. In O-Lattice theory, both lattices are considered to be strained to form an intermediate reference state with a coherent interface. The references state, in combination with an assumed set of dislocation directions, allows one to determine the Burgers vectors of the dislocations. Vattre and Demkowicz [60] postulated that the proper reference state was that which results in no far-field strain, essentially balancing the strain that was applied to form the reference state with that from the dislocations, a function of the elastic moduli. Following this formulation, and using the line directions that are determined unambiguously from the atomistic simulations, the Burgers vectors are found to have magnitudes of 4.26 Å for the dislocation with line direction ½110a and 2.51 Å for the ½111a line direction. As would be expected, their spacing match our previous estimates from simple lattice mismatch arguments (Table 5). The continuum model also predicts displacements in the interface plane as well as interfacial energies. Following the methods outlined in [60], we compute the displacements at the interfaces, Fig. 9. These displacements compare well with the atomistic observations, Fig. 4a, with the largest differences arising due to the spreading of the dislocation cores. The interfacial energy is determined by taking the dot product of these displacements along with the tractions predicted by this formulation and integrating them over the area between the sets of dislocations. However, due to the singularity that occurs in singular dislocation theory, a core region around the dislocation must be excluded. We use a fairly standard approximation for this value, which is half the Burgers vector of each dislocation. This yields a interfacial energy of 0.57 J/m2, which is well within the range of values predicted by atomistic simulations. There are limitations to this method, notably it cannot differentiate between terminating planes and, in this formulation, dislocation core spreading. However, it is useful in reinforcing the observations made in atomistic simulations as the continuum model results are largely independent from values calculated by atomistic simulations.

4. Continuum model

5. Conclusions

The existence of interfacial dislocations observed in our atomistic simulations follow long standing classical descriptions of

Atomistic simulations and a continuum model, based on O-lattice theory and anisotropic elasticity theory, are used to

Fig. 9. Displacement map of the Bagaryatskii OR predicted by Vattre and Demkowicz O-lattice formulation.

M. Guziewski et al. / Acta Materialia 119 (2016) 184e192

characterize the interface of pearlite in the Bagaryatski OR. The interface structure, from atomistic simulations, are found to be composed of a rectangular network of interfacial dislocations, with line directions along ½111a and ½110a . This rectangular array of dislocations is found to be present regardless of the terminating planes that make up the interface. The dislocations are observed to spread in the plane anisotropically with the amount of spreading depending on the terminating planes and the line directions of the dislocations. From energetic analysis, it is found that the FeC-Fe was the lowest energy interface followed by the Fe-FeC and Fe-Fe. Additionally, while there has been some discussion in other works as to whether these interfacial dislocations arise from the local structure of the ferrite and cementite or phase transformation process, the results discussed in this work suggest that it is the local structure that dominates the formation of these dislocations. Analysis of the interface structure also suggests why there is a strong dependence on the terminating plane, or chemistry, of the cementite structure. The interfacial energy is dominated by the atomic fit of the layer of ferrite and cementite that form the interface, with higher coherency leading to a lower interfacial energy which is related to the splitting or spreading of the dislocations. The FeC-Fe layer has the lowest generalized stacking fault energy in the cementite lattice and dislocations more easily able to spread and lower the interfacial energy. The Fe-FeC and Fe-Fe interfaces have higher energies because the dislocations are not able to spread out as much due to the higher stacking fault energies. However, the Fe-FeC interface is consistently lower than the Fe-Fe, which is attributed to its ability to better continue the cementite structure when it forms an interface with ferrite. The continuum model, which is based on O-lattice theory combined with anisotropic elasticity theory, predicts dislocation spacing in good agreement with the atomistic simulations. Displacement maps at the interface plane also match relatively well with atomistic results, except where dislocation core spreading occurs. The continuum theory is also able to predict the Burgers vectors of 4.26 Å for the dislocation with line direction ½110a and 2.51 Å for the ½111a as well as an interfacial energy of 0.57 J/m2. Acknowledgments This research was supported though a grant from the Petroleum Research Fund, PRF # 54697-DNI10. Many of the simulations were run using Drexel University's PROTEUS computing cluster, part of the University Computing Facility. Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.actamat.2016.08.017. References [1] E.M. Taleff, J.J. Lewandowski, B. Pourladian, Microstructure-property relationships in pearlitic eutectoid and hypereutectoid carbon steels, JOM 54 (2002) 25e30. [2] W.D. Callister, D.G. Rethwisch, Materials Science and Engineering: an Introduction 7, Wiley, New York, 2007. [3] S.R. Phillpot, D. Wolf, S. Yip, Effects of atomic-level disorder at solid interfaces, MRS Bull. 15 (1990) 38e45. [4] A. Elwazri, P. Wanjara, S. Yue, The effect of microstructural characteristics of pearlite on the mechanical properties of hypereutectoid steel, Mater. Sci. Eng. A 404 (2005) 91e98. [5] J. Hyzak, I. Bernstein, The role of microstructure on the strength and toughness of fully pearlitic steels, Metall. Trans. A 7 (1976) 1217e1224. [6] D. Zhou, G. Shiflet, Ferrite: cementite crystallography in pearlite, Metall. Trans. A 23 (1992) 1259e1269. [7] P.R. Howell, The pearlite reaction in steels mechanisms and crystallography: Part I. from HC Sorby to RF Mehl, Mater. Charact. 40 (1998) 227e260. [8] M.-X. Zhang, P. Kelly, Accurate orientation relationships between ferrite and

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