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cementite interface in pearlitic steel: An atomistic simulation study
Accepted Manuscript Characterization of the misfit dislocations at the ferrite/cementite interface in pearlitic steel: An atomistic simulation study Jaemin Kim, Keonwook Kang, Seunghwa Ryu PII:
S0749-6419(16)30067-5
DOI:
10.1016/j.ijplas.2016.04.016
Reference:
INTPLA 2053
To appear in:
International Journal of Plasticity
Received Date: 24 February 2016 Revised Date:
29 April 2016
Accepted Date: 30 April 2016
Please cite this article as: Kim, J., Kang, K., Ryu, S., Characterization of the misfit dislocations at the ferrite/cementite interface in pearlitic steel: An atomistic simulation study, International Journal of Plasticity (2016), doi: 10.1016/j.ijplas.2016.04.016. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Characterization of the misfit dislocations at the ferrite/cementite interface in pearlitic steel: An atomistic simulation study
a
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Jaemin Kima, Keonwook Kangb,**, Seunghwa Ryua,* Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea
Department of Mechanical Engineering, Yonsei University, Seoul 03722, Republic of Korea
*
Co-corresponding author. Tel.: +82-42-350-3019; fax: +82-42-350-3059.
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b
Email address: [email protected] (Seunghwa Ryu) **
Co-corresponding author. Tel.: +82-2-2123-2825; fax: +82-2-312-2159.
Abstract
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Email address: [email protected] (Keonwook Kang)
The characteristics of the misfit dislocations at ferrite/cementite interfaces (FCIs) for
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various orientation relationships (ORs) have important implications for the mechanical behavior and the phase transformation of pearlitic steels; however, the detailed characteristics
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of these misfit dislocations have not been thoroughly elucidated to date. Using the extended atomically informed Frank-Bilby (xAIFB) method and atomistic simulation, we characterized the structures of misfit dislocations and calculated the interface energies of five ORs (Bagaryatsky, Isaichev, Pitsch-Petch, Near Bagaryatsky and Near Pitsch-Petch), respectively. Atomistic calculations of the interface energies of five ORs reveal that (1) the Isaichev OR has the lowest interface formation energy and (2) Near Bagaryatsky and Near Pitsch-Petch ORs are energetically more favorable than exact Bagaryatsky and Pitsch-Petch ORs in spite of small misorientation angle. The interface formation energy of each OR is qualitatively well
ACCEPTED MANUSCRIPT explained by the structure and spacing of FCI dislocations, which demonstrate the importance of the characterization of misfit dislocations. Keywords: pearlitic steel, ferrite, cementite, misfit dislocation, molecular statics/dynamics
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1. Introduction
Pearlitic steel is widely used in steel wire applications, for instance, as bridge cable and tire cord, because of its high strength and good ductility (Elices, 2004; Raabe et al., 2010).
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Mechanical strength of pearlitic steel is enhanced by severe plastic deformation, such as cold
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drawing or ball milling, of which strain hardening mechanisms are associated with grain refinement, cementite dissolution and carbon segregation to grain boundaries under extremely large strain (Herbig et al., 2014; Li et al., 2014; Li et al., 2011; Raabe et al., 2010). On the other hand, in the early stage of plastic deformation, the on-set of strain hardening is caused by Orowan looping or other locking mechanisms of dislocations in ferrite, followed
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by their reaction with the phase boundaries between ferrite and cementite in pearlite colony as transformed from eutectoid reaction. These boundaries are also regarded as good
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dislocation source and sink to point defects (Demkowicz and Thilly, 2011; Kolluri and Demkowicz, 2012; Sutton and Balluffi, 2007), playing additional role for subsequent strain
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hardening. Thus, to modulate the strength and ductility of pearlitic steel, it is crucial to understand both the structure of homo/heterophase boundary, which strongly influences the mechanical behavior of various defects (Charmers, 1949; Chu et al., 2013; Demkowicz et al., 2008; Hirth, 1972; Hirth et al., 2006; Li et al., 2012; Mishin et al., 2010; Takahashi et al., 2007), and the complex evolution of internal microstructure associated with the confined lamellar phase geometry (Li et al., 2014; Raabe et al., 2010). In this study, our interests lie on (1) the characterization of the phase boundary structure in terms of misfit dislocation array at the interface and (2) the development of characterization methodology that can be generally
ACCEPTED MANUSCRIPT applied to any metallic composites other than pearlitic steel. Over the past several decades, many researchers have studied to characterize the structure of interphase boundary (Bollmann, 1970; Hirth et al., 2013; Sun et al., 2016; Sutton
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and Balluffi, 2007; Vattré and Demkowicz, 2013; Wang et al., 2013; Wang et al., 2014). Among these studies, the AIFB method (Wang et al., 2013; Wang et al., 2014) are one of the most promising characterization method, which constructs the reference lattice satisfying
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continuum and discrete theory. However, it needs to be extended to characterize the structure of ferrite/cementite interface (FCI) because the original AIFB method does not consider the
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in-plane shear transformation from cementite to ferrite lattice. FCI in pearlite colony is the outcome of diffusive eutectoid reaction by which parent austenite (face-centered cubic, fcc) is decomposed into product ferrite (body-centered cubic, bcc) and cementite (orthorhombic) in lamellar structure (Soffa and Laughlin, 2014). Numerous researchers have worked to
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understand the diffusive eutectoid reaction mechanism of pearlitic steel (Andrews, 1963; Hackney and Shiflet, 1987; Hillert, 1962; Hull and Mehl, 1942; S. A. Hackney, 1987; Zhang and Kelly, 1998, 2005, 2009; Zhang et al., 2015; Zhou and Shiflet, 1991). Among these
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researchers, Zhang and Kelly (Zhang and Kelly, 1998, 2005) proposed the edge-to-edge matching condition, which states that the heterophase boundary generally forms a semi-
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coherent interface to minimize the interface free energy. The researchers suggested that the atoms in different phases of the heterophase boundary (or habit plane) tend to match each other to minimize lattice disregistry and keep the strain energy as low as possible from a thermodynamics point-of-view. Therefore, the interface free energy and the strain energy act as energy barriers to the progression of the diffusive eutectoid decomposition reaction (Zhang et al., 2007). Thus, the FCI has specific orientation relationship (OR) and habit plane to have low energy barriers.
ACCEPTED MANUSCRIPT The three conventional ORs at the FCI - the Bagaryatsky (BA) (Bagaryatsky, 1950), Isaichev (IS) (Isaichev, 1947) and Pitsch-Petch (PP) (Petch, 1953; Pitsch, 1962) ORs - have been consistently reported in the literature. These conventional ORs have been used in
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various theoretical studies to explain the mechanical behavior of pearlitic steel and the mechanism of diffusive eutectoid reaction (Andrews, 1963; Dippenaar and Honeycombe, 1973; Karkina et al., 2015; Nematollahi et al., 2013; Ruda et al., 2009; Zhang et al., 2015).
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However, the frequent use of these conventional ORs is controversial because it remains unclear whether the exact BA and PP ORs exist. Zhou and Shiflet (Zhou and Shiflet, 1992)
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first suggested a theoretical analysis indicating that the BA and PP ORs do not exist in pearlitic steel. Unfortunately, this study did not provide the exact OR and habit plane at the FCI because the angular resolution of the experimental equipment was around 3 to 5 degrees (Zhang and Kelly, 1997). Later, Zhang and Kelly (Zhang and Kelly, 1997) conducted an
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experiment to accurately characterize the ORs at the FCI in pearlitic steel samples. These researchers observed new ORs close to the PP OR and the exact IS OR, but the exact BA OR and PP OR were not found. In addition, Zhang et al. (Zhang et al., 2007) carried out a
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detailed inspection of this system using both high-resolution transmission electron microscope (TEM) and an atomic arrangement analysis on the habit plane from the
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crystallographic point-of-view to reveal the exact ORs at the FCI. In that end, the researchers found four types of ORs which excluded the exact BA and PP ORs. From a thermodynamics point-of-view, the FCI is expected to prefer specific ORs
that promote a low energy interface. For a given OR, the structure of the FCI contains misfit dislocations and the areas of low or moderate local strain to compensate for the lattice disregistry and the misfit strain, caused by the different atomic arrangements and lattice parameters between ferrite and cementite phases. Therefore, it is necessary to analyze the structure of the FCI at the atomic level, which affects the mechanical response of pearlitic
ACCEPTED MANUSCRIPT steel because it governs the mechanical behavior of dislocations such as interface crossings (Karkina et al., 2015), confined layer slips (Phillips et al., 2003) and dislocation pile-ups (Embury and Fisher, 1966; Langford, 1977). The type of ORs present at the FCI has not yet
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been rigorously established. Hence, the aims of this study are to analyze the structure of the FCI at a given OR on the atomic level and to test the existence of conventional ORs in pearlitic steel. We investigated three conventional ORs (i.e., BA (Bagaryatsky, 1950), IS
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(Isaichev, 1947) and PP (Petch, 1953; Pitsch, 1962)) and two newly reported ORs (i.e., Near BA and Near PP). Near BA and Near PP correspond to the Near Bag OR and the P-P 1 I OR
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in reference (Zhang et al., 2007), respectively, and they are classified as major ORs in the experimental results. We employed atomistic simulation and the extended atomically informed Frank-Bilby (xAIFB) method. Detailed information pertaining to the ORs is summarized in Table 1.
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2. Simulation Methods
Molecular statics/dynamics (MS/MD) simulations were carried out to model the structure of the FCI. We used the modified embedded-atom method (MEAM) with
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interatomic potentials to model the interatomic interaction between Fe and C (developed by
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Liyanage et al. (Liyanage et al., 2014)). While the EAM potential considers only radial dependence in the electron density and its applicability is limited to the metallic bond, the MEAM potential contains additional terms in electron density which represent the symmetry of s, p, d, f orbitals to account for the directionality of interatomic bond such as covalent bonding in carbon. Table 1 lists the information of ORs and habit planes used for construction of the FCI structure. Ferrite has a BCC structure with a lattice parameter of af = 2.851 Å, and cementite has an orthorhombic structure with lattice parameters of ac = 4.470, bc = 5.088 and cc = 6.670 Å (Figure 1). To verify the existence of the ORs and characterize the structure of
ACCEPTED MANUSCRIPT the FCI, we computed the interface energies and conducted an xAIFB analysis after performing the MS/MD simulation. 2.1. Modeling of the ferrite/cementite interface
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In order to conduct the atomistic simulation, we consider flat ferrite/cementite bilayers of the five ORs. To construct the simulation cell with periodic boundaries along the interface, we adjust the orientation of ORs around an arbitrary axis by 0.5 degrees or less,
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which is within the resolution of typical experimental equipment. The size of the initial ferrite and cementite layer was determined by minimizing the misfit strain for each OR, satisfying
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the geometric compatibility and maintaining mechanical equilibrium. Finally, we assembled the ferrite and cementite into a ferrite/cementite bilayer (an un-relaxed structure or a natural dichromatic pattern, NDP). A detailed procedure can be found in the literature (Demkowicz and Thilly, 2011; Kang et al., 2012a; Wang et al., 2013). Other energy minimum structures
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can be obtained by changing the simulation conditions such as the rotation angle of interface (tilt and/or twist), applied strain level and equilibrium lattice parameter as pointed out in the literature (Kang et al., 2012b; Salehinia et al., 2015; Shao et al., 2014). The current work does
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observed ORs.
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not consider the change of simulation conditions but only focus on the experimentally
2.2. Molecular statics/dynamics simulation We always placed the habit plane perpendicular to the y-axis and periodic boundary
conditions (PBC) were applied to the x and z directions (Figure 2). In order to model the structure of the FCI for each OR, we obtained the minimum energy state of the ferrite/cementite bilayers by employing MS and MD simulations. First, all of the simulations were carried out at 0 K and a minimum energy state was found using a MS simulation with the conjugate gradient method. Then, MD simulations with simulated annealing were carried
ACCEPTED MANUSCRIPT out for 100 ps from 800 K to 10 K using the Nosé-Hoover isobaric-isothermal (NPT) ensemble. Finally, we conducted a MS simulation with the conjugate gradient method again to obtain the fully relaxed structure of the FCI.
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2.3. Calculation of the interface energy Once we completed the MS/MD simulations, we computed the interface energies of the atomic structures to verify the existence of the five ORs from a thermodynamics point-of-
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view. The true ORs are most likely to have lower interface energies compared to ones that do
can be expressed as Ncf
Nc
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not exist. For the relaxed structure of the ferrite/cementite bilayer, the total potential energy
Nf
c f cf c f Ecf i = Ei + Ei + γ + γ + γ ·A i=1
i=1
(1)
i=1
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th c f where A is the cross-sectional area, Ecf i , Ei and Ei are the potential energies of i atom in
the ferrite/cementite bilayer, the initial cementite layer and the initial ferrite layer, respectively, Ncf, Nc and Nf are the number of atoms in the ferrite/cementite bilayer, the initial
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cementite layer and the initial ferrite layer, respectively, γcf, γc and γf are the interface energies for the FCI and the surface energies of the cementite and ferrite phases, respectively. If we
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assume that the effect of the free surface is negligible at the interfacial region defined by the half-thickness, t, in Figure 2, then Equation 1 can be simplified to c f cf Ecf i ≅ Ei + Ei + γ · A i∈g
i∈g
i∈g
(2)
where g = { i | -t ≤ yk ≤ t, k = 1 to Ncf } where yk is the y-coordinate of the kth atom and g denotes the interface group. Hence, the interface energy can be obtained using Equation 2, which does not require the knowledge of
ACCEPTED MANUSCRIPT the surface energies. 2.3. Characterization of misfit dislocations Misfit dislocations are formed to compensate for the atomic mismatches between the
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ferrite and cementite phases. In order to characterize the misfit dislocation network at the FCI of each OR, we extended the original atomically informed Frank-Bilby method (Wang et al., 2013; Wang et al., 2014) to consider the in-plane shear transformation from cementite to
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ferrite. In the xAIFB method, we assume the zig-zag atomic array on the habit plane is a straight line using an edge-to-edge matching condition (Zhang and Kelly, 1998, 2005) and
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only consider the in-plane misfit dislocations on the habit plane. The overall procedure for completing the xAIFB method is summarized in the flow chart in Figure 3. To construct the reference lattice (or a commensurate/coherent dichromatic pattern (CDP)), it is necessary to compute the transformation matrix Scf from cementite to ferrite.
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The transformation matrix Scf can be expressed as
(3)
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xf = Scf · xc
where xf and xc are the atomic positions of the NDPs for ferrite and cementite (Figure 4),
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respectively. The original AIFB method (Wang et al., 2013; Wang et al., 2014) considered the transformation to be a combination of rotation and dilatation because it was developed to analyze the square or hexagonal arrays between two cubic crystals; FCC and BCC . Because we want to consider the interface between BCC and orthorhombic crystals in this study, it is necessary to involve an additional shear transformation to construct the general reference lattice structure (Hirth et al., 2013). The transformation matrix Scf can be expressed as
ACCEPTED MANUSCRIPT Scf = Q · T · M where Q = cosθ sinθ
-sinθ , T = 1 0 cosθ
m γ and M = x 0 1
(4)
0 mz
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where Q, T and M are the rotation, shear and dilatation transformation matrices, respectively. Further, we can build the trial reference lattice xr by using the geometric factors, χx, χz, χɵ and
xr = Sc · xc = Sf · xf
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Sc = Qc · Tc · Mc
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χγ. The trial reference lattice can be computed as
cosχθ θ -sinχθ θ 1 where Qc = , Tc = sinχθ θ cosχθ θ 0
χ mx χγ γ and Mc = x 0 1
0 χz mz
(5)
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where Sf and Sc are the transformation matrices from the ferrite/cementite to the trial
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reference lattices, and Qc, Tc and Mc are the rotation, shear and dilatation transformation matrices from the cementite to the trial reference lattices, respectively. Utilizing Equations 3 through 6, we compute the net Burgers vector, BFB, at the continuum level using the Frank-
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Bilby equation (Sutton and Balluffi, 2007). The Frank-Bilby equation can be expressed as BFB p = invSf − invSc · p
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where p, invSf and invSc represent the probe vectors, the inverse of Sf and Sc, respectively. We identified the possible set of Burgers vectors, bipossible , in the trial reference lattice as described in Figure 5 and conducted the disregistry analysis between the un-relaxed and relaxed structures of the FCI using the reference lattice. To conduct the disregistry analysis, UNR we computed the relative displacements, rCDP and rRLX ij , rij ij . The relative displacements,
, rUNR and rRLX , represent the relative displacement between the ith atom in rCDP ij ij ij
ACCEPTED MANUSCRIPT ferrite/cementite and jth atom in the cementite/ferrite pair from the initial structure to the trial reference lattice, the un-relaxed and relaxed structure, respectively. After computing the rc ru uc rc relative displacements, we computed the disregistries, ruc ij , rij and rij . rij and rij , which
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represent the relative displacements from the trial reference lattice to the un-relaxed and relaxed structures, respectively, and rru ij represents the relative displacement from the unrelaxed structure to the relaxed structure. Based on the disregistry information, we
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characterized the geometry of the misfit dislocations, including the line orientation (ξi) and the line spacing (di). In this study, the line orientation is represented by the angle measured
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counterclockwise from the x-axis. Finally, the net Burgers vector, BKN, at the discrete level can be calculated using the Knowles equation. The Knowles equation (Knowles, 2006) can be expressed as
N
B
KN
p = n × ξi · p bi / di
(8)
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i=1
where n and N represent the normal vector of the FCI (the y-axis in this work) and the number of misfit dislocation types at the FCI, respectively, and bi represents the Burgers
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vector of each dislocation. Once we obtained the net Burgers vector, BFB, at the continuum
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level and the net Burgers vector, BKN, at the discrete level, we checked to see if they were identical. If they did not match, we changed the geometric factors and conducted the xAIFB analysis again until it reached convergence. Once we obtained the converged reference lattice, we characterized the detailed information pertaining to the misfit dislocations at the FCI. More information on the original AIFB method can be found in the literature (Wang et al., 2013; Wang et al., 2014). 2.4. Atomic structure of Pitsch-Petch orientation relationship
ACCEPTED MANUSCRIPT Taking the PP OR as a representative example, we will explain how the misfit dislocations were characterized. First, within a maximum of 0.5 degrees of rotation of the ferrite around the normal vector of the habit plane, we constructed the atomic structure of the PP OR with periodic boundary conditions along the x and z directions. The constructed PP
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10 35]f, Yf = [5 2 1]f and Zf = [10 31 OR and its habit plane can be represented by Xf = [11
]f for the ferrite layer and Xc = [0 1 0]c, Yc = [0 0 1]c and Zc = [1 0 0]c for the cementite 12
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layer. In this study, the subscripts f and c represent the crystal coordinates of ferrite and cementite, respectively; the real Cartesian coordinate does not include a subscript. Yf and Yc are perpendicular to each habit plane. The angle between Xf and [1 1 3]f is 2.49 degrees, and
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the angle between Zf and [1 3 1 ]f is 2.73 degrees. Hence, even after a slight adjustment in orientation, the constructed PP OR is consistent with the conventional PP OR in Table 1 within the tolerance of the experimental apparatus. From the refined PP OR, we determined
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the number of periodic unit cells satisfying the minimum misfit strain: Nfx = 2, Nfy = 6 and Nfz = 2 for ferrite and Ncx = 43, Ncy = 14 and Ncz = 44 for cementite. The in-plane misfit strain is ɛxx = 8.95 × 10-3 and ɛzz = 6.42 × 10-3 for the x and z direction, respectively. In order
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to make-up for the misfit strain, we iteratively applied the local strain to the ferrite and cementite until both geometric compatibility and mechanical equilibrium were attained, and
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we arrived at the un-relaxed structure as described in Figure 2. Once the un-relaxed structure of the FCI was constructed, we conducted a MS/MD simulation (described in Section 2.2). When the MS/MD simulation was performed, the final dimension of the un-relaxed and relaxed structure of the FCI were found to be Lcx = Lfx = 216.71 Å and Lcz = Lfz = 198.09 Å in the x and z directions, respectively. We obtained the un-relaxed and relaxed structures as described in Figure 7. In addition, we computed the interface energy using Equation 2. The interface energy of the PP OR was 660.0 mJ/m2. We repeated the above procedure to construct the ferrite/cementite bilayers for the other ORs. The geometric information for the
ACCEPTED MANUSCRIPT five ORs is summarized in Table 2, and the interface energies are plotted in Figure 6. The atomic arrangement of the habit plane of the PP OR involves rotation, shear and dilatation transformations from cementite to ferrite. One must apply the xAIFB method to
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characterize the misfit dislocations in the PP OR. To this end, we constructed the trial reference lattice using the NDPs and the initial geometric factors (χx, χz, χɵ, χγ) = (0.5, 0.5, 0.5, 0.5) in Figure 4. After that, we computed the net Burgers vector, BFB, at the continuum level
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using Equation 7. The computed net Burgers vectors are BFB(p1) = [-0.075, 0.041] Å and BFB(p2) = [0.051, 0.054] Å for the given probe vectors, p1 = [1, 0] Å and p2 = [0, 1] Å,
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respectively. Since we considered the in-plane transformation in this work, we present only the x and z components of each vector. The atomic configurations of the un-relaxed and relaxed structures on the habit plane are presented in Figure 7. For the un-relaxed and relaxed structures, we observe Moiré patterns and semi-coherency at the FCI, respectively, providing
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little information on the geometry of the misfit dislocations such as the line orientation and spacing. In order to analyze the misfit dislocations in a quantitative manner, disregistry analyses were carried out. From the Moiré patterns and the disregistry analysis, we clearly
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identified the line orientations and spacings. The line orientations are ξ1 = 58.34 degrees and ξ2 = 140.55 degrees with respect to the positive x-axis, and the line spacings are d1 = 52.51 Å
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and d2 = 67.46 Å, respectively (see Figure 5). After the disregistry analysis, we selected the possible set of Burgers vectors from the trial reference lattice (see Figure 5). The selected Burgers vectors for each dislocation are b1 = [-4.91, -0.11] Å and b2 = [0.11, 4.59] Å. Next, we computed the net Burgers vector, BKN, at the discrete level using Equation 8. The computed net Burgers vectors are BKN(p1) = [-0.079, 0.042] Å and BKN(p2) = [0.050, 0.054] Å for the given probe vectors. The net Burgers vectors determined using the Frank-Bilby analysis and the Knowles equation were not identical. So, we changed the geometric factors and repeated the xAIFB analysis until both net Burgers vectors converged within a reasonable
ACCEPTED MANUSCRIPT tolerance. In this study, we used 10-4 as the tolerance. As a result, the geometric factors are (χx, χz, χɵ, χγ) = (0.14, 0.23, 0.13, 0.85), the Burgers vectors are b1 = [-4.76, -0.18] Å and b2 = [0.11, 4.67] Å and the net Burgers vectors are BKN(p1) = BFB(p1) = [-0.079, 0.042] Å and
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BKN(p2) = BFB(p2) = [0.046, 0.055] Å for the given probe vectors. In Figure 8, we see that the disregistry analysis and the computed Burgers vectors give identical values. In the disregistry analysis, rrc ij represents the non-uniform displacement from CDP to relaxed structure. It
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shows displacement jump when we have dislocation line along the probe vector. So, we ru utilize rrc ij as a measure of Burgers vectors. rij describes atomic displacement during the
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formation of misfit dislocation in the relaxation process. It gives information on the stationary points which are candidates for the positions of dislocations. Combining information of rrc ij and rru ij , we can determine the position of dislocation lines. In figure 8, we used green arrow to indicate the position of dislocation lines. We repeated the above procedures to characterize
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the misfit dislocations for the other ORs. The characteristics of the misfit dislocations for each OR computed using the xAIFB method are summarized in Table 3, and the atomic configurations of the un-relaxed and relaxed structures, the idealized dislocations, 2D
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disregistry maps for five ORs and additional information such as dislocation density maps are
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summarized in the Supplementary Material. 3. Results
3.1. Interface energies for the five orientation relationships In the diffusive eutectoid reaction, the lamellar structure forms along the direction
which forms small interface energy at FCI. The existence of the five ORs can be tested by comparing the interface energies for the five ORs calculated using Equation 2 and increasing the half-thickness of the interface region, t, as described in Figure 2. The results are given in Figure 6 and summarized in Table 2. Figure 6 shows that all the interface energies converge
ACCEPTED MANUSCRIPT to specific values over 15 Å, meaning the effect of the interface becomes negligible over 15 Å from the FCI. Among the five ORs, the IS OR has the lowest interface energy (499.9 mJ/m2) and shows rapid convergence compared to the others. The interface energies of BA
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and Near BA OR are 628.2 and 536.1 mJ/m2, respectively. Although the BA and Near BA ORs have small differences in orientation, their interface energies are very different. Likewise, the interface energies of PP and Near PP OR are 660.0 and 555.2 mJ/m2, respectively. From
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these results, we see that small changes in orientation can give rise to large differences in interface energy. Additionally, we suspect that of the similar ORs, the Near BA OR is
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preferred to the BA OR and the Near PP OR is preferred to the PP OR.
3.2. The characteristics of the misfit dislocations for the five orientation relationships The characteristics of misfit dislocation networks obtained by the xAIFB method are summarized in Table 3. Unlike the others, the IS OR has dislocation lines along only one
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direction. For the IS OR, the Burgers vector is b1 = [-2.51, 0.00] Å, the line orientation is paralleled to the z-axis and line spacing is d1 = 83.95 Å. Thus, the atomic mismatch between ferrite and cementite along the z-axis is almost zero. For BA and Near BA OR, two
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dislocation lines are formed. For BA OR, the two dislocation lines have the following Burgers vectors: b1 = [0.00, 4.20] and b2 = [-2.47, 0.00] Å. The line orientations are parallel
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to the x- and z-axis, and the line spacings are d1 = 40.32 Å and d2 = 81.48 Å, respectively. For Near BA OR, the dislocation lines have the following Burgers vectors: b1 = [0.84, -2.09] and b2 = [-2.51, 0.00] Å. The line orientations are parallel to the x- and z-axis, and the line spacings are d1 = 18.18 Å and d2 = 86.42 Å, respectively. Each component of the Burgers vectors for BA OR is larger than the respective vectors for Near BA OR, meaning that the atomic mismatches in the BA OR are large compared with those in the Near BA OR. Likewise, both PP and Near PP OR have two dislocation lines. For PP OR, the Burgers
ACCEPTED MANUSCRIPT vectors of each dislocation are b1 = [-4.76,-0.18] and b2 = [0.11, 4.67] Å, and their line orientations are ξ1 = 58.34 degrees and ξ2 = 140.55 degrees, respectively. The line spacings are d1 = 52.51 Å and d2 = 67.46 Å, respectively. For Near PP OR, the Burgers vector of each
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dislocation is b1 = [2.18, 1.06] and b2 = [-4.91, 0.00] Å, the line orientations are ξ1 = 0 degrees and ξ2 = 41.30 degrees and the line spacing distances are d1 = 22.35 Å and d2 = 43.66 Å, respectively. For the Near BA and Near PP OR, the dislocation lines along the x-axis are
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generated by the out-of-plane steps formed by small misorientation angle of the ferrite and the other are developed by the in-plane atomic mismatches.
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4. Discussion
The different in-plane atomic arrangements of ferrite and cementite result in atomic mismatches. Hence, each atom undergoes non-uniform displacement to satisfy the crystallographic compatibility (i.e., the edge-to-edge matching condition). In this study, we
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verify the existence of the reported five ORs and identify the structure of the FCI. The interface energies of the FCI have been obtained from experimental and theoretical studies (Das et al., 1993; Deb and Chaturvedi, 1982; Kilchner et al., 1977; Kramer et al., 1958;
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Martin and Sellars, 1970; Ruda et al., 2009; Zhang et al., 2015). The interface energies
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reported in the literature are summarized in Table 4. For BA and PP OR, the interface energies obtained here are larger than those previously reported. In addition, the experimental observation of the FCI (Zhang and Kelly, 1997; Zhou and Shiflet, 1992) noted that the exact BA and PP ORs may not exist in pearlitic steel. Based on the interface energies obtained in this work and the experimental observations of the FCI, we conclude that the BA and PP ORs reported in experimental studies performed before the 1990s (Bagaryatsky, 1950; Isaichev, 1947; Petch, 1953; Pitsch, 1962) are indeed the Near BA and Near PP ORs. In short, the exact BA and PP ORs are less likely to exist in pearlitic steel. Small differences between the Near
ACCEPTED MANUSCRIPT BA and BA ORs (as well as the Near PP and PP ORs) were not differentiated due to the limited resolution of the apparatuses used during this time period. From the detailed information on the misfit dislocations shown in Table 3, we see
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that IS OR has a single dislocation line parallel to the z-axis. This component of the Burgers vector of IS OR is relatively small compared with that of the other ORs, meaning that the atoms at the FCI in the IS OR undergo small non-uniform displacements. Accordingly, the
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potential energy of each atom at the interface region may not change as much as that of atoms in the other ORs. The interface energy of the IS OR is the smallest of five ORs (Figure 6).
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Although BA and Near BA OR have small misorientation angles, the difference between the interface energies of the BA OR and the Near BA OR is quite large (the same is observed with the PP OR and the Near PP OR). Kang et al. (Kang et al., 2012b) showed that the change in the interface energy over the misorientation angle can be explained using the strain
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energy from the misfit dislocations, which is proportional to the square of the Burgers vector, ||b||2, and inversely proportional to its periodic length, L. For the dislocations with b1 Burgers vectors, the periodic length of the BA OR (L = d1 = 40.32 Å) is approximately twice that of
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the Near BA OR (L = d1 = 18.18 Å); however, the square of the Burgers vector of the BA OR (||b1||2 = 17.64) is more than three times that of the Near BA OR (||b1||2 = 5.07). Hence, we
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conclude that the BA OR has a higher interface energy than the Near BA OR because of the larger strain energy needed to form misfit dislocation networks. These qualitative analyses suggest that small misorientation angles can cause significant changes in the strain energies by changing the magnitude of the Burgers vectors of the dislocations and their spacing at the heterophase boundary. Conversely, it is interesting to note that the true ORs (IS, Near BA and Near PP) satisfy the parallelism of the close-packed plane {103}c//{110}f from a crystallographic point-of-view. The interplanar spacings of {103}c (= 1.991 Å) and {110}f (= 2.016 Å) are similar in value. The {103}c and {110}f parallelism of the FCI guarantees the
ACCEPTED MANUSCRIPT edge-to-edge matching conditions along the z-axis at the FCI. Moreover, the true ORs also satisfy the parallelism of the close-packed direction [010]c//[111]f for the IS OR, [010]c//[111]f for the Near BA OR and [010]c//[113]f for the Near PP OR. The interatomic
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spacings of [010]c (= 2.544 Å), [111]f (= 2.469 Å) and [113]f (= 2.364 Å) are not much closer in value, but they satisfy the edge-to-edge matching condition. So, each atom of the IS, Near BA and Near PP ORs does not need to undergo a large displacement compared with each one
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in the BA and PP ORs. As a result, the structure of the FCI for the IS, Near BA and Near PP ORs show lower interface energies compared with the BA and PP ORs. Hence, we conclude
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that the exact BA and PP ORs are less likely to exist in pearlitic steel. 5. Conclusions
Atomistic simulation and xAIFB analysis were performed to characterize the FCI misfit dislocations in five ORs and to verify the existence of each OR. We present the atomic
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structure of the FCIs for the five ORs. We find that the parallelism of both the close-packed plane {103}c//{110}f and the close-packed direction [010]c//[111]f (or [1 13]f) is related to the formation of low interfacial free energies, and may govern the overall diffusive eutectoid
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reaction from a thermodynamics point-of-view. We conclude that the true ORs of pearlitic
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steel are the IS, Near BA and Near PP ORs, supported by the calculated interface energies and experimental results. To summarize, -
The xAIFB method was utilized to analyze the general interface structure.
The existence of the IS, Near BA and Near PP ORs was verified based on interface energies.
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The atomic structure of pearlitic steel was identified using the xAIFB method for the first time.
ACCEPTED MANUSCRIPT Acknowledgments We acknowledge support from the Basic Science Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future
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Planning (2013R1A1A1010091) and the Ministry of Education (2013R1A1A2063917). References
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Andrews, K.W., 1963. The structure of cementite and its relation to ferrite. Acta Metallurgica 11,
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Liyanage, L.S.I., Kim, S.G., Houze, J., Kim, S., Tschopp, M.A., Baskes, M.I., Horstemeyer, M.F., 2014. Structural, elastic, and thermal properties of cementite (Fe3C) calculated using a modified embedded atom method. Physical Review B 89. Martin, A.G., Sellars, C.M., 1970. Measurement of Interracial Energy from Extraction Replicas of Particles on Grain Boundaries. Metallography 3, 259-273. Mishin, Y., Asta, M., Li, J., 2010. Atomistic modeling of interfaces and their impact on microstructure and properties. Acta Materialia 58, 1117-1151.
ACCEPTED MANUSCRIPT Nematollahi, G.A., von Pezold, J., Neugebauer, J., Raabe, D., 2013. Thermodynamics of carbon solubility in ferrite and vacancy formation in cementite in strained pearlite. Acta Materialia 61, 1773-1784. Petch, N.J., 1953. Acta Cryst. 6, 96. Phillips, M.A., Clemens, B.M., Nix, W.D., 2003. A model for dislocation behavior during deformation of Al/Al3Sc (fcc/L12) metallic multilayers. Acta Materialia 51, 3157-3170.
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Ruda, M., Farkas, D., Garcia, G., 2009. Atomistic simulations in the Fe C system. Computational
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Shao, S., Wang, J., Misra, A., 2014. Energy minimization mechanisms of semi-coherent interfaces. Journal of Applied Physics 116, 023508.
Sun, X.-Y., Taupin, V., Fressengeas, C., Cordier, P., 2016. Continuous description of the atomic structure of grain boundaries using dislocation and generalized-disclination density fields. International Journal of Plasticity 77, 75-89.
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Sutton, A.P., Balluffi, R.W., 2007. Interfaces in crystalline materials. Clarendon Press, Oxford. Takahashi, T., Ponge, D., Raabe, D., 2007. Investigation of Orientation Gradients in Pearlite in Hypoeutectoid Steel by use of Orientation Imaging Microscopy. steel research inc. Vattré, A.J., Demkowicz, M.J., 2013. Determining the Burgers vectors and elastic strain energies of interface dislocation arrays using anisotropic elasticity theory. Acta Materialia 61, 5172-5187.
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Wang, J., Zhang, R.F., Zhou, C., Beyerlein, I.J., Misra, A., 2013. Characterizing interface dislocations by atomically informed Frank-Bilby theory. Journal of Materials Research 28, 1646-1657.
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Wang, J., Zhang, R.F., Zhou, C.Z., Beyerlein, I.J., Misra, A., 2014. Interface dislocation patterns and dislocation nucleation in face-centered-cubic and body-centered-cubic bicrystal interfaces. International Journal of Plasticity 53, 40-55. Zhang, M.-X., Kelly, P.M., 1997. Accurate Orientation Relationships Between Ferrite and Cementite in Pearlite. Scripta Materialia 37, 2009-2015. Zhang, M.-X., Kelly, P.M., 1998. Crystallography and morphology of Widmanstatten cementite in austenite. Acta Materialia 46, 4617-4628. Zhang, M.-X., Kelly, P.M., 2005. Edge-to-edge matching model for predicting orientation relationships and habit planes
the improvements. Scripta Materialia 52, 963-968.
Zhang, M.-X., Kelly, P.M., 2009. The morphology and formation mechanism of pearlite in steels. Materials Characterization 60, 545-554. Zhang, X., Hickel, T., Rogal, J., Fähler, S., Drautz, R., Neugebauer, J., 2015. Structural transformations
ACCEPTED MANUSCRIPT among austenite, ferrite and cementite in Fe C alloys: A unified theory based on ab initio simulations. Acta Materialia 99, 281-289. Zhang, Y.D., Esling, C., Calcagnotto, M., Zhao, X., Zuo, L., 2007. New insights into crystallographic correlations between ferrite and cementite in lamellar eutectoid structures, obtained by SEM FEG/EBSD and an indirect two-trace method. Journal of Applied Crystallography 40, 849-856. Metallurgical Transactions A 22A, 1349-1365.
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Zhou, D.S., Shiflet, G.J., 1991. Interfacial steps and growth mechanism in ferrous pearlites. Zhou, D.S., Shiflet, G.J., 1992. Ferrite: Cementite crystallography in pearlite. Metallurgical
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Transactions A 23A, 1259-1269.
ACCEPTED MANUSCRIPT Figure Captions
Figure 1. Unit-cell structure of cementite (orthorhombic) and ferrite (body-centered cubic).
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Figure 2. An atomic model of the ferrite/cementite bilayer of PP OR.
Figure 3. A flow chart of the extended atomically informed Frank-Bilby method.
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Figure 4. The in-plane lattice structure of cementite (left), the reference lattice (center) and ferrite (right) for PP OR.
Figure 5. The possible set of Burgers vectors in the reference lattice and the idealized misfit dislocation for PP OR.
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Figure 6. The interface energies for the five ORs with increasing half-thickness of the interface layer of thickness, t.
Figure 7. The atomic arrangement of the un-relaxed (top) and relaxed (middle) structure and a 2D disregistry (rru ij ) plot (bottom) for PP OR (scaling factor for 2D disregistry = 1.0).
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Figure 8. Disregistry plots along the (a) line AB and (b) line CD for PP OR; disregistry rrc ij is used for the measure of the Burgers vectors and green arrows indicate the position of dislocations.
ACCEPTED MANUSCRIPT Table 1. The orientation relationships between the ferrite and cementite phases. Name
Orientation relationship
Habit plane
[100]c || [11ത 0]f, [010]c || [111]f and (001)c || (112ത)f
(001)c||(112ത)f
IS
(103ത)c || (101ത )f, [010]c || [111]f and (1ത 01)c || (21ത 1ത)f
(1ത 01)c||(21ത 1ത)f
PP
[100]c ∠ [1ത 31ത]f = 2.6°, [010]c ∠ [1ത 13]f = 2.6° and (001)c || (521)f
(001)c||(521)f
Near BA
(103ത)c || (101ത )f, [010]c || [111]f and (1ത 01)c || (21ത 1ത)f
(001)c ∠ (112ത)f = 3°
Near PP
(103)c || (110)f, [010]c || [1ത 13]f and (001)c ∠ (521)f = 1°
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BA
(001)c ∠ (521)f = 1°
ACCEPTED MANUSCRIPT Table 2. The geometry of the ferrite/cementite bilayer and the interface energy for each orientation relationship. Name BA
Lx
Lcy
Lfy
Lz
εxx
εzz
167.21
80.04
83.80
166.02
1.82 × 10-5
4.31 × 10-4
628.2
-3
-4
γcf [mJ/m2]
152.42
81.69
83.80
160.57
2.92 × 10
4.34 × 10
499.9
PP
216.71
93.38
93.69
198.09
8.95 × 10-3
6.42 × 10-3
660.0
Near BA
167.18
93.38
94.47
162.66
1.82 × 10-5
1.68 × 10-2
536.1
Near PP
131.54
73.37
74.45
134.96
7.31 × 10-4
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IS
4.47 × 10-3
555.2
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*Unit: Å (Angstrom)
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Burgers vector
Line orientation
2
1
Line spacing
2
b [bx, bz]
b [bx, bz]
ξ [deg.]
ξ [deg.]
d1
d2
BA
[ 0.00, 4.20]
[-2.47, 0.00]
0.00
90.00
40.32
81.48
IS
[-2.51, 0.00]
N/A
90.00
N/A
83.95
N/A
PP
[-4.76,-0.18]
[-0.11, 4.67]
58.34
140.55
52.51
67.46
Near BA
[ 0.84,-2.09]
[-2.51, 0.00]
0.00
90.00
18.18
86.42
Near PP
[ 2.18, 1.06]
[-4.91, 0.00]
0.00
41.30
22.35
43.66
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*Unit: Å (Angstrom), line orientation represents the angle between positive x-axis and dislocation line around negative y-axis
ACCEPTED MANUSCRIPT Table 4. Summary of the interface energies reported in the literature.
Coarsening rate Coarsening rate Interface enthalpy Interface enthalpy Atomic simulation Atomic simulation Dihedral angle
γcf [mJ/m2] 560 248 - 417 700 ± 300 500 ± 360 615 450 520 ± 130
Note
Reference (Das et al., 1993) (Deb and Chaturvedi, 1982) (Kramer et al., 1958) (Kilchner et al., 1977)
BA OR BA OR
(Ruda et al., 2009)
(Zhang et al., 2015)
(Martin and Sellars, 1970)
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861 903 - 963 1000 973
Method
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Temperature [K]
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Figure 1
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Figure 1. Unit-cell structure of cementite (orthorhombic) and ferrite (body-centered cubic).
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Figure 2
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Figure 2. An atomic model of the ferrite/cementite bilayer of PP OR.
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Figure 3
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Figure 3. A flow chart of the extended atomically informed Frank-Bilby method.
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Figure 4
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Figure 4. The in-plane lattice structure of cementite (left), the reference lattice (center) and ferrite (right) for PP OR.
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Figure 5
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Figure 5. The possible set of Burgers vectors in the reference lattice and the idealized misfit dislocation for PP OR.
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Figure 6
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Figure 6. The interface energies for the five ORs with increasing half-thickness of the interface layer of thickness, t.
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Figure 7
Figure 7. The atomic arrangement of the un-relaxed (top) and relaxed (middle) structure and a 2D disregistry (rru ij ) plot (bottom) for PP OR (scaling factor for 2D disregistry = 1.0).
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Figure 8
Figure 8. Disregistry plots along the (a) line AB and (b) line CD for PP OR; disregistry rrc ij is used for the measure of the Burgers vectors and green arrows indicate the position of dislocations.
ACCEPTED MANUSCRIPT An atomistic-to-continuum method is proposed to study the heterophase boundary.
2.
The method is applied to investigate the ferrite/cementite interface in pearlitic steel.
3.
Misfit dislocations and interface energies of five orientation relationships are analyzed.
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Near BA, IS and Near PP ORs are energetically more favorable than BA and PP ORs.
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The interface energies are explained by the characteristics of misfit dislocations.
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1.
S0749-6419(16)30067-5
DOI:
10.1016/j.ijplas.2016.04.016
Reference:
INTPLA 2053
To appear in:
International Journal of Plasticity
Received Date: 24 February 2016 Revised Date:
29 April 2016
Accepted Date: 30 April 2016
Please cite this article as: Kim, J., Kang, K., Ryu, S., Characterization of the misfit dislocations at the ferrite/cementite interface in pearlitic steel: An atomistic simulation study, International Journal of Plasticity (2016), doi: 10.1016/j.ijplas.2016.04.016. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Characterization of the misfit dislocations at the ferrite/cementite interface in pearlitic steel: An atomistic simulation study
a
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Jaemin Kima, Keonwook Kangb,**, Seunghwa Ryua,* Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea
Department of Mechanical Engineering, Yonsei University, Seoul 03722, Republic of Korea
*
Co-corresponding author. Tel.: +82-42-350-3019; fax: +82-42-350-3059.
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Email address: [email protected] (Seunghwa Ryu) **
Co-corresponding author. Tel.: +82-2-2123-2825; fax: +82-2-312-2159.
Abstract
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Email address: [email protected] (Keonwook Kang)
The characteristics of the misfit dislocations at ferrite/cementite interfaces (FCIs) for
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various orientation relationships (ORs) have important implications for the mechanical behavior and the phase transformation of pearlitic steels; however, the detailed characteristics
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of these misfit dislocations have not been thoroughly elucidated to date. Using the extended atomically informed Frank-Bilby (xAIFB) method and atomistic simulation, we characterized the structures of misfit dislocations and calculated the interface energies of five ORs (Bagaryatsky, Isaichev, Pitsch-Petch, Near Bagaryatsky and Near Pitsch-Petch), respectively. Atomistic calculations of the interface energies of five ORs reveal that (1) the Isaichev OR has the lowest interface formation energy and (2) Near Bagaryatsky and Near Pitsch-Petch ORs are energetically more favorable than exact Bagaryatsky and Pitsch-Petch ORs in spite of small misorientation angle. The interface formation energy of each OR is qualitatively well
ACCEPTED MANUSCRIPT explained by the structure and spacing of FCI dislocations, which demonstrate the importance of the characterization of misfit dislocations. Keywords: pearlitic steel, ferrite, cementite, misfit dislocation, molecular statics/dynamics
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1. Introduction
Pearlitic steel is widely used in steel wire applications, for instance, as bridge cable and tire cord, because of its high strength and good ductility (Elices, 2004; Raabe et al., 2010).
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Mechanical strength of pearlitic steel is enhanced by severe plastic deformation, such as cold
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drawing or ball milling, of which strain hardening mechanisms are associated with grain refinement, cementite dissolution and carbon segregation to grain boundaries under extremely large strain (Herbig et al., 2014; Li et al., 2014; Li et al., 2011; Raabe et al., 2010). On the other hand, in the early stage of plastic deformation, the on-set of strain hardening is caused by Orowan looping or other locking mechanisms of dislocations in ferrite, followed
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by their reaction with the phase boundaries between ferrite and cementite in pearlite colony as transformed from eutectoid reaction. These boundaries are also regarded as good
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dislocation source and sink to point defects (Demkowicz and Thilly, 2011; Kolluri and Demkowicz, 2012; Sutton and Balluffi, 2007), playing additional role for subsequent strain
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hardening. Thus, to modulate the strength and ductility of pearlitic steel, it is crucial to understand both the structure of homo/heterophase boundary, which strongly influences the mechanical behavior of various defects (Charmers, 1949; Chu et al., 2013; Demkowicz et al., 2008; Hirth, 1972; Hirth et al., 2006; Li et al., 2012; Mishin et al., 2010; Takahashi et al., 2007), and the complex evolution of internal microstructure associated with the confined lamellar phase geometry (Li et al., 2014; Raabe et al., 2010). In this study, our interests lie on (1) the characterization of the phase boundary structure in terms of misfit dislocation array at the interface and (2) the development of characterization methodology that can be generally
ACCEPTED MANUSCRIPT applied to any metallic composites other than pearlitic steel. Over the past several decades, many researchers have studied to characterize the structure of interphase boundary (Bollmann, 1970; Hirth et al., 2013; Sun et al., 2016; Sutton
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and Balluffi, 2007; Vattré and Demkowicz, 2013; Wang et al., 2013; Wang et al., 2014). Among these studies, the AIFB method (Wang et al., 2013; Wang et al., 2014) are one of the most promising characterization method, which constructs the reference lattice satisfying
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continuum and discrete theory. However, it needs to be extended to characterize the structure of ferrite/cementite interface (FCI) because the original AIFB method does not consider the
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in-plane shear transformation from cementite to ferrite lattice. FCI in pearlite colony is the outcome of diffusive eutectoid reaction by which parent austenite (face-centered cubic, fcc) is decomposed into product ferrite (body-centered cubic, bcc) and cementite (orthorhombic) in lamellar structure (Soffa and Laughlin, 2014). Numerous researchers have worked to
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understand the diffusive eutectoid reaction mechanism of pearlitic steel (Andrews, 1963; Hackney and Shiflet, 1987; Hillert, 1962; Hull and Mehl, 1942; S. A. Hackney, 1987; Zhang and Kelly, 1998, 2005, 2009; Zhang et al., 2015; Zhou and Shiflet, 1991). Among these
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researchers, Zhang and Kelly (Zhang and Kelly, 1998, 2005) proposed the edge-to-edge matching condition, which states that the heterophase boundary generally forms a semi-
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coherent interface to minimize the interface free energy. The researchers suggested that the atoms in different phases of the heterophase boundary (or habit plane) tend to match each other to minimize lattice disregistry and keep the strain energy as low as possible from a thermodynamics point-of-view. Therefore, the interface free energy and the strain energy act as energy barriers to the progression of the diffusive eutectoid decomposition reaction (Zhang et al., 2007). Thus, the FCI has specific orientation relationship (OR) and habit plane to have low energy barriers.
ACCEPTED MANUSCRIPT The three conventional ORs at the FCI - the Bagaryatsky (BA) (Bagaryatsky, 1950), Isaichev (IS) (Isaichev, 1947) and Pitsch-Petch (PP) (Petch, 1953; Pitsch, 1962) ORs - have been consistently reported in the literature. These conventional ORs have been used in
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various theoretical studies to explain the mechanical behavior of pearlitic steel and the mechanism of diffusive eutectoid reaction (Andrews, 1963; Dippenaar and Honeycombe, 1973; Karkina et al., 2015; Nematollahi et al., 2013; Ruda et al., 2009; Zhang et al., 2015).
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However, the frequent use of these conventional ORs is controversial because it remains unclear whether the exact BA and PP ORs exist. Zhou and Shiflet (Zhou and Shiflet, 1992)
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first suggested a theoretical analysis indicating that the BA and PP ORs do not exist in pearlitic steel. Unfortunately, this study did not provide the exact OR and habit plane at the FCI because the angular resolution of the experimental equipment was around 3 to 5 degrees (Zhang and Kelly, 1997). Later, Zhang and Kelly (Zhang and Kelly, 1997) conducted an
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experiment to accurately characterize the ORs at the FCI in pearlitic steel samples. These researchers observed new ORs close to the PP OR and the exact IS OR, but the exact BA OR and PP OR were not found. In addition, Zhang et al. (Zhang et al., 2007) carried out a
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detailed inspection of this system using both high-resolution transmission electron microscope (TEM) and an atomic arrangement analysis on the habit plane from the
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crystallographic point-of-view to reveal the exact ORs at the FCI. In that end, the researchers found four types of ORs which excluded the exact BA and PP ORs. From a thermodynamics point-of-view, the FCI is expected to prefer specific ORs
that promote a low energy interface. For a given OR, the structure of the FCI contains misfit dislocations and the areas of low or moderate local strain to compensate for the lattice disregistry and the misfit strain, caused by the different atomic arrangements and lattice parameters between ferrite and cementite phases. Therefore, it is necessary to analyze the structure of the FCI at the atomic level, which affects the mechanical response of pearlitic
ACCEPTED MANUSCRIPT steel because it governs the mechanical behavior of dislocations such as interface crossings (Karkina et al., 2015), confined layer slips (Phillips et al., 2003) and dislocation pile-ups (Embury and Fisher, 1966; Langford, 1977). The type of ORs present at the FCI has not yet
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been rigorously established. Hence, the aims of this study are to analyze the structure of the FCI at a given OR on the atomic level and to test the existence of conventional ORs in pearlitic steel. We investigated three conventional ORs (i.e., BA (Bagaryatsky, 1950), IS
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(Isaichev, 1947) and PP (Petch, 1953; Pitsch, 1962)) and two newly reported ORs (i.e., Near BA and Near PP). Near BA and Near PP correspond to the Near Bag OR and the P-P 1 I OR
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in reference (Zhang et al., 2007), respectively, and they are classified as major ORs in the experimental results. We employed atomistic simulation and the extended atomically informed Frank-Bilby (xAIFB) method. Detailed information pertaining to the ORs is summarized in Table 1.
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2. Simulation Methods
Molecular statics/dynamics (MS/MD) simulations were carried out to model the structure of the FCI. We used the modified embedded-atom method (MEAM) with
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interatomic potentials to model the interatomic interaction between Fe and C (developed by
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Liyanage et al. (Liyanage et al., 2014)). While the EAM potential considers only radial dependence in the electron density and its applicability is limited to the metallic bond, the MEAM potential contains additional terms in electron density which represent the symmetry of s, p, d, f orbitals to account for the directionality of interatomic bond such as covalent bonding in carbon. Table 1 lists the information of ORs and habit planes used for construction of the FCI structure. Ferrite has a BCC structure with a lattice parameter of af = 2.851 Å, and cementite has an orthorhombic structure with lattice parameters of ac = 4.470, bc = 5.088 and cc = 6.670 Å (Figure 1). To verify the existence of the ORs and characterize the structure of
ACCEPTED MANUSCRIPT the FCI, we computed the interface energies and conducted an xAIFB analysis after performing the MS/MD simulation. 2.1. Modeling of the ferrite/cementite interface
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In order to conduct the atomistic simulation, we consider flat ferrite/cementite bilayers of the five ORs. To construct the simulation cell with periodic boundaries along the interface, we adjust the orientation of ORs around an arbitrary axis by 0.5 degrees or less,
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which is within the resolution of typical experimental equipment. The size of the initial ferrite and cementite layer was determined by minimizing the misfit strain for each OR, satisfying
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the geometric compatibility and maintaining mechanical equilibrium. Finally, we assembled the ferrite and cementite into a ferrite/cementite bilayer (an un-relaxed structure or a natural dichromatic pattern, NDP). A detailed procedure can be found in the literature (Demkowicz and Thilly, 2011; Kang et al., 2012a; Wang et al., 2013). Other energy minimum structures
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can be obtained by changing the simulation conditions such as the rotation angle of interface (tilt and/or twist), applied strain level and equilibrium lattice parameter as pointed out in the literature (Kang et al., 2012b; Salehinia et al., 2015; Shao et al., 2014). The current work does
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observed ORs.
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not consider the change of simulation conditions but only focus on the experimentally
2.2. Molecular statics/dynamics simulation We always placed the habit plane perpendicular to the y-axis and periodic boundary
conditions (PBC) were applied to the x and z directions (Figure 2). In order to model the structure of the FCI for each OR, we obtained the minimum energy state of the ferrite/cementite bilayers by employing MS and MD simulations. First, all of the simulations were carried out at 0 K and a minimum energy state was found using a MS simulation with the conjugate gradient method. Then, MD simulations with simulated annealing were carried
ACCEPTED MANUSCRIPT out for 100 ps from 800 K to 10 K using the Nosé-Hoover isobaric-isothermal (NPT) ensemble. Finally, we conducted a MS simulation with the conjugate gradient method again to obtain the fully relaxed structure of the FCI.
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2.3. Calculation of the interface energy Once we completed the MS/MD simulations, we computed the interface energies of the atomic structures to verify the existence of the five ORs from a thermodynamics point-of-
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view. The true ORs are most likely to have lower interface energies compared to ones that do
can be expressed as Ncf
Nc
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not exist. For the relaxed structure of the ferrite/cementite bilayer, the total potential energy
Nf
c f cf c f Ecf i = Ei + Ei + γ + γ + γ ·A i=1
i=1
(1)
i=1
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th c f where A is the cross-sectional area, Ecf i , Ei and Ei are the potential energies of i atom in
the ferrite/cementite bilayer, the initial cementite layer and the initial ferrite layer, respectively, Ncf, Nc and Nf are the number of atoms in the ferrite/cementite bilayer, the initial
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cementite layer and the initial ferrite layer, respectively, γcf, γc and γf are the interface energies for the FCI and the surface energies of the cementite and ferrite phases, respectively. If we
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assume that the effect of the free surface is negligible at the interfacial region defined by the half-thickness, t, in Figure 2, then Equation 1 can be simplified to c f cf Ecf i ≅ Ei + Ei + γ · A i∈g
i∈g
i∈g
(2)
where g = { i | -t ≤ yk ≤ t, k = 1 to Ncf } where yk is the y-coordinate of the kth atom and g denotes the interface group. Hence, the interface energy can be obtained using Equation 2, which does not require the knowledge of
ACCEPTED MANUSCRIPT the surface energies. 2.3. Characterization of misfit dislocations Misfit dislocations are formed to compensate for the atomic mismatches between the
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ferrite and cementite phases. In order to characterize the misfit dislocation network at the FCI of each OR, we extended the original atomically informed Frank-Bilby method (Wang et al., 2013; Wang et al., 2014) to consider the in-plane shear transformation from cementite to
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ferrite. In the xAIFB method, we assume the zig-zag atomic array on the habit plane is a straight line using an edge-to-edge matching condition (Zhang and Kelly, 1998, 2005) and
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only consider the in-plane misfit dislocations on the habit plane. The overall procedure for completing the xAIFB method is summarized in the flow chart in Figure 3. To construct the reference lattice (or a commensurate/coherent dichromatic pattern (CDP)), it is necessary to compute the transformation matrix Scf from cementite to ferrite.
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The transformation matrix Scf can be expressed as
(3)
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xf = Scf · xc
where xf and xc are the atomic positions of the NDPs for ferrite and cementite (Figure 4),
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respectively. The original AIFB method (Wang et al., 2013; Wang et al., 2014) considered the transformation to be a combination of rotation and dilatation because it was developed to analyze the square or hexagonal arrays between two cubic crystals; FCC and BCC . Because we want to consider the interface between BCC and orthorhombic crystals in this study, it is necessary to involve an additional shear transformation to construct the general reference lattice structure (Hirth et al., 2013). The transformation matrix Scf can be expressed as
ACCEPTED MANUSCRIPT Scf = Q · T · M where Q = cosθ sinθ
-sinθ , T = 1 0 cosθ
m γ and M = x 0 1
(4)
0 mz
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where Q, T and M are the rotation, shear and dilatation transformation matrices, respectively. Further, we can build the trial reference lattice xr by using the geometric factors, χx, χz, χɵ and
xr = Sc · xc = Sf · xf
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Sc = Qc · Tc · Mc
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χγ. The trial reference lattice can be computed as
cosχθ θ -sinχθ θ 1 where Qc = , Tc = sinχθ θ cosχθ θ 0
χ mx χγ γ and Mc = x 0 1
0 χz mz
(5)
(6)
where Sf and Sc are the transformation matrices from the ferrite/cementite to the trial
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reference lattices, and Qc, Tc and Mc are the rotation, shear and dilatation transformation matrices from the cementite to the trial reference lattices, respectively. Utilizing Equations 3 through 6, we compute the net Burgers vector, BFB, at the continuum level using the Frank-
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Bilby equation (Sutton and Balluffi, 2007). The Frank-Bilby equation can be expressed as BFB p = invSf − invSc · p
(7)
where p, invSf and invSc represent the probe vectors, the inverse of Sf and Sc, respectively. We identified the possible set of Burgers vectors, bipossible , in the trial reference lattice as described in Figure 5 and conducted the disregistry analysis between the un-relaxed and relaxed structures of the FCI using the reference lattice. To conduct the disregistry analysis, UNR we computed the relative displacements, rCDP and rRLX ij , rij ij . The relative displacements,
, rUNR and rRLX , represent the relative displacement between the ith atom in rCDP ij ij ij
ACCEPTED MANUSCRIPT ferrite/cementite and jth atom in the cementite/ferrite pair from the initial structure to the trial reference lattice, the un-relaxed and relaxed structure, respectively. After computing the rc ru uc rc relative displacements, we computed the disregistries, ruc ij , rij and rij . rij and rij , which
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represent the relative displacements from the trial reference lattice to the un-relaxed and relaxed structures, respectively, and rru ij represents the relative displacement from the unrelaxed structure to the relaxed structure. Based on the disregistry information, we
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characterized the geometry of the misfit dislocations, including the line orientation (ξi) and the line spacing (di). In this study, the line orientation is represented by the angle measured
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counterclockwise from the x-axis. Finally, the net Burgers vector, BKN, at the discrete level can be calculated using the Knowles equation. The Knowles equation (Knowles, 2006) can be expressed as
N
B
KN
p = n × ξi · p bi / di
(8)
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i=1
where n and N represent the normal vector of the FCI (the y-axis in this work) and the number of misfit dislocation types at the FCI, respectively, and bi represents the Burgers
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vector of each dislocation. Once we obtained the net Burgers vector, BFB, at the continuum
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level and the net Burgers vector, BKN, at the discrete level, we checked to see if they were identical. If they did not match, we changed the geometric factors and conducted the xAIFB analysis again until it reached convergence. Once we obtained the converged reference lattice, we characterized the detailed information pertaining to the misfit dislocations at the FCI. More information on the original AIFB method can be found in the literature (Wang et al., 2013; Wang et al., 2014). 2.4. Atomic structure of Pitsch-Petch orientation relationship
ACCEPTED MANUSCRIPT Taking the PP OR as a representative example, we will explain how the misfit dislocations were characterized. First, within a maximum of 0.5 degrees of rotation of the ferrite around the normal vector of the habit plane, we constructed the atomic structure of the PP OR with periodic boundary conditions along the x and z directions. The constructed PP
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10 35]f, Yf = [5 2 1]f and Zf = [10 31 OR and its habit plane can be represented by Xf = [11
]f for the ferrite layer and Xc = [0 1 0]c, Yc = [0 0 1]c and Zc = [1 0 0]c for the cementite 12
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layer. In this study, the subscripts f and c represent the crystal coordinates of ferrite and cementite, respectively; the real Cartesian coordinate does not include a subscript. Yf and Yc are perpendicular to each habit plane. The angle between Xf and [1 1 3]f is 2.49 degrees, and
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the angle between Zf and [1 3 1 ]f is 2.73 degrees. Hence, even after a slight adjustment in orientation, the constructed PP OR is consistent with the conventional PP OR in Table 1 within the tolerance of the experimental apparatus. From the refined PP OR, we determined
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the number of periodic unit cells satisfying the minimum misfit strain: Nfx = 2, Nfy = 6 and Nfz = 2 for ferrite and Ncx = 43, Ncy = 14 and Ncz = 44 for cementite. The in-plane misfit strain is ɛxx = 8.95 × 10-3 and ɛzz = 6.42 × 10-3 for the x and z direction, respectively. In order
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to make-up for the misfit strain, we iteratively applied the local strain to the ferrite and cementite until both geometric compatibility and mechanical equilibrium were attained, and
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we arrived at the un-relaxed structure as described in Figure 2. Once the un-relaxed structure of the FCI was constructed, we conducted a MS/MD simulation (described in Section 2.2). When the MS/MD simulation was performed, the final dimension of the un-relaxed and relaxed structure of the FCI were found to be Lcx = Lfx = 216.71 Å and Lcz = Lfz = 198.09 Å in the x and z directions, respectively. We obtained the un-relaxed and relaxed structures as described in Figure 7. In addition, we computed the interface energy using Equation 2. The interface energy of the PP OR was 660.0 mJ/m2. We repeated the above procedure to construct the ferrite/cementite bilayers for the other ORs. The geometric information for the
ACCEPTED MANUSCRIPT five ORs is summarized in Table 2, and the interface energies are plotted in Figure 6. The atomic arrangement of the habit plane of the PP OR involves rotation, shear and dilatation transformations from cementite to ferrite. One must apply the xAIFB method to
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characterize the misfit dislocations in the PP OR. To this end, we constructed the trial reference lattice using the NDPs and the initial geometric factors (χx, χz, χɵ, χγ) = (0.5, 0.5, 0.5, 0.5) in Figure 4. After that, we computed the net Burgers vector, BFB, at the continuum level
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using Equation 7. The computed net Burgers vectors are BFB(p1) = [-0.075, 0.041] Å and BFB(p2) = [0.051, 0.054] Å for the given probe vectors, p1 = [1, 0] Å and p2 = [0, 1] Å,
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respectively. Since we considered the in-plane transformation in this work, we present only the x and z components of each vector. The atomic configurations of the un-relaxed and relaxed structures on the habit plane are presented in Figure 7. For the un-relaxed and relaxed structures, we observe Moiré patterns and semi-coherency at the FCI, respectively, providing
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little information on the geometry of the misfit dislocations such as the line orientation and spacing. In order to analyze the misfit dislocations in a quantitative manner, disregistry analyses were carried out. From the Moiré patterns and the disregistry analysis, we clearly
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identified the line orientations and spacings. The line orientations are ξ1 = 58.34 degrees and ξ2 = 140.55 degrees with respect to the positive x-axis, and the line spacings are d1 = 52.51 Å
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and d2 = 67.46 Å, respectively (see Figure 5). After the disregistry analysis, we selected the possible set of Burgers vectors from the trial reference lattice (see Figure 5). The selected Burgers vectors for each dislocation are b1 = [-4.91, -0.11] Å and b2 = [0.11, 4.59] Å. Next, we computed the net Burgers vector, BKN, at the discrete level using Equation 8. The computed net Burgers vectors are BKN(p1) = [-0.079, 0.042] Å and BKN(p2) = [0.050, 0.054] Å for the given probe vectors. The net Burgers vectors determined using the Frank-Bilby analysis and the Knowles equation were not identical. So, we changed the geometric factors and repeated the xAIFB analysis until both net Burgers vectors converged within a reasonable
ACCEPTED MANUSCRIPT tolerance. In this study, we used 10-4 as the tolerance. As a result, the geometric factors are (χx, χz, χɵ, χγ) = (0.14, 0.23, 0.13, 0.85), the Burgers vectors are b1 = [-4.76, -0.18] Å and b2 = [0.11, 4.67] Å and the net Burgers vectors are BKN(p1) = BFB(p1) = [-0.079, 0.042] Å and
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BKN(p2) = BFB(p2) = [0.046, 0.055] Å for the given probe vectors. In Figure 8, we see that the disregistry analysis and the computed Burgers vectors give identical values. In the disregistry analysis, rrc ij represents the non-uniform displacement from CDP to relaxed structure. It
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shows displacement jump when we have dislocation line along the probe vector. So, we ru utilize rrc ij as a measure of Burgers vectors. rij describes atomic displacement during the
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formation of misfit dislocation in the relaxation process. It gives information on the stationary points which are candidates for the positions of dislocations. Combining information of rrc ij and rru ij , we can determine the position of dislocation lines. In figure 8, we used green arrow to indicate the position of dislocation lines. We repeated the above procedures to characterize
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the misfit dislocations for the other ORs. The characteristics of the misfit dislocations for each OR computed using the xAIFB method are summarized in Table 3, and the atomic configurations of the un-relaxed and relaxed structures, the idealized dislocations, 2D
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disregistry maps for five ORs and additional information such as dislocation density maps are
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summarized in the Supplementary Material. 3. Results
3.1. Interface energies for the five orientation relationships In the diffusive eutectoid reaction, the lamellar structure forms along the direction
which forms small interface energy at FCI. The existence of the five ORs can be tested by comparing the interface energies for the five ORs calculated using Equation 2 and increasing the half-thickness of the interface region, t, as described in Figure 2. The results are given in Figure 6 and summarized in Table 2. Figure 6 shows that all the interface energies converge
ACCEPTED MANUSCRIPT to specific values over 15 Å, meaning the effect of the interface becomes negligible over 15 Å from the FCI. Among the five ORs, the IS OR has the lowest interface energy (499.9 mJ/m2) and shows rapid convergence compared to the others. The interface energies of BA
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and Near BA OR are 628.2 and 536.1 mJ/m2, respectively. Although the BA and Near BA ORs have small differences in orientation, their interface energies are very different. Likewise, the interface energies of PP and Near PP OR are 660.0 and 555.2 mJ/m2, respectively. From
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these results, we see that small changes in orientation can give rise to large differences in interface energy. Additionally, we suspect that of the similar ORs, the Near BA OR is
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preferred to the BA OR and the Near PP OR is preferred to the PP OR.
3.2. The characteristics of the misfit dislocations for the five orientation relationships The characteristics of misfit dislocation networks obtained by the xAIFB method are summarized in Table 3. Unlike the others, the IS OR has dislocation lines along only one
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direction. For the IS OR, the Burgers vector is b1 = [-2.51, 0.00] Å, the line orientation is paralleled to the z-axis and line spacing is d1 = 83.95 Å. Thus, the atomic mismatch between ferrite and cementite along the z-axis is almost zero. For BA and Near BA OR, two
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dislocation lines are formed. For BA OR, the two dislocation lines have the following Burgers vectors: b1 = [0.00, 4.20] and b2 = [-2.47, 0.00] Å. The line orientations are parallel
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to the x- and z-axis, and the line spacings are d1 = 40.32 Å and d2 = 81.48 Å, respectively. For Near BA OR, the dislocation lines have the following Burgers vectors: b1 = [0.84, -2.09] and b2 = [-2.51, 0.00] Å. The line orientations are parallel to the x- and z-axis, and the line spacings are d1 = 18.18 Å and d2 = 86.42 Å, respectively. Each component of the Burgers vectors for BA OR is larger than the respective vectors for Near BA OR, meaning that the atomic mismatches in the BA OR are large compared with those in the Near BA OR. Likewise, both PP and Near PP OR have two dislocation lines. For PP OR, the Burgers
ACCEPTED MANUSCRIPT vectors of each dislocation are b1 = [-4.76,-0.18] and b2 = [0.11, 4.67] Å, and their line orientations are ξ1 = 58.34 degrees and ξ2 = 140.55 degrees, respectively. The line spacings are d1 = 52.51 Å and d2 = 67.46 Å, respectively. For Near PP OR, the Burgers vector of each
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dislocation is b1 = [2.18, 1.06] and b2 = [-4.91, 0.00] Å, the line orientations are ξ1 = 0 degrees and ξ2 = 41.30 degrees and the line spacing distances are d1 = 22.35 Å and d2 = 43.66 Å, respectively. For the Near BA and Near PP OR, the dislocation lines along the x-axis are
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generated by the out-of-plane steps formed by small misorientation angle of the ferrite and the other are developed by the in-plane atomic mismatches.
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4. Discussion
The different in-plane atomic arrangements of ferrite and cementite result in atomic mismatches. Hence, each atom undergoes non-uniform displacement to satisfy the crystallographic compatibility (i.e., the edge-to-edge matching condition). In this study, we
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verify the existence of the reported five ORs and identify the structure of the FCI. The interface energies of the FCI have been obtained from experimental and theoretical studies (Das et al., 1993; Deb and Chaturvedi, 1982; Kilchner et al., 1977; Kramer et al., 1958;
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Martin and Sellars, 1970; Ruda et al., 2009; Zhang et al., 2015). The interface energies
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reported in the literature are summarized in Table 4. For BA and PP OR, the interface energies obtained here are larger than those previously reported. In addition, the experimental observation of the FCI (Zhang and Kelly, 1997; Zhou and Shiflet, 1992) noted that the exact BA and PP ORs may not exist in pearlitic steel. Based on the interface energies obtained in this work and the experimental observations of the FCI, we conclude that the BA and PP ORs reported in experimental studies performed before the 1990s (Bagaryatsky, 1950; Isaichev, 1947; Petch, 1953; Pitsch, 1962) are indeed the Near BA and Near PP ORs. In short, the exact BA and PP ORs are less likely to exist in pearlitic steel. Small differences between the Near
ACCEPTED MANUSCRIPT BA and BA ORs (as well as the Near PP and PP ORs) were not differentiated due to the limited resolution of the apparatuses used during this time period. From the detailed information on the misfit dislocations shown in Table 3, we see
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that IS OR has a single dislocation line parallel to the z-axis. This component of the Burgers vector of IS OR is relatively small compared with that of the other ORs, meaning that the atoms at the FCI in the IS OR undergo small non-uniform displacements. Accordingly, the
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potential energy of each atom at the interface region may not change as much as that of atoms in the other ORs. The interface energy of the IS OR is the smallest of five ORs (Figure 6).
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Although BA and Near BA OR have small misorientation angles, the difference between the interface energies of the BA OR and the Near BA OR is quite large (the same is observed with the PP OR and the Near PP OR). Kang et al. (Kang et al., 2012b) showed that the change in the interface energy over the misorientation angle can be explained using the strain
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energy from the misfit dislocations, which is proportional to the square of the Burgers vector, ||b||2, and inversely proportional to its periodic length, L. For the dislocations with b1 Burgers vectors, the periodic length of the BA OR (L = d1 = 40.32 Å) is approximately twice that of
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the Near BA OR (L = d1 = 18.18 Å); however, the square of the Burgers vector of the BA OR (||b1||2 = 17.64) is more than three times that of the Near BA OR (||b1||2 = 5.07). Hence, we
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conclude that the BA OR has a higher interface energy than the Near BA OR because of the larger strain energy needed to form misfit dislocation networks. These qualitative analyses suggest that small misorientation angles can cause significant changes in the strain energies by changing the magnitude of the Burgers vectors of the dislocations and their spacing at the heterophase boundary. Conversely, it is interesting to note that the true ORs (IS, Near BA and Near PP) satisfy the parallelism of the close-packed plane {103}c//{110}f from a crystallographic point-of-view. The interplanar spacings of {103}c (= 1.991 Å) and {110}f (= 2.016 Å) are similar in value. The {103}c and {110}f parallelism of the FCI guarantees the
ACCEPTED MANUSCRIPT edge-to-edge matching conditions along the z-axis at the FCI. Moreover, the true ORs also satisfy the parallelism of the close-packed direction [010]c//[111]f for the IS OR, [010]c//[111]f for the Near BA OR and [010]c//[113]f for the Near PP OR. The interatomic
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spacings of [010]c (= 2.544 Å), [111]f (= 2.469 Å) and [113]f (= 2.364 Å) are not much closer in value, but they satisfy the edge-to-edge matching condition. So, each atom of the IS, Near BA and Near PP ORs does not need to undergo a large displacement compared with each one
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in the BA and PP ORs. As a result, the structure of the FCI for the IS, Near BA and Near PP ORs show lower interface energies compared with the BA and PP ORs. Hence, we conclude
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that the exact BA and PP ORs are less likely to exist in pearlitic steel. 5. Conclusions
Atomistic simulation and xAIFB analysis were performed to characterize the FCI misfit dislocations in five ORs and to verify the existence of each OR. We present the atomic
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structure of the FCIs for the five ORs. We find that the parallelism of both the close-packed plane {103}c//{110}f and the close-packed direction [010]c//[111]f (or [1 13]f) is related to the formation of low interfacial free energies, and may govern the overall diffusive eutectoid
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reaction from a thermodynamics point-of-view. We conclude that the true ORs of pearlitic
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steel are the IS, Near BA and Near PP ORs, supported by the calculated interface energies and experimental results. To summarize, -
The xAIFB method was utilized to analyze the general interface structure.
The existence of the IS, Near BA and Near PP ORs was verified based on interface energies.
-
The atomic structure of pearlitic steel was identified using the xAIFB method for the first time.
ACCEPTED MANUSCRIPT Acknowledgments We acknowledge support from the Basic Science Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future
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Planning (2013R1A1A1010091) and the Ministry of Education (2013R1A1A2063917). References
939-946. Bagaryatsky, Y.A., 1950. Dokl. Akad. Nauk SSSR 73, 1161.
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Andrews, K.W., 1963. The structure of cementite and its relation to ferrite. Acta Metallurgica 11,
Bollmann, W., 1970. Crystal defects and crystalline interfaces. Springer-Verlag, Berlin. Society of London A 196, 64-73.
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Charmers, B., 1949. Some crystal-boundary phenomena in metals. Proceedings of the Royal Chu, H.J., Wang, J., Beyerlein, I.J., Pan, E., 2013. Dislocation models of interfacial shearing induced by an approaching lattice glide dislocation. International Journal of Plasticity 41, 1-13. Das, S.K., Biswas, A., Ghosh, R.N., 1993. Volume fraction dependent particle coarsening in plain carbon steel. Acta Metallurgica Materials 41, 777-781.
Deb, P., Chaturvedi, M.C., 1982. Coarsening Behavior of Cementite Particles in a Ferrite Matrix in
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10B30 steel. Metallography 15, 341-354.
Demkowicz, M.J., Thilly, L., 2011. Structure, shear resistance and interaction with point defects of interfaces in Cu Nb nanocomposites synthesized by severe plastic deformation. Acta Materialia 59, 7744-7756. 14, 141-205.
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Demkowicz, M.J., Wang, J., Hoagland, R.G., 2008. Interfaces Between Dissimilar Crystalline Solids. Dippenaar, R.J., Honeycombe, R.W.K., 1973. The crystallography and nucleation of pearlite.
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ACCEPTED MANUSCRIPT among austenite, ferrite and cementite in Fe C alloys: A unified theory based on ab initio simulations. Acta Materialia 99, 281-289. Zhang, Y.D., Esling, C., Calcagnotto, M., Zhao, X., Zuo, L., 2007. New insights into crystallographic correlations between ferrite and cementite in lamellar eutectoid structures, obtained by SEM FEG/EBSD and an indirect two-trace method. Journal of Applied Crystallography 40, 849-856. Metallurgical Transactions A 22A, 1349-1365.
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ACCEPTED MANUSCRIPT Figure Captions
Figure 1. Unit-cell structure of cementite (orthorhombic) and ferrite (body-centered cubic).
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Figure 2. An atomic model of the ferrite/cementite bilayer of PP OR.
Figure 3. A flow chart of the extended atomically informed Frank-Bilby method.
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Figure 4. The in-plane lattice structure of cementite (left), the reference lattice (center) and ferrite (right) for PP OR.
Figure 5. The possible set of Burgers vectors in the reference lattice and the idealized misfit dislocation for PP OR.
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Figure 6. The interface energies for the five ORs with increasing half-thickness of the interface layer of thickness, t.
Figure 7. The atomic arrangement of the un-relaxed (top) and relaxed (middle) structure and a 2D disregistry (rru ij ) plot (bottom) for PP OR (scaling factor for 2D disregistry = 1.0).
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Figure 8. Disregistry plots along the (a) line AB and (b) line CD for PP OR; disregistry rrc ij is used for the measure of the Burgers vectors and green arrows indicate the position of dislocations.
ACCEPTED MANUSCRIPT Table 1. The orientation relationships between the ferrite and cementite phases. Name
Orientation relationship
Habit plane
[100]c || [11ത 0]f, [010]c || [111]f and (001)c || (112ത)f
(001)c||(112ത)f
IS
(103ത)c || (101ത )f, [010]c || [111]f and (1ത 01)c || (21ത 1ത)f
(1ത 01)c||(21ത 1ത)f
PP
[100]c ∠ [1ത 31ത]f = 2.6°, [010]c ∠ [1ത 13]f = 2.6° and (001)c || (521)f
(001)c||(521)f
Near BA
(103ത)c || (101ത )f, [010]c || [111]f and (1ത 01)c || (21ത 1ത)f
(001)c ∠ (112ത)f = 3°
Near PP
(103)c || (110)f, [010]c || [1ത 13]f and (001)c ∠ (521)f = 1°
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BA
(001)c ∠ (521)f = 1°
ACCEPTED MANUSCRIPT Table 2. The geometry of the ferrite/cementite bilayer and the interface energy for each orientation relationship. Name BA
Lx
Lcy
Lfy
Lz
εxx
εzz
167.21
80.04
83.80
166.02
1.82 × 10-5
4.31 × 10-4
628.2
-3
-4
γcf [mJ/m2]
152.42
81.69
83.80
160.57
2.92 × 10
4.34 × 10
499.9
PP
216.71
93.38
93.69
198.09
8.95 × 10-3
6.42 × 10-3
660.0
Near BA
167.18
93.38
94.47
162.66
1.82 × 10-5
1.68 × 10-2
536.1
Near PP
131.54
73.37
74.45
134.96
7.31 × 10-4
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IS
4.47 × 10-3
555.2
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*Unit: Å (Angstrom)
ACCEPTED MANUSCRIPT Table 3. Detailed information regarding misfit dislocation for each orientation relationship. Name
Burgers vector
Line orientation
2
1
Line spacing
2
b [bx, bz]
b [bx, bz]
ξ [deg.]
ξ [deg.]
d1
d2
BA
[ 0.00, 4.20]
[-2.47, 0.00]
0.00
90.00
40.32
81.48
IS
[-2.51, 0.00]
N/A
90.00
N/A
83.95
N/A
PP
[-4.76,-0.18]
[-0.11, 4.67]
58.34
140.55
52.51
67.46
Near BA
[ 0.84,-2.09]
[-2.51, 0.00]
0.00
90.00
18.18
86.42
Near PP
[ 2.18, 1.06]
[-4.91, 0.00]
0.00
41.30
22.35
43.66
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*Unit: Å (Angstrom), line orientation represents the angle between positive x-axis and dislocation line around negative y-axis
ACCEPTED MANUSCRIPT Table 4. Summary of the interface energies reported in the literature.
Coarsening rate Coarsening rate Interface enthalpy Interface enthalpy Atomic simulation Atomic simulation Dihedral angle
γcf [mJ/m2] 560 248 - 417 700 ± 300 500 ± 360 615 450 520 ± 130
Note
Reference (Das et al., 1993) (Deb and Chaturvedi, 1982) (Kramer et al., 1958) (Kilchner et al., 1977)
BA OR BA OR
(Ruda et al., 2009)
(Zhang et al., 2015)
(Martin and Sellars, 1970)
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861 903 - 963 1000 973
Method
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Temperature [K]
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Figure 1
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Figure 1. Unit-cell structure of cementite (orthorhombic) and ferrite (body-centered cubic).
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Figure 2
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Figure 2. An atomic model of the ferrite/cementite bilayer of PP OR.
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Figure 3
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Figure 3. A flow chart of the extended atomically informed Frank-Bilby method.
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Figure 4
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Figure 4. The in-plane lattice structure of cementite (left), the reference lattice (center) and ferrite (right) for PP OR.
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Figure 5
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Figure 5. The possible set of Burgers vectors in the reference lattice and the idealized misfit dislocation for PP OR.
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Figure 6
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Figure 6. The interface energies for the five ORs with increasing half-thickness of the interface layer of thickness, t.
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Figure 7
Figure 7. The atomic arrangement of the un-relaxed (top) and relaxed (middle) structure and a 2D disregistry (rru ij ) plot (bottom) for PP OR (scaling factor for 2D disregistry = 1.0).
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Figure 8
Figure 8. Disregistry plots along the (a) line AB and (b) line CD for PP OR; disregistry rrc ij is used for the measure of the Burgers vectors and green arrows indicate the position of dislocations.
ACCEPTED MANUSCRIPT An atomistic-to-continuum method is proposed to study the heterophase boundary.
2.
The method is applied to investigate the ferrite/cementite interface in pearlitic steel.
3.
Misfit dislocations and interface energies of five orientation relationships are analyzed.
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Near BA, IS and Near PP ORs are energetically more favorable than BA and PP ORs.
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The interface energies are explained by the characteristics of misfit dislocations.
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