A first principles study of cementite (fe3c) and its alloyed counterparts: elastic constants, elastic anisotropies, and isotropic elastic moduli

A first principles study of cementite (Fe3C) and its alloyed counterparts Elastic constants, elastic anisotropies, and isotropic elastic moduli A first principles study of cementite (Fe3C) and its all[.]

A first-principles study of cementite (Fe3C) and its alloyed counterparts: Elastic constants, elastic anisotropies, and isotropic elastic moduli G Ghosh Citation: AIP Advances 5, 087102 (2015); doi: 10.1063/1.4928208 View online: http://dx.doi.org/10.1063/1.4928208 View Table of Contents: http://aip.scitation.org/toc/adv/5/8 Published by the American Institute of Physics AIP ADVANCES 5, 087102 (2015) A first-principles study of cementite (Fe3C) and its alloyed counterparts: Elastic constants, elastic anisotropies, and isotropic elastic moduli G Ghosha Department of Materials Science and Engineering, Robert R McCormick School of Engineering and Applied Science, Northwestern University, 2220 Campus Drive, Evanston, IL 60208-3108, USA (Received 25 May 2015; accepted 22 July 2015; published online 10 August 2015) A comprehensive computational study of elastic properties of cementite (Fe3C) and its alloyed counterparts (M3C (M = Al, Co, Cr, Cu, Fe, Hf, Mn, Mo, Nb, Ni, Si, Ta, Ti, V, W, Zr, Cr2FeC and CrFe2C) having the crystal structure of Fe3C is carried out employing electronic density-functional theory (DFT), all-electron PAW pseudopotentials and the generalized gradient approximation for the exchangecorrelation energy (GGA) Specifically, as a part of our systematic study of cohesive properties of solids and in the spirit of materials genome, following properties are calculated: (i) single-crystal elastic constants, Cij, of above M3Cs; (ii) anisotropies of bulk, Young’s and shear moduli, and Poisson’s ratio based on calculated Cijs, demonstrating their extreme anisotropies; (iii) isotropic (polycrystalline) elastic moduli (bulk, shear, Young’s moduli and Poisson’s ratio) of M3Cs by homogenization of calculated Cijs; and (iv) acoustic Debye temperature, θD, of M3Cs based on calculated Cijs We provide a critical appraisal of available data of polycrystalline elastic properties of alloyed cementite Calculated single crystal properties may be incorporated in anisotropic constitutive models to develop and test microstructureprocessing-property-performance links in multi-phase materials where cementite is a constituent phase C 2015 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4928208] I INTRODUCTION Cementite (Fe3C) is a metastable phase in the binary Fe-C system Its alloyed counterparts (here, represented as M3C with M = Al, Co, Cr, Cu, Fe, Hf, Mn, Mo, Nb, Ni, Si, Ta, Ti, V, W, Zr, Cr2FeC and CrFe2C), though unstable or metastable, is ubiquitous in low-, medium- and high-carbon steels It is well known that the morphology and kinetics of cementite precipitates have a tremendous influence on the mechanical properties of steels Historically, these factors have been manipulated by adding one or more alloying elements in steels to obtain desired microstructures, and hence properties The precipitation of cementite can significantly alter the subsequent precipitation of more stable carbides.1–4 Besides structural steels, thermo-physical properties of iron carbides (including cementite) have received considerable attention as they are important constituents in earth’s core.5–7 Recently, the Materials Genome Initiative (MGI)8–10 and genomic approaches are being actively pursued to facilitate materials design and to promote Integrated Computational Materials Engineering (ICME) It has also been realized, at least by some, that MGI approaches need highly reliable thermodynamic and kinetic databases within CALPHAD formalism11 to achieve lofty goals of ICME Many important issues relevant to both MGI and ICME, especially first-principles calculations and their integration within CALPHAD formalism, were also underscored in our previous publication,12 but in the context of modeling phase stability of multi-component, multi-phase systems a Corresponding author Tel.: +1-847-467-2594; fax: +1-847-491-7820 E-mail address: g-ghosh@northwestern.edu 2158-3226/2015/5(8)/087102/19 5, 087102-1 © Author(s) 2015 087102-2 G Ghosh AIP Advances 5, 087102 (2015) Usually, diffusion-controlled transformations lead to partitioning of alloying elements between phases but the martensitic transformation is a composition invariant transformation and it is driven by chemical forces Hence, it is crucial to quantify the magnitude of driving force to model and predict martensitic transformation kinetics In this context, a highly successful example is the development of kMART101 (kinetics of MARtensitic Transformation) database13,15 that integrates solution thermodynamics within CALPHAD formalism and empirical modeling (temperature and composition dependence) of elastic properties14 to assist the design of secondary hardening ultrahigh strength steels, an effort initiated by the Steel Research Group9,16 at Northwestern University Notwithstanding its technological importance, until recently relatively little effort has been made to investigate intrinsic mechanical properties of cementite For example, the hardness and isotropic (polycrystalline) elastic moduli of cementite have been reported by several authors.17–20,22–29 Inoue et al.30,31 reported deformation and fracture behavior of cementite Umemoto and co-workers24,28 reported the effect of V, Cr, Mo and Mn on hardness and polycrystalline elastic moduli of Fe3C Single-phase, thin films of cementite have been synthesized by PVD method on a Si substrate,21,22 and their Young’s modulus and Poisson’s ratio have been measured The thickness of such films was about 2.5 µm and a grain size of about 50 nm Umemoto et al.23–25,27,28 synthesized bulk samples of cementite using a combination of mechanical alloying followed by spark plasma sintering at 900oC and achieved a density of 98% of theoretical density They reported hardness and polycrystalline elastic moduli of cementite Ledbetter29 used Fe-C alloys with various C-contents and measured elastic-stiffness tensor Cijkl using pulse-echo and resonance ultrasound spectroscopy (RUS) techniques, and then results were extrapolated to 25 at.% C As reported by Ledbetter,29 above experimental techniques allow quasi-isotropic moduli to be extracted even from specimens with texture Besides Young’s modulus, Ledbetter reported several other thermo physical properties of Fe3C, such as Debye temperature, shear modulus, Poisson’s ratio Inoue et al.30,31 studied fracture behavior of cementite lamellae embeded in an iron matrix To overcome experimental difficulties in synthesizing single-phase cementite and large specimens are required for experimental studies of its mechanical properties, attempts have been made to calculate elastic properties using first-principles methods.7,32–42 To the best of our knowledge, all first-principles calculations have been carried out only for Fe3C, and a very few attempts have been made to predict its stability and understand mechanical properties of multi-component cementite Only recently, Jang et al.40 reported the effect of Si on the stability of Fe3C The crystal structure of cementite, Fe3C, is well established It is orthorhombic wit the space group Pnma (no 62) The Wyckoff positions of atoms are: C at 4c (x 1,0.25,z1); FeI at 4c (x 2,0.25,z2); FeII at 8d (x 3, y3,z3) Also, it is well known that Fe3C is metallic and ferromagnetic with a Curie temperature around 488 K To bridge the current knowledge gaps due to lack of sufficient mechanical property data of cementite in real alloys, we have undertaken a systematic study of elastic properties of M3Cs (M = Al, Co, Cr, Cu, Fe, Hf, Mn, Mo, Nb, Ni, Si, Ta, Ti, V, W and Zr) The alloying elements, M, are chosen to represent a wide variety of commercial alloys Among these, only Fe3C and Mn3C with the structure of cementite are present in respective binary phase diagrams Other alloying elements, M, may dissolve, to a greater or lesser extent in Fe3C depending on the kinetic mode of precipitation.43–45 In general, crystalline materials are mechanically anisotropic but the extent of anisotropies cannot be quantified without prior knowledge of single-crystal elatic constants Hence, in the spirit of materials genome and to enhance our knowledge and fundamental understanding mechanical properties of solids, our goals are to calculate following properties of M3C using first-principles methods: (i) single-crystal elastic constants (Cijs), (ii) the extent of elastic anisotropies based on calculated Cijs, (iii) isotropic (polycrystalline) elastic moduli using the homogenization technique, (iv) rationalize available experimental data of polycrystalline cementites, and (v) acoustic Debye temperature We treat M3Cs (besides Fe3C and Mn3C) as virtual intermetallic and calculate their properties These may provide us valuable information regarding the chemical trend when we try to rationalize the properties of alloyed cementite.24,28 087102-3 G Ghosh AIP Advances 5, 087102 (2015) This paper is organized as follows Section II describes the computational methodology employed in the current study Section III presents single-crystal elastic constants, elastic anisotropies, homogenization of elastic constants to compute isotropic (polycrystalline) elastic moduli (bulk, Young’s, shear moduli and Poisson’s ratio), elastic anisotropies of Young’s and shear moduli, and acoustic Debye temperature Finally, Section IV provides a summary of the findings of the present study II COMPUTATIONAL METHODOLOGY A Ab initio total energy calculations All calculations were carried out using the ab initio total-energy code, VASP (Vienna ab initio simulation package),46–48 the details have been described elsewhere.49 Briefly, we have used pseudopotentials constructed by projector-augmented wave (PAW) method50 with a kinetic energy cut-off of 500 eV for the expansion of electronic wave functions, generalized gradient approximation (GGA) for the exchange-correlation functional due to Perdew and Wang51 with the Vosko-Wilk-Nussair52 interpolation of the correlation energy Due to the importance of magnetism, relevant calculations for cementite containing Co, Cr, Fe, Mn or Ni were done employing spin polarized Hamiltonian; otherwise, non-spin polarized Hamiltonian was employed Cementites, k-point meshes used in our calculations, along with the number of k-points in the irreducible Brillouin zone (IBZ), are listed in Table I B Equation of state and formation energy We have used the equation of state (EOS) due to Vinet et al.53 defining equilibrium energy (Eo), corresponding volume (Vo), bulk modulus (Bo), its pressure derivative (Bo) and a scaled distance (x), as described in a recent publication.49 C Calculation of single-crystal elastic constants of cementite (M3C) For an orthorhombic crystal, there are nine independent single-crystal elastic constants, C11, C12, C13, C22, C23, C33, C44, C55 and C66 We have utilized the generalized Hooke’s law (due to TABLE I k-mesh and the corresponding number of k-points in the irreducible Brillouin zone (IBZ) used in our firstprinciples calculations of equation of state and calculation of C ijs of cementite (M3C) Cementite M3C Al3C Co3C Cr3C Cu3C Fe3C Hf3C Mn3C Nb3C Ni3C Si3C Ta3C Ti3C V3C W3C Zr3C CrFe2C Fe2CrC k-mesh IBZ-kpoints 16 × 13 × 17 11 × × 12 11 × × 12 10 × × 11 11 × × 12 15 × 10 × 22 11 × × 13 11 × × 15 14 × 10 × 15 16 × 13 × 19 13 × × 17 15 × 10 × 22 14 × 10 × 19 19 × 13 × 22 14 × × 20 12 × × 14 12 × × 14 952 144 144 120 144 440 1144 192 144 560 315 440 350 770 350 210 210 087102-4 G Ghosh AIP Advances 5, 087102 (2015) Cauchy), σij = Cijklεkl, at small deformations to account for multiaxial loading conditons and the elastic anisotropy, where σ ij is the stress tensor, ε kl is the Lagrangian strain tensor, and Cijkl is the elastic constant tensor which is × matrix (36 elements in general cases) with the independent elements of elastic constant tensor are determined by the crystal symmetry First-principles calculation of stress tensors was introduced by Nielsen and Martin,54,55 and subsequently implemented in many DFT packages Using Voigt notations, the generalized Hooke’s law for an orthorhombic crystal may be written as: σ1 C11 C12 C13 0 ε1 * +/ * +* + 0 // ε 2// σ2// C21 C22 C23 / / σ3// C31 C32 C33 0 // ε 3// // = / / 0 C44 0 // γ4// τ4 // / / τ5 // 0 0 C55 // γ5// 0 0 C66- ,γ6, τ6 - , (1) where σi and τi are normal and shear stress, respectively, while, ε i and γi are normal and shear strains, respectively Ab initio techniques for calculating single crystal elastic constants may be conveniently classified as energy density, stress-tensor and phonon methods In an energy density method, the elastic constants are derived by calculating first derivatives of energy density (defined as the total energy per volume) as a function of appropriately selected strains In the stress-tensor method, elastic constants are derived by calculating first derivative of stresses with respect to strain In the phonon method, the elastic constants are derived by calculating first derivative of phonon branch with respect to an appropriate wave vector Here, we have used the stress-tensor method to calculate single-crystal elastic constants of cementite Optimal lattice deformations needed to derive Cijs in various crystal systems has been proposed elsewhere.56–58 Compared to energy density and phonon methods, the stress-tensor method significantly reduces both time and computational resources Starting with the fully relaxed geometry of M3C unit cell, we have applied following four distortions allowing calculation of all nine Cijs: *.1 + δ D1 = , 0 0+ 0/// 1- *1 D2 = ,0 1+δ *.1 D3 = ,0 (2) 0+ 0/// 1- (3) + /// + δ- (4) γ y x γz x 2 + γ y z /// (5) // / γyz It is obvious that in first three distortions the symmetry of the unit cell is conserved (remains orthorhombic) but the volume changes, while in the last distortion the symmetry is not conserved (becomes monoclinic) but it is volume-invariant For each type of distortion, we have used strains (δ or γ in Eq (2) to (5)) of ±0.25 and ±0.5% Under imposed distortions, the stresses are calculated after relaxing the unit cell-internal degrees of freedom, and then the Cijs are extracted by a least-square method.56–58 * γx y D4 = γz x , 087102-5 G Ghosh AIP Advances 5, 087102 (2015) Within the range of imposed strains, we find that, indeed, the linear relationship between stress and strain is obeyed Golesorkhtabar et al.58 have discussed numerical accuracies associated with Cijs calculated by the stress tensor method Specifically, considering TiSi2 as a model system with orthorhombic structure, Golesorkhtabar et al.58 noted up to 45% discrepancy between experimental and calculated (using a FP-LAPW code) Cijs Our calculated Cijs of M3Cs are listed in Table II, where we note that Cijs reported by Mookherjee7 are consistently higher than calculated by others We attribute this to the fact that Mookherjee7 carried out non-spin polarized calculations while others performed spin polarized calculations III RESULTS AND DISCUSSIONS A Equation of state (EOS) of binary of M3C The EOS parameters of M3C have been summarized in a recent publication.49 Experimentally, the EOS parameters have been determined only for Fe3C using diamond-and-anvil apparatus26,5,59–61 TABLE II Calculated C ijs (at 0K, in GPa) of M3C, and the acoustic Debye temperature (θD in K) Our calculated values are compared, where available, with similar previously reported results Cementite C 11 C 12 C 13 C 22 C 23 C 33 C 44 C 55 C 66 Debye temperature θD Al3C Co3C 219 416 186 169 184 189 320 345 81 170 344 364 -592 119 124 135 117 Cr3C 522 211 216 438 224 382 114 188 195 Cu3C Fe3C 164 375 81 161 108 144 174 339 54 172 174 298 -33 13 60 132 49 30 Hf3C Mn3C 388b 393c 385d 480g 265 544 156 144 157 237 56 241 162 141 162 236 84 167 345 340 341 443 240 504 162 149 167 188 90 187 322 319 316 480 303 432 15 -60 13 -6 80 62 134 145 131 149 73 200 134 114 131 153 179 Elastically unstable 265.5 386a 696.9 514a Elastically unstable 331.9 394a,468-475e, 501f 477b Mo3C Nb3C Ni3C 468 347 388 217 144 130 204 131 177 333 376 271 218 137 139 351 335 315 37 86 -9 125 68 115 59 85 90 Si3C Ta3C Ti3C V3C 255 402 57 365 106 138 30 130 155 51 130 215 397 228 338 89 161 82 128 179 382 332 405 78 103 75 93 71 70 74 30 90 -7 104 W3C Zr3C CrFe2C Cr2FeC 526 231 421 469 258 48 125 197 230 79 164 231 406 194 341 496 242 76 163 191 422 271 387 469 57 63 14 125 118 61 128 125 32 -14 153 177 a Entropic 147.7 644.5 440a 351.5 429.3 Elastically unstable 338 352.2 321.9 Elastically unstable 588.4 554a 231.5 Elastically unstable 340.7 685.4 Debye temperature as defined in Ref 99 and critical assessment of thermal properties: Ref 100 GGA at K: Ref 37 c CASTEP, LDA at K: Ref 39 dVASP, GGA at K: Ref 38 e VASP, GGA at K: Ref 26 f VASP, GGA at K: Ref 29 gVASP, GGA at K; non-spin polarized calculations: Ref bVASP, 087102-6 G Ghosh AIP Advances 5, 087102 (2015) and computationally using36,39 employing FP-LAPW and CASTEP codes In general, our Bo is higher than experimental values that could be due to the fact that experiments were conducted at room temperature or higher Nevertheless, our calculated Bo of Fe3C shows a good agreement with both FP-LAPW and CASTEP data Also, our Bo value is in reasonably good agreement with experimental data The zero-temperature phase stability of M3C, defined by the energy of formation, ∆Ef , have been summarized in a recent publication.49 We find that ∆Ef values are negative for Cr, Hf, Mn, Nb, Ta, Ti, V and Zr While these are well known carbide formers but none with the structure of cementite Comparing lattice parameters, we find49 that the calculated values for Cr3C, Fe3C and Mn3C agree to within ±1.5% of experimental values.24,62–66 Furthermore, our calculated lattice parameters of Fe3C agree to within ±1% of FP-LAPW values.36,40 As discussed in a recent publication,49 we have observed a good agreement between calculated and measured (when available) Wyckoff positions: agreement to within two significant figures is obtained for all non-symmetry-constrained Wyckoff positions between our calculations and reported experimental data B Single-crystal elastic constants and isotropic (polycrystalline) elastic moduli Elastic constants and elastic anisotropy are fundamental properties in understanding mechanical properties ranging from stress-strain behavior, dislocation motion, crack nucleation, crack propagation etc The necessary conditions for elastic stability in an orthorhombic crystal are:67 C11 > 0,C22 > 0,C33 > 0,C44 > 0,C55 > 0,C66 > 0; (C11 + C22 − 2C12) > 0; (C11 + C33 − 2C13) > 0; (C22 + C33 − 2C23) > 0; (C11 + C22 + C33 + 2C12 + 2C13 + 2C23) > 0; (6) Calculated Cijs of M3Cs are listed in Table II Among M3Cs considered here, the above criterion is not satisfied for Al3C, Cu3C, Ni3C, Ti3C and Zr3C; hence, these are found to be elastically unstable (see, Table II) Consistent with this result, they are also found to be dynamically unstable, as one or more phonon branches yield imaginary frequencies Our calculated values are in good agreement with previously calculated Cijs of Fe3C.37,38 However, Lv et al.39 reported that C44 is negative Barring any typographical error, their Cij values of predict that Fe3C is elastically unstable, which is inconsistent with both present and previous calculations.37,38 Recently, Grimvall et al.68 have discussed the issue of elastic instability and its relation to phonon instability in a number of materials Besides classic cases of bcc-Ti, Hf and Zr, authors have discussed other cases as well As noted by Grimvall et al.,68 in addition to elastic stability the thermodynamic stability of a phase is also influenced by temperature, pressure, magnetism etc via anharmonic effects At present, experimental Cij data of none of M3Cs listed in Table II is available, presumbly due to difficulties in preparing large single crystals necessry to measure individual elastic constants Nevertheless, isotropic (polycrystalline) elastic moduli of Fe3C have been reported several times With the knowledge of Cijs (as in Table II), there are several approaches to estimate the isotropic elastic moduli These include, among others, methods due to Voigt,69 Reuss,70 and Hill.71 Voigt’s69 method is based on the assumption of uniform local strain, ∂εij/∂xk = 0, where εij is the strain tensor and x is the spatial coordinate, while Reuss’s70 method is based on the assumption of uniform local stress, ∂σij/∂xk = 0, where σij is the stress tensor Hill showed that Reuss and Voigt methods yield lower and upper bound, respectively, on effective moduli The arithmatic mean of Voigt bound and Reuss bound has been termed as Voigt-Reuss-Hill (VRH) bound by Chung.72 For anisotropic materials, geometric mean73 or harmonic mean74 has also been suggested to be useful approximation to effective moduli Hashin and Shtrikman (HS)75–77 proposed a method based on variational principles 087102-7 G Ghosh AIP Advances 5, 087102 (2015) leading to tighter bounds for isotropic moduli than the Voigt and Reuss averages Here, the lower and upper HS bounds of elastic moduli are represented as HS(-) and HS(+), respectively Golesorkhtabar et al.58 have discussed numerical accuracies associated with Cijs calculated by the stress tensor method Their calculations show that calculated Cijs of Al and Ti differ from experimental Cijs by as much as 17% We have also calculated Cijs of relevant pure metals in their ground state and compared them with experimental data, especially those available at low temperatures.78 Specifically, we have calculated Cijs (also by stress tensor method) of cubic elements (Al, Cr, Cu, Fe, Nb, Ni, Si, Ta, V and W) and hexagonal elements (Co, Ti, Hf and Zr) relevant to this study Considering inherent limitations of computational methods we conclude that our computed Cijs of M3Cs are accurate and highly reliable but will have to await for relevant experimental data Table III lists the Reuss, Voigt, VRH averages, and HS bounds for the effective bulk (Bo) and isotropic shear (µ) moduli calculated from single-crystal stiffnesses of M3C Usually, these are chosen as the two independent isotropic (polycrystalline) elastic constants Other elastic constants, Young’s modulus (E) and Poisson’s ratio (ν), are readily expressable in terms of Bo and µ Calculated bounds for E and ν are listed in Table IV For all M3Cs, we find that the HS bounds lie within the Voigt bound and Reuss bound The self-consistency between EOS and Cij results is demonstrated in fig 1, showing an excellent correlation between BEOS, BVRH , BHS(-) and BHS(+) This is important, because BEOS is based on the total energy calculations followed by defining an EOS,49 whereas BVRH , BHS(-) and BHS(+) is based on the calculation of Cijs by the stress-tensor method followed by their homogenization, as mentioned above Values of bulk moduli (BEOS) are listed in a recent publication,49 and values of BVRH , BHS(-) and BHS(+) are listed in Table IV In each case, the best fit line in Fig passes nearly through the origin TABLE III Calculated isotropic (polycrystalline) bulk modulus of M3C (based on Cijs at K, (see, values in Table II): bulk modulus (Bo, in GPa) and shear modulus (µ, in GPa), according to Voigt,69 Reuss,70 VRH,72 Hashin and Shtrikman75–77 (lower bound: (HS(−) and upper bound: HS(+)), schemes Our calculated values are compared, where available, with reported experimental data Bo µ M3C Reuss Voigt VRH HS(-) HS (+) Reuss Voigt VRH HS(-) HS(+) Co3C Cr3C Fe3C 240.53 289.90 216.61 242.54 293.72 218.42 240.53 289.90 216.61 241.66 292.32 217.76 90.79 133.42 43.85 99.19 145.62 90.83 96.20 140.60 89.41 138.17 291.08 265.05 208.49 104.88 232.02 208.25 308.71 225.93 297.18 140.91 296.98 269.99 209.24 115.69 232.02 209.26 312.59 228.27 297.21 138.17 291.08 265.05 208.49 104.89 232.02 208.25 308.71 225.93 297.18 139.66 294.46 267.86 208.89 112.49 232.02 208.79 310.87 227.37 297.19 31.61 123.29 64.74 88.47 23.01 97.11 98.56 64.89 48.90 136.63 71.15 147.17 78.52 91.10 53.42 101.32 102.14 82.99 114.41 139.81 94.99 139.95 67.34 74a 70b 69, 72c 72d 90e 51.38 135.22 71.63 89.79 38.22 99.22 100.35 73.94 81.66 138.22 90.79 133.45 43.85 Hf3C Mn3C Mo3C Nb3C Si3C Ta3C V3C W3C CrFe2C Cr2FeC 241.54 291.81 217.52 237a 125b 105, 125c 175±4d 168e 139.54 294.03 267.52 208.87 110.29 232.02 208.76 310.65 227.10 297.19 31.61 123.29 64.74 88.47 23.01 97.11 98.56 64.98 48.90 136.63 63.46 139.52 73.48 90.00 45.82 99.61 100.67 77.43 100.88 138.37 a Resonance technique: Ref 17 technique: Ref 22 c Pulse-echo technique: Ref 26 dPulse-echo technique: Ref 27 e Pulse-echo technique: Ref 29 bResonance 087102-8 G Ghosh AIP Advances 5, 087102 (2015) TABLE IV Calculated isotropic (polycrystalline) Young’s modulus of M3C (based on C ijs at K, see Table II): Young’s modulus (E, in GPa) and Poisson’s ratio (υ), according to Voigt,69 Reuss,70 VRH,72 Hashin and Shtrikman75–77 (lower bound: (HS(−) and upper bound: HS(+)), schemes Our calculated values are compared, where available, with reported experimental data E υ M3C Reuss Voigt VRH HS(-) HS (+) Reuss Voigt VRH HS(-) HS(+) Co3C 241.93 261.87 241.93 254.79 0.3324 0.3200 0.3261 0.3324 0.3243 Cr3C 347.02 347.90 347.02 363.58 0.3005 0.2873 0.2933 0.3005 0.2973 Fe3C 123.23 239.32 251.94 202a 361.98 320a 183.12 200a 180b 216c 180d 140e 168f 175-176g 190h 230±12i 137.29 351.74 245a 197.28 235.61 155a 102.79 260.52 259.47 352a 205.51 218.76 359.00 123.23 214.79 0.4052 0.3174 0.3597 0.4052 0.3356 Hf3C Mn3C 88.11 324.11 182.69 378.92 Mo3C Nb3C Ni3C Si3C Ta3C V3C 179.59 232.52 214.74 238.66 64.32 255.66 255.39 138.88 265.34 263.54 W3C CrFe2C Cr2FeC 181.92 136.83 355.42 228.73 294.01 362.58 0.361b 0.460c 88.11 324.11 165.34 361.47 0.3937 0.3144 0.2839 0.2873 0.3360 0.3006 0.26f 0.223- 0.226g 0.32h 0.275i 0.3937 0.3144 179.59 232.52 201.97 236.09 0.3871 0.3141 0.3674 0.3099 0.3771 0.3199 0.3871 0.3141 0.3743 0.3163 64.33 255.66 255.39 121.03 261.42 260.19 0.3978 0.3164 0.2956 0.2999 0.3094 0.2901 0.3447 0.3128 0.2928 0.3978 0.3164 0.2956 0.3207 0.3122 0.2923 181.92 136.83 355.42 214.48 263.64 359.34 0.4018 0.3991 0.3007 0.3780 0.2853 0.2967 0.3897 0.3394 0.2987 0.4018 0.3991 0.3007 0.3850 0.3067 0.2985 0.3027 0.2954 a Critical assessment of thermal properties: Ref 100 technique: Ref 17 c X-ray technique: Ref 18 dResonance technique: Ref 19 e Resonance technique: Ref 20 f Resonance technique: Ref 22 gPulse-echo technique: Ref 26 hPulse-echo technique: Ref 27 i Pulse-echo technique: Ref 29 bResonance C Bulk modulus and its anisotropy All crystalline materials are anisotropic, hence, the orientation-dependent properties are very important to analyze, predict and design new materials with prescribed properties Below, we provide the several orientation-dependent properties (bulk, Young’s, shear moduli and Poisson ratio) of cementite With the knowledge of elastic stiffness (or compliance) constants presented above it is possible to calculate orientation dependence of bulk modulus (B) given by79 = (s11 + s12 + s13)l 12 + (s12 + s22 + s23)l 22 + (s13 + s23 + s33)l 32 B where sij (= Ci j −1) are elastic compliance constants, and l 1, l and l are direction cosines (7) 087102-9 G Ghosh AIP Advances 5, 087102 (2015) FIG Correlation between BEOS (see, Ref 49) and BVRH or BHS(−) or BHS(+) (see, values in Table IV), all based on our calculated results, of M3Cs listed in Table II Values of BEOS, BVRH, BHS(−) and BHS(+) are reported in a previous publication49 and in Table III, respectively Note that the best fit straight lines pass nearly through the origin From the above relation, directional dependence of M3Cs are calculated As an example, the orientation dependence of B for Mo3C is shown in Fig Clearly, it exhibits nonspherical 3-D shape due to its intrinsic anisotropy We have also calculated linear bulk moduli along orthogonal axes (Ba, Bb, Bc) of M3Cs, and these are listed in Table V It is equally interesting to investigate anisotropy of bulk modulus along the a axis (BBa = Ba/Bb) and c (BBc = Bc/Bb) axis with respect to the b axis which is related to linear compressibility and expressable in terms of Cijs, as has been worked out by Ravindran et al.80 The anisotropy of bulk modulus of M3Cs, mentioned above, is listed in Table V A value of one corresponds to elastic isotropy, while any departure from one corresponds to a degree of anisotropy It is interesting to note that none of experimental bulk modulus value of M3Cs (see Table III in our previous publication),49 especially Fe3C where experimental data is the most, exceeds calculated bulk modulus along any of the orthogonal axes Therefore, these results demonstrate self-consistency between available experimental data and our calculated values D Young’s modulus and its anisotropy It has long been recognized that the Young’s modulus of Fe3C is highly anisotropic.20 The extent of anisotropy of Young’s modulus (E) can be graphically demonstrated by calculating its orientation dependence For an orthorhombic crystal, the Young’s modulus in any direction is given by79 = s11l 14 + s22l 24 + s33l 34 + (2s12 + s66)l 12l 22 + (2s23 + s44)l 22l 32 + (2s13 + s55)l 12l 32 E (8) We now present the orientation dependence of Young’s modulus in single-crystal cementite For the ab-plane with the tensile axis rotated from [100] to [010], the direction cosines are l = cos θ, l = sin θ, l = 0, for the bc-plane with the tensile axis rotated from [010] to [001], the direction cosines are l = 0, l = cos θ, l = sin θ, and for the ca-plane with the tensile axis rotated from 087102-10 G Ghosh AIP Advances 5, 087102 (2015) FIG Depiction of directional dependence of bulk modulus of the virtual phase Mo3C having the structure of cementite (Fe3C) [001] to [100], the direction cosines are l = sin θ, l = 0, l = cos θ Calculated Young’s moduli of selected M3Cs along different crystallographic directions are listed in Table VI Figure shows the orientation dependence of calculated Young’s modulus, E, of selected M3Cs in the ab-, bc- and ca-plane in Fig 3(a), 3(b) and 3(c), respectively An important result to be noted is that unlike Fe3C, Cr3C does not exhibit extreme elastic anisotropy as shown (see, Fig 3(b)) Also, our calculated results for Fe3C are essentially the same as those reported by Nikolussi et al.38 TABLE V Directional bulk modulus (in GPa) along orthorhombic crystallographic axes a (B100), b (B010), and c (B 001) and the compressibility anisotropic factors BBa and BBc in M3Cs Directional bulk modulus M3C Co3C Cr3C Fe3C Hf3C Mn3C Mo3C Nb3C Si3C Ta3C V3C W3C CrFe2C Cr2FeC Linear compressibility Ba Bb Bc BBa BBc 913.9 1171.7 739.6 382.8 1064.7 1260.5 612.6 277.3 693.8 623.5 1340.0 742.4 914.9 600.8 899.6 755.9 343.4 1037.6 641.9 720.9 1416.8 694.9 550.1 803.3 541.4 871.5 714.9 673.7 515.0 583.7 652.5 703.4 562.9 191.5 699.5 724.6 801.1 811.7 889.3 1.521 1.302 0.978 1.114 1.026 1.964 0.849 0.196 0.998 1.134 1.668 1.371 1.049 1.278 1.739 1.436 0.656 1.631 1.792 1.088 1.448 0.992 0.861 1.672 0.915 1.029 087102-11 G Ghosh AIP Advances 5, 087102 (2015) TABLE VI Calculated Young’s modulus (E hk l ) of selected M3Cs along different crystallographic directions M3C E 100 E 010 E 001 E 110 E 011 E 101 E 111 Cr3C Fe3C CrFe2C Cr2FeC 376.6 280.3 339.9 329.5 289.1 220.1 263.7 388.6 239.1 197.7 279.6 333.7 420.3 304.9 346.3 416.3 310.1 49.4 51.9 323.9 426.9 298.8 393.5 342.9 434.8 85.2 92.4 364.6 An implication of a large anisotropy of Young’s modulus is that the measured values will depend on the type of specimen (e.g., void free, texture free) and the measurement technique A combination of these two factors determines the number of orientations being sampled during measurement For example, in pulse-echo technique the Young’s modulus is evaluated by probing bulk specimens containing numerous randomly oriented grains In contrast, in the nanoindentation technique81 the Young’s modulus is evaluated, typically, by probing a single grain in the specimen, unless the grain size is very small (in the nanoscale regime) Therefore, in the latter technique the measured Young’s modulus will most likely correspond to the directional property of cementite grain (plane normal vs loading direction) being probed In such situations, the measured property is expected to exhibit a large variation in Young’s modulus (a) (b) (c) FIG Orientation dependence of Young’s modulus (E, in GPa) of selected M3Cs, demonstrating its extreme anisotropy: (a) ab-plane with the tensile axis rotated from [100] to [010], (b) bc-plane with the tensile axis rotated from [010] to [001], and (c) ca-plane with the tensile axis rotated from [001] to [100] 087102-12 G Ghosh AIP Advances 5, 087102 (2015) TABLE VII Calculated elastic anisotropy factors and Cauchy pressures (in GPa) in M3Cs M3C Co3C Cr3C Fe3C Hf3C Mn3C Mo3C Nb3C Si3C Ta3C V3C W3C CrFe2C Cr2FeC Single-crystal shear Polycrystal anisotropy factor anisotropy factor Cauchy pressure A1 A2 A3 A B(%) A G(%) C 12 −C 66 C 13 −C 55 C 23 −C 44 0.1053 1.4559 0.2044 1.7096 0.6217 0.5398 1.5579 0.9188 1.6298 1.5843 0.7412 0.2166 2.5509 1.3444 2.0245 1.8017 0.8080 1.4233 2.0109 0.6272 0.1359 0.6218 0.6092 1.3708 1.2727 0.8542 1.1009 1.4510 1.3301 0.0915 1.2651 0.6448 0.7835 0.4695 0.6869 0.9354 0.3113 1.1949 1.2425 0.49 0.58 0.41 0.96 0.98 0.89 0.17 4.89 0.0 0.24 0.36 0.53 0.0 4.50 4.74 34.88 38.49 8.80 9.64 0.63 39.79 2.07 1.74 2.84 40.11 1.16 51.5 15.32 30.7 47.1 62.7 157.48 59.14 75.6 48.04 25.95 225.2 -28.44 20.14 64.6 27.84 11.88 10.28 -33.06 79.62 62.56 -6.28 83.86 55.52 112.38 36.62 107.04 164.9 109.9 158.56 9.6 125.48 180.26 50.5 10.66 57.54 35.62 185.26 149.58 65.4 TABLE VIII Calculated Poisson’s ratios (υij) of selected M3C M3C Cr3C Fe3C CrFe2C Cr2FeC υ21 υ21 υ13 υ31 υ23 υ32 0.2734 0.3251 0.2032 0.2464 0.2099 0.2553 0.1576 0.2906 0.4051 0.2959 0.3389 0.3934 0.2572 0.2087 0.2788 0.3985 0.4687 0.4536 0.3554 0.2635 0.3878 0.4074 0.3768 0.2262 Encouraged by the results shown in Fig 3(b), we have investigated the effect of site substitution by Cr on the elastic properties of Fe3C At first, we assume that all 4c or 8d sites in Fe3C are occupied by Cr These are designated as CrFe2C and Cr2FeC, respectively, in Tables I to VIII As the next step, all unit cell-internal and cell-external degrees of freedom are optimized, and then the elastic constants of CrFe2C and Cr2FeC are calculated by the procedure mentioned above As seen in Fig 3(b), the site substitution of Fe by Cr at 4c has a dramatic effect on the Young’s modulus of Fe3C Even though Fe3C, in equilibrium with ferrite in the Fe-Cr-C system, is known to dissolve a substantial amount of Cr,82,83 the nature of site substitution and its effect on the elastic properties are not known Umemoto and co-workers24,27 reported the effect of V, Cr, Mo and Mn on polycrystalline Young’s modulus and hardness of Fe3C Up to about 20 at.% Cr and Mn, at.% V and Mo were added Umemoto’s results indicate that these elements indeed increase, almost linearly, Young’s modulus of Fe3C As listed in Table III, our calculated values of Young’s modulus (VRH) of Fe3C, V3C, Cr3C, Mo3C and Mn3C are 183, 259, 362, 197, and 352 GPa, respectively A linear interpolation of calculated Young’s modulus between Fe3C and M3C (M = V, Cr, Mo, Mn) suggests the increase in Young’s modulus of Fe3C with the addition of other alloying elements should be in the following order: Mo < V < Mn < Cr On the other hand, their experimental data24,27 suggest the increase in Young’s modulus of Fe3C should follow in the sequence of: Mo ≈ Mn < Cr < V Additional experimental data is needed to check if these alloyed carbides exhibit nonlinear effects (moduli vs composition) E Shear modulus and its anisotropy Inoue et al30,31 studied deformation and fracture behavior of cementite (θ) in steels by Vickers indentation followed by TEM examination of thin foils The plastic deformation in cementite is found 087102-13 G Ghosh AIP Advances 5, 087102 (2015) to be highly non-uniform, and based on the trace analysis they concluded that glide dislocations lie on (100)[010]θ and (001)[010]θ Also, the slip behavior is strongly affected by temperature of deformation At room temperature, both (100)θ and (010)θ operate, and above 400oC {110} planes become active and multiple glide becomes operational The fracture of cementite at room tempearture is of cleavage type, and the indices of fracture surface was determined to be (110)θ, (100)θ and (210)θ To understand plastic deformation in cementite, it is useful to investigate the dependence of the shear modulus due to ooperational shear system Here, we choose a shear plane (hkl) and we vary the shear stress direction [uvw] within that plane Let [HKL] be the vector normal to the (hkl) shear plane, and we make a coordinate transformation such that the the [uvw] and [HKL] directions become the x and y axes, respectively, then the shear modulus on the (hkl) shear plane with shear stress applied along [uvw] direction is given by G = (s66)−1 = (4s1212)−1 (9) Let l 11, l 12 and l 13 are direction cosines of the shear stress direction [uvw], and l 21, l 22 and l 23 are direction cosines of the shear plane normal [HKL], then the shear modulus is given by ′ 2 2 2 = s66 = 4s11l 11 l 21 + 4s22l 12 l 22 + 4s33l 13 l 23 + 8s12l 11l 21l 12l 22 + 8s23l 112l 22l 13l 23 G +8s13l 112l 21l 13l 23 + s44(l 12l 23 + l 22l 13)2 + s55(l 11l 23 + l 21l 13)2 + s66(l 11l 22 + l 21l 12)2 (10) Following the experimental studies on deformation and fracture of cementite by Inoue et al.,31 we present the orientation dependence of shear modulus in single-crystal M3Cs having three shear systems: (001)[010], (110)[001], and (111)[110] For the (001) shear plane with the shear stress direction rotated from [100] to [010], the direction cosines are l 11 = cosθ, l 12 = sinθ, l 13 = 0, l 21 = l 22 = 0, l 23 = For the (110) shear plane with the shear stress direction rotated from [001] to √ θ , l 13 = cosθ, l 21 = √1 , l 22 − √1 , and l 23 = For the [110], the direction cosines are l 11 = l 12 = sin 2 (111) shear plane with the shear stress direction rotated from [110] to [112],the direction cosines are l 11 = √12 cosθ − √16 sinθ, l 12 = − √12 cosθ − √16 sinθ, l 13 = √26 sinθ, l 21 = l 22 = l 23 = √13 Figure shows calculated shear modulus of selected M3Cs Once again, the results are calculated for Cr3C, Fe3C, CrFe2C and Cr2FeC In fig 4(a) and 4(b), considering the case of Fe3C relatively low values of shear modulus for (001)[100], (110)[110] may be seen In fact, Inoue et al30,31 noted that these two shear systems exhibit a strong anisotropy in shear modulus, and also reported that the cleavage fracture of Fe3C occurs on the (011) plane Figure implies that the elastic response of Cr3C under shear is quite different from Fe3C These results demonstrate that the site occupancy of Cr in Fe3C has a strong influence on the shear response of corresponding M3C In an orthorhombic crystal, there are more than one way to quantify the anisotropy of shear modulus If we consider simple shear, C44 corresponds to (001) ⟨010⟩ shear, C55 corresponds to (100) ⟨001⟩ shear, and C66 corresponds to (010) ⟨100⟩ shear With the knowledge of Cij of M3Cs presented in Table II, we have calculated their single-crystal shear anisotropy factors as these are related to dislocation motion, crack initiation and propagation In an orthorhombic crystal, the shear anisotropy factors are defined as A1 = 4C44 (C11 + C33 − 2C13) (11) for {100} shear planes in ⟨010⟩ and ⟨011⟩ directions, and A2 = 4C55 (C22 + C33 − 2C23) (12) for {100} shear planes in ⟨001⟩ and ⟨101⟩ directions, and A3 = 4C66 (C11 + C22 − 2C12) for {100} shear planes in ⟨010⟩ and ⟨110⟩ directions (13) 087102-14 G Ghosh AIP Advances 5, 087102 (2015) (a) (b) (c) FIG Orientation dependence of shear modulus (G, in GPa) of selected M3Cs, demonstrating its exreme anisotropy, (a) the (001) shear plane as the shear direction is rotated from [100] to [010], (b) the (110) shear plane as the shear direction is rotated from [001] to [110], and (c) the (111) shear plane as the shear direction is rotated from [110] to [112] The deviation of Ai, in Eq 11-13, from unity is a measure of shear anisotropy Calculated values of Ai for M3Cs are listed in Table VII We find that in most cases, one (or two) are greater (or lesser) than unity Cr3C is the only cementite whose all Ais are greater than unity Once again, this is an unforeseen result, and could not have been guessed without calculating or prior knowledge of Cijs Clerc and Ledbetter84 have shown that hardness varies linearly with elastic stiffness, especially with the shear dominated modes Consistent with our calculated shear anisotropy factors listed in Table VIII and anisotropy of shear modulus shown in fig 4, Kagawa and Okamoto85,86 observed that hardness of cementite is also anisotropic In polycrystalline materials, Chung and Buessem87 have defined percentage anisotropy in compressibility and shear, AB and AG, respectively, as given below AB = (BV − BR ) (BV + BR ) (14) AG = (GV − G R ) (GV + G R ) (15) Here, a value zero corresponds to elastic isotropy, while a value 100% corresponds to the largest possible anisotropy All AB and AG values M3Cs are listed in Table VII We now examine the trend of experimental data of M3C Referring to Table III, our calculated values of shear modulus (VRH) of Fe3C, V3C, Cr3C, Mo3C and Mn3C are 67, 100, 140, 79, and 147 GPa, respectively Considering these values, the isotropic shear modulus of alloyed 087102-15 G Ghosh AIP Advances 5, 087102 (2015) cementite, (Fe, M)3C, should increase in the following order: Mo < V < Mn < Cr Umemoto and co-workers24,27 did not report polycrystalline shear modulus of (Fe, M)3Cs However, they found that the hardness of (Fe, M)3Cs increase, almost linearly with the concentration of M (V, Cr, Mo, Mn) Due to a linear relationship between hardness and shear elastic stiffness,84 it is reasonable to expect that the polycrystalline shear modulus of (Fe, M)3C increases in the following order: Mo < Cr ≈ Mn < V Even though the observed sequence (both Young’s modulus and hardness or shear modulus) is at variance with that expected from our calculated properties, it is important to note that the experimental results could be influenced by porosity, texture etc In literature, various criteria have been proposed to establish ductile/brittle character of intermetallics For example, Pugh88 proposed a simple relationship, empirically linking the plastic properties of materials with their elastic moduli The bulk modulus B represents a measure of fracture stress, while the shear modulus µ represents a measure of resistance for dislocation motion leading to plastic deformation A high B/µ ratio may be associated with ductility whereas a low value may correspond to brittle nature The critical value separating brittle and ductile materials, though debatable, is believed to be around 1.75, i.e., if B/µ > 1.75 the material behaves in a ductile manner, otherwise in a brittle manner Pettifor and co-workers89,90 have suggested that angular character of atomic bonding in metals and intermetallics, which also relates to the brittle or ductile characteristics, could be described by the Cauchy pressure Specifically, they suggested that brittleness could originate from a peculiar property of elastic constants, namely a negative Cauchy pressure (e.g., C12 − C44 < for cubic materials, or C13 − C44 < and C12 − C66 < for hexagonal crystals) For metallic bonding the Cauchy pressure is typically positive, while for directional bonding with angular character it is negative, with larger negative pressure representing a more directional character These correlations have been verified for ductile materials such as Ni and Al having typical metallic bonding, as well as brittle semiconductors such as Si with directional bonding In orthorhombic crystals, the Cauchy pressures are defined by C12 − C66, C13 − C55 and C23 − C44 These values for M3Cs are listed in Table VII Except for Mn3C and Si3C, all Cauchy pressures in binary M3Cs are found to be positive implying the dominance of metallic bonding F Poisson’s ratio and its anisotropy For an isotropic homogeneous material Poisson’s ratio is a constant and lies in the range (-1, 21 ) For anisotropic homogeneous materials Poisson’s ratio is not constant but depends on the direction n of the applied stress and on m, orthogonal to n, the direction in which lateral contraction is measured Then, the Poisson’s ratio ν (n; m) is defined as the ratio of the lateral contraction −eijmimj to longitudinal extension ersnrns ν(n; m) = − ei j m i m j er s nr n s (16) For homogeneous anisotropic materials with orthorhombic symmetry, Poisson’s ratio need not be bounded either above or below It may be arbitrarily large, positive or negative, as has been shown by Ting and Chen.91 Also, for materials with orthorhombic symmetry, Rovati92 has discussed the orientation dependence of Poisson’s ratio in cases of plane of elastic mirror symmetry Figure shows calculated Poisson’s ratio of selected M3Cs Poisson’s ratio provides more information about the characteristics of the bonding forces than any other elastic constant.93 It has been proved that υ = 0.25 is the lower limit for central force solids and 0.5 is the upper limit, which corresponds to infinite elastic anisotropy.94 Selected values of Poisson’s ratio are listed in Table VIII, and once again they exhibit anisotropy In general, these values (νi j , ν j i ) imply a non-central character of bonding forces in M3C This is also consistent, as seen in Table II, that the ratios C23/C44, C13/C55, C12/C66 deviate considerably from unity Ravindran et al80 noted similar results in TiSi2 Frantsevich95 suggested to distinguish between brittleness and ductility by Poisson’s ratio According to Frantsevich the critical value of Poisson’s ratio of a material is 1/3 For brittle materials such as ceramics, the Posson’s ratio of less than 1/3 Our calculated Poisson’s ratio of lies between 087102-16 G Ghosh AIP Advances 5, 087102 (2015) (b) (a) (c) FIG Orientation dependence of Poisson’s ratio of selected M3Cs in planes of elastic mirror symmetry and demonstrates its extreme anisotropy: (a) ab-plane where the tensile direction is rotated from [100] to [010], (b) bc-plane where the tensile direction is rotated from [010] to [001], and (c) ac-plane where the tensile directionis rotated from [100] to [001] 0.29 and 0.41 For Fe3C, the Poisson’s ratio (isotropic limit) is predicted to be 0.36, while experimentally it varies from 0.22 to 0.46 (see, Table IV) Such a large variation of υ in cementite most likely to have originated from several factors ranging from specimen size (i.e., number of grains sampled), experimental technique, porosity and possible texture G Debye temperature The Debye temperature (θD) is a fundamental yet simplest attribute relating to many useful properties of solids, such as specific heat, melting point, vibrational entropy, transport properties etc For example, θD is used to distinguish between high- and low-temperature regimes in a solid If T > θD we expect all modes to have energy k BT, and if T < θD one expects high-frequency modes to be frozen In addition, when T/θD ≥ 12 the electrical resistivity is known to be governed by electron-phonon scattering Strictly speaking, the Debye temperature is not an experimentally determined quantity There are various methods to estimate θD A usual method is to estimate θD from the values of Cijs and the sound velocities such as average longitudinal and transverse elastic wave velocities Generally speaking, the Debye temperature calculated from elastic constant is the same as that determined from specific heat measurements at low temperatures The acoustic Debye temperature θD of M3Cs may be calculated as ( ) 1/3 h 9N θD = ρ−1/2ao −1/3 (17) k B 4πV 087102-17 G Ghosh AIP Advances 5, 087102 (2015) where h is Planck’s constant, kB is Bolzmann’s constant, N is the number of atoms in the unit cell, V is volume of the unit cell, ρ is the density (taken as its theoretical value), and ao is a function √ √ √ √ elastic constants ([100], [010], [001],[110], [011], [101], [ 310], [1 30], [01 3], [10 3]) has been proposed by Joardar et al.96 Maxisch and Ceder97 also adopted a similar approach to calculate Debye temperature of FePO4 and LiFePo4 having an orthorhombic structure Calculated θD, as defined in Eq (17), for M3Cs are listed in Table II An increase in θD for Cr3C, Mn3C and Mo3C, compared with Fe3C, is associated with an increase in both B and µ (see, Table III), as they affect both longitudinal and transverse sound velocities via Navier’s equation98 as well as ao via Cijs On the other hand, a decrease in θD for cementites with 5d elements such as Hf3C, Ta3C and W3C, is due to their much higher theoretical densities compared with Fe3C In other cases, an increase or a decrease in θD is influenced by respective Cijs affecting ao in Eq (17) Guillermet and Grimvall99 derived entropic Debye temperature, θS, based on a logarithmic average of the frequencies of the individual phonon modes of Co3C, Cr3C, Fe3C, Mn3C, Ni3C and V3C Their values are based on systematic analysis of cohesive properties transition metal carbides along with certain empirical relations In a subsequent study, Miodownik100 used these θS values to predict elastic properties of carbides Nevertheless, considering the same M3C the entropic θS value99 shows a modest agreement with our calculated acoustic θD presented in Table II IV CONCLUSIONS We have carried out a comprehensive and systematic study of elastic properties of M3C (M = Al, Co, Cr, Cu, Fe, Hf, Mn, Mo, Nb, Ni, Si, Ta, Ti, V, W, Zr, Cr2FeC and CrFe2C) having the crystal structure of cementite Our systematic and comprehensive study make use of electronic density functional theory in conjunction with pseudopotentials constructed by all-electron projector-augmented wave (PAW) method and GGA for the exchange-correlation functional The following conclusions are drawn: (i) Single-crystal elastic constants (Cij) of M3C are calculated by the stress tensor method Among M3Cs investigated here, we find that Al3C, Cu3C, Ni3C, Ti3C and Zr3C are elastically unstable (ii) Employing homogenization method of Cijs, isotropic (polycrystalline) elastic moduli (bulk, Young’s, shear moduli and Poisson’s ratio) of M3Cs are calculated (iii) Using calculated Cijs, anisotropies of bulk, Young’s, shear moduli and Poisson’s ratio are demonstrated These in turn help us understand a large scatter in experimental data, a critical appraisal of experimental data In general orientation-dependent properties are important in all engineering materials Specific to this study, orientation-dependent properties (bulk, shear, Young’s moduli and Poisson ratio) are important to engineers to analyze deformation and fracture behavior of alloys where Fe3C is a constituent phase (iv) An important result of this computational study is the effect of site substitution of Cr (4c vs 8d) in Fe3C This is important as substantial amount of Cr dissolves in Fe3C and an important alloying element in commercial alloys Therefore, the effect of site substitution of Cr in Fe3C should be experimentally investigated, as our computational results predict a strong influence leading to an improvement of extreme anisotropic properties These are discussed in section III C to III F, and also shown in Fig to (v) Using calculated Cijs, the acoustic Debye temperature (θD) of M3Cs are derived showing a modest agreement with previously reported entropic Debye temperature (θS) ACKNOWLEDGEMENTS This research was supported by the U S Department of Energy, under Award No DE-FG3608GO18131 through Eaton Corporation, Southfield, MI Supercomputing resources were provided by the National Energy Research 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and its predecessors named MART, MART1, MART2, MART3, MART4, MART5 and mart GG are unpublished and intellectual properties of the author.13,15 53 ...AIP ADVANCES 5, 087102 (2015) A first- principles study of cementite (Fe3C) and its alloyed counterparts: Elastic constants, elastic anisotropies, and isotropic elastic moduli G Ghosha Department... temperature, θD, of M3Cs based on calculated Cijs We provide a critical appraisal of available data of polycrystalline elastic properties of alloyed cementite Calculated single crystal properties may be... Metals, Metals Park, OH, 1983) 95 I.N Frantsevich, F.F Voronov, and S .A Bokuta, in Elastic Constants and Elastic Moduli of Metals and Insulators Handbook, edited by I.N Frantsevich (Naukova Dumka,