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Atomistic study on the structure and thermodynamic properties of Cr7C3, Mn7C3, Fe7C3
Acta Materialia 53 (2005) 2727–2732 www.actamat-journals.com
Atomistic study on the structure and thermodynamic properties of Cr7C3, Mn7C3, Fe7C3 Jia-ying Xie a,d, Nan-xian Chen a,b,c, Jiang Shen c, Lidong Teng d, Seshadri Seetharaman d,* a
d
Institute of Materials Science and Engineering, University of Science and Technology, Beijing, 10083, China b Department of Physics, Tsinghua University, Beijing 10084, China c Institute of Applied Physics, University of Science and Technology, Beijing 10083, China Department of Materials Science and Engineering, Royal Institute of Technology (KTH), 10044 Stockholm, Sweden Received 13 December 2004; received in revised form 23 February 2005; accepted 25 February 2005 Available online 7 April 2005
Abstract The crystal structures and stabilities of Cr7C3, Mn7C3, Fe7C3 have been investigated using the interatomic potentials obtained by the lattice inversion method. The calculated structures of Cr7C3, Mn7C3 and Fe7C3 are proposed to be hexagonal with P63mc space group and the calculated lattice constants are in basic agreement with the experimental data. The calculated cohesive energies indicate that the increase in the atomic number of the metal is accompanied by the decrease in the stability of its carbides. The phonon density of states and vibrational entropy related to dynamic phenomena are also evaluated. This work provides a new method for studying the properties of carbides with complex structure. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Carbides; Lattice inversion; Crystal structure; Dynamic phenomena; Thermodynamics
1. Introduction The transition metal carbides are of considerable scientific and technical importance because of their extensive applications in the heat-resistant and hard materials industries. The precipitation of transition metal carbides in steels can dramatically influence the mechanical properties of the steels. Generally, the mechanical properties are related to the components, crystal structure, thermodynamic properties and stability of the transition metal carbides. Numerous experimental and theoretical investigations have over the years been published on the properties of carbides [1– 10]. Theoretical metallurgists and materials scientists have developed an elaborate combination of databases *
Corresponding author. Tel.: +46 8 790 83 55; fax: +46 8 510 511 39. E-mail address: [email protected] (S. Seetharaman).
with computer programs and assessed thermodynamic information to calculate phase diagrams of binary and ternary alloy systems [11]. Such calculations are very sensitive to the accuracy of the input information, which usually is taken from experiments. Compared with experimental work, the ab initio calculations are very quick and inexpensive, but only for elemental solids and simple compounds. On the other hand, the atomistic simulation is simpler, faster and cheaper than ab initio calculation. It can calculate the properties of complex compounds whereas the ab initio method is yet to be developed fully. In this work, the lattice inversion technique [16,17] is applied to obtain a series of interatomic potentials related to M7C3 (M = Cr, Mn, Fe) from the first-principle cohesive energy calculation of some simple structures. Based on these calculated potentials, the crystal structure, stability and thermodynamic properties of M7C3 are investigated. This paper
1359-6454/$30.00 Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.02.039
J.-y. Xie et al. / Acta Materialia 53 (2005) 2727–2732
provides a promising method for the research of transition metal carbides with complex structure. 2. Methodology 2.1. Derivation of pair potentials One of the key problems for atomistic simulation is how to determine the interatomic potentials. Many empirical interatomic pair potentials have been widely used in various kinds of materials [12–14], but these empirical potentials include adjustable parameters, which are determined by fitting some experimental data of the systems involved, for instance, the lattice constant, cohesive energy, elastic constants, formation energy, vacancy formation energy and so on. Sometimes, it is hard to obtain some reliable experimental data such as the elastic modulus of the brittle materials and the single vacancy formation that may differ with the different experimental techniques employed. One of the effective solutions for the uncertainty of multiple-parameter fittings may be the lattice inversion method, which was first presented to determine the pair potentials from ab initio calculated cohesive energy by Carlsson, Gelatt, and Ehrenrchich (CGE) [15]. Chen et al. [16,17] used the Mobius-inversion formula in number theory to obtain the pair potentials not only for the pure metals, but for the intermetallic compounds with faster convergence than the CGE method. The method has been applied successfully to study the properties of ionic crystal and rare-earth interatomic compounds [18,19], in the analysis of the field-ion microscopy image of Fe3Al [20]. The method was also applied to the phonon dispersion of diamond-type materials [21]. In the present work, the lattice inversion method is used for the first time to study properties of carbides. In accordance with lattice inversion [16,17], the aim in the present work was to calculate the total energy curves in the body-centred cubic structure to extract the pair potential of same type atoms and the B2(CsCl) structure to extract the pair potential of the different type atoms. These calculations are performed on the basis of the augmented-spherical-wave (ASW) method [22,23] within the local density functional theory. It is found that the obtained interatomic pair potentials can be approximately expressed as Morse function UðxÞ ¼ D0 fexp ½aðx=R0 1Þ 2 exp ½a=2ðx=R0 1Þg; ð1Þ where x is the distance between two atoms, D0, a and R0 are potential parameters. The calculated potential parameters are given in Table 1. The pair potential curves are shown in Fig. 1.
Table 1 Potential parameters acquired by the lattice inversion method Potential types
D0 (eV)
a
˚) R0 (A
C–C Cr–Cr Mn–Mn Fe–Fe C–Cr C–Mn C–Fe
0.278 0.690 0.668 0.624 0.779 0.767 0.721
6.243 6.942 7.083 7.245 7.785 8.069 7.938
3.185 3.102 2.979 2.903 2.564 2.496 2.490
0.6 0.4 0.2
Pair Potnetial (eV)
2728
0.0
C-C Fe-Fe Mn-Mn Cr-Cr C-Fe C-Mn C-Cr
-0.2 -0.4 -0.6 -0.8
2
3
4
5
7
6
Interatomic distance (A)
Fig. 1. Potentials of C–C, Fe–Fe, Mn–Mn, Cr–Cr, C–Fe, C–Mn, and C–Cr.
2.2. Calculation method for DOS and thermodynamic parameters In the harmonic approximation of the lattice dynamics, the secular equation of the lattice vibration can be written as * det jDab;kl q x2 dab dkl j ¼ 0; ð2Þ *
where x is the angular frequency and q is the wave vec* tor. Dab ðq Þ is a dynamical matrix and its elements are written as X * * * Uab;ik;jl exp½i q ðRi Rj Þ; ð3Þ Dab;kl ¼ ðM k M l Þ1=2 Ri Rj *
*
where Uab,ik,jl is the force constant, Ri ; Rj are positions in the ith and jth cell, and k, l represent different atoms in the same cell. Using the interatomic pair potential, the force constant can be expressed as: Uab;ik;jl i 8 hR R 0 Ra Rb 00 1 a b 1 0 > / ðRÞ / ðRÞ / ðRÞd if R 6¼ 0; ab 3 2 < 2 R k;l k;l R k;l R ¼ * * PP P * * > Uab;ik:jl if R ¼ 0 ðRi ¼ Rj ;rk ¼ rl Þ; : Uab;il;jl k6¼l
i6¼j k
ð4Þ
J.-y. Xie et al. / Acta Materialia 53 (2005) 2727–2732 *
*
*
*
*
*
*
where R ¼ j R j ¼ j Ri Rj þ rk rl j; rl ; rk are the relative positions of the different atoms in the same cell, /kl(R) is the pair potential between the kth and lth atoms, and / 0 and / 0 0 are the first and second derivative of /, respectively. * The phonon dispersion xðq Þ is given by Eq. (2). Then the density of states (DOS) g(x) can be acquired from * xðq Þ. The specific heat and vibrational entropy can be written as: Z 1 2 ðhx=k B T Þ ehx=kB T C V ðT Þ ¼ 3Nk B gðxÞ dx; ð5Þ ðehx=kB T 1Þ2 0 SðT Þ ¼
Z
T 0
C V ðT 0 Þ dT 0 ; T0
2729
influence of the interatomic potentials. The results are presented in Tables 2–4. The structures of Cr7C3 and Mn7C3 proposed by Westgren are relaxed to the structure proposed by Herbstein and Snyman with a toler˚ . The tolerance range indicates the ance of 0.01 A atomic derivation distance, which can be viewed as the error in the process of determining the space group of compound. The relaxed structures of Cr7C3, Mn7C3 and Fe7C3 with the lowest energy are hexagonal with P63mc space group. Hence, it is proposed that the structure of M7C3 is hexagonal with P63mc space group which is consistent with the structure proposed by Herbstein and Snyman [25].
ð6Þ 3.2. Lattice constants and cohesive energies of Cr7C3, Mn7C3, Fe7C3
where g(x) is the normalized DOS.
3. Results and discussion 3.1. Crystal structure of Cr7C3, Mn7C3, Fe7C3 The carbides of type M7C3 are important components of chromium steels, high-alloyed cast irons and other engineering materials. Three structures have been proposed for description of these carbides: trigonal by Westgren (1935), orthorhombic by Bouchaud and Fruchart (1964) and hexagonal by Herbstein and Snyman (1964). All the structures are very similar and can be considered as being built up from the same structural elements called ÔtriadsÕ; they differ by the arrangement of these triads in the elementary unit cell. In the present work, the M7C3 with three different structures [24–27] are constructed and then the energy minimization is performed using the conjugate gradient method under the
In the calculation procedure, the initial lattice constants of Cr7C3 are chosen arbitrarily in a certain range. By the energy minimization, the final structure stabilizes to hexagonal with space group P63mc with a tolerance ˚ (Table 5). The lattice constants are of 0.01 A ˚ , c = 4.337 A ˚ , which are close to the experia = 7.371 A mental data [25]. A certain randomness in range of the initial structure and the stability of the final structure illustrate that the interatomic potentials are valid for studying the crystal structure properties. According to the stabilized structure of M7C3 obtained from the calculation above, the supercell (M7C3)64 is applied to evaluate the lattice parameters and cohesive energy for M7C3. In the calculation, the configuration average is taken from 30 samples. The calculated results are presented in Table 6. It can be seen that the calculated lattice constants of M7C3 are consistent with the experimental results [25,26]. The lattice
Table 2 Relaxed structural parameters and cohesive energy of Cr7C3 with different structures Initial lattice parameter ˚) ˚) a (A b (A
Space group
Relaxed lattice parameter ˚) ˚) ˚) a (A b (A c (A
Space group
˚) Tolerance (A
Cohesive energy (eV/atom)
˚) c (A
13.98 [24] 4.526 [27] 6.910 [25]
4.523 [24] 12.142 [27] 4.495 [25]
P31c Pnma P63mc
7.371 4.190 7.371
P63mc Pnma P63mc
0.01 0.01 0.01
8.541 8.570 8.626
13.98 [24] 7.010 [27] 6.910 [25]
7.371 7.739 7.371
4.337 12.626 4.337
Table 3 Relaxed structural parameters and cohesive energy of Mn7C3 with different structures Initial lattice parameter ˚) ˚) a (A b (A
˚) c (A
13.90 [27] 4.546 [27] 6.944 [26]
4.54 [27] 11.976 [27] 4.542 [26]
13.90 [27] 6.959 [27] 6.944 [26]
Space group
Relaxed lattice parameter ˚) ˚) ˚) a (A b (A c (A
Space group
˚) Tolerance (A
Cohesive energy (eV/atom)
P31c Pnma P63mc
7.263 4.034 7.263
P63mc Pnma P63mc
0.01 0.01 0.01
8.010 7.947 8.022
7.263 7.020 7.263
4.262 13.840 4.262
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J.-y. Xie et al. / Acta Materialia 53 (2005) 2727–2732
Table 4 Relaxed structural parameters and cohesive energy of Fe7C3 with different structures Initial lattice parameter ˚) ˚) a (A b (A
˚) c (A
4.540 [27] 6.882 [26]
11.942 [27] 4.540 [26]
6.879 [27] 6.882 [26]
Space group
Relaxed lattice parameter ˚) ˚) ˚) a (A b (A c (A
Space group
˚) Tolerance (A
Cohesive energy (eV/atom)
Pnma P63mc
4.991 7.187
6.953 7.187
Pnma P63mc
0.01 0.01
7.286 7.356
13.740 4.235
Table 5 Determination of the lattice parameters of Cr7C3 Initial lattice parameter (unrelaxed) ˚) ˚) ˚) a (A b (A c (A a
b
c
Final lattice parameter (relaxed) ˚) ˚) ˚) a (A b (A c (A
a
b
c
7 6 5 8 7.371 7.371 7 9
90 90 90 90 80 75 80 75
120 120 120 120 115 125 125 130
7.371 7.371 7.371 7.371 7.371 7.371 7.371 7.371
90 90 90 90 90 90 90 90
90 90 90 90 90 90 90 90
120 120 120 120 120 120 120 120
6 5 4 9 7.371 7.371 6 6
4 5 2 6 4.337 4.337 5 3
90 90 90 90 85 80 75 85
7.371 7.371 7.371 7.371 7.371 7.371 7.371 7.371
Table 6 Comparison between the calculated and experimental lattice parameters for M7C3 ˚) a (A
Cr7C3 Mn7C3 Fe7C3
P63mc P63mc P63mc P63mc P63mc P63mc P63mc P63mc
˚) c (A
Calculated
Experimental
Error (%)
Calculated
Experimental
Error (%)
7.371 7.263 7.187
6.910 [25] 6.944 [26] 6.882 [25]
6.671 4.805 4.432
4.337 4.262 4.235
4.495 [25] 4.542 [26] 4.540 [25]
3.515 6.123 6.718
constants, both a and c, decrease with the increase in the atomic number of the metal, and the former is larger while the latter is smaller than the experimental data. The average deviation for a is 5.303% from the experimental data and that for c is 5.452%.
Fig. 2 shows the dependence of the cohesive energy for M7C3 on the atomic number of the metal. With increasing atomic number of the metal, the cohesive energy of its carbides increases, this means a decrease in the stability of its carbides. This agrees with the experimental results [1,3,28]. 3.3. Lattice vibration of Cr7C3, Mn7C3
-7.2 -7.4
Cohesive energy (eV/atom)
4.337 4.337 4.337 4.337 4.337 4.337 4.337 4.337
Space group
-7.6 -7.8 -8.0 -8.2 -8.4 -8.6 -8.8
24
25
26
Atomic number of the metal
Fig. 2. The dependence of the cohesive energy for M7C3 on the atomic number of the metal.
The phonon DOS reflects the lattice dynamic properties, from which some important thermodynamic parameters can be derived, for example, specific heat, vibrational entropy and Debye temperature. Generally, the phonon spectra are measured by an inelastic neutron scattering technique then fit the interatomic force constants to the experimental data by constructing an empirical model or be determined by the first-principle calculations. In this section, with the inverted interatomic potentials, the total DOS and the partial DOS of different elements for M7C3 are evaluated in a crystal cell based on lattice theory. Figs. 3 and 4 show the results computed considering the contribution to the DOS of the distinct atoms. The ratio of the modes contributed by M and C is 7:3 in the total frequency range. However, metal
J.-y. Xie et al. / Acta Materialia 53 (2005) 2727–2732
35
Cr C
Partial DOS
30 25 20 15 10 5 0 0
5
10
15
20
25
Frequency (THz) Fig. 3. Phonon density of states of Cr7C3.
40
Mn 7 C 3
35
Mn C
Partial DOS
30 25 20 15 10 5 0 0
5
10
15
20
25
Frequency (THz) Fig. 4. Phonon density of states of Mn7C3.
atoms contribute a major part to acoustic modes and the carbon atoms contribute a major part to optic modes. In the present work, the localized modes have been analyzed qualitatively from interatomic potentials in Fig. 1. For Cr7C3, there are many Cr atoms at a distance of ˚ from C atoms. From Fig. 1, it can be seen that 2.5 A the Cr atom reacts strongly with the C atom at these distances. The mass of the Cr atom is approximately four times that of the C atom, so the Cr atoms are assumed motionless relative to the C atoms. Then the C atoms are restricted by the Cr atoms in the Cr–C potential well and excite the local modes which correspond to the higher frequencies. The optic modes are separated by a very large gap from the acoustic ones and have very high frequencies, which are due to not only the large mass difference of the atoms but the strong force constants between nearest neighbor metal atoms and carbon atoms [29]. The maximal frequencies of acoustic modes for Cr7C3 are higher than those for Mn7C3 and the highest frequencies of optic modes for Cr7C3 are lower than those for Mn7C3, which indicates that the interaction between Cr and C is stronger than Mn and C.
Furthermore, the vibrational entropy of a material is a suitable parameter to describe the phenomena of solidstate physics associated with lattice vibrations. In this paper, the dependences of the vibrational entropy on the temperature for Cr7C3 and Mn7C3 are derived from the calculated phonon DOS based on Eqs. (5) and (6). The results are shown in Figs. 5 and 6. Fig. 5 gives the comparison between our calculated results and the calculated results based on experimental results [30]. To enable the comparison with the calculated results based on the experimental results, the calculated values are given on the basis of 1 g-transition metal atom. In our calculation, we only consider the vibration entropy but not the configurative entropy. So the calculated vibrational entropies of Cr7C3 are a little smaller than the calculated entropies based on experimental results [30], which shows that the dominating contribution to entropy is made by vibrational entropy [9]. Fig. 6 shows the comparison of entropy between Cr7C3 and Mn7C3. It can be seen that the vibrational entropies of Mn7C3 are
Vibrational Entropy (cal/k.gm-atom Me)
Cr 7 C 3
22 20 18 16 14 12 10 8
Calculated vibrational entropy of Cr7C 3
6
Calculated entropy based on experimental results of Cr7C 3 400
600
800
1000
1200
1400
Temperature (K) Fig. 5. Comparison between our calculated vibrational entropy and the calculated entropy based on experimental results.
Vibratory Entropy (cal/K.gm-atom Me)
40
2731
22
Cr7C3 20
Mn7C3
18 16 14 12 10 8 6 400
600
800
1000
1200
1400
Temperature (K) Fig. 6. Calculated vibrational entropy of Cr7C3, Mn7C3.
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J.-y. Xie et al. / Acta Materialia 53 (2005) 2727–2732
higher than Cr7C3, which is also agreement with the experimental results.
4. Conclusion In this work, the crystal structure and cohesive energy of M7C3 have been calculated based on interatomic potentials acquired by the lattice inversion method. Based on the calculated cohesive energy, the order of stability for Cr7C3, Mn7C3 and Fe7C3 are deduced. The increase in atomic number of the metal is accompanied by the decrease in stability of its carbides, which is in agreement with the experimental results. The phonon densities of states and vibrational entropy for Cr7C3, Mn7C3 are also calculated and the calculated results were consistent with results based on experiment. All the results above show that the interatomic potentials achieved by the lattice inversion method are effective in investigating structural and some other simple thermodynamic properties for these carbides with complex structures. Although generally pair potentials have inherent drawbacks, which induce a false Cauchy relation and almost hardly deal with the anisotropies, they can reproduce cohesive energy, lattice constants, bulk modulus and some other simple properties very well. On the other hand, the interatomic potentials used in this paper are obtained from the simplified model, so its applicability to carbides should be further tested.
Acknowledgements The authors express their deep gratitude to Professor Wenchao Li and Benfu Hu for their valuable help. This work is supported by Special Funds for Major State Basic Research of China (Grant Nos. G2000067101, G2000067106).
References [1] Dawson WD, Sale FR. Metall Trans A 1980;11:1849. [2] Sichen D, Seetharaman S, Staffansson L-I. Metall Trans B 1989;20:911. [3] Kleykamp H, Bunsenges B. Phys Chem 1969;73:354. [4] Teng LD, Aune RE, Seetharaman S. Intermetallics 2003; 11:1229. [5] Teng LD, Lu XG, Aune RE, Seetharaman S. Metall Mater Trans A 2004;35:3673. [6] Moattar F, Anderson JS. Trans Faraday Soc 1971;67:2303. [7] Rajagopalan PK, Krishnan TS, Bose DK. J Alloy Compd 2000;297:L1. [8] Meschel SV, Kleppa OJ. J Alloy Compd 1997;257:227. [9] Guillermet AF, Grimvall G. J Phys Chem Solids 1992;53:105. [10] Small M, Ryba E. Metall Trans A 1981;12:1389. [11] Guillermet AF, Haglund J, Grimvall G. Phys Rev B 1992;45:11557. [12] Torrens I. Interatomic potentials. New York (NY): Academic Press; 1972. [13] Chen SP, Yan M, Gale JD, Grimes RW, Devanathan R, Sickafus KE, Yu N, Nastasi M. Philos Mag Lett 1996;73:51. [14] Grimes RW, Binks DJ, Lidiard AB. Philos Mag A 1995;72:651. [15] Carlsson AE, Gelatt CD, Ehrenreich H. Philos Mag A 1980;41:241. [16] Chen NX. Phys Rev Lett 1990;6:1193. [17] Chen NX, Chen ZD, Wei YC. Phys Rev E 1997;55:5. [18] Zhang S, Chen NX. Phys Rev B 2002;66:064106. [19] Chang H, Chen NX, Liang JK, Rao GH. Eur Phys J B 2003;33:55. [20] Chen NX, Ge XJ, Zhang WQ, Zhu FW. Phys Rev B 1998;57:14203. [21] Li M, Chen NX. Phys Rev B 1995;52:997. [22] Monkhorst HJ, Pack JD. Phys Rev B 1976;13:5188. [23] Williams AR, Kuebler J, Gelatt JR. Phys Rev B 1979;19:6049. [24] Westgren A. Jerkontorets Ann 1935;119:231. [25] Herbstein FH, Snyman JA. Inorg Chem 1964;3:894. [26] Hajek B, Karen P, Brozek V, Czech C. Chem Commun 1983;48:2740. [27] Villars P, Calvert LD. PearsonÕs handbooks of crystallographic data for intermetallic phase. Vitznau, Switzerland: MPDS; 1991. [28] Dawson WM, Sale FR. Metall Trans A 1977;8:15. [29] Weber W. Phys Rev B 1973;8:5082. [30] Toth LE. Transition metal carbides and nitrides. New York (NY): Academic Press; 1971.
Atomistic study on the structure and thermodynamic properties of Cr7C3, Mn7C3, Fe7C3 Jia-ying Xie a,d, Nan-xian Chen a,b,c, Jiang Shen c, Lidong Teng d, Seshadri Seetharaman d,* a
d
Institute of Materials Science and Engineering, University of Science and Technology, Beijing, 10083, China b Department of Physics, Tsinghua University, Beijing 10084, China c Institute of Applied Physics, University of Science and Technology, Beijing 10083, China Department of Materials Science and Engineering, Royal Institute of Technology (KTH), 10044 Stockholm, Sweden Received 13 December 2004; received in revised form 23 February 2005; accepted 25 February 2005 Available online 7 April 2005
Abstract The crystal structures and stabilities of Cr7C3, Mn7C3, Fe7C3 have been investigated using the interatomic potentials obtained by the lattice inversion method. The calculated structures of Cr7C3, Mn7C3 and Fe7C3 are proposed to be hexagonal with P63mc space group and the calculated lattice constants are in basic agreement with the experimental data. The calculated cohesive energies indicate that the increase in the atomic number of the metal is accompanied by the decrease in the stability of its carbides. The phonon density of states and vibrational entropy related to dynamic phenomena are also evaluated. This work provides a new method for studying the properties of carbides with complex structure. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Carbides; Lattice inversion; Crystal structure; Dynamic phenomena; Thermodynamics
1. Introduction The transition metal carbides are of considerable scientific and technical importance because of their extensive applications in the heat-resistant and hard materials industries. The precipitation of transition metal carbides in steels can dramatically influence the mechanical properties of the steels. Generally, the mechanical properties are related to the components, crystal structure, thermodynamic properties and stability of the transition metal carbides. Numerous experimental and theoretical investigations have over the years been published on the properties of carbides [1– 10]. Theoretical metallurgists and materials scientists have developed an elaborate combination of databases *
Corresponding author. Tel.: +46 8 790 83 55; fax: +46 8 510 511 39. E-mail address: [email protected] (S. Seetharaman).
with computer programs and assessed thermodynamic information to calculate phase diagrams of binary and ternary alloy systems [11]. Such calculations are very sensitive to the accuracy of the input information, which usually is taken from experiments. Compared with experimental work, the ab initio calculations are very quick and inexpensive, but only for elemental solids and simple compounds. On the other hand, the atomistic simulation is simpler, faster and cheaper than ab initio calculation. It can calculate the properties of complex compounds whereas the ab initio method is yet to be developed fully. In this work, the lattice inversion technique [16,17] is applied to obtain a series of interatomic potentials related to M7C3 (M = Cr, Mn, Fe) from the first-principle cohesive energy calculation of some simple structures. Based on these calculated potentials, the crystal structure, stability and thermodynamic properties of M7C3 are investigated. This paper
1359-6454/$30.00 Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.02.039
J.-y. Xie et al. / Acta Materialia 53 (2005) 2727–2732
provides a promising method for the research of transition metal carbides with complex structure. 2. Methodology 2.1. Derivation of pair potentials One of the key problems for atomistic simulation is how to determine the interatomic potentials. Many empirical interatomic pair potentials have been widely used in various kinds of materials [12–14], but these empirical potentials include adjustable parameters, which are determined by fitting some experimental data of the systems involved, for instance, the lattice constant, cohesive energy, elastic constants, formation energy, vacancy formation energy and so on. Sometimes, it is hard to obtain some reliable experimental data such as the elastic modulus of the brittle materials and the single vacancy formation that may differ with the different experimental techniques employed. One of the effective solutions for the uncertainty of multiple-parameter fittings may be the lattice inversion method, which was first presented to determine the pair potentials from ab initio calculated cohesive energy by Carlsson, Gelatt, and Ehrenrchich (CGE) [15]. Chen et al. [16,17] used the Mobius-inversion formula in number theory to obtain the pair potentials not only for the pure metals, but for the intermetallic compounds with faster convergence than the CGE method. The method has been applied successfully to study the properties of ionic crystal and rare-earth interatomic compounds [18,19], in the analysis of the field-ion microscopy image of Fe3Al [20]. The method was also applied to the phonon dispersion of diamond-type materials [21]. In the present work, the lattice inversion method is used for the first time to study properties of carbides. In accordance with lattice inversion [16,17], the aim in the present work was to calculate the total energy curves in the body-centred cubic structure to extract the pair potential of same type atoms and the B2(CsCl) structure to extract the pair potential of the different type atoms. These calculations are performed on the basis of the augmented-spherical-wave (ASW) method [22,23] within the local density functional theory. It is found that the obtained interatomic pair potentials can be approximately expressed as Morse function UðxÞ ¼ D0 fexp ½aðx=R0 1Þ 2 exp ½a=2ðx=R0 1Þg; ð1Þ where x is the distance between two atoms, D0, a and R0 are potential parameters. The calculated potential parameters are given in Table 1. The pair potential curves are shown in Fig. 1.
Table 1 Potential parameters acquired by the lattice inversion method Potential types
D0 (eV)
a
˚) R0 (A
C–C Cr–Cr Mn–Mn Fe–Fe C–Cr C–Mn C–Fe
0.278 0.690 0.668 0.624 0.779 0.767 0.721
6.243 6.942 7.083 7.245 7.785 8.069 7.938
3.185 3.102 2.979 2.903 2.564 2.496 2.490
0.6 0.4 0.2
Pair Potnetial (eV)
2728
0.0
C-C Fe-Fe Mn-Mn Cr-Cr C-Fe C-Mn C-Cr
-0.2 -0.4 -0.6 -0.8
2
3
4
5
7
6
Interatomic distance (A)
Fig. 1. Potentials of C–C, Fe–Fe, Mn–Mn, Cr–Cr, C–Fe, C–Mn, and C–Cr.
2.2. Calculation method for DOS and thermodynamic parameters In the harmonic approximation of the lattice dynamics, the secular equation of the lattice vibration can be written as * det jDab;kl q x2 dab dkl j ¼ 0; ð2Þ *
where x is the angular frequency and q is the wave vec* tor. Dab ðq Þ is a dynamical matrix and its elements are written as X * * * Uab;ik;jl exp½i q ðRi Rj Þ; ð3Þ Dab;kl ¼ ðM k M l Þ1=2 Ri Rj *
*
where Uab,ik,jl is the force constant, Ri ; Rj are positions in the ith and jth cell, and k, l represent different atoms in the same cell. Using the interatomic pair potential, the force constant can be expressed as: Uab;ik;jl i 8 hR R 0 Ra Rb 00 1 a b 1 0 > / ðRÞ / ðRÞ / ðRÞd if R 6¼ 0; ab 3 2 < 2 R k;l k;l R k;l R ¼ * * PP P * * > Uab;ik:jl if R ¼ 0 ðRi ¼ Rj ;rk ¼ rl Þ; : Uab;il;jl k6¼l
i6¼j k
ð4Þ
J.-y. Xie et al. / Acta Materialia 53 (2005) 2727–2732 *
*
*
*
*
*
*
where R ¼ j R j ¼ j Ri Rj þ rk rl j; rl ; rk are the relative positions of the different atoms in the same cell, /kl(R) is the pair potential between the kth and lth atoms, and / 0 and / 0 0 are the first and second derivative of /, respectively. * The phonon dispersion xðq Þ is given by Eq. (2). Then the density of states (DOS) g(x) can be acquired from * xðq Þ. The specific heat and vibrational entropy can be written as: Z 1 2 ðhx=k B T Þ ehx=kB T C V ðT Þ ¼ 3Nk B gðxÞ dx; ð5Þ ðehx=kB T 1Þ2 0 SðT Þ ¼
Z
T 0
C V ðT 0 Þ dT 0 ; T0
2729
influence of the interatomic potentials. The results are presented in Tables 2–4. The structures of Cr7C3 and Mn7C3 proposed by Westgren are relaxed to the structure proposed by Herbstein and Snyman with a toler˚ . The tolerance range indicates the ance of 0.01 A atomic derivation distance, which can be viewed as the error in the process of determining the space group of compound. The relaxed structures of Cr7C3, Mn7C3 and Fe7C3 with the lowest energy are hexagonal with P63mc space group. Hence, it is proposed that the structure of M7C3 is hexagonal with P63mc space group which is consistent with the structure proposed by Herbstein and Snyman [25].
ð6Þ 3.2. Lattice constants and cohesive energies of Cr7C3, Mn7C3, Fe7C3
where g(x) is the normalized DOS.
3. Results and discussion 3.1. Crystal structure of Cr7C3, Mn7C3, Fe7C3 The carbides of type M7C3 are important components of chromium steels, high-alloyed cast irons and other engineering materials. Three structures have been proposed for description of these carbides: trigonal by Westgren (1935), orthorhombic by Bouchaud and Fruchart (1964) and hexagonal by Herbstein and Snyman (1964). All the structures are very similar and can be considered as being built up from the same structural elements called ÔtriadsÕ; they differ by the arrangement of these triads in the elementary unit cell. In the present work, the M7C3 with three different structures [24–27] are constructed and then the energy minimization is performed using the conjugate gradient method under the
In the calculation procedure, the initial lattice constants of Cr7C3 are chosen arbitrarily in a certain range. By the energy minimization, the final structure stabilizes to hexagonal with space group P63mc with a tolerance ˚ (Table 5). The lattice constants are of 0.01 A ˚ , c = 4.337 A ˚ , which are close to the experia = 7.371 A mental data [25]. A certain randomness in range of the initial structure and the stability of the final structure illustrate that the interatomic potentials are valid for studying the crystal structure properties. According to the stabilized structure of M7C3 obtained from the calculation above, the supercell (M7C3)64 is applied to evaluate the lattice parameters and cohesive energy for M7C3. In the calculation, the configuration average is taken from 30 samples. The calculated results are presented in Table 6. It can be seen that the calculated lattice constants of M7C3 are consistent with the experimental results [25,26]. The lattice
Table 2 Relaxed structural parameters and cohesive energy of Cr7C3 with different structures Initial lattice parameter ˚) ˚) a (A b (A
Space group
Relaxed lattice parameter ˚) ˚) ˚) a (A b (A c (A
Space group
˚) Tolerance (A
Cohesive energy (eV/atom)
˚) c (A
13.98 [24] 4.526 [27] 6.910 [25]
4.523 [24] 12.142 [27] 4.495 [25]
P31c Pnma P63mc
7.371 4.190 7.371
P63mc Pnma P63mc
0.01 0.01 0.01
8.541 8.570 8.626
13.98 [24] 7.010 [27] 6.910 [25]
7.371 7.739 7.371
4.337 12.626 4.337
Table 3 Relaxed structural parameters and cohesive energy of Mn7C3 with different structures Initial lattice parameter ˚) ˚) a (A b (A
˚) c (A
13.90 [27] 4.546 [27] 6.944 [26]
4.54 [27] 11.976 [27] 4.542 [26]
13.90 [27] 6.959 [27] 6.944 [26]
Space group
Relaxed lattice parameter ˚) ˚) ˚) a (A b (A c (A
Space group
˚) Tolerance (A
Cohesive energy (eV/atom)
P31c Pnma P63mc
7.263 4.034 7.263
P63mc Pnma P63mc
0.01 0.01 0.01
8.010 7.947 8.022
7.263 7.020 7.263
4.262 13.840 4.262
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Table 4 Relaxed structural parameters and cohesive energy of Fe7C3 with different structures Initial lattice parameter ˚) ˚) a (A b (A
˚) c (A
4.540 [27] 6.882 [26]
11.942 [27] 4.540 [26]
6.879 [27] 6.882 [26]
Space group
Relaxed lattice parameter ˚) ˚) ˚) a (A b (A c (A
Space group
˚) Tolerance (A
Cohesive energy (eV/atom)
Pnma P63mc
4.991 7.187
6.953 7.187
Pnma P63mc
0.01 0.01
7.286 7.356
13.740 4.235
Table 5 Determination of the lattice parameters of Cr7C3 Initial lattice parameter (unrelaxed) ˚) ˚) ˚) a (A b (A c (A a
b
c
Final lattice parameter (relaxed) ˚) ˚) ˚) a (A b (A c (A
a
b
c
7 6 5 8 7.371 7.371 7 9
90 90 90 90 80 75 80 75
120 120 120 120 115 125 125 130
7.371 7.371 7.371 7.371 7.371 7.371 7.371 7.371
90 90 90 90 90 90 90 90
90 90 90 90 90 90 90 90
120 120 120 120 120 120 120 120
6 5 4 9 7.371 7.371 6 6
4 5 2 6 4.337 4.337 5 3
90 90 90 90 85 80 75 85
7.371 7.371 7.371 7.371 7.371 7.371 7.371 7.371
Table 6 Comparison between the calculated and experimental lattice parameters for M7C3 ˚) a (A
Cr7C3 Mn7C3 Fe7C3
P63mc P63mc P63mc P63mc P63mc P63mc P63mc P63mc
˚) c (A
Calculated
Experimental
Error (%)
Calculated
Experimental
Error (%)
7.371 7.263 7.187
6.910 [25] 6.944 [26] 6.882 [25]
6.671 4.805 4.432
4.337 4.262 4.235
4.495 [25] 4.542 [26] 4.540 [25]
3.515 6.123 6.718
constants, both a and c, decrease with the increase in the atomic number of the metal, and the former is larger while the latter is smaller than the experimental data. The average deviation for a is 5.303% from the experimental data and that for c is 5.452%.
Fig. 2 shows the dependence of the cohesive energy for M7C3 on the atomic number of the metal. With increasing atomic number of the metal, the cohesive energy of its carbides increases, this means a decrease in the stability of its carbides. This agrees with the experimental results [1,3,28]. 3.3. Lattice vibration of Cr7C3, Mn7C3
-7.2 -7.4
Cohesive energy (eV/atom)
4.337 4.337 4.337 4.337 4.337 4.337 4.337 4.337
Space group
-7.6 -7.8 -8.0 -8.2 -8.4 -8.6 -8.8
24
25
26
Atomic number of the metal
Fig. 2. The dependence of the cohesive energy for M7C3 on the atomic number of the metal.
The phonon DOS reflects the lattice dynamic properties, from which some important thermodynamic parameters can be derived, for example, specific heat, vibrational entropy and Debye temperature. Generally, the phonon spectra are measured by an inelastic neutron scattering technique then fit the interatomic force constants to the experimental data by constructing an empirical model or be determined by the first-principle calculations. In this section, with the inverted interatomic potentials, the total DOS and the partial DOS of different elements for M7C3 are evaluated in a crystal cell based on lattice theory. Figs. 3 and 4 show the results computed considering the contribution to the DOS of the distinct atoms. The ratio of the modes contributed by M and C is 7:3 in the total frequency range. However, metal
J.-y. Xie et al. / Acta Materialia 53 (2005) 2727–2732
35
Cr C
Partial DOS
30 25 20 15 10 5 0 0
5
10
15
20
25
Frequency (THz) Fig. 3. Phonon density of states of Cr7C3.
40
Mn 7 C 3
35
Mn C
Partial DOS
30 25 20 15 10 5 0 0
5
10
15
20
25
Frequency (THz) Fig. 4. Phonon density of states of Mn7C3.
atoms contribute a major part to acoustic modes and the carbon atoms contribute a major part to optic modes. In the present work, the localized modes have been analyzed qualitatively from interatomic potentials in Fig. 1. For Cr7C3, there are many Cr atoms at a distance of ˚ from C atoms. From Fig. 1, it can be seen that 2.5 A the Cr atom reacts strongly with the C atom at these distances. The mass of the Cr atom is approximately four times that of the C atom, so the Cr atoms are assumed motionless relative to the C atoms. Then the C atoms are restricted by the Cr atoms in the Cr–C potential well and excite the local modes which correspond to the higher frequencies. The optic modes are separated by a very large gap from the acoustic ones and have very high frequencies, which are due to not only the large mass difference of the atoms but the strong force constants between nearest neighbor metal atoms and carbon atoms [29]. The maximal frequencies of acoustic modes for Cr7C3 are higher than those for Mn7C3 and the highest frequencies of optic modes for Cr7C3 are lower than those for Mn7C3, which indicates that the interaction between Cr and C is stronger than Mn and C.
Furthermore, the vibrational entropy of a material is a suitable parameter to describe the phenomena of solidstate physics associated with lattice vibrations. In this paper, the dependences of the vibrational entropy on the temperature for Cr7C3 and Mn7C3 are derived from the calculated phonon DOS based on Eqs. (5) and (6). The results are shown in Figs. 5 and 6. Fig. 5 gives the comparison between our calculated results and the calculated results based on experimental results [30]. To enable the comparison with the calculated results based on the experimental results, the calculated values are given on the basis of 1 g-transition metal atom. In our calculation, we only consider the vibration entropy but not the configurative entropy. So the calculated vibrational entropies of Cr7C3 are a little smaller than the calculated entropies based on experimental results [30], which shows that the dominating contribution to entropy is made by vibrational entropy [9]. Fig. 6 shows the comparison of entropy between Cr7C3 and Mn7C3. It can be seen that the vibrational entropies of Mn7C3 are
Vibrational Entropy (cal/k.gm-atom Me)
Cr 7 C 3
22 20 18 16 14 12 10 8
Calculated vibrational entropy of Cr7C 3
6
Calculated entropy based on experimental results of Cr7C 3 400
600
800
1000
1200
1400
Temperature (K) Fig. 5. Comparison between our calculated vibrational entropy and the calculated entropy based on experimental results.
Vibratory Entropy (cal/K.gm-atom Me)
40
2731
22
Cr7C3 20
Mn7C3
18 16 14 12 10 8 6 400
600
800
1000
1200
1400
Temperature (K) Fig. 6. Calculated vibrational entropy of Cr7C3, Mn7C3.
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higher than Cr7C3, which is also agreement with the experimental results.
4. Conclusion In this work, the crystal structure and cohesive energy of M7C3 have been calculated based on interatomic potentials acquired by the lattice inversion method. Based on the calculated cohesive energy, the order of stability for Cr7C3, Mn7C3 and Fe7C3 are deduced. The increase in atomic number of the metal is accompanied by the decrease in stability of its carbides, which is in agreement with the experimental results. The phonon densities of states and vibrational entropy for Cr7C3, Mn7C3 are also calculated and the calculated results were consistent with results based on experiment. All the results above show that the interatomic potentials achieved by the lattice inversion method are effective in investigating structural and some other simple thermodynamic properties for these carbides with complex structures. Although generally pair potentials have inherent drawbacks, which induce a false Cauchy relation and almost hardly deal with the anisotropies, they can reproduce cohesive energy, lattice constants, bulk modulus and some other simple properties very well. On the other hand, the interatomic potentials used in this paper are obtained from the simplified model, so its applicability to carbides should be further tested.
Acknowledgements The authors express their deep gratitude to Professor Wenchao Li and Benfu Hu for their valuable help. This work is supported by Special Funds for Major State Basic Research of China (Grant Nos. G2000067101, G2000067106).
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