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Journal of Alloys and Compounds 470 (2009) 85–89 Contents lists available at ScienceDirect Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom Phase stability determination of the Mg–B binary system using the CALPHAD method and ab initio calculations Sungtae Kim a,∗ , Donald S. Stone a , Jae-Ik Cho b , Chang-Yeol Jeong b , Chang-Seog Kang b , Jung-Chan Bae b a b Department of Materials Science and Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA Korea Institute of Industrial Technology, Gwangju Research Center, Gwangju, Republic of Korea a r t i c l e i n f o Article history: Received 7 January 2008 Received in revised form 19 February 2008 Accepted 27 February 2008 Available online 10 April 2008 Keywords: Thermodynamics Ab initio electron theory CALPHAD Mg–B phase diagram a b s t r a c t Knowledge of the Mg–B binary phase diagram is important in for the synthesis of superconducting MgB2 and of potential, new Mg–B alloys. However, the literature phase diagram is highly uncertain because it was originally constructed by guessing the decomposition temperatures of magnesium boride phases. We therefore utilized the CALPHAD method and ab initio calculations to compute thermodynamic model parameters to determine the decomposition temperatures of magnesium borides and re-assess the Mg–B binary phase diagram. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Magnesium diboride (MgB2 ) is a superconductor whose critical temperature is the highest of any conventional superconductor, so its thermodynamic properties and stability are of interest ranging from low temperatures, where it is used, to high temperatures, where it is usually synthesized [1–5]. At the same time, MgB2 is being considered for introduction into magnesium alloys [6], the latter of which are attractive to the transportation industries because of the low density, abundance and relative environmental friendliness of magnesium [7]. To aid in the development of Mg–MgB2 -based alloys it is necessary to develop a fluent database for describing the phase stability in the Mg–B binary and multicomponent systems so that alloy design can proceed without arbitrary selections of composition and solidification processing conditions. The existing Mg–B phase diagram (Fig. 1) is not reliable [8]. Several decades ago, the Mg–B binary phase diagram was first tentatively constructed by guessing the decomposition temperatures of magnesium boride phases—MgB2 , MgB4 and MgB7 [9]. There has been little theoretical or experimental underpinning for that diagram. Recent efforts [10,11] that improved the Mg–B phase diagram by means of a calculation of phase diagram (CALPHAD) method have referred the originally guessed phase diagram due to a deficient database of experimentally analyzed phase stability information. The literature decomposition temperatures of the MgB2 , MgB4 and ∗ Corresponding author. Tel.: +1 608 263 9642; fax: +1 608 262 8353. E-mail address: [email protected] (S. Kim). 0925-8388/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2008.02.099 MgB7 phases in the existing Mg–B phase diagram are 1550 ◦ C, 1830 ◦ C and 2150 ◦ C, respectively [8]. The reported decomposition reaction temperatures of boride phases: 2MgB2 ⇒ MgB4 + Mg(g) and 7MgB4 ⇒ 4MgB7 + 3Mg(g) are 850 ◦ C ≤ Tdecomp (MgB2 ) ≤ 1550 ◦ C and 1020 ◦ C ≤ Tdecomp (MgB4 ) ≤ 1827 ◦ C, respectively [8–17]. The literature decomposition temperatures exhibit big uncertainty. These issues require that the phase diagram of the Mg–B system be re-examined. Therefore, the present study estimates theoretically thermodynamic properties of magnesium boride phases, and subsequently the Mg–B phase diagram is assessed using the estimated thermodynamic data. 2. Computational methodologies The high volatility of magnesium and strong tendency of boron to oxidize have made the synthesis of high-purity MgB2 a problem that remains to be solved. As a result of air exposure, the Mg and B element ingredients of magnesium borides contain MgO and B2 O3 oxide impurity [18]. For this reason we forego for now an experimental effort to construct the phase diagram. Instead, we utilize ab initio calculations to compute the enthalpies of magnesium borides referenced against the enthalpies of elements in their standard reference state at 298.15 K (HSER). In addition, we employ additive entropy formulation [19] to estimate the entropy of mag- 86 S. Kim et al. / Journal of Alloys and Compounds 470 (2009) 85–89 Fig. 1. The literature binary Mg–B phase diagram [8]. nesium borides as functions of temperature and therefore the heat capacities of the magnesium borides. Based on the estimated thermodynamic database, the CALPHAD method can be used to assess the Mg–B binary phase diagram using PANDAT [20], a package for calculating multicomponent phase diagrams. A limited number of experimental data are available for the compounds in the Mg–B system, and these allow for independently checking the results of our theoretical analyses. 3. Results and discussions Three magnesium boride phases in the Mg–B binary system are all considered as stoichiometric compounds due to their ordered crystal structure [12,21–25], which allows for assuming that all Mg atoms occupy the Mg sublattice sites and all B atoms occupy the B sublattice sites. Accordingly, there is no mixing entropy contribution to the entropy of a magnesium boride. It has been shown that the entropies of such phases can often be estimated from a simple rule of mixtures of the entropies of component elements [19]. In this case, the entropy of a magnesium boride is described as [19] ◦ SMg 1−x Bx ◦ = (1 − x)SMg + xSB◦ Fig. 2. Estimated entropies of magnesium boride phases at 298.15 K, compared to those in the NIST-JANAF table [13]. internal energy, in which case the enthalpy of the system at 0 K becomes coincident with the internal energy, Etot , estimated from ab initio calculations [26]. In this regard, the formation enthalpy of ◦ (Mg Mg1−x Bx at 0 K, Hf,0 1−x Bx ) can be re-written [27,28]: ◦ (Mg Hf,0 1−x Bx ) = H0◦ (Mg1−x Bx ) − (1 − x)H0◦ (Mg) − xH0◦ (B) (4) = Etot (Mg1−x Bx ) − (1 − x)Etot (Mg) − xEtot (B) Ab initio calculations were performed by means of ABINIT code [29], which is a Density Functional Theory (DFT) code that uses pseudopotentials and a planewave basis. The total internal energy, Etot , is determined by finding the k point grids and the kinetic energy cutoff that gives the total energy converged to within 1 meV/atom, where the kinetic energy cutoff controls the number of planewaves at given k point grids [29]. Atomic crystal structures of magnesium borides in Fig. 3 are employed to compute the total internal energy Etot . In Fig. 3(a) the crystal structure of MgB2 [3,21,25] is hexagonal close packed with a space group of P6/mmm and the lattice param- (1) SB◦ ◦ SMg and are the entropies of Mg and B elements. To check where the validity of this analysis, Fig. 2 compares the entropies of magnesium boride phases at 298.15 K estimated by Eq. (1) with entropies from the literature [13]. The estimated entropy of a magnesium boride agrees closely with the literature entropy. When Mg and B atoms are assumed to stay on their own sublattice sites up to the melting temperature of a magnesium boride intermetallic compound, Eq. (1) allows for establishing the entropy of the boride as a function of temperature. Based on definition of the heat capacity, Cp , the entropy and enthalpy with respect to HSER can be established: ST◦ = ◦ S298.15 C T p + HT◦ T 298.15 = ◦ H298.15 dT (2) Cp dT (3) T + 298.15 By assuming the standard form Cp = a + bT + cT−2 + dT2 and fitting Eq. ◦ , a, b, c and (2) to the estimated entropy data the coefficients— S298.15 d were optimized. In thermodynamics, the enthalpy for a closed system at constant pressure P consists of the internal energy of the system and P–V work, where V is the volume change against the external pressure P. For solids at 1 atm the P–V work is negligible compared to the Fig. 3. Crystal structure of (a) MgB2 , (b) MgB4 , (c) MgB7 and (d) B12 icosahedra. S. Kim et al. / Journal of Alloys and Compounds 470 (2009) 85–89 87 Table 1 Optimized constant-pressure molar heat capacities of magnesium borides and their standard entropies (J/K/g-mol) S298.15 (J/K/g-mol) a b c d MgB2 MgB4 MgB7 14.78 20.13 7.67 × 10−3 −6.39 × 105 6.80 × 10−7 11.20 18.66 9.96 × 10−3 −7.22 × 105 −1.18 × 10−6 9.19 17.88 11.19 × 10−3 −7.72 × 105 −2.20 × 10−6 eters of a = 3.084 Å and c = 3.522 Å. MgB2 consists of close packed B layers alternating with Mg layers. B layer is a honeycomb net by arranging B atoms at the corners of a hexagon with three nearest neighbor B atoms. Mg atoms in each B layer are located at the body center of the B hexagonal prism that is formed between adjacent B layers. The crystal structure of MgB4 [22] (Fig. 3(b)) is orthorhombic with a space group of Pnam and the lattice parameters of a = 5.464 Å, b = 7.472 Å and c = 4.428 Å. Mg atoms are located in the tunnel that is formed by chains of boron pentagonal pyramids in which the averaged B–B bond is 1.787 Å. The crystal structure of MgB7 [12,23,24] (Fig. 3(c)) is orthorhombic with a space group of Imam and the lattice parameters of a = 5.970 Å, b = 8.125 Å and c = 10.480 Å. Chains of B12 icosahedra shown in Fig. 3(d) form a three-dimensional boron network via B–B bonds involving non-icosahedral atoms. For calculations reported here, the local density approximation (LDA) [28,30] pseudopotentials are employed. The enthalpy of Mg1−x Bx at 298.15 K with respect to HSER can be described: 298.15 ◦ (Mg1−x Bx )=H0◦ (Mg1−x Bx )+ H298.15 Cp (Mg1−x Bx ) dT (5) 0 ◦ ◦ where H298.15 (Mg1−x Bx ) ≡ H298.15 (Mg1−x Bx ) − (1 − ◦ ◦ (Mg) − xH298.15 (B), which is equivalent to the formax)H298.15 ◦ tion enthalpy of Mg1−x Bx at 298.15 K, Hf,298.15 (Mg1−x Bx ). The enthalpy of Mg1−x Bx at 0 K with respect to HSER can be described as ◦ ◦ (Mg)−xH298.15 (B) (6) H0◦ (Mg1−x Bx )=H0◦ (Mg1−x Bx )−(1 − x)H298.15 H0◦ (Mg1−x Bx ) = {H0◦ (Mg1−x Bx ) − (1 − x)H0◦ (Mg) − xH0◦ (B)} ◦ (Mg) − H0◦ (Mg)} − (1 − x){H298.15 ◦ (B) − H0◦ (B)} −x{H298.15 (7) Fig. 4. Calculated formation enthalpies of magnesium borides at 298.15 K, compared to experimental data in the literature [13,14,31,32]. PANDAT software package [20] was utilized to thermodynamically re-assess the Mg–B binary phase diagram (Fig. 5), in which liquid and gas phases were assumed to be ideally mixed. The re-assessed Mg–B phase diagram (Fig. 5) reveals significant differences in the decomposition temperatures of the magnesium boride phases compared to the literature temperatures. The literature decomposition temperatures of the MgB2 , MgB4 and MgB7 phases that were originally guessed to construct the Mg–B phase diagram are 1550 ◦ C, 1830 ◦ C and 2150 ◦ C, respectively [8]. For the decomposition reaction of boride phases: 2MgB2 ⇒ MgB4 + Mg(g) and 7MgB4 ⇒ 4MgB7 + 3Mg(g), upper and lower bounds for the literature decomposition temperatures are 850 ◦ C ≤ Tdecomp (MgB2 ) ≤ 1550 ◦ C and 1020 ◦ C ≤ Tdecomp (MgB4 ) ≤ 1827 ◦ C, respectively [8,10–17]. The present study estimated the decomposition temperatures at 1174 ◦ C for MgB2 , 1273 ◦ C for MgB4 and 2509 ◦ C for MgB7 (Fig. 5). For comparison, the Mg vapor pressures for the MgB2 /MgB4 equilibrium reaction are plotted together with the literature data in Fig. 6. Our calculated Mg vapor pressures are approximately 0.5–2.7 orders of magnitude higher than the literature data. This indicates that at a given temperature pure MgB2 can be more stable at higher where {H0◦ (Mg1−x Bx ) − (1 − x)H0◦ (Mg) − xH0◦ (B)} can be estimated ◦ ◦ from Eq. (4). Terms {H298.15 (Mg) − H0◦ (Mg)} and {H298.15 (B) − H0◦ (B)} are obtained from Eq. (3) and the literature heat capacity data [13] of Mg and B elements. The enthalpy of Mg1−x Bx at 0 K with respect to HSER is −25.91 kJ/g-mol for MgB2 , −27.64 kJ/g-mol for MgB4 and −27.68 kJ/g-mol for MgB7 . Eq. (5) and ab initio calculations estimated the formation enthalpies for MgB2 , MgB4 , and MgB7 at −23.44 kJ/g-mol, −25.67 kJ/g-mol, and −25.99 kJ/g-mol, respectively, at 298.15 K (Fig. 4). The literatures establish upper and lower bounds for the formation enthalpy at 298.15 K, −51.97 kJ/g-mol ≤ ◦ Hf,298.15 (MgB2 ) ≤ −17.17 kJ/g-mol and −21.0 kJ/g-mol ≤ ◦ Hf,298.15 (MgB4 ) ≤ −14.39 kJ/g-mol [13,14,31,32]. The estimated formation enthalpy for the MgB2 phase falls within the bounds for the literature MgB2 formation enthalpy, while that for the MgB4 phase lays near the literature bounds for MgB4 . This result indicates that the estimated formation enthalpies for MgB2 , MgB4 , and MgB7 are reliable to use for a thermodynamic assessment of the Mg–B phase diagram. Based on the estimated thermodynamic property data of magnesium boride phases as shown in Table 1 and Fig. 4, Fig. 5. Re-assessed phase diagram of the Mg–B system. 88 S. Kim et al. / Journal of Alloys and Compounds 470 (2009) 85–89 Fig. 6. Plot of log PMg (Pa) versus 10,000/T [10,11,13,14]. Mg pressures than that would be suggested by the literature data. In thermodynamics a vapor pressure is the saturated pressure of a system at the temperature T that equals to the external pressure (atmosphere pressure) acting on the system + system vapor. In this regard, Fig. 6 reveals that the decomposition temperature for the MgB2 /MgB4 equilibrium reaction increases with the increasing atmosphere pressure. Fig. 5 is the Mg–B binary phase diagram at an external pressure of 101.325 kPa (1 atm). To examine the stability of the MgB2 /MgB4 equilibrium reaction, the Mg–B binary phase diagram is assessed with the changing external pressures. In thermodynamics, the Gibbs free energy is dependent of pressure: ∂G ∂P =V (8) T where G is the Gibbs free energy, V is the volume and T is the temperature. For a solid substance, the volume compressibility is described [33]: dP d ln V = −B (9) where B is the bulk modulus. Based on an assumption of the bulk modulus independent of pressure, inserting Eq. (9) into Eq. (8) becomes: GT = −B V (10) where V is the volume change due to the changing pressure P and can be estimated from Eq. (9): P −P 2 1 V = V1 exp − B −1 (11) When the external pressure increases from 1 atm (P1 = 101.325 kPa) to 100 MPa (P2 ), the volume change of MgB2 is −0.00082V1 . Using the known lattice parameters (a = b = 3.084 Å, c = 3.522 Å [3,21,25]) of MgB2 and its bulk modulus of B = 122 GPa [34], the free energy change becomes 0.52 kJ/g-mol (Table 2). For Mg and B, their free energy changes due to the external Table 2 Free energy changes (kJ/g-mol) of Mg, B and MgB2 when the external pressure increases from 101.325 kPa (1 atm) to 100 MPa Bulk modulus (GPa) Free energy change (kJ/g-mol) Mg B MgB2 45 1.40 320 0.60 122 0.52 The bulk modulus is obtained from Refs. [34,35]. Fig. 7. Assessed Mg–B binary phase diagrams at the external pressures of (a) 1 MPa, (b) 10 MPa and (c) 100 MPa. pressure increase are 1.40 kJ/g-mol and 0.60 kJ/g-mol, respectively. However, the free energy change of the ideally mixed gas phase is RT ln(P2 /P1 ) = 17.20 kJ/g-mol at 25 ◦ C [13]. Based on these estimations, the free energy change of solid substance under the external pressure in ranges of 0.1–100 MPa is small enough to be ignorable, compared to the free energy change of the ideally mixed gas phase. Therefore, the gas phase is the only phase affected by the external pressure change: Ggas = i Xi Gi◦ + RT i Xi ln(Xi ) + RT ln P P1 (12) S. Kim et al. / Journal of Alloys and Compounds 470 (2009) 85–89 where Xi is the mole fraction of i (i = Mg, B), Gi◦ is the standard free energy of i at 1 atm, P is the external pressure and P1 is 101.325 kPa. Fig. 7 shows the assessed Mg–B phase diagrams at the external pressures of 1 MPa, 10 MPa and 100 MPa, respectively. When the external pressure is 1 MPa (Fig. 7(a)), Mg(l) phase exhibits a B solubility of 4.8% at 1465 ◦ C. The decomposition temperature of the MgB2 /MgB4 reaction increases from 1174 ◦ C at 101.325 kPa to 1514 ◦ C at 1 MPa. When the external pressure is 10 MPa (Fig. 7(b)), the gas phase appears at T > 2121.5 ◦ C. In addition, two peritectic reactions, L + MgB4 ⇔ MgB2 and L + MgB7 ⇔ MgB4 , arise at 1740.5 ◦ C and 2201 ◦ C, respectively. The Mg-rich liquid phase has the highest B solubility of 41.7% at 2374.5 ◦ C. Further increase in the external pressure increases the boiling temperature of Mg. In the Mg–B binary phase diagram at the external pressure of 100 MPa (Fig. 7(c)), gas phase does not exist below 3000 ◦ C. There are two peritectic reactions, L + MgB4 ⇔ MgB2 and L + MgB7 ⇔ MgB4 , at 1740.5 ◦ C and 2201.1 ◦ C as well as the congruent melting of MgB7 at 2810.2 ◦ C. Based on the Mg–B binary phase diagrams shown in Fig. 7, it is worthy to note that at high ambient pressure magnesium boride phases can be solidified from liquid without loss of Mg through evaporation. Since Mg evaporation can be suppressed, the high ambient pressure limits loss of Mg during syntheses of magnesium boride alloys, indicating that the alloy composition becomes controllable. In practical, the high ambient pressure allows for casting magnesium boride alloys. 4. Conclusion The Mg–B binary phase diagram is thermodynamically reconstructed by means of ab initio calculations and CALPHAD methods. The estimated formation enthalpies of magnesium boride phases by ab initio calculations reveal that they are reliable to use for thermodynamic assessments of the Mg–B phase diagram and their decomposition temperatures. The Mg–B phase diagrams at higher external pressure illustrate that Mg loss in the form of gas phase can be avoided during the syntheses of the Mg–B alloys. Acknowledgement The financial support of Korea Institute of Industrial Technology (KITECH) is gratefully acknowledged. 89 References [1] D.K. Finnemore, J.E. Ostenson, S.L. Bud’ko, G. Lapertot, P.C. Canfield, Condens. Mater. (2001) 0102114. [2] D.G. Hinks, H. Claus, J.D. Jorgensen, Nature 411 (2001) 457. [3] P. Ravindran, P. Vajeeston, R. Vidya, A. Kjekshus, H. Fjellvåg, Phys. Rev. B 64 (2001) 224509. [4] H.J. Choi, D. Roundy, H. Sun, M.L. Cohen, S.G. Louie, Nature 418 (2002) 758. [5] L. Jansen, Physica A 315 (2002) 583. [6] K. Nishiyama, T. Ogawa, K. Takagi, M. Omori, S. Umekawa, J. Jpn. Soc. Powder Powder Metall. 41 (2) (1994) 156. [7] L. Zhang, T. Dupont, Mater. Sci. Forum 546–549 (2007) 25. [8] T.B. Massalski, H. Okamoto, P.R. Subramanian, L. Kacprzak, Binary Alloy Phase Diagrams, 2nd ed., ASM International, Ohio, 1990. [9] K. E. Spear, SP-496, National Bureau of Standards, Maryland, 1977, p. 10. [10] Z.-K. Liu, Y. Zhong, D.G. Schlom, X.X. Xi, Q. Li, CALPHAD 25 (2001) 299. [11] Z.-K. Liu, D.G. Schlom, Q. Li, X.X. Xi, Appl. Phys. Lett. 78 (2001) 3678. [12] R. Naslain, A. Guette, P. Hagenmuller, J. Less-Common Met. 47 (1976) 1. [13] M.W. Chase Jr., NIST-JANAF Thermochemical Tables, 4th ed., American Institute of Physics for the National Institute of Standards and Technology, New York, 1998. [14] L.P. Cook, R. Klein, W. Wong-Ng, Q. Huang, R.A. Ribeiro, P.C. Canfield, IEEE Trans. Appl. Supercond. 15 (2005) 3227. [15] P. Duhart, Ann. Chim. (Paris) 7 (1962) 339. [16] S. Lee, A. Yamamoto, H. Mori, Yu. Eltsev, T. Masui, S. Tajima, Physica C 378–381 (2002) 33. [17] S. Lee, T. Masui, H. Mori, Yu. Eltsev, A. Yamamoto, S. Tajima, Supercond. Sci. Technol. 16 (2003) 213. [18] J. Jiang, B.J. Senkowicz, D.C. Larbalestier, E.E. Hellstrom, Supercond. Sci. Technol. 19 (2006) L33–L36. [19] M.S. Borovikova, V.V. Fesenko, J. Less-Common Met. 117 (1986) 287. [20] PANDAT, CompuTherm, LLC, Madison, WI, USA. http://www.computherm.com. [21] M.E. Jones, R.E. Marsh, J. Am. Chem. Soc. 76 (1954) 1434. [22] R. Naslain, A. Guette, M. Barret, J. Solid State Chem. 8 (1973) 68. [23] V.I. Matkovich, J. Economy, Acta Crystallogr. B26 (1970) 616. [24] A. Guette, M. Barret, R. Naslain, P. Hagenmuller, J. Less-Common Met. 82 (1981) 325. [25] V. Russell, R. Hirst, F.A. Kanda, A.J. King, Acta Crystallogr. 6 (1953) 870. [26] W. Zhang, J.R. Smith, X.-G. Wang, Phys. Rev. B 70 (2004) 024103. [27] M. Kim, A.J. Freeman, S. Kim, J.H. Perepezko, G.B. Olson, Appl. Phys. Lett. 87 (2005) 261908. [28] C. Wolverton, X.-Y. Yan, R. Vijayaraghavan, V. Ozolins, Acta Mater. 50 (2002) 2187. [29] X. Gonze, ABINIT, Lab PCPM at Universite Catholique de Louvain, Louvain-laNeuve, Belgium. http://www.abinit.org. [30] L. Hedin, B.I. Lundqvist, J. Phys. C 4 (1971) 2064. [31] S.M. Ariya, M.P. Morozova, G.A. Semenov, G.A. Rybakova, Russ. J. Phys. Chem. 45 (1971) 102. [32] G. Kaptay, Mater. World 2 (3) (2001). [33] G.E. Dieter, Mechanical Metallurgy, 3rd ed., McGraw Hill, New York, 1986. [34] A.K.M.A. Islam, F.N. Islam, S. Kabir, J. Phys.: Condens. Matter 13 (2001) L641. [35] WebElementsTM Periodic Table, The University of Sheffield and WebElements Ltd., UK. http://www.webelements.com.