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Journal of Alloys and Compounds 470 (2009) 85–89
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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-
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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.
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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
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