Magnesium Technology 2007 Edited by Randy S. Beals, Alan A. Luo, Neale R. Neelameggham, and Mihriban O. Pekguleryuz
TMS (The Minerals, Metals & Materials Society), 2007
Title of Publication Edited by
TMS (The Minerals, Metals & Materials Society), 2007
Enthalpies of Formation of Magnesium Compounds from First-Principles Calculations
Hui Zhang, James Saal, Arkapol Saengdeejing, Yi Wang, Long-Qing Chen, Zi-Kui Liu
Department of Materials Science and Engineering, The Pennsylvania State University, 304 Steidle Building,
University Park, PA 16802, USA
Keywords: magnesium, first-principles, enthalpy of formation
magnetic properties, such as Dy and Lu, spin-polarized
calculations were performed. For consistency, a 5,000 k-point per
reciprocal atom mesh was used with an energy cut-off of 360 eV.
Some rare earth elements, such as Pr and Eu, were not included in
this work due to the difficulty of calculating their energies
accurately in DFT. This is due to the highly localized nature of
the f-electrons [8].
Abstract
An energetics database of binary magnesium alloys has been
developed from first-principles calculations. The systems
investigated include Mg-As, -Ba, -Ca, -Cd, -Cu, -Dy, -Ga, -Ge, La, -Lu, -Ni, -Pb, -Sb, -Si, -Sn and –Y. The lattice parameters and
the enthalpies of formation of binary compounds in these systems
are presented and compared with available experimental data.
B. Crystal structures: Pure elements and compounds
Introduction
The Mg-X binary systems were chosen based on the availability
of crystal structure data for all the phases in the systems. For pure
elements, the energy of each element in its stable structure at
room temperature is calculated. The observed ground state
structures for each element X are listed in Table 1 [10]. For the
ordered compounds, the energies of the most Mg-X compounds
are calculated in the present work. The crystal structures of these
compounds are listed in Table 2 [10].
Table 1: Crystal structures of pure elements
Strukturbericht
Phase Structure Pearson Symbol
Designation
As
α -As
hR6
A7
Ba
bcc
cI2
A2
Bi
α -As
hR2
A7
Ca
fcc
cF4
A1
Cd
hcp
hP2
A3
Cu
fcc
cF4
A1
Dy
hcp
hP2
A3
Ga
α-Ga
oC8
A11
Ge
diamond
cF8
A4
La
hcp
hP2
A3
Mg
hcp
hP2
A3
Ni
fcc-FM
cF4
A1
Pb
fcc
cF4
A1
Sb
α -As
hR2
A7
Si
diamond
cF8
A4
Sr
bcc
cI2
A2
Sn
tI4
A5
β-Sn
Y
hcp
hP2
A3
Zn
hcp
hP2
A3
Magnesium alloys are of great importance due to their low
density. A greater understanding of the thermodynamics of these
alloys will allow better control in designing materials with desired
properties. The CALPHAD (CALculation of PHAse Diagrams)
approach, coupled with first-principles calculations, provides an
efficient route to model thermodynamic properties of multicomponent materials [1]. One group of the critically needed input
data in CALPHAD modeling is the enthalpy of formation of
compounds, which can be reliably obtained through firstprinciples calculations as demonstrated in the literature [2-5]. In
first-principles calculations, by inputting only the crystal structure
and composition of the phase, the total energy at 0K of a
compound in its relaxed equilibrium state can be calculated, from
which the enthalpy of formation can be derived. In our previous
study of the Mg-based systems, the thermodynamic database of
the Mg-Zr [3] and the Mg-Ca [5] systems were developed by this
method. The thermodynamic description of the Mg-Ca-Sn [4] and
the Ca-Mg-Sr [6] ternary systems were predicted as well. The
present work further extends the calculations of the enthalpies of
formation of a number of binary magnesium alloys and compares
them with available experimental data by the first-principle
method. The systems considered include Mg-As, -Ba, -Ca, -Cd, Cu, -Dy, -Ga, -Ge, -La, -Lu, -Ni, -Pb, -Sb, -Si, -Sn and –Y.
Methodology
A. First-principles method
First-principles calculations, based on the density functional
theory (DFT), were performed using the projected augmented
wave (PAW) pseudo-potentials as implemented in VASP (Vienna
Ab-initio Simulation Package) [7, 8] with the generalized gradient
approximation refined by Perdew, Burke and Ernzerhof (PBE)[9].
All the structures were fully relaxed with respect to volume and
the atomic coordinates. For calculations of elements involving
345
Table 2: Crystal structures of compounds considered in this paper
System
Formula
Compound
Pearson Symbol
Strukturbericht
Designation
Prototype
Mg3As2-ht
hP5
D519
La2O3
Mg3As2-rt
cI80
---
Mn2O3
MgAs4
tP20
---
MgAs4
Mg17Ba2
hR57
---
Zn17Th2
Mg23Ba6
cF116
D84
Th6Mn23
Mg2Ba
hP12
C14
MgZn2
Mg-Ca
Mg2Ca
hP12
C14
MgZn2
Mg-Cd
MgCd
oP4
B19
AuCd
MgCd3
hP8
D019
Ni3Sn
Mg3Cd
hP8
D019
Ni3Sn
Mg-Cu
MgCu2
cF24
C15
MgCu2
Mg-Dy
MgDy
cP2
B2
CsCl
Mg2Dy
hP12
C14
MgZn2
Mg3Dy
cF16
D03
BiFe3
Mg24Dy5
cI58
A12
Ti5Re24
MgGa
tI32
---
MgGa
MgGa2
hP6
B82
CaIn2
Mg2Ga5
tI28
---
Mg2Ga5
Mg-Ge
Mg2Ge
cF12
C1
CaF2
Mg-La
MgLa
cP2
B2
CsCl
Mg2La
cF24
C15
MgCu2
Mg3La
cF16
D03
BiF3
Mg17La2
hP38
---
Th2Ni17
Mg12La
oI338
---
---
MgLu
cP2
B2
CsCl
Mg2Lu
hP12
C14
MgZn2
Mg24Lu5
cI58
A12
Ti5Re24
Mg2Ni
hP18
Ca
Mg2Ni
MgNi2
hP24
C36
MgNi2
Mg-As
Mg-Ba
Mg-Ga
Mg-Lu
Mg-Ni
Mg-Sb
Mg3Sb2
hP5
D519
Al3Ni2
Mg-Si
Mg2Si
cF12
C1
CaF2
Mg-Sn
Mg2Sn
cF12
C1
CaF2
Mg-Pb
Mg2Pb
cF12
C1
CaF2
Table 2: Crystal structures of compounds considered in this paper
(continued)
System
Mg-Sr
Mg-Y
Formula
Compound
Strukturbericht
Pearson Symbol
Designation
Prototype
Mg2Sr
hP12
C14
MgZn2
Mg23Sr6
cF116
D84
Mn23Th6
Mg38Sr9
hP94
---
Mg38Sr9
Mg17Sr2
hP38
---
Th2Ni17
MgY
cP2
B2
CsCl
Mg2Y
hP12
C14
MgZn2
Mg24Y5
cI58
A12
Ti5Re24
∆ f E (Mg x X y ) = E (Mg x X y ) −
x
y
E (Mg) −
E(X )
x+ y
x+ y
(1)
where E:s are the total energy of the compound and the pure
elements in their stable structures at room temperature. Since the
influences of pressure on the condensed phases are ignored and
the energies are calculated at 0 K without any entropy
contribution, the energy of formation is approximated as the
enthalpy of formation as shown below.
∆ f H (Mg x X y ) ≈ ∆ f E (Mg x X y )
(2)
One method to determine the accuracy of the calculations of the
pure elements is to compare the calculated lattice parameters with
those determined experimentally. Such a comparison is given in
Table 3. The calculated data shows good agreement with
experiments, all within 1 or 2% difference. A similar assessment
of lattice parameters is presented for the binary compounds in
Table 4 [11]. Again, there is good agreement between the
calculated and experimental data.
The differences in the calculated enthalpies of formation of the
binary compounds compared with the available experimental data
are given in Table 5 [12, 13]. There is generally better agreement
in the lattice parameters than in the enthalpies of formation. The
magnitude of the differences in the enthalpies of formation can be
easily seen in Figure 1, in which the calculated and experimental
values are plotted in the x- and y-axis, respectively, with the
dotted lines representing ±5kJ/mole-atom. The top right corner of
Figure 1 is enlarged in Figure 2. The calculated enthalpies of
formation compare favorably with experiment for most
compounds, keeping in mind the large uncertainly in experimental
data. There are seven compounds showing difference between the
calculated and the experimental data over 40%.
Enthalpies of formation can vary greatly depending on the
experimental methods used, more so than lattice parameter
determination. Consequently, larger error can be expected for the
enthalpies of formation. As an example, the enthalpy of
formation of Mg3Sb2 has been experimentally determined
numerous times, with values ranging from -320 to -240 kJ/mol
formula [12]. Furthermore, the number of enthalpy of formation
data is fewer than that of lattice parameter data, so the error in a
given experiment is hard to ascertain. For instance, there is only
Results and Discussion
The enthalpy of formation of a compound can be defined as the
difference in total energy of the compound and the energies of its
constituent elements in their stable state:
346
0
]
rm
of
lo
j/m
k[
no-10
it
a
m
ro
Ff
o
yp
la
ht
nE-20
la
tn
e
m
ir
ep
xE
-10
]
m
orf
lo
-30
m
j/k
[
no
it
a -50
rm
oF
fo
yp -70
la
ht
nE
la
tn -90
e
im
re
px -110
E
MgGa
Mg Ga
Mg Ca Mg Ge
Mg Dy
2
2
2
2
2
Mg Ga
5
Mg Ga
2
Mg Sb
5
3
2
Mg Sn
2
2
MgAs
4
-110
3
MgCd
3
MgCu
2
MgNi
2
MgDy
Mg Pb
2
Mg Ni
2
MgGa
Mg Si
2
-30
-130
-130
Mg Cd
MgCd
-90
-70
-50
-30
-30
-10
Calculated Enthalpy of Formation [kj/mol form]
-20
-10
0
Calculated Enthalpy of Formation [kj/mol form]
Figure 1: Comparison of experimental and calculated enthalpy of
formation for the binary compounds in the Mg-X systems that are
greater than -130 kJ/ mol-formula. The solid line shows unity
(x=y) while the dashed lines …. present an error range of ±5
kJ/mol-formula. The region inside the dashed lines - - - is enlarged
in Figure 2.
Figure 2: Comparison of experimental and calculated enthalpy of
formation for the binary compounds in the Mg-X systems that are
greater than -30 kJ/ mol-formula. The solid line shows unity (x=y)
while the dashed lines present an error range of ±5 kJ/mol-formula.
Dongwon Shin in our Phases Research Lab for stimulating
discussions.
one experimental value of the enthalpy of formation of As2Mg3
[12] measured by acid solution calorimetry.
References
Summary
1. Z.-K. Liu and L.-Q. Chen, in Applied Computational
Materials Modeling: Theory, Experiment and Simulations G.
First-principles calculations are performed for the phases in
various Mg-X binary systems. The calculated lattice parameters
and enthalpies of formation are compared with experimental data
in the literature. There is good agreement in lattice parameter
data. However, agreement in enthalpies of formation is less
satisfactory. Possible sources of error include the unknown
uncertainty of experimental data.
Bozzolo, Ed. (Springer, 2006).
2. S. Curtarolo, D. Morgan and G. Ceder, CALPHAD, 29,
(2005) 163-211.
3. R. Arroyave, D. Shin and Z. K. Liu, CALPHAD, 29, (2005)
230-238.
4. R. Arroyave, M. Ohno, R. Schmid-Fetzer and Z.-K. Liu, in
Magnesium Technology 2006, A. Luo, Ed., San Francisco
(Minerals, Metals and Materials Society/AIME, 184 Thorn
Hill Road, Warrendale, PA, 2006), pp. 175-180.
5. Y. Zhong, K. Ozturk, J. O. Sofo and Z. K. Liu, J. Alloy.
Compd., 420, (2006) 98-106.
6. Y. Zhong, J. O. Sofo, A. A. Luo and Z. K. Liu, J. Alloy.
Compd., 421, (2006) 172-178.
7. G. Kresse and J. Furthmuller, Phys. Rev. B, 54, (1996)
8. G. Kresse and J. Furthmuller, Comput. Mater. Sci., 6, (1996)
9. J. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 77,
(1996) 3865.
10. P. P. Villars, Pearson's handbook of crystallographic data
for intermetallic phases (American Society for Metals,
1985).
11. "Pauling File", available on the web
(http://crystdb.nims.go.jp/) or CD, (2002)
12. A. A. Nayeb-Hashemi and J. B. Clark, ASM International,
(1988)
13. Selected values of chemical thermodynamic properties
(United States National Bureau of Standards, Washington,
1965).
Acknowledgements
This work is supported in part by the National Science Foundation
(NSF) through the grants DMR-0510180 and DMR-0205232.
The authors would also like to thank the United States
Automotive Materials Partnership (USAMP) consisting of
General Motors Corporation, Ford Motor Company and DaimlerChrysler Corporation, for their financial support. The authors
acknowledge that this research was supported, in part, by
Department of Energy Cooperative Agreement No. DE-FC05
02OR22910. Such support does not constitute an endorsement by
the Department of Energy of the views expressed herein. The
Materials Simulation Center (MSC) at Penn State is
acknowledged for the computer resources to perform first
principles calculations implemented in VASP, and the LION
cluster supported in part by the NSF grants (DMR-9983532, DMR
0122638, and DMR-0205232,) is also used. We would also like
to thank Dr. Raymundo Arroyave, Dr. Shunli Shang and Mr.
347
Table 3: Calculated lattice parameters for the elements in the Mg-X systems in comparison with experimental data
Element
a (Å)
b (Å)
c (Å)
Calculated Experimental % Difference Calculated Experimental % Difference Calculated Experimental % Difference
As
Ba
Bi
Ca
Cd
Cu
Dy
Ga
Ge
La
Lu
Mg
Ni
Pb
Sb
Si
Sn
Y
Zn
3.823
5.025
4.583
5.537
3.033
3.634
3.615
4.593
5.724
3.761
3.494
3.193
3.514
5.024
4.38
5.47
6.657
3.655
2.7
3.760
5.025
4.557
5.599
2.979
3.615
3.590
4.514
5.658
3.765
3.500
3.213
3.524
4.9503
4.3
5.4303
6.483
3.655
2.6655
1.681
0.000
0.571
-1.107
1.813
0.520
0.696
1.750
1.166
-0.106
-0.171
-0.607
-0.295
1.489
1.860
0.731
2.684
0.000
1.294
3.823
5.025
4.583
5.537
3.033
3.634
3.615
7.767
5.724
3.761
3.494
3.193
3.514
5.024
4.38
5.47
6.657
3.655
2.7
3.760
5.025
4.557
5.599
2.979
3.615
3.590
7.644
5.658
3.765
3.500
3.213
3.524
4.9503
4.3
5.4303
6.483
3.655
2.6655
348
1.681
0.000
0.571
-1.107
1.813
0.520
0.696
1.609
1.166
-0.106
-0.171
-0.607
-0.295
1.489
1.860
0.731
2.684
0.000
1.294
10.683
5.025
12.186
5.537
5.680
3.634
5.639
4.590
5.724
12.085
5.473
5.179
3.514
5.024
11.466
5.47
6.644
5.685
4.765
10.548
5.025
11.862
5.599
5.617
3.615
5.640
4.526
5.658
12.150
5.505
5.213
3.524
4.9503
11.251
5.4303
6.483
5.751
4.9488
1.285
0.000
2.731
-1.107
1.122
0.520
-0.018
1.414
1.166
-0.535
-0.581
-0.656
-0.295
1.489
1.911
0.731
2.483
-1.148
-3.714
Table 4: Calculate lattice parameters for the binary compounds in the Mg-X systems in comparison with experimental data
System
Phase
Mg-As
As2Mg3-ht
As2Mg3-rt
MgAs4
Mg17Ba2
Mg23Ba6
Mg2Ba
Mg2Ca
MgCd
MgCd3
Mg3Cd
MgCu2
MgDy
a (Å)
b (Å)
c (Å)
Cal.
Exp.
% Diff.
Cal.
Exp.
% Diff.
Cal.
Exp.
% Diff.
Mg2Dy
Mg3Dy
Mg24Dy5
Mg5Ga2
Mg2Ga
MgGa
MgGa2
Mg2Ga5
4.294
12.455
5.472
10.625
15.220
6.665
6.234
5.049
6.353
6.313
7.065
3.786
6.049
10.3
11.255
7.036
7.805
10.690
6.868
8.725
4.264
12.355
5.385
10.650
15.263
6.636
6.230
5.005
6.234
6.310
7.04
3.776
6.02
10.33
11.246
7.017
7.794
10.530
6.802
8.627
0.704
0.809
1.616
-0.235
-0.282
0.437
0.064
0.869
1.917
0.048
0.355
0.265
0.482
-0.290
0.080
0.271
0.141
1.519
0.970
1.136
4.294
12.455
5.472
10.625
15.220
6.665
6.234
3.213
6.353
6.313
7.065
3.786
6.049
10.3
11.255
13.747
7.805
10.690
16.457
8.725
4.264
12.355
5.385
10.650
15.263
6.636
6.230
3.222
6.234
6.310
7.04
3.776
6.02
10.33
11.246
13.708
7.794
10.530
16.346
8.627
0.704
0.809
1.616
-0.235
-0.282
0.437
0.064
-0.272
1.917
0.048
0.355
0.265
0.482
-0.290
0.080
0.285
0.141
1.519
0.679
1.136
6.768
12.455
16.083
15.564
15.220
10.577
10.093
5.266
4.949
5.037
7.065
3.786
9.788
5.94
11.255
6.028
6.941
5.555
4.139
7.178
6.738
12.355
15.798
15.587
15.263
10.655
10.120
5.270
5.045
5.080
7.04
3.776
9.76
5.96
11.246
6.020
6.893
5.530
4.111
7.111
0.445
0.809
1.804
-0.148
-0.282
-0.732
-0.267
-0.067
-1.903
-0.846
0.355
0.265
0.287
-0.336
0.080
0.133
0.696
0.452
0.681
0.942
Mg-Ge
Mg-La
Mg2Ge
MgLa
Mg2La
Mg3La
Mg17La2
Mg12La
6.423
3.966
8.776
7.5
10.351
10.337
6.385
3.97
8.809
7.494
10.36
10.34
0.597
-0.101
-0.375
0.080
-0.087
-0.029
6.423
3.966
8.776
7.5
10.351
10.337
6.385
3.97
8.809
7.494
10.36
10.34
0.597
-0.101
-0.375
0.080
-0.087
-0.029
6.423
3.966
8.776
7.5
10.156
5.911
6.385
3.97
8.809
7.494
10.24
7.74
0.597
-0.101
-0.375
0.080
-0.820
-23.630
Mg-Lu
MgLu
Mg2Lu
Mg24Lu5
3.703
5.973
11.162
3.727
5.96
11.185
-0.644
0.218
-0.206
3.703
5.973
11.162
3.727
5.96
11.185
-0.644
0.218
-0.206
3.703
9.684
11.162
3.727
9.71
11.185
-0.644
-0.268
-0.206
Mg-Ni
Mg2Ni
MgNi2
5.199
4.815
5.212
4.825
-0.244
-0.207
5.199
4.815
5.212
4.825
-0.244
-0.207
13.188
15.821
13.254
15.790
-0.496
0.196
Mg-Pb
Mg-Si
Mg2Pb
Mg2Si
6.944
6.358
6.836
6.351
1.580
0.110
6.944
6.358
6.836
6.351
1.580
0.110
6.944
6.358
6.836
6.351
1.580
0.110
Mg-Sn
Mg2Sn
MgY
Mg2Y
Mg24Y5
6.825
3.803
6.049
11.26
6.759
3.797
6.037
11.28
0.971
0.158
0.199
-0.177
6.825
3.803
6.049
11.26
6.759
3.797
6.037
11.28
0.971
0.158
0.199
-0.177
6.825
3.803
9.831
11.26
6.759
3.797
9.752
11.28
0.971
0.158
0.810
-0.177
Mg-Ba
Mg-Ca
Mg-Cd
Mg-Cu
Mg-Dy
Mg-Ga
Mg-Y
349
Table 5: Calculated enthalpies of formation of the binary compounds in comparison with experimental data
System
phase
Mg-As
Enthalpy of formation (kJ/mol-formula)
Calculated
Experimental
Difference
% Difference
Mg3As2-ht
Mg3As2-rt
-297.560
-306.302
---401.230
---94.928
---23.659
Mg As4
Mg17Ba2
-112.025
-216.334
-122.860
---
-10.835
---
-8.819
---
Mg23Ba6
Mg2Ba
-125.042
-101.668
-----
-----
-----
Mg2Ca
MgCd
-36.412
-21.026
-36.681
-16.076
-0.269
4.95
-0.733
30.791
MgCd3
Mg3Cd
-25.744
-30.144
-25.344
-29.700
0.4
0.444
1.578
1.495
MgDy
Mg2Dy
-24.356
-21.609
-24.020
-48.840
0.336
-27.231
1.398
-55.756
Mg3Dy
Mg24Dy5
-30.160
-139.616
-64.280
-397.880
-34.12
-258.264
-53.080
-64.910
MgCu2
Mg5Ga2
-14.252
-80.563
-12.853
-76.300
1.399
4.263
10.885
5.587
Mg2Ga
MgGa
-37.783
-27.416
-35.100
-26.000
2.683
1.416
7.644
5.446
Mg Ga2
Mg2Ga5
-34.750
-75.495
-34.200
-69.300
0.55
6.195
1.608
8.939
Mg2Ge
MgLa
-68.287
-23.274
-115.560
---
-47.273
---
-40.908
---
Mg2La
Mg3La
-37.655
-53.769
-----
-----
-----
Mg17La2
Mg12La
-146.251
-75.328
-----
-----
-----
MgLu
Mg2Lu
-6.879
-12.407
-----
-----
-----
-81.254
-59.607
---60.291
---
Mg-Ni
Mg24Lu5
Mg2Ni
-0.684
---1.134
Mg-Pb
MgNi2
Mg2Pb
-77.547
-8.538
-77.871
-20.513
-0.324
-11.975
-0.416
-58.378
Mg-Sb
Mg-Si
Mg3Sb2
Mg2Si
-179.882
-47.165
-300.200
-80.130
-40.079
-41.139
Mg-Sn
Mg-Y
Mg2Sn
MgY
-51.511
-21.271
-----
-120.318
-32.965
-----
Mg2Y
Mg24Y5
-27.524
-169.383
-----
-----
-----
Mg-Ba
Mg-Ca
Mg-Cd
Mg-Dy
Mg-Cu
Mg-Ga
Mg-Ge
Mg-La
Mg-Lu
350
-----