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The optimized composition of Mg–Al–Cu metallic
glass investigated by thermodynamic calculations
and an atomistic approach
S. Zhao, J. H. Li* and B. X. Liu
Issues related to the glass formation of ternary Mg–Al–Cu metallic glass are investigated by thermodynamic
calculations and an atomistic approach. Based on the extended Miedema model, thermodynamic
calculations show that Mg–Al–Cu metallic glasses are favored over a large composition range, and that
the sub-region of MgxAlyCu1xy (x ¼ 20–30; y ¼ 30–40; 1 x y ¼ 35–45) shows a better glass
formation ability than the other compositions. Then, a realistic interatomic potential was constructed for
the Mg–Al–Cu system and applied in Monte Carlo simulations to predict the favored composition at the
atomic level, and even pinpoint the optimized composition. The simulation not only predicted
a quadrilateral region, within which Mg–Al–Cu metallic glass formation is energetically favored, but also
pinpointed an optimized sub-region within which the amorphization driving force (ADF), i.e. the energy
difference between the solid solution and the disordered state, is larger than that outside. The
Received 14th July 2016
Accepted 7th October 2016
predictions of the atomistic approach are consistent with the thermodynamic calculation. The
DOI: 10.1039/c6ra17942h
simulations not only provided predictions for producing Mg–Al–Cu metallic glasses, but also revealed
the physical origin of the crystal–amorphous transition, which could be of great help for designing
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ternary glass formers.
1. Introduction
Bulk metallic glasses (BMGs) have been a hot topic ever since
they were rst found.1–3 The properties of BMGs, such as high
yield strength, hardness, elastic strain limit, and corrosion
resistance4–7 have led people to spare no effort in researching
them. To date, BMGs have been obtained in many multicomponent alloy systems, such as Mg–Cu–Y,8,9 Al–Cu–Zr10,11 and
Ca–Al–Mg–Cu.12 In the eld of BMGs, a basic and interesting
issue is to clarify the glass forming ability (GFA), so that one
could easily design appropriate compositions for producing
metallic glasses. To characterize the GFA, some experimental
criteria13–15 have been proposed based on the glass transition
temperature Tg, melting temperature Tm and crystallization
onset temperature Tx. However, these criteria are not able to
predict the GFA, because they are obtained aer the glass has
been produced. Besides, it takes plenty of work to pinpoint the
optimized compositions. Thus, several criteria have been
proposed to predict the glass formation range and the optimized compositions, such as deep eutectic,16 size difference17 or
structural difference18 rules. Although these empirical rules
have been widely used for producing BMGs, the predictions are
not accurate enough. Besides, these rules are restricted in
Key Laboratory of Advanced Materials (MOE), School of Materials Science and
Engineering, Tsinghua University, Beijing 100084, China. E-mail: lijiahao@mail.
tsinghua.edu.cn
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reecting the internal characteristics and mechanisms. Therefore, it is of signicance to seek a more valid way to clarify
the underlying physics of glass formation and predict more
accurately.
Among various thermodynamic calculation schemes, Miedema's model and Alonso's method19,20 are widely used to
quantitatively explain and predict the formation of metallic
glass from a semi-empirical perspective. Furthermore, once the
interaction between the atoms is conrmed, atomistic simulations, including molecular dynamics (MD) and Monte Carlo
(MC) simulations, can be a powerful tool for investigating
fundamental issues relating to metallic glasses.
Recently, bulk metallic glasses including the elements of Mg,
Al and Cu have been widely studied,8,9,11,12,21 due to their good
GFA, low cost, high specic strength and biocompatibility.
However, metallic glasses which consist of Mg, Al and Cu are
less researched. Therefore, we selected ternary Mg–Al–Cu as
a model system, and investigated it in two ways, i.e. through
thermodynamic calculations and MC simulations.
2.
Thermodynamic calculations
Miedema's model and Alonso's method are proposed based on
the macroscopic atom picture, of which the basic assumption is
the choice of the reference as the atoms embedded in the metal
instead of the free atoms, and it has been veried widely in
predicting glass formation.19,20 When applying these methods,
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the Gibbs free energy of the competing alloy phases, including
the solid solution, intermetallic compounds and the amorphous phase should be calculated, which can be expressed as
DG ¼ DH TDS, where DH and DS are the enthalpy and entropy
terms, respectively. The entropy term DS for a concentrated
solid solution or an amorphous phase can be approximated as
that of an ideal solution. For a ternary system consisting of the
elements of A, B and C, DS can be expressed as
DS ¼ R[cA ln cA + cB ln cB + cC ln cC]
(1)
where R is the gas constant, and cA, cB and cC is the atomic
concentration of A, B and C, respectively.
According to the work by Miedema and Alonso, the enthalpy
of formation of a ternary solid solution of transition metals A, B
and C is given by:
c
e
s
DHss
ABC ¼ DHABC + DHABC + DHABC
(2)
where DHcABC, DHeABC and DHsABC is the chemical, elastic and
structural contribution, respectively.
The chemical term DHcABC is related to the electron redistribution generated at the boundary of the Wigner–Seitz unit cell
when the alloy is formed and can be divided into three binary
sub-systems as:
DHcABC
¼
DHcAB
+
DHcAC
+
DHcBC
(3)
For a binary subsystem consisting of A and B, DHcAB is given
by:
DHcAB ¼ cAcB[cBDH inter
A in
B
inter
+ cADH
A in B]
(6)
For the A–B binary subsystem, DHeAB is given by:
elastic
DHeAB ¼ cAcB[cBDH elastic
A in B + cADH B in A]
102330 | RSC Adv., 2016, 6, 102329–102335
(8)
, ZA, ZB and ZC is the average number of valence elecwhere Z
trons of the alloy phase and valence electrons of the pure metals
), Es(ZA), Es(ZB) and Es(ZC) is the
A, B and C, respectively. Es(Z
lattice stability parameter of the alloy phase and the pure metals
A, B and C, respectively.
For the amorphous phase, the enthalpy difference between
the amorphous and crystalline states of the pure metals should
be considered, while the elastic term and the structural term are
absent. Thus, the formation enthalpy of the amorphous phase
can be calculated as:
c
a–s
a–s
a–s
DHam
ABC ¼ DHABC + cADHA + cBDHB + cCDHC
(9)
where DHAa–s, DHBa–s and DHCa–s is the enthalpy difference
between the amorphous and crystalline states of pure metals A,
B and C, respectively. As proposed by van der Kolk et al.,22 DHa–
s
i can be calculated by:
¼ aTm,i
DHa–s
i
(10)
where a ¼ 3.5 J mol1 K1 and Tm,i is the melting temperature of
component i.
According to Xia's proposal23 and Wang's modication,24 the
GFA can be evaluated as follows:
g*ABC ¼ GFAf
am
DHABC
inter
DHABC
am
DHABC
(11)
(5)
where csA and csB are the cell surface concentrations, which can
cA VA 2=3
and csB ¼ 1 csA. The
be calculated by csA ¼
2=3
cA VA þ cB VB 2=3
parameter g describes the short-range ordering in different
alloys and is an empirical constant which is usually taken as 0, 5
and 8 for the solid solution, amorphous phase and ordered
compound, respectively. Similarly, the chemical terms of
DHcAC and DHcBC can be calculated by eqn (4), thus the chemical
term can be obtained by eqn (3).
The elastic term DHeABC is the contribution of the atomic size
mismatch and can also be divided into three binary subsystems:
DHeABC ¼ DHeAB + DHeAC + DHeBC
[cAEs(ZA) + cBEs(ZB) + cCEs(ZC)]
DHsABC ¼ Es(Z)
(4)
inter
inter
where DH
A in B and DH B in A are the electron redistribution
contributions to the enthalpy of A dissolved in B and that of B
dissolved in A. Considering the possible chemical short range
ordering of the alloy phases, the right-hand side of eqn (4)
should be multiplied by a factor
f ¼ 1 + g(csAcsB)2
elastic
elastic
where DH
A in B and DH B in A are the elastic contributions to the
enthalpy of A dissolved in B and that of B dissolved in A.
Similarly, the calculation of DHeAC and DHeBC are the same, thus
DHeABC of the ternary system can be calculated by eqn (6).
The structural contribution DHsABC is the correlation between
the number of valence electrons and the crystal structure of the
) of each
metals. It can be deduced from the lattice stability Es(Z
crystal structure s (s ¼ bcc, fcc or hcp) as a function of the
number of valence electrons Z of the metal:
(7)
Thus, by calculating the parameter g*ABC , the composition
dependence of the GFA can be predicted. The parameters used
in the thermodynamic calculations of the Mg–Al–Cu system are
listed in Table 1. The glass formation range (GFR), within which
the Gibbs free energy of the amorphous phase is lower than that
Table 1 The parameters used in the thermodynamic calculations for
A),
the Mg–Al–Cu system. Tm,i is expressed in K, and Es(Z
inter
elastic
elastic
1
inter
DH
,
D
H
,
D
H
,
and
D
H
are
expressed
in
kJ
mol
A in B
B in A
A in B
B in A
Tm,i
A)
Es(Z
inter
DH
A in B
inter
DH
B in A
elastic
DH
A in B
elastic
DH
B in A
Mg
Al
Cu
922
0
933
0
1357
1.5
Mg–Al
Mg–Cu
Al–Cu
6.35
6.98
8.23
1.40
22.58
17.11
36.22
42.40
47.17
37.61
16.19
12.34
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A glass formation stoichiometry diagram, obtained from thermodynamic calculations at 300 K for the Mg–Al–Cu ternary system. (a) The
glass formation range bounded by dashed lines and (b) the distribution of g*ABC of each composition within the glass formation range.
Fig. 1
of the solid solution, is shown in Fig. 1(a) and is represented by
black dots and bounded by dashed lines. Fig. 1(b) shows the
distribution of the GFA, in which the sub-region (colored red)
with the composition MgxAlyCu1xy (x ¼ 20–30; y ¼ 30–40; 1 x y ¼ 35–45) shows a better GFA than other compositions. In
the following sections, we proceed to investigate glass formation at the atomic level and compare the results with those of
thermodynamic calculations.
3.
Mg–Al–Cu interatomic potential
To perform the atomistic simulations, a starting base is the
construction of a realistic interatomic potential, which can
reect the internal interactions in a certain system and thus
clarify the glass formation and predict the favored compositions
of the glassy alloys, and even pinpoint the optimized
compositions.
Therefore, we rst constructed the atomic interactions of the
Mg–Al–Cu system and applied the framework of smoothed and
long-range second-moment-approximation of tight-binding
(TB-SMA).25,26 The formalism can be written as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X 1X (12)
4 rij j rij
Ei ¼
2 jsi
jsi
4 rij ¼
8
rij
>
>
1
;
2A
1 exp p1
>
<
r
0
Table 2 Lattice constants (a and c), cohesive energies Ec, elastic
constants Cij and bulk moduli B0 of Al, Mg, Cu fitted by the potential
and obtained from experimental data or ab initio calculations
rij # rm1
n1
>
>
r
rc1 rij
>
: 2A1m exp p1m ij 1
;
r0
r0
r0
between atom i and j. In the expression, P1, A1, rm1, A1m, rc1, P1m,
P2, A2, rm2, A2m, rc2 and P2m are the potential parameters to be
tted, while n1 and n2 are selected as 4 and 5, respectively.
Readers can refer to a previous paper25 for more details about
the potential parameters. It is easy to see that the pair item and
density item, as well as their higher derivatives, can go to zero
continuously and smoothly at the cutoff distance, thus
removing the leaps of energy and force, as well as avoiding some
non-physical behaviors.27
Generally, six kinds of interaction should be considered for
a ternary system. In the Mg–Al–Cu system, the interactions are
Mg–Mg, Al–Al, Cu–Cu, Mg–Al, Mg–Cu and Al–Cu. The potential
parameters were obtained by tting to the lattice constants,
cohesive energies, elastic constants and bulk moduli of
elements as well as the stable or meta-stable compounds in
each of the binary systems.28,29 In the tting process, ab initio
calculations using the Cambridge serial total energy package
(CASTEP)30 were applied to derive the relevant physical properties of the compounds.
The tted potential parameters of the Mg–Al–Cu system are
listed in Table 3. Tables 2 and 4 give the cohesion energies,
elastic constants and bulk moduli of Mg, Al and Cu and their
Hcp-Mg
rm1\rij # rc1
(13)
j rij ¼
8
rij
>
>
A
exp
p
1
;
2
> 2
<
r
0
rij # rm2
n2
>
>
r
rc2 rij
>
: A2m exp p2m ij 1
; rm2 \rij # rc2
r0
r0
r0
(14)
where Ei is the total potential energy of atom i, 4 and j is the
pair item and n-body item respectively, and rij is the distance
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a (Å)
c (Å)
Ec (eV)
C11 (Mbar)
C12 (Mbar)
C13 (Mbar)
C33 (Mbar)
C44 (Mbar)
B0 (Mbar)
a
Fcc-Al
Fcc-Cu
Fitted
Exp.
Fitted
Exp.
Fitted
Exp.
3.209
5.235
1.508
0.591
0.270
0.223
0.642
0.112
0.362
3.209a
5.21a
1.510b
0.595a
0.261a
0.218a
0.616a
0.164a
0.354a
4.051
4.050a
3.611
3.615a
3.387
0.821
0.705
3.390b
1.067a
0.605a
3.502
1.688
1.225
3.490b
1.683a
1.221a
0.289
0.743
0.283a
0.722a
0.745
1.361
0.757a
1.370a
Ref. 27. b Ref. 28.
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Table 3
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The fitted potential parameters of the Mg–Al–Cu system
p1
A1 (eV)
rm1 (Å)
n1
p1m
A1m (eV)
rc1 (Å)
p2
A2 (eV2)
rm2 (Å)
n2
p2m
A2m (eV2)
rc2 (Å)
r0 (Å)
Mg
Al
Cu
Mg–Al
Mg–Cu
Al–Cu
10.37307
0.145780
3.522308
4
3.850843
0.538535
5.487015
4.375061
0.951887
2.588516
5
0.000378
1.130393
6.250000
3.203567
8.776460
0.402184
2.764394
4
2.588558
2.917212
4.607023
5.249466
4.738155
3.786874
5
0.000477
1.114067
6.515324
2.864321
9.625372
0.315491
2.124148
4
2.860823
8.049825
3.634148
4.903930
3.854734
3.611361
5
0.000602
0.458708
6.215324
2.553618
10.23828
0.190712
2.654430
4
3.447698
3.457160
4.421070
3.439821
1.890899
2.636021
5
0.000438
0.441637
6.996059
2.999131
7.831151
0.257021
2.100661
4
3.867485
0.723304
5.005640
3.735459
2.685112
2.555986
5
0.000781
0.948748
6.409856
2.878592
4.575749
0.977351
2.162239
4
1.107382
1.112537
5.286440
4.620459
9.720919
3.618114
5
0.000619
1.388810
6.550000
2.708969
Table 4 The properties reproduced from the interatomic potential (first line) and calculated via ab initio methods (second line) of the Mg–Al,
Mg–Cu and Al–Cu compounds
Compounds
Space group
a or a, c or a, b, c (Å)
Ec (eV)
B0 (Mbar)
MgAl3
m
Pm3
4.168
4.156
2.913
2.913
0.6364
0.6390
MgAl
m
Pm3
3.424
3.402
2.379
2.378
0.4902
0.4717
Mg17Al12
3m
I4
10.59
10.57
2.305
2.305
0.4875
0.4912
MgCu2
m
Pm3
7.118
7.048
2.984
2.984
0.806
0.954
compounds which are obtained by tting experiments or ab
initio calculations.28,29 It can be seen that the physical properties
tted by the potential match quite well with the experimental
results or ab initio calculations.
We further evaluated the tted potentials in a non-equilibrium state. The equation of state (EOS) derived from the
potentials was compared with the Rose equation.31 The pair
terms, n-body parts and the total energies reproduced from the
potential, together with the corresponding Rose equations, are
shown in Fig. 2. It can be seen that the pair terms, n-body parts
and the total energy of these structures are smooth and
continuous over the entire range. Meanwhile, the EOSs derived
from the proposed potential agree well with the corresponding
Rose equations, indicating that the constructed Mg–Al–Cu
potential could be applied to describe the atomic interactions of
the system even far from the equilibrium state. In conclusion,
the newly constructed Mg–Al–Cu potential is adequate to
describe the atomic interactions in both equilibrium and nonequilibrium environments.
4. Simulation models and
characterization methods
During the process of forming metallic glasses, kinetic conditions were always limited, and thus complicated intermetallic
compounds are not able to nucleate and grow. The phase
competing against the metallic glass is therefore the solid
102332 | RSC Adv., 2016, 6, 102329–102335
MgCu
m
Pm3
3.194
3.159
2.595
2.594
0.532
0.698
Mg2Cu
Fddd
9.156, 5.333, 18.55
9.062, 5.283, 18.35
2.291
2.292
0.497
0.540
Al3Cu
m
Pm3
4.009
3.939
3.494
3.493
0.590
0.611
Al2Cu
I4/mcm
5.997, 5.083
6.067, 4.877
3.623
3.623
0.681
0.792
AlCu3
m
Pm3
3.700
3.696
3.660
3.658
0.862
1.253
solution, frequently of a simple structure. Consequently, the
issue related to the glass formation is converted into one
comparing the relative stability of the solid solution to its
amorphous counterpart.18,32–34
Based on the Mg–Al–Cu n-body potential, Monte Carlo
methods were employed for the following simulations. Since the
stable crystalline structure of Mg, Al and Cu is hcp, fcc and fcc,
respectively, two types of solid solution model, i.e. the hcp and
fcc solid solution models, were constructed according to which
type of atom is dominant in the alloy composition. For fcc
models, the [100], [010] and [001] crystalline direction is parallel
to the x, y and z axes, respectively, while for the hcp model, the
[100], [001] and [120] crystalline direction is parallel to the x, y
and z axes, respectively. Periodic boundary conditions were
applied in the three directions. The hcp and fcc solid solution
models consist of 2912 (13 8 7 4) atoms and 2916 (9 9 9 4) atoms, respectively. In constructing the solid solution
models, the solvent atoms were substituted randomly by
a certain number of solute atoms to obtain a desired composition. The initial solid solutions were annealed at zero pressure
and 300 K in an isothermal–isobaric ensemble for a sufficient
simulation time in order to reach a relatively stable state.
4.1
Glass formation region of the Mg–Al–Cu system
Based on the solid solution models, MC simulations were
carried out over the entire composition triangle. For a solid
solution of MgxAlyCu1xy, the values of x and y were varied with
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Fig. 2 Total energies, pair terms and n-body parts as a function of lattice constant calculated from the interatomic potential and the Rose
equation for Mg, Al, Cu, Mg17Al12, MgCu and Al3Cu.
a composition interval of 5% to cover the range of 0 to 100%,
and the initial lattice parameter of each solid solution was set
according to Vegard's law.
Aer an adequate MC simulation time, the initially constructed solid solution models reach two different stable states
with varying compositions, i.e., either keeping the initial crystalline state or collapsing into a disordered state. Taking the
Mg50Al50 and Mg5Al70Cu25 alloys as examples, Fig. 3 shows the
total pair-correlation functions S(q) 35 and atomic position
projections of these two alloys. In Fig. 3(a), it is obvious that the
S(q) curve of Mg50Al50 shows crystalline peaks, suggesting
a long-range ordered state, and in Fig. 3(c) the atoms are
arranged regularly. For Mg5Al70Cu25, as seen in Fig. 3(b), all the
peaks beyond the second have disappeared, exhibiting typical
long-range disordered and short-range ordered features. This is
also shown by the fact that the crystalline lattice has collapsed
into an amorphous state, as shown in Fig. 3(d).
According to S(q) and atomic position projections, we dealt
with the entire Mg–Al–Cu composition as shown in Fig. 4(a).
The composition triangle was divided into three regions by two
critical solubility lines, i.e., the lines AB and CD. When an alloy
composition was located beyond line CD and toward the Al–Mg
side, or beyond line AB and toward the Cu corner, the crystalline
structure could remain stable. When the composition falls in
the central quadrilateral region, enclosed by ABCD, the crystalline structure would become unstable and spontaneously
collapse into an amorphous state. This quadrilateral region is
thus dened as the glass formation region (GFR). To further
validate the predicted GFR, experimental observations36–40 were
extensively collected for the Mg–Al–Cu system and shown in
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Fig. 4(a) by the red dots. It can be seen that these experimental
data mostly fall within the predicted quadrilateral region. In
comparison to the results obtained by thermodynamic calculations, they agree well with each other, suggesting that the
results predicted by the MC simulations are quite reasonable for
the Mg–Al–Cu system.
Meanwhile, the procedure of the MC simulations conrms
that the formation mechanism of metallic glasses is the
competition between the metallic glass and the solid solutions.
Besides, the obtained GFR is derived directly from the interatomic potential, so it could be considered as the intrinsic GFR
of the Mg–Al–Cu system, which is determined by its internal
characteristics.
4.2
Composition optimization for glass formation
Based on the obtained glass formation range, one can conveniently predict the glass formation at a given composition.
However, the issues related to the GFA of each Mg–Al–Cu alloy,
i.e. the ease or difficulty in obtaining the metallic glass, still
remain. To depict the GFA, the energy differences between the
amorphous phase and the solid solutions of each MgxAlyCu1xy alloy were calculated and dened as the amorphization
driving force (ADF), represented by the term DEam, which can be
expressed as:
DEam ¼ Eam [xEMg + yEAl + (1 x y)ECu]
(15)
where Eam is the energy per atom of the MgxAlyCu1xy amorphous phase and EMg, EAl and ECu is the lattice energy of Mg, Al
and Cu respectively. The ADF of each MgxAlyCu1xy alloy was
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Fig. 3 The total pair-correlation functions S(q) and atomic position projections for (a and c) the crystalline state of Mg50Al50 and (b and d) the
amorphous state of Mg5Al70Cu25. Green circles represent Mg, orange circles represent Al and blue circles represent Cu.
Fig. 4 The glass formation stoichiometry diagram derived from the MC simulations at 300 K for the Mg–Al–Cu ternary system. (a) The glass
formation range bounded by ABCD and (b) the distribution of the ADF of each composition within the glass formation range.
thus obtained and shown in Fig. 4(b). It is clear to see that the
ADF is negative over the whole GFR and energetically favored,
suggesting that the energy of the amorphous phase is lower
than that of the solid solution. Moreover, the larger the ADF is,
the stronger the GFA is. Thus, a sub-region with the composition of MgxAlyCu1xy (x ¼ 20–30; y ¼ 30–40; 1 x y ¼ 35–45)
shows a larger ADF than those alloys located outside the subregion. It follows that the Mg–Al–Cu amorphous alloys within
the sub-region are expected to be more attainable or
more thermally stable, thus providing basic guidelines for
designing appropriate alloy compositions for producing Mg–
Al–Cu metallic glasses. Besides, the sub-region is consistent
102334 | RSC Adv., 2016, 6, 102329–102335
with the thermodynamic calculations, in that they both show
similar compositions for the best GFA. The relevance of the
optimal compositions of the Mg–Al–Cu system, as well as the
validity of the constructed n-body potential, has been further
conrmed.
5.
Conclusion
In conclusion, the glass formation range and ability of ternary
Mg–Al–Cu metallic glass are obtained by thermodynamic
calculations and an atomistic approach. Firstly, thermodynamic calculations were applied and show that the Mg–Al–Cu
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metallic glasses are favored over a large composition range, and
that the sub-region of MgxAlyCu1xy (x ¼ 20–30; y ¼ 30–40; 1 x y ¼ 35–45) shows a better GFA than the other compositions.
Then, we further investigated the system from an atomic level
and a realistic interatomic potential was constructed for the
Mg–Al–Cu system under a proposed modied tight-binding
formalism, and applied in Monte Carlo simulations to predict
the favored composition, and even pinpoint the optimized
composition. The simulation not only predicted a quadrilateral
region, within which Mg–Al–Cu metallic glass formation is
energetically favored, but also pinpointed an optimized subregion (MgxAlyCu1xy (x ¼ 20–30; y ¼ 30–40; 1 x y ¼ 35–
45)) within which the amorphization driving force (ADF), i.e. the
energy difference between the solid solution and the disordered
state, is larger than that outside. The GFR and the optimized
compositions are fairly consistent with the thermodynamic
calculations. The simulations not only provided predictions for
producing Mg–Al–Cu metallic glasses, but also revealed the
physical origin of the crystal–amorphous transition, which
could be of great help for guiding the composition design of
ternary glass formers.
Acknowledgements
The authors are grateful for the nancial support from the
National Natural Science Foundation of China (51131003,
51571129), and the Administration of Tsinghua University. The
authors appreciate Dr Shunning Li and Dr Simin An for their
collection of experimental data and providing some advice for
the present work.
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