Journal of Non-Crystalline Solids 388 (2014) 23–31
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids
journal homepage: www.elsevier.com/ locate/ jnoncrysol
Predicted atomic arrangement of Mg67Zn28Ca5 and Ca50Zn30Mg20 bulk
metallic glasses by atomic simulation
Shin-Pon Ju a,⁎, Hsin-Hong Huang a, Jacob Chih-Ching Huang b
a
b
Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
Department of Materials and Optoelectronic Science, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
a r t i c l e
i n f o
Article history:
Received 1 November 2013
Received in revised form 1 January 2014
Available online xxxx
Keywords:
Molecular dynamics simulation;
Bulk metallic glasses;
Tight-binding;
Honeycutt–Anderson
a b s t r a c t
The microstructures of Mg-based Mg67Zn28Ca5 and Ca-based Ca50Zn30Mg20 bulk metallic glasses are predicted by
the simulated-annealing basin-hopping method with the tight-binding potential function. The parameters of Mg,
Zn, and Ca, and the cross-element pairs of the TB potential are first fitted by the force-matching method with the
reference data from experimental results and the density functional theory calculation. The structures from the
SABH methods reveal that the average bond lengths of different atomic pairs are almost the same for these
two BMGs. However, the microstructures found by the Honeycutt–Anderson pair analysis are very different.
For Mg67Zn28Ca5, the perfect icosahedral local structure occupies the highest fraction, while for Ca50Zn30Mg20
the distorted icosahedral local structures are predominant.
© 2014 Elsevier B.V. All rights reserved.
1. Introduction
The past decade has seen the progression of glass-fabricating techniques such as rapid casting [1–3], severe plastic deformation (SPD)
[4] and spark plasma sintering (SPS) [5] for the production of various
bulk metallic glasses (BMGs) in binary, ternary, and multicomponent
metal systems. These BMGs have therefore attracted considerable interest for their special properties [6,7]. These improved characteristics include high (tension or compression) strength up to 5 GPa [8], a large
limit of elastic deformation of about 2% [9], and excellent corrosion resistance [10], which are sometimes superior to their crystalline counterparts, suggesting new applications.
Some Zr-, Ti-, Ta-based BMGs have promising biocompatibility and
good corrosion resistance [11–13], and they have been commonly
used as permanent implants. On the other hand, Mg-based or Cabased BMGs, due to their degradable character, can be applied as biodegradable implants [14]. Other than Mg, Zn and Ca are the two other
metal elements constituting the human bone, so MgZnCa BMGs are suitable choices for biocompatibility [15,16]. Another important reason for
using MgZnCa BMGs in orthopedic biodegradable implants is that the
Young's moduli of MgZnCa BMGs (41–45 GPa) are close to that of the
human bone (3–30 GPa) [11]. When the MgZnCa BMGs are fabricated
into porous foams, the moduli can be further reduced to 1–20 GPa,
preventing the occurrence of the stress filter effect or a serious stress
concentration at the BMG/bone interface.
Since investigating atomic arrangement of BMGs directly by experiment is difficult due to the very small scale involved, numerical
methods are commonly preferred because these methods can assist in
⁎ Corresponding author. Tel.: +886 7 5252000x4231; fax: +886 7 5252132.
E-mail address: [email protected] (S.-P. Ju).
0022-3093/$ – see front matter © 2014 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.jnoncrysol.2014.01.005
clarifying the physical insights gained from experimental results. Molecular dynamics (MD) and Monte Carlo (MC) simulations are two powerful tools to predict the atomic arrangement of alloys by using accurate
potential functions. Simulating the annealing process from a temperature higher than the melting point back to room temperature at a proper cooling rate has been used to determine stable BMG structures. For
example, Hui and Pederiva [17] used MD simulation to study the local
structural variations for the Ni3Al alloy during an annealing process
from 2000 to 300 K by using the tight-binding potential. From the pair
correlation profiles and Honeycutt–Anderson (HA) analysis, they
found that the local structures of Ni3Al alloy display a prominent change
as the temperature dropped below 800 K. In Li et al.'s [18] study, MD
and MC simulations revealed that the Al–Zr metallic glass can be obtained within the composition range of 24–66 at.% Zr, and they found the
glass-forming ability of the Al56Zr44 alloy to be the strongest. In Kaban
et al.'s [19] study, MD and MC techniques were applied to elucidate
the atomic structure of Ni–Zr and Cu–Zr alloys in glassy and crystalline
states and to find differences in the case of Cu65Zr35, which is assumed
to be the most decisive factor increasing its bulk GFA.
From the discussions above, it is clear that molecular simulation
methods provide a powerful tool for investigating the atomic arrangement within the BMGs at the atomic level once the potentials that accurately describe the interatomic interaction are available. Among all of
the MgZnCa BMGs, Mg67Zn28Ca5 has been proven to have the highest
corrosion resistance [14], whereas Ca50Zn30Mg20 has the lowest [20].
In literature, there have been very few reports on the atomic packing
in the Mg-based or Ca-based BMGs. To further understand their atomic
arrangements, the simulated-annealing basin-hopping method [21]
was used to predict the atomic arrangements of Mg, Zn, and Ca atoms
in these two BMGs. The fitting process for many-body tight-binding potential functions was first carried out on the basis of experimental data
24
S.-P. Ju et al. / Journal of Non-Crystalline Solids 388 (2014) 23–31
and density functional theory calculation. Then a system with 30,000
atoms was used to study the local structure distributions of Mg67Zn28Ca5 and Ca50Zn30Mg20 by the HA analysis.
2. Simulation model
All atomic interactions are modeled by the many-body, tightbinding potential (TB) [22–25] with the potential form shown as
Eq. (1):
9
8
=1=2 X
<X
r ij
r ij
2
ξ exp −2q
−1
þ
A exp −p
−1
ð1Þ
Ei ¼ −
;
:
r0
r0
j
M
X
Nk
3
k¼1
Element
Function
a (Å)
c/a
E (eV/atom)
Mg (HCP)
Exp.
DFT
Error
Exp.
DFT
Error
Exp.
DFT
Error
3.21
3.11
3%
2.66
2.66
0%
5.58
5.47
2%
1.62
1.69
4%
1.86
1.98
6%
−1.51
−1.50
1%
−1.35
−1.27
6%
−1.84
−1.72
6%
Zn (HCP)
Ca (FCC)
j
where ξ is an effective hopping integral, rij is the distance between
atoms i and j, and r0 is the first neighbor distance. The first part in the
potential function is the summary of the band energy, which is characterized by the second moment of the d-band density of state. Meanwhile, the second part is a Born–Mayer type repulsive form. In
Karolewski's studies [26], the parameters ξ, A, p, q, and r0 of several transition metals were determined on the basis of the experimentally obtained values of cohesive energy, vacancy formation energy, lattice
parameter and elastic constants [27,28].
To get accurate cross-element interactions of Mg–Zn, Ca–Mg and
Zn–Ca pairs for MgZnCa BMG, the force-matching method (FMM) was
used to determine the tight-binding potential parameters for Mg–Zn,
Ca–Mg and Zn–Ca interactions [29]. Although TB parameters of Mg,
Zn, and Ca elements can be found in previous studies [23,26], these parameters were also modified by FMM to minimize the errors between
the reference data and those from TB potentials for two-element or
three-element systems. FMM is based on the variable optimization process of an objective function, which is constructed by the summation of
squares of differences between the atomic forces obtained by a potential
function and the corresponding atomic forces by ab initio or density
functional theory (DFT) calculations. The FMM minimizes the following
objective function [30]:
Z ðα Þ ¼
Table 1
Comparisons between experimental [22] and DFT results for the HCP Mg, HCP Zn and FCC
Ca. The a and c are lattice constants, c/a is the ratio for lattice constant and E is binding
energy.
!−1
Nk M X
X
0 2
F ki ðα Þ− F ki ð2Þ
k¼1 i¼1
where α, M and Nk are the entire set of potential parameters, the number of atomic configurations, and the number of atoms in a configuration k. Fki(α) is the force acting on atom i of the configuration k, which
is computed from the potential parameters α. F0ki is the corresponding
referenced force calculated from the ab initio or DFT approach. Except
for atomic forces of all optimized structures, the binding energy, bulk
moduli, and elastic constants of crystal reference structures were also
included in our object function. To improve the many-body effect in
TB potential [26], the binding energies of structures with one-atom defects were used for the object function.
All required reference data for FMM were prepared by DFT calculation. For binding energies, the Dmol3 program [31] was used and
CASTEP [32] was adopted to obtain the elastic constants and bulk moduli. The generalized gradient approximation (GGA) with parameterization of Perdew–Burke–Ernzerhof (PBE) function [33] was used for
Dmol3 and CASTEP. For Dmol3 settings, all electron calculations were
carried out with a double numeric plus polarization (DNP) basis set
[34]. The energy tolerance in the self-consistent field calculations was
set at 10− 5 Ha and the energy and force tolerances for the ionic
step were 2.0 × 10−5 Ha and 4.0 × 10−3 Ha/Å. For CASTEP, the convergent conditions were set as 5 × 10−4 Ǻ for the atomic displacement,
5 × 10−6 eV/atom for the energy change, 380 eV for the plane-wave
cutoff energy, and 0.01 for the k-point separations. Table 1 lists the lattice constants and binding energies from Dmol3 calculations and the
corresponding experimental values for hexagonal closed packed
(HCP) Mg, HCP Zn and face-centered cubic (FCC) Ca. Data from Dmol3
calculations are clearly very close to the experimental values, indicating
these Dmol3 settings are accurate enough to predict the binding energies for MgZnCa systems. For CASTEP results, Table 2 shows lattice constants, elastic constants, bulk modulus and shear modulus for Mg, Zn,
and Ca. The calculated lattice constants of Mg, Zn, and Ca are very
close to the experimental values listed in Table 1. In terms of mechanical
properties, it can be seen the calculation results are in fair agreement
with the experimental values, and the relative values of elastic constants do not change for the calculation results. Accordingly, CASTEP
settings are also expected to be accurate enough to obtain the mechanical properties of the MgZnCa ternary alloy systems.
Fifteen crystal configurations were used to prepare the reference
data for FMM: these structures include pure elements (Mg, Zn
and Ca), binary (Mg1Zn1, Mg3Zn1, Mg1Zn3, Ca1Mg1, Ca3Mg1, Ca1Mg3,
Zn1Ca1, Zn3Ca1 and Zn1Ca3), and ternary (Mg2Zn1Ca1, Mg1Zn2Ca1 and
Mg1Zn1Ca2) metal systems, with the corresponding space symmetry arrangements shown in the second column of Table 3. To obtain the binding energies of the defect configurations, another fifteen structures
were used. These were constructed by taking one atom from the crystal
structures with a supercell of 2 times the unit cell of the crystal structure, as shown in the first column of Table 3. The atom type taken
from the crystal configuration is shown in the parentheses of the sixth
column of Table 3.
In the fitting process, the optimization of TB parameters was applied
to the objective function by the general utility lattice program (GULP)
engine [35], with the basin-hopping method used to randomly change
Table 2
Values of the lattice constant, elastic constants (C11, C12, C13, C33, and C44), bulk
modulus (B) and shear modulus (S) for experimental [23] and CASTEP.
Element
Property
Exp.
Castep
Error (%)
Mg (HCP)
a (Å)
c/a
C11 (GPa)
C12 (GPa)
C13 (GPa)
C33 (GPa)
C44 (GPa)
B (GPa)
S (GPa)
a (Å)
c/a
C11 (GPa)
C12 (GPa)
C13 (GPa)
C33 (GPa)
C44 (GPa)
B (GPa)
S (GPa)
a (Å)
C11 (GPa)
C12 (GPa)
C13 (GPa)
B (GPa)
S (GPa)
3.21
1.62
63.20
26.30
21.60
65.70
18.40
36.90
19.30
2.66
1.86
179.10
37.50
55.40
68.80
45.90
80.40
51.10
5.58
27.80
18.20
16.30
20.00
8.90
3.21
1.62
56.36
26.87
22.74
57.64
16.97
35.00
16.27
2.64
1.90
178.84
38.35
42.59
65.29
35.78
74.45
48.32
5.54
21.68
14.01
14.12
16.57
10.00
0
0
11
2
5
12
8
5
15
1
2
0
2
23
5
22
7
5
1
22
23
13
17
12
Zn (HCP)
Ca (FCC)
S.-P. Ju et al. / Journal of Non-Crystalline Solids 388 (2014) 23–31
Table 3
Binding energies and vacancy formation energy are predicted by TB and DFT, marked E
(eV/atom) and EC (eV/atom), respectively.
Phase
Space group
E_TB
E_DFT
ΔE
(%)
EC_TB
EC _DFT
ΔEC
(%)
Mg
Zn
Ca
Mg3Zn1
Mg1Zn1
Mg1Zn3
Ca3Mg1
Ca1Mg1
Ca1Mg3
Zn3Ca1
Zn1Ca1
Zn1Ca3
Mg2Zn1Ca1
Mg1Zn2Ca1
Mg1Zn1Ca2
P63/MMC
−1.59
−1.1
−1.55
−1.58
−1.54
−1.39
−1.61
−1.68
−1.63
−1.19
−1.42
−1.34
−1.51
−1.23
−1.47
−1.55
−1.27
−1.72
−1.49
−1.44
−1.32
−1.59
−1.59
−1.51
−1.34
−1.59
−1.48
−1.47
−1.24
−1.46
2
13
10
6
7
5
1
5
8
11
10
9
3
1
1
−1.51(Mg)
−1.08(Zn)
−1.45(Ca)
−1.37(Zn)
−1.53(Zn)
−1.22(Mg)
−1.49(Mg)
−1.63(Mg)
−1.52(Ca)
−1.12(Ca)
−1.42(Zn)
−1.34(Zn)
−1.47(Mg)
−1.35(Mg)
−1.14(Ca)
−1.44
−1.21
−1.48
−1.15
−1.46
−1.18
−1.42
−1.51
−1.31
−1.16
−1.56
−1.36
−1.41
−1.36
−1.32
5
11
2
19
5
3
5
8
16
3
9
1
4
1
13
FM-3M
PM-3M
P4/MMM
the TB parameter values after each optimization process. The Monte
Carlo method was adopted to find the parameter set which produced
the global minimal value of the objective function [36]. After the fitting
process, a set of potential parameters with the accuracy comparable to
the DFT calculations can be obtained. The fitted parameters of TB potential are listed in Table 4, and a comparison between some DFT reference
data and those calculated from the TB potential with parameters in
Table 4 are shown in Table 3. The errors of most reference data are
lower than 10%, indicating the predicted material properties by TB potentials are in good agreement with the corresponding data from DFT
calculations. It should be noted that the largest errors of binding energies of crystal and defect configurations occur in HCP Zn, because the
TB potential is a little inaccurate for the HCP structure, with an a/c
ratio of only 1.633 [23]. However, for BMG systems, the percentage of
perfectly packed HCP Zn is negligible, so the TB parameter set in
Table 4 can still predict stable material properties for the MgZnCa BMG.
Before determining the mechanical properties of the MgZnCa BMG,
the most important step is to obtain a stable amorphous MgZnCa configuration. Many global optimization algorithms have been successfully
developed to obtain the lowest-energy configuration, such as the
genetic algorithm (GA) [37], the big-bang method (BB) [38], and the
simulated-annealing basin-hopping (SABH) method. In the traditional
BH method, a conjugate gradient method is used to reach the stable
configuration with the local energy minimum. In our BH method, the
conjugate gradient method was replaced by the limited memory BFGS
method (LBFGS) [39], which can be used to simulate a system consisting
of a large number of atoms. Also, using the LBFGS method in the BH
method is faster than the conjugate gradient method for a larger system.
Furthermore, the simulated-annealing (SA) method was also implemented with the BH method to become the SABH method, which includes a wider search within the energy space. In the original BH
method, the searching direction for the structure with the lower energy
is preferable, so it will eventually lead to a crystalline structure. Thus, for
the current study to find the BMG amorphous configurations, the
searching criterion of SABH is to find the stable local-minimal structure
with a higher energy. To make each optimization efficient, the LBFGS
Table 4
Parameters of the tight-binding potential for Mg–Mg, Zn–Zn, Ca–Ca, Mg–Zn, Ca–Mg, and
Zn–Ca.
Type
A
ζ
p
q
r0
Mg–Mg
Zn–Zn
Ca–Ca
Mg–Zn
Ca–Mg
Zn–Ca
0.0135
0.0953
0.0058
0.0135
0.0437
0.0397
0.4641
0.6276
0.4242
0.4921
0.6349
0.5317
14.5115
11.7371
16.7684
19.6370
11.5320
13.6599
2.1220
4.3535
1.8408
2.9367
2.8799
3.3259
3.2053
2.6499
4.1060
2.9237
3.5992
3.2185
25
was used until the root-mean-square of all atomic forces was lower
than 5.0 × 10− 3 eV/Å (and then the fast inertial relaxation engine
algorithm followed until the tolerance of energy was 1.0 × 10−5 eV).
Initially, 20,100 Mg, 8400 Zn and 1500 Cu atoms, a total of 30,000
atoms, were randomly distributed in a periodic box of 84.85 ×
84.60 × 84.68 Å3 in the x-, y-, and z-dimensions. During each LBFGS
or fast inertial relaxation engine algorithm optimization step, the system volume was adjusted according to the system pressure of 0 GPa,
controlled by Andersen barostat [40]. The MgZnCa BMG with the
highest total energy obtained by the above process was characterized
by the X-ray diffraction (XRD) simulation module in Materials Studio,
REFLEX [41–43]. The simulated XRD profiles shown in Fig. 1(a) and
(b) confirm both Mg67Zn28Ca5 and Ca50Mg30Zn20 BMGs are amorphous.
3. Result and discussions
The radial distribution function (RDF) and partial radial distribution
function (PRDF) were used to check if the fitted TB parameters could reflect the relative atomic sizes of different elements. After the complete
optimization, a periodic boundary condition system with 30,000
atoms was used to investigate the Mg67Zn28Ca5 and Ca50Zn30Mg20 metallic glass structural properties. Figs. 2 and 3 show the RDF and PRDF
profiles for total, Mg-total, Zn-total, Ca-total, and their individual
atom pairs in the Mg67Zn28Ca5 BMG. Here, “total” refers to all of the
six atom pairs: Mg–Mg, Zn–Zn, Ca–Ca, Mg–Zn, Zn–Ca, and Ca–Mg.
“Mg-total” refers to the three Mg-related pairs of Mg–Mg, Mg–Zn, and
Mg–Ca, with the same reference logic for the others. The metallic radii
for Mg, Zn, and Ca atoms are 1.60, 1.33, and 1.97 Å [44], Ca and Zn
being the largest and smallest ones, respectively. In Fig. 2(a), the height
of the first peak reflects the number of the first neighbor atoms, and the
splitting of the second peak of the RDF for total atoms indicates the
amorphous configuration for the Mg67Zn28Ca5 BMG, which is consistent
with the inference of the XRD profile shown in Fig. 1(a). There is only a
short range order of atomic arrangement for Mg, Zn and Ca atoms, and
there is no corresponding specific lattice structure.
The first peaks of total, Mg-total, Zn-total, and Ca-total RDFs are located at 2.95, 3.05, 2.82, and 3.75 Å (Table 5), which reflects the relative
atomic sizes of Mg, Zn, and Ca. For the largest Ca atoms, the larger first
neighbor distance from the referenced Ca atom results in the larger
shell volume for the first neighbor atoms. It allows more atoms with stable distances among all first neighbor atoms surrounding Ca. On the
contrary, a Zn atom has a smaller size and a shorter first neighbor distance, leading to smaller shell volume for its first neighbor atoms. Consequently, fewer first neighbor atoms can stably stay within the shell
volume around a Zn atom, resulting in a narrower first PRDF peak. For
the Ca50Zn30Mg20 BMG, except for the heights of the first peaks, the
RDF and PRDF profiles are very similar to those shown in Figs. 2 and 3
for the Mg67Zn28Ca5 BMG and therefore are not shown. The reason
might be that PDFs and PRDFs show the integral properties of the structure, so they can hardly present the differences in the local structures
between Mg67Zn28Ca5 and Ca50Zn30Mg20. For this reason, some parameters are used to explore different local structure in the following text.
A comparison of the atomic radii of pure Mg, Zn and Ca metals (1.60,
1.33, and 1.97 Å [42]) and the first peak distance in Figs. 2 and 3 or
Table 5 shows that the distance for the first peak of the “total”
(2.95 Å) for Mg67Zn28Ca5 is smaller than the predicted value based on
simple rule of mixture (3.09 Å). This is postulated to be a result of the
relative heat of mixing, with values for Mg–Zn, Mg–Ca and Zn–Ca
being −4 kJ/mol, −6 kJ/mol and −22 kJ/mol, respectively [45]. Since
they are all negative heats of mixing, the three kinds of atoms attract
each other, reducing the overall atom spacing. Furthermore, the more
negative heat of mixing Zn–Ca tends to form stronger bonding, causing
the average PRDF first distance for Zn–Ca to shrink from 3.30 Å (which
is the sum of Zn and Ca atomic radii, 1.33 + 1.97 Å) to 3.25 Å. The
stronger attractive force for the Ca atoms causes the all first distances
of the Ca-related pairs to shrink with respect to their sums of atomic
26
S.-P. Ju et al. / Journal of Non-Crystalline Solids 388 (2014) 23–31
a
b
Fig. 1. Simulated XRD patterns of the (a) Mg67Zn28Ca5 and (b) Ca50Zn30Mg20 BMGs.
radii, as in Ca-total, Mg–Ca, Zn–Ca, and Ca–Ca. On the other hand, with
less negative heat of mixing, the atomic pairs expand, such as is the case
for Mg–Zn or Zn–Zn.
For Mg67Zn28Ca5 and Ca50Zn30Mg20 BMGs, Table 6 lists the average
coordination numbers (CNs) for Mg, Zn, and Ca atoms as well as the
partial coordination numbers of different atomic pairs, which were calculated by counting the amount of neighbor atoms within the distances
of the first minimums of the corresponding RDF and PRDF profiles. The
first subscript for atomic pair indicates the type of the reference atom
and the second subscript stands for the atom type of the first neighbor
a
b
Zn-Total
G(r)
Total
c
d
Mg-Total
Ca-Total
r(Å)
Fig. 2. The RDF profiles for (a) total, (b) Mg, (c) Zn and (d) Ca atoms in the Mg67Zn28Ca5 BMG.
S.-P. Ju et al. / Journal of Non-Crystalline Solids 388 (2014) 23–31
27
b
a
Mg-Mg
Zn-Zn
d
c
Mg-Zn
G(r)
Zn-Ca
f
e
Mg-Ca
Ca-Ca
r(Å)
Fig. 3. The PRDF profiles of different pairs in the Mg67Zn28Ca5 BMG.
of the reference atom. For Mg67Zn28Ca5 BMG, the average total CN is
12.56, among which the Ca and Zn atoms have the highest and lowest CNs of 16.07 and 11.32, respectively. These results are consistent
with the wider and narrower first peaks of Ca and Zn RDF profiles
shown in Fig. 2. For Ca 50Zn30 Mg20 BMG, the CNs of total, Mg,
Zn, and Ca atoms are smaller than the corresponding CNs of
Mg 67 Zn28 Ca 5 , with the Ca and Zn atoms again having the highest
and lowest CNs of 13.36 and 9.85, respectively. This should also be
a result of the relative of heat of mixing values for Mg–Zn, Mg–Ca
and Zn–Ca. Ca has more negative values with Mg and Zn, and therefore Ca tends to be located at the center of a bigger atomic cluster
with a higher CN.
28
S.-P. Ju et al. / Journal of Non-Crystalline Solids 388 (2014) 23–31
Table 5
Distances of all first peaks of RDF and PRDF.
Table 7
CSRO parameters (αij) for all atomic pairs of Mg67Zn28Ca5 and Ca50Zn30Mg20.
Type
Total
Mg-total
Zn-total
Ca-total
Mg–Mg
CSRO
Mg67Zn28Ca5
Ca50Zn30Mg20
Distance (Å)
2.95
3.05
2.82
3.75
3.12
αMg–Mg
αMg–Zn
αMg–Ca
αZn–Mg
αZn–Zn
αZn–Ca
αCa–Mg
αCa–Zn
αCa–Ca
0.0857
0.0387
−0.3314
−0.0954
0.2740
−0.2564
−0.0442
0.1442
−0.2146
0.1734
0.1833
−0.1793
0.0276
0.2915
−0.1859
0.0099
0.1861
−0.1156
Type
Mg–Zn
Mg–Ca
Zn–Zn
Zn–Ca
Ca–Ca
Distance (Å)
2.95
3.50
2.75
3.25
3.83
On the basis of CN information in Table 6, the chemical affinities
of a referenced atom for its first neighbor atoms are evaluated by the
Warren–Cowley chemical short-range-order CSRO parameter. The definition of this parameter is as the following equation:
α i j ¼ 1−
Ni j
ð3Þ
c j Ni
where αij is the CSRO parameter of the i-type referenced atom related to
j-type atom; Nij is the partial CN calculated from BMG configuration for
the i-type referenced atom related to j-type atom; cj and Ni are the fraction of j-type atom within the alloy and the average CN of i-type atom,
respectively. The value of cj by Ni is an ideal partial CN for the referenced
i-type atom related to the first neighbor j-type atom, with this value
completely depending on the respective atomic composition fraction
of Mg67Zn28Ca5. For an ideal solution, within which different element
types do not have any bonding preference to a specific atom, the value
of Nij should be very close to cjNi, and therefore the value of αij should
be very close to 0, where a perfectly random alloy is characterized.
A positive αij means the affinity of the referenced i-type atom to
the j-type first neighbor atom is much less than that of pairs with
negative αij values. According to Eq. (3), one can see the αij value is
different from the αji value, indicating the many-body effect on the
αij value, where the CSRO parameter depends on the atomic type distribution of the first neighbor atoms around the referenced atom.
Table 7 illustrates the CSRO parameters for all atomic pairs of
Mg67Zn28Ca5 and Ca50Zn30Mg20; these parameters can be either positive (low affinity), negative (high affinity) or very close to 0. Note
that the CSRO analysis results are similar in physical meaning as
the heat of mixing. The Zn atoms in the Mg67Zn28Ca5 BMG tend to
have higher affinity to the largest atom in the system, Ca, and have
negative affinity to Zn itself, the smallest atom. The very small CSRO
value for Zn–Mg pair indicates Mg atoms distribute randomly around
a Zn atom with the fraction close to the average Mg system composition. The Mg atom also has a higher affinity to the largest atom Ca and
negative affinity to itself. The small CSRO value of the Mg–Zn pair indicates Zn can randomly distribute around Mg atoms. The Ca atom tends
to have good affinity with itself and negative affinity with the smallest
atom Zn, while Ca has no preference to its first neighbor Mg
atoms. After the CSRO analysis, one can see that Mg, Zn, and Ca atoms
Table 6
Average coordination numbers (CNs) for Mg, Zn, and Ca atoms and each pair.
CNs
Mg67Zn28Ca5
Ca50Zn30Mg20
NMg–Mg
NMg–Zn
NMg–Ca
NMg
NZn–Mg
NZn–Zn
NZn–Ca
NZn
NCa–Mg
NCa–Zn
NCa–Ca
NCa
Ntotal
8.51
3.45
0.85
12.81
8.31
2.30
0.71
11.32
11.24
3.85
0.98
16.07
12.56
1.84
2.77
6.67
11.28
1.91
2.11
5.83
9.85
2.65
3.27
7.44
13.36
11.89
of Mg67Zn28Ca5 BMG tend to be surrounded by the largest atom, Ca,
which can be seen by the negative CSRO values of Zn–Ca, Mg–Ca, and
Ca–Ca pairs. For the two smallest atoms, Zn and Mg atoms, the CSRO
of Zn–Zn and Mg–Mg are both positive with very small Zn–Mg and
Mg–Zn CSRO values, indicating that the low atomic affinity between
the Zn and Mg atoms and negative affinity of Zn–Zn and Mg–Mg pairs
produce the larger Ca fraction in Zn and Mg first neighbor atoms. Note
that the trends observed from the CSRO analysis are also consistent
with the trends predicted by the mutual heat of mixing. The highest affinity will always belong to Ca–Zn, and the lowest affinity to Mg–Zn.
The CSRO analysis of Ca50Zn30Mg20 BMG, shown in Table 7, indicates
the values of Zn and Ca atoms are very similar to those of Mg67Zn28Ca5
BMG. For Mg atoms, there is no CSRO value close to 0 and it seems the
negative affinities for Mg–Zn and Mg–Mg pairs indicate Mg has high affinity to the Ca atom.
A further study into the local microstructural distribution was conducted by using the Honeycutt–Anderson (HA) pair analysis [46,47].
HA pair analysis is an effective technique to filter out the numbers of different local structures within the simulation system. The HA indexes
consist of four numbers (i, j, k, and l), each with a different function in
describing the local structure. If one atom is located within the first
neighbor of the other atom, these two atoms are considered as the
bonded root pair and the first HA index is labeled as 1, while the
index for a non-bonded root pair is designated as 2. The non-bonded
root pair does not describe any particular local structure and therefore
has not been used often. The second number in the HA index represents
the number of first neighbor atoms around both atoms comprising the
bonded root pair. The third number in the HA index represents the
bond number between these first neighbor atoms. However, since
some local structures have the same corresponding first three index
values, the fourth index must be used to label these different local structures. The fourth index does not have any topological information about
the local structures and is artificially designated to distinguish two local
structures with the same first three HA indexes. For example, 1421 and
1422 represent the fact that there are four first neighbor atoms around
the bonded root pair (represented as the first index value “1”), and
there are two bonds formed between these neighbor atoms. The local
atomic arrangements for FCC and HCP crystal structures are therefore
labeled by the same first three indexes as “142,” so the fourth index
must be used to label different local structures, where the fourth indexes of 1 and 2 represent FCC and HCP crystal structures, indicating different topological arrangement of two bonds between the four neighbors.
In the current study, the HA indexes 1431, 1541, and 1551, (which occupy the largest fraction in the amorphous or liquid state) are used to
search the icosahedral local structures. The 1551 pair is particularly
characteristic of icosahedra ordering; the 1541 and 1431 are indexes
for the icosahedra defect and FCC defect local structures, respectively.
The indexes 1421 and 1422 are used to find the local structures of FCC
and HCP crystal structures, respectively. HA indexes 1661 and 1441
are employed to identify the local body-centered cubic (BCC) structure.
Finally, the indexes 1321 and 1311 are the packing related to rhombohedra, pairs which tend to evolve when the 1551 packing forms and
can be viewed as the side product of icosahedra atomic packing. The
Fraction (%)
S.-P. Ju et al. / Journal of Non-Crystalline Solids 388 (2014) 23–31
HA index
Fig. 4. The HA fractions for the Mg67Zn28Ca5 BMG.
schematic diagrams for the HA indexes above are shown in Hui and
Pederiva's study [17] and are therefore not shown here.
The HA fractions of these local structures for Mg67Zn28Ca5 BMG are
illustrated in Fig. 4. The vertical axis represents the fraction of the number of one particular HA pair divided by the total number of all HA pairs
which begin with 1. The HA pairs lower than 1% are ignored and are not
discussed. For the icosahedral local structures, the sum of 1431, 1541,
and 1551 occupies about 67% of all HA indexes, and the sum of FCC
and HCP fractions is about 17% in the Mg67Zn28Ca5 BMG. Both the BCC
local (1441 and 1661) and the rhombohedral (1321 and 1311) structures occupy a fraction of about 6%. It is apparent the icosahedral local
structures are predominant and the configuration of Mg67Zn28Ca5 is
amorphous.
Since the atomic radii of Mg, Zn, and Ca are different, the HA indexes
for different atom type pairs could be very different. Fig. 5(a)–(f) shows
the HA index distributions for different atom pairs of Mg67Zn28Ca5.
a
Because the fractions of Mg and Zn are the highest two atomic components of Mg67Zn28Ca5, the HA fractions of Mg–Mg and Mg–Zn in
Fig. 5(a) and (c) are relatively higher than those of the other four atomic
pairs. Although the Zn fraction is as high as 28% in Mg67Zn28Ca5, the occupancy of HA fractions of the Zn–Zn pair is relatively low, because the
average bond length of Zn–Zn is longer than the sum of Zn metallic radii,
resulting in a lower probability of a Zn–Zn bonded pair. The average
bond lengths of Ca–Ca and Mg–Ca pairs are the longest and the distances between the first neighbor atoms of Ca–Ca and Mg–Ca pairs are
relatively longer. Consequently, the fraction of 1541, the icosahedra defect local structure or the distorted icosahedra, is relatively higher, as
displayed in Fig. 5(e) and (f). On the contrary, the Zn–Zn bond length
is the shortest and the atomic size of Zn is the smallest; in Fig. 5(b),
the fraction of 1541 is relatively lower than those of 1551 and 1431. In
Fig. 5(a), (c), and (d), for Mg–Mg, Mg–Zn, and Zn–Ca pairs with the medium average bond lengths among all atomic pairs, the HA index distributions are relatively similar to that shown in Fig. 4.
Fig. 6 shows the HA index distribution of Ca50Zn30Mg20 BMG,
confirming that the fraction of icosahedra-like (1551, 1541, and 1431)
local structures are predominant. Among these icosahedra-like local
structures, the 1431 fraction is the highest. The HCP local structure
(1422) and 1311 occupy about 8% and 7.8%, which are both higher
than in Mg67Zn28Ca5 BMG. For BCC local structures (1441 and 1661),
the sum of 1441 and 1661 of Ca50Zn30Mg20 BMG is relatively lower
than that of Mg67Zn28Ca5 BMG.
The HA indexes of different atomic pairs of Ca50Zn30Mg20 BMG are
shown in Fig. 7(a)–(f). Because the fractions of Ca, Mg, and Zn are
more comparable in Ca50Zn30Mg20 BMG, the HA indexes of all atomic
pairs occupy certain amounts of fractions. For the Zn–Zn and Mg–Zn
pairs with the shortest two bond lengths, the 1431 fraction is the most
Mg-Mg
b
Zn-Zn
Mg-Zn
-Zn
d
Zn-Ca
Fraction (%)
c
29
e
Mg-Ca
f
HA index
Fig. 5. The HA indexes for different pairs of the Mg67Zn28Ca5 BMG.
Ca-Ca
30
S.-P. Ju et al. / Journal of Non-Crystalline Solids 388 (2014) 23–31
Fraction(%)
PRDF profiles as well as the average bond lengths of different atomic
pairs of Mg67Zn28Ca5 and Ca 50Zn30Mg20 BMG are very similar, the
local structures of these two BMG are very different as shown by
the HA analysis. For Mg67Zn28Ca5, the fraction of 1551, the icosahedral local structure, is the highest, while in Ca50Zn30Mg20 the 1431,
the distorted icosahedral local structure, is the highest. This study
provides detailed information about atomic arrangements for Mg67Zn28Ca5 and Ca50Zn30Mg20, and this DFT calculation could be used for
future studies of their corrosion processes.
HA index
Acknowledgments
Fig. 6. The HA fractions for the Ca50Zn30Mg20 BMG.
prominent among all HA indexes, while for the Ca–Ca pair with the longest bond distance, the fraction of 1541 is the highest, the same as for
Ca–Ca in Mg67Zn28Ca5.
We thank (1) the National Science Council of Taiwan, under
Grant No. NSC101-2628-F-110-003-MY3, (2) National Center for
High-performance Computing, Taiwan, and (3) partial funding from
the Ministry of Education, for supporting this study.
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4. Conclusion
Fraction(%)
The SABH method has been carried out to predict the Mg67Zn28Ca5
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