A model of atom dense packing for metallic glasses with high-solute
concentration
Bao-Chen Lu, Jia-Hao Yao, Jian Xu, and Yi Li
Citation: Appl. Phys. Lett. 94, 241913 (2009); doi: 10.1063/1.3157136
View online: http://dx.doi.org/10.1063/1.3157136
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APPLIED PHYSICS LETTERS 94, 241913 共2009兲
A model of atom dense packing for metallic glasses with high-solute
concentration
Bao-Chen Lu,1 Jia-Hao Yao,1 Jian Xu,1,a兲 and Yi Li2,a兲
1
Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy
of Sciences, 72 Wenhua Road, Shenyang 110016, People’s Republic of China
2
Department of Materials Science and Engineering, Faculty of Engineering, National University
of Singapore, Singapore 117576, Singapore
共Received 31 March 2009; accepted 29 May 2009; published online 19 June 2009兲
Taking the chemical ordering in the metallic glasses into consideration, we have extended the
efficient cluster packing model to predict the composition in high-solute concentration alloys with
atomic dense packing. Its validity is supported by the good agreements between the predicted
compositions with the maximum packing efficiency and the experimentally optimized bulk metallic
glass formers in the Cu–Zr, Cu–Hf, and Ni–Nb binary systems. Despite its simplicity, it seems that
the structure predicted by our model reflects in certain way the averaged structural configuration of
metallic glasses. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3157136兴
It has been well documented that metallic glasses 共MGs兲
exhibit a structural feature of atom dense packing.1,2 Since
the ease of glass formation correlates with highly dense
packing of atoms, it is possible to predict the alloy composition to favor the glass formation by virtue of model of these
structures through calculating the coordination number 共CN兲
of atoms in clusters.3–6 However, the capability of this approach currently remains limited as it is only applied to the
alloys with low solute concentration 共⬍30 at. %兲 in most
cases, based on the scheme of the solute-center clusters.1,3–6
One such model, the efficient cluster packing 共ECP兲 model
developed by Miracle et al.1,3–5 captures the essence of the
basic packing mode, i.e., the efficient packing of atoms with
solute-centered structure entities. However with the solutecentered clusters containing only solvent atoms in the first
coordination shell 关as shown in Figs. 1共a兲 and 1共c兲兴, this
model limits itself to the scenario of low concentration. It
therefore has been a challenge to predict MG-forming alloys
with high-solute concentration 共e.g., 30– 50 at. %兲 since the
homoatomic pairs in the first coordinate shell are unavoidable. Furthermore, as presented, the bulk MGs 共BMGs兲 can
be formed at high-solute concentration in a number of alloys,
including the binary systems of the Cu–Zr,7–9 Cu–Hf,9–11 and
Ni–Nb.12,13 Currently, apart from a “cluster plus glue atom”
model,14 which still essentially is with a solute-centered
structure, no models are available on the compositional prediction of dense packing at the range of high-solute concentration in MG-forming systems.
In this work, through extending the ECP model and taking into account the short range order in MGs with highsolute concentration in the range of 30– 50 at. %, we developed a highly ECP model for binary MGs. This is
accomplished by starting with solute-centered clusters with
solvent atoms only in the first coordinate shell and then systematically replacing solvent atoms with solute atoms, one at
a time. Alloy compositions predicted using this atomic structural model match very well with the best glass formers exa兲
Authors to whom correspondence should be addressed. Electronic addresses: [email protected] and [email protected].
0003-6951/2009/94共24兲/241913/3/$25.00
perimentally obtained in several typical BMG-forming systems.
It can be reasonably deduced from the results of Park et
al.15 that for the effect of atomic size mismatch 共␭兲 in the i-j
binary, there exists only one maximum value ␭max, which is
obtained at x j max = 1 / 共1 + R3兲, where R represents the ratio of
the radius of j atom to that of i atom. It is obvious that x j max,
partitions the entire composition range into two parts. As
indicated,16 the compositions favoring the glass formation
are usually located on the side of component with larger
atomic size and based on Ref. 6, effectively the boundaries
共x j兲 for solute concentration in our study should be only restricted as 30 at. % ⱕ x j ⬍ 1 / 共1 + R3兲.
For a given i-j binary system, it can also be assumed that
there should be two different kinds of atomic clusters considering the first neighbor atoms, existing at the i- or j-rich
composition side, i.e., i- and j-centered clusters, respectively.
For simplification, the atom with larger atomic radius is denoted as i, the other atom with a smaller atomic radius as j.
With the ECP model, the CN Niij 共or Niji兲 can be predicted
when the atoms at first coordinate shell are completely occupied by j 共or i兲 in the i-centered 共or j-centered兲 cluster, as
FIG. 1. 共Color online兲 Two-dimensional schematic diagrams of only solvent
atoms packing 共a兲 i-centered cluster and 共c兲 j-centered cluster; and unlike
atoms simultaneously packing 共b兲 i-centered cluster and 共d兲 j-centered cluster in the first coordinate shells of solute-centered clusters.
94, 241913-1
© 2009 American Institute of Physics
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241913-2
Appl. Phys. Lett. 94, 241913 共2009兲
Lu et al.
shown in Figs. 1共a兲 and 1共c兲. In the case of high-solute concentration, the solute-solute 共i-i or j-j兲 pairs in the alloys are
inevitable; therefore, i and j atoms should be simultaneously
involved in the packing in the first coordinate shell, as shown
in Figs. 1共b兲 and 1共d兲. Since the i and j atoms are different in
size, the sum 关designated as Ni 共or N j兲兴 of partial CNs Nij and
Nii 共or N jj and N ji兲, representing the individual number of j
and i atoms in the first coordinate shell, respectively, will be
different from Niij 共or N jji兲. A credible model should have the
capability to provide Nij and Nii 共or N jj and N ji兲 successfully.
Pelton et al.17 proposed a “modified quasichemical model”
and introduced the composition-dependent CNs in liquid.
Based on this model and combining with the description of
chemical ordering in MGs by Cargill III and Spaepen,18 we
have
Nij
Niij
+
Nii
Niii
= 1,
or
N ji
N jji
+
N jj
N jjj
共1兲
= 1,
which is used to determine the integer Nij and Nii for
i-centered cluster 共or N ji and N jj for j-centered cluster兲,
where Niii 共=N jjj兲 represents i 共or j兲 atom in i-centered 共or
j-centered兲 cluster. In fact, the above equation is just the
ultimate case of unity state of inequalities for the total local
packing efficiencies given by Miracle,4 which means that the
closer is the total local packing efficiency to unity, the higher
is the local packing efficiency.4 All of Niij, Niii, N jji, and N jjj are
the theoretical maximum CNs and can be estimated as NT,3
NT =
再
4␲
冋 冉冊
␲共2 − n兲 + 共2n兲arccos sin
FIG. 2. 共Color online兲 Two-dimensional schematic diagram of an ECP structure in 兵100其 plane of a single sc cluster unit cell. The features of interpenetrating j-centered clusters, efficient atomic packing around each j atom,
ordering of j atoms, and randomness of surrounding atoms are illustrated in
these schematic figures. The dashed circles show overlapping j-centered
clusters.
␲
冑R共R + 2兲
n
共R + 1兲
册冎
,
共2兲
where R represents the ratio of the radius of center atom to
that of coordinate atom; n = 5 when R ⱖ 0.902 and n = 4 when
0.414ⱕ R ⬍ 0.902. Based on this equation, Niii and N jjj is
13.33 for R = 1 and n = 5. Then, Niij and N jji can be calculated
based on the actual atomic size ratio. Furthermore, the condition of Nij 共or N ji兲 is always larger than Nii 共or N jj兲 based
on the principle stated in Refs. 17 and 18 should be satisfied.
With these data, the Eq. 共1兲 will then permit the calculation
of the corresponding Nij 共N jj兲 and Nii 共N ji兲 necessary for a
dense packing cluster.
Even with the proper Niij and N jji in a cluster, two issues
need to be determined before a composition of dense packing
is predicted. First it is necessary to select the packing mode
of clusters. Herewith, we adopt the face-centered-cubic 共fcc兲
or simple cubic 共sc兲 as the basic packing mode for the clusters in MGs, the same as those in Ref. 5. For the sc packing
mode of the clusters, only hexahedral interstitial sites can be
filled with the interstitial atoms to destabilize the sc packing
mode. For the fcc packing mode of the clusters, each interstitial atom is only placed at octahedral interstitial site, which
is obviously larger than the tetrahedral interstitial site. For
metal-metal glass-forming system, it is unpractical for the
tetrahedral interstice to be filled with one interstitial atom
because the size of solute atom is usually much larger than
the size of the tetrahedral interstice.4 Like the ECP model,
only one atom is placed in each interstice in our model 共as
shown in Fig. 2兲.
Second, due to the inter-penetrating and overlapping of
the clusters in the real cases 共as shown in Fig. 2兲, the actual
CN ⍀⍀ in the first coordinate shell for each cluster has to be
modified as ⍀⍀ = NS / 共1 + ␾ / NS兲,4 where ␾ is 6 共or 12兲 for sc
共or fcc兲 packing mode of the clusters2 and Ns represents the
sum 关Ni 共or N j兲兴 of Nii and Nij 共or N jj and N ji兲.
With all the above, the alloy composition with dense
packing for i-centered or j-centered cluster can be calculated
as following:
xi = 共Nii⍀⍀/Ni + 2兲/共⍀⍀ + 2兲,
or x j = 共N jj⍀⍀/N j + 2兲/共⍀⍀ + 2兲.
共3兲
To testify our model, we selected the binary Cu–Zr, Cu–
Hf, and Ni–Nb systems consisting of the transition metals.
The sc packing mode was applied for the Cu–Zr and Cu–Hf
as it was reported that the packing mode for the Zr–Ni glass
is sc,4 while Ni–Nb binary is reported to be fcc packing
mode of the clusters.4 As the representative, the calculated
compositions for the Cu–Zr binary are listed in Table I. According to the ideal high packing efficiency 关Eq. 共1兲兴, only
the calculated compositions which possess both integer 共or
nearly integer兲 Nii and Nij 共or N jj and N ji兲 are valid in terms
of physical meaning. Then, according to the boundaries 共x j兲
for solute concentration in our study, the valid predicted
compositions are obtained to be Zr54.9Cu45.1 共A兲, Zr49.8Cu50.2
共B兲, and Zr36.3Cu63.7 共C兲, respectively, as shown in Table I. It
is of interest to note that these calculated compositions match
very well with the experimentally optimized BMG formers
at Zr55Cu45,9 Zr49Cu51,19 and Zr36Cu64,7 respectively.
Moreover, the average total CNs NCu of the cluster are
also roughly evaluated for A, B, and C compositions in the
Cu–Zr binary. The NCu for three alloys of A, B, and C are
calculated to be around 11.5, 11.7, and 12.2, respectively. It
is noticed that the NCu for A and C alloys is quite well in
agreement with the experimental values, 11.5 共Ref. 20兲 and
12.0,21 respectively. It implies that quasiequivalent icosahedra clusters are dominant structural units in the MG with the
compositions of A, B, and C, as proven by the findings that
the icosahedra clusters determine the nature of atomic denser
packing.2 Also, the icosahedra clusters should be more
prevalent in alloy C than in alloys A and B.
Besides the Cu–Zr binary, the validity of the model was
examined also in the binary Cu–Hf and Ni–Nb systems. Figure 3 shows a comparison of the composition predicted by
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241913-3
Appl. Phys. Lett. 94, 241913 共2009兲
Lu et al.
Cu
Cu
Zr
TABLE I. Calculated and valid clusters in Zr–Cu binary system 共NCuCu
= 13.33, NCuZr
= 10.06, NZrZr
= 13.33,
Zr
NZrCu = 16.94, xmin = 30 at. %, and xmax = 65.8 at. %兲.
Zr centered cluster
Cu centered cluster
NZrZr
NZrCu
NZr
Calculated composition
NCuCu
NCuZr
NCu
Calculated composition
0
1
2
3
4
5
6
7
16.94
15.67
14.40
13.13
11.86
10.59
9.32
8.04
16.94
16.67
16.40
16.13
15.86
15.59
15.32
15.04
Out of range
Invalid
Invalid
Out of range
Zr36.3Cu64.7 共C兲
Invalid
Invalid
Zr54.9Cu45.1 共A兲
0
1
2
3
4
5
10.06
9.30
8.55
7.79
7.04
6.28
10.06
10.30
10.55
10.79
11.04
11.28
Out of range
Invalid
Invalid
Invalid
Zr49.8Cu50.2 共B兲
Invalid
the present model with the experimentally obtained BMG
formers for the three binary systems. As indicated, reasonable matches are presented also for Cu–Hf 共Refs. 9–11兲 and
Ni–Nb 共Refs. 12 and 13兲 systems. Consequently, it means
that the present simple model approximately describes the
averaged structural configuration of MGs consisting of early
transition metal and late transition metal. Furthermore, the
coincidence of the dense packing and the best glass former
may suggest the intrinsic correlation between packing efficiency and glass-forming ability,22 especially in high-solute
concentration alloys. Finally the model we propose is the
sufficient condition at best, not the necessary condition for
the dense packing. This is to say that the composition predicted by our model should be found in the experiments,
while the results obtained in the experiments may not necessarily be able to be predicted by our model just like the case
for the one miss in the Zr–Cu system, as shown in Fig. 3.
In summary, extending the ECP model, we proposed a
simple model of atomic dense packing for metallic glasses
with high-solute concentration. Its validity is supported by
the good agreements between predicted compositions with
the maximum local packing efficiencies and experimentally
optimized BMF formers in the Cu-Zr, Cu-Hf, and Ni-Nb
FIG. 3. 共Color online兲 Comparison of dense packing compositions predicted
by the model with experimentally optimized glass former for Cu–Zr, Cu–Hf,
and Ni–Nb binary alloys.
binary alloys. It suggests that the optimized glass-forming
composition intrinsically correlates with maximum packing
efficiencies of atoms.
This work was supported by the National Basic Research
Program of China 共973 Program兲 under Contract No.
2007CB613906. B.-C.L. gratefully acknowledges stimulating discussions with Dr. Qing-Song Mei. The authors are
also grateful for the referee’s valuable comments and suggestions.
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