Supplementary Materials for
"Metallic liquids and glasses: atomic order and global packing"
by Liu et al.
1. Molecular Dynamics (MD) Simulations
The classical molecular dynamics (MD) simulations were conducted to produce
three dimensional (3D) atomic configurations for both metallic liquids and glasses.
Their pair distribution functions (PDFs) were then obtained by analyzing these 3D
atomic configurations. Tight-binding potentials [1] have been utilized to describe the
interatomic interactions for Ni, Cu, and Al pure metals, and the embedded atom
method (EAM) potential developed recently by Sheng et al. [2] has been used to
model the Zr-Cu-Al ternary system. Simulations were carried out in the NPT
isobaric-isothermal ensemble with constant particle number, pressure (P) and
temperature (T), which was implemented with 5th Gear prediction-correction
algorithm. The temperature and pressure of modeling system were controlled through
the volume-scaled method and the velocity-scaled method, respectively. The MD time
step was set as 1 fs to integrate Newton’s equations of motion. The modeling cells
with periodic boundary conditions contain 4,000 atoms for the pure metal systems and
10,000 atoms for the Zr-Cu-Al ternary system. During simulation, the system was
firstly heated from 300 K to 2000 K under a zero external pressure and then relaxed
for 30000 MD steps (30 ps) at 2000 K to get a full liquid. Subsequently, the system
was quenched from 2000 K to 300 K at cooling rates ranging from 1010 K/s to 1013
K/s to get glassy states. The pair distribution function (PDF) can be obtained by
1
following formula:
g (r ) 
L3
N
 N

  n (r )  4r 2 dr ,
 1

(S1)
where L is the box size and N is the number of particles.
2. Materials Fabrication
Master ingots with nominal compositions were prepared by arc melting under a
Ti-gettered argon atmosphere with pure metals (≥99.9%, wt%) as charge materials.
Zr38.2Cu61.8, Zr44Cu56, Zr66.7Ni33.3 and Zr40Cu55Al5 (at. %) amorphous ribbons with a
cross-section of 0.044 mm2 used for PDF analyses were produced by a
melt-spinning technique under argon atmosphere. Amorphous nature of the samples
was ascertained by X-ray diffraction (XRD) (see Fig. S1).
3. Pair distribution function (PDF) analysis
Synchrotron X-ray scattering experiments were performed at beamline 11-ID-C
of Advanced Photon Source at Argonne National Laboratory, using synchrotron
radiation of energy of 115.381 keV (λ=0.10748 Å). A two dimensional image plate
detector (MAR345) was used to collect data over a wide Q range (Q=4πsinθ/λ). The
measurement of each sample was repeated five times. X-ray diffraction data of an
empty capillary were also collected at the same experimental condition for
background corrections. The corresponding images were subjected to geometrical
corrections, integrated, and reduced to one-dimensional scattering intensity, I(Q)
using the program FIT2D, which were further converted into absolute electron units,
2
and reduced to the total scattering structure function, S(Q), using the program
PDFGETX2 [3]. The Fourier transformation of S(Q) yields the pair distribution
function (PDF), g(r) (Fig. S2):

g (r )  1  1 2 2 0  Q[S (Q)  1]sin( Qr )d (Q) ,
0
(S2)
where ρ0 is the average number density and Q is the magnitude of the scattering vector.
The peak position (Ri) was obtained by fitting the corresponding peak using Gaussian
function. For instance, the R2 and R3 (see Fig. S2) were determined by fitting the
splitting second peak in the g(r) with two overlapping Gaussian functions.
4. Effects of different radiation resources on PDF
The experimental PDFs in literature were obtained by Fourier transformation of
total interference functions derived from raw scattering data. Therefore, magnitude for
each hidden partial PDF (in the case of multicomponent systems) depends on atomic
scattering factor for the individual radiation source. In other words, weighting factors
related to atomic scattering factors should be considered to analyze the experimental
total PDFs. To clarify effects of different radiation resources on the total PDFs, we
have compared the PDFs by introducing atomic scattering factors in a Zr46Cu46Al8
model MG. Firstly, we obtained the partial PDFs of the Zr46Cu46Al8 MG by MD
simulations (Fig. S3). Secondly, we calculated the atomic scattering factors for X-ray,
electron and neutron radiation sources, respectively. Based on the atomic scattering
factor data, the weighting factors (Table S2) for each radiation source by the
following formula were calculated [4]:
3
wij  ci c j f i f j   ci f i 
 i

2
(S3),
where ci is the concentration of element i, fi is the atomic scattering factor. Then, the
total PDF can be obtained by the weighted sum of the partial PDFs:
g ( r )   wij gij ( r )
i
(S4).
j
The PDF profile (Fig. S4) is slightly changed with different radiation resources,
but in principle, R2 and R3 are not strongly affected by the radiation resources. This is
because R2 and R3 are atomic pair distances for short-range orders (SROs), which are
not affected by radiation resources. Nevertheless, due to the broadening effect of the
peak width, to a certain degree, experimentally measured Ri may be influenced by the
radiation resources. However, this influence should not strongly affect the global
feature, i.e., effects of the radiation sources on the ratios of Ri/R1 are ignorable. For
example, there is an additional peak locating at ~3.10 Å on the PDF curve from the
electron radiation in addition to the first peak locating at ~2.84 Å observed for all the
three radiation resources (Fig. S4). Taking the partial PDF curves (Fig. S3) and the
weight partial functions (Table S2) into account, we know that the Zr-Cu atomic pair
(the distance of 2.83 Å, which is determined from the Fig. S3) is the major
contribution to the first peak, and the Zr-Zr atomic pair (3.17 Å) is related to the
additional peak. From the SRO viewpoint, the Zr-Cu atomic pair should be more
predominant than Zr-Zr atomic pair in the Zr46Cu46Al8 MG. Therefore, the first peak,
which represents the predominant SROs and reflects the global feature of MGs, is
selected as R1 in this case. Accordingly, we can determine the R2 and R3 value. Indeed,
4
the similar results can also be found in Ref. 5.
5
Table S1. The first peak position R1 on the PDF patterns and the Ri/R1 (i=1,2,…,5)
ratios for a variety of metallic glasses determined from diffraction measurements
and molecular dynamic simulations.
Metallic glass
R1
R2/R1
R3/R1
R4/R1
R5/R1
References
Cu*
Ni*
Al*
Zr38.2Cu61.8†
Zr44Cu56†
Zr66.7Ni33.3*
Zr40Cu55Al5†
Zr50Cu50†
Zr35.5Cu64.5*
Zr66Ni34*
Zr65Cu27.5Al7.5#
Zr55Cu35Al10†#
Zr46Cu46Al8†
Zr65Ni25Al15*
Zr52.5Cu17.9Ni14.6Al10Ti5†#
Zr52Al6Cu14Ni8Fe20†
Zr44.5Al10Cu20Ni8 Ti7.5†
Zr57Al10Cu20Ni8 Ti5†
Zr58Al10Cu20Ni8 Ti4†
Zr59Al10Cu20Ni8 Ti3†
Zr60Al10Cu20Ni8 Ti2†
Zr41Ti14Cu12.5Ni10Be22.5**
Cu40Ag60*
Cu30Ag70*
Cu60Hf30Ti10†
Cu60Zr30Ti10†
Cu60Ti20Zr20*
Cu55Hf25Ti15Pd5†
Cu47Ti33Zr11Ni8Si1†
Fe50Al50*
Fe75B25†#
Fe81B19†#
Fe86B14†#
Fe83P17*
Fe84B16*
Fe82B18†
Fe82Nb18†
Fe82B15Nb3†
Fe82B12Nb6†
Fe82B9Nb9†
Fe82B6Nb12†
Fe82B3Nb15†
Fe90Zr7B3#
Fe80P13C7†
2.51
2.47
2.85
2.76
2.78
2.93
2.79
2.94
2.75
2.78
3.07
2.91
2.94
2.98
2.85
2.97
3.04
3.11
3.11
3.11
3.12
2.72
2.73
2.81
2.84
2.69
2.66
2.74
2.75
2.53
2.42
2.48
2.55
2.54
2.54
2.57
2.57
2.57
2.57
2.57
2.57
2.54
2.50
2.55
1.73
1.73
1.75
1.73
1.74
1.73
1.74
1.69
1.73
1.74
1.74
1.68
1.71
1.72
1.69
1.70
1.68
1.66
1.68
1.66
1.67
1.73
1.73
1.71
1.70
1.74
1.71
1.73
1.74
1.70
1.70
1.71
1.71
1.71
1.69
1.65
1.68
1.66
1.67
1.67
1.66
1.67
1.71
1.72
1.98
1.99
1.96
1.91
1.90
1.93
1.90
1.97
1.96
1.95
1.91
2.00
1.95
1.95
1.83
1.96
1.91
1.89
1.94
1.90
1.89
2.00
1.97
1.95
1.93
1.96
1.96
1.97
2.02
1.92
1.97
1.97
1.94
1.97
1.92
1.96
1.94
1.95
1.94
1.92
1.95
1.96
1.94
1.96
2.64
2.63
2.58
2.62
2.62
2.63
2.60
2.49
2.55
2.65
2.55
2.60
2.50
2.56
2.63
2.59
2.47
2.45
2.43
2.45
2.45
2.67
2.56
2.55
2.47
2.59
2.66
2.62
2.62
2.64
2.62
2.55
2.53
2.60
2.61
2.53
2.62
2.53
2.59
2.61
2.61
2.64
2.60
2.57
3.47
3.46
3.46
3.42
3.45
3.46
3.44
3.40
3.40
3.51
3.38
3.43
3.36
Present work
Present work
Present work
Present work
Present work
Present work
Present work
[6]
[7]
[8]
[9]
[10, 11]
[6]
[12]
[13]
[14]
[14]
[14]
[14]
[14]
[14]
[15]
[16]
[17]
[18]
[18]
[19]
[20]
[21]
[22]
[23]
[23]
[23]
[24]
[24]
[25]
[25]
[25]
[25]
[25]
[25]
[25]
[26]
[27]
3.39
3.32
3.27
3.29
3.30
3.28
3.57
3.39
3.57
3.51
3.50
3.49
3.41
3.38
3.39
3.43
3.34
3.47
3.38
3.42
3.44
3.45
3.48
3.48
3.46
6
Metallic glass
R1
R2/R1
R3/R1
R4/R1
R5/R1
References
Ni73.8P26.2†
Ni76P24†
Ni77.2P22.8†
Ni78.9P21.1†
Ni81.4P18.6†
Ni50Al50*
Ni66.7Al33.3*
Pd80Si20†
Pd30Ni50P20†
Pd40Ni40P20†
Pd50Ni34P16†
Pd40Cu30Ni10P20†
Al89La6Ni5#
Al90Fe5Nb5†
Al77.5Mn22.5†
Al56Si30Mn14†
Mn75P15C10†
Re82Tb18†
Re93V7*
Re*
2.47
2.47
2.47
2.41
2.41
2.51
2.50
2.79
2.64
2.65
2.70
2.75
2.75
3.15
2.75
2.58
2.57
2.76
2.73
2.72
1.70
1.68
1.67
1.70
1.74
1.76
1.74
1.71
1.70
1.69
1.70
1.68
1.70
1.74
1.72
1.70
1.73
1.68
1.68
1.69
1.93
1.94
1.93
1.96
2.00
1.96
1.94
1.94
1.91
1.91
1.96
1.92
1.92
2.04
1.95
2.02
1.95
1.93
1.95
1.95
2.61
2.61
2.59
2.63
2.63
2.64
2.62
2.58
2.55
2.54
2.51
2.54
2.69
2.47
2.46
2.52
2.59
2.57
2.62
2.62
3.46
3.46
3.44
3.51
3.51
3.49
3.44
[27]
[27]
[27]
[27]
[27]
[28]
[29]
[27]
[30]
[30]
[30]
[31]
[32]
[33]
[34]
[34]
[35]
[36]
[36]
[36]
3.37
3.36
3.38
3.37
3.64
3.33
3.47
3.44
3.37
3.41
3.43
†:
Synchrotron X-ray diffraction;
Electron diffraction;
†#: Neutron diffraction;
*: Molecular dynamics;
**: ab initio molecular dynamics.
#:
Table S2: Weight partial functions wij for different radiation sources in the
Zr46Cu46Al8 model system
Partial function
X-ray
Electron Neutron
wZr-Zr
0.31499 0.41899
0.214
wZr-Cu
0.45688 0.38602 0.46135
wZr-Al
0.03562 0.07059 0.03585
wCu-Cu
0.16567 0.08891 0.24865
wCu-Al
0.02583 0.03252 0.03865
wAl-Al
0.00101 0.00297
0.0015
7
Figure S1: The typical XRD patterns for as-cast amorphous ribbons.
Intensity (a.u.)
Zr38.2Cu61.8
Zr44Cu56
Zr66.7Ni33.3
30
40
50
60
2 (deg.)
70
80
8
Figure S2: Typical g(r) curves obtained from synchrotron X-ray diffraction
experiments for Zr-Cu-(Al) MGs. The dash lines illustrate schematically the
peak positions.
Zr38.2Cu61.8
Zr40Cu55Al5
R2 R
3
R5
g(r)
R4
R1
2
3
4
5
6
7
r (Å)
8
9
10
11
9
Figure S3: Calculated partial PDF curves of the Zr46Cu46Al8 model system by
MD simulations.
Partial PDF
Zr-Zr
Zr-Cu
Zr-Al
Cu-Cu
Cu-Al
Al-Al
0
2
4
6
8
r (Å)
10
Figure S4: PDF curves obtained from different radiation sources in the
Zr46Cu46Al8 model system.
Zr46Cu46Al8
PDF
X-ray
Electron
Neutron
2
4
6
8
r (Å)
10
12
14
10
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