PHYSICAL REVIEW B
95
, 024103 (2017)
Ordering and dimensional crossovers in metallic glasses and liquids
David Z. Chen,
1
,
*
Qi An,
2
William A. Goddard III,
2
and Julia R. Greer
1
,
3
1
Division of Engineering and Applied Sciences, California Institute of Technology, Pasadena, California, 91125, USA
2
Materials and Process Simulation Center, California Institute of Technology, Pasadena, California, 91125, USA
3
The Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California, 91125, USA
(Received 1 February 2016; revised manuscript received 8 November 2016; published 4 January 2017)
The atomic-level structures of liquids and glasses are amorphous, lacking long-range order. We characterize the
atomic structures by integrating radial distribution functions (RDF) from molecular dynamics (MD) simulations
for several metallic liquids and glasses: Cu
46
Zr
54
,Ni
80
Al
20
,Ni
33
.
3
Zr
66
.
7
,andPd
82
Si
18
. Resulting cumulative
coordination numbers (CN) show that metallic liquids have a dimension of d
=
2
.
55
±
0
.
06 from the center
atom to the first coordination shell and metallic glasses have d
=
2
.
71
±
0
.
04, both less than 3. Between the
first and second coordination shells, both phases crossover to a dimension of d
=
3, as for a crystal. Observations
from discrete atom center-of-mass position counting are corroborated by continuously counting Cu glass- and
liquid-phase atoms on an artificial grid, which accounts for the occupied atomic volume. Results from Cu grid
analysis show short-range d
=
2
.
65 for Cu liquid and d
=
2
.
76 for Cu glass. Cu grid structures crossover to
d
=
3at
ξ
∼
8
̊
A(
∼
3 atomic diameters). We study the evolution of local structural dimensions during quenching
and discuss its correlation with the glass transition phenomenon.
DOI:
10.1103/PhysRevB.95.024103
I. INTRODUCTION
The viscosities and relaxation times of glasses and liquids
across the glass transition temperature (T
g
) are separated
by many orders of magnitude [
1
]. This large increase in
viscosity over a short temperature range is not accompanied
by significant changes in the long-range atomic structure,
which remains amorphous. Metallic glasses are locally more
ordered in the short- and medium-range than their liquid
counterparts [
2
,
3
], but this ordering plays an ambiguous role in
the glass transition [
4
]. A structural model that captures both
liquids and glasses is useful for understanding the amorphous
structure and the subtle changes, if any, that occur across T
g
and
their potential connection to the glass transition phenomenon.
The local dimension, d, describes how, on average, the mass
of atoms within a spherical section of material with radius r
scales, M(r)
∝
r
d
[
5
]. In relating the positions of the first sharp
x-ray diffraction peaks (q
1
) to sample volume (
V
), several
groups have reported a scaling relationship in metallic glasses,
with exponent, d
∼
2
.
31–2
.
5, which deviates from the d
=
3
expected under the assumption that q
1
∝
1
/
a, where a is the
interatomic spacing [
6
–
9
]. Recent real-space measurements on
Ti
62
Cu
38
also revealed a dimensionality of roughly 2.5 [
10
].
Experiments on electrostatically levitated metallic liquids
show a non-cubic power law exponent of d
∼
2
.
28 [
11
], albeit
with a limited range in data and a significant amount of
scatter [
12
]. Without translational symmetry, the connection
between diffraction peak positions and interatomic distances
in amorphous materials is not simple [
12
]. Nonetheless,
the estimated power law exponents are related to the local
dimension of the atomic structure, and observations of an
exponent/dimension less than 3 have led to suggestions of an
underlying fractal structure in metallic glasses [
6
,
8
]. However,
the long-range scaling relationship in metallic glass structure
is not fractal over all length scales because macroscopic pores
*
Corresponding author: dzchen@caltech.edu
or voids are absent in their microstructure, and such pores are
a defining characteristic of fractals that maintain their scaling
relationships over long ranges (e.g. the Sierpinski triangle).
In response to this inconsistency, Chen
et al.
proposed
that metallic glasses at the atomic-level can be described
using percolation [
8
], a model that captures the intercon-
nectivity of sites on a lattice or spheres in a continuum [
5
].
Three-dimensional percolation models, such as hard sphere
and overlapping sphere continuum models, exhibit a fractal
dimension of d
∼
2
.
52 at lengths below a correlation length,
ξ
, and a crossover to a dimension d
∼
3 above
ξ
, where
ξ
is roughly the diameter/length of finite, non-percolating
clusters [
5
]. Using molecular dynamics (MD) simulations,
Chen
et al.
found that two distinct metallic glasses have short-
range dimensions of d
∼
2
.
5 below
ξ
∼
2 atomic diameters
and a dimension of 3 occurs over longer lengths. This
suggested that metallic glasses are structurally similar to
a continuum percolation (i.e. spatially-random coalescence)
of spherical particles [
8
]. This crossover at
ξ
may explain
the anomalous non-cubic scaling exponents in q
1
vs. V
observed experimentally in macroscopically homogeneous
and fully dense metallic glasses and liquids [
6
–
8
,
11
]. Such a
connection between percolation structure and glasses has also
been suggested by Orbach, who applied percolation theory to
describe high frequency (short wavelength) vibrational states
in glassy systems and also suggested that amorphous materials
may exhibit fractal properties at short length scales [
13
].
The question remains whether liquids exhibit a crossover in
dimension from d
<
3tod
=
3. Percolation structure has been
studied in hard spheres [
14
,
15
], overlapping spheres [
16
,
17
],
and recently metallic glasses [
8
], suggesting a possible connec-
tion to metallic liquids, which share structural similarities with
both metallic glasses and hard sphere systems [
18
]. It would be
interesting to study the development of this ordering as a func-
tion of temperature, across the glass transition. One previous
method to measure dimension utilized hydrostatic pressures
to induce peak shifts in radial distribution functions (RDF)
that were compared to corresponding volume changes [
8
].
2469-9950/2017/95(2)/024103(8)
024103-1
©2017 American Physical Society
CHEN, AN, GODDARD III, AND GREER
PHYSICAL REVIEW B
95
, 024103 (2017)
However, this pressure-induced peak shift method is not well
suited for studying liquids, in which atomic rearrangement and
exchange of neighbors lead to significant structural changes
under pressure. The correlation lengths,
ξ
, can only be inferred
based on the scaling of various peaks. Moreover, the broadness
of the RDF peaks leads to results that are sensitive to the
specific method of generating and measuring the RDF [
19
]. To
overcome these challenges, we integrated the RDFs to obtain
cumulative coordination numbers (CN). This integral method
estimates the local dimension of the structure without the need
for applying hydrostatic pressures or measuring small shifts
in broad amorphous peak positions, which are methods that
were used previously [
8
]. With this CN analysis, we observe
acrossoverindimensionfromd
=
2
.
55
±
0
.
06 in metallic
liquids and d
=
2
.
71
±
0
.
04 in metallic glasses, to d
=
3for
the second coordination shell and beyond, suggesting that
ξ
∼
3 atomic diameters.
II. DIMENSION AND CROSSOVER
One measure of dimension comes from the scaling of
extensive properties with size such as mass, i.e. M(r)
∝
r
d
,
where M(r) is the mass contained in a sphere of radius r. M(r)
is calculated as an average over the entire system by choosing
different atoms as the center of the sphere [
5
]. In our analysis,
we use the value CN
+
1 to represent the average number
of atoms within a sphere of radius r (1 added to account for
the center atom), an extensive property that is proportional
to average mass. In percolation, the scaling relationship for
a system above the percolation threshold,
φ
c
, exhibits a
crossover in dimension from d
∼
2
.
52 to d
∼
3at
ξ
, where
ξ
∝
(
φ
−
φ
c
)
−
ν
[
5
]. The parameter definitions are as follows:
φ
is the packing fraction,
ν
=
0
.
8764 is the critical exponent
for the correlation length [
20
], and
φ
c
is the percolation
threshold in 3-dimensional continuum percolation [
5
]. From
percolation theory, the expected crossover point for several of
the metallic systems studied here has been roughly estimated
to be
ξ
∼
2[
8
]. This value represents the average size of
non-percolating clusters in units of atomic diameters and
suggests that the crossover occurs around the first atomic
coordination shell. To avoid inaccuracies that may arise from
determining precise peak shifts in broad amorphous peaks, we
obtain the dimension of each atomic structure by measuring
d(ln(CN
grid
))
/
d(ln(r)) for Cu
46
Zr
54
,Ni
80
Al
20
,Ni
33
.
3
Zr
66
.
7
, and
Pd
82
Si
18
metallic liquids and glasses. We find that a crossover
from d
<
3tod
=
3 occurs in all cases beyond the first to
second coordination shell. We compare these results to those
for pure Cu and Zr (SI) in liquid and crystalline phases.
A. Metallic glasses
We measure d by performing a linear fit between the radius
of the center atom, r
avg
, and the outer radius of the first
coordination shell, r
1s
.Ther
avg
is defined as the weighted
average radii of the atoms in the binary systems (i.e. for
Cu
46
Zr
54
,r
avg
=
0
.
46r
Cu
+
0
.
54r
Zr
, refer to SI). There is on
average one atom (i.e. the center atom) within this radius,
making it an appropriate first point in the analysis of the
dimension. Using this approach, we establish the following
estimates of dimensions: d
=
2
.
68 for Ni
80
Al
20
,d
=
2
.
73 for
Ni
33
.
3
Zr
66
.
7
,d
=
2
.
66 for Pd
82
Si
18
, and d
=
2
.
74 or 2.73 for
Cu
46
Zr
54
using FF
1
[
21
]orFF
2
[
22
], respectively (Fig.
2
),
all at 300 K. The average dimension for metallic glasses of
d
=
2
.
71
±
0
.
04 is
∼
0
.
19 higher than what would be expected
from percolation theory, where d
∼
2
.
52 [
5
], and is higher than
previous estimates of
∼
2
.
3–2
.
5[
6
,
7
] (diffraction experiments)
and
∼
2
.
5[
8
] (MD with hydrostatic pressure). In the region
between the center atom and first coordination shell, r
avg
−
r
1s
,
CN rises sharply due to the discrete nature of the atom counting
procedure, which bins the atoms according to their center
of mass position, providing no information on their physical
volume (i.e. from excluded volume interaction) and precluding
the counting of fractions of atoms. A continuous measure of
the CN that captures this missing structural information might
give a smooth, filled-in curve between the center atom and first
coordination shell and a more accurate estimate of short-range
dimension (refer to Section C). Between the outer radii of the
first and second coordination shells, r
1s
−
r
2s
, the dimension
crosses over to 3 for all cases, suggesting that these metallic
glasses have a correlation length of
ξ
∼
3
.
56 atom diameters,
FIG. 1. (a) Diagram of expected crossover in log-log plot of mass versus radius. Short-range fractal dimension d
f
crosses over to long-range
dimension d at the correlation length
ξ
. b) Radial distribution functions for Cu
46
Zr
54
(FF
2
) in the glass and liquid phase. Dashed lines indicate
positions for the first peak, r
1
, and coordination shells, r
is
,where
i
=
1–4.
024103-2
ORDERING AND DIMENSIONAL CROSSOVERS IN . . .
PHYSICAL REVIEW B
95
, 024103 (2017)
FIG. 2. Log-log plots of total atom number (CN
+
1) versus radius, r, showing local dimension in metallic glasses of Cu
46
Zr
54
(a) FF
1
,
(b) FF
2
, (c) Ni
80
Al
20
,(d)Ni
33
.
3
Zr
66
.
7
,and(e)Pd
82
Si
18
. Short-range dimension, d
=
2
.
71
±
0
.
04, is measured through a linear fit between the
radius of the center atom and the outer radius of the first coordination shell. Long-range dimension, d
=
3, is measured from a linear fit of
points beyond the outer radius of the second coordination shell.
which is slightly higher than previous estimates [
8
]. Here
ξ
is estimated using (r
1s
+
r
2s
)
/
2r
avg
[
23
]. Within the first to
second coordination shell, free volume arising from packing
inefficiencies contributes to a reduced dimensionality in the
structure. This reduced (
<
3) dimension cannot proliferate to
greater lengths because the free volume necessarily remains
smaller than the volume occupied by atoms, whose relative
positions are dictated by long-range attraction and low kinetic
energy. At longer length scales, where free volume is less
significant and the atom clusters appear closely packed, we
find that the dimension of the structure is 3.
B. Metallic liquids
Applying the same method to metallic liquids, we measure
d
=
2
.
57 for Cu
46
Zr
54
FF
1
at 2500 K, d
=
2
.
55 for FF
2
at 2000 K, d
=
2
.
48 for Ni
80
Al
20
at 3000 K, d
=
2
.
64
for Ni
33
.
3
Zr
66
.
7
at 2500 K, and d
=
2
.
53 for Pd
82
Si
18
at 2000 K (Fig.
3
) from r
avg
to r
1s
. These estimates are
dependent on temperature, as the position of r
1s
changes
due to thermal expansion (see Section D). The average
value of d
=
2
.
55
±
0
.
06 is in line with the value of
∼
2
.
52
from percolation theory [
5
] and is
∼
0
.
16 lower than the
average value in our metallic glasses. This difference in local
dimensions in liquid and glassy phases may be related to the
accumulation of dense ordered clusters, such as icosahedra,
across the glass transition, which pack more efficiently and
reduce local free volume [
3
,
24
,
25
]. A crossover in dimension
from d
<
3tod
=
3 occurs in roughly the same region as in the
metallic glasses, which suggests that the liquids are structurally
analogous to percolation structures with a correlation length
of
ξ
∼
3
.
68 atomic diameters, slightly higher than previous
suggestions [
8
]. In percolation theory, the correlation
length is inversely related to the atomic packing fraction
(
ξ
∝
(
φ
−
φ
c
)
−
ν
), and more loosely packed liquid structures
may exhibit longer
ξ
. Metallic liquids are dense, possessing
packing fractions of around
φ
∼
0
.
67 (FF
2
at 2000 K), a value
that is only
∼
8% lower than their glassy counterparts (
φ
∼
0
.
73 for FF
2
glass at 300 K). To observe structures with
ξ
∼
4
diameters or longer, we estimate that we would need to study
liquids and glasses with packing fractions in the neighborhood
of
φ
∼
0
.
5, which is not feasible for our metallic systems,
as a first-order phase transition to the gaseous phase would
likely precede such a low packing fraction in the liquid phase.
C. Comparison to copper and grid analysis
We compare our results to those for crystalline Cu at 300 K,
which has a dimension of 2.93 between the center atom and
024103-3
CHEN, AN, GODDARD III, AND GREER
PHYSICAL REVIEW B
95
, 024103 (2017)
FIG. 3. Log-log plots of total atom number (CN
+
1) versus radius, r, showing local dimension for metallic liquids of Cu
46
Zr
54
(a) FF
1
at
2500 K, (b) FF
2
at 2000 K, (c) Ni
80
Al
20
at 3000 K, (d) Ni
33
.
3
Zr
66
.
7
at 2500 K, and (e) Pd
82
Si
18
at 2000 K. Short-range dimension, d
=
2
.
55
±
0
.
06,
is measured through linear fit between the radius of the center atom and the outer radius of the first coordination shell. Long-range dimension,
d
=
3, is measured from a linear fit of points beyond the outer radius of the second coordination shell.
the minimum after the first peak (measured at the midpoint
between the first and second peak) [
23
]. Beyond the first peak,
the dimension is
∼
3 [Fig.
4(a)
]. We expect the long-range
crystal dimension to be exactly 3 owing to its close-packed
cubic structure. In the short-range, the crystal dimension
should be slightly less than 3, owing to finite-temperature
fluctuations and presence of defects.
Comparison of the crystalline (300 K), glassy (300 K),
and liquid (2500 K) phases of Cu shows that the major
contribution to
<
3 dimensionality in the liquid and glassy
phases is the short-range structure, which, due to fluctuations
in free volume, can be locally more open. The overall CN
curve is shifted toward higher radii for the liquid phase, which
reduces its short-range dimension. The short-range structure
in the glass phase appears denser and more ordered compared
to the liquid—the CN rises more steeply in the first shell,
increasing d towards a close-packed, crystalline value.
The discrete nature of our atom-counting procedure intro-
duces error into the estimates for local dimension and makes
the measurements of short-range dimension in these structures
delicate, as the fitting is performed over only two points.
This motivates a method to count the atoms continuously by
modeling them as spheres that occupy a volume based on their
atomic radii. For this purpose, we represented our Cu system
as points on a grid, which occupy the physical volume of each
Cu atom with a 0.3-
̊
A resolution [Fig.
4(b)
]. To generate the
grid, we impose a mesh onto the entire system with a specified
spacing. We select a grid spacing of 0.3
̊
A in order to optimize
spatial resolution while weighing computation time. We keep
the nodes on the mesh that lie within r
Cu
of the center of mass
of each Cu atom, where r
Cu
is the radius of Cu
,
∼
1
.
28
̊
A, and
we reject nodes that do not meet this criterion. The remaining
nodes are the grid points that occupy the physical space of our
Cu atoms. To perform the atom counting, we take the partial
RDFs of each atom center of mass position with respect to the
grid points and normalize by the average number of grids per
atom.
With the grid method, we find that the short-range di-
mension of the Cu liquid is 2.65 and that of the Cu glass
is 2.76 [Fig.
4(c)
]. We estimate
ξ
∼
3using
ξ
=
r
c
/
r
Cu
,
where r
c
∼
4
̊
A and r
Cu
=
1
.
278
̊
A, for both phases. The
correlation length is slightly longer for the liquid phase than
the glass phase, owing to a lower global density, which leads
to a longer r
c
. This lower global density (higher local free
volume concentration) also contributes to a lower short-range
dimension in the Cu liquid compared to the glass,
∼
2
.
65 vs.
024103-4
ORDERING AND DIMENSIONAL CROSSOVERS IN . . .
PHYSICAL REVIEW B
95
, 024103 (2017)
FIG. 4. Comparison of crossovers in pure Cu systems using discrete and grid counting methods. (a) d
∼
2
.
90 in Cu crystal (300 K),
d
∼
2
.
69 in Cu glass (300 K), and d
∼
2
.
51 in Cu liquid (2500 K) below
ξ
with CN counted by atom center positions. (b) Schematic of the grid
procedure. Cu atoms in the simulation box (left) are replaced by effective grid points representing their physical volume. Grid points capture
the overall atomic structure (see 1
̊
A slice, right). (c) Crossovers in dimension from d
∼
2
.
65 and d
∼
2
.
76 to d
∼
3 for Cu liquid and glass,
respectively, using a grid method for continuous counting. Here CN
grid
is the normalized coordination number based on counting grids within
each atom. Secondary axis (right side): d(ln(CN
grid
))
/
d(ln(r)) versus r showing a distinct crossover near
ξ
∼
8
̊
A.
2.76. This effect dominates over the averaging effect due to
temperature fluctuations, which may serve to increase local
dimension (see Section D about d
v
). Our observations on the
relative dimensions from the grid method of counting CN
corroborate those from the two-point analysis involving r
avg
and r
1s
. The estimates for local dimension are more accurate
in the grid analysis, as the fitting is performed over a longer
range of r, rather than two points. Even so, it is likely more
pertinent to compare the relative dimensions of identical
systems under various conditions rather than consider their
absolute values. This is evident from the observation that the
local dimensionality is r-dependent [see secondary-axis plot
in Fig.
4(c)
]. The local 1st derivatives of the ln(CN
grid
) plots,
d(ln(CN
grid
))
/
d(ln(r)) vs. r, show that the local dimensions of
these systems vary depending on the real-space region of the
structure. In the Cu glass, the local dimension is close to 3
in a narrow peak between r
avg
(1.278
̊
A) and r
1
(
∼
2
.
5
̊
A). In
the Cu liquid, this peak is lower and shifted toward longer r.
Interestingly, there is a stable real-space region between r
1
and
r
2
where the
<
3 dimensionality reliably occurs for both glassy
and liquid phases.
D. Temperature effects on atomic structure during quench
We examine the evolution of local dimensions within
real-space regions of interest in our Cu systems as a function
of temperature during quenching from the liquid state to
the glassy state (Fig.
5
). Each temperature snapshot is taken
via quenching from the immediately higher temperature. The
short-range dimension, d
s
, which we define heuristically as
ranging from
∼
1
.
2r
Cu
to
∼
3
.
2r
Cu
, increases roughly linearly
with decreasing temperatures. This is somewhat unexpected,
as the global volume change during cooling is linear in the
liquid and glassy regions, while strictly nonlinear near the
glass transition [
23
]. The short-range dimensional changes
indicated by d
s
do not reflect the same trend as that from the
global volume, showing instead a lack of an inflection point
near T
g
. This suggests that d
s
is mostly temperature-dependent
and is not sensitive to the glass transition. It also suggests
that the local dimension in a real-space segment within d
s
must be decreasing very rapidly near and beyond T
g
during
cooling. The valley in d(ln(CN
grid
))
/
d(ln(r)) versus r from
∼
r
1
to
∼
3
.
2r
Cu
, corresponding roughly to the center of the first
nearest neighbor to the edge of the second nearest neighbor, has
a local dimension, denoted d
v
, that is very sensitive to the glass
transition. The d
v
hovers around 2.6 at temperatures above
1500 K and dips abruptly below 2.55 on cooling past 1200 K,
close to the glass transition temperature of T
g
∼
1150 K. This
abrupt shift corresponds also to the appearance of a shoulder in
the first minimum, which indicates the development of ordered
clusters. The d
v
region relates to the amount of free volume in
the system around the first neighbor. The liquid phase has
a higher d
v
due to stronger thermal fluctuations occurring
at higher temperatures, which play an averaging role on the
local dimension. A reduction in d
v
indicates increased local
free volume just beyond the nearest neighbor, which suggests,
somewhat counterintuitively, increased local order via ordered
clusters. This is analogous to the development of interstitial
volume, which greatly decreases local density within a narrow
region in r during crystallization.
We consider the evolution of d
v
as a function of volume
fraction,
φ
, and the effects of global volume change on the local
dimensionality (Fig.
6
). We calculate
φ
using N
grid
V
grid
/
V
s
,
where N
grid
is the number of occupied grids in the system,
V
grid
is the volume of each grid voxel, and V
s
is the total
system volume. We observe an inflection point in d
v
versus
φ
around
φ
∼
0
.
64–0
.
66. Notably, this value corresponds to the
random close packed (RCP) value and the maximally random
024103-5
CHEN, AN, GODDARD III, AND GREER
PHYSICAL REVIEW B
95
, 024103 (2017)
FIG. 5. Temperature effects on local dimensions in Cu glass
and liquids using grid counting. (a) ln(CN
grid
) versus r and
d(ln(CN
grid
))
/
d(ln(r)) versus r plots at temperatures from 600–2100 K
at 300 K intervals. We define heuristically intervals for short-range
dimension, d
s
,
∼
1
.
2–3
.
2r
Cu
, and valley dimension, d
v
,
∼
2–3
.
2r
Cu
,
where the local dimension is largely temperature insensitive but
appears to be sensitive to the liquid/glass phase. r
Cu
=
1
.
28
̊
A.
(b) d
s
and d
v
versus temperature; d
s
increases roughly linearly
as temperature is decreased, and d
v
appears sensitive to the glass
transition (T
g
∼
1150 K).
FIG. 6. d
v
versus
φ
showing a connection between the maximally
random jammed (MRJ
,φ
∼
0
.
64) state and onset of T
g
.
jammed (MRJ) value in monodisperse hard spheres [
26
,
27
].
This inflection point also occurs close to the packing fraction
at the glass transition,
φ
∼
0
.
66. Other methods of calculating
φ
, such as taking
φ
=
N
Cu
V
Cu
/
V
s
, where N
Cu
and V
Cu
are,
respectively, the number and volume of Cu atoms, yield similar
results.
III. DISCUSSION
The glass transition may be related to the densifica-
tion/ordering that occurs in the local glass structure, but
the connection is not clear. Previous analyses comparing
amorphous and crystalline structures have emphasized that
radii ratios of
∼
0
.
6–0
.
95 in binary systems favor formation of
amorphous phases [
28
], and local icosahedral structure in the
first shell plays an important role in driving glass formation
for Cu-Zr-Al metallic glasses [
21
,
29
]. In our analysis, we
observe an increase in d from
∼
2
.
55–2
.
65 to
∼
2
.
71–2
.
76
from the liquid to glass phases, suggesting that some ordering
occurs across the glass transition in these metallic alloys
and metals. This ordering can be seen more clearly in the
grid analysis of Cu liquid and glass structures [Fig.
4(c)
],
where the two main observations are: 1) the short-range
dimension, d(ln(CN
grid
))
/
d(ln(r)) vs. r plot from
∼
1
.
5
̊
Ato
∼
4
̊
A, is d
Cuglass
≈
2
.
76 for the Cu glass,
∼
0
.
11 higher than
that of the liquid phase, which has d
Culiquid
≈
2
.
65; and 2)
d(ln(CN
grid
))
/
d(ln(r)) vs. r shows sharpening in the first peak of
the Cu glass, reaching a slope of around 3, indicating ordering
in the first nearest neighbors, and a shoulder appears near the
first minimum, indicating the development of ordered clusters.
Absolute changes in d,
∼
0
.
11–0
.
16, across the glass transition
are small, representing only a
∼
4–6% increase. However,
keeping in mind that the values for d are roughly constrained to
be from 2 to 3, as these structures occupy 3-dimensional space,
the relative changes in slopes are actually closer to
∼
20–30%.
The liquid-glass transition appears to be a universal phe-
nomenon in that any liquid can vitrify with sufficiently fast
cooling [
4
,
30
]. Diverging relaxation time and viscosity can
happen with or without accompanying structural changes. For
example, symptoms of the glass transition such as the jump
in heat capacity and logarithmic increase of T
g
with quench
rate can be explained without invoking phase transitions and
thermodynamics by considering that the systems stop relaxing
within the experimental timescale [
31
]. In these metallic
systems, the structural changes that appear across the glass
transition may be unique—other common glasses such as
covalent network glasses or molecular glasses have not yet
been studied in this way, although the methods presented
here can be extended to study those systems. Nonetheless,
the structural effects observed in this study on metallic
glasses may be instructive for a more general understanding
of the liquid-glass transition (refer to SI for additional
discussion).
The short-range dimension in our metallic glasses, d
∼
2
.
71–2
.
76, in contrast to the metallic liquids, deviates consid-
erably from percolation models, where the fractal dimension
is
∼
2
.
52. In simple percolation models, the constituent units
occupy lattice sites or are allowed to overlap one another [
5
]
such that no limit exists for the site occupancy probability
or volume fraction of overlapped spheres. In real systems
024103-6
ORDERING AND DIMENSIONAL CROSSOVERS IN . . .
PHYSICAL REVIEW B
95
, 024103 (2017)
and hard sphere percolation models, the constituent spherical
particles (e.g. metallic atoms) have excluded volume. This
gives rise to fundamental limits in the RCP fraction of hard
spheres, which is
φ
∼
0
.
637 for monodisperse spheres [
26
]
and
∼
0
.
64–0
.
83 for bi-disperse spheres, depending on their
radii ratios and compositions [
32
]. Stable binary metallic
glasses, while not perfectly represented by hard spheres,
have high packing fractions:
∼
0
.
73 for our Cu
46
Zr
54
(FF
2
)
and above
∼
0
.
7 for other binary alloys [
33
]. Interestingly,
our monatomic Cu system exhibits sensitive changes in d
v
near T
g
and at a volume packing fraction of
φ
∼
0
.
64. This
corresponds closely with RCP and MRJ states in monodisperse
hard spheres. The densification/ordering that occurs in these
systems at the atomic level may be due to the frustration
and jamming of the atoms, which approach and exceed the
maximal packing fractions allowed by the random packing
of spheres, arresting molecular motion. A similar idea has
been explored in granular materials; for example, Xia
et al.
found that polytetrahedra serve as structural elements to
glassy order in hard-sphere particle glasses, forming a globally
jammed fractal structure [
34
]. The mechanism for geometrical
constraint in our systems may be similar to ideas in jamming
or rigidity percolation [
35
,
36
].
IV. SUMMARY
We find that the cumulative CN analysis shows a crossover
in dimension for both metallic glasses and liquids. We observe
that the short-range dimension is less than 3, d
∼
2
.
55–2
.
71
for both liquids and glasses using two methods: 1) two-
point analysis from linear fit between r
avg
and r
1s
in binary
systems, and 2) grid analysis of continuously counting grid
points representing monatomic Cu systems. The long-range
dimension crosses over to 3 beyond the first coordination
shell. Analysis of the structural evolution during quenching
suggests that ordering develops across the glass transition
as short-range dimension increases roughly linearly with
decreasing temperatures. Observations of local dimensions
between
∼
2–3
.
2r
Cu
in Cu shows sensitivity to the glass
transition and a correlation with the packing fraction around
RCP and MRJ states, suggesting that densification during
cooling of metallic liquids may be arrested by fundamental
packing limits near the glass transition.
ACKNOWLEDGMENTS
The authors would like to acknowledge Jun Ding and Mark
Asta for pointing out the sensitivity in measuring precise
RDF peak positions. The authors gratefully acknowledge the
financial support of the U.S. Department of Energy, Office
of the Basic Energy Sciences (DOE-BES) under Grant DE-
SC0006599 and NASA’s Space Technology Research Grants
Program through J.R.G.’s Early Career grants. Parts of the
computations were carried out on the SHC computers (Caltech
Center for Advanced Computing Research) provided by the
Department of Energy National Nuclear Security Administra-
tion PSAAP project at Caltech (DE-FC52-08NA28613) and by
the National Science Foundation (NSF) DMR-0520565 CSEM
computer cluster. Q.A. and W.A.G. received support from NSF
(DMR-1436985). This material is based upon work supported
by the NSF Graduate Research Fellowship under Grant No.
DGE-1144469. Any opinion, findings, and conclusions or
recommendations expressed in the material are those of the
authors and do not necessarily reflect the views of the NSF.
APPENDIX: MOLECULAR DYNAMICS METHODS
All molecular dynamics simulations of the metallic liquids
and glasses discussed here used embedded atom model (EAM)
potentials:
(1) The Cu
46
Zr
54
systems (54 000 atoms) were prepared
using two potentials, Cheng
et al.
[
21
]. (FF
1
) and Mendelev
et al.
[
22
]. (FF
2
).
(2) The Ni
80
Al
20
systems (32 000 atoms) were prepared
using Pun
et al.
[
37
],
(3) The Ni
33
.
3
Zr
66
.
7
systems (32 000 atoms) were prepared
using Mendelev
et al.
[
38
], and
(4) The Pd
82
Si
18
systems (32 000) were prepared using
Ding
et al.
[
39
].
Cutoff distance: FF
1
–6
.
5
̊
A
,
FF
2
–7
.
6
̊
A
,
NiAl–6
.
3
̊
A
,
NiZr–
7
.
6
̊
A
,
Cu–7
.
6
̊
A, and PdSi–6
.
5
̊
A. Cooling procedure: cooled
from melt to room temperature over 1 000 000 ps (steps of
0.001 ps). Thermalization at the end of cooling: fixed NPT at
300 K and 0 Pa for 100 000 ps.
We selected four binary metallic glasses and liquids:
Cu
46
Zr
54
,Ni
80
Al
20
,Ni
33
.
3
Zr
66
.
7
, and Pd
82
Si
18
. Among these
four MGs, Cu-Zr, Ni-Zr, and Pd-Si belong to metal-metal
MGs, and Pd-Si belongs to metal-metalloid MGs. The binary
Cu-Zr and Pd-Si MGs have been synthesized in experi-
ments [
40
]. Although bulk metallic glasses have not been
formed in binary Ni-Zr and Ni-Al systems, they are interesting
to study in simulations because they have good (simulated)
glass forming ability [
41
,
42
].
In all cases, the binary metallic glasses were quenched
from the liquid phase (2000–3000 K) at a rate of
∼
10
12
K
/
sto
room temperature (300 K). The Cu crystal (13 500 atoms),
liquid (2048 atoms), and glass (2048 atoms) are prepared
from FF
2
. The Cu metallic glass was quenched at a rate of
∼
10
14
K
/
s.
For the grid analysis, we first mapped the whole space onto
grid sites on a cubic lattice with spacing
∼
0
.
3
̊
A. We remove
grid points outside the average radius of the atoms by marking
all of the grid points within one atomic radius from an atomic
center. The remaining grid points fill the excluded volume of
our systems. The total number of grid points is 657 545 for the
Cu glass and 642 106 for the Cu liquid.
The RDFs were calculated by binning the atomic structure
(100 000 bins for binary systems and Cu crystal, 5000 bins
for Cu liquid and glass). Coordination numbers are obtained
by integrating the total RDF. The CN
grid
value is taken from
the partial RDF from the Cu atom center positions to the
grid points. We normalize the final CN
grid
value by the
average number of grid points within each atom (CN
grid
at
r
=
r
avg
).
Supporting Information: coordination number dimension
analysis for Zr crystal, different RDF binning conditions, and
applied hydrostatic pressures (30 GPa).
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, 024103 (2017)
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