1
Ordering and d
imensional
crossover
s
in metallic
glass
es
and
liquids
David Z. Chen
1
,*
, Qi An
2
, William A. Goddard III
2
, Julia R. Greer
1,3
1
Division of Engineering and Applied Sciences,
California Institute of Technology, Pasadena CA, 91125, USA
2
Materials and Process Simulation Center,
California Institute of Technology, Pasadena CA, 91125, USA
3
The Kavli Nanoscience Institute, California Institute of Technology, Pasadena CA, 91125, USA
*Corresponding author, Email
: dzchen@caltech.edu
Abstract: The atomic
-
level structure
s
of liquids
and
glass
es are
amorphous, lacking long
-
range
order
.
W
e characterize
the
atomic
struc
t
ures
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
,
and
Pd
82
Si
18
.
Resulting cumulative coordination numbers (CN) show
that
metallic
liquid
s
ha
ve
a dimension
of
d
= 2.55
± 0.06
from the center atom to the
first
coordination shell
and
metal
lic
glass
es
ha
ve
d
=
2.71
± 0.04
, both less than 3.
Between
the
first
and second
coo
rdination shell
s
,
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 s
hort
-
range d = 2.65 for Cu
liquid
and d = 2.76 for Cu gla
ss
. Cu
grid
structures crossover to d = 3 at
ξ
~8
Å (
~
3 atomic
diameters).
We study the evolution of local structural dimensions during quenching and
discuss
its
correlation
with
the glass transition
phenomenon
.
KEYWORDS
:
Molecular dynamics,
dimension, meta
llic glass
, percolation
, glass transition,
jamming
2
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 visc
osity
over a short temperature range is not accompanied by significant changes in the long
-
range
atomic structure, wh
ich 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 tran
sition
.
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.
T
he 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 position
s
of
the first sharp
X
-
ray diffraction peaks
(q
1
)
to
sample
volume
(V)
,
several
groups have reported a
scaling
r
elationship in metallic glasses
,
with
exponent
,
d
~2.31
-
2.5
,
which
deviat
es
from
the d = 3
expected under
the assumption that
q
1
∝
1/a, whe
re a is the interatomic spacing
.
6
-
9
R
ecent
r
eal
-
space measurements on Ti
62
Cu
3
8
also revealed
a
dimensionality of roughly 2.5
.
10
E
xperiments on
electrostatically levitated metallic liquid
s
a
lso show a
non
-
cubic
power law exponent of
d
~2.28
,
11
albeit with a limi
ted 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, t
he 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
be
cause
macroscopic pores or voids are
absent
in
their microstructure
,
and such pores are
a
defining characteristic of fractals that maintain their scaling relationship
s over long ranges
(e.g.
the
S
ierpinski triangle)
.
In response to
this
inconsistency
,
Chen
et al.
proposed
that metallic glasses at the atomic
-
level
can be described
using
percolatio
n
,
8
a model that
captures
the interconnectivity of sites on a
lattice or spheres
in a continuum
.
5
T
hree
-
dimensional
percolation models
,
such
as
hard sphere
and overlapping sp
here continuum models
,
exhibit
a fractal dimension
of
d~2.5
2
at lengths
below a
correlation length,
ξ
,
and
a crossover to
a dimension d~3 above
ξ
, where
ξ
is roughly the
3
diameter/length of finite, non
-
percolating clusters
.
5
Using molecular dynamics (MD)
simulations, Chen et al.
f
ound
that
two
distinct
m
etallic glasses
have short
-
range
dimension
s of
d
~2.5
below
ξ
~2 atomic diam
eters
and
a
dimension
of 3
occurs
over longer lengths
.
This
suggest
ed
that metallic glasses are structurally similar to a
continuum percolation
(i.e.
spatiall
y
-
random coalescence)
of spherical particles
.
8
This crossover at
ξ
may expla
in
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 perco
lation structure
and glasses has
also been suggested by
Orbach
, who
applied percolation theory to describe high
frequency (short
wave
length) 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
<
3 to d =
3
.
P
ercolation
structure
has been studied in hard spheres,
14
,
15
overlapping spheres,
16
,
17
and
recently
metallic glasses,
8
sug
gesting
a
possible
connection to
metallic
liquid
s
, 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 function of temperature, across the glass transition.
One
p
revious
method
to measure
dimension
utilize
d
hydrostatic pressures to induce peak shifts in
radial d
istribution functions (RDF) that
we
re
compared
to corresponding volume changes.
8
However, t
his pressure
-
induced
peak
shift
method
is
not
well suited
for
studying
liquid
s,
in
which atomic rearrangement
and exchange
of
neighbors
leads to significant structural
changes
under pressure. T
he c
orrelation lengths,
ξ
, can only be inferred based on the scaling of various
peaks
.
Moreover,
the broadness
of
the
RDF
peak
s
leads to
results that are
sensitiv
e
to
the specific
method of
generati
ng
and measur
ing the RDF
.
19
T
o overcome
these
challenges
, we
integrat
e
d
the
RDFs to obtain cumulative coordination numbers (CN). This i
ntegral method
estimates
the local
dimension of the structure without
the need for
applying
hydrostatic pressure
s
or
measuring
small shifts in
broad amorphous peak positions
,
which are
methods
that we
re
used previously
.
8
With this
CN
analysis
, w
e observe
a crossover in dimension from d
= 2.55
± 0.06
in
metallic
liquids
and
d
= 2.71
± 0.04 in metallic
glasses
, to d = 3
for
the second coordination shell
and
beyond
,
suggesting that
ξ
~3
atomic diameters
.
II.
DIMENSION
AND CROSSOVER
4
One measure of
dimension
come
s from
the scaling of extens
ive 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, t
he scaling relationship for a
system
above the percolation
threshold,
φ
!
, exhibi
ts a crossover in dimension
from d~2.52 to d~3 at
ξ
, where
ξ
∝
(
φ
−
φ
!
)
!
!
.
5
The parameter definitions are:
φ
is
the packing fraction,
ν
= 0.8764 is the critical
exponent for the correlation length,
20
and
φ
!
is the percolation threshold in 3
-
dimensional
continuum percolation.
5
From percolation theory, t
he 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
suggest
s
that the
crossover occurs
around
the first
atomic
coordination shell.
To avoid inaccuracies
that may
ari
se
from determi
ning precise
peak
shifts in
broad amorphous peaks
, w
e obtain the dimension of each
at
omic structure by measuring
d(ln(CN
gri
d
))/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
<
3
to d
=
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
. The r
avg
is
defined as
the average
r
adii
of the at
oms
in the binary systems (i.e.
for Cu
46
Zr
54
, r
avg
= 0.46r
Cu
+ 0.54
r
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
f
or
Pd
82
Si
18
, and d
= 2.74
or
2.
73
for Cu
46
Zr
54
using FF
1
21
or FF
2
22
, respectively
(Figure 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 p
ercolation
theory
, where d~2.52,
5
and
is
higher than previo
us
estimates
of ~2.3
-
2.5
6
,
7
(diffraction experiments) and ~2.5
8
(molecular dynamics
with hydrostatic pressure
).
In the reg
ion
5
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
volu
me
(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, fill
ed
-
in curve between the
center atom and first coordinati
on 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
,
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 contribute
s
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 at
om clusters appear closely packed,
we find that
the
dimension of the structure is 3.
Figure
1
.
a)
Diagram
of e
xpected crossover in log
-
log plot of mass versus radius.
Short
-
range
fractal dimension d
f
crosse
s over to long
-
range di
mension 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
.
a
b
6
Figure
2
.
Log
-
log
plots of total atom number (CN+1) vers
us 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.
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
(Figu
re 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
~0.1
6
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
a
b
c
e
d
7
volume.
3
,
24
,
25
A crossover in dimension from d
< 3
to d
= 3
occurs in
roughly
the same region as
in the metallic glasses,
which
suggests
that the liquids are structu
rally analogous to
percolation
structures with a correlation length
of
ξ
~3.6
8
atomic diameters
,
slightly higher than
previous
suggestions
.
8
In percolation theory, t
he correlation length is inversely related to
the atomic
packing fraction
(
ξ
∝
(
φ
−
φ
!
)
!
!
)
, and mor
e loosely packed liquid structures may exhibit longer
ξ
.
M
etallic 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).
T
o
observe st
ructures 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
.
Figure
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
a
b
c
e
d
8
coordination shell. Long
-
range dimension
,
d
= 3,
is measured from a linear fit of points beyond
the outer radius of the second coordination shell.
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 the minimum after the first peak
(
measure
d
at
the midpoint
between
the first and second peak
)
.
23
Beyond the
first peak, the dimension is ~3 (Figure 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
fluctu
ations 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 fluc
tuations in free volume,
can be
locally more open
.
The
overall coordination number curve is shifted toward higher radi
i
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 coordination number rises more steeply in the first
shell, increasing d towards a close
-
packed, crystalline value.
The discrete nature of our atom
-
counting procedure introduces error into the estimates for local
dimens
ion and makes the
measurements of
short
-
range dimension in these structures
delicate
, as
the fitting is performed over only two points.
This
motivate
s
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
-
Å
resolution
(Figure 4b)
.
To generate the grid, we impose a
mesh onto the entire system with a
specified spacing.
We
select
a
grid spacing of
0.3 Å
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 Å,
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 normali
ze
by the
average
number of grids per at
om
.
9
With the grid method, we find that the short
-
range
dimension of
t
he Cu liquid is 2.65 and that
of
the Cu glass is 2.76
(Figure 4c)
.
We estimate
ξ
~3 using
ξ
=
r
c
/r
Cu
, where r
c
~4
Å
and r
Cu
= 1.278
Å, 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 contribut
es to a lower short
-
range dimension in the
Cu liquid compared to the glass, ~2.65 vs. 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 observati
ons
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 ran
ge
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 the
ir
absolute
value
s
. This is evident from the observation that the local dimension
ality is r
-
dependent (see
secondary
-
axis plot in
Figure 4c). The local 1
st
derivatives of the ln(CN
grid
) plots
,
d(ln(CN
grid
))/d
(ln(
r
))
vs. r,
show that the local di
mensions of these systems vary
depending on the
real
-
space region of the structure.
I
n the Cu glass, the local dimension is
close to
3 in a narrow
peak
between r
avg
(1.278 Å) and r
1
(~2.5 Å).
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.