S
1
Supporting Information
for
First Order Phase Transition in
L
iquid Ag to the Heterogeneous G
-
phase
Qi An
1
, William L. Johnson
2
, Konrad Samwer
3
,
Sydney L. Corona
2
, and William A. Goddard
III
4
1
Department of Chemical and Materials Engineering,
University of Nevada
-
Reno, Reno, Nevada
89557, USA
2
Keck Engineering Laboratories, California Institute of Technology, Pasadena, CA 91125,
USA
3
I.
Physi
kalisches
Institut,
University of Goettingen, 37077 Goettingen, Germany
4
Materials and Process
Simulation Center, California Institute of Technology, Pasadena, CA
91125
,
USA.
Contents
Simulation Details
Supporting Tables
Table S1.
The HA index of G
-
phase and L
-
phase
at 800 K.
Supporting Figures
Figure S1. Two
-
phase simulation model (liquid + cryst
al) to determine the equilibrium melting
temperature (T
X,M
) of FCC Ag.
Figure S2. (A) The RDF analyses of various phases (L, X, G) extended to 2 nm
;
and
(B) The
structure factor of various phases (L, X, G)
for
L, G and X
phases
.
Figure S3. 30 multiple isothermal simulations at various temperatures: (A) 750 K, (B) 800 K, (C)
825 K, (D) 850 K, (E) 875 K.
Figure S4. Remelting of the glass phase (850 K) at various temperature.
S
2
Simulation
Details
We employed molec
ular dynamics (MD) simulations with
an
Ag
Embedded Atom Model
(
EAM
)
potential
1
to examine
liquid
quenching
and glass formation
in
elemental
Ag. All MD simulations
were performed using the Large
-
scale Atomic/Molecular Massively Parallel Simulator
(LAMMPS)
.
2
A
ll simulations
used the velocity Verlet algorithm for integrati
ng
the equations of
motion
with
a timestep of 1 fs.
P
eriodic boundary cond
itions were applied along all three directions
to avoid the surface effects.
To simulate the liquid to glass (or crystal) transition, we first created liquid structures by melting
FCC Ag single crystals at 2000 K using system sizes, N, of 32000, and 256,0
00 Ag atoms. Then
the liquid was fast quenched to room temperature at zero pressure with various quenching rates
from 3.4 × 10
11
to 3.4 × 10
12
K/second to produce either a glass or crystalline FCC
-
Ag. The glass
phase was remelted by heating from 300 K to 2
000 K with various heating rates from 3.4 × 10
11
to 1.7 × 10
12
K/second. The isothermal
–
isobaric (NPT) ensemble (constant temperature, constant
pressure, constant number of particles) was applied in all quenching and heating simulations to
adopt the total
energy and volume changes during phase transition. The Nose
-
Hoover thermostat
and barostat were used with the damping constants of 200 fs and 2000 fs for temperature and
pressure, respectively. The NPT ensemble was subsequently applied under isothermal con
ditions
to obtain
the thermodynamics properties (volume, potential energy, entropy, and free energy) of
various phases (liquid (L), glass (G), and crystal (X)) at various temperatures.
To obtain the equilibrium crystal melting temperature (T
X,
M
) and glass
melting temperature (T
G,M
),
we performed two
-
phase simulations by combining a liquid phase with
the
crystal (or glass) phase
,
as shown in Fig. S1 of the
supporting
information (SI).
The dimensions for the liquid structure
were adjusted to match the cross section of
the
crystal (or glass) phase before two phases were
combined. The two
-
phase model was equilibrated at various temperature
s
using the NPT
ensemble. When the temperature is
above T
X,M
(or T
G,M
), the potential energy increases, and the
two
-
phase model will eventually become
entirely
liquid.
By contrast,
the system become
s a
crystal
(or glass) and
the
potential energy decreases when the temperature is below T
X,M
(or T
G,M
).
To o
btain the glass (or crystal) nucleation time from the liquid, we performed multiple simulations
(30 runs) at each of various fixed temperatures of 750 K, 800 K, 825 K, 850 K, and 875 K,
respectively. All initial structures at a given temperature were obtai
ned from the quench simulation
with the fastest quench rate of 3.4 × 10
12
K/s. At fixed temperatures, additional statistics
simulations were performed with the same initial quenched liquid structure by assigning different
velocity distributions. All the ve
locity distributions were generated using the different random
seeds to
initiate
the Maxwell
-
Boltzmann distribution.
The Honeycut
-
Anderson (HA) analysis was applied to identify atomic types in X, L and G phases
.
3
An
atom was identified as the FCC if it has
12 pairs 1421 HA index
or HCP type
if it has
6 pairs
1421 and 6 pairs of 1422 HA index.
A
toms
w
ith other type of HA indexes were identified as liquid
atoms. We also identified the icosahedral atomic type using 12 pairs
with
1551 HA index.
However, very few icosahedral atoms (less than 0.1%) were found during the L
-
G, and L
-
X
S
3
transitions. The first minimum of
radial distribution function (
RDF
),
was applied as the cutoff
distance in the HA analysis.
To derive the elastic rigidi
ty, we computed the shear modulus of
the
glass and liquid using the
shear
stress
-
shear
strain relationship from finite shear deformations
. The strain rate used was 2.5
× 10
8
per second which is slow enough for a shear sound wave to cross the simulation box
multiple
times thereby achieving elastic equilibrium.
The canonical ensemble (NVT) was applied
in these
shear simulations.
To validate our Ag force field, we computed the elastic constants of single
-
crystal Ag at room
temperature by
measur
ing
the change
in average stress tensor
as
the cell volume undergoes a finite
deformation.
The NVT ensemble was used and strain was applied up to 2% for each finite
deformation. We performed a 10 ps equilibrium run before measuring the average stress tensor
over 30 ps at
room temperature.
References
(1)
Williams, P. L.
;
Mishin, Y.
;
Hamilton, J. C. An embedded
-
atom potential for the Cu
–
Ag system
. Model
.
Simul. Mater, Sci. Eng.
2006
,
14
, 817
–
833
.
(2)
Plimpton, S.
Fast parallel algorithms for short
-
range molecular dynamics.
J. Comp.
Phys.
1995
,
117
, 1
–
19
.
(3)
Honeycutt, J. D.
;
Andersen, H. C. Molecular dynamics study of melting and freezing
of small lennard
-
jones clusters.
J. Phys. Chem.
1987
,
91
, 4950
–
4963
.
S
4
Table S1
The HA index of G
-
phase and Liquid at 800 K. The HA index 1421 is characteristic of
FCC structure.
E
qual
amounts of 1421 and 1422 are characteristic of HCP structure. On the other
hand, the index 1551 indicate icosahedral packing characteristic of liquid or glassy metals; also
1331 and 1541 indices show the structure of liquid state.
G
-
phase (32000
atoms)
L
iquid (32000 atoms)
G
-
phase (256,000
atoms)
1551
0.43
%
19.03%
0.57%
1421
58.41
%
6.38%
60.8
1%
1422
13.79
%
8.45%
13.82%
1541
6.41
%
23.17%
8.05%
1311
8.13
%
3.20%
4.92%
Others
12.83
%
39.77%
11.83%
FCC+HCP ratio
(32.15+10.01)%
(1.97+1.37)%
(
35.87+13.04
) %
S
5
Supporting Figures
Figure S
1
. Two
-
phase simulation model (liquid + crystal) to determine the equilibrium melting
temperature (T
X,M
) of FCC Ag. (
A
) Potential energy curves at various temperatures, suggesting
that the T
X,M
is ~1250 K; (
B
) the snapshot of two
-
phase simulation cell at 1250 K in the end of
200 ps simulations. The arrows represent the interfaces between liquid and crystal. The L and X
represent liquid and crystal phases, respectively.
S
6
Figure S2. (A) The RDF analyses of various phases
(L, X, G)
extended to 2 nm
, and
(B) The
structure factor of various phases
(L, X, G)
.
S
7
Figure S
3
. 30 multiple isothermal simulations at various temperatures: (
A
) 750 K, (
B
) 800 K, (
C
)
825 K, (
D
) 850 K, (
E
) 875 K.
S
8
Figure S
4
. Remelt
ing of the
glass phase (850
K
) at various temperature. The glass is stable below
1100 K and will melt to liquid at 1200 K.