Observation of an apparent first-order glass transition
in ultrafragile Pt
–
Cu
–
P bulk metallic glasses
Jong H. Na
a
, Sydney L. Corona
b
, Andrew Hoff
b
, and William L. Johnson
a,b,1
a
Glassimetal Technologies, Pasadena, CA 91107; and
b
Keck Laboratory of Engineering, Department of Applied Physics and Materials Science, California
Institute of Technology, Pasadena, CA 91125
Contributed by William L. Johnson, December 15, 2019 (sent for review September 20, 2019; reviewed by John H. Perepezko and Frans Spaepen)
An experimental study of the configurational thermodynamics for
a series of near-eutectic Pt
80-
x
Cu
x
P
20
bulk metallic glass-forming
alloys is reported where 14
<
x
<
27. The undercooled liquid alloys
exhibit very high fragility that increases as
x
decreases, resulting in
an increasingly sharp glass transition. With decreasing
x
, the ex-
trapolated Kauzmann temperature of the liquid,
T
K
, becomes in-
distinguishable from the conventionally defined glass transition
temperature,
T
g
. For
x
<
17, the observed liquid configurational
enthalpy vs.
T
displays a marked discontinuous drop or latent heat
at a well-defined freezing temperature,
T
gm
. The entropy drop for
this first-order liquid/glass transition is approximately two-thirds
of the entropy of fusion of the crystallized eutectic alloy. Below
T
gm
, the configurational entropy of the frozen glass continues to
fall rapidly, approaching that of the crystallized eutectic solid in
the low T limit. The so-called Kauzmann paradox, with negative
liquid entropy (vs. the crystalline state), is averted and the liquid
configurational entropy appears to comply with the third law of
thermodynamics. Despite their ultrafragile character, the liquids at
x
=
14 and 16 are bulk glass formers, yielding fully glassy rods up
to 2- and 3-mm diameter on water quenching in thin-wall silica
tubes. The low Cu content alloys are definitive examples of glasses
that exhibit first-order melting.
glass transition
|
metallic glass
|
melting
|
phase transition
A
liquid near its glass transition is a metastable state of matter
that ultimately crystallizes given adequate time to explore its
available configurational phase space. Crystallization is triggered
by a relatively improbable fluctuation whereby the liquid crosses
a crystal nucleation barrier. If the time required for this improb-
able fluctuation sufficiently exceeds the time required for the liq-
uid to explore its available noncrystalline configurations, one may
define a specific metastable configurational entropy for the liquid
as
s
C
=
k
B
ln(
W
C
), where
W
C
enumerates the noncrystalline con-
figurations per atom available. Stable liquid configurations or in-
herent states are defined as atomic arrangements that minimize
the specific liquid configurational potential energy per atom,
φ
.
Interpreting
W
C
(
φ
) as a density of inherent states (per atom) vs.
φ
leads to
s
C
(
φ
)
=
k
B
ln
W
C
(
φ
) as the definition of liquid configu-
rational entropy. Since the early work of Goldstein (1), this picture
has been the basis for liquid theories based on the potential
energy landscape (PEL) approach (2
–
7). PEL theory maintains
that thermodynamic state functions (e.g., entropy, enthalpy, free
energy, etc.) of a liquid are approximately separable into addi-
tive configurational and vibrational contributions that are weakly
coupled via anharmonic effects (2, 3).
Defining and/or measuring
s
C
(
φ
)or
s
C
(
T
) requires that the
liquid explore all noncrystalline inherent states of given
φ
over
the relevant experimental timescale. In short, the liquid should
behave ergodically within the noncrystalline configurational
phase space. Crystallization of a metastable undercooled liquid
always limits the available experimental configurational equilibra-
tion time. The transformation to a fully crystallized sample involves
both crystal nucleation and growth. The former is statistical, and the
latter is often kinetically sluggish (8), so that progression of the
liquid
–
crystal transformation is o
ften spread over timescales
spanning orders of magnitude (9). To further complicate mea-
surements of
s
C
(
T
), liquids near
T
g
relax toward equilibrium
following
“
stretched
”
exponential behavior (10), where
Δ
s
C
(
t
)
∼
exp[
−
(
t/
τ
)
β
] with a stretching exponent
β
<
1. Typically,
β
∼
1/2
or less for fragile undercooled liquids. Achieving metastable
configurational equilibrium may therefore require times exceeding
the Maxwell relaxation time
τ
Μ
by orders of magnitude. Configu-
rational relaxation involves a spectrum of activated processes that
include a slow
α
-relaxation and a fast
β
-relaxation. These appear to
merge into a single process above a dynamic merging or crossover
temperature,
T
C
(11, 12).
In this report, we present an experimental study of the config-
urational thermodynamic state functions for a series of high-
fragility Pt
80-
x
Cu
x
P
20
bulk metallic glass-forming liquids (13). The
bulk metallic glasses form along a binary eutectic line that termi-
nates at a ternary eutectic composition. The eutectic solid is a
mixture of three crystalline phases, each having a distinct com-
position that differs from that of the liquid. We use this multiphase
crystallized eutectic solid as a well-defined thermodynamic refer-
ence state. The crystalline eutectic solid exhibits a sharp melting
transition to a homogeneous single-phase liquid at the eutectic
composition and temperature,
c
E
and
T
E
, with a well-defined en-
thalpy and entropy of fusion.
For the metallic glasses investigated, crystal nucleation from
the undercooled liquid is a transient process whereby the observed
nucleation rate rises very steeply following extended configura-
tional relaxation of the liquid over a relatively long incubation time
t
LX
(
T
). Nucleation is followed by rapid coupled eutectic growth
where the liquid-to-crystal transformation is completed in a rela-
tively shorter timescale,
Δ
t
LX
<<
t
LX
(
T
). We show that
t
LX
(
T
)
exceeds the configurational
α
-relaxation time (Maxwell re-
laxation time),
τ
α
, of the liquid by orders of magnitude; typi-
cally,
t
LX
(
T
)/
τ
α
∼
10
3
to 10
6
. This ratio quantifies the extent of
Significance
The glass transition is ubiquitous among materials and is com-
monly characterized by the rapid kinetic arrest of atomic or
molecular rearrangements in a liquid as temperature is lowered.
The possible existence of an underlying thermodynamic phase
transition has been a subject of continuing debate. Here, we
present compelling evidence that the glass transition evolves
into a first-order melting transition in the limit of very high-
fragility metallic glass-forming liquids. These Pt-rich glasses melt
in a discontinuous manner similar to melting of a crystal.
Author contributions: J.H.N. and W.L.J. designed research; J.H.N., S.L.C., and A.H. per-
formed research; J.H.N., S.L.C., A.H., and W.L.J. analyzed data; and J.H.N., S.L.C., and
W.L.J. wrote the paper.
Reviewers: J.H.P., University of Wisconsin
–
Madison; and F.S., Harvard University.
The authors declare no competing interest.
Published under the
PNAS license
.
1
To whom correspondence may be addressed. Email: wlj@caltech.edu.
This article contains supporting information online at
https://www.pnas.org/lookup/suppl/
doi:10.1073/pnas.1916371117/-/DCSupplemental
.
First published January 28, 2020.
www.pnas.org/cgi/doi/10.1073/pnas.1916371117
PNAS
|
February 11, 2020
|
vol. 117
|
no. 6
|
2779
–
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APPLIED PHYSICAL
SCIENCES
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configurational equilibration of the liquid prior to the de-
tectable onset of crystallization, and thereby the extent to which
the liquid achieves metastable configurational equilibrium. The
temporally sharp release of enthalpy associated with the short
Δ
t
LX
permits an accurate calorimetric determination of the total heat of
crystallization at ambient pressure
P
0
and fixed
T
,
h
LX
(
T
,
P
0
,
c
E
).
We argue that
h
LX
(
T
,
P
0
,
c
E
) is an accurate measure of the liquid
configurational enthalpy, i.e., that
h
C
(
T
,
P
0
,
c
E
)
∼
h
LX
(
T
,
P
0
,
c
E
). This
follows since the difference in vibrational enthalpy and entropy
between the liquid and crystallized samples is observed to be quite
small and essentially neglectable i
n bulk metallic glasses as reported
recently for Cu
–
Zr and Al
–
Cu
–
Zr bulk glasses (14). To simplify
notation, we henceforth drop ambient pressure
P
0
and liquid com-
position
c
E
as independent thermodynamic variables, since both are
fixed in each of our experiments, and report
h
LX
(
T
) data.
It is common in the glass literature to measure the configura-
tional contribution to the liquid heat capacity. For metallic glasses,
this is commonly assumed to be well approximated by difference in
liquid and crystal heat capacity
c
LX
(
T
)(15
–
17). This heat capacity
difference is often assumed to follow a
“
1/
T
2
”
temperature de-
pendence as originally proposed by Kubaschewski et al. (18). In
PEL theory, this
T
dependence follows directly from assuming a
Gaussian distribution of inherent states,
W
C
(
φ
)(2
–
4). By inte-
grating
dh
LX
/
T
=
c
LX
(
T
)
dT
/
T
, one obtains a specific configura-
tional enthalpy that follows a
“
1/T
”
temperature dependence
with an integration constant
h
LX
ð
∞
Þ
representing
h
LX
(
T
) in the
limit
T
→
∞
. We describe our data using a more general em-
pirical form for the liquid configurational enthalpy:
h
C
ð
T
Þ
h
C
ð
∞
Þ
=
1
−
θ
h
T
n
,
[1]
where
θ
h
is an alloy-dependent characteristic
“
isenthalpic
”
tem-
perature and further introduce the exponent
n
, which takes the
value
n
=
1 for the special case of a Gaussian distribution of
W
(
φ
). We refer to
n
as a thermodynamic fragility index. As
shown in
SI Appendix
, section B), Eq.
1
is equivalent to a micro-
canonical configurational entropy
s
C
(
φ
)
=
k
B
ln[
W
C
(
φ
)] of the
following form:
s
C
ð
φ
Þ
∝
s
C
ð
φ
0
Þ
−
C
ð
φ
0
−
φ
Þ
n
+
1
n
,
[2]
where
C
is a normalization constant and
φ
0
is the limiting value
of configurational potential energy in the high
T
limit. For
n
=
1,
the expression reduces to a Gaussian distribution for
W
C
(
φ
)(2
–
7).
Fitting our data yields
n
values spanning a surprisingly broad
range, 4
<
n
K
50, in stark contrast to the prediction
n
=
1as-
sociated with a Gaussian distribution. For the ternary Pt
80-
x
Cu
x
P
20
alloys, the experimental
n
values increase systematically with
decreasing Cu content from
n
∼
4at
x
=
27 to
n
>
13 for
x
≤
20.
The exponent
n
appears to diverge and changes sign at the lowest
Cu contents,
x
=
14 and 16. For such large
n
, the normalized
configurational enthalpy,
h
C
(
T
)/
h
C
(
∞
)inEq.
1
, approaches a
Heaviside step function that abruptly jumps from 0 to 1 on
passing through
T
∼
θ
h
. The glass transition approaches a first-
order phase transition.
From the fits to
h
C
(
T
)
∼
h
LX
(
T
) using Eq.
1
, we compute the
specific liquid configurational entropy,
s
C
(
T
), by integration. For
the Pt
–
Cu
–
P alloys, we find
s
C
(
T
)
→
0, at a
“
Kauzmann
”
tem-
perature (19),
T
K
, that lies near or even slightly above the labo-
ratory glass transition. While the configurational entropy at
T
K
vanishes, the configurational enthalpy (relative to the crystallized
eutectic solid) remains finite. This residual enthalpy,
h
R
=
h
C
(
T
K
),
isfoundtobe30to38%of
h
C
(
∞
) over the alloy series. For
x
=
14
and 16, the bulk glasses display a distinct latent heat and a dis-
continuous first-order transition at an apparent melting tempera-
ture
T
gm
. In addition to first-order melting, the
x
=
14 and 16
glasses display 1) ultrahigh Angell fragility parameter (16)
m
>
90,
and non-Newtonian viscosity behavior at very low strain rates
(
∼
10
−
6
s
−
1
); and 2) a systematic reduction in crystal nucleation
time
t
LX
(
T
) as strain rate increases. Despite their ultrahigh
fragility, the alloys at
x
=
14 and 16 form bulk glasses (
>
2-mm
diameter glassy rods) if quenc
hed under relatively quiescent
conditions.
Results and Analysis
Table 1 lists the alloy compositions studied and summarizes
important properties of the Pt
80-
x
Cu
x
P
20
bulk glass-forming al-
loys. The crystalline alloys exhibit sharp near-eutectic melting.
Instrumental broadening of the melting transition depends on
the heating rate used in the differential scanning calorimetry
(DSC) melting scan. This rate dependence was quantified by
observing the melting transition of a pure metal (e.g., Sn) as
discussed in
SI Appendix
, Fig. S7
. Table 1 gives the liquidus and
solidus temperatures at the lowest heating rate used for a given
alloy. Based on the melting data, the composition
x
=
23 appears
to be a ternary eutectic composition with a melting transition of
width
≤
4 K at a heating rate of 0.2 K/m, essentially indistin-
guishable from that observed for pure Sn. The binary Pt
–
Psystem
displays a eutectic at the composition Pt
80
P
20
with
T
E
=
588 °C
(20). Substitution of Cu for Pt results in a binary eutectic line in the
ternary system that appears to terminate at the ternary eutectic
composition Pt
57
Cu
23
P
20
,where
T
E
∼
548 °C. The DSC melting
curves taken at a commonly used heating rate of 20 K/m are shown
in
SI Appendix
, Fig. S1
.
The undercooled liquid alloys all display a single very sharp
crystallization peak in both isothermal and constant heating rate
DSC scans done throughout the temperature range from
T
g
to
T
E
. The observed glass-forming ability vs. Cu content (13) is
included in
SI Appendix
, section A and Fig. S2
. The bulk glasses
at all compositions display an extended metastable liquid region
with
T
X
−
T
g
>
50 K. Table 1 lists the calorimetric onset of the
glass transition at 10 K/min, the measured total heat of fusion
(averaged over five or more measurements) at the indicated
heating rates, the measured melting onset temperature (solidus)
T
S
, and melting completion temperature (liquidus)
T
L
, both at
the lowest heating rate used. Finally, the table lists the critical
casting thickness,
d
C
, for rods of bulk glass determined by water
quenching from the high temperature melt in a thin walled (1 mm)
silica tube (21, 22). The
d
C
values provide one measure of alloy
glass-forming ability that can be related to
t
LX
(
T
nose
); the mini-
mum crystallization time at the
“
nose
”
of the TTT-diagram (21,
22) is discussed in ref. 21.
Fig. 1
A
shows isothermal DSC scan
s at different tempera-
tures above
T
g
for the most extensively studied ternary eutectic
Pt
57
Cu
23
P
20
alloy. The scans were obtained using disks (
∼
60 to
80 mg) cut from a 2-mm glassy rod prepared by water quenching.
To bring the as-cast glass samples to a well-defined initial state,
all samples were first equilibrated by preannealing in the DSC
at the calorimetric onset
T
g
(
∼
230°C)for15h.At
T
g
,thetime
for the onset of crystallization is observed to be
t
LX
(
T
)
>
10
6
s
(
∼
1 wk). Following the preanneal, each sample was slowly
heated at 0.5 K/min from
T
g
to a designated holding temper-
ature
T
. Crystallization is indicated in Fig. 1
A
by a single sharp
exothermic peak in the isothermal scans as plotted on a loga-
rithmic timescale. When the exothermic peak is normalized by
its height (in watts per gram) and the curves at different tem-
peratures are shifted to align at the peak time,
t
peak
,asingle
peak shape vs. log(
t
/
t
peak
)isseen(Fig.1
B
). The plot illustrates
key features of the liquid/crystal transformation: 1) there is a
long incubation period
t
LX
(
T
)
∼
10
4
to 10
6
s preceding the de-
tectable onset of exothermic heat release, 2) crystallization emerges
abruptly with an exothermic signal
rising sharply to a peak and then
decaying rapidly, and 3) crystallization is completed over a rel-
atively short time
Δ
t
LX
(
T
)
<<
t
LX
(
T
). The peak time
t
peak
(
T
)and
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Na et al.
Downloaded at California Institute of Technology on February 11, 2020
transformation time
Δ
t
LX
follow simple scaling behavior, implying a
single crystallization mechanism. Fig. 1
C
shows an Arrhenius plot
of log(
t
peak
)vs.1/
T
, essentially a TTT-diagram for crystallization.
By extrapolating the data for log(
t
peak
) vs. 1/
T
to the apparent
nucleation
“
nose
”
temperature of the TTT-diagram (
∼
350 °C),
one estimates a
“
nose time
”
of
∼
1 s. Here, nose time refers to the
minimum time to crystal nucleation in the C-shaped TTT-
diagram as described in ref. 22. This extrapolated time roughly
agrees with that inferred from the critical casting thickness of the
glass,
d
C
=
15 mm (Table 1) of
∼
1 s (22). To assess the extent of
liquid configurational relaxation prior to crystallization, we carried
out viscosity measurements. The viscosity data are provided in
SI
Appendix
,Fig.S3
. We used the viscosity data to estimate a Max-
well configurational relaxation time for the undercooled liquid,
τ
α
=
η
(
T
)/
G
, where
G
∼
32 GPa is the shear modulus of the glass
as measured ultrasonically. Fig. 1
C
compares log(
τ
α
) with log(
t
peak
)
and shows that the ratio
t
peak
/
τ
α
ranges from 10
3
to 10
5
as T in-
creases above
T
g
. This ratio quantifies the extent of liquid re-
laxation toward metastable configurational equilibrium prior to
crystallization. It establishes that the measured
h
LX
(
T
)isrepre-
sentative of a configurationally relaxed undercooled liquid in
metastable equilibrium.
To ensure the liquid/crystal transformation is completed (no
glass remains) following the isothermal segment, the samples
were subsequently heated (at 5 K/m
in) through the melting tran-
sition. For Pt
57
Cu
23
P
20
, a relatively small heat release (
∼
4 J/g) is
consistently observed during heating in the range of 320 to
340 °C. This is believed to result fr
om relaxation and/or coarsening
of the initially crystallized structure. No other calorimetric events
are observed through the eutectic melting transition. The heat of
fusion was measured during each heating segment and found to
have an average value of 68.5
±
1 J/g.
High-resolution scanning electron microscopy (SEM) and
chemical mapping were used to determine that the as-cast glass
was chemically homogeneous on the length scale of nanometers
and lacked any observable microstructure as shown in
SI Ap-
pendix
, Fig. S5
. X-ray diffraction was used to establish that the
as-quenched, preannealed (at
T
g
for 15 h), and isothermally held
sample prior to the exothermic event, were all fully amorphous.
A series of diffraction scans on a single 3-mm diameter disk
sample were taken at various steps in the thermal history as
shown in Fig. 1
D
for a sample isothermally crystallized at 240 °C.
The X-ray sample was processed in the DSC to ensure its ther-
mal history was identical to that of the other calorimetric sam-
ples. The DSC segment scans were interrupted at each respective
step and the sample cooled to room temperature for X-ray. The
diffraction scans indicate a fully amorphous structure for the
initial as-cast sample (scan 1), the preannealed state for 15 h at
230 °C (scan 2), and the isothermally equilibrated sample at
240 °C following a holding time of 6 h and prior to the onset of
the crystallization event (scan 3). Following the crystallization
event after 26 h at 240 °C, the X-ray scan indicates a fully
crystallized alloy (scan 4). After subsequent heating to 350 °C
and holding for 2 h (scan 5), the pattern is essentially identical
with scan 4. The crystalline phases produced during the exo-
thermic event at 240 °C (scan 4) are unchanged on heating to
higher temperature and following the small heat release at 320 to
340 °C. Since no new diffraction peaks are associated with the
small exothermic heat release at 320 to 340 °C, the heat release
in the isothermal segment at 240 °C corresponds to a trans-
formation from a homogeneous and equilibrated undercooled
liquid to a fully crystallized sample.
The total measured heat of crystallization during all iso-
thermal segments (including the small excess contribution at
∼
320 to 340 °C) are compiled for reference in
SI Appendix
, Table
S1. For temperatures above
∼
270 °C, it was not possible to carry
out accurate isothermal measurements of
h
LX
(
T
)as
t
LX
(
T
)
becomes too short to permit stabilization of the DSC control
loop prior to the onset of crystallization. Data above 270 °C were
acquired by preannealing the sample at 230 °C (15 h), then
continuous heating at varying rates ranging from 0.1 to 10 K/min.
These constant heating scans also exhibit a single very sharp
crystallization event as shown in Fig. 1
E
. With increasing heating
rate, crystallization occurs at progressively higher temperatures
and the small exothermic heat release between 320 and 340 °C
disappears, apparently merging with the single sharp exothermic
event at heating rates above 1 K/min. No other detectible heat
release is observed during continued heating through the eu-
tectic melting transition. To quantify the width of the crystalli-
zation event, consider the difference between the onset and the
completion temperatures. At low heating rates (below 5 K/min),
the peak width is of order 2 K. The crystallization peak tempera-
ture thus accurately reflects the temperature at which the heat
h
LX
(
T
) is released. At the highest heating rates, the exothermic
peak becomes broadened and slightly shifted by instrumental ef-
fectsasdescribedin
SI Appendix
, section F and Fig. S7
. Corrections
for this rate-dependent peak shift are listed there (
SI Appendix
,
Table S1
). The constant heating rate data were combined with the
isothermal data to give an
h
LX
(
T
) curve over a broader range of
T
(from 230 °C up to
∼
290 °C) as displayed in Fig. 2
A
.
Fig. 2
A
also shows a third set of DSC results obtained by
constant cooling the equilibrium melt from 900 °C, far above the
liquidus temperature, and into the undercooled region below
T
E
(543 °C) at 10 K/min. The undercooling results were obtained
using a single sample of Pt
57
Cu
23
P
20
cycled 40 times between 900
and 200 °C at a fixed cooling/heating rate of 10 K/min. Examples
of these cyclic DSC undercooling scans are provided in
SI Ap-
pendix
, section C and Fig. S4
. The sample exhibited increased
undercooling with increasing number of cooling
–
reheating cycles.
This is presumed due to dissolution of heterogeneous nucleants in
the melt following repeated remelting and overheating to 900 °C.
Each cycle represents roughly 2 h in
real time for a total processing
time (40 cycles) of roughly 3 d. The undercooling samples were
unfluxed. Following repeated cycling, maximum liquid undercooling
Table 1. Summary of basic properties of Pt
80-
x
Cu
x
P
20
bulk metallic glass-forming alloys in the present study
Alloy, at.%
T
g
, K, onset DSC
T
S
, K, solidus
(rate, K/m)
T
L
, K, liquidus
(rate, K/m)
Heat of
fusion, J/g
Critical glass
thickness,
d
C
,mm
Pt
66
Cu
14
P
20
504
828.9 (1 K/m)
879.4 (1 K/m)
67.7
1.5
–
2*
Pt
64
Cu
16
P
20
504
827.8 (1 K/m)
877.5 (1 K/m)
68.7
3
Pt
62
Cu
18
P
20
505
816.9 (0.5 K/m) 864.1 (0.5 K/m)
71.8
4
Pt
60
Cu
20
P
20
505
825.6 (1 K/m)
833.7 (1 K/m)
67.8
6
–
7
Pt
57
Cu
23
P
20 eutectic composition
505
823.8 (0.2 K/m) 827.9 (0.2 K/m)
68.5
15
Pt
53
Cu
27
P
20
506
819.5 (0.1 K/m) 840.1 (0.1 K/m)
67.5
27
Table includes onset of laboratory
T
g
at 10 K/m, melting data for solidus
T
S
and liquidus
T
L
temperatures at the indicated
heating rates, heat of fusion, and the critical casting thickness (maximum rod diameter) for glass formation by water quenching
in thin-wall silica tubes.
*Glass formed when quenched without mechanical agitation (see text) in a thin-wall silica capillary tube.
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to
∼
365 °C (
∼
180 °C below
T
E
) was attained. The undercooling
scans display a single sharp exothermic crystallization peak provided
that liquid undercooling
>
50 K is achieved. The undercooling data
provided a measure of
h
LX
(
T
) from just above the nucleation
nose to within 50 K of
T
E
as shown in Fig. 2
A
. Following each
crystallization event, continued cooling to 200 °C resulted in no
further detectable calorimetric events. The sample was then
reheated through melting and back to 900 °C to complete the
cycle. The heat of fusion during the reheating was measured and
was repeatable with an average value of 68.5
±
1 J/g.
Finally, Fig. 2
A
shows two additional measurements done
using the rapid discharge heating (RDH) method previously
described (23, 24). Here, a glassy sample rod of 4-mm diameter
was heated rapidly and uniformly using ohmic dissipation at a
rate of
∼
10
5
K/s to a target temperature and held there until
crystallization occurred. The rod temperature at its center was
measured using a high-speed infrared pyrometer (5-
μ
s response
time). Following rapid heating to 350 °C, the sample configura-
tionally relaxes, and the temperature drops and stabilizes at
T
∼
312 °C as shown (
SI Appendix
, Fig. S8
)in
SI Appendix
. Fol-
lowing an incubation time of
t
LX
(
T
∼
312 °C) of
∼
0.6 s, the
sample abruptly crystallized accompanied by a sharp reca-
lescence event and temperature rises to
∼
515 °C, below but near
the alloy eutectic temperature. This implies that the sample was
constitutionally undercooled at 312 °C (25). A high-speed in-
frared video camera was employed to image the eutectic crys-
tallization front as described in
SI Appendix
, Fig. S8
. A coupled
eutectic growth front propagates from a single nucleation site
over the entire sample at an average speed of
∼
1.5 cm/s. This
eutectic growth speed is quite high [see Orava and Greer (8)]. It
suggests that the short-range chemical order of the crystalline
phases must already be present in the liquid, thereby mitigating
AB
D
C
E
Fig. 1.
(
A
) Isothermal DSC scans vs. log(t-s) at various fixed temperatures illustrating the sharp exothermic crystallization event for eutectic Pt
57
Cu
23
P
20.
All iso-
thermal scans were done following a preanneal of the as-cast sample for 15 h at
T
g
=
230 °C. (
B
) Normalized signal vs. ln(
t
/
t
peak
). (
C
)Plotoflog(
t
peak
)vs.
T
g
/
T
compared with the logarithm of the Maxwell relaxation time, log(
τ
M
)
=
log(
η
(
T
)/
G
) as described in the text. The vertical dashed lines indicate the configurational
equilibration region. (
D
) X-ray diffraction scans of Pt
57
Cu
23
P
20
sample beginning in the as-cast state and at various steps employed for isothermal crystallization at
240 °C. The sample remains fully glassy prior to the exothermic heat release at 240 °C (scans 1 to 3), while it is fully crystallized following this event
(scan 4). On
further heating to 350 °C for 2 h (scan 5), the diffraction pattern remains essentially identical to that of scan 4. (
E
) Constant heating rate scans for Pt
57
Cu
23
P
20
from
T
g
=
230 °C through melting over a wide range of heating rates. All constant heating rate samples were preannealed for 15 h at 230 °C prior to the scan as in the
isothermal scans in
A
.
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the need for chemical partitioning along the advancing crystal-
lization front. The high growth speed explains why the liquid
–
crystal transformation in the Pt
–
Cu
–
P alloys (at and below
312 °C) is completed in very short timescales (as in Fig. 1
E
). The
temperature rise of
∼
200 K on recalescence was used to estimate
a heat release of
h
LX
(312 °C)
∼
52 J/g as displayed in Fig. 2
A
.
The complete set of
h
LX
(
T
) data obtained using the four dif-
ferent methods above are included in Fig. 2
A
. The collective data
were fitted using Eq.
1
yielding fitting parameters
n
=
8.04,
θ
h
=
484 K, and a normalization constant
h
C
(
∞
)
=
70.2 J/g as shown
by the solid curve in the figure. The fit gives an excellent cor-
relation of 99.8%. This fit was used to analytically compute the
liquid specific configurational entropy
s
C
(
T
) and specific con-
figurational Gibb
’
s free energy function
g
C
(
T
). By integration,
one obtains an expression for
s
C
(
T
)
=
s
LX
(
T
), where the in-
tegration constant is taken to be the alloy entropy of fusion
Δ
s
F
(
T
E
) at the eutectic temperature
T
E
:
s
C
ð
T
Þ
=
Δ
s
F
ð
T
E
Þ
+
Z
T
T
E
h
C
ð
∞
Þ
n
θ
n
h
dT
T
n
+
2
=
Δ
s
F
ð
T
E
Þ
−
nh
C
ð
∞
Þ
θ
n
h
n
+
1
1
T
n
+
1
T
T
E
.
[3]
The analytic expressions for both
s
C
(
T
) and
g
C
(
T
) for the Cu23
alloy are shown in Fig. 2
B
and
C
. The entropy of fusion (Table
1) was evaluated at
T
E
(taken as the average of the experimental
T
S
and
T
L
). From the computed
s
C
(
T
), one obtains an expression
for the Kauzmann temperature
T
K
by setting
s
C
(
T
K
)
=
0. Using a
dimensionless
Δ
s
F
(in units of [
h
C
(
∞
)/
θ
h
]),
T
K
is given by the
following:
T
K
=
T
E
1
+
n
+
1
n
Δ
s
F
T
E
θ
h
n
+
1
1
n
+
1
.
[4]
For Pt
57
Cu
23
P
20
, we find
T
K
=
505.5 K, very close to the calori-
metric onset
T
g
=
505 K. From the
η
(
T
) data given in
SI Appen-
dix
, Fig. S3
, the rheological glass transition temperature defined
by
η
(
T
g
)
=
10
12
Pa-s, is
T
g
=
501.5 K. The experimentally de-
termined
T
K
is essentially indistinguishable from the rheologically
defined laboratory
T
g
. From the viscosity data, an Angell fragility
parameter (16) for the alloy was determined to be
m
=
72
±
3;
the alloy is rheologically very fragile. At ambient pressure, one
may ignore the
“
Pv
”
term in the configurational Gibbs free energy,
i.e.,
g
C
(
T
)
∼
h
C
(
T
)
−
Ts
C
(
T
). To calculate
g
C
(
T
)inFig.2
C
,we
assume that at
T
K
, the fully ordered glass configurationally freezes.
Below
T
K
,wetake
s
C
(
T
)
=
0and
g
C
(
T
)
=
h
C
(
T
K
). The apparent
entropy from our fit would become negative (subensemble) be-
low
T
K
, suggesting the frozen system effectively runs out of con-
figurational states. A similar picture was introduced by Derrida
(26) to describe freezing of spin glasses. Notice that
g
C
(
T
)remains
linear vs.
T
down to deep undercooling temperatures, thereby
following the Turnbull
–
Spaepen approximation (27) to surprisingly
high accuracy. Configurational freezing apparently sets in only at
very deep liquid undercooling.
A
BC
Fig. 2.
(
A
) Summary of all data for heat of crystallization of Pt
57
Cu
23
P
20
eutectic alloy vs.
T
. Data in red were obtained by using rapid discharge heating (RDH)
as described in the text. The solid is the fit obtained using Eq.
1
in the text. Fitting parameters
n
=
8.04,
θ
h
=
484 K, and a normalization constant
h
C
(
∞
)
=
70.2
J/g for the high-temperature limit value are obtained. (
B
) Results of calculation of the configurational entropy
s
LX
(
T
) using Eq.
3
showing both the Kauzmann
temperature from Eq.
4
and eutectic temperature. (
C
) The configurational free energy
g
LX
(
T
) (crystallization driving force) for Pt
57
Cu
23
P
20
. The Turnbull linear
approximation for
g
LX
(
T
) (26) is indicated by the dotted line and provides an excellent approximation to the driving force for crystallization to deep
undercooling. This indicates the configurational entropy of the liquid is nearly constant to very deep undercoolings.
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Following the same methods described above for the Cu23 alloy,
we determined the
h
LX
(
T
) curves for the other metallic glasses in
the Pt
80-
x
Cu
x
P
20
alloy series. Both the isothermal and constant
heating data for all alloys of the series are compiled for reference in
SI Appendix
,TableS1
. Table 2 summarizes the fitting parameters
h
C
(
∞
),
θ
h
,and
n
obtained using Eq.
1
for
x
=
18, 20, 23, and 27. Fig.
3
A
compares
h
LX
(
T
) plots for three representative compositions
and illustrates their variation with Cu content. For the two cases
x
=
14 and 16 with the lowest Cu content, a separate discussion and
analysis is given below. Table 2 includes the Kauzmann tempera-
tures,
T
K
,obtainedfromEq.
4
. While the computed configurational
entropy of the liquids vanishes at
T
K
, the configurational enthalpy
does not. The ordered glasses at their respective
T
K
possess a finite
residual enthalpy
h
R
=
h
C
(
T
K
) vs. the crystallized eutectic reference
state. This residual enthalpy was evaluated using Eqs.
3
and
1
.Table
2 gives the ratio
h
R
/
h
LX
(
∞
), which varies from 0.28 to 0.36. Note
that
h
R
represents the heat of crystallization of a fully ordered or
ideal glass at
T
K
. The configurational enthalpy or equivalently
potential energy of the ordered glass lies
h
R
above that of the
crystalline state. Fig. 3
B
shows the overall variation of 1/
n
vs. Cu
content
x
. The solid curve in Fig. 3
B
is a simple linear fit vs.
composition and suggests that
n
diverges at
x
∼
17. For reference,
the X-ray diffraction data for
x
=
16 before and following crys-
tallization and on subsequent heating are included in
SI Appendix
,
section E and Fig. S6
. The scans are analogous to those for the
x
=
23 case in Fig. 1
D
.AsseeninTable2,
T
K
increases as
x
decreases
from
T
K
=
472
±
10 K at
x
=
27 to a value of
T
K
=
517.4 K at
x
=
18.
For
x
≤
18, determining
T
K
requires a further discussion as
presented below.
The diverging
n
near
x
∼
17 suggests the glass transition
becomes a first-order freezing transition. On heating at a fixed
rate, the calorimeter response always lags behind the measured
calorimeter temperature. This lag effect was assessed and
quantified by measuring the melting transition of pure Sn (
T
m
=
501 K) at heating rates varying from 0.5 to 20 K/min. The
melting of Sn (
T
m
=
232 °C) was chosen for its proximity to the
present temperature range of interest. The correction is de-
scribed and tabulated in
SI Appendix
, Fig. S7 and Table S1
. The
peak shift ranges from less than
∼
1 K at low heating rates up to
∼
10 K at the highest heating rates. The corrected
h
LX
(
T
) data for
x
=
14 and 16 are plotted in Fig. 4
A
. For
x
=
14, the data display a
clear vertical step at a glass melting temperature of
T
gm
=
533 K.
For
x
=
16, one observes a smaller vertical step at
T
gm
=
548 K.
Both curves imply a latent heat and a first-order glass
–
liquid
melting transition. For the
x
=
14 sample, we obtain a latent heat
of
∼
27 J/g at 533 K and
T
gm
lies
∼
30 K above the nominal lab-
oratory onset
T
g
. For
x
=
16, the latent heat of
∼
20 J/g is
somewhat smaller but occurs at a somewhat higher
T
gm
=
547 K.
By contrast with the
x
=
20, 23, and 27 curves in Figs. 2
A
and
3
A
, the
h
C
(
T
) curves for
x
=
14 and 16 in Fig. 4
A
show opposite
curvature and exhibit a clear discontinuous enthalpy jump. Be-
low
T
gm
, one approaches the apparent first-order transition from
the solid-like or
“
glass
”
side of the enthalpy discontinuity. It
follows that Eq. (
1
) is no longer appropriate. Recognizing that
the residual enthalpy of the glass,
h
R
, is an additive constant to
the enthalpy of the liquid
–
glass freezing transition, it is natural to
describe the glass below
T
gm
using a modified version of Eq.
1
that describes the approach to the discontinuity from below the
step. To analyze these glass data, we introduce a power law form:
ð
h
C
ð
T
Þ
−
h
R
Þ
ð
h
C
ð
∞
Þ
−
h
R
Þ
=
T
θ
h
n
,
[5]
to describe the approach to melting of the glass. Notice that
θ
h
now represents the temperature where
h
C
(
T
) for the low-
temperature glass phase crosses
h
C
(
∞
), or equivalently (in the
case of very large
n
) where
h
C
(
T
) for the solid-like glass phase
crosses the enthalpy of fusion of the eutectic crystalline solid.
Note that the sign of
n
is reversed compared with the
n
values
obtained using Eq.
1
, which describes the freezing of the liquid
phase to the glass. Finally, note that
h
C
(
T
) is referenced to the
residual enthalpy of the ideal glass
h
R
, as this is the natural zero
for the glass enthalpy. With increasing
n
, Eq.
5
describes the
approach to an appropriately normalized Heaviside step func-
tion from below
θ
h
. Using Eq.
5
to fit the
x
=
14 and 16 data
below
T
gm
gives the solid curves in Fig. 4. The fits yield very large
values of the exponent;
n
=
−
19.2 and
−
29.2 for Cu14 and Cu16,
respectively. The fitting parameters for
x
=
14, 16 using Eq.
5
are
summarized in Table 2 together with those for
x
>
16. Using the
fits from Eq.
5
, one may compute the entropy of the glass by
integration of
dh
/
T
from low temperature where
s
C
=
0toa
temperature
T
K
*
where
s
C
(
T
K
*
)
=
Δ
s
F
, the entropy of fusion
of the crystallized eutectic.
T
K
*
is now an
“
inverse
”
Kauzmann
temperature, i.e., the temperature where the glass configura-
tional entropy equals the entropy of fusion of the crystalline
eutectic alloy. The analytic expression for
T
K
*
is as follows:
T
p
K
=
θ
h
ð
n
−
1
Þð
h
C
ð
∞
Þ
−
h
R
Þ
θ
h
nh
C
ð
∞
Þ
T
E
1
=
ð
n
−
1
Þ
.
[6]
T
K
*
is effectively an upper bound for the glass melting temper-
ature. Note that
T
K
*
for the
x
=
14 alloy is slightly lower than for
x
=
16, implying a more restrictive upper bound.
The data for Cu14 (Cu16) compositions in the glass region
below
T
gm
=
533 K (546 K) are generally obtained from iso-
thermal crystallization scans. The glass enthalpy approaches a
residual value of
h
R
∼
26.0 J/g (24.5 J/g). The
T
dependence of
the glass enthalpy below
T
gm
suggests that the equilibrium glass is
in a configurationally excited state. While slow kinetics could
limit relaxation and equilibration of the glass below
T
gm
, this
seems unlikely as the data were obtained under isothermal
conditions whereby the glass relaxes for timescales far exceeding
the configurational Maxwell relaxation time as discussed earlier.
Assuming the data represent a glass in configurational equilib-
rium, one may use the enthalpy fits from Eq.
5
to compute the
T
-
dependent configurational entropy of the glass below
T
gm
.
Combining the fits with the measured latent heat of glass melting
at
T
gm
and the measured entropy of fusion of the crystalline
eutectic, one may generate a piecewise continuous configura-
tional
s
C
(
T
) plot for the glass/liquid system as displayed in Fig.
4
B
. The figure shows the configurational entropy vs.
T
(entropy
Table 2. Compilation of Pt
80-
x
Cu
x
P
20
metallic-glass fitting
parameters obtained from Eq. 1 for
x
≥
18 (Eq. 5 for
x
=
14, 16),
along with Angell fragility parameter
m
from viscosity data
(
SI Appendix
,
Supplemental Materials
), the calculated Kauzmann
temperature
T
K
from Eq. 4 or
T
K
* from Eq. 6 as described in
the text, and the normalized residual configurational enthalpy,
h
R
/h
C
(
∞
), for the fully ordered glass
Alloy
h
C
(
∞
), J/g
θ
h
,K
n
m
fragility
(Angell)
T
K
(*), K
h
R
h
C
ð
∞
Þ
Pt
66
Cu
14
P
20
67.9 561.1*
−
19.2*
>
90
†
T
K
*
=
562.1 0.383
Pt
64
Cu
16
P
20
68.5 560.5*
−
28.3*
>
90
†
T
K
*
=
560.7 0.360
Pt
62
Cu
18
P
20
72.4 508.3 25.7
>
90
†
517.4 0.360
Pt
60
Cu
20
P
20
68.3 492.2 13.3
>
82
†
511.4 0.341
Pt
57
Cu
23
P
20
70.2 484.8
8.36
73
505.5 0.334
Pt
53
Cu
27
P
20
69.9 433.6
4.30
—
472
±
10
‡
0.266
*Values obtained using Eqs.
5
and
6
. See text for discussion of fitting proce-
dures used for samples with Cu content
x
=
14, 16.
†
Viscosity is highly non-Newtonian and strain rate sensitive. Crystallization is
induced by flow. It was not possible to determine an accurate value for
Angell parameter
m
. Values given are a lower bound.
‡
Value has large error due to large uncertainty in Eq.
1
fitting parameters.
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Na et al.
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in units of the gas constant R) for both the
x
=
14 and 16 glass
–
liquid systems. Within the accuracy of the Eq.
5
and assumption
of configurational equilibrium, for
x
=
14 one finds the config-
urational entropy at
T
gm
=
533 K to be 0.0308 J/g-K or 0.533 R.
The Eq.
5
fit then provides an extrapolation of the glass entropy
from
T
gm
to T
=
0 K. The predicted entropy at
T
=
0 K vanishes
within an uncertainty of approximately
±
0.03 R; and
s
C
(
T
) obeys
the third law of thermodynamics within the stated uncertainties.
This resolves Kauzmann
’
s apparent paradox. The equilibrium
glass configurational entropy never becomes negative but instead
rapidly approaches that of the crystallized eutectic solid as
T
falls
below
T
gm
. In this respect, the glass behaves much like a crystalline
solid but with a relatively smaller heat of formation for configura-
tional excitations (or defects in the case of crystals).
It should be noted that we have ignored possible vibrational
contributions to the entropy throughout our analysis. In fact, our
data for the heat of crystallization implicitly include any contri-
bution arising from differences in anharmonicity between the
glass and crystalline eutectic since we measure the total heat
release during crystallization. In this regard, the form of Eq.
5
and the high values of the exponent
n
obtained strongly suggest
that anharmonic contributions are relatively small. This follows
since anharmonicity, to leading order in
T
, would display a
“
T
2
”
contribution to the heat of crystalliz
ation. This contrasts sharply with
the large values of
n
indicated from fitting the data. Configurational
degrees of freedom must therefore dominate the
T
dependence
of
h
C
(
T
)below
T
gm
. The ordered glass might be usefully viewed
as a configurationally ordered solid, albeit with an infinite unit cell.
It should not be surprising that this configurationally ordered glass
has a unique configurational ground state enthalpy,
h
R
, that differs
from the eutectic crystalline solid state. Allotropic or polymorphic
crystalline phases also have characteristic ground state energies that
differ by phase.
Discussion and Summary
The Pt
80-
x
Cu
x
P
20
bulk metallic glass-forming alloys investigated
are unique in several respects. The undercooled homogeneous
liquid crystallizes in a single sharp step by coupled eutectic
growth producing a well-defined crystallized reference state. This
enables an accurate direct measurement of the liquid enthalpy
referenced to the crystallized eutectic solid. The onset of crys-
tallization follows a relatively long incubation time
t
LX
(
T
) that
far exceeds the Maxwell relaxation time
τ
M
(
T
) of the liquid by
orders of magnitude as seen in Fig. 1
C
. The liquid undergoes
extensive configurational relaxation, thereby achieving meta-
stable equilibrium prior to the onset of crystallization. These
features enable accurate assessment of the metastable equilib-
rium configurational enthalpy of the liquid as referenced to the
crystalline eutectic state.
The
T
dependence of the
h
C
(
T
) curves differs dramatically
from that expected for a conventional glass having a Gaussian
distribution
W
(
φ
) of inherent configurational states where
n
=
1.
According to Eq.
2
, the large and increasing
n
values obtained as
Cu content decreases imply a microcanonical density of config-
urational inherent states that approaches simple exponential
behavior of the form
W
(
φ
)
∼
e
K
φ
vs. configurational potential
energy
φ
. The large
n
values imply a diverging heat capacity,
c
LX
(
T
)
=
nh
(
∞
)/
θ
h
at
T
=
θ
h
,as
n
increases. Equivalently, the glass
transition approaches a first-order melting transition as
n
−
1
→
0.
At
x
=
14 and 16, Eq.
1
fails to provide a suitable description of
the data since the curvature of the
h
C
(
T
) changes sign. For these
cases, we introduced Eq.
5
instead to describe the equilibrium
configurational excitation of the solid glass phase as it approaches
melting at
T
gm
. By analogy to Eq.
2
, it is straightforward to show
Eq.
5
is also equivalent to a microcanonical density of configu-
rational states of the form
W
(
φ
)
∼
exp[
φ
(
n
−
1)/
n
]. Within measure-
ment capability, we observe a latent heat of melting at a well-defined
T
gm
for the
x
=
14 and 16 samples. This signals unambiguous first-
order melting. The glasses apparently melt in a similar manner to
a crystal, but very differently than expected for a traditional glass
transition. The ordered glasses melt to a fully disordered liquid in its
high temperature limit. Finally, the equilibrium glass configurational
entropy remains positive and finite below
T
gm
and approaches zero
in the limit of low
T
. The Kauzmann paradox is averted, and the
configurational entropy contribution appears to obey the third law.
Based on molecular dynamic (MD) simulations, Berthier and
coworkers (28, 29) and others (30) have recently reported that
ultrastable (configurationally relaxed) atomic glasses with a poly-
disperse atomic size distribution ex
hibit first-order melting behavior
on rapid heating through the glass transition. Such ultrastable
glasses are prepared experimentally (and in simulations) using
layer-by-layer atomic deposition onto substrates held near
T
g
,
thereby allowing a degree of configur
ational relaxation not achievable
on cooling a monolithic liquid (28, 29). In the MD simulations,
such glasses melt by propagation of a first-order melting front in
the same manner as crystals melt. The authors attribute this first-order
melting behavior to the highly ordered, low configurational enthalpy
state associated with the ultrastable glasses. In the ultrafragile
Pt
–
Cu
–
P alloys, we have shown that extensive configurational
relaxation of the liquid state is achieved near
T
gm
prior to the
onset of crystallization. We th
ereby achieve a low enthalpy
“
ideal-glass
”
state of zero configurational entropy lying near the
lowest achievable glass enthalpy
h
C
(
T
)
=
h
R
. We suggest that the
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0
5
10
15
20
25
30
35
1/n
Cu content (%)
First
Order
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
n = 1
Gaussian
n
= 4.3
8.4
13.3
x = 20
x = 23
x = 27
∞
θ
A
B
Fig. 3.
(
A
) Plot of dimensionless configurational enthalpy data [
h
C
(
T
)/
h
C
(
∞
)]
vs. dimensionless temperature,
θ
/
T
, for alloys with Cu contents
x
=
27, 23,
and 20 (
Left
to
Right
) as described in the text. The curves illustrate evolution
of the glass transition toward a first-order phase transition with decreasing
Cu content. The black line shows prediction for a Gaussian density of in-
herent states. The solid curves (green, red, and blue) are fits using Eq.
1
.(
B
)
Variation of the effective glass transition width,
n
−
1
, vs. Cu content for all
samples (see text for discussion of Cu14 and Cu16 cases).
Na et al.
PNAS
|
February 11, 2020
|
vol. 117
|
no. 6
|
2785
APPLIED PHYSICAL
SCIENCES
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