Thermodynamics and kinetics of the undercooled liquid
and the glass
transition of the Zr41.2Ti13.8Cu,2.5Ni10.0Be22.5 alloy
R. Busch,a) Y. J. Kim, and W. L. Johnson
W M. Keck Luboratoty of Engineering Materials, California Institute oj’ Technology,
Pasadena, California 91125
(Received 29 August 1994; accepted for publication 20 December 1994)
Differential scanning calorimetry (DSC) was used to determine the thermodynamic functions of the
undercooled liquid and the amorphous phase with respect to the crystalline state of the
~41.2Ti13.8CU125Nilo.OBe22.5
bulk mltallic glass forming alloy. The specific heat capacities of this
alloy. in the undercooled liquid, the amorphous state and the crystal were determined. The
differences in enthalpy, AH, entropy, AS, and Gibbs free energy, AG, between crystal and the
undercooled liquid were calculated using the measured specific heat capacity data as well as the heat
of fusion. The results indicate that the Gibbs free energy difference between metastable undercooled
liquid and crystalline solid, AG, stays small compared to conventional metallic glass forming alloys
even for large undercoolings. Furthermore, the Kauzmann temperature,
TK,
where the entropy of
the undercooled liquid equals to that of the crystal, was determined to be 560 K. The Kauzmann
temperature is compared with the experimentally observed rate-dependent glass transition
temperature,
TR.
Both onset and end temperatures of the glass transition depend linearly on the
logarithm of the heating rate based on the DSC experiments. Those characteristic temperatures for
the kinetically observed glass transition become equal close to the Kauzmann temperature in this
alloy, which suggests an underlying thermodynamic glass transition as a lower bound for the
kinetically observed freezing process.
0 1995 American Institute
of
Physics.
I. INTRODUCTION
Recently, new families of multicomponent glass forming
alloys such as
La-Al-Ni, ’ Zr-Ni-Al-Cu,’ and
Zr-Ti-Cu-Ni-Be3 have been found which exhibit extraordi-
nary glass forming ability. Cooling rates of less than 100 K/s
are usually sufficient to suppress nucleation of crystalline
compounds and thus form a bulk metallic glass (BMG)
in these
alloy
systems. For the
particular
~41.2Ti13.8CU12.5Ni10.OBe22.5
alloy, it was even shown that the
melt freezes to a glass applying cooling rates lower than 2
K/s if heterogeneous nucleation is avoided by containerless
processing.4 Sample sizes range up to 25 mm in the smallest
dimension. This brings BMG close to technical applicability
because of their unique features, such as high processability
in the undercooled liquid regime and significantly increased
yield strength below the glass transition.5
Due to the high thermal stability of undercooled liquids
of BMG formers, detailed studies of the thermophysical
properties of metallic melts in the whole undercooled liquid
region become possible. These thermophysical properties in-
clude specific heat capacity, viscosity, surface tension, and
thermal expansion- coefficient. Furthermore, in contrast to
most conventional metallic glass formers, the glass transition
can be investigated in a broad range of time scales.
In this article we focus on thermodynamics and the glass
transition of. the Zr41.2Ti13.sCu115Ni10.0Be22,5 alloy, which is
the best bulk glass former known so far. In order to deter-
mine the thermodynamic properties of the undercooled liquid
and the amorphous phase in this alloy we used differential
scanning calorimetry (DSC) to measure specific heat capaci-
-~
3Electronic mail: busch@hyperfine.cakech.edu
ties and heats of transformation close to the glass transition
temperature,
Tg ,
and at the melting point,
T,
. From the
measurements, the thermodynamic functions are calculated
as a function of temperature. This leads to the determination
of the Kauzmann temperature6 which is shown to be the
lower bound for the kinetically observed glass transition.
II. EXPERIMENTAL METHODS
Amorphous alloy ingots, with the nominal composition
~41.2~13.8CUl~Nilo.oBe22.5 9
were prepared from a mixture
of the elements of purity ranging from 99.5% to 99.9% by
induction melting on a water cooled silver boat under a Ti-
gettered argon atmosphere. Samples of 20-40 mg were in-
vestigated in a Perkin-Elmer DSC7. Prior to every experi:
ment, the DSC was evacuated to a pressure of 1 Pa and
purged several times with 99.9999% Ar. Each sample was
heated up above the glass transition using a rate of 0.33 K/s
and cooled with the rate of 3.3 K/s back to room temperature
to ensure the same thermal history for all samples. The calo-
rimeter was recalibrated for each heating rate with indium
and zinc standards. A second run for each specimen was
directly carried out after the first cycle without changing the
conditions of the measurement to construct a baseline. Mea-
surements of absolute values of the specific heat capacity in
the amorphous alloys up to 593 K and the crystallized
samples up to 893 K were undertaken. To do this, the sample
was heated with 0.33 K/s to a certain temperature and held
there for 180 s. The resulting step in heat flux was compared
with the signal of a sapphire standard and the empty Cu pan.
This procedure was done every 20 K. In addition, DSC mea-
J. Appl. Phys. 77 (8), 15 April 1995
0021-8979/95/77(8)/4039/5/$6.00
0 1995 American Institute of Physics
4039
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60, a , . , . , . , . , I
‘p
g 30
g
G= 0
ii
2
ii 30
a,
T
-60
500 600 700 600 900 1000
temperature
(K)
FIG. 1. DSC
thermogram
of
the amorphous
Zr,,3Ti,3,sCu,2.sNi,o.0Be22.5 sample ahoy at a heating rate of 0.167 K/s indi-
cating the crystallization with the heat release, AH,, and subsequent melting
with the heat of fusion, AHf. Additionally marked are the onset of the glass
transition, qme’,
the onsets of the three crystallization steps TX, (i= l-3),
the eutectic temperature, T,, , and the liquidus temperature, Tti, .
surements were also carried out in a high-temperature calo-
rimeter of type Seteram DSC 2000 K using 99.999% Ar.
Runs of the empty crucibles taken prior to the experiments
were subtracted from the measurements to obtain a straight
baseline. Care was taken to remove the air prior to the runs.
III. EXPERIMENTAL RESULTS
The glass transition and crystallization behavior of the
Zr41.2n13.8CUlZ.5Ni10.0Be225
alloy were monitored in DSC
scans for various heating rates. Figure 1 shows the DSC
thermogram for a sample heated up with 0.167 K/s. It exhib-
its the endothermal heat effect due to the glass transition and
three characteristic steps of heat release, indicating the suc-
cessive stepwise transformations at T+,
Tx2,
and
TX3
from
the metastable. undercooled liquid state into the crystalline
compounds at 680, 720, and 746 K, respectively. Upon fur-
ther heating, the crystallized sample finally starts to melt at
the eutectic temperature T,,, =937 K, followed by complete
melting at the liquidus temperature 7’,1,=993 K.
be attributed to the fact that relaxation times which are re-
lated to structural relaxations in the glass transition region
are within the time of the experiment. The temperature at
which the glass begins to respond to the temperature increase
is defined by the onset of the glass transition
Tyt. The
end
temperature,
TFd,
above which the sample can fully equili-
brate into the metastable undercooled liquid state during
heating, is rate dependent as well. All the samples reach the
undercooled liquid regime at different end temperatures.
From there on they follow the same specific heat capacity
curve. It can be clearly seen in Fig. 2 that cp in the under-
cooled liquid is a well defined decreasing function with in-
creasing temperature. The temperature range in which the
undercooled liquid can be observed in a DSC experiment is
limited by the crystallization of the sample.
The crystallization of the sample is also rate dependent.
This is caused by the fact that nucleation is a thermally ac-
tivated process, whereas the rate dependence of the kinetic
glass transition is due to the relaxation processes in the glass
transition region.
The onset temperatures of the glass transition,
Tg
, and
the transformation peaks,
Tx(1,2,3),
appear to be strongly de-
pendent on the heating rates. The heats of transformation
determined by integrating the peak areas show a slight heat-
ing rate dependence as well. In Fig. 2 the specific heat ca-
I
pacity (c,) of the amorphous phase throughout the glass tran-
sition into the undercooled liquid is presented with respect to
the crystalline solid for different heating rates. The absolute
values for cp of the crystal and the amorphous alloy were
measured with respect to sapphire standards. Below the onset
With increasing heating rate the crystallization peaks are
shifted to higher temperatures. Since the total heat release
during crystallization measures the enthalpy difference be-
tween undercooled liquid and the crystalline state, we expect
an increasing overall heat of crystallization with increasing
heating rate. Table I summarizes the different characteristic
temperatures and the measured enthalpies of transformation
depending on heating rate as determined with the Perkin-
Elmer DSC7. In particular, the heat release during the second
transformation step is slightly dependent on the heating rate.
I
temperature of the glass transition, the specific heat capacity
The heat of fusion is- determined by the Seteram DSC
of the amorphous phase does not change for different heating
2000 K (see Fig.
lj,
at a rate of 0.167 K/s, and amounts to
rates and it is only approximately 1 J/g-atom-K larger than
8.2 kJ/g-atom that is raised in two steps of 5.4 and 2.8 kJ/g-
the specific heat capacity of the crystal, as Pig. 2 indicates.
atom at the eutectic and liquidus temperature, respectively.
Upon undergoing the glass transition, the track of the specific
The total heat release during crystallization was 5.5 kJ/g-
heat capacity becomes dependent on heating rate. This can
atom for this heating rate.
60
"PO.33 K/s
-1.65WS
+ 6.66 Iv.5
l
steps
liquid
O~Ooo
-
I I
1
I
550 600 650 700
temperature
(K)
0
FIG. 2. Course of the specific heat capacity from the amorphous phase
throughout the glass transition into the undercooled liquid for different heat-
ing rates. The specific heat capacity data of the crystallized samples were
measured in steps of 20 K in reference to a sapphire standard.
4040 J. Appl. Phys., Vol. 77, No. 8, 15 April 1995
Busch, Kim, and Johnson
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TABLE I. Onset and end temperature Tt
and pznd of the glass transition depending on heating rate measured
with the Perkin-Elmer DSC7. In addition, me onset temperatures and heat releases of the three crystakation
steps depending on heating rate are listed.
Temperatures and enthalpies
y (K)
Td 0-Q
T,, (K)
AHx, (W/g-atom)
Txz K)
AH.+ &I/g-atom)
Tx3 (K)
Akr,j(kJ/g-atom)
0.0167
602
628
644
1.1
694
1.2
710
2.8
Heating rates k (K/s)
0.083
0.33 1.67
614
620 631
643
653 668
666
692 712
1.2
1.2 1.2
710
731 750
2.6
3.0 3.4
736
763 793
1.4
1.4 1.3
6.67
636
680
726
1.2
769
3.6
827
1.4
IV. DISCUSSION
A. The thermodynamic functions of the undercooled
liquid
The thermodynamic
functions of the
Zr41.2Til3.8CU12.5Ni10.OBe22.5
alloy as a function of tempera-
ture are ,calculated based on the measured specific heat ca-
pacity data, which are shown in Fig. 3. The specific heat
capacity of the amorphous phase is marked with triangles.
The specific heat capacity data of the undercooled liquid
above the glass transition (circles), obtained before crystalli-
zation upon heating the alloy, show about twice the value of
the amorphous phase. They are a decreasing function with
increasing temperature. The specific heat capacities that were
obtained throughout the glass transition are not presented in
Fig. 3, because they do not represent a thermodynamic equi-
librium or metastable equilibrium state. The specific heat ca-
pacity above the eutectic temperature was measured with a
modified Per&-Elmer DSC7 by Fecht7 and does not exceed
41 J/g-atom-K (diamond). Finally, the specific heat capacity
of the crystallized ahoy is also marked (squares).
I * I * I - I - I
60
10
0
500
600 700 800 900
1000
temperature
(K)
PlG. 3. Measured specific heat capacity of the undercooling liquid (0 +),
the amorphous phase (A), and the crystal (Cl). The specific heat capacity
curves of the undercooled liquid and the crystal are fitted to the data.
According to Kubaschewski et al.,” the temperature de-
pendence of the specific heat capacity of the undercooled
liquid far above the Debye temperature can be expressed
mainly as a l/T2 law as follows:
cp=3R+b.T+c.T-2.
(1)
This has been successfully applied in various
calculation-of-phase-diagram (CALPHAD) calculations to
describe the temperature dependence of the specific heat ca-
pacity in the undercooled liquid.‘,” The fits to the specific
heat capacity data for the crystal and the undercooled liquid
are added in Fig. 3. The specific heat capacity of the under-
cooled liquid obeys the following equation:
cF=3R+
i
7.5X10-“.T+
3.17X106
J
K
T’
‘K’ g-atom-K’
1
(2)
The specific heat capacity difference between the liquid and
the crystal close to the melting point is less than 5 J/g-
atom. K.
Since the temperature dependence of the specific heat
capacity in the Zr41,2Ti13.sCu12,5Ni10,0BeU.5 alloy is known
for the noncrystalline and crystalline state as well, the Gibbs
free energy of undercooled liquid with respect to the crystal,
AGI-,(T),
can be calculated by integrating the specific heat
capacity difference according to
I
=0
AG&T)=+AHf-A+T,- T
Acf;n(T’)dT’
+T
AC;-“<
T’) dT,
T’ ’
in which
AH~
and ASf are the enthalpy and entropy of fu-
sion, respectively, at the temperature
To. To
is the tempera-
ture where the Gibbs free energy of the crystal is equal to the
Gibbs free energy of the liquid. A$” is the difference in
specific heat capacity between liquid and solid. Even though
To
is not exactly known for our alloy, from the pattern of the
DSC track on melting we can assume that the alloy is quite
far away from the eutectic composition. This means that
To
is located between
T,,,
and
T,iq
and is likely to be very close
J. Appl. Phys., Vol. 77, No. 8, 15 April 1995
Busch, Kim, and Johnson
4041
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~4,.2Ti13.8Cu,2.5Ni,o.oBe22.5
8
5‘8
E
Tf 4
P
32
3
0
-2
400
500 600 700 800 900
1000
8
6
g4
3
&I 2
5
-2 -
TK
T cut
-4-
, ,,*I* , , 8,
400 500 600 700 800 900 I(
temperature (K)
temperature (K)
FIG. 4. Entropy of the undercooled liquid w&h respect to the crystal, in-
cluding the entropy of fusion, ASf , and the Kauzmann temperature, TK .
to the eutectic temperature. In our .calculations we use a
value of To=948 K that corresponds to the maximum of the
first melting peak. For the determination of the total As(T),
m(T), and AG(T), the error will be small, especially for
large undercooling (20.1 kJ/g-atom) because the specific
heat capacity difference between crystal and melt is small at
the melting point.
Figure 4 shows the calculated entropy of the under-
cooled Zr,,,2Ti13,sCu,25Ni10,0B822.5 melt with respect to the
crystal. The entropy of the undercooled liquid decreases with
increasing undercooling until it reaches the entropy of the
crystal at the Kauzmann temperature,
TK .
The existence of the undercooled liquid below this tem-
perature would violate the Kauzmann paradox,6 suggesting
TK
to be the lower bound for the glass transition for thermo-
dynamic reasons. This is due to the fact that the liquid should
not have a smaller entropy than the crystal. An alloy, which
could be kept as an undercooled liquid from
T,,,
down to
T, ,
has to undergo the transition into the amorphous state by a
sudden drop of the specific heat capacity to the value of the
glass. For our glass, we obtain a Kauzmann temperature of
560 K, which is below the kinetically observed glass transi-
tion temperatures.
The enthalpy difference between the undercooled liquid
and the crystal, which decreases with increasing undercool-
ing, is plotted in Fig. 5. The specific heat capacity is inte-
grated from
To
down to the Kauzmann temperature. Since
the difference in specific heat capacity between amorphous
phase and crystal is only 1 J/g-atom.K the enthalpy differ-
ence remains virtually constant for temperatures below
TK.
However, the extrapolated enthalpy below about 620 K’can
hardly be achieved in the real experiment. The curve below
620 K belongs to an ideal undercooled liquid and glass, re-
spectively, that requires extremely slow cooIitrg rate to form
(see next paragraph). In the real experiment, the liquid
freezes to a glass throughout the glass transition, and the
larger the cooling rate, the more residual enthalpy and en-
tropy are frozen in. In Fig. 5, the measured heats of crystal-
lization are included (triangles), indicating a good agreement
DO
FIG. 5. Enthalpy of the undercooled liquid with respect to the crystal, in-
cluding the overall heats of crystalli&ion depending on heating rates (A).
The paths of crystallization and melting are marked according to the differ-
ent transformation steps for a rate of 0.33 K/s.
with the independently obtained
AH
curve. With increasing
heating rates, the samples start to crystallize at higher tem-
peratures from the undercooled liquid, which is already in
metastable equilibrium. This explains the observed heating
rate dependence of the overall heat of crystallization.
The calculated Gibbs free energy function with respect
to the crystalline state is plotted in Fig. 6. Since there is a
finite and increasing difference in specific heat capacity be-
tween the melt and the crystal on undercooling, the Tumbull
approximation (AcbVx= 0, for ‘lX
T,,,) ‘I
is only valid for un-
dercoolings of about 100 K. For larger undercoolings, the
real Gibbs free energy difference becomes smaller due to the
relative stabilization of the undercooled melt. This stabiliza-
tion is caused by the increasing specific heat capacity that is
attributed to a decreasing free volume, and most likely a
gradual gain of short range order in the alloy melt. The ob-
served Gibbs free energy difference is, for example, 1.5 kJ/
g-atom at 0.8
T,
. This value is relatively small compared to
4
I * ,I I. I., *
Zr4i.2Ti13.8C"12.5Ni10.0Be22.5
3-
i amorphous
-1 -
TK TO
-2 8 , I , I , , , , ,
400 500 600 700 800 900 I(
temperature (K)
30
FIG. 6. Gibbs free energy of the undercooled liquid with respect to the
crystal as a function of temperature.
4042 J. Appl. Phys., Vol. 77, No. 8, 15 April 1995
Busch, Kim, and Johnson
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1000
100
E 10
g 1
al
3 0.1
F
'E
2
0.01
0.001
0.0001!
.;'. .- . , : I , . ) . t
550 600 650 700 750
temperature
(K)
FIG. 7. Kinetical map of the glass transition and the first two crystallization
peaks. Onset and end temperatures of the glass transition are fitted linearly
to the logarithm of the heating rate. The heating rate dependence of the
crystallization is fitted according to an Arrhenius law.
conventional binary glass forming alloys like Ni5,,TiZ0 or
Nb50Nis0 at 0.8 T,n, where Gibbs free energy differences of
2.5 k.J/g-atom’” and 3.2 kJ/g-atom,” respectively, are found.
The small Gibbs free energy difference turns out to be one
crucial point in understanding the high glass forming ability
in BMG formers, and will be the subject of further investi-
gations.
B. Glass transition and Kauzmann temperature
In the following, the calculated Kauzmann temperature
is compared with the kinetically observed glass transition.
Onset and end temperatures of the glass transition, as well as
crystallization, as measured in the DSC experiments for heat-
ing rates between 0.0167 and 6.667 K/s are plotted in Fig. 7.
We find that both yt
and pgnd depend linearly on the loga-
rithm of the heating rate of the experiment. This behavior of
the glass transition temperature was also found by other au-
thors for metallic and nonmetallic systems as wel113Y’4 and is
discussed by Jackle.15
The slopes of the curves describing the heating rate de-
pendence of yt and Trd
in Fig. 7 are different, which is
due to the fact that the width of the glass transition region
becomes smaller with decreasing heating rate. The extrapo-
lation of both curves to lower temperatures and heating rates
leads to a point of intersection. Here, the width of the glass
transition region becomes zero, suggesting that the specific
heat capacity would step from the value of the glass to the
undercooled liquid. The temperature that corresponds to the
point of intersection is 562 K. This is about the value of the
Kauzmann temperature within the experimental error. The
heating rate to reach this lower limit for the glass transition
temperature would be 1.67X 10m5 K/s. It would be extremely
difficult to observe this ideal glass transition experimentally,
although it might be theoretically accessible since the ex-
trapolated onset temperature of the primary crystallization
also intersects both
TFt
and Fgnd close to their point of
intersection (see Fig. 7).
V. SUMMARY AND CONCLUSIONS
The specific heat capacities of the undercooled liquid,
the amorphous
state and the crystal of the
Zr41.2Ti13.8CU12.5Nilo.oBe22.5
bulk metallic glass former were
measured by differential scanning calorimetry. In addition,
the glass transition temperatures, crystallization tempera-
tures, heats of crystallization, and heats of fusion were mea-
sured as a function of heating rate. Based on the thermody-
namic data the thermodynamic functions of the undercooled
liquid were calculated using a I./T2 dependence of the spe-
cific heat capacity in the undercooled liquid. The calculations
show that the Gibbs free energy difference between liquid
and solid state stays small even for large undercoolings. For
example, AG is 1.5 kJ/g-atom at 0.8 T, . This relatively
small Gibbs free energy difference appears to be a contrib-
uting factor in the high glass forming ability of the alloy. The
Kauzmann temperature of the Zr41.2Ti13.8Cu12,5Ni10~OBe22,5
alloy is calculated to be 560 K, representing the lower bound
for the kinetically observed glass transition. This is deduced
from extrapolating the rate dependence of onset and end tem-
perature to low heating rates. Both onset and end tempera-
tures depend linearly on the logarithm of heating rate and
intersect at the Kauzmann temperature.
ACKNOWLEDGMENTS
The authors would like to thank D. Isheim and S.
Friedrichs for their assistance on the DSC experiments and
U. Geyer, S. Schneider, E. Bakke, and H. Fecht for valuable
help
and fruitful discussions.
This work was
supported by the
German Alexander von Humboldt Foundation via the Feodor
Lynen Program, the Department of Energy (Grant No.
DEFG-03-86ER45242) and the National Aeronautics and
Space Administration (Grant No. NAG8-954).
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J. Appl. Phys., Vol. 77, No. 8, 15 April 1995
Busch, Kim, and Johnson 4043
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