Correlation between viscous-flow activation energy and phase diagram in four systems of Cu-based alloys

Article abstract of DOI:10.1016/j.physb.2010.05.055

Activation energy is obtained from temperature dependence of viscosities by means of a fitting to the Arrhenius equation for liquid alloys of CuSb, CuTe, CuSn and CuAg systems. We found that the changing trend of activation energy curves with concentration is similar to that of liquidus in the phase diagrams. Moreover, a maximum value of activation energy is in the composition range of the intermetallic phases and a minimum value of activation energy is located at the eutectic point. The correlation between the activation energy and the phase diagrams has been further discussed.

Full text of DOI:10.1016/j.physb.2010.05.055

Physica B 405 (2010) 3633–3637
Contents lists available at ScienceDirect
Physica B
journal homepage: www.elsevier.com/locate/physb
Correlation between viscous-flow activation energy and phase diagram
in four systems of Cu-based alloys
Shuang Ning, Xiufang Bian n, Zhenfeng Ren
Key Laboratory of Liquid Structure and Heredity of Materials, Ministry of Education, Shandong University, Jinan 250061, China
a r t i c l e i n f o
a b s t r a c t
Article history:
Activation energy is obtained from temperature dependence of viscosities by means of a fitting to the
Arrhenius equation for liquid alloys of Cu–Sb, Cu–Te, Cu–Sn and Cu–Ag systems. We found that
the changing trend of activation energy curves with concentration is similar to that of liquidus in the
phase diagrams. Moreover, a maximum value of activation energy is in the composition range of
the intermetallic phases and a minimum value of activation energy is located at the eutectic point. The
correlation between the activation energy and the phase diagrams has been further discussed.
& 2010 Elsevier B.V. All rights reserved.
Received 2 January 2010
Received in revised form
12 April 2010
Accepted 18 May 2010
Keywords:
Viscosity
Activation energy
Phase diagram
Cu-based alloys
1. Introduction
2. Materials and methods
Cu-based alloys are applied extensively in material, machine,
electricity, and especially in the field of functional composite
materials because of their excellent mechanical, electrical and
chemical properties. Much attention has been paid to the physical
properties and structures of Cu-based alloys. It is well known that
the characteristic of liquid metals or melts will have a strong
impact on the qualities of the final materials [1]. Recently,
some studies have investigated the particular structures and
corresponding properties of several Cu-based alloy systems, e.g.
Cu–Sb, Cu–Sn, Cu–Ge and Cu–Si, through the measurements of
dynamic viscosity, electrical resistivity, diffusion coefficients, and
so on [2–6].
Activation energy, in viscous activation picture, as an energy
barrier for the movement of atom in liquids, was experimentally
proposed by Glasstone [7]. For strong liquid, it has been found
that activation energy is temperature independent [8,9]. For some
metals, the activation energy of different melts has a linear
relationship with the corresponding melting points [10]. In
addition, it has been revealed that the activation energy is
proportional to the high-frequency shear modulus by studying
the viscosity of glass-forming molecular liquids [11]. In the
present work, we will investigate the relationship between the
activation energy and phase diagram of Cu-based alloys.
Samples of Cu-based alloys used in the present work were
prepared from ingot Cu (99.999 wt%), Sb (99.98 wt%) and Te
(99.98 wt%). They were melted in a graphite clay crucible using a
medium frequency induction electric furnace and cast ingots in a
graphite mould.
The measurements of viscosity were carried out using an
oscillating cup viscometer. The torsion oscillation of a cylindrical
crucible suspended on a wire is damped by the friction with the
melt. Based on the Shvidkovskii equation [12], the viscosity of
the liquid can be calculated from the periodic time and the
logarithmic damping decrement of the harmonic oscillation. After
the chamber was cleaned with a vacuum of 7 ꢀ 10ꢁ6 Pa, it was
filled with high purity argon (99.999%) of 1.1 ꢀ 105 Pa [13].
Samples are placed in an Al2O3 crucible and heated at a rate of
4 K/min to 300–400 K above the liquidus temperature. After
maintaining for 3 h at the given temperatures, the samples were
cooled down to the experimental temperatures with a rate of
4 K/min and held for 1 h before each measurement. The viscosity
measurements were repeated four times at each temperature
and at least ten temperature points above the liquidus
temperature were taken for each alloy system. The error bar of
data was no more than 73% at the same temperature in different
measurements.
In a wide temperature range above the liquidus temperature
TL, the viscosity behavior of the melts generally obeys the
Arrhenius equation [14]:
n Corresponding author. Tel.: +86 531 88392748; fax: +86 531 88395011.
Z
¼
Z0expðEv=RTÞ
ð1Þ
E-mail address: xfbian@sdu.edu.cn (X. Bian).
0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.physb.2010.05.055

3634
S. Ning et al. / Physica B 405 (2010) 3633–3637
where Z0 is a pre-exponential constant, which is associated with
the nature of the liquid, R is the molar gas constant and Ev is the
viscous-flow activation energy.
3. Results
Viscosity data for the melts of Cu–Sb and Cu–Te are obtained
through the oscillating cup viscometer. Fig. 1 shows that the
viscosities of Cu–Sb alloys with different compositions increase
with decreasing temperature according to the Arrhenius-type
empirical equation. Moreover, the analogous viscosity trend with
temperature can also be revealed in the Cu–Te alloys, as shown in
Fig. 2. The fitting parameters of Cu–Sb and Cu–Te alloy systems
are shown in Tables 1 and 2, respectively, where R2 is the square
of correlation coefficient, which reflects the error of the Arrhenius
relationship for the viscosities of Cu–Sb and Cu–Te systems. The
fitted results show that the temperature dependence of the
viscosity obeys the Arrhenius relationship well.
Figs. 3 and 4 show the viscosities of Cu–Sb and Cu–Te alloys as
a function of the atomic concentrations at the same superheated
temperature, respectively. In comparison with the phase
diagrams [15], it can be seen from Fig. 3 that the maximum of
viscosity of Cu–Sb system is located at 25 at% Sb, corresponding to
Fig. 2. Temperature dependence of viscosities of molten Cu–Te alloys during
cooling process. Solid lines are the Arrhenius plots.
Table 1
Parameters of the viscosity fitted by the Arrhenius equation in Cu–Sb system.
the range of
b-Cu–Sb intermetallic phase in the solid state. Fig. 4
Alloys
Z0 (mPa s)
Ev/R (K)
R2
shows the maximum of viscosities of Cu–Te system located at
DZ0 (mPa s)
D(Ev/R) (K)
36.5 at% Te, corresponding to the range of Cu2ꢁxTe (
the solid state.
b-Cu–Te) in
Cu84.5Sb15.5
Cu81Sb19
Cu75Sb25
Cu67Sb33
Cu45Sb55
Cu37Sb63
Cu25Sb75
1.32
1.75
1.27
1.13
1.43
1.25
0.98
0.03
0.10
0.03
0.04
0.05
0.04
0.05
1574.5
1229.8
1658.9
1479.1
845.7
28.2
62.6
24.6
34.7
34.8
28.4
52.9
0.994
0.968
0.997
0.993
0.976
0.983
0.951
Values of activation energy for the melts of Cu–Sb, Cu–Te,
Cu–Sn [5] and Cu–Ag [16] systems have been summarized. The
phase diagrams coupled with the corresponding activation energy
curve with concentration are listed and compared in Fig. 5 for
each system. It can be observed that the changing trend of
activation energy curves with concentration similar to that of
liquidus in both compound forming systems (Cu–Sb, Cu–Te and
Cu–Sn systems) and simple eutectic systems (Cu–Ag system).
Interestingly, activation energy reaches a minimum at eutectic
point for all alloys mentioned above and the maximum of the
activation energy covers the range of the intermetallic phases for
all compound forming systems. Fig. 5(a) shows the maximum of
activation energy located at Sb-content of 25 at%, corresponding
803.2
861.6
Table 2
Parameters of the viscosity fitted by the Arrhenius equation in Cu–Te system.
Alloys
Z0 (mPa s)
Ev/R (K)
R2
D
Z0 (mPa s)
D(Ev/R) (K)
Cu63.5Te36.5
Cu58.5Te41.5
Cu50Te50
1.25
0.75
0.93
0.60
0.33
0.11
0.03
0.02
0.03
0.02
2601.8
2464.9
1649.4
1340.0
1654.7
126.9
44.4
22.5
33.8
33.4
0.976
0.997
0.998
0.995
0.996
to the range of
b-Cu–Sb phase in the solid state. Fig. 5(b)
Cu27Te73
shows the maximum of activation energy of Cu–Te system is
Cu20Te80
Fig. 1. Temperature dependence of viscosities of molten Cu–Sb alloys during
Fig. 3. Experimental viscosities of Cu–Sb alloy melts at the same superheated
cooling process. Solid lines are the Arrhenius plots.
temperature.

S. Ning et al. / Physica B 405 (2010) 3633–3637
3635
located at a Te-content of 36.5 at%, corresponding to the range of
Cu2ꢁxTe ( -Cu–Te) in the solid state. Fig. 5(c) indicates that the
maximum of activation energy is located at Cu84.9Sn15.1 alloy,
corresponding to the range of -Cu–Sn of solid state in Cu–Sn
system. Strikingly, the value of activation energy at the range of
the intermetallic phases is much larger than that at the range of
eutectic and near-eutectic composition.
Cu-based alloys. Expressed in algebraic form, the relationship is
b
Ev ¼ a ꢀ TL þb
ð2Þ
b
The fitting parameters a and b for different systems are
showed in Table 3, where R is the correlation coefficient, which
reflects the relevance of the values of activation energy and
liquidus temperature. Based on Fig. 6 and Table 3, it can be found
that different categories of Cu-based alloys have different slope
and different relevance of Ev and TL.
As shown in Fig. 6, the values of Ev exhibit a general linear
relationship with the TL in an individual alloy system for the
Fig. 4. Experimental viscosities of Cu–Te alloy melts at the same superheated
Fig. 6. Activation energy vs liquidus temperatures for different Cu-based alloys.
temperature.
Fig. 5. Comparison of the phase diagram with the activation energy curve of liquid Cu–Sb system (a), Cu–Te system (b), Cu–Sn system (c) and Cu–Ag system (d). Symbols
stand for the fitting data, and the solid lines are guides to the eye.

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S. Ning et al. / Physica B 405 (2010) 3633–3637
Table 3
Fitting parameters according to Eq. (2) for different systems.
Alloys
a
b
R
Cu–Sb system
Cu–Te system
Cu–Sn system
Cu–Ag system
0.0306
0.0133
0.0183
0.0150
ꢁ17.7
3.7
0.8992
0.9344
0.9083
0.9463
ꢁ6.5
ꢁ12.6
4. Discussion
It is believed that the change of activation energy is related to the
microstructure of liquid. In the intermetallic-containing alloy
systems, some clusters of unlike atoms can be formed in the liquid
at the range of intermetallic phases owing to the strong interplay
between the electronic and the atomic structure [4,17]. According to
the studies of the liquid structure of Cu–Sn, Cu–Ge, Cu–Sb, Cu–Te,
Fe–Al and Fe–Si alloys [4,18–26], it has been revealed that the
clusters of unlike atoms exist. Simple eutectic alloys of Cu–Ag
system are micro-homogeneous in the liquid state similar to their
solid state [27]. In the light of the heat by mixing the liquid copper
alloys, it has been also proved that the liquid copper alloys
containing electron compounds of complex crystal structures in
solid state have some atoms associating into small groups of atoms
similar to the type AmB molecules and are bound by a sort of
covalent. While in a simple eutectic structure, atoms are randomly
arranged in the liquid alloys [28,29]. Clusters of unlike atoms, which
correspond to the solid structure of compound, cause the atoms to
move from one position to another more difficultly on account of
enhancing interaction of atoms [27]. Thus the clusters of unlike
atoms induce Ev to increase significantly [29]. Within each interval
melts, it will constitute two kinds of clusters. The variation of
composition within a certain region does not influence the existence
of clusters in a certain type, but only change their volume [3,18].
Accordingly, the volume fraction of the unlike atom clusters is bigger
at the range of intermetallic phases. It means that the influence of
the unlike atom clusters on the activation energy of eutectic alloys
or near-eutectic alloys is not as strong as that on the activation
energy of Cu-rich alloys containing intermetallic phase in Cu–Sb,
Cu–Te and Cu–Sn systems. Thus the maximum of the activation
energy covers the composition range in which the intermetallic
compounds will exist in the solid state. And the value of activation
energy at the range of the intermetallic phases is much larger than
that at the range of eutectic and near-eutectic composition. In the
case of the liquid copper alloys (i.e. Cu–Sb, Cu–Ge, Cu–Sn and Cu–In),
it is also observed that the heat of mixing exhibits a deep valley in
the heat of evolution in the ranges of the intermetallic phases [28].
The maximum of viscosity at the same superheated temperature is
located at the same composition as the activation energy in Cu–Sb
and Cu–Te systems, respectively. On the other hand, these results
have also proved the influence of the unlike atom clusters in liquids.
For the eutectic alloy, there exist no dominant correlations of
atomic bonds in the eutectic liquid alloy. And thus, the clusters
show a continuous and random fluctuation in the liquids [27].
Obviously, the interaction between atoms is weak in the liquid
eutectic alloys, which induces activation energy to reach a
minimum at the eutectic point in all systems. Liquidus tempera-
ture is relative to the atomic bonds. Accordingly, the general
similarity of the shape of activation energy curves to that of
liquidus can be observed. The maximum of electrical resistivity of
molten Cu–Sb alloys is located at an Sb-content of 25 at%, while
the minimum of electrical resistivity of molten alloys is located at
eutectic point [30]. It can be reasonable to conclude that the
clusters in liquid have influenced properties of liquid.
Fig. 7. Viscosity
Z in liquid Al100ꢁxCux (x¼18, 25, 35) alloys as a linear function of
correlation radius rc at the same temperature. Solid lines are fitting curves.
Experimental data are taken from Refs. [32,33].
Table 4
Equilibrium lattice parameters for different CuxTe structures. The numbers in
parenthesis are experimental data [34].
x
Structure
˚
˚
˚
a0 (A)
b0 (A)
c0 (A)
2
Trigonal
8.24
7.79
3.18
8.24
7.79
3.88
7.13
6.03
6.88
1.5
1
Tetragonal
Orthorhombic
in liquid Al100ꢁxCux (x¼18, 25 and 35) alloys [32] shows linear
characteristic with the correlation radius, rc [33] at the same
temperature. The correlation between rc and viscosity of melts
indicates that viscous behavior of melts is related with the
clusters of unlike atoms in liquid intuitively.
Correlation radius rc is influenced by the structure of solid
state [33]. It is thought that the activation energy is related to the
structure of solid state in nature. The lattice parameters of Cu–Te
alloys with Te content between 33 and 50 at% are listed in Table 4
[34]. Obviously, the unit cell of the Cu2Te cluster is larger than
that of Cu1.5Te cluster, and the latter is larger than the unit cell of
the CuTe cluster. The change of Ev with Te content increasing has
the same tendency as that of the unit cell volume in Cu–Te
system. The correlation between Ev and lattice parameters in
Cu–Te system proves that the structure of solid state is related to
activation energy.
The crystal structures of some elements or space configuration of
the solid state persist in the liquid state [10]. It is deemed that the
different slope in Fig. 6 is responsible for different atomic structure
and space configuration. Moreover, because of the complicated
structure configuration in Cu-rich composition range for Cu–Sn and
Cu–Sb system in solid state, their relevance of Ev and TL in Table 3 is
much lower. While for simple eutectic Cu–Ag system, the relevance
of Ev and TL is the highest. This fact indicates that activation energy is
related to the structure of solid state.
5. Conclusions
Viscosities of Cu–Sb and Cu–Te alloys with different composi-
tions increase with decreasing temperature. The temperature-
related viscosity curves obey the Arrhenius relationship.
The correlation radius of clusters in liquids rc is influenced by
clusters of unlike atoms [27,31]. Fig. 7 shows that the viscosity
Z

S. Ning et al. / Physica B 405 (2010) 3633–3637
3637
It is found that the changing trend of activation energy curves
with concentration is similar to that of liquidus in Cu–Sb, Cu–Te,
Cu–Sn and Cu–Ag systems. Moreover, a maximum of activation
energy covers the ranges of the intermetallic phases for all
compound forming systems and a minimum of activation energy
is located at the eutectic point for all alloys. According to the
discussion of the correlation between the activation energy and
the phase diagrams, it is deemed that activation energy is related
to clusters of unlike atoms and the structure of solid state.
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We would like to acknowledge financial support from the
National Basic Research Program of China (973 Program
2007CB613901), the National Natural Science Foundation of
China (Grant nos. 50631010 and 50871062) and the Natural
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