Experimental determination of thermal conductivity and interfacial energies of solid Pb solution in the Pb-Sb eutectic system

Article abstract of DOI:10.1016/j.cplett.2011.01.006

The equilibrated grain boundary groove shapes of solid Pb solution in equilibrium with the Pb-Sb eutectic liquid have been observed from quenched sample. The Gibbs-Thomson coefficient, solid-liquid interfacial energy and grain boundary energy of solid Pb solution have been determined. The thermal conductivity values for Pb-5.8 at.% Sb and Pb-17.5 at.% Sb and the thermal conductivity ratio of eutectic liquid phase to eutectic solid for Pb-17.5 at.% Sb alloy at the melting temperature have also been measured with a radial heat flow apparatus and Bridgman type growth apparatus, respectively.

Full text of DOI:10.1016/j.cplett.2011.01.006

Chemical Physics Letters 503 (2011) 220–225
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Chemical Physics Letters
journal homepage: www.elsevier.com/locate/cplett
Experimental determination of thermal conductivity and interfacial energies of solid
Pb solution in the Pb–Sb eutectic system
c
c
L. Karabulut a, S. Aksöz b, K. Kesßliog˘lu c, , N. Marasßlı , Y. Ocak
⇑
a Department of Physics, Institute of Science and Technology, Erciyes University, Kayseri 38039, Turkey
b Department of Physics, Faculty of Arts and Sciences, Nevßsehir University, Nevßsehir 50300, Turkey
c Department of Physics, Faculty of Science, Erciyes University, Kayseri 38039, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history:
The equilibrated grain boundary groove shapes of solid Pb solution in equilibrium with the Pb–Sb eutec-
tic liquid have been observed from quenched sample. The Gibbs–Thomson coefficient, solid–liquid inter-
facial energy and grain boundary energy of solid Pb solution have been determined. The thermal
conductivity values for Pb–5.8 at.% Sb and Pb–17.5 at.% Sb and the thermal conductivity ratio of eutectic
liquid phase to eutectic solid for Pb–17.5 at.% Sb alloy at the melting temperature have also been mea-
sured with a radial heat flow apparatus and Bridgman type growth apparatus, respectively.
Ó 2011 Elsevier B.V. All rights reserved.
Received 3 November 2010
In final form 4 January 2011
Available online 6 January 2011
ꢂ
ꢃ
rSL
1
1
1. Introduction
D
Tr ¼
þ
ð2Þ
D
Sf r1 r2
The solid–liquid interfacial energy,
rSL, is recognised to play a
where r1 and r2 are the principal radii of the curvature. For the case
of a planar grain boundary intersecting a planar solid–liquid inter-
face, r2 = 1 and the Eq. (2) becomes
key role in a wide range of metallurgical and materials phenomena
from wetting [1] and sintering through to phase transformations
and coarsening [2]. Thus, a quantitative knowledge of rSL values
is necessary. However, the determination of rSL is difficult. Since
1985, a technique for the quantification of solid–liquid interfacial
free energy from the grain boundary groove shape has been estab-
lished [3–18]. Observation of groove shape in a thermal gradient
can be used to determine the interfacial energy, in dependent of
the grain boundary energy because the interface near the groove
must everywhere satisfy
rSL
D
C
¼ r
D
Tr ¼
ð3Þ
Sf
where
C
is the Gibbs–Thomson coefficient. This equation is called
the Gibbs–Thomson relation [13].
At present the most powerful method to measure solid–liquid
interfacial energy experimentally is the grain boundary groove
method. This method is based on the direct application of the
Gibbs–Thomson equation and can be applied to measure rSL for
multi-component systems as well as pure materials, for opaque
materials as well as transparent materials, for any observed grain
boundary groove shape and for any value of the thermal conductiv-
ity ratio of the equilibrated liquid phase to solid phase, R = KL/KS.
Lead–antimony solders are one of the most familiar materials
used for various microelectronic connections in computer industry.
Different compositions of lead–antimony solders have consider-
able potential for advanced structural and electronic applications.
The intensive interest in these solder alloys is attributed to their
low cost and unique material properties including the high super
plastic properties, low melting temperature, wettability. At this
time, it is very interesting to study the some thermo–physical
properties such as solid–liquid interfacial energy, Gibbs–Thomson
coefficient, grain boundary energy and thermal conductivity of so-
lid and liquid phases of Pb–Sb alloy. These thermo-physical prop-
erties could be used to people doing comparisons between
experimentally observed solidification morphology and predic-
tions from theoretical models. Thus the aim of the present work
"
!
!
#
ꢀ
ꢁ
1
d
dn2
2rSL
d
dn2
2rSL
D
Tr ¼
rSL
þ
j1
þ
rSL
þ
j2
ð1Þ
D
Sf
1
2
where
DTr, is the curvature undercooling, DSf is the entropy of fu-
sion per unit volume, n (nx, ny, nz) is the interface normal, j1 and
j2 are the principal curvatures, and the derivatives are taken along
the directions of principal curvature. Thus, the curvature undercool-
ing is a function of curvature, interfacial free energy and the second
derivative of the interfacial free energy. Eq. (1) is valid only if the
interfacial free energy per unit area is equal to surface tension per
unit length, rSL
¼
c. When interfacial free energy differs from sur-
face tension, the problem is more complicated and the precise mod-
ification of the Gibbs–Thomson equation is not yet established [19].
When the solid–liquid interfacial free energy is isotropic, Eq. (1)
becomes
⇑
Corresponding author. Fax: +90 352 437 49 33.
E-mail address: kesli@erciyes.edu.tr (K. Keßsliog˘lu).
0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2011.01.006

L. Karabulut et al. / Chemical Physics Letters 503 (2011) 220–225
221
was to determine the thermal conductivity, Gibbs–Thomson coef-
ficient, solid–liquid interfacial energy and grain boundary energy
for solid Pb solution in the Pb–Sb alloy.
0.01 K at 283 K (10 °C) by using a Poly Science digital 9102 model
heating/refrigerating circulating bath to maintain a constant radial
temperature gradient on the sample. A thin liquid layer (1–2 mm
thick) was melted around the central heating wire and the speci-
men was annealed in a very stable temperature gradient for a long
time. In this condition, the solid and liquid phases were mixed to-
gether. During the annealing period, the liquid droplets move up
towards the hot zone of the sample by temperature gradient zone
melting and single solid phase can grow on the eutectic cast phase.
If the temperature difference between solidus and liquidus lines is
high, the freezing range would be larger. As can be seen from the
phase diagram of Pb–Sb system in Ref. [20], the temperature differ-
ence between solidus and liquidus lines for Pb–5.8 at.% Sb alloy is
approximately 60 °C. Because of this larger temperature difference
the annealing time will be long. We have experimentally found the
optimum annealing period for Pb–5.8 at.% Sb alloy is 17 days after
several experiments. During the annealing period, the temperature
in the specimen and the vertical temperature variations on the
sample were continuously recorded by the stationary thermocou-
ples and a moveable thermocouple, respectively by using a data
logger via computer. The input power was also recorded periodi-
cally. The temperature in the sample was stable to about of
0.025 K for hours and 0.05 K for up to 17 days. At the end of
the annealing time the specimen was rapidly quenched by turning
off the input power which was sufficient to get a well defined so-
lid–liquid interface, because the liquid layer around the central
heating wire was very thin (typically less than 0.5–1 mm).
2. Materials and methods
2.1. Sample production
It is necessary to consider what is happening during the anneal-
ing period. Consider a binary eutectic system. If the alloy composi-
tion (C0) is near the eutectic composition (C0 ffi CE) above the
eutectic temperature, a binary eutectic system consists of liquid.
If this system is held in a very stable temperature gradient there
will be no liquid droplets behind the solid phase and two or more
solid phases (a, b and c) can grow together on the eutectic struc-
ture. Since the composition of Pb–Sb alloy in our experiment is
far from the eutectic composition only one solid phase can grow
on the cast structure. As can be seen from Figure 1, the tempera-
ture gradient on the sample is radial and single solid Pb solution
phase grows from eutectic liquid on the cast structure.
From the phase diagram of Pb–Sb alloy, solid solubility of Sb in
solid Pb is 5.8 at.% Sb at the eutectic temperature, 525 K and the
eutectic liquid composition is Pb–17.5 at.% Sb [20]. Thus, the com-
position of alloy was chosen to be Pb–5.8 at.% Sb to grow single so-
lid Pb solution on the cast structure. Pb–5.8 at.% Sb alloy was
prepared in a vacuum furnace by using 99.99% lead and 99.99%
antimony. After stirring, the molten alloy was poured into a graph-
ite crucible held in a specially constructed casting furnace at
approximately 50 K above the melting temperature of alloy. The
molten alloy was then directionally solidified from bottom to top
to ensure that the crucible was completely full. The sample was
then placed in the radial heat flow apparatus.
In order to observe the equilibrated grain boundary groove
shapes in opaque materials, Gündüz and Hunt [13] designed a ra-
dial heat flow apparatus. Maraßslı and Hunt [14] improved the
experimental apparatus for higher temperature. The details of the
apparatus and experimental procedures are given in Refs. [13–
17]. In the present work, a similar apparatus was used to observe
the grain boundary groove shapes in the Pb–Sb eutectic system.
The sample was heated from its centre by a single heating wire
and the outside of the sample was kept constant to an accuracy of
2.2. Measurements of the coordinates of equilibrated grain boundary
groove shapes
The quenched sample was cut transversely into lengths typi-
cally of 25 mm, and transverse sections were ground flat with
180 grit SiC paper. Grinding and polishing were then carried out
by following a standard route. After polishing, the samples were
etched with a 30 ml acetic acid and 20 ml hydrogen peroxide for
5 s.
The equilibrated grain boundary groove shapes were then pho-
tographed with an Olympus DP12 type CCD digital camera placed
on top of an Olympus BX51 type light optical microscope. A grati-
cule (200 Â 0.01 = 2 mm) was also photographed using the same
Central alumina tube
Cast structure
Quenched eutectic liquid
Solid Pb solution
(b)
(a)
30 µm
(c)
(d)
Figure 1. (a) Transverse section of sample. (b) Schematic representation of sample. (c) Solid–liquid interface of sample. (d) Typical grain boundary groove shape for solid Pb
solution.

222
L. Karabulut et al. / Chemical Physics Letters 503 (2011) 220–225
objective. The photographs of the equilibrated grain boundary
groove shapes and the graticule were superimposed on one an-
other using Adobe PhotoShop 8.0 version software so that accurate
measurement of the groove coordinate points on the groove shapes
could be made.
full. The specimen was then placed in the radial heat flow
apparatus.
The specimen was heated from the centre using a single heating
wire (140–190 mm in length and 2.5 mm in diameter, Kanthal A–1)
in steps of 50 K up to 10 K below the melting temperature of the
alloy and the outside of the specimen was cooled to maintain a ra-
dial temperature gradient. To obtain a reliable value of thermal
conductivity in the thermal conductivity measurement, a larger ra-
dial temperature gradient is desired. For this purpose, the gap be-
tween the cooling jacket and the specimen was filled with free
running sand or graphite dust and the outside of the specimen
was kept at 283 K using a heating/refrigerating circulating bath.
The length of central heating wire was chosen to be 10 mm longer
than the length of specimen to make isotherms parallel to the ver-
tical axis.
2.3. Geometrical correction for the groove coordinates
The coordinates of the cusp, x, y should be measured using the
coordinates x, y, z where the x axis is parallel to the solid–liquid
interface, the y axis is normal to the solid–liquid interface and
the z axis lies at the base of the grain boundary groove. Marasßlı
and Hunt [14] devised a geometrical method to make appropriate
corrections to the groove shapes and the details of the geometrical
method are given in Ref. [14].
The coordinates of equilibrated grain boundary groove shapes
were measured with an optical microscope to an accuracy of
10 lm by following Maraßslı and Hunt’s geometrical method so
that appropriate corrections to the shape of the grooves could be
deduced [14]. The uncertainty in the measurements of equilibrated
grain boundary groove coordinates is 0.1%.
The specimen was kept at steady-state condition for at least
two hours for a setting temperature. At steady state, the total input
power and the stationary thermocouple temperatures were re-
corded with a Hewlett Packard 34401 type multimeter and a Pico
TC–08 data–logger. The temperatures on the different parts of the
specimen were measured with mineral insulated metal sheathed,
0.5 mm in diameter K type thermocouples. The zero or minimum
vertical temperature gradient is desired in thermal conductivity
measurements. The vertical temperature for each setting was tried
to be made as parallel as possible to the vertical axis by moving the
central heater up and down. After all desired power settings and
temperature measurements had been completed during the heat-
ing procedure, the cooling procedure was started in same steps
down to room temperature.
2.4. Measurements of the thermal conductivity of solid and liquid
phases
The thermal conductivity ratio of equilibrated eutectic liquid
phase (Pb–17.5 at.% Sb) to solid Pb solution (Pb–5.8 at.% Sb) phase,
R = KL(eutectic liquid)/KS(Pb) must be known or measured to evaluate
the Gibbs–Thomson coefficient with the present numerical meth-
od. The radial heat flow method is an ideal technique for measur-
ing the thermal conductivity of the solid phase. The thermal
conductivity of the solid Pb solution phase is also needed to eval-
uate the temperature gradient in the solid phases. In the radial heat
flow method, a cylindrical sample was heated by using a single
heating wire along the axis at the centre of the sample and the
sample was kept in a very stable temperature gradient for a period
to achieve a steady-state condition. At the steady–state condition,
the temperature gradient in the cylindrical specimen is given by
Fourier’s law,
Then the sample was removed from the furnace and cut trans-
versely near to the measurement points, after that the specimen
was ground and polished for the measurements of r1 and r2. The
positions of the thermocouples were then photographed with an
Olympus DP12 CCD digital camera placed in conjunction with an
Olympus BX51 type light optical microscope.
A
graticule
(200 Â 0.01 = 2 mm) was also photographed using the same objec-
tive. The photographs of the positions of the thermocouples and
the graticule were superimposed on one another using Adobe
PhotoShop 8.0 software so that accurate measurement of the dis-
tances of stationary thermocouples could be made to an accuracy
dT
dr
Q
of 10 lm. The transverse and longitudinal sections of the speci-
GS ¼
¼ À
ð4Þ
2pr‘KS
men were examined for the porosity, crack and casting defects to
make sure that these would not introduce any error to the mea-
surements. The experimental value of a0 for solid Pb solution and
Pb–Sb eutectic solid were 1.470, 1.487 mÀ1, respectively in the
present work.
where Q is the total input power, r is the distance of the solid–liquid
interface to the centre of the sample, ‘ is the length of the heating
wire which is constant and KS is the thermal conductivity of the so-
lid phase. Integration of the Eq. (4) gives
The thermal conductivities of solid Pb solution and eutectic so-
lid versus temperature are shown in Figure 2. A comparison of
thermal conductivities of solid Pb solution and Pb–Sb eutectic solid
with the thermal conductivity of Pb [21] and Sb [22] are also given
in Figure 2. The values of KS for the solid Pb solution and Pb–Sb eu-
tectic solid at the eutectic melting temperature were obtained to
be 28.7 and 26.8 W/Km, respectively by extrapolating to the eutec-
tic temperature as shown in Figure 2. The thermal conductivities of
solid and liquid phases for solid Pb solution and Pb–Sb eutectic so-
lid and their ratios are given in Table 1.
Q
KS ¼ a0
ð5Þ
T1 À T2
where a0 ¼ lnðr2=r1Þ=2
p
‘ is an experimental constant, r1 and r2
(r2 > r1) are fixed distances from the central axis of the specimen,
T1 and T2 are the temperatures at the fixed positions, r1 and r2.
Eq. (5) could be used to obtain the thermal conductivity of the solid
phase by measuring the difference in temperature between two
fixed points for a given power level provided that the vertical tem-
perature variation are minimum or zero.
The thermal conductivities of solid Pb solution (Pb–5.8 at.% Sb)
and eutectic solid (Pb–17.5 at.% Sb) were measured with the radial
heat flow apparatus. Sufficient amount of metallic materials were
melted to produce an ingot of approximately 100 mm in length
and 30 mm in diameter in a vacuum furnace by using 99.99% pure
Pb and 99.99% pure Sb. After stirring, the molten metallic alloy was
poured into a graphite crucible held in a specially constructed hot
filling furnace at approximately 100 K above the melting tempera-
ture of alloy. The molten metallic alloy was then directionally fro-
zen from bottom to top to ensure that the crucible was completely
It is not possible to measure the thermal conductivity of the li-
quid phase with the radial heat flow apparatus since a thick liquid
layer (10 mm) is required. A layer of this size would certainly have
led to convection. If the thermal conductivity ratio of the liquid
phase to the solid phase is known and the thermal conductivity
of the solid phase is measured at the melting temperature, the
thermal conductivity of the liquid phase can then be evaluated.
The thermal conductivity ratio can be obtained during directional
growth with a Bridgman type growth apparatus. The detail of the
experimental procedure was given in Refs. [13–17].

L. Karabulut et al. / Chemical Physics Letters 503 (2011) 220–225
223
2.5. Measurement of temperature gradient in the solid phase
The average temperature gradient of the solid phase must be
determined for each grain boundary groove shape. This was done
by measuring the input power, the length of heating wire, the po-
sition of the solid–liquid interface and the value of KS for solid Pb
solution phase at the eutectic melting point. By using these mea-
sured values in Eq. (4), temperature gradient can be determined
for each grain boundary groove shape. The total fractional uncer-
tainty in the measurement of temperature gradient is about 6.5%
[14].
3. Results and discussions
3.1. Determination of Gibbs–Thomson coefficient
If the thermal conductivity ratio of equilibrated liquid phase to
solid phase, the coordinates of the grain boundary groove shape
and the temperature gradient of the solid phase are known, the
Figure 2. Thermal conductivities of solid Pb solution and eutectic solid versus
temperature in the Pb–Sb system.
Gibbs–Thomson coefficient (C) can be obtained using the numeri-
cal method described in detail Ref. [13]. The experimental error in
the determination of Gibbs–Thomson coefficient is the sum of
experimental errors in the measurement of the temperature gradi-
ent, thermal conductivity and groove coordinates. Thus the total
error in the determination of Gibbs–Thomson coefficient is esti-
mated to be about 7% [14].
Table 1
The thermal conductivity of solid and liquid phases and their ratios at the eutectic
melting temperature in the Pb–Sb eutectic system.
Alloy
Phases
Melting temperature
(K)
K
R = KL/KS
(W/Km)
The Gibbs–Thomson coefficients for solid Pb solution in equilib-
rium with the eutectic liquid (Pb–17.5 at.% Sb) were determined
with the present numerical model by using ten equilibrated grain
boundary groove shapes. Typical grain boundary groove shape
for solid Pb solution in equilibrium with the eutectic liquid (Pb–
17.5 at.% Sb) is shown in Figure 4. As can be seen from Figure 4, so-
lid Sb solution phase first nucleates on the surface of the Pb solid
solution phase, then both solid Sb solution and the Pb solid solu-
tion phases grow together to form a eutectic grain and this allows
a well defined solid–liquid interface to be observed during the
quench and also the phases, grains and interfaces of the system
are very clear.
Pb–Sb
Eutectic liquid
(Pb–17.5 at.% Sb)
Eutectic solid
(Pb–17.5 at.% Sb)
Eutectic liquid
(Pb–17.5 at.% Sb)
Solid Pb solution
(Pb–5.8 at.% Sb)
525.15
21.7
26.8
21.7
28.7
0.81
525.15
0.76
The thermal conductivity ratio of the eutectic liquid (Pb–17.5
at.% Sb) to the eutectic solid (Pb–17.5 at.% Sb), R = KL(eutectic)/KS(eutec-
tic) was measured to be 0.81 as shown in Figure 3. The thermal con-
ductivity of eutectic solid at the eutectic melting temperature was
also measured to be 26.8 W/Km. Thus, the thermal conductivity of
eutectic liquid was determined to be 21.7 W/Km. The value of
R = KL(eutectic)/KS(Pb) is also found to be 0.76 by using the values of
KL(eutectic) and KS(Pb). The values of KL and KS used in the determina-
tion of Gibbs–Thomson coefficient are also given in Table 1.
The values of
average value of
Pb solution in equilibrium with the Pb–Sb eutectic liquid.
C
C
for solid Pb solution are given in Table 2. The
from Table 2 is (13 1.0) Â 10À8 Km for solid
3.2. Determination of entropy of fusion per unit volume
In order to determine the solid–liquid interfacial energy it is
also necessary to know the entropy of fusion per unit volume,
DSf for the solid phase. The entropy of fusion per unit volume for
an alloy is given by [13],
Figure 4. Typical grain boundary groove shape for solid Pb solution in equilibrium
Figure 3. The cooling rate of Pb–17.5 at.% Sb eutectic alloy.
with the Pb–17.5 at.% Sb eutectic liquid.

224
L. Karabulut et al. / Chemical Physics Letters 503 (2011) 220–225
lid Pb solution is 1.82 Â 10À5 m3 [13]. The values of the relevant
constant used in the determination of entropy of fusion per unit
volume were obtained from the phase diagram [20] and are given
in Table 3.
Comparisons of thermo-physical properties of solid Pb phases
in the different binary systems at their eutectic melting tempera-
ture are given in Table 3. As can be seen from Table 3, the value
of entropy of fusion for solid Pb solution in the Pb–Sb system
agrees with the values of entropy of fusion for Pb solution in the
different binary systems except the value of entropy of fusion for
Pb solution in the Pb–Sn eutectic system. The error in the determi-
nation of entropy of fusion per unit volume is estimated to be
about 5% [24].
Table 2
The Gibbs–Thomson coefficients for solid Pb solution in equilibrium with the Pb–Sb
eutectic liquid. The subscripts LHS and RHS refer to left hand side and right hand side
of the groove respectively.
Grove no.
GK Â 102
a
°
b°
Gibbs–Thomson coefficient
(K/m)
CLHS Â 10À8
CRHS Â 10À8
(Km)
(Km)
a
b
c
d
e
f
g
h
i
30.4
31.4
31.7
30.6
31.3
31.5
33.1
29.8
32.0
31.0
15.4
15.8
17.8
14.5
11.2
13.1
15.2
16.7
18.6
16.9
13.3
11.2
14.1
13.3
14.1
14.1
12.3
15.0
14.3
15.4
13.6
14.4
14.0
13.3
14.0
13.7
14.5
13.2
13.4
13.2
13.6
13.8
13.5
13.8
14.0
13.9
13.2
13.2
13.5
3.3. Evaluation of the solid–liquid interfacial energy
j
13.3
C
= (13 1.0) Â 10À8 Km
ꢀ
If the values of
C and DSf are known, the value of solid–liquid
interfacial energy, rSL can be evaluated from Eq. (3). The solid–li-
quid interfacial energy of the solid Pb solution in equilibrium with
the Pb–Sb eutectic liquid (Pb–17.5 at.% Sb) was evaluated to be
ð1 À CSÞðSL À SS Þ þ CSðSL À SS Þ
A
A
B
B
D
Sf ¼
ð6Þ
(50 6.1) Â 10À3 J mÀ2 by using the values of
C and DSf . The exper-
VS
where SLA, SAS ; SBL and SSB are partial molar entropies for A and B mate-
rials and CS is the concentration of solid phase. Since the entropy
terms are generally not available, for convenience, the undercooling
at constant composition may be related to the change in composi-
tion at constant temperature. For a solid sphere of radius r [23]
imental error in the determination of solid–liquid interfacial en-
ergy is the sum of experimental errors of the Gibbs–Thomson
coefficient and the entropy change of fusion per unit volume. Thus,
the total experimental error of the solid–liquid interfacial energy
evaluation in present work is estimated to be about 12%.
2
r
SLVSð1 À CLÞCL
3.4. Grain boundary energy
D
Cr ¼
ð7Þ
rRTMðCS À CLÞ
If the grains on either side of the grain boundary are the same
phase then the grain boundary energy can be expressed by
where R is the gas constant, TM is the melting temperature and VS is
the molar volume of solid phase. For small changes
rgb ¼ 2
where h ¼
r
SL cos h
ð12Þ
2mL
r
SLVSð1 À CLÞCL
D
Tr ¼ mLDCr ¼
ð8Þ
hAþhB
is the angle that the solid–liquid interfaces make
rRTMðCS À CLÞ
2
with the y axis. The angles, hA and hB were obtained from the cusp
coordinates, x, y using a Taylor expansion for parts at the base of the
groove. According to Eq. (12), the value of rgb should be smaller or
where mL is the slope of liquidus. For spherical solids, the curvature
undercooling is
equal to twice of solid–liquid interfacial energy, i.e. rgb 6 2rSL
.
2
rDSf
rSL
D
Tr ¼
ð9Þ
The value of the grain boundary energy for the solid Pb solution
was found to be (99 13.0) Â 10À3 J mÀ2 by using the values of the
rSL and h into Eq. (12). The estimated error in determination of h
angles was found to be 1%. Thus the total experimental error in
the resulting grain boundary energy is about 13%.
From Eqs. (8) and (9), the entropy of fusion for an alloy is writ-
ten as
RTM CS À CL
D
Sf ¼
ð10Þ
ð11Þ
As mentioned above, interfacial energy anisotropy is considered
to play a critical role in many phase transformations. The determi-
nation of effects of anisotropy on the interfacial energy is difficult.
In literature, there are no theoretical and experimental available
data for anisotropy of interfacial energy of the solid Pb phase. Thus
the solid–liquid interfacial energy is assumed to be isotropic.
The comparisons of the values of Gibbs–Thomson coefficient
mLVS ð1 À CLÞCL
The molar volume, VS is expressed as
1
VS ¼ VcN
a n
where VC is the volume of the unit cell, Na is the Avogadro’s number
and n is the number of atoms per unit cell. The molar volume of so-
(C), solid–liquid interfacial energy (rSL) and grain boundary energy
Table 3
Comparisons of thermo-physical properties of solid Pb phases in the different binary systems at their eutectic melting temperature.
System
Pb–Sn
Pb–Sb
Pb–Pd
Pb
Pb–Pt
Pb
Pb–Bi
Pb–Ag
Pb–Au
Pb
Solid phase
Pb–29 at.% Pb solution (Pb–5.8
Sn
2.32 [13]
456 [13]
Pb solution (Pb–21.9
at.% Bi)
Pb solution (Pb–0.2
at.% Ag)
1.00 [20]
at.% Sb)
0.81 [20]
525.15 [20]
The value of f(C)a for solid phase
Eutectic melting temperature, Tm (K)
1.09 [20] 1.06 [20] 0.68 [20]
533.15
[20]
1.82 [13] 1.82 [13] 1.82 [13]
871.43
[20]
1.18 [20]
485.65
[20]
1.82 [13]
816.50
[20]
563.15
[20]
460.15 [20]
577.15 [20]
Molar volume of solid phase  10À5 (m3) 1.82 [13]
1.82 [13]
527.20 [20]
1.82 [13]
952.00[20]
Liquidus slope, mL (K/at.fr)
408 [13]
895.06
[20]
472.59 [20]
Entropy of fusion for solid phase,
D
11.8 [13]
3.7
3.0
3.0
3.0
2.8
3.2
Sf  105 (J/K m3)
a
CS ÀCL
fðCÞ ¼
.
ð1ÀCLÞCL

L. Karabulut et al. / Chemical Physics Letters 503 (2011) 220–225
225
Table 4
Comparisons of the values of C, rSL, rgb for solid Pb solution in the Pb–Sb eutectic system measured in present work with the values of C, rSL, rgb for solid Pb phases in the
different binary systems obtained in previous works.
System
Pb
Solid phase
Liquid phase
Melting
temperature
(K)
Gibbs–Thomson
Solid–liquid interface energy rSL Â 10À3
Grain boundary
coefficient
(Km)
C
 10À8
(J mÀ2
)
energy rgb  10À3
(J mÀ2
–
)
Theoretical
Experimental
Pb
Pb
600.6
–
50.2 [25]
69 [26]
66 [27]
49 [28]
55 [29,6]
62 [31]
42.7 [32]
40 7 [33]
46 [5]
40 [34]
50 [31]
49 [31]
Pb–Bi
Pb–Ag
Pb–Au
Pb–Pt
Pb–Pd
Pb–Sn
Pb–Sb
Pb
Pb
Pb
Pb
Pb
Pb
Pb
Pb–38.0 at.% Bi
Pb–4.69 at.% Ag
Pb–15.24 at.% Au
Pb–5.29 at.% Pt
Pb–8.40 at.% Pd
Sn–26.1 at.% Pb
Pb–17.5 at.% Sb
460.15
577.15
485.65
563.15
533.15
456
14.7*
19.3*
14.5*
17.0*
16.8*
44.1 [30]
53.9 [30]
46.6 [30]
51.2 [30]
50.5 [30]
52.7 [28]
53 [30]
–
–
–
–
–
–
–
–
–
–
4
0.45 [13]
56 8.15 [13]
50 6.1 [PW]
111 15.61 [13]
99 13.0 [PW]
525.15
13 1.0 [PW]
PW: Present work.
*
Calculated from Eq. (3) by using the values of rSL and
DSf .
(r
gb) for solid Pb solution measured in present work with the val-
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This project was supported by Erciyes University Scientific Re-
search Project Unit under Contract No: FBY-09-976. The authors
are grateful to Erciyes University Scientific Research Project Unit
for their financial support.
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