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RARE METALS
Vol. 30, No. 4, Aug 2011, p. 424
DOI: 10.1007/s12598-011-0408-0
Effect of sample diameter on primary and secondary dendrite arm spacings
during directional solidification of Pb-26wt.%Bi hypo-peritectic alloy
HU Xiaowua, YAN Honga, CHEN Wenjingb, LI Shuangmingc, and FU Hengzhic
a
School of Mechanical-Electrical Engineering, Nanchang University, Nanchang 330031, China
Department of Applied Physics, Northwestern Polytechnical University, Xi’an 710072, China
c
State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China
b
Received 27 September 2010; received in revised form 11 December 2010; accepted 13 December 2010
© The Nonferrous Metals Society of China and Springer-Verlag Berlin Heidelberg 2011
Abstract
The microstructure scales of dendrites, such as primary and secondary dendrite arm spacings, control the segregation profiles and the formation of secondary phases within interdendritic regions, which determine the properties of solidified structures. Investigations on primary and
secondary dendrite arm spacings of primary α-phase during directionally solidified Pb-26wt%Bi hypo-peritectic alloy were carried out in this
research, and systematic studies were conducted using cylindrical samples with different diameters (Φ = 1.8 and 7.0 mm) in order to analyze
the effects of sample diameter on the primary and secondary dendrite arm spacings. In this work, the dependence of dendrite arm spacings on
growth velocity was established. In addition, the experimental data concerning the primary and secondary dendrite arm spacings were compared with the main predictive dendritic models from the literatures. A comparison between experimental results for dendrite arm spacings of
the 1.8-mm-diameter sample and 7.0-mm-diameter sample was also conducted.
Keywords: lead bismuth alloys; dendrite arm spacing; directional solidification; dendrites
1. Introduction
Peritectic solidification has attracted more attention in
experimental and theoretical studies [1-17]. Since many
technologically important materials are peritectic, studies on
directional solidification of peritectic alloys, such as Sn-Cd
[2-3], Sn-Sb [4], Pb-Bi [5-6], Zn-Cu [7-8], Zn-Ag [9], high
temperature intermetallics Ti-Al [10] and Ni-Al [11], superconducting materials YBCO [12], magnetic materials
Nd-Fe-B [13], and structural materials Fe-Ni [14-15] and
Fe-Cr-Ni [16] have been carried out.
It is a well-known fact that the dendrite arm spacings can
affect not only microsegregation profiles but also the formation of the secondary phase within interdendritic regions,
which influences mechanical properties of cast structures.
Recently, some investigations on microstructure scales during directional solidification of peritectic alloys were carried
out, for example, Ma et al. [7] studied the relationship between the microstructural length scales, the solidification
conditions and alloy composition in Zn-Cu peritectic alloys.
For Fe-Ni peritectic alloys, the spacing selection of cellular
peritectic coupled growth during directional solidification
Corresponding author: HU Xiaowu
E-mail: [email protected]
was investigated [18], it is found that the cellular spacing of
CPCG first decreases and then increases with increasing velocity. Secondary dendrite arm coarsening in peritectic solidification was studied through considering the effects of
peritectic reaction and transformation during directional solidification in Zn-Cu, Pb-Bi [19] and Nd-Fe-B [20] peritectic alloys.
However, beside the effects of the peritectic reaction and
transformation, the effect of sample diameter should be
considered due to the intensity of melt convection which
varies with different sample diameters that influences
the peritectic solidification. Thus, the effects of sample diameter on the primary and secondary dendrite arm spacings
of the primary phase during directionally solidified peritectic
alloys need to be investigated. In this paper, the Pb-26wt.%Bi
hypo-peritectic alloy was chosen for study. In order to
investigate the effect of sample diameter on the microstructural scales, two kinds of alumina tubes with different
inner diameters of 1.8 mm and 7.0 mm were used in directional solidification experiments. And the experimental results were compared with current theoretical models suggested for primary spacing and secondary spacing, respec-
Hu X.W. et al., Effect of sample diameter on primary and secondary dendrite arm spacings during directional…
range, ΔT0 and ΔT ′ can be evaluated by
tively.
ΔT ′ = m(C0 − Clm )
2. Dendrite arm spacing models
⎛ DΓ ⎞
⎟
⎝ ΔT 0 k ⎠
1/ 4
λ1 = 4.3ΔT ′1/ 2 ⎜
V −1/ 4G −1/ 2
(Kurz-Fisher model [22])
λ1 = 2 2G
−1/ 2
V
−1/ 4
(2)
( Lk ΔT0 Γ D )
⎡
(3)
Γ
⎤
⎥
−
mC
(
k
1)
0
⎣
⎦
0.41
⎛D⎞
⎜ ⎟
⎝V ⎠
0.59
(Hunt-Lu model [24])
(4)
λ
is
the
primary
dendrite
arm
spacing,
k
is
the
equiwhere 1
librium distribution coefficient, D is the diffusion coefficient
in the liquid, Г is the Gibbs-Thomson coefficient, V is the
growth velocity, G is the temperature gradient, C0 is the alloy composition, m is the liquidus slope, and L is a constant
that depends on harmonic perturbations. According to
Trivedi’s analysis [23], for dendritic growth, the value of L
is equal to 28. ΔT0 and ΔT ′ are the equilibrium and nonequilibrium solidification temperature range and are given
as follows:
m(k −1)C0
ΔT0 =
(5)
k
GD ⎞ ΔT0
⎛
ΔT ′ = ⎜ 1 −
⎟
⎝ V ΔT 0 ⎠ 1 − k
(6)
For secondary dendrite arm spacing, Feurer and Wunderlin [25] predicted that the secondary dendrite arm spacing,
λ2, was proportional to the cube root of solidification time (tf)
and gave
λ 2 = 5.5( Mt f )1/ 3
(7)
with
Γ D ln(C lm / C 0 )
m(k − 1)(C lm − C 0 )
(8)
and
ΔT ′
(9)
GV
where ΔT ′ is the nonequilibrium solidification range,
which is usually larger than the equilibrium solidification
tf =
is the maximum concentration
For a peritectic system,
in the final drop of liquid, can be approximately estimated
simply as
C lm = C p
(11)
where Cp is the peritectic composition in the liquid at peritectic temperature. This estimate is reasonable on the basis
that the arm coarsening for the primary phase should be
suppressed by the formation of the peritectic phase surrounding the primary phase if the liquid composition exceeds the peritectic composition Cp.
1/ 4
(Trivedi model [23])
λ1 = 8.18k −0.335 ⎢
(10)
Clm
The theoretical models proposed for determining the
primary dendrite arm spacing [21-24] are shown as Eqs.
(1)-(4):
2.83(k ΔT0 DΓ ) 1/ 4
λ1 =
(Hunt model [21])
(1)
V 1/ 4G 1/ 2
M =
425
3. Experimental
The Pb-26wt.%Bi alloy was used, and the master alloy
ingot was prepared from 99.95% purity lead and 99.99%
purity bismuth in an induction furnace under an argon atmosphere. The samples were machined to 1.8 and 7.0 mm in
diameter and 100 mm in length from the as-cast ingot.
A vertical Bridgman furnace, which was described in detail elsewhere [26], was used to produce directionally solidified rods of the Pb-26wt.%Bi alloy as follows. The machined rods were inserted into a high purity alumina tube
with 1.8- and 7.0-mm inner diameter, respectively, placed
into a graphite cylinder of the furnace. After being heated to
stabilization at 375°C for 30 min, the sample was then withdrawn into a water-cooled cylinder containing liquid
Ga-In-Sn metal (LMC) at a constant velocity (from 5 to 500
μm/s) for 80 mm to ensure a steady-state solidification condition. At the end of each experiment, the alumina tube was
dropped into the LMC to quench the solid/liquid interface.
The sample temperature measurements were carried out using NiCr-NiSi thermocouple. The measured temperature
gradient around the peritectic temperature is about 20
K/mm.
The quenched specimens were sectioned transversely and
longitudinally, respectively. The microstructures of the polished specimens were revealed using the reagent consisting
of 100 mL distilled water (H2O), 10 mL of nitric acid
(HNO3) and 4 g of ammonium molybdate ((NH4)6MO7O24⋅
4H2O). An Olympus TG-3 optical microscope was employed to observe the microstructures of the specimens. The
values of the primary dendrite arm spacing (λ1) and secondary dendrite arm spacing (λ2) were measured on the transverse and longitudinal sections, respectively. For accuracy,
the measurements were carried out at least on six different
regions for each specimen. Figs.1(a) and 1(b) show the
schematic diagrams for measuring λ1 and λ2, respectively.
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RARE METALS, Vol. 30, No. 4, Aug 2011
Fig. 1. Schematic diagrams for measuring λ1 (a) and λ2 (b).
4. Results and discussion
Figs. 2 and 3 shows the observed typical microstructures
of Pb-26wt.%Bi alloy on the transverse sections with increasing growth velocity in the range of 5-500 μm/s at G =
20 K/mm. The left-hand side is the microstructures of
specimens with a 1.8-mm diameter, and the right-hand side
is the microstructures of specimens with a 7.0-mm diameter,
respectively. Fig. 4 corresponds to the longitudinal microstructures. The formation of dendritic primary α-phase in a
matrix of peritectic β-phase and secondary dendrite arms
from primary dendrites can be clearly noticed.
Fig. 2. Typical microstructures observed at the transverse section of Pb-26wt.%Bi hypo-peritectic alloy with 1.8-mm-diameter
samples (left) and 7.0-mm-diameter samples (right).
Hu X.W. et al., Effect of sample diameter on primary and secondary dendrite arm spacings during directional…
427
Fig. 3. Typical microstructures observed at the transverse section of Pb-26wt.%Bi hypo-peritectic alloy with 1.8-mm-diameter
samples (left) and 7.0-mm-diameter samples (right).
The primary and secondary dendrite arm spacings were
measured accurately through the transverse and longitudinal
sections, respectively. Figs. 5 and 6 present the average experimental values of primary and secondary dendrite arm
spacings (λ1 and λ2) as a function of tip growth velocity (V),
respectively. In Fig. 5, circles present the experimental results of the primary dendrite arm spacing of α-phase in the
7.0-mm-diameter sample, and the λ1 values in the 7.0-mm-
diameter sample were published before in Ref. [6]. Triangles
correspond to the results of the 1.8-mm-diameter sample.
In Fig. 6, squares represent the experimental results of the
secondary dendrite arm spacing of α-phase in the 7.0-mmdiameter sample, and the dot line represents a linear fit of
these experimental results. Diamonds present the results of
the 1.8-mm-diameter sample, and the dash line represents a
linear fit for experimental data.
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RARE METALS, Vol. 30, No. 4, Aug 2011
Fig. 5. Comparison of observed primary dendrite arm spacing of primary α-phase as a function of V with the predictions
of the Hunt model, Kurz-Fisher model, Trivedi model and
Hunt-Lu model, respectively.
Fig. 6. Comparison of the measured λ2 of primary α-phase
for Pb-26wt.%Bi hypo-peritectic alloy as a function of V with
predictions of the Feurer-Wunderlin (F-W) model
Fig. 4. Typical microstructures observed at the longitudinal
section of Pb-26wt.%Bi hypo-peritectic alloy with 1.8-mm-diameter samples (left) and 7.0-mm-diameter samples (right).
Fig. 5 also shows comparisons between the present experimental results of primary dendrite arm spacing and
theoretical predictions furnished by the Hunt model,
Kurz-Fisher model, Trivedi model and Hunt-Lu model,
given by Eqs. (1-4), respectively. The physical parameters
used are given in Table 1. The predictions of the Hunt model
and Trivedi model show reasonable agreement, whereas the
prediction of the Kurz-Fisher model overestimates the primary spacing, and the prediction of the Hunt-Lu model is far
smaller than the experimental results. Because the Hunt-Lu
model is used to predict the primary dendrite spacing or
cellular spacing with high growth rate, the spacing is thus
small. Analyzing the experimental measurements in Fig. 5,
it can be found that the primary dendrite arm spacing in both
7.0-mm- and 1.8-mm-diameter samples decreases with increasing growth velocity. Further, Fig. 5 indicates that, for
any comparison examined, the experimental scatter of the
Hu X.W. et al., Effect of sample diameter on primary and secondary dendrite arm spacings during directional…
7.0-mm-diameter sample lies between the calculation values
obtained by the Trivedi model and Hunt model. A similar
observation was reported recently concerning directionally
solidified hypoeutectic Al-Cu alloys [27]. Comparing with
the experimental results of the 7.0-mm-diameter sample, the
results of the 1.8-mm-diameter sample do not completely lie
between the calculated ranges by the Trivedi model and
Hunt model, as shown Fig. 5. In addition, the exponent values of growth rate (V) for average primary dendrite arm
spacing (λ) were found to be 0.359 and 0.333 for 7.0-mmand 1.8-mm-diameter samples, respectively. In detail, the
relationships between λ1 and V can be given as λ1V0.359 =
468.61 and λ1V0.333 = 308.32 for 7.0-mm- and 1.8-mm-diameter samples, respectively.
Table 1. Physical parameters regarding peritectic reaction
L + α → β at 187°C for the Pb-Bi system
Peritectic temperature, Tp / °C
187
Bi content of α at Tp, Cα / wt.%
22.1
Bi content of β at Tp, Cβ / wt.%
29.2
Bi content of L at Tp, Cp / wt.%
38.2
α-liquidus slope, mα / (°C/wt.%)
−5.01
β-liquidus slope, mβ / (°C/wt.%)
−2.17
Distribution coefficient of α, kα
0.579
Distribution coefficient of β, kβ
0.764
Gibbs-Thomson coefficient, Г / mK
1.3 × 10−7
Diffusion coefficient in liquid, D / (cm2⋅s−1)
1.3 × 10−5
−1
Temperature gradient, G / (K⋅mm )
20
Fig. 6 shows comparisons between the theoretical predictions for secondary spacing furnished by the F-W model,
given by Eqs. (7)-(11) with the present experimental results.
A good agreement can be observed for the 7.0-mm-diameter
sample, but a slight overestimation for the 1.8-mm-diameter
sample. In both cases, the secondary dendrite arm spacing
decreases with increasing growth velocity. It can be seen in
Fig. 6 that exponent values of 0.231 and 0.18 are suggested
to relate the secondary spacing of primary α-phase variation
with the growth velocity for 7.0-mm- and 1.8-mm-diameter
samples, respectively. Thus, the relationships between λ2
and V can be given as λ2V0.231 = 77.50 and λ2V0.18 = 41.37
for 7.0-mm- and 1.8-mm-diameter samples. The exponent
values of V for primary and secondary spacings of the previous theoretical models and the present experimental results are shown in Table 2.
In order to investigate the effect of sample diameter on the
primary and secondary dendrite arm spacings of primary
α-phase during directionally solidifying of Pb-26wt.%Bi
hypo-peritectic alloy, the samples with diameters of 1.8 mm
and 7.0 mm were unidirectionally solidified at various
429
Table 2. Exponent values of V for primary and secondary
spacings
Model
K1*
K2*
Ref.
Hunt
0.25
—
[21]
Kurz-Fisher
0.25
—
[22]
Trivedi
0.25
—
[23]
Hunt-Lu
0.59
—
[24]
Feurer-Wunderin
—
0.33
[25]
Bouchard-Kirkaldy
0.5
0.66
[28]
Trivedi-Sombounsok
0.5
0.5
[29]
This work for 1.8-mm diameter
0.333
0.18
—
This work for 7-mm diameter
0.359
0.231
—
* K1 is the exponent value of V for λ1 and K2 is the exponent value
of V for λ2.
growth velocities (ranging from 5 to 500 μm/s). When
comparing the average values of λ1 obtained in the 1.8-mmdiameter samples with those from the 7.0-mm-diameter
samples, a tendency of reduction of primary dendrite arm
spacing can be observed for all the growth velocities. For
the same V, λ1 is reduced (about 30%) in the 1.8-mm-diameter sample. The same tendency will occur for secondary
dendrite arm spacing, too.
For the Pb-Bi peritectic system, the dendrite arm spacings
are influenced by the peritectic reaction and transformation
[30]. The peritectic reaction is controlled by the solute diffusion through the liquid, while the peritectic transformation is
controlled by the solute diffusion through the peritectic
phase. In the case of 7.0-mm-diameter samples, the solute
convection in the interdendritic regions is stronger than that
in 1.8-mm-diameter samples. This stronger solute convection results in a faster diffusion of the Bi component in the
primary α-phase interdendritic regions, which controls the
rate and degree of peritectic reaction and transformation of
primary α-phase to peritectic β-phase. And the thin-arm
dissolution is performed by the peritectic reaction and
transformation. As a result, it is expected that stronger solute
convection in a larger sample diameter can enhance the dissolution of thin-arms of primary α-phase, which thus leads
to larger dendrite arm spacings of α-phase, including primary and secondary dendrite arm spacings.
5. Conclusions
Directional solidification experiments were conducted on
Pb-26wt.%Bi hypo-peritectic alloy using samples with diameters of 7.0 mm and 1.8 mm to investigate the effect of
sample diameter on the primary and secondary dendrite arm
spacings of primary α-phase. The major conclusions derived
from the present study are shown as follows.
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RARE METALS, Vol. 30, No. 4, Aug 2011
(1) The primary (λ1) and secondary (λ2) dendrite arm
spacings of primary α-phase were found to be dependent on
growth velocity and observed to decrease as the growth velocity increased.
(2) Using the method of linear fitting experimental data,
the relationships between λ1 and V can be given as λ1V0.359 =
468.61 μm1.359/s0.359 and λ1V0.333 = 308.32 μm1.333/s0.333 for
7.0-mm- and 1.8-mm-diameter samples, respectively. And
the relationships between λ2 and V are given as follows:
λ2V0.231 = 77.50 μm1.231/s0.231 for the 7.0-mm-diameter sample and λ2V0.18 = 41.37 μm1.18/s0.18 for the 1.8-mm-diameter
sample.
(3) Both the Hunt model and Trivedi model match the
experimental results for primary dendrite arm spacing in
both 7.0-mm- and 1.8-mm-diameter samples. However, the
Kurz-Fisher model predictions overestimate the primary
spacings. The prediction of the F-W model is compared with
the experimental results of secondary dendrite arm spacing
in 7.0-mm- and 1.8-mm-diameter samples, a reasonable
agreement is obtained for the former case, and a little overestimation is generated for the latter case.
(4) Comparing the values of λ1 and λ2 obtained in the
1.8-mm-diameter sample with those from the 7.0-mm-diameter sample, a tendency of reduction of dendrite arm
spacings can be observed. It seems that the solute convection currents induced inside the interdendritic regions of a
sample with a larger diameter (7.0 mm) are responsible for
the reduction in these dendritic spacings.
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
Acknowledgements
This research was financially supported by the China
Postdoctoral Science Foundation (No. 20110491492), National Natural Science Foundation of China (No. 50765005)
and the Innovative Group of Science and Technology of
College of Jiangxi Province, China (No. 00008713).
[15]
[16]
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