METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 34A, OCTOBER 2003—2365
A Mushy-Zone Rayleigh Number to Describe Interdendritic
Convection during Directional Solidification of Hypoeutectic
Pb-Sb and Pb-Sn Alloys
S.N. TEWARI and R. TIWARI
Based on measurements of the specific dendrite surface area (S
v
), fraction of interdendritic liquid
(
), and primary dendrite spacing (
1
) on transverse sections in a range of directionally solidified
hypoeutectic Pb-Sb and Pb-Sn alloys that were grown at thermal gradients varying from 10 to
197Kcm
1
and growth speeds ranging from 2 to 157
m s
1
, it is observed that S
v
1
1
S*
0.33
(3.38 3.29
8.85
2
), where S* D
l
G
eff
/Vm
1
C
o
(k 1)/k, with D
l
being the solutal diffu-
sivity in the melt, G
eff
being the effective thermal gradient, V being the growth speed, m
l
being the
liquidus slope, C
o
being the solute content of the melt, and k being the solute partition coefficient.
Use of this relationship in defining the mushy-zone permeability yields an analytical Rayleigh
number that can be used to describe the extent of interdendritic convection during directional solid-
ification. An increasing Rayleigh number shows a strong correlation with the experimentally observed
reduction in the primary dendrite spacing as compared with those predicted theoretically in the
absence of convection.
I. INTRODUCTION
N
ATURAL
convection in the dendritic mushy zone is
responsible for the nucleation of spurious grains
[1]
and the
formation of channel segregates
[2–9]
in directionally solidi-
fied alloys. It produces radial
[10,11,12]
and longitudinal
macrosegregations
[13,14]
and alters the cellular-dendritic and
planar-cellular transitions.
[15]
It also appears to reduce the
primary dendrite spacing.
[13–18]
Numerical simulations of the transport phenomena
during alloy solidification have shown some success in
predicting some of the solidification defects in metal
alloys,
[19–22]
but they have two serious limitations. First,
the two-order-of-magnitude difference between the thermal
and solutal length scales makes the three-dimensional com-
putation very time consuming, and, second, the interdepen-
dence of the mushy-zone morphology, its permeability, and
its convection does not allow an exact analysis of the prob-
lem. For example, the extent of convection depends upon
the permeability of the mushy zone, but the convection alters
the primary dendrite spacing and, hence, affects the perme-
ability itself. Therefore, several attempts have been made to
describe convection in terms of a nondimensional mushy-
zone Rayleigh number.
[4,7,23–26]
The mushy-zone Rayleigh
number proposed by Beckermann et al.
[26]
(R
aB
) has been
successful in describing the freckle initiation in nickel-based
superalloy castings. However, the critical Rayleigh number
for the onset of freckling has not been tested against the
extensive freckling data available in the literature on other
alloy systems, such as Pb-Sn,
[3–7,9]
Pb-Sb,
[4]
and Al-Mg.
[8]
Describing convection through the mushy zone requires
knowledge of the relationship between mushy-zone perme-
ability (K) and its morphology, such as primary dendrite
spacing (
1
), fraction of interdendritic liquid (
), etc. Many
experimental
[27–34]
and numerical
[35,36,37]
investigations have
been carried out to describe such a relationship. Experiments
involving molten lead, to measure permeability in an Al-Cu
alloy;
[27]
water, to determine permeability in a partially solid-
ified Al-Si alloy;
[28]
and a transparent borneol-parafin sys-
tem
[29]
showed that permeability is proportional to (
)
n
, with
n varying between 2 and 3.3. Others
[31–34]
used eutectic liquid
to measure permeability in aluminum-copper alloys and
observed it to follow the Kozeny–Carman relationship for
flow through a porous bed:
[38]
K
3
/k
c
S
v
2
(1
)
2
, where
S
v
, the area of the solid-liquid interface per unit volume of
the solid, is replaced by the dendrite surface area per unit
volume, and k
c
, a constant that depends upon the charac-
teristic of the porous medium, was found to be about 5.
Bhat et al.
[37]
analyzed the aforementioned experimental data
and used numerical analysis to describe the permeability
parallel to the primary dendrite array (K
y
) and that perpendi-
cular to the array (K
x
) in the following manner:
Schneider et al.
[20]
used the experimental data reported by
Bhat
[32]
to determine the following mushy-zone perme-
ability: K 6 10
4
1
2
3
/(1
)
2
. Beckermann et al.
[26]
have used this permeability relationship in describing their
mushy-zone Rayleigh number. Yang et al.
[24]
have used
K 3.75 10
4
2
1
2
to describe their nondimensional
mushy-zone Rayleigh number.
for 0.75 f 1
(6.49 10
2
5.43 10
2
(f/1 f)
0.25
)l
1
˛
2
4.04 10
6
(f/1f)
6.7336
l
1
˛
2
for 0.65 f 0.75
K
X
1.09 10
3
f
3.32
l
1
˛
2
for f 0.65
0.5(1f)
2
) l
1
˛
2
for 0.75
1
0.074 (ln(1f) 1.49 2(1f)
2.05 10
7
(f/1f)
10.739
l
1
˛
2
for 0.65 f 0.75
K
Y
3.75 10
4
f
2
l
1
˛
2
for f 0.65
S.N. TEWARI, Professor, and R. TIWARI, Graduate Student, are with
the Chemical and Biomedical Engineering Department, Cleveland State
University, Cleveland, OH 44115. Contact e-mail: [email protected]
Manuscript submitted February 4, 2003.
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2366—VOLUME 34A, OCTOBER 2003 METALLURGICAL AND MATERIALS TRANSACTIONS A
An examination of the previously described permeability
relationships shows that they all assume permeability to
depend only upon the primary dendrite spacing and fraction
of interdendritic liquid. They do not take into account the
role of side branching in determining the mushy-zone per-
meability. For example, one can visualize two dendrite arrays
having equal
1
and
values, but one array growing at a
smaller gradient of constitutional supercooling (just beyond
the cell-dendrite transition, having less-developed side
branches) and the other growing at a larger gradient of con-
stitutional supercooling (well-developed side branches). It is
obvious that the permeability of the second array would be
smaller than the first array, even though their
1
and
val-
ues are identical.
The purpose of this study is to identify a mushy-zone
Rayleigh number that accounts for the role of side branch-
ing. We will first carry out a detailed examination of the
growth-parameter dependence of the specific dendrite sur-
face area in directionally solidified Pb-Sb and Pb-Sn alloys,
in order to obtain its dependence on the primary dendrite
spacing, fraction of interdendritic liquid, and the parameter
S* D
l
G
eff
/(Vm
l
C
o
(k1)/k), where D
l
is the solutal
diffusivity in the melt, G
eff
is the effective thermal gradi-
ent, V is the growth speed, m
l
is the liquidus slope, C
o
is
the solute content of the melt, and k is the solute parti-
tion coefficient. The parameter S* indicates the extent of
side branching of dendrite arrays; it is equal to unity for
a planar liquid-solid interface and decreases to zero with
increasing side-branching tendency of dendrites. This value
of S
v
will be used to define a new mushy-zone Rayleigh
number (R
aM
) in a manner identical to that used by
Beckermann et al.
[26]
An appropriately defined mushy-zone Rayleigh number
must be able to describe three important experimental obser-
vations. It should be able to correlate the extent of natural
convection with the reduction in primary dendrite spacing;
it should be able to relate the intensity of mushy-zone
convection during directional solidification and the result-
ing longitudinal macrosegregation; and it should be useful
in predicting the onset of channel-segregate formation.
In this article, we will examine the first effect: primary-
dendrite-spacing reduction due to mushy-zone convection.
In a subsequent article, we will show that the Rayleigh
number R
aM
can also be used to predict the extent of lon-
gitudinal macrosegregation and onset of channel-segregate
formation in a range of alloys.
II. EXPERIMENTAL
This study is mostly based on our earlier-reported and
ongoing research on directionally solidified hypoeutectic
Pb-Sb and Pb-Sn alloys. These samples were direction-
ally solidified in flowing argon atmosphere, mostly in
0.7-cm-i.d. quartz ampoules, in a furnace arrangement
where the furnace was translated and the sample was kept
stationary to avoid the mechanical vibrations. Most of the
samples were quenched after 9 to 10 cm of directional
solidification (initial melt-column length at the onset of
directional solidification was 18 to 20 cm) by a blast of
helium gas circulating through a liquid nitrogen tank. Some
samples were quenched by spraying water on the ampoule
surface. A steady-state thermal gradient was maintained
during the entire directional solidification process.
[9]
A range
of alloy compositions (Pb-2.2 and 5.8 wt pct Sb, Pb-10 to
54.7 wt pct Sn) were directionally solidified at several ther-
mal gradients (ranging from 10 to 197 K cm
1
) and growth
speeds (ranging from 2 to 157
m s
1
). Transverse sec-
tions through the mushy zone as a function of distance
from the quenched array tips were utilized for measuring
S
v
; S
v
is the ratio of the perimeter of the dendrite and its
cross-sectional area. The sample-sectioning and image-
analysis techniques have been described earlier.
[39]
The
transverse images were also used to measure the corres-
ponding fraction of interdendritic liquid. The primary-
dendrite-spacing data used in this article correspond to
where N is the number of dendrites on a sam-
ple cross-sectional area of A.
III. RESULTS AND DISCUSSION
A. Experimental Characterization of the Mushy-Zone
Morphology
1. Typical microstructures
Figure 1 shows typical transverse mushy-zone microstruc-
tures as a function of distance from the quenched array tips
in a Pb-5.8 wt pct Sb sample grown at 1.5
m s
1
and
40Kcm
1
. The dendrites (light-colored features) in this
sample are not well branched, because the growth condition
is just beyond the cell-to-dendrite transition.
Figure 2 shows similar transverse sections through the
quenched mushy zone of a Pb-23 wt pct Sn alloy sample
that was directionally solidified at 2
m s
1
with a thermal
gradient of 52 K cm
1
. In this figure, the darker features are
the lead dendrites, and the lighter features are the quenched
interdendritic liquid. This sample has well-branched primary
dendrites.
The typical microstructures shown in Figures 1 and 2
are from the as-milled sample surfaces without any subse-
quent polishing. The dendrite boundaries were traced from
such micrographs for determining the area and the perimeter
of the individual dendrites and the corresponding fraction
of solid.
2. Mushy-zone morphology
Figure 3 shows a typical variation in the mean dendrite
perimeter, mean dendrite area, and fraction of solid as a
function of distance from the dendrite tip for a Pb-23 wt pct
Sn alloy sample that was directionally solidified at 2
ms
1
with a thermal gradient of 52 K cm
1
.
Data, such as those shown in Figure 3, were used to obtain
the variation in S
v
as a function of fraction of liquid (
),
shown in Figure 4. The scatter bars correspond to a 1
standard deviation. Figure 4(a) contains data from Pb-5.8
wt pct Sb alloys grown at 1, 1.5, and 3
ms
1
at a ther-
mal gradient of 140 K cm
1
. The 1
m s
1
sample had a
cellular morphology, the 1.5
m s
1
sample was grown
at the cell-to-dendrite transition, and the 3
m s
1
sample
had a dendritic morphology. Figure 4(b) contains data
from Pb-5.8 wt pct Sb alloy samples grown at a thermal
gradient of 40 K cm
1
at 3, 10, and 30
m s
1
. All these
three samples correspond to the well-branched primary den-
drites. Figure 4(c) contains data from several Pb-Sn alloys
1
(A/N 1),
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METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 34A, OCTOBER 2003—2367
Fig. 1—Typical transverse microstructures of the mushy zone at varying distances from the quenched array tip in a directionally solidified Pb-5.8 wt pct Sb
alloy grown at 1.5
m s
1
with a thermal gradient of 40 K cm
1
: (a) 24
m from the tip, (b) 71
m from the tip, (c) 375
m from the tip, and (d) 1065
m
from the tip.
(c)(d)
(a)(b)
grown with a well-branched dendrite morphology at vary-
ing thermal gradients and growth speeds. The scatter band
is available for only the Pb-23 wt pct Sn sample grown at
2
m s
1
with a thermal gradient of 53 K cm
1
and the
Pb-30 wt pct Sn sample grown at 0.35
m s
1
and
8Kcm
1
. It is not available for the other three Pb-Sn alloy
samples that were examined some time ago. This figure
shows that, generally, the dendrite specific surface area is
low near the base of the mushy zone and increases toward
the array tips. The side-branch coarsening, impingement,
and coalescence that occur away from the array tips are
responsible for this behavior.
B. Mushy-Zone Permeability
Let us compare the porous-bed permeability, defined by the
Kozeny–Carman relationship K
3
/k
c
S
v
2
(1
)
2
,
[38]
with
the permeability relationships that have been used to define
the mushy-zone Rayleigh numbers. Beckermann et al.
[26]
have used K 6 10
4
1
2
3
/(1
)
2
, and Yang et al.
[24]
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2368—VOLUME 34A, OCTOBER 2003 METALLURGICAL AND MATERIALS TRANSACTIONS A
(a)
(b)
(c)(d)
Fig. 2—Typical transverse microstructures of the mushy zone at varying distances from the quenched array tip in a directionally solidified Pb-23 wt pct Sn
alloy grown at 2
m s
1
with a thermal gradient of 52 K cm
1
: (a) 75
m from the tip, (b) 222
m from the tip, (c) 3562
m from the tip, and (d) 17200
m
from the tip.
have used K 3.75 10
4
2
1
2
. The first one would sug-
gest that S
v
2
1
2
is constant and does not depend upon
, and
the second one would suggest that S
v
2
1
2
is proportional to
/(1
)
2
. However, as shown in Figure 5, which plots the
parameter S
v
1
2
as a function of fraction of liquid for the Pb-Sb
(Figure 5(a)) and Pb-Sn (Figure 5(b)) alloys, the experimen-
tal data do not support either of these deductions. The
1
values
used here are those obtained experimentally from
as indicated earlier in the experimental section. In the 20 to
90 pct fraction-of-liquid regime included in Figure 5, the para-
meter S
v
1
appears to have a linear dependence on the frac-
tion of liquid; the solid lines represent the linear regressions.
An examination of the various growth conditions represented
in this figure shows that the samples with a higher side-branch-
ing tendency (higher growth speeds for a constant thermal gra-
dient, or lower thermal gradients for a constant growth speed)
tend to have higher slopes.
The slopes of the S
v
1
vs
plots obtained from Fig-
ure 5 are plotted as a function of the parameter S*
D
l
G
eff
/(Vm
l
C
o
(k1)/k) on a log-log scale in Figure 6.
1
(A/N 1),
Let us recall that S* is very small for well-branched den-
drites and increases toward unity as the side-branching
tendency decreases. The physical properties
[40–45]
used for
calculating S* values are listed in Table I. The straight
line represents the linear regression and the curved lines
show the 95 pct confidence interval. The fit is not all
that great: R
2
is only 0.44, but the tendency for the slope
to decrease with the increasing S* is evident. The slope
of the linear-regression line in Figure 6 is 0.33. It sug-
gests that instead of looking for a unique dependence
between S
v
1
and the fraction of liquid in the mushy zone,
as we have been doing so far, we should examine the pos-
sibility of such a relationship between (S
v
1
S*
0.33
) and
the fraction of liquid.
Figure 7 plots (S
v
1
S*
0.33
) vs fraction of liquid for all
the Pb-Sb and Pb-Sn alloy samples examined in this
study. It represents 12 different directional solidifica-
tion experiments and a total of 343 different mushy-zone
cross sections. The best-fit polynomial to the data, indi-
cated by the solid line, yields the following relationship:
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METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 34A, OCTOBER 2003—2369
Fig. 3—Mushy-zone morphology in a Pb-23 wt pct Sn alloy directionally solidified at 2
m s
1
with a thermal gradient of 52 K cm
1
: (a) mean dendrite
perimeter vs distance from the array tip, (b) mean dendrite area vs distance from the array tip, and (c) fraction solid vs distance from the array tip.
(a)(b)
(c)
(S
v
1
S*
0.33
) (3.38 0.33) (3.29 1.29)
(8.85
1.2)
2
. Using this relationship to determine mushy-
zone permeability, however, assumes that the dendrite
morphology (perimeter and surface area) does not change
during quenching. Thermal models of quenching may be
developed to extract the dendrite morphology that existed
during directional solidification from the quenched
microstructures, but the analysis is expected to be quite
complex and we have not made any attempt to do that.
The extent of dendrite growth/coarsening during quench-
ing would be expected to be dependent on the alloy com-
position and the growth conditions. We believe that this
quenching artifact is the major source of scatter in Figure 7,
which combines data from many experiments.
C. Mushy-zone Rayleigh Number
During directional solidification of hypoeutectic lead-
antimony or lead-tin alloys, with melt on the top and solid
below, the thermal profile in the mushy zone is stabilizing
against natural convection. However, the solute content of
the melt in the mushy zone increases as one moves from
the array tips toward the eutectic isotherm at the mushy-zone
bottom. Since the volumetric coefficient of expansion of the
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2370—VOLUME 34A, OCTOBER 2003 METALLURGICAL AND MATERIALS TRANSACTIONS A
(c)
(b)
(a)
Fig. 4—Dependence of dendrite surface area per unit volume (S
v
) on fraction
liquid in the mushy zone. Error bars correspond to 1 standard deviation:
(a) Pb-5.8 wt pct Sb alloys grown at 140 K cm
1
, (b) Pb-5.8 wt pct Sb
alloy grown at 40 K cm
1
, and (c) Pb-Sn alloys solidified at varying growth
(b)
(a)
Fig. 5—Dependence of S
v
on the interdendritic fraction liquid (
). The
corresponding alloy composition, thermal gradient, and growth speeds are
given in these figures. The solid lines denote the linear regression. Only the
data from
0.2 to 0.9 are included in these figures: (a) Pb-5.8 wt pct
Sb alloys and (b) Pb-Sn alloys.
melt due to the increasing solute content (
c
), for example,
5.83 10
3
(wt pct Sb)
1 [40,41]
is several orders of magni-
tude larger than its thermal coefficient of expansion (
t
),
1.22 10
4
K
1 [40,41]
the solute contribution dominates and
is responsible for the density inversion and convection in the
mushy zone. Beckermann et al.
[26]
have defined a Rayleigh
number, R
aB
g(
/
o
)
y/
v, to represent the extent of
this mushy-zone convection. Here, g is the acceleration due
to gravity, y is the distance from the array tip into the mushy
zone,
/
o
is the relative density inversion in the melt at y
with respect to that at the tip,
is the mean permeability
averaged over the distance y,
is the melt thermal diffu-
sivity, and v is the melt kinematic viscosity. They assume
that the mean permeability is related to the mean solid fraction
(
s
):
6.10 10
04
1
2
(1
s
)
3
/
s
2
, where
1
is the
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METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 34A, OCTOBER 2003—2371
Table I. Physical Properties of Pb-Sb and Pb-Sn Alloys
Physical Property Pb-Sb Reference Pb-Sn Reference
Thermal volumetric expansion coefficient (
T
), K
1
1.22 10
4
40, 41 1.15 10
4
4
Solutal volumetric expansion coefficients (
c
), wt pct
1
5.83 10
3
40, 41 5.20 10
3
4
Thermal diffusivity (
), m
2
s
1
1.70 10
5
41, 45 1.10 10
5
4
Kinemetic viscosity (v), m
2
s
1
2.50 10
7
42 2.47 10
7
4
Solute diffusivity in melt (D
l
), cm
2
s
1
3.00 10
5
43 3.00 10
5
4
Capillarity (), m K 9.89 10
8
40, 44, 45 assumed same as Pb-Sb —
Thermal conductivity (solid) 2.97 10
1
41 assumed same as Pb-Sb —
Thermal conductivity (liquid) 1.54 10
1
41 assumed same as Pb-Sb —
Heat of fusion, J m
3
2.79 10
8
45 assumed same as Pb-Sb —
Fig. 6—Dependence of the slope of the S
v
versus
plots on parameter
S*, where S* D
l
G
eff
/[Vm
l
C
o
(k1)/k]. Here, G
eff
is the effective thermal
gradient, V the growth speed, m
l
the liquidus slope, C
o
the alloy solute con-
tent, and k the solutal partition coefficient. The parameter S* indicates the
extent of side branching; it is equal to unity for plane front solidification
and decreases to zero with increasing side-branching tendency of dendrites.
The figure shows the linear regression and the corresponding 95 pct
confidence interval.
Fig. 7—Dependence of S
v
S*
1/3
on the volume fraction interdendritic
liquid,
. The graph includes all the Pb-Sb and Pb-Sn alloys examined in
this study, a total of 343 different mushy zone cross sections. The solid
line represents the best polynomial fit through the experimental data:
S
v
S*
1/3
(3.38 0.33) (3.29 1.29)
(8.85 1.2)
2
.
primary dendrite spacing, and
s
is the mean of the solid
fraction from the tip to the distance, y. The mean solid frac-
tion (
s
) is obtained from the following relationship:
where, using the Scheil equation,
s
(y) is assumed to be equal
to (1 (1 (yG
l
/m
l
C
o
))
(1/(k1)
). However, the assumption
that permeability is equal to 6.10 10
4
1
2
(1
s
)
3
/
s
2
is
not supported by the experimental data, presented in Sec-
tion III–B, because S
v
2
1
2
is not constant; it varies with S*
and
. We will, therefore, incorporate the experimentally
determined processing-parameter dependence of the dendrite
specific surface area, S
v
1
1
S*
0.33
(3.383.29
8.85
2
), into the Kozeny–Carman permeability of a packed
bed,
3
/4.2 S
v
2
(1
)
2
,
[38]
in order to represent the per-
meability of a mushy zone during directional solidification.
This permeability will then be used in the previously described
relationship, R
a
g (
/
o
)
y/
v, to define the new mushy-
zone Rayleigh number, R
aM
.
Two opposing effects come into play as one moves away
from the array tips into the mushy zone; the extent of the den-
s
y
1
y
0
s
(y) dy
sity inversion increases with the increasing distance, but
the mush also becomes less permeable. It was suggested by
Beckermann et al.
[26]
that the mushy-zone Rayleigh number
increases as a function of distance from the array tips, it reaches
a maximum at some depth, and then it begins to decrease
again. However, in a recent publication, Frueh et al.
[46]
have
shown that the maximum Rayleigh number is at the array tip
and not in the mushy zone. They have argued that it is the
convection in the destabilizing melt layer immediately ahead
of the array tips that is the main source of “channel” nucle-
ation, and not the convection deep within the mushy zone.
The “channel segregates” form only if the solidification con-
ditions permit the growth of these “channel nuclei” deeper
into the mushy zone. Since the interdendritic convection is
localized in the immediate vicinity of the array tips,
[46,47]
we
decided to calculate the two Rayleigh numbers, R
aB
and R
aM
,
at y equal to 30 times the corresponding dendrite-tip radius,
to represent the extent of mushy-zone convection during direc-
tional solidification of Pb-Sb and Pb-Sn alloys. However, it
should be pointed out that the observations described subse-
quently were also found to be valid when we used y equal to
50 times the radius to calculate R
aB
and R
aM
. We will use the
tip radius and the primary-dendrite-spacing values predicted
from the dendrite model due to Hunt and Lu
[48]
for determining
R
aB
and R
aM
. This method does not require a prior knowledge
of primary dendrite spacing to calculate the mushy-zone
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2372—VOLUME 34A, OCTOBER 2003 METALLURGICAL AND MATERIALS TRANSACTIONS A
Rayleigh number. Following the procedure described by Beck-
ermann et al.,
[26]
we also use the following relationships for
calculating R
aM
.
1. (
/
o
)
T
(T
y
T
t
) (C
y
C
t
), where
T
and
C
are
the thermal and solutal volumetric coefficients of expan-
sion for the melt, respectively; T
t
and C
t
are the tem-
perature of the melt and its solute content at the dendrite
tip (as calculated from the Hunt–Lu
[48]
model), respect-
ively; and T
y
and C
y
are the temperature and solute con-
tent at the distance y, respectively.
2. The assumption of a constant mushy-zone thermal gradient
and local liquid-solid equilibrium: C
y
C
t
yG
eff
/m
l
.
3. A Scheil relationship to describe the fraction of liquid:
y
(C
y
/C
t
)
(1/(k1))
.
D. Mushy-Zone Convection and Reduction in Primary
Dendrite Spacing
Table II lists C
o
, G
l
, V, and
1(experiment)
data for all the
directionally solidified Pb-Sb and Pb-Sn alloy samples to
be examined in the following analysis, together with the
corresponding G
eff
, and S* values. It includes the theor-
etically predicted (Hunt–Lu model
[48]
) primary-spacing
(
1 Theory
) values, and the corresponding R
aB
and R
aM
values
for all the samples grown with a dendritic-array mor-
phology. In the following discussion, we will also include
other literature-reported data on directionally solidified
Pb-Sb and Pb-Sn alloys.
[13,49–52]
However, we will consider
only the dendritic-morphology samples and exclude those
with a cellular morphology. All the data, except for those
from Sarazin and Hellawell,
[4,49]
are for the steady-state
directional solidification experiments, where either the
cylindrical sample or the furnace setup was translated at
a constant speed and constant thermal gradient to achieve
directional solidification. The Sarazin and Hellawell
[4,49]
data are from the end-quench type directional solidification
experiments, where the melt was poured into a ceramic
mold, heated from the top and cooled from the bottom,
and directional solidification was achieved by slow con-
trolled cooling of the furnace. Since the thermal gradients
and growth speeds did not remain constant during
solidification of these samples, we have used their reported
average values.
Let us first examine only those experiments where the alloy
composition was kept constant and the growth speed was
varied for several constant G
l
values. Figure 8 plots the ratio
of the experimentally observed and the theoretically predicted
primary dendrite spacing as a function of R
aB
, the mushy-zone
Rayleigh number defined by Beckermann et al.,
[26]
for several
Pb-Sb alloys. Figure 8(a) contains data for the Pb-5.8 wt pct
Sb alloy generated during this research and also data marked
as “JD” from Spittel and Lloyed
[50]
for similar alloy compos-
itions (Pb-5.7 and Pb-5.2 wt pct Sb). This figure shows that
an increasing mushy-zone convection, as indicated by an
increasing R
aB
, results in greater reduction in the primary den-
drite spacing as compared with those predicted theoretically
in the absence of convection. However, it is interesting to
note that the data become segregated along various lines that
represent constant thermal gradients. As the thermal gradient
decreases from 197 to 20 K cm
1
, the linear-regression lines
through the data shift to the right-hand side. A similar reduction
in the primary dendrite spacing due to the mushy-zone con-
vection requires a significantly higher R
aB
for the samples
grown at a lower thermal gradient than that grown at a higher
thermal gradient. The same observations are also valid for the
Pb-2.2 wt pct Sb alloy samples grown at various thermal
gradients (Figure 8(b)).
Figure 9 combines all the available primary-dendrite-ratio
data on the Pb-Sb alloys, including those listed as mixed
(not a constant G
l
, but a combination of G
l
and V ), and plots
them together as a function of R
aB
. The data marked as JD
are from Reference 50. In this plot, the data from the high-
antimony-content alloy, Pb-5.2 to 5.8 wt pct Sb (filled
symbols), and those from the low-antimony-content alloy,
Pb-2.2 wt pct Sb (open symbols), become segregated in two
different regions. A similar reduction in the primary dendrite
spacing due to the mushy-zone convection corresponds to
a significantly higher R
aB
for the high-antimony alloy than
that for the low-antimony alloy.
Figure 10 plots the same primary-dendrite-ratio data
(shown in Figure 9) as a function R
aM
, the mushy-zone
Rayleigh number defined in this article that includes the
side-branching contribution in defining the mushy-zone
permeability. Figure 10 also includes the Pb-Sb (mixed)
data from Reference 50, marked as JD, and from Refer-
ence 49, marked as “Sarazin,” for directionally solidified
samples where the antimony content varied from 0.53 to
8.37 wt pct. Now the spacing-ratio data come together,
irrespective of the thermal gradient or the antimony con-
tent, and show a definite linear trend of decrease in the
primary-dendrite-spacing ratio as a function of increasing
mushy-zone Rayleigh number. This suggests that including
the side-branch contribution into the mushy-zone perme-
ability relationship allows us to define a more consistent
Rayleigh number.
Figure 11 combines all of our Pb-Sb and Pb-Sn alloy
primary-spacing data with those available in the literature
and plots them as a function of the two Rayleigh num-
bers, R
aB
(Figure 11(a)) and R
aM
(Figure 11(b)). The sym-
bol “SH” indicates data from Reference 4, JD indicates
data from reference 50, and “KVT” indicates data from
Reference 51. This figure includes data from 164 different
directional solidification experiments (80 for Pb-Sn alloys
and 84 for Pb-Sb alloys). Figure 11(a) shows that the
primary-dendrite-spacing ratios for the Pb-Sb and the
Pb-Sn alloys become segregated into two different regimes.
A similar decrease in the primary dendrite spacing due
to the mushy-zone convection corresponds to an order-
of-magnitude higher R
aB
for the Pb-Sb alloys as compared
with the Pb-Sn alloys. However, when the same primary-
dendrite-spacing ratios are plotted as a function of R
aM
(Figure 11(b)), not only is the scatter decreased, but, also,
the data from the two alloys are pulled together into a
single straight line. Figure 11(b) shows a very clear trend
of primary-dendrite-spacing decease due to an increasing
mushy-zone convection.
However, at this stage, the exact mechanism by which the
mushy-zone convection brings about the decrease in the pri-
mary dendrite spacing is not clear. It may affect the dendrite-
tip radius and, hence, alter the primary dendrite spacing; in
this case, one would expect the spacing decrease to be accom-
panied by an increase in the tip radius. It is also possible that
convection simply melts the side branches, especially near the
31_03-52A-10.qxd 11/9/03 5:11 PM Page 2372
METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 34A, OCTOBER 2003—2373
Table II. Growth Parameters and Rayleigh Numbers for Directionally Solidified Pb-Sb and Pb-Sn Alloys
C
o
G
l
V
1 exptal
G
eff
1 Theory
Sample (wt pct) (K cm
1
)
m s
1
m(K cm
1
) S*(
m) R
aB
R
aM
Ding (6Sb4) Pb-5.8 Sb 140 2 200 95.6 0.1647 590.3 0.1046 0.37493
Ding (6Sb9) Pb-5.8 Sb 140 3 163 95.6 0.1098 527.4 0.08453 0.1868
Ding (6Sb8) Pb-5.8 Sb 140 3.5 176 95.6 0.0941 502.8 0.07708 0.14386
Ding (6Sb6) Pb-5.8 Sb 140 4 193 95.6 0.0824 481.5 0.0709 0.11505
Ding (6Sb5) Pb-5.8 Sb 140 8 173 95.6 0.0412 378.3 0.04427 0.03720
Ding (6Sb3) Pb-5.8 Sb 140 10 170 95.6 0.033 348.6 0.0377 0.02619
Ding (6Sb2b) Pb-5.8 Sb 140 20 158 95.6 0.0165 268.4 0.0225 0.00903
Yushi Pb-5.8 Sb 120 10 166 81.9 0.0283 369.4 0.04957 0.02987
Yushi Pb-5.8 Sb 120 30 153 81.9 0.0095 243.4 0.02171 0.00578
Kunal Pb-5.8 Sb 82 2.5 202 56 0.0772 696.0 0.2552 0.37149
Kunal Pb-5.8 Sb 82 30 199 56 0.0065 282.0 0.04276 0.00852
Ding/Neck Pb-5.8 Sb 155 10 142 105.8 0.0365 335.6 0.03147 0.02397
Hui Pb-5.8 Sb 40 3 227.1 27.3 0.0314 873.1 0.8364 0.51997
Hui Pb-5.8 Sb 40 10 209.9 27.3 0.0095 565.8 0.3532 0.08929
Hui Pb-5.8 Sb 40 30 205.1 27.3 0.0032 376.8 0.1571 0.01897
Hui Pb-5.8 Sb 40 70 168.9 27.3 0.0014 274.7 0.08366 0.00594
Hui Pb-5.8 Sb 40 157 121.1 27.3 7.0000 203.4 0.04588 0.00212
Dave/DrG Pb-5.8 Sb 10 80 213 6.8 3.0000 488.8 1.067 0.02678
Dave/DrG Pb-5.8 Sb 20 20 265 13.6 0.0024 587.1 0.76405 0.07443
Dave/DrG Pb-5.8 Sb 20 50 197 13.6 9.5000 420.8 0.3929 0.02108
Dave/DrG Pb-5.8 Sb 20 80 195 13.6 6.0000 354.7 0.2791 0.01123
Dave/DrG Pb-5.8 Sb 40 20 199 27.3 0.0048 438.0 0.2121 0.03356
Wu Pb-5.8 Sb 40 5 219 27.3 0.0189 728.1 0.5834 0.24339
Weng Pb-2.2 Sb 164 4 109.8 112 0.2543 388.2 0.00489 0.04666
Weng Pb-2.2 Sb 164 5 124.8 112 0.2035 369.7 0.00454 0.03091
Sb(4) Pb-2.2 Sb 140 4 142 95.6 0.2171 420.3 0.00690 0.04990
Sb(6) Pb-2.2 Sb 140 10 149 95.6 0.0869 316.0 0.00415 0.00930
Sb(7a) Pb-2.2 Sb 140 15 136 95.6 0.0579 271.0 0.00310 0.00454
Sb(8a) Pb-2.2 Sb 140 20 126 95.6 0.0435 241.7 0.00249 0.00276
Sb(9) Pb-2.2 Sb 140 30 125 95.6 0.029 204.7 0.00181 0.0014
Ding Pb-2.2 Sb 86 2.1 217 58.7 0.2539 612.2 0.02429 0.19880
Ding Pb-2.2 Sb 86 2.5 223 58.7 0.2133 591.4 0.02298 0.14618
Ding Pb-2.2 Sb 86 2.5 204 58.7 0.2133 591.4 0.02298 0.01461
Ding Pb-2.2 Sb 86 3 201 58.7 0.1777 565.7 0.02129 0.10481
Ding Pb-2.2 Sb 86 6 199 58.7 0.0889 454.9 0.01425 0.03013
Ding Pb-2.2 Sb 86 10 180 58.7 0.0533 376.6 0.00995 0.01248
Ding Pb-2.2 Sb 86 14 184 58.7 0.0381 330.3 0.00772 0.00709
Ding Pb-2.2 Sb 86 18.2 171 58.7 0.0293 297.4 0.0063 0.00461
Ding Pb-2.2 Sb 86 30 165 58.7 0.0178 242.6 0.00421 0.00207
Ding Pb-2.2 Sb 40 1.25 203 27.3 0.1985 999.3 0.1467 0.74141
Ding Pb-2.2 Sb 40 3 264 27.3 0.0827 773 0.09067 0.1607
Ding Pb-2.2 Sb 40 5 239 27.3 0.0497 644.1 0.06373 0.06839
Ding Pb-2.2 Sb 40 7 220 27.3 0.0355 567.6 0.04972 0.03795
Ding Pb-2.2 Sb 40 10 212 27.3 0.0249 494.6 0.03803 0.02248
Hui Pb-2.2 Sb 40 3 258.4 27.3 0.0827 773 0.09067 0.1608
Hui Pb-2.2 Sb 40 10 235.4 27.3 0.0249 494.6 0.03803 0.02248
Hui Pb-2.2 Sb 40 30 189.3 27.3 0.0083 319.6 0.01607 0.00420
Hui Pb-2.2 Sb 40 70 133.1 27.3 0.0036 226.8 0.00813 0.00121
Hui Pb-2.2 Sb 40 157 111.8 27.3 0.0016 163.4 0.00424 0.00038
RAJESH Pb-10 Sn 110 3.5 125 75.1 0.207 537.9 0.00698 0.44389
RAJESH Pb-10 Sn 110 4 134 75.1 0.1811 519.3 0.00587 0.35689
RAJESH Pb-10 Sn 110 5 130 75.1 0.1449 485.9 0.00433 0.2469
RAJESH Pb-10 Sn 110 6 130 75.1 0.1208 457.6 0.00333 0.1826
RAJESH Pb-10 Sn 110 8 123 75.1 0.0906 413.1 0.00216 0.1133
SN001 Pb-10 Sn 110 10 116 75.1 0.0725 379.7 0.00152 0.07854
SN15 Pb-10 Sn 110 20 116 75.1 0.0363 287.3 0.00049 0.02548
(Ojha)DS3 Pb-23 Sn 52 10 150 35.5 0.0174 526.5 0.00736 0.43766
(Ojha)DS4 Pb-23 Sn 52 5 160 35.5 0.0347 683.9 0.02699 1.2268
(Ojha)DS5 Pb-23 Sn 52 2 188 35.5 0.0867 943.3 0.1543 4.9542
(Song, 18mm)L5 Pb-41.9 Sn 20 6 224 13.6 0.0048 1058.6 0.00969 7.35
31_03-52A-10.qxd 11/9/03 5:11 PM Page 2373
2374—VOLUME 34A, OCTOBER 2003 METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 9—A combined plot for the high antimony (Pb-5.2, 5.7, 5.8 wt pct
Sb) and the low antimony (Pb-2.2 wt pct Sb) alloys that shows the ratio
of the experimentally observed and theoretically predicted primary den-
drite spacing vs R
aB
.
(b)
(a)
Fig. 8—Ratio of the experimentally observed and theoretically predicted pri-
mary dendrite spacings vs R
aB
, the mushy-zone Rayleigh number defined
by Beckermann et al.,
[26]
for Pb-Sb alloys directionally solidified at various
growth speeds at several constant thermal gradients. The solid lines repre-
sent the corresponding linear regressions: (a) Pb-5.8, 5.7 and 5.2 wt pct Sb
alloys (as indicated in Table I, the data identified as JD are from Ref. 49);
and (b) Pb-2.2 wt pct Sb alloy.
Fig. 10—Ratio of the experimentally observed and theoretically predicted
primary dendrite spacing for all the Pb-Sb alloys examined in this study
vs the mushy-zone Rayleigh number defined in this paper, R
aM
.
Table II. Continued. Growth Parameters and Rayleigh Numbers for Directionally Solidified Pb-Sb and Pb-Sn Alloys
C
o
G
l
V
1 exptal
G
eff
1 Theory
Sample (wt pct) (K cm
1
)
m s
1
m(K cm
1
) S*(
m) R
aB
R
aM
(Song, 8mm)L6 Pb-32 Sn 60 8 159 40.9 0.0140 568 0.011 0.892
Rajesh 4a Pb-33.4 Sn 75 8 166 51.2 0.0168 523.5 0.00975 0.75572
4b Pb-38.7 Sn 75 8 137 51.2 0.0097 558.9 0.00701 1.075
4c Pb-34 Sn 17 30 172 11.6 0.001 609.9 0.00068 0.6782
3g Pb-27.1Sn 17 1 240 11.6 0.03 2014.6 0.2387 54.58
l Pb-10 Sn 110 10 115 75.1 0.0725 379.7 0.00152 0.07825
lA Pb-16.5 Sn 101 4 172 68.9 0.225 532.8 0.01724 1.84
5a Pb-57.9 Sn 105 10 234 71.7 0.0066 498.0 0.00416 1.05971
5b Pb-54.7 Sn 67 40 177 45.7 0.0012 365.6 0.00040 0.25595
3b Pb-23.7 Sn 81 24 164 55.3 0.011 318.9 0.00165 0.08049
3c Pb-23.4 Sn 77 6 185 52.6 0.0421 553.8 0.02271 0.66169
3d Pb-27 Sn 59 64 155 40.3 0.0025 251.4 0.00023 0.03057
3f Pb-30.3 Sn 20 6 208 13.7 0.0072 975.4 0.01207 4.072
3e Pb-30.0 Sn 8 0.35 194 5.5 0.0495 3842 1.47 625.4
*In this table, C
o
is solute content of the alloy, G
l
is thermal gradient in the liquid at the dendrite tips, V is growth speed,
1 exptl
is experimentally
measured primary dendrite spacing, G
eff
is effective thermal gradient,
1 Theory
is primary spacing predicted from the Hunt–Lu model,
[48]
R
aB
is Rayleigh
number following the procedure presented by Beckerman et al.,
[26]
and R
aM
is mushy-zone Rayleigh number, as defined in this article.
31_03-52A-10.qxd 11/9/03 5:11 PM Page 2374
