j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 145–155
journal homepage: www.elsevier.com/locate/jmatprotec
Effects of growth rate and temperature gradient on the
microstructure parameters in the directionally solidified
succinonitrile–7.5 wt.% carbon tetrabromide alloy
N. Maraşlı a , K. Keşlioğlu a,∗ , B. Arslan a , H. Kaya b , E. Çadırlı c
a
b
c
Erciyes University, Faculty of Arts and Sciences, Department of Physics, 38039 Kayseri, Turkey
Erciyes University, Faculty of Education, Department of Science Education, 38039 Kayseri, Turkey
Niğde University, Faculty of Arts and Sciences, Department of Physics, 51200 Niğde, Turkey
a r t i c l e
i n f o
a b s t r a c t
Article history:
Succinonitrile (SCN)–7.5 wt.% carbon tetrabromide (CTB) alloy was unidirectionally solid-
Received 8 August 2006
ified with a constant growth rate (V = 33 ␮m/s) at five different temperature gradients
Received in revised form
(G = 4.1–7.6 K/mm) and with a constant temperature gradient (G = 7.6 K/mm) at five different
18 May 2007
growth rates (V = 7.2–116.7 ␮m/s). The primary dendrite arm spacings, secondary dendrite
Accepted 5 September 2007
arm spacings, dendrite tip radius and mushy zone depths were measured. Theoretical
models for the microstructure parameters have been compared with the experimental
observations, and a comparison of our results with the current theoretical models and
Keywords:
previous experimental results have also been made.
© 2007 Elsevier B.V. All rights reserved.
Organic compounds
Crystal growth
Solidification
Microstructures
Phase transitions
1.
Introduction
Over the last 40 years, the formation of dendrite arms during
solidification has been studied extensively and several studies (Glicksman and Koss, 1994; Han and Trivedi, 1994; Warren
and Langer, 1993; Makkonen, 2000; Walker and Mullis, 2001) of
directionally solidification under steady-state conditions have
been applied to dendritic growth in alloy systems. Dendritic
growth is the ubiquitous form of crystal growth encountered
when metals, alloys and many other materials solidify under
low-thermal gradients, a situation which typically occurs in
most industrial solidification processes (Glicksman and Koss,
1994). A dendrite structure is characterized by the microstructure parameters. The microstructure parameters 1 , 2 , R and
∗
(d) are shown in Fig. 1. Numerous solidification studies have
been reported with a view to characterizing microstructure
parameters as a function of solidification parameters (Co , V
and G) (Çadırlı et al., 1999, 2000, 2003; Kaya et al., 2005; Üstün
et al., 2006).
Recent empirical (Çadırlı et al., 1999, 2000, 2003; Kaya et
al., 2005; Üstün et al., 2006) and theoretical (Lu and Hunt,
1996; Langer and Müller-Krumbhaar, 1978; Hunt, 1979; Kurz
and Fisher, 1981; Trivedi, 1984; Bouchard and Kirkaldy, 1996,
1997) studies have claimed the existence of an allowable
range of stable spacings. This has been interpreted in such
a way that no unique spacing selection criterion operates
for , and an array with a band of spacing is stable under
given experimental conditions. A literature survey shows
Corresponding author. Tel.: +90 352 4374901x33128; fax: +90 352 4374933.
E-mail address: [email protected] (K. Keşlioğlu).
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmatprotec.2007.09.011
146
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 145–155
Fig. 1 – Schematic illustrations of methods used for measurements of (a) primary dendrite arm spacing, secondary dendrite
arm spacing and the mushy zone depth and (b) dendrite tip radius.
several theoretical models (Lu and Hunt, 1996; Langer and
Müller-Krumbhaar, 1978; Hunt, 1979; Kurz and Fisher, 1981;
Trivedi, 1984; Bouchard and Kirkaldy, 1996, 1997; Trivedi and
Somboonsuk, 1984) used to examine the influence of solidification parameters on the microstructure parameters. The
majority of results in the literature show a decrease in
spacing with increasing growth rate for a given alloy composition and with increasing solute concentration for a given
growth rate (Çadırlı et al., 1999, 2000, 2003; Kaya et al., 2005;
Üstün et al., 2006; Sharp and Hellawell, 1969; Spittle and
LIoyd, 1979). Besides, the primary arm spacings have been
reported to decrease as temperature gradient or growth rate
increases. Sharp and Hellawell (1969) concluded that Co has
little effect on primary spacings and Spittle and LIoyd (1979)
have found that in the case of steady-state growth with
low G, 1 decreased as Co increases and was independent
of Co for high G. Nevertheless, in many cases it has been
assumed that 1 increases as Co increases for any growth condition (Bouchard and Kirkaldy, 1997; Okamoto and Kishitake,
1975).
The aim of present work is to experimentally investigate the effect of the temperature gradient and growth
rate on the microstructure parameters in the directionally solidified SCN–7.5 wt.% CTB alloy and to compare
the results with theoretical models (Lu and Hunt, 1996;
Langer and Müller-Krumbhaar, 1978; Hunt, 1979; Kurz and
Fisher, 1981; Trivedi, 1984; Bouchard and Kirkaldy, 1996,
1997; Trivedi and Somboonsuk, 1984) and previous experimental results (Han and Trivedi, 1994; Çadırlı et al.,
1999, 2000, 2003; Kaya et al., 2005; Üstün et al., 2006;
Mullins and Sekarka, 1965; Glicksman and Singh, 1986;
Schmidbauer et al., 1993; De Cheveigne et al., 1985; Huang
et al., 1993; Taha, 1979; Grugel and Zhou, 1989; Dey and
Sekhar, 1993; Liu and Kirkaldy, 1994a; Seetharaman et al.,
1989; Trivedi and Mason, 1991; Rutter and Chalmers, 1953;
Tiller et al., 1953; Kurz and Fisher, 1989; Esaka and Kurz,
1985).
The theoretical models for microstructure parameters are
briefly described as follows.
1.1.
Theoretical models for primary dendrite arm
spacing
Hunt (1979) and Kurz and Fisher (1981) have proposed the theoretical models to characterize cells/primary dendrite spacing
(1 ) as a function of growth rate (V), temperature gradients (G)
and alloy composition (Co ) during steady-state growth conditions. At high-growth rate, the results predicted by these two
theories differ only by a constant. The equations representing
these two theories can be expressed, respectively, as
Hunt model:
1 = 2.83[m (k − 1) D ]
0.25
C0.25
V −0.25 G−0.5
0
(1)
Kurz and Fisher model:
1 = 4.3
m(k − 1)D
k2
0.25
C0.25
V −0.25 G−0.5
0
(2)
where m is the liquidus slope, the Gibbs–Thomson coefficient, k the solute partition coefficient and D is the liquid
solute diffusivity.
Trivedi (1984) has modified the Hunt’s model by using the
marginal stability criterion to characterize the dentritic primary arm spacing, 1 as function of G, V and Co and it can be
expressed as
Trivedi model:
1 = 2.83[m(k − 1)DL]
0.25
C0.25
V −0.25 G−0.5
0
(3)
where L is a constant depends on the harmonic of perturbation.
Lu and Hunt (1996) have proposed a numerical model to
characterize the dentritic primary arm spacing, 1 . The model
describes steady- or unsteady-state of an ax symmetric cell or
dendrite and it can be expressed as
Lu and Hunt model:
= 0.07798 V (a−0.75)
(V − G )
0.75
G
−0.6028
(4)
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 145–155
where
=
147
Hunt model (Hunt, 1979):
To
k
G =
G k
(To )
2
V =
V k
D To
To =
m Co (k − 1)
k
R=
2D 0.5
m(k − 1)
C−0.5
V −0.5
o
(8)
and
according to the Kurz and Fisher model (Kurz and Fisher, 1981):
a = −1.131 − 0.1555 log G − 0.007589 (log G )
2
Bouchard and Kirkaldy (1996, 1997) have also proposed a
numerical model to characterize the dentritic primary arm
spacing (1 ) for unsteady- and steady-state heat-flow conditions. A heuristically derived steady-state formula, after
modification, has been recommended by these authors for
purposes of predicting primary dendritic spacing in the
unsteady regime and is given by
Bouchard and Kirkaldy model:
1 = a1
1/2
16Co Go εD
(1 − k)mGV
1/2
Secondary dendrite arm spacing
8DL 0.5
(6)
kVTo
For secondary dendrite arm spacing, Bouchard and Kirkaldy
(1997) have also derived an expression which is very similar to
Mullins and Sekarka (1965). This expression independents on
temperature gradient and is given by
Bouchard and Kirkaldy model:
2 = 2a2
D 2 1/3
4
2
Co (1 − k) TF
V
(7)
where a2 is the secondary dendrite-calibrating factor, which
depends on alloy composition and TF is the fusion temperature
of the solvent. The Bouchard and Kirkaldy model additionally
depends on empirical dimensionless calibration parameters,
a1 for 1 and a2 for 2 as shown by Eqs. (5) and (7). These
authors have proposed different a1 values for different alloys
(Bouchard and Kirkaldy, 1997).
1.3.
D
m(k − 1)
0.5
C−0.5
V −0.5
o
(9)
and according to the Trivedi model (Trivedi, 1984):
R=
2kDL 0.5
m(k − 1)
C−0.5
V −0.5
o
(10)
As can be seen from Eqs. (8)–(10) the theoretical models for
dendritic tip radius, R are also very similar and the difference
among them is a constant only.
1.4.
Langer and Müller-Krumbhaar (1978) have carried out a
detailed numerical analysis of the wavelength of instabilities
along the sides of a dendrite and have predicted a scaling
law as 2 /R = 2. Using the scaling law 2 /R = 2, the variation in
2 for small peclet number conditions given by Trivedi and
Somboonsuk (1984) as
Trivedi and Somboonsuk model:
2 =
(5)
where Go ε is a characteristic parameter (600 × 6 K/cm) and
a1 is the primary dendrite-calibrating factor (Bouchard and
Kirkaldy, 1997).
1.2.
R = 2
Dendrite tip radius
As mentioned in the previous section, the Hunt model (Hunt,
1979), the Kurz and Fisher model (Kurz and Fisher, 1981) and
Trivedi model (Trivedi, 1984) have been applied to find the relationships between R as a function V and Co . According to the
Mushy zone depth
The mushy zone depth (d) is defined as the distance between
the tip and the root of a dendrite trunk. Using constitutional
supercooling criterion (Trivedi and Mason, 1991; Rutter and
Chalmers, 1953) for binary alloy systems in the absence of
convection, the mushy zone depth (d) is given by
d≈m
CE − Co
G
(11)
where CE is the eutectic composition. The mushy zone depth
(d) assumed to be equal to the distance between the liquidus
temperature (TL ) corresponding to Co and the solidus temperature (TS ) corresponding to CL which is equal to CE when the
composition of alloy is over the composition of single solid
phase (Co > CS ). The temperature difference between the liquidus and the solidus is given (Kurz and Fisher, 1989) as
To = TL − TS = −m(CE − Co )
(12)
By using Eqs. (11) and (12), (d) can be expressed as
d=
2.
To
G
(13)
Experimental procedure
SCN–7.5 wt.% CTB alloy was prepared from 99.9% purity of
SCN and 99.9% purity of CTB supplied by Sigma–Aldrich
Chemical Company. The specimen was contained in a glass
cell made from two glass cover slips (50 mm long, 24 mm
wide and 0.05 mm thick). The slides were stuck together
with a silicone elastomer. The slides were placed with their
largest surface in the x–y plane and spaced a distance of
about 100–120 ␮m apart in the z direction to observe the
dendrite in x–y plane (2D). Organic materials usually react
with this type of glue. Before filling the cell with alloy, the
cell was annealed at 523 K to prevent the reaction with the
glue.
After filling the cell with alloy, the specimen cell was placed
in the temperature gradient stage. The detail of the experi-
148
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 145–155
Table 1a – Dependency of the microstructure parameters for the directionally solidified SCN–7.5 wt.% CTB alloy on the
temperature gradients
Microstructure parametersa
Solidification parameters
G (K/mm)
V (␮m/s)
4.1
4.8
5.6
6.8
7.6
33.0
33.0
33.0
33.0
33.0
1 (␮m)
311.4
290.4
266.4
248.8
230.0
±
±
±
±
±
15.2
15.8
14.2
11.6
12.5
Constant (k)
2 (␮m)
68.4
63.6
57.2
49.5
44.5
±
±
±
±
±
7.1
3.5
1.9
4.6
3.3
34.0
31.2
28.3
25.1
22.6
±
±
±
±
±
2.5
2.2
1.9
2.0
1.7
d (␮m)
2 /R
±
±
±
±
±
2.01
2.04
2.02
1.97
1.97
655.8
610.0
570.4
528.4
475.6
15.5
19.2
23.8
17.0
10.6
Correlation coefficients (r)
r1 = −0.999
r2 = −0.995
r3 = −0.996
r4 = −0.989
k1 = 22.4 (␮m0.53 K0.47 )
k2 = 2.24 (␮m0.42 K0.58 )
k3 = 0.96 (␮m0.35 K0.65 )
k4 = 13.18 (␮m0.47 K0.53 )
a
R (␮m)
The relationships: 1 = k1 G−0.47 ; 2 = k2 G−0.58 ; R = k3 G−0.65 ; d = k4 G−0.53 ; (2 /R)ave = 2.0.
mental system was given by Trivedi (1984). When one side of
the cell was heated, the other side of the cell was kept cool
with a water cooling system. The temperature of heater was
controlled to be ±0.1 K with a Eurotherm 905S type controller.
The temperatures in the specimen were measured with the
insulated K type four thermocouples, 50 ␮m thick which were
placed perpendicular to heat flow on the sample. The temperature gradient in front of the solid–liquid interface on the
specimen during the solidification was observed to be constant.
The SCN–7.5 wt.% CTB alloy was solidified in a horizontal directional solidification apparatus to directly observe
the microstructures using a transmission optical microscope.
The SCN–7.5 wt.% CTB alloy was solidified with a constant growth rate (V = 33 ␮m/s) at five different temperature
gradients (G = 4.1–7.6 K/mm) and with a constant temperature gradient (G = 7.6 K/mm) at five different growth rates
(V = 7.2–116.7 ␮m/s). During the solidification, the photographs
of the microstructures were taken with a CCD digital camera
placed on a transmission Olympus BH2 optical microscope
by using ×5, ×10, and ×20 objectives and the photographs
of a graticula (100 × 0.01 = 1 mm) were also taken with same
objectives.
2.1.
Measurements of temperature gradient and
growth rate
The specimen was slowly melted until the solid–liquid interface passed through the second thermocouple by driving
the specimen cell toward to the heating system. When the
solid–liquid interface was between the second and third thermocouples, the synchronous motor was stopped and the
specimen was left to reach thermal equilibrium. After the
specimen reached the steady-state conditions, the solidification was started by the driving of the specimen toward to
cooling system by synchronous motor. While the interface
was passing the distance between two thermocouples the
solidification time (t) and temperature difference between
two thermocouples (T) was recorded and the photographs
of the thermocouple positions and solidification microstructures were taken with a CCD digital camera. Thus the values
of T, t and x were measured accurately and then the tem-
Table 1b – Dependency of the microstructure parameters for the directionally solidified SCN–7.5 wt.% CTB alloy on the
growth rates
Microstructure parametersa
Solidification parameters
G (K/mm)
V (␮m/s)
7.6
7.6
7.6
7.6
7.6
7.2
14.5
33.0
58.9
116.7
Constant (k)
a
352.4
279.8
230.0
190.4
176.2
±
±
±
±
±
10.4
12.0
12.5
14.8
13.1
2 (␮m)
85.1
66.0
44.5
34.7
24.2
±
±
±
±
±
R (␮m)
6.4
5.8
3.3
3.3
2.1
43.5
32.2
22.6
15.5
11.8
±
±
±
±
±
1.9
1.8
1.7
0.2
0.7
Correlation coefficients (r)
−0.25
k5 = 5.62 × 10 (␮m
s
)
k6 = 2.21 × 102 (␮m1.46 s−0.46 )
k7 = 1.12 × 102 (␮m1.48 s−0.48 )
k8 = 10.96 × 102 (␮m1.33 s−0.33 )
2
1 (␮m)
1.25
r5 = −0.996
r6 = −0.992
r7 = −0.989
r8 = −0.995
The relationships: 1 = k5 V−0.25 ; 2 = k6 V−0.46 ; R = k7 V−0.48 ; d = k8 V−0.33 ; (2 /R)ave = 2.07.
d (␮m)
674.8
596.0
475.6
418.6
343.8
±
±
±
±
±
33.4
29.8
10.6
26.2
19.1
2 /R
1.96
2.12
1.97
2.24
2.05
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 145–155
149
Fig. 2 – Typical microstructures of the directionally solidified SCN–7.5 wt.% CTB alloy (a) V = 33 ␮m/s; G = 4.1 K/mm, (b)
V = 33 ␮m/s; G = 4.8 K/mm, (c) V = 33 ␮m/s; G = 5.6 K/mm, (d) V = 33 ␮m/s; G = 6.8 K/mm, (e) V = 33 ␮m/s; G = 7.6 K/mm, (f)
G = 7.6 K/mm; V = 7.2 ␮m/s, (g) G = 7.6 K/mm; V = 14 ␮m/s, (h) G = 7.6 K/mm; V = 33.0 ␮m/s, (i) G = 7.6 K/mm; V = 58.9 ␮m/s, and (j)
G = 7.6 K/mm; V = 116.7 ␮m/s.
150
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 145–155
perature gradient, G = (T/x) and the growth rate, V = (x/t)
were determined by using the values of t, T and x.
2.2.
Measurements of microstructure parameters
Schematic representation of the microstructure parameters
are shown in Fig. 1. The primary dendrite arm spacing (1 )
was obtained by measuring the distance between nearest two
dendrite tips. The secondary dendrite arm spacing (2 ) was
measured by averaging the distance between adjacent side
branches of a primary dendrite as a function of the distance
from the dendrite tip. The dendrite tip radius (R) was measured
by fitting a suitable circle to the dendrite tip side. The mushy
zone depth (d) is also defined by means of region between tip
side and root side of the dendrites (Çadırlı et al., 1999, 2000;
Kaya et al., 2005; Üstün et al., 2006; Lu and Hunt, 1996). This
parameter was measured as far from the steady-state condition in the dendrites as possible. In the measurements of
microstructure parameters, 30–40 values of 1 , 2 , R and (d) for
each growth rates and each temperature gradients were measured to increase statistical sensitivity. Thus the solidification
parameters and microstructure parameters were measured
and the average values of 1 , 2 , R and (d) with their standard
deviations are given in Tables 1a and 1b.
3.
Results and discussion
SCN–7.5 wt.% CTB alloy was solidified with a constant
growth rate (V = 33 ␮m/s) at five different temperature
gradients (G = 4.1–7.6 K/mm) and with a constant temperature gradient (G = 7.6 K/mm) at five different growth rates
(V = 7.2–116.7 ␮m/s) in order to investigate the dependency of
microstructures parameters on the growth rate and temperature gradient and to find the relationship between them.
Typical microstructures of SCN–7.5 wt.% CTB alloy are shown
in Fig. 2 and the mean experimental values of the microstructure parameters with the standard variations as a function
of growth rates and temperature gradients are given in
Tables 1a and 1b. The dependency of the microstructure
parameters on the growth rate and temperature gradient was
obtained by linear regression analysis and the results are also
given in Tables 1a and 1b.
Fig. 3a and b present the experimental values of primary
dendritic arm spacing (1 ), secondary dendritic arm spacing
(2 ), tip radius (R) and mushy zone depth (d) as a function of
temperature gradients (G) and growth rates (V), respectively.
As can be seen from Fig. 3a and Table 1a, the values of 1 ,
2 , R and (d) decrease as the value of (G) increases at a constant
Co V and the average exponent value of 1 , 2 , R and (d) in the
directionally solidified SCN–7.5 wt.% CTB alloy with a constant
growth rate (V = 33 ␮m/s) at different temperature gradients
(G = 4.1–7.6 K/mm) was found to be −0.47, −0.58, −0.65 and
−0.53, respectively.
From Fig. 3b and Table 1b, the values of 1 , 2 , R and (d)
also decrease as the growth rate increase at a constant Co G
and the average exponent value of 1 , 2 , R and (d) in the
directionally solidified SCN–7.5 wt.% CTB alloy with a constant
temperature gradient (G = 7.6 K/mm) at different growth rates
(V = 7.2–116.7 ␮m/s) was found to be −0.25, −0.46, −0.48 and
−0.33, respectively.
Fig. 3 – (a) Microstructure parameters as a function of
temperature gradients for the directionally solidified
SCN–7.5 wt.% CTB alloy with a constant growth rate
(V = 33.0 ␮m/s). (b) Microstructure parameters as a function
of growth rates for the directionally solidified SCN–7.5 wt.%
CTB alloy with a constant temperature gradient
(G = 7.6 K/mm).
A number of experimental studies have been reported in
the literature to characterize the variations in the 1 , 2 , R and
(d) as a function of (V) and (G). Available experimental results
related to the values of 1 , 2 , R and (d) can be found in Table 2.
As can be seen from Tables 1a, 1b and 2, the exponent values of
1 , 2 , R and (d) in the directionally solidified SCN–7.5 wt.% CTB
151
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 145–155
Table 2 – A comparison of the experimental results of the microstructure parameters with previous experimental works
Pure or alloy system
SCN–7.5 wt.% CTB
SCN–7.5 wt.% CTB
SCN–13 wt.% ACE
SCN–(5–20) wt.% Salol
PVA
PVA
Camphene
Camphene
SCN–(5,10,20,40) wt.% CTB
SCN–3.6 wt.% ACE
SCN–3.6 wt.% ACE
KCl–5 mol.% CsCl
CBr4
SCN–25 wt.% ETH
SCN–2.5 wt.% Benzyl
SCN–(0.15–5) wt.% ACE
SCN–1.4 wt.% H2 O
Salol
SCN–(0.001–0.004) mol.% Salol
SCN–5.5 mol.% ACE
SCN–4 wt.% ACE
SCN
SCN–ACE
SCN–%2.5 ETH
SCN–salol
SCN–7.5 wt.% CTB
SCN–7.5 wt.% CTB
PVA
PVA
Camphene
SCN–(5,10,20,40) wt.% CTB
SCN–3.6 wt.% ACE
SCN–3.6 wt.% ACE
CBr4 –(8–10.5) wt.% C2 Cl6
PVA–0.82 wt.% ETH
SCN–1.3 wt.% ACE
SCN–5.5 mol.% ACE
SCN–4 wt.% ACE
SCN–(5–20) wt.% Salol
SCN–7.5 wt.% CTB
SCN–7.5 wt.% CTB
PVA
PVA
Camphene
Camphene
SCN–(5,10,20,40) wt.% CTB
SCN–3.6 wt.% ACE
SCN–3.6 wt.% ACE
CBr4 –(8–10.5) wt.% C2 Cl6
PVA–0.82 wt.% ETH
SCN–1.3 wt.% ACE
SCN–2 wt.% H2 O
SCN–5.5 mol.% ACE
CBr4 –Hexacloroethane
CBr4 –C2 Cl6
SCN–%2.5 ETH
NaCl
PVA
SCN–7.5 wt.% CTB
SCN–7.5 wt.% CTB
PVA
PVA
Temperature gradient
Temperature
gradient G
Growth rate V
(␮m/s)
4.1–7.6
4.1–7.6
2
4.5
1.64
1.64–4.86
6.94
2.25–6.94
7.5
3.5–5.7
3.5–5.7
3
7
4.8–10.8
1.6–9.5
3.8
6.24
5.4
7.2–116.7
7.2–116.7
7.25–11.35
6.7–112.4
0.7–85.8
9.6
6.6–116.5
6.6
6.5–103.4
6.5–113
6.5–113
13–130
7–100
3–54
56–92
48–225
140
5–75
6–15
–
6.7
50
5.6–35
48
26–35
4.1–7.6
4.1–7.6
1.64
1.644.86
6.94
7.5
3.5–5.7
3.5–5.7
3
0.85–2.26
1.6–9.7
6.7
6.7
4.5
4.1–7.6
4.1–7.6
1.6
1.644.86
6.94
2.25–6.94
7.5
3.5–5.7
3.5–5.7
3
0.85–2.26
1.6–9.7
2.4–3.3
6.7
3
3
48
–
0.001–1.1 ◦ C/s
4.1–7.6
4.1–7.6
1.64
1.64–0.864
60–160
–
1–100
45–500
8.57–194
3–54.2
1–25
7.2–116.7
7.2–116.7
.7–85.8
9.6
6.6–116.5
6.5–103.4
6.5–113
6.5–113
0.2–20
0.3–80
1.6–250
0.4–100
1–100
6.7–112.4
7.2–116.7
7.2–116.7
6.7–85.8
9.6
6.6–116.5
6.6
6.5–103.4
6.5–113
6.5–113
0.2–20
0.3–80
1.6–250
0.76–105
0.4–100
0.2–20
0.2–20
3–54.2
30–100
1–100
7.2116.7
7.2116.7
0.7–85.8
19.6
Relationships
References
1 = 22.4G−0.47
1 = 562V−0.25
1 = kV−0.58
1 = kV−0.26
1 = kV−0.32
1 = kG−0.36
1 = kV−0.25
1 = kG−0.47
1 = 584.7V−0.25
1 = 9.7G−0.50
1 = 240.1V−0.25
1 = kV−0.42
1 = kV−0.55
1 = 470V−0.33
1 = kG−0.50 V−0.25
1 = kG−0.50 V−0.25
1 = kG−0.50
1 = k(GV)−0.50
This work
This work
Han and Trivedi (1994)
Çadırlı et al. (2003)
Çadırlı et al. (1999)
Çadırlı et al. (1999)
Çadırlı et al. (2000)
Çadırlı et al. (2000)
Kaya et al. (2005)
Üstün et al. (2006)
Üstün et al. (2006)
Schmidbauer et al. (1993)
De Cheveigne et al. (1985)
Huang et al. (1993)
Taha (1979)
Taha (1979)
Grugel and Zhou (1989)
Dey and Sekhar (1993)
1 = 0.16G−1/3 V −1/3¦ Xo
1 ˛V−0.27
1 = kV−0.37
˛V−0.20 G−3/4
1 = 535.2G−0.81
1 = 679.2V−0.40
1 = 60(GV)−0.33
2 = 2.24G−0.58
2 = 221V−0.46
2 = kV−0.43
2 = kG−0.29
2 = kV−0.24
2 = 177V−0.46
2 = 1.4G−0.50
2 = 49.5V−0.48
2 ˛V−0.44
2 ˛V−0.58
2 ˛V−0.51
2 ˛V−0.56
2 = kV−0.56
R = kV−0.45
R = 0.96G−0.65
R = 112V−0.48
R = kV−0.50
R = kG−0.46
R = kV−0.24
R = kG−0.50
R = 090V−0.46
R = 0.5G−0.50
R = 20.5V−0.50
R˛V−0.53
R˛V−0.54
R˛V−0.53
R˛V−0.43
R˛V−0.53.
R˛V−0.53
R˛V−0.47
R = 22V−0.50
R = 59.34V−0.531
R = 1.41V−0.5
d = 13.2G−0.53
d = 1096V−0.33
d = kV−0.45
d = kG−0.33
−1/3
Liu and Kirkaldy (1994a)
Somboonsuk et al. (1984)
Billia and Trivedi (1993)
Shangguan (1988)
Ma (2004)
Ma (2002)
Trivedi et al. (2003)
This work
This work
Çadırlı et al. (1999)
Çadırlı et al. (1999)
Çadırlı et al. (2000)
Kaya et al. (2005)
Üstün et al. (2006)
Üstün et al. (2006)
Seetharaman et al. (1989)
Trivedi and Mason (1991)
Esaka and Kurz (1985)
Somboonsuk et al. (1984)
Huang and Glicksman (1981)
Çadırlı et al. (2003)
This work
This work
Çadırlı et al. (1999)
Çadırlı et al. (1999)
Çadırlı et al. (2000)
Çadırlı et al. (2000)
Kaya et al. (2005)
Üstün et al. (2006)
Üstün et al. (2006)
Seetharaman et al. (1989)
Trivedi and Mason (1991)
Esaka and Kurz (1985)
Cattaneo et al. (1994)
Somboonsuk et al. (1984)
Somboonsuk et al. (1984)
Somboonsuk et al. (1984)
Ma (2002)
Gorbunov (1992)
Glicksman (1984)
This work
This work
Çadırlı et al. (1999)
Çadırlı et al. (1999)
152
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 145–155
Table 2 (Continued )
Pure or alloy system
Temperature gradient
Temperature
gradient G
PVA
Camphene
Camphene
Camphene
SCN–3.6 wt.% ACE
SCN–3.6 wt.% ACE
Relationships
References
Growth rate V
(␮m/s)
1.64–0.86
6.94
2.25–6.94
2.25–6.94
3.5–5.7
3.5–5.7
0.7–85.8
6.6–116.5
6.6–116.5
6.6–116.5
6.5–113
6.5–113
d = k(GV)−0.44
d = kV−0.22
d = kG−0.65
d = k(GV)−0.34
d = 18.4G−0.49
d = 351.7V−0.25
Çadırlı et al. (1999)
Çadırlı et al. (2000)
Çadırlı et al. (2000)
Çadırlı et al. (2000)
Üstün et al. (2006)
Üstün et al. (2006)
PVA: pivalic acid, SCN: succinonitrile, ACE: acetone, ETH: ethanol, and CTB: carbon tetrabromide.
Table 3 – A comparison of 2 /R values obtained in present work with previous theoretical and experimental works
Pure or alloy system
–
SCN–7.5 wt.% CTB
PVA
Camphene
SCN–5 wt.% CTB
SCN–10 wt.% CTB
SCN–20 wt.% CTB
SCN–40 wt.% CTB
SCN–3.6 wt.% ACE
SCN–ACE
SCN
CBr4 –10.5 wt.% C2 Cl6
CBr4 –7.9 wt.% C2 Cl6
PVA–0.82 wt.% ETH
H2 O–NH4 Cl
SCN–5.6 wt.% H2 O
NH4 Cl–70 wt.% H2 O
SCN
SCN
2 /R values
References
2.10 (theoretical value)
2.07
4.64
3.33
2.12
2.10
2.07
2.03
2.6
2.0
2.5
3.18
3.47
3.8
4.68
2.8
4.02
3.0
3.0
alloy with a constant growth rate at different temperature gradients and with a constant temperature gradient at different
growth rates obtained in the present work are in good agreement with the exponent values of 1 , 2 , R and (d) for the same
alloys and different organic materials obtained by previous
workers (Çadırlı et al., 1999, 2000, 2003; Kaya et al., 2005; Üstün
et al., 2006; Taha, 1979; Grugel and Zhou, 1989; Dey and Sekhar,
1993; Liu and Kirkaldy, 1993, 1994b; Seetharaman et al., 1989;
Trivedi and Mason, 1991; Rutter and Chalmers, 1953; Tiller et
al., 1953; Kurz and Fisher, 1989; Esaka and Kurz, 1985; Cattaneo
et al., 1994; Somboonsuk et al., 1984; Billia and Trivedi, 1993;
Shangguan, 1988; Ma, 2002, 2004; Trivedi et al., 2003; Gorbunov,
1992; Glicksman, 1984; Honjo and Sawada, 1982; Liu et al., 2002;
Hansen et al., 2002; Huang and Glicksman, 1981).
Langer and Müller-Krumbhaar (1978)
This work
Çadırlı et al. (1999)
Çadırlı et al. (2000)
Kaya et al. (2005)
Kaya et al. (2005)
Kaya et al. (2005)
Kaya et al. (2005)
Üstün et al. (2006)
Trivedi and Somboonsuk (1984)
Glicksman and Singh (1986)
Seetharaman et al. (1989)
Seetharaman et al. (1989)
Trivedi and Mason (1991)
Honjo and Sawada (1982)
Liu et al. (2002)
Hansen et al. (2002)
Hansen et al. (2002)
Huang and Glicksman (1981)
The comparisons of the experimentally obtained 1 values in present work with the calculated 1 values by Hunt
(1979), Kurz and Fisher (1981), Trivedi (1984), Lu and Hunt
(1996) and Bouchard and Kirkaldy (1996, 1997) models for
SCN–7.5 wt.% CTB alloy are given in Fig. 4. As can be seen from
Fig. 4, the calculated line of 1 with Kurz and Fisher (1981),
Trivedi (1984), Lu and Hunt (1996) and Hunt (1979) models are
slightly upper, slightly lower and fairly below our experimental values, respectively, and the calculated values of 1 with
Bouchard and Kirkaldy (1996) model are in good agreement
with our experimental results at high-growth rates but slightly
lower than the experimental values at low-growth rates. It
can be seen from Fig. 4 that the values of 1 experimentally
obtained in present work are very close the calculated values
Table 4 – Physical properties of SCN–CTB alloys used in the calculations
Liquidus slope (m)
Liquid diffusion coefficient (D)
Equilibrium partition coefficient (k)
2.25 (K/wt.%) or 0.225 × 103 (Kmol−1 fr−1 )
2 × 10−5 (cm2 /s)
0.2
The Gibbs–Thomson coefficient ( )
The harmonic perturbations
Equilibrium melting point of SCN (Te )
5.56 × 10−8 (Km)
10 (mJ/m2 )
331.24 (K)
Rai and Rai (1998)
Venugopalan and Kirkaldy (1984)
Liu and Kirkaldy (1993, 1994a,b) and
Gandin et al. (1996)
Maraşlı et al. (2003)
Trivedi (1984)
Venugopalan and Kirkaldy (1984) and
Gandin et al. (1996)
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 145–155
153
Fig. 5 – Comparison of the experimental and theoretical
secondary dendrite arm spacing as a function of growth
rates for directionally solidified SCN–7.5 wt.% CTB alloy
with a constant temperature gradient.
Fig. 4 – Comparison of the experimental and theoretical
primary dendrite arm spacing as a function of growth rates
for directionally solidified SCN–7.5 wt.% CTB alloy with a
constant temperature gradient.
of 1 with Kurz and Fisher (1981) model for SCN–7.5 wt.% CTB
alloy.
The variations in the values of 2 experimentally obtained
with a constant G at different V in present work have been
compared with the values of 2 calculated from Trivedi and
Somboonsuk (1984) and Bouchard and Kirkaldy (1996, 1997)
models and the comparisons are given in Fig. 5. As it can be
seen from Fig. 5, the calculated line of 2 with Trivedi and
Somboonsuk (1984) model as a function of V is in good agreement with our experimental values and the calculated line of
2 with Bouchard and Kirkaldy (1996, 1997) model as function
of V is slightly lower than the our experimental values.
Fig. 6a shows the comparisons of the experimentally
obtained R values as a function of V with the calculated R val-
154
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 145–155
the present work, the average value of 2 /R for directionally
solidified SCN–7.5 wt.% CTB alloy with a constant temperature gradient has been found to be 2.07 (see Table 1b). As it
can be seen from Tables 1b and 3, the average value of 2 /R
obtained in present work for SCN–7.5 wt.% CTB alloy is in good
agreement with the value of 2 /R estimated by Langer and
Müller-Krumbhaar (1978).
A comparison of 2 /R values obtained in present work with
the previous experimental works (Çadırlı et al., 1999, 2000;
Kaya et al., 2005; Üstün et al., 2006; Trivedi and Somboonsuk,
1984; Glicksman and Singh, 1986; Seetharaman et al., 1989;
Trivedi and Mason, 1991; Billia and Trivedi, 1993; Shangguan,
1988; Ma, 2002, 2004) is also given in Table 3. The average value
of 2 /R obtained in the present work for SCN–7.5 wt.% CTB alloy
is approximately equal to the value of 2 /R estimated by Langer
and Müller-Krumbhaar (1978) for pure substances. In addition,
the experimental value (2.07) is also very close to 2.1, 2.0, 2.6,
2.5 and 2.8 values reported by Kaya et al. (2005) for SCN–CTB
alloys, Trivedi and Somboonsuk (1984) for SCN–ACE, Üstün et
al. (2006) for SCN–2.6 wt.% ACE, Glicksman and Singh (1986) for
SCN and Liu et al. (2002) for SCN–5.6 wt.% H2 O alloy, respectively. As can bee seen from Table 3 the experimental values
are quite different from some of the other experimental results
given by Çadırlı et al. (1999, 2000), Glicksman and Singh (1986),
Seetharaman et al. (1989), Trivedi and Mason (1991), Honjo and
Sawada (1982), Hansen et al. (2002) and Huang and Glicksman
(1981). The physical parameters of SCN–CTB alloy used in 1 ,
2 , R and d calculations with the theoretical models are given
in Table 4.
4.
Fig. 6 – (a) Comparison of the experimental and theoretical
dendrite tip radius as a function of growth rates for
directionally solidified SCN–7.5 wt.% CTB alloy with a
constant temperature gradient. (b) Comparison of the
experimental and Ruther Chalmers theoretical dendrite
mushy zone depth as a function of inverse of temperature
gradients for the directionally solidified SCN–7.5 wt.% CTB
alloy with a constant growth rate.
ues from Hunt (1979), Kurz and Fisher (1981) and Trivedi (1984)
the models. It can be seen from Fig. 6a, the calculated values
of R with Trivedi (1984) model for SCN–7.5 wt.% CTB alloy is in
good agreement with our experimental values.
A comparison of the experimentally obtained mushy zone
depth (d) values as an inverse function of G in present work
with the calculated d values by Rutter and Chalmers (1953)
model is given in Fig. 6b and the calculated line of d with
Rutter and Chalmers (1953) model is quite upper than our
experimental results.
The value of 2 /R for undercooled dendrites was estimated to be 2.1 by Langer and Müller-Krumbhaar (1978). In
Conclusions
(a) The dependency of 1 , 2 , R and (d) on the (G) and (V) for the
directionally solidified SCN–7.5 wt.% CTB alloy was investigated. Our experimental observations show that the
values of 1 , 2 , R and (d) decrease as the values of (G) and
(V) increase. The relationships between the microstructure parameters and the solidification parameters with
a constant solute composition have been obtained to
be 1 = k1 G−0.47 , 2 = k2 G−0.58 , R = k3 G−0.65 , d = k4 G−0.53 ,
1 = k5 V−0.25 , 2 = k6 V−0.46 , R = k7 V−0.48 and d = k8 V−0.33 by
linear regression analysis. These exponent values show
that the dependencies of 2 , R and (d) on the (G) and (V)
are stronger than 1 .
(b) A comparison of the exponent values of 1 , 2 , R and (d) for
directionally solidified SCN–7.5 wt.% CTB with a constant
growth rate at different temperature gradients or with a
constant temperature gradient at different growth rates
obtained in present work with the theoretical models and
previous experimental works have been made. From the
comparison, it can be seen that the exponent value of 1 is
in a good agreement with the theoretical exponent values
of 1 and the average exponent values of 2 , R and (d) are
slightly lower than the theoretical values.
(c) The values of 1 , 2 R and (d) for directionally solidified SCN–7.5 wt.% CTB with a constant growth rate at
different temperature gradient or with a constant temperature gradient at different growth rates measured in
present work have been compared with the calculated
values of 1 , 2 , R and (d) from Kurz and Fisher (1981),
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 2 ( 2 0 0 8 ) 145–155
Trivedi (1984), Bouchard and Kirkaldy (1996, 1997), Lu and
Hunt (1996), Trivedi and Somboonsuk (1984) and Rutter
and Chalmers (1953) models and it was seen that the
experimental results are mostly in good agreement with
the calculated values from Kurz and Fisher (1981), Trivedi
(1984), Bouchard and Kirkaldy (1996, 1997), Trivedi and
Somboonsuk (1984) and Rutter and Chalmers (1953) models.
(d) Langer and Müller-Krumbhaar (1978) have carried out a
detailed numerical analysis of the wavelength of instabilities along the sides of a dendrite according to undercooling
and have predicted the values of 2 /R to be 2.1. In the
present work, the average value of 2 /R for SCN–7.5 wt.%
CTB alloy was found to be 2.07 and this value is very close
to 2.1 predicted by Langer and Müller-Krumbhaar (1978). A
comparison of 2 /R values obtained in present work with
the values of 2 /R obtained from theoretical and previous
experimental works have also been made.
Acknowledgements
This project was supported by Erciyes University Scientific
Research Project Unit. The authors are grateful to Erciyes
University Scientific Research Project Unit for their financial
supports.
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