Eutectic growth of unidirectionally solidified bismuth–cadmium alloy

Eutectic growth of unidirectionally solidified bismuth–cadmium alloy

Journal of Materials Processing Technology 183 (2007) 310–320 Eutectic growth of unidirectionally solidified bismuth–cadmium alloy H. Kaya a , E. C ¸...

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Journal of Materials Processing Technology 183 (2007) 310–320

Eutectic growth of unidirectionally solidified bismuth–cadmium alloy H. Kaya a , E. C ¸ adırlı a , M. G¨und¨uz b,∗ a

b

Ni˘gde University, Faculty of Arts and Sciences, Department of Physics, Ni˘gde, Turkey Erciyes University, Faculty of Arts and Sciences, Department of Physics, Kayseri, Turkey

Received 7 November 2005; received in revised form 15 October 2006; accepted 18 October 2006

Abstract Bismuth (Bi)–Cadmium (Cd) eutectic alloy was melted in a graphite crucible under vacuum atmosphere. This alloy was directionally solidified upward with a constant growth rate, V (8.33 ␮m/s), and different temperature gradient ranges, G (1.93–4.74 K/mm), and also with a constant G (4.74 K/mm), and different V (8.33–167.32 ␮m/s) in the Bridgman type directional solidification furnace. The lamellar spacings λi (λa , λ∗a , λ∗m , λ∗M ), were measured from both transverse section (λa ) and longitudinal section (λ∗a , λ∗m , λ∗M ) of the specimens. The variations of λi with respect to G and V were determined by using linear regression analysis. The dependence of lamellar spacings λi on undercooling (T) was also analysed. The variations of T with G at constant V and with V at constant G were investigated. According to these results, it has been found that λi decreases with increasing values of G and V. Also T increases with increasing V for a constant G and with increasing G for a given V, respectively. λ2 V, Tλ, TV−0.5 and λ2 G values were determined by using λi , T, V and G values. The results obtained in this work have been compared with the Jackson–Hunt eutectic theory and the similar experimental results © 2006 Published by Elsevier B.V. Keywords: Bi–Cd alloys; Eutectic growth; Lamellar spacing; Undercooling

1. Introduction A eutectic reaction can be defined as the instance where two (or more) distinctively different solid phases simultaneously solidify from the parent liquid, i.e. liquid → solid ␣ + solid ␤. This is a complex process involving interactions of, among others, heat and mass transfer. Our knowledge and appreciation of eutectic solidification has advanced considerable as may be attested to by the great number of references cited in texts, review articles and books for last 60 years [1–9]. With the advent of directional solidification techniques in late 1940s, and their subsequent refinement, it was possible to accurately relate the eutectic phase spacing to the solidification processing parameters of temperature gradient (G) growth rate (V), and composition (Co ). Not only was this a powerful investigation approach, but the consequent alignment of the alternating phases resulted in superior mechanical properties and initiated [10]. Eutectic alloys are the basis of many engineering materials [1–15]. This has led to an extensive theoretical and experimental



Corresponding author. Tel.: +90 352 4374937x33126; fax: +90 352 4374933. E-mail address: [email protected] (M. G¨und¨uz).

0924-0136/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.jmatprotec.2006.10.022

study of the relationship between microstructure and solidification conditions [16–60]. Directional solidification of binary or pseudo-binary eutectics, may result in regular structures of fibrous or lamellar type. When two solid phases ␣ and ␤ growing from a liquid of eutectic composition CE , the average undercooling T at the interface results from three contributions. T = TE − TL = Tc + Tr + Tk

(1)

where T is the average interface undercooling, TE is the eutectic temperature and TL is the local interface temperature, and Tc , Tr , and Tk are the chemical, capillary, and kinetic undercoolings, respectively. For regular metallic eutectic systems, however, Tk can usually be neglected compared to Tc and Tr . The ␣ and ␤ lamellae grow under steady state conditions with a build up of B atoms in the liquid ahead of the ␣ phase and the lateral transfer of solute to ensure steady state growth. One of the most significant theoretical studies is the Jackson and Hunt (J–H) model of the eutectic structures [29]. The J–H model [29] gives the following relationship between the undercooling T, the growth rate V and the lamellar spacing λ for an isothermal solidification front as, T = K1 Vλ +

K2 λ

(2)

H. Kaya et al. / Journal of Materials Processing Technology 183 (2007) 310–320

λ2e V =

K2 K1

311

(3.a)

Tλe = 2K2

(3.b)

T 2 = 4K1 K2 V

(3.c)

where K1 and K2 can be calculated from phase diagram and thermodynamic data (Appendix). They are given by K1 =

mPCo f␣ f␤ D

and



K2 = 2mδ

(4)

(Γi sin θi /mi fi );

i = ␣, ␤

(5)

i

where m = m␣ m␤ /(m␣ + m␤ ) in which m␣ and m␤ are the slopes of the liquidus lines of the ␣ and ␤ phases at the eutectic temperature, Co the difference between the composition in the ␤ and the ␣ phase, f␣ and f␤ the volume fractions of ␣ and ␤ phases, respectively. Γ i is the Gibss–Thompson coefficient, D solute diffusion coefficient for the melt, θ ␣ and θ ␤ are the groove angles of ␣/liquid phases and ␤/liquid phases at the three-phase conjunction point. These parameters concerning Bi–Cd eutectic alloy are given in Appendix. The parameter δ is unity for the lamellar growth. For lamellar eutectic the parameter P is defined as [29], P = 0.3383(f␣ f␤ )1.661

(6)

A well known conjecture of this criterion is the minimum supercooling arguments. This indicates that the spacing λi , as indicated in Fig. 1, will be the operating point of spacing selection [26]. Analysis of the stability of the solidifying interface shows that this argument coincides with the marginal stability principle [50,51]. The experimentally confirmed inter-relationship between the lamellar spacing (λi ), growth rate (V) and the undercooling (T) in eutectic system implies that a mechanism is available for changing the lamellar spacing when the growth rate and/or T varied. Fig. 1a shows the variation of the undercooling with lamellar spacing according to the minimum undercooling criterion. In Fig. 1 λe is extremum lamellar spacing, λm is minimum lamellar spacing, λa average lamellar spacing and λM is maximum lamellar spacing for the minimum undercooling condition. When λi becomes greater than the λM , tip splitting occurs (Fig. 1b). When λi is smaller than λe , the growth will be unstable and when λ is smaller λm overgrowth (Fig. 1c) will always occur [26]. So that lamellar spacing λi for the steady growth must satisfy λm < λa < λM conditions. For eutectic growth, the T–V–λ relationships can be predicted by the Jackson–Hunt (J–H) [29] and Trivedi–Magnin–Kurz (TMK) [30] models. It is clear that the maximum spacing must be greater than twice the minimum spacing (λM ≥ 2λm ), otherwise the new lamella can not catch up [26]. Most studies [29–38] have shown that lamellar terminations are constantly created and move through the structure during eutectic growth. The presence and movement of faults and fault

Fig. 1. (a) The schematic plot of average undercooling T vs. lamellar spacing λi for a given growth rate V. The stable and unstable regions, as predicted by the Jackson–Hunt analysis, are also shown. (b) The readjustment of local spacing by the positive terminations. (c) The readjustment of local spacing by the negative terminations [26].

lines provide a means by which lamellar spacing changes can occur in response to growth rate fluctuations or a small growth rate change. As can be seen from Fig. 1b and c, in this respect, the role of lamellar faults, and in particular lamellar terminations (positive and negative terminations), has been emphasised [39–41]. The aim of the present work is to experimentally investigate the dependence of the lamellar spacing λ on the solidification parameters (G, V, T) and also find out the effect of G and V on T and to compare the results with the previous experimental results and the existing theoretical model. 2. Experimental procedure The eutectic samples were prepared by melting weighed quantities of Bi and Cd of (>99.9%) high purity metals in a graphite crucible which was placed into the vacuum melting furnace [42]. After allowing time for melt homogenisation, the molten alloy was poured into 13 graphite crucibles (6.35 mm o.d., 4 mm i.d. and 200 mm in length) in a hot filling furnace. Then, each specimen was positioned in a Bridgman type furnace in a graphite cylinder (40 mm o.d., 10 mm i.d. and 300 mm in length). After stabilizing the thermal conditions in the furnace under an argon atmosphere, the specimen was grown by pulling it downwards at various temperature gradients (1.93–4.74 K/mm, V constant) and various growth rates (8.33–167.32 ␮m/s, G constant) by means of different speed synchronous motors. After 100–120 mm steady state growth of the

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Table 1 The values of lamellar spacings λi and undecooling T for the directionally solidified Bi–Cd eutectic system and experimental relationships (a) for different G at a constant V and (b) for different V at a constant G Solidification parameters G (K/mm)

V (␮m/s)

Part (a): G, variable; V, constant 1.93 8.33 2.64 8.33 3.40 8.33 4.11 8.33 4.74 8.33 The relationships

Lamellar spacings T (K)

λe (␮m)

λa (␮m)

λ∗a (␮m)

λ∗m (␮m)

λ∗M (␮m)

0.078 0.093 0.105 0.122 0.137

3.05 2.52 2.17 1.94 1.78

3.15 ± 0.09 2.65 ± 0.15 2.30 ± 0.03 2.07 ± 0.11 1.89 ± 0.09

4.52 ± 0.12 4.22 ± 0.03 3.72 ± 0.08 3.22 ± 0.14 3.05 ± 0.09

3.35 2.81 2.54 2.25 2.05

6.33 5.62 4.88 4.22 3.78

k1 G0.61

k2 G−0.58

k3 G−0.47

k4 G−0.45

k5 G−0.53

k6 G−0.58

Constant (k) k1 = 3.58 K0.39 ␮m0.61 k2 = 0.07 ␮m0.42 K0.58 k3 = 0.09 ␮m0.53 K0.47 k4 = 0.28 ␮m0.45 K0.55 k5 = 0.12 ␮m0.47 K0.53 k6 = 0.18 ␮m0.42 K0.58 Part (b): V, variable; G, constant 4.74 8.33 4.74 16.52 4.74 40.99 4.74 81.66 4.74 167.32 The relationships

Correlation coefficients (r) r1 = 0.996 r2 = −0.999 r3 = −0.997 r4 = −0.979 r5 = −0.995 r6 = −0.988 0.137 0.188 0.276 0.622 0.706

1.78 1.35 0.88 0.63 0.44

1.89 ± 0.09 1.48 ± 0.11 1.05 ± 0.16 0.72 ± 0.07 0.50 ± 0.02

3.05 ± 0.09 2.44 ± 0.12 1.76 ± 0.09 1.22 ± 0.18 0.88 ± 0.12

2.15 1.6 1.19 0.89 0.6

3.78 3.18 2.46 1.69 1.22

k7 V0.56

k8 V−0.47

k9 V−0.46

k10 V−0.43

k11 V−0.42

k12 V−0.40

Constant (k) k7 = 0.037 K ␮m−0.56 s0.56 k8 = 4.91 ␮m1.47 s−0.47 k9 = 5.03 ␮m1.46 s−0.46 k10 = 7.73 ␮m1.43 s−0.43 k11 = 5.21 ␮m1.42 s−0.42 k12 = 9.02 ␮m1.40 s−0.40

Correlation coefficients (r) r7 = 0.994 r8 = −0.999 r9 = −0.995 r10 = −0.997 r11 = −0.992 r12 = −0.987

λ: the values of the lamellar spacing obtained from the transverse section of the samples. λ* : The values of the lamellar spacing obtained from the longitudinal section of the samples.

samples, they were quenched by pulling them rapidly into the water reservoir. After metallographic process which including mechanical and electropolishing techniques, the microstructures of the specimens were revealed. Microstructures of the specimens were photographed from both transverse and longitudinal sections by means of optical microscopy (OM) (Fig. 2) and scanning electron microscopy (SEM) (Fig. 3).

2.1. The measurement of lamellar spacings λi , temperature gradient,G, growth rates,V and The calculation of undercooling, T The lamellar spacings were measured from Figs. 2 and 3 with a linear intercept method [43]. The values of λi (λa , λ∗a , λ∗m and λ∗M ) are given in Table 1 and Fig. 4 as a function of G and V. The details of measurements of G and V are given in ref [23,42,54–56]. The minimum undercooling values, T, were obtained from the detailed T–λe curves (Fig. 5) which were plotted by using experimental V and G values with the system parameters K1 , K2 .

3. Result and discussion Bi–Cd eutectic specimens were unidirectionally solidified with a constant V (8.33 ␮m/s) and different G (1.93–4.74 K/mm), and also, with a constant G (4.74 K/mm),

and different V (8.33–167.32 ␮m/s) in order to see the effect of V and G on the lamellar spacings, λi and the undercooling, T. As can be seen from Figs. 2 and 3 during eutectic growth, a large number of eutectic grains can be formed. All grains seem to be oriented parallel to growth direction but usually differed in rotation about the growth axis. The normal of the ␣ and ␤ planes must be parallel to the polished longitudinal plane [42], however these are not always possible. When the normal of the ␣ and ␤ planes are not parallel to the longitudinal plane, the lamellar spacings λ* observed on the longitudinal plane give larger value than the lamellar spacings λ from the transverse polished plane. As can be seen from Table 1a and b, even some of the λ∗m values can be higher than the average λa values. In a longitudinal view, the lamellar spacing seems to be different in each grain because they were cut under different angles θ, to the polished surface. θ a values can be obtained by using the measured λ∗a and λa values from Table 1a and b (θ a = 51.73 ± 4.7◦ ). For that reason, longitudinal sections are inadequate for evaluation of the lamellar spacing without the geometrical correction. It was observed that some of the eutectic grains increased with the increasing growth rate for the longitudinal section but the transverse section. So λa values measured on the transverse section of the sample are more

H. Kaya et al. / Journal of Materials Processing Technology 183 (2007) 310–320

313

Fig. 2. Variation of lamellar spacings with G and V of the directionally solidified Bi–Cd eutectic alloy (a1 ) longitudinal section, (a2 ) transverse section for G = 1.93 K/mm and V = 8.33 ␮m/s, (b1 ) longitudinal section, (b2 ) transverse section for G = 3.40 K/mm and V = 8.33 ␮m/s, (c1 ) longitudinal section, (c2 ) transverse section for G = 4.74 K/mm and V = 8.33 ␮m/s, (d1 ) longitudinal section, (d2 ) transverse section for G = 4.74 K/mm and V = 40.99 ␮m/s, (e1 ) longitudinal section, (e2 ) transverse section for G = 4.74 K/mm and V = 167.32 ␮m/s.

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boundary tilts toward the ␤ lamella side and a pocked range will appear in liquid in the front of the ␣–L interface with finally a new ␤ lamella growing in the pocked and a positive termination forming. By this dynamic mechanism, the local spacing will decrease (λ∗m ). The ␣–␤ boundary tilts toward the ␣ lamella side and the local ␣–L interface disappear with the lamella being overlapped by the two neighbour ␤ lamella (a negative termination) [39–41]. Despite this microstructures that changed by positive and negative termination mechanism λ∗m , λ∗M values were measured as accurately as possible on each specimens. 3.1. The effect of the temperature gradient and the growth rate on the lamellar spacings The variation of the lamellar spacings, λi as a function of the temperature gradients is given in Table 1a and Fig. 4a. It can be observed that an increase in the temperature gradient leads to decrease in the lamellar spacings for a given V (8.33 ␮m/s). Thus we can describe the mathematical relationship between λi and G by linear regression analysis as, λi = k1 G−m

(for the constant V )

(7)

As can be seen from Table 1a and Fig. 4a, dependence of λi (λe , λa , λ∗a , λ∗m and λ∗M ) on the temperature gradient exponents were found to be 0.58, 0.47, 0.45, 0.53 and 0.58, respectively (ma = 0.47). The influence of temperature gradient G on λi has not been considered in theoretical studies for the regular eutectic growth. But the influence of G cannot be ignored for regular and irregular eutectic systems. The influence of temperature gradient on the lamellar spacings was investigated by several authors [23,42,46–48,54–56]. As a result λ2 V is no more constant, i.e. λ decreases with the increasing G for a constant V (Table 1a). If Eq. (7) is used in Eq. (3) and applying the condition of growth at minimum undercooling [(∂T/∂G)V = 0], to Eq. (3) yields   K2 V = G2m (8) K1 k12 Using Eq. (7) in Eq. (2) gives T = K1 λ G2m +

Fig. 3. Scanning electron micrographs of the directionally solidified Bi–Cd eutectic alloy (G = 4.74 K/mm and V = 8.33 ␮m/s) (a) longitudinal section, (b) transverse section, (c) higher magnification views of transverse section.

reliable. In this work λa values have been compared with results of the similar works and the J–H theory [23,37,42,44,45,55,56]. In addition to the above microstructural characteristics, several solidification faults like layer mismatches and lamellar termination were observed. As can be seen from Fig. 1b, the ␣–␤

K2 λ

(9)

where K1 = K2 /k12 . Eq. (9) gives the relationship between the average undercooling T, the temperature gradient G and the lamellar spacing λi for an isothermal solidification front. As can be seen from Eqs. (3) and (9), the temperature gradient, G makes similar relationships with T and λ as the growth rate V. Applying the condition of growth at minimum undercooling [(∂T/∂λ)G = 0], to Eq. (9) gives λe Gm = k2 = constant1

(10.a)

2K2 Te = = constant2 Gm k1

(10.b)

Tλe = 2K2 = constant3

(10.c)

H. Kaya et al. / Journal of Materials Processing Technology 183 (2007) 310–320

315

Fig. 4. (a) Variation of lamellar spacings with G at a constant V (V = 8.30 ␮m/s), (b) variation of lamellar spacings with V at a constant G (V = 4.74 K/mm), (c) comparison of the experimental values with the values obtained by J–H eutectic theory in the Bi–Cd eutectic alloy.

As can be seen from Eqs. (3.b) and (10.c), λe T values are exactly the same for both growths with different G at constant V and growth with different V at constant G. The relationship between λi and G for constant V, gives similar result with the λi and V for constant G and also the relationships between T and G is similar to the relationships between T and V for the both cases (Tables 1 and 2). The exponent value, m (0.47) obtained for transverse section in this work (Table 1a) is in good agreement with (0.49) value obtained by C¸adırlı and G¨und¨uz [23] for Pb–Sn eutectic alloy, but 0.47 value higher than the results (0.30), (0.33), (0.37), (0.37) and (0.28) obtained by C¸adırlı et al. [42] for Al–Cu eutectic alloy, Toloui and Hellawell [47], G¨und¨uz et al. [54] for Al–Si alloy, C ¸ adırlı et al. [55] for Pb–Cd eutectic alloy, Kaya et al. [56] for Sn–Zn eutectic alloy, respectively. In the limit of the exper-

imental uncertainties, if m is taken as 0.5, Eq. (10.a) becomes λ2e G = constant. As can be seen from Table 2, average λ2e G value is constant with increasing G for a constant V, and also, average λ2e V value is constant with increasing V for a constant G. Variation in lamellar spacings, λi with V at constant G (4.74 K/mm) are given Table 1b and shown in Fig. 4b. The variations of λ versus V is essentially linear on the logarithmic scale. As can be seen from Table 1b and Fig. 4b, the data form straight lines, the linear regression analysis gives the proportionality equation as, λi = k2 V −n

(for the constant G)

(11)

The values of n exponent for λi (λe , λa , λ∗a , λ∗m and λ∗M ) are equal to 0.47, 0.46, 0.43, 0.42 and 0.40, respectively

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Table 2 Comparison of the experimental results with the theoretical predictions for the directionally solidified Bi–Cd eutectic alloy (a) for different G and constant V, (b) for different V and constant G Dependence of λi on T and G

Dependence of T on G

λa T (K ␮m)

λ∗a T (K ␮m)

λ2e G (K ␮m2 )

λ2a G (K ␮m2 )

2 λ2∗ a G (K ␮m )

TG−0.61 (K1.61 ␮m−0.61 )

Part (a): V, constant; G, variable 0.24 0.25 0.23 0.25 0.23 0.24 0.24 0.25 0.24 0.26 0.240 ± 0.006 0.250 ± 0.007

0.35 0.39 0.39 0.39 0.42 0.390 ± 0.020

0.018 0.017 0.016 0.015 0.015 0.016 ± 0.001

0.019 0.019 0.018 0.018 0.017 0.018 ± 0.001

0.039 0.047 0.047 0.043 0.044 0.044 ± 0.003

3.531 3.478 3.365 3.483 3.585

λe T (K ␮m)

λ∗m T = 0.267 (constant) λ∗M T = 0.512 (constant)

TG−0.61 = 3.488 ± 0.08 (constant)

2 λ2∗ m G = 0.021 ± 0.0008 K ␮m 2∗ λM G = 0.077 ± 0.0061 K ␮m2

Dependence of λi on T and V λe T (K ␮m)

λa T (K ␮m)

Part (b): G, constant; V, variable 0.24 0.26 0.25 0.28 0.24 0.29 0.39 0.45 0.31 0.35 0.289 ± 0.060 0.326 ± 0.080 (λ∗m T = 0.380) (λ∗M T = 0.741)

Dependence of T on V λ∗a T

(K ␮m)

0.42 0.46 0.49 0.76 0.62 0.548 ± 0.120

λ2e V

(␮m3 /s)

26.39 30.11 31.74 32.41 32.39 30.609 ± 2.530

λ2a V

(␮m3 /s)

29.76 36.19 45.19 42.33 41.83 39.059 ± 6.130

λ2∗ e V

(␮m3 /s)

77.49 98.35 126.97 121.54 129.57 110.77 ± 22.32

3.2. Effect of temperature gradients and growth rates on the minimum undercooling Fig. 5 shows the minimum undercooling T, of the solidifying interface was obtained from T–λe curves which were plotted by using the experimental V and G values with Eqs. (3) and (10). Fig. 5a shows the relationship between T and λe for the Bi–Cd eutectic system at different G, in a constant V. As can

0.042 0.039 0.035 0.053 0.040 TV−0.50 = 0.052 TV−0.56 = 0.042 ± 0.007 (constant)

λ2m V = 52.752 ± 3.45 λ2M V = 203.28 ± 35.22

(na = 0.46). It is apparent that the dependence of λi values on the growth rate exponent (0.46) was found to be close to the value predicted by eutectic theory (0.50). The experimental measurements in the Bi–Cd eutectic system obey the relationships λ2 V = constant for a given G {λ2e V = 30.609 ␮m3 /s 3 λ2∗ (calc.), λ2a V = 39.059 ␮m3 /s, a V = 110.770 ␮m /s, 2∗ 2∗ 3 3 λm V = 52.752 ␮m /s, λM V = 203.281 ␮m /s}. The variation of the lamellar spacings as a function of the inverse square root of the growth rate is given in Fig. 4c. The λ2a V value (39.059 ␮m3 /s) and the λ2e V value (30.609 ␮m3 /s) in this work are slightly higher than results (21.1, 21.8, 23.7, 19.6, and 29.75 ␮m3 /s) obtained by Trivedi et al. [37], Moore and Elliot [44], C ¸ adırlı et al. [55] for Pb–Cd eutectic system, Whelan and Haworth [57] for Bi–Cd eutectic system and J–H eutectic theory [29], respectively. The value of λ2a V (39.059 ␮m3 /s) is smaller than the value of 156 ␮m3 /s obtained by C¸adırlı et al. [42] for Al–Cu eutectic. The 39.059 ␮m3 /s value in this work is very close to values of (40.61 and 35.10 ␮m3 /s) obtained by Kaya et al. [56] for Sn–Zn eutectic and obtained by C ¸ adırlı and G¨und¨uz. [23] for Pb–Sn eutectic, respectively.

TV−0.56 K ␮m−0.56 s0.56

be seen from Fig. 5a, T increases with the increasing G while as the extremum spacing λe decreases. Although G values increased approximately 2.5 times, T value increased approximately 1.75 times. Fig. 5b shows the T–λe curves for different V in a constant G. The influence of V is certain on the lamellar spacing, λ, and T. T also increases with the increasing V, whereas λe decreases. Although V values increased approximately 20 times, T value increased approximately 5.5 times. Fig. 6a shows the dependence of T on G for a constant V. As can be seen from Tables 1 and 2 and Fig. 6a, the relationship between T and G, T and λe can be expressed as: T = k1 G0.61

(for constant V )

Tλe = constant3

(for constant V )

(12a) (12b)

(3.58 K0.39 ␮m0.61 ).

where k1 is a constant The exponent value (0.61) of the temperature gradient for the T–G relationship is slightly higher than the exponent value (0.47) of G for the λa –G relationship. As can be seen from Figs. 5 and 6, temperature gradients affect the lamellar spacings and the undercoolings in the same way with the growth rates. Fig. 6b shows the variation of T as function of V in a constant G. T increases with the increasing V. As can be seen from Tables 1 and 2 and Fig. 6b, the dependence of T on V and λe can be given as: Te = k7 V 0.56

(for constant G)

Tλe = constant3

(for constant G) = 0.037 K ␮m−0.56 s0.56 ).

(13a) (13b)

where k7 is a constant (k7 The exponent value (0.56) is in good agreement with 0.53, 0.48, 0.50

H. Kaya et al. / Journal of Materials Processing Technology 183 (2007) 310–320

Fig. 5. (a) Calculated average undercooling T values vs. extremum lamellar spacing λe for the Bi–Cd eutectic alloy (a) at a constant V (8.33 mm/s) and (b) at a constant G (4.74 K/mm).

and 0.50 obtained by G¨und¨uz et al. [54] for Al–Si eutectic alloy, C ¸ adırlı et al. [55] for Pb–Cd eutectic alloy, Kaya et al. [56] for Sn–Zn eutectic alloy and J–H eutectic theory Eq. (3.a), respectively. Fig. 7a and b shows the variation λ2 V, as function of G for constant V and λ2 G, as function of V for constant G, respectively. The values of λ2 V decreases as the values G increase (Fig. 7a), λ2e V = k1 G−1.19 λ2a V

=

k2 G−1.12

(k1 = 0.044 K1.19 ␮m1.81 s−1 )

(14.a)

(k2

(14.b)

= 0.071 K

1.12

␮m

1.88 −1

s

)

317

Fig. 6. (a) Variation of the minimum undercooling with G at a constant V (V = 8.33 ␮m/s) and (b) the variation of the undercooling with V at a G (G = 4.74 K/mm).  −0.93 λ2∗ a V = k3 G

(k3 = 0.570 K0.93 ␮m2.07 s−1 )

(14.b)

and the values of λ2 V decreases as the values of G increase. The values of λ2 G decreases as the values of V increase (Fig. 7b) λ2e G = k1 V −0.91

(k1 = 0.110 ␮m2.91 K s−0.91 )

(15.a)

λ2a G = k2 V −0.96

(k2 = 0.142 ␮m2.96 K s−0.96 )

(15.b)

 −0.82 λ2∗ e G = k3 V

(k3 = 0.265 ␮m2.82 K s−0.82 )

(15.c)

From Eq. (15) λa can be expressed as following, λa = k V −1/2 G−1/2

(16)

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obtained by binary regression analysis as follows: λa = k3 G−0.47 (for a constant V ) (k3 = 0.09 ␮m0.53 K0.47 )

λa = k9 V −0.46 (for a constant G) (k9 = 5.03 ␮m1.46 s−0.46 ) 2. Effects of temperature gradients G and growth rates V on the undercooling T were examined and the relationships between them were obtained as follows: T = k1 G0.61 (for constant G)

T = k7 V 0.56

(k1 = 3.58 K0.39 ␮m0.61 )

(for constant V )

(k7 = 0.037 K ␮m−0.56 s0.56 ) It can be seen from these relationships, effects of G and V on the T are nearly the same. 3. The bulk growth rate λ2a V was obtained by using experimental values and found to be 39.059 ␮m3 /s which is slightly higher than the J–H theoretical value (29.75 ␮m3 /s). λ2e V , 2 λ2m V , λ2M V , λ2∗ a V and λa V values have been found to be constant for a given G with the increasing V according to Eqs. (2) and (3) (physical parameters [61–65] used for calculations 2∗ 2∗ are given in Appendix).λ2e G, λ2a V , λ2∗ a G, λm G and λM G values have been found to be constant with the increasing G for the constant V according to Eq. (10). 4. T increases with the increasing temperature gradient G for a given V, whereas λe decreases, similarly, the T increases with the increasing growth rate V for a given G, whereas λe decreases. On the other hand, T␭e value was found to be constant for both cases. 5. It can be seen from exponent values of G and V, effects of temperature gradient and growth rate on the lamellar spacings and the undercoolings are similar. Fig. 7. (a) Variation of λ2 V as a function of G at a constant V (V = 8.33 ␮m/s), (b) variation of λ2 G as a function of V at a constant G (G = 4.74 K/mm).

As can be seen from Eqs. (14.b) and (15.b) λ values decreases both increasing G and V values. The average value of exponent (1.12) in this work is higher than (0.67) obtained by Toloui and Hellawell [47], (0.73) obtained by G¨und¨uz et al. [54], (0.81) obtained by Fisher and Kurz [60] for Al–Si eutectic alloy, (0.63) obtained by Kaya et al. [56] for Sn–Zn eutectic alloy, (0.7) obtained by Sch¨urman and L¨oblich [58] for cast iron and (0.84) obtained by Fisher [59] for borneol-succinonitrile. 4. Conclusions 1. The change of lamellar spacings λi , according to the solidification parameters V, G and T for the Bi–Cd eutectic alloy was investigated and relationships between them were

Appendix The physical parameters used for Bi–Cd eutectic alloy TE (K) 418.5 [61] −3.173 [61] m␣ (K (wt.%)−1 ) 2.91 [61] m␤ (K (wt.%)−1 ) CE (wt.%) 39.70[61] 100 [61] Co∗ (wt.%) f␣ 0.43 [62] 0.57 [62] f␤ 0.105 [63] Γ ␣ (K ␮m) Γ ␤ (K ␮m) 0.106 [63] 9.57 [63] θ ␣ (◦ ) 24.38 [63] θ ␤ (◦ ) D (␮m2 /s) 5000 [64,65] K1 (K s/␮m2 ) 0.004 (calculated from the physical parameters) 0.119 (calculated from the physical parameters) K2 (␮m K) λ2 V = K2 /K1 = 29.75 ␮m3 /s (calculated from the physical parameters)

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