Eutectic growth of unidirectionally solidified bismuth–cadmium alloy

of 11/11
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 Ni˘ gde University, Faculty of Arts and Sciences, Department of Physics, Ni˘ gde, Turkey b 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 λ i (λ a , λ a , λ m , λ M ), were measured from both transverse section (λ a ) and longitudinal section (λ a , λ m , λ M ) of the specimens. The variations of λ i with respect to G and V were determined by using linear regression analysis. The dependence of lamellar spacings λ i 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 λ i 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. λ 2 V, Tλ, TV 0.5 and λ 2 G values were determined by using λ i , 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 simultane- ously 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 appreci- ation 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 process- ing parameters of temperature gradient (G) growth rate (V), and composition (C o ). Not only was this a powerful investiga- tion 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). study of the relationship between microstructure and solidifi- cation 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 C E , the average undercool- ing T at the interface results from three contributions. T = T E T L = T c + T r + T k (1) where T is the average interface undercooling, T E is the eutec- tic temperature and T L is the local interface temperature, and T c , T r , and T k are the chemical, capillary, and kinetic undercoolings, respectively. For regular metallic eutectic sys- tems, however, T k can usually be neglected compared to T c and T r . The and lamellae grow under steady state con- ditions 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 = K 1 + K 2 λ (2) 0924-0136/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.jmatprotec.2006.10.022
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Transcript of Eutectic growth of unidirectionally solidified bismuth–cadmium alloy

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    Journal of Materials Processing Technology 183 (2007) 310320

    Eutectic growth of unidirectionally solidified bismuthcadmium alloy

    H. Kaya a, E. Cadrl a, M. Gunduz b,a Nigde University, Faculty of Arts and Sciences, Department of Physics, Nigde, Turkey

    b 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

    bstract

    Bismuth (Bi)Cadmium (Cd) eutectic alloy was melted in a graphite crucible under vacuum atmosphere. This alloy was directionally solidifiedpward with a constant growth rate, V (8.33 m/s), and different temperature gradient ranges, G (1.934.74 K/mm), and also with a constant G4.74 K/mm), and different V (8.33167.32 m/s) in the Bridgman type directional solidification furnace. The lamellar spacings i (a, a ,

    m,

    M), were measured from both transverse section (a) and longitudinal section (

    a ,

    m,

    M) of the specimens. The variations of i with respect to G

    nd V were determined by using linear regression analysis. The dependence of lamellar spacings i on undercooling (T) was also analysed. Theariations of T with G at constant V and with V at constant G were investigated. According to these results, it has been found that decreasesiith 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. 2V,T, TV0.5 and 2G values were determined by using i, T, V and G values. The results obtained in this work have been compared with the

    acksonHunt eutectic theory and the similar experimental results 2006 Published by Elsevier B.V.

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    wt

    utad

    eywords: BiCd alloys; Eutectic growth; Lamellar spacing; Undercooling

    . Introduction

    A eutectic reaction can be defined as the instance wherewo (or more) distinctively different solid phases simultane-usly solidify from the parent liquid, i.e. liquid solid + solid. This is a complex process involving interactions of, amongthers, heat and mass transfer. Our knowledge and appreci-tion of eutectic solidification has advanced considerable asay be attested to by the great number of references cited in

    exts, review articles and books for last 60 years [19]. Withhe advent of directional solidification techniques in late 1940s,nd their subsequent refinement, it was possible to accuratelyelate the eutectic phase spacing to the solidification process-ng parameters of temperature gradient (G) growth rate (V),nd composition (Co). Not only was this a powerful investiga-ion approach, but the consequent alignment of the alternatinghases resulted in superior mechanical properties and initiated

    10].

    Eutectic alloys are the basis of many engineering materials115]. 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. Gunduz).

    gJTt

    924-0136/$ see front matter 2006 Published by Elsevier B.V.oi:10.1016/j.jmatprotec.2006.10.022tudy of the relationship between microstructure and solidifi-ation conditions [1660]. Directional solidification of binaryr pseudo-binary eutectics, may result in regular structures ofbrous or lamellar type. When two solid phases and growingrom a liquid of eutectic composition CE, the average undercool-ng T at the interface results from three contributions.

    T = TE TL = Tc + Tr + Tk (1)here T is the average interface undercooling, TE is the eutec-

    ic temperature and TL is the local interface temperature, andTc, Tr, and Tk are the chemical, capillary, and kinetic

    ndercoolings, respectively. For regular metallic eutectic sys-ems, however, Tk can usually be neglected compared to Tcnd Tr. The and lamellae grow under steady state con-itions with a build up of B atoms in the liquid ahead of thephase and the lateral transfer of solute to ensure steady state

    rowth. One of the most significant theoretical studies is theackson and Hunt (JH) model of the eutectic structures [29].he JH model [29] gives the following relationship betweenhe undercooling T, the growth rate V and the lamellar spacingfor an isothermal solidification front as,

    T = K1V + K2

    (2)

    mailto:[email protected]/10.1016/j.jmatprotec.2006.10.022

  • rocessing Technology 183 (2007) 310320 311

    wt

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    a

    K

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    P

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    Fig. 1. (a) The schematic plot of average undercooling T vs. lamellar spacingi for a given growth rate V. The stable and unstable regions, as predicted by theJtt

    lortt[

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    H. Kaya et al. / Journal of Materials P

    2eV =

    K2

    K1(3.a)

    Te = 2K2 (3.b)T 2

    V= 4K1K2 (3.c)

    here K1 and K2 can be calculated from phase diagram andhermodynamic data (Appendix). They are given by

    1 = mPCoffD

    (4)

    nd

    2 = 2m

    i

    (i sin i/mifi); i = , (5)

    here m = mm/(m + m) in which m and m are the slopesf the liquidus lines of the and phases at the eutectic tem-erature, Co the difference between the composition in the nd the phase, f and f the volume fractions of and hases, respectively. i is the GibssThompson coefficient, Dolute diffusion coefficient for the melt, and are the groovengles of /liquid phases and /liquid phases at the three-phaseonjunction point. These parameters concerning BiCd eutec-ic alloy are given in Appendix. The parameter is unity forhe lamellar growth. For lamellar eutectic the parameter P isefined as [29],

    = 0.3383(ff)1.661 (6)A well known conjecture of this criterion is the minimum

    upercooling arguments. This indicates that the spacing i, asndicated in Fig. 1, will be the operating point of spacing selec-ion [26]. Analysis of the stability of the solidifying interfacehows that this argument coincides with the marginal stabilityrinciple [50,51].

    The experimentally confirmed inter-relationship between theamellar spacing (i), growth rate (V) and the undercoolingT) in eutectic system implies that a mechanism is availableor changing the lamellar spacing when the growth rate and/orT varied. Fig. 1a shows the variation of the undercoolingith lamellar spacing according to the minimum undercooling

    riterion. In Fig. 1 e is extremum lamellar spacing, m is min-mum lamellar spacing, a average lamellar spacing and M is

    aximum lamellar spacing for the minimum undercooling con-ition. When i becomes greater than the M, tip splitting occursFig. 1b). When i is smaller than e, the growth will be unstablend when is smaller m overgrowth (Fig. 1c) will always occur26]. So that lamellar spacing i for the steady growth must sat-sfy m < a < M conditions. For eutectic growth, the TVelationships can be predicted by the JacksonHunt (JH) [29]nd TrivediMagninKurz (TMK) [30] models. It is clear thathe maximum spacing must be greater than twice the minimumpacing (M 2m), otherwise the new lamella can not catch up

    26].

    Most studies [2938] have shown that lamellar terminationsre constantly created and move through the structure duringutectic growth. The presence and movement of faults and fault

    1tiae

    acksonHunt analysis, are also shown. (b) The readjustment of local spacing byhe positive terminations. (c) The readjustment of local spacing by the negativeerminations [26].

    ines provide a means by which lamellar spacing changes canccur in response to growth rate fluctuations or a small growthate change. As can be seen from Fig. 1b and c, in this respect,he role of lamellar faults, and in particular lamellar termina-ions (positive and negative terminations), has been emphasised3941].

    The aim of the present work is to experimentally investigatehe dependence of the lamellar spacing on the solidificationarameters (G, V, T) and also find out the effect of G and V onT and to compare the results with the previous experimental

    esults and the existing theoretical model.

    . Experimental procedure

    The eutectic samples were prepared by melting weighed quantities of Bind Cd of (>99.9%) high purity metals in a graphite crucible which was placednto the vacuum melting furnace [42]. After allowing time for melt homogeni-ation, the molten alloy was poured into 13 graphite crucibles (6.35 mm o.d.,mm i.d. and 200 mm in length) in a hot filling furnace. Then, each specimenas positioned in a Bridgman type furnace in a graphite cylinder (40 mm o.d.,

    0 mm i.d. and 300 mm in length). After stabilizing the thermal conditions inhe furnace under an argon atmosphere, the specimen was grown by pullingt downwards at various temperature gradients (1.934.74 K/mm, V constant)nd various growth rates (8.33167.32 m/s, G constant) by means of differ-nt speed synchronous motors. After 100120 mm steady state growth of the

  • 312 H. Kaya et al. / Journal of Materials Processing Technology 183 (2007) 310320

    Table 1The values of lamellar spacings i and undecooling T for the directionally solidified BiCd eutectic system and experimental relationships (a) for different G at aconstant V and (b) for different V at a constant G

    Solidification parameters Lamellar spacings

    G (K/mm) V (m/s) T (K) e (m) a (m) a (m) m (m) M (m)

    Part (a): G, variable; V, constant1.93 8.33 0.078 3.05 3.15 0.09 4.52 0.12 3.35 6.332.64 8.33 0.093 2.52 2.65 0.15 4.22 0.03 2.81 5.623.40 8.33 0.105 2.17 2.30 0.03 3.72 0.08 2.54 4.884.11 8.33 0.122 1.94 2.07 0.11 3.22 0.14 2.25 4.224.74 8.33 0.137 1.78 1.89 0.09 3.05 0.09 2.05 3.78The relationships k1G0.61 k2G0.58 k3G0.47 k4G0.45 k5G0.53 k6G0.58

    Constant (k) Correlation coefficients (r)k1 = 3.58 K0.39 m0.61 r1 = 0.996k2 = 0.07 m0.42 K0.58 r2 = 0.999k3 = 0.09 m0.53 K0.47 r3 = 0.997k4 = 0.28 m0.45 K0.55 r4 = 0.979k5 = 0.12 m0.47 K0.53 r5 = 0.995k6 = 0.18 m0.42 K0.58 r6 = 0.988

    Part (b): V, variable; G, constant4.74 8.33 0.137 1.78 1.89 0.09 3.05 0.09 2.15 3.784.74 16.52 0.188 1.35 1.48 0.11 2.44 0.12 1.6 3.184.74 40.99 0.276 0.88 1.05 0.16 1.76 0.09 1.19 2.464.74 81.66 0.622 0.63 0.72 0.07 1.22 0.18 0.89 1.694.74 167.32 0.706 0.44 0.50 0.02 0.88 0.12 0.6 1.22The relationships k7V0.56 k8V0.47 k9V0.46 k10V0.43 k11V0.42 k12V0.40

    Constant (k) Correlation coefficients (r)k7 = 0.037 K m0.56 s0.56 r7 = 0.994k8 = 4.91 m1.47 s0.47 r8 = 0.999k9 = 5.03 m1.46 s0.46 r9 = 0.995k10 = 7.73 m1.43 s0.43 r10 = 0.997k11 = 5.21 m1.42 s0.42 r11 = 0.992k12 = 9.02 m1.40 s0.40 r12 = 0.987

    ampleo

    sAttsm

    2g

    cFg

    w

    3

    i(

    aVAltrmtpstActtvfg

    : the values of the lamellar spacing obtained from the transverse section of the sf the samples.

    amples, they were quenched by pulling them rapidly into the water reservoir.fter metallographic process which including mechanical and electropolishing

    echniques, the microstructures of the specimens were revealed. Microstruc-ures of the specimens were photographed from both transverse and longitudinalections by means of optical microscopy (OM) (Fig. 2) and scanning electronicroscopy (SEM) (Fig. 3).

    .1. The measurement of lamellar spacings i, temperatureradient,G, growth rates,V and The calculation of undercooling,T

    The lamellar spacings were measured from Figs. 2 and 3 with a linear inter-ept method [43]. The values of i (a, a , m and M) are given in Table 1 andig. 4 as a function of G and V. The details of measurements of G and V areiven in ref [23,42,5456].

    The minimum undercooling values, T, were obtained from the detailedTe curves (Fig. 5) which were plotted by using experimental V and G valuesith the system parameters K1, K2.

    . Result and discussionBiCd eutectic specimens were unidirectionally solid-fied with a constant V (8.33 m/s) and different G1.934.74 K/mm), and also, with a constant G (4.74 K/mm),

    ssrv

    s. *: The values of the lamellar spacing obtained from the longitudinal section

    nd different V (8.33167.32 m/s) in order to see the effect ofand G on the lamellar spacings, i and the undercooling, T.s can be seen from Figs. 2 and 3 during eutectic growth, a

    arge number of eutectic grains can be formed. All grains seemo be oriented parallel to growth direction but usually differed inotation about the growth axis. The normal of the and planesust be parallel to the polished longitudinal plane [42], however

    hese are not always possible. When the normal of the and lanes are not parallel to the longitudinal plane, the lamellarpacings * observed on the longitudinal plane give larger valuehan the lamellar spacings from the transverse polished plane.s can be seen from Table 1a and b, even some of the m values

    an be higher than the average a values. In a longitudinal view,he lamellar spacing seems to be different in each grain becausehey were cut under different angles , to the polished surface. aalues can be obtained by using the measured a and a valuesrom Table 1a and b (a = 51.73 4.7). For that reason, lon-itudinal sections are inadequate for evaluation of the lamellar

    pacing without the geometrical correction. It was observed thatome of the eutectic grains increased with the increasing growthate for the longitudinal section but the transverse section. So aalues measured on the transverse section of the sample are more

  • H. Kaya et al. / Journal of Materials Processing Technology 183 (2007) 310320 313

    Fig. 2. Variation of lamellar spacings with G and V of the directionally solidified BiCd eutectic alloy (a1) longitudinal section, (a2) transverse section forG = 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) trans-verse 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) longitudinalsection, (e2) transverse section for G = 4.74 K/mm and V = 167.32 m/s.

  • 314 H. Kaya et al. / Journal of Materials Pro

    Fig. 3. Scanning electron micrographs of the directionally solidified BiCdeutectic alloy (G = 4.74 K/mm and V = 8.33 m/s) (a) longitudinal section, (b)t

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    baanwsonpw

    3r

    tbtTa

    A(w(

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    ransverse section, (c) higher magnification views of transverse section.

    eliable. In this work a values have been compared with results

    f the similar works and the JH theory [23,37,42,44,45,55,56].

    In addition to the above microstructural characteristics, sev-ral solidification faults like layer mismatches and lamellarermination were observed. As can be seen from Fig. 1b, the cessing Technology 183 (2007) 310320

    oundary tilts toward the lamella side and a pocked range willppear in liquid in the front of the L interface with finallynew lamella growing in the pocked and a positive termi-

    ation forming. By this dynamic mechanism, the local spacingill decrease (m). The boundary tilts toward the lamella

    ide and the local L interface disappear with the lamella beingverlapped by the two neighbour lamella (a negative termi-ation) [3941]. Despite this microstructures that changed byositive and negative termination mechanism m, M valuesere measured as accurately as possible on each specimens.

    .1. The effect of the temperature gradient and the growthate on the lamellar spacings

    The variation of the lamellar spacings, i as a function of theemperature gradients is given in Table 1a and Fig. 4a. It cane observed that an increase in the temperature gradient leadso decrease in the lamellar spacings for a given V (8.33 m/s).hus we can describe the mathematical relationship between ind G by linear regression analysis as,

    i = k1Gm (for the constant V ) (7)s can be seen from Table 1a and Fig. 4a, dependence of i

    e, a, a , m and M) on the temperature gradient exponentsere 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 i has not been

    onsidered in theoretical studies for the regular eutectic growth.ut the influence of G cannot be ignored for regular and irreg-lar eutectic systems. The influence of temperature gradientn the lamellar spacings was investigated by several authors23,42,4648,5456]. As a result 2V is no more constant, i.e.decreases with the increasing G for a constant V (Table 1a).If Eq. (7) is used in Eq. (3) and applying the condition of

    rowth at minimum undercooling [(T/G)V = 0], to Eq. (3)ields

    =(

    K2

    K1k21

    )G2m (8)

    sing Eq. (7) in Eq. (2) gives

    T = K1 G2m +K2

    (9)

    here K1 = K2/k21. Eq. (9) gives the relationship between theverage undercooling T, the temperature gradient G and theamellar spacing i for an isothermal solidification front. Asan be seen from Eqs. (3) and (9), the temperature gradient, Gakes similar relationships with T and as the growth rate. Applying the condition of growth at minimum undercooling

    (T/)G = 0], to Eq. (9) gives

    eGm = k2 = constant1 (10.a)Te

    Gm= 2K2

    k1= constant2 (10.b)

    Te = 2K2 = constant3 (10.c)

  • H. Kaya et al. / Journal of Materials Processing Technology 183 (2007) 310320 315

    F /s), (bc theo

    AeVbaGc

    tva(taf

    i

    va

    (aAle

    ig. 4. (a) Variation of lamellar spacings with G at a constant V (V = 8.30 momparison of the experimental values with the values obtained by JH eutectic

    s can be seen from Eqs. (3.b) and (10.c), eT values arexactly the same for both growths with different G at constant

    and growth with different V at constant G. The relationshipetween i and G for constant V, gives similar result with the ind V for constant G and also the relationships between T and

    is similar to the relationships between T and V for the bothases (Tables 1 and 2).

    The exponent value, m (0.47) obtained for transverse sec-ion in this work (Table 1a) is in good agreement with (0.49)alue obtained by Cadrl and Gunduz [23] for PbSn eutecticlloy, but 0.47 value higher than the results (0.30), (0.33), (0.37),

    0.37) and (0.28) obtained by Cadrl et al. [42] for AlCu eutec-ic alloy, Toloui and Hellawell [47], Gunduz et al. [54] for AlSilloy, Cadrl et al. [55] for PbCd eutectic alloy, Kaya et al. [56]or SnZn eutectic alloy, respectively. In the limit of the exper-

    a

    ) variation of lamellar spacings with V at a constant G (V = 4.74 K/mm), (c)ry in the BiCd eutectic alloy.

    mental uncertainties, if m is taken as 0.5, Eq. (10.a) becomes2eG = constant. As can be seen from Table 2, average 2eGalue is constant with increasing G for a constant V, and also,verage 2eV value is constant with increasing V for a constant G.

    Variation in lamellar spacings, i with V at constant G4.74 K/mm) are given Table 1b and shown in Fig. 4b. The vari-tions of versus V is essentially linear on the logarithmic scale.s can be seen from Table 1b and Fig. 4b, the data form straight

    ines, the linear regression analysis gives the proportionalityquation as,i = k2Vn (for the constant G) (11)

    The values of n exponent for i (e, a, a , m and M)re equal to 0.47, 0.46, 0.43, 0.42 and 0.40, respectively

  • 316 H. Kaya et al. / Journal of Materials Processing Technology 183 (2007) 310320

    Table 2Comparison of the experimental results with the theoretical predictions for the directionally solidified BiCd eutectic alloy (a) for different G and constant V, (b) fordifferent V and constant G

    Dependence of i on T and G Dependence of T on G

    eT (K m) aT (K m) aT (K m) 2eG (K m2) 2aG (K m2) 2a G (K m2) TG0.61 (K1.61 m0.61)

    Part (a): V, constant; G, variable0.24 0.25 0.35 0.018 0.019 0.039 3.5310.23 0.25 0.39 0.017 0.019 0.047 3.4780.23 0.24 0.39 0.016 0.018 0.047 3.3650.24 0.25 0.39 0.015 0.018 0.043 3.4830.24 0.26 0.42 0.015 0.017 0.044 3.5850.240 0.006 0.250 0.007 0.390 0.020 0.016 0.001 0.018 0.001 0.044 0.003mT = 0.267 (constant) 2m G = 0.021 0.0008 K m2 TG0.61 = 3.488 0.08 (constant)MT = 0.512 (constant) 2M G = 0.077 0.0061 K m2

    Dependence of i on T and V Dependence of T on V

    eT (K m) aT (K m) aT (K m) 2eV (m3/s) 2aV (m3/s) 2e V (m3/s) TV0.56 K m0.56 s0.56

    Part (b): G, constant; V, variable0.24 0.26 0.42 26.39 29.76 77.49 0.0420.25 0.28 0.46 30.11 36.19 98.35 0.0390.24 0.29 0.49 31.74 45.19 126.97 0.0350.39 0.45 0.76 32.41 42.33 121.54 0.0530.31 0.35 0.62 32.39 41.83 129.57 0.0400.289 0.060 0.326 0.080 0.548 0.120 30.609 2.530 39.059 6.130 110.77 22.32

    3.4535.2

    (tvmt(

    i

    taEats[iKG

    3t

    ipa

    B

    bwimVlVm

    Ab

    w(srgt

    GTc

    (mT = 0.380) 2mV = 52.752 (MT = 0.741) 2MV = 203.28

    na = 0.46). It is apparent that the dependence of i values onhe growth rate exponent (0.46) was found to be close to thealue predicted by eutectic theory (0.50). The experimentaleasurements in the BiCd eutectic system obey the rela-

    ionships 2V = constant for a given G {2e V = 30.609 m3/scalc.), 2aV = 39.059 m3/s, 2a V = 110.770 m3/s,2m V = 52.752 m3/s, 2M V = 203.281 m3/s}.

    The variation of the lamellar spacings as a function of thenverse square root of the growth rate is given in Fig. 4c. The2aV value (39.059 m

    3/s) and the 2eV value (30.609 m3/s) in

    his work are slightly higher than results (21.1, 21.8, 23.7, 19.6,nd 29.75 m3/s) obtained by Trivedi et al. [37], Moore andlliot [44], Cadrl et al. [55] for PbCd eutectic system, Whelannd Haworth [57] for BiCd eutectic system and JH eutecticheory [29], respectively. The value of 2aV (39.059 m

    3/s) ismaller than the value of 156 m3/s obtained by Cadrl et al.42] for AlCu eutectic. The 39.059 m3/s value in this works very close to values of (40.61 and 35.10 m3/s) obtained byaya et al. [56] for SnZn eutectic and obtained by Cadrl andunduz. [23] for PbSn eutectic, respectively.

    .2. Effect of temperature gradients and growth rates onhe minimum undercooling

    Fig. 5 shows the minimum undercooling T, of the solid-fying interface was obtained from Te curves which were

    lotted by using the experimental V and G values with Eqs. (3)nd (10).

    Fig. 5a shows the relationship between T and e for theiCd eutectic system at different G, in a constant V. As can

    wn

    TV0.50 = 0.0522 TV0.56 = 0.042 0.007 (constant)

    e seen from Fig. 5a, T increases with the increasing Ghile as the extremum spacing e decreases. Although G values

    ncreased approximately 2.5 times, T value increased approxi-ately 1.75 times. Fig. 5b shows the Te curves for differentin a constant G. The influence of V is certain on the lamel-

    ar spacing, , and T. T also increases with the increasing, whereas e decreases. Although V values increased approxi-ately 20 times, T value increased approximately 5.5 times.Fig. 6a shows the dependence of T on G for a constant V.

    s can be seen from Tables 1 and 2 and Fig. 6a, the relationshipetween T and G, T and e can be expressed as:

    T = k1 G0.61 (for constant V ) (12a)Te = constant3 (for constant V ) (12b)here k1 is a constant (3.58 K0.39 m0.61). The exponent value

    0.61) of the temperature gradient for the TG relationship islightly higher than the exponent value (0.47) of G for the aGelationship. As can be seen from Figs. 5 and 6, temperatureradients affect the lamellar spacings and the undercoolings inhe same way with the growth rates.

    Fig. 6b shows the variation ofT as function of V in a constant. T increases with the increasing V. As can be seen fromables 1 and 2 and Fig. 6b, the dependence of T on V and ean be given as:

    Te = k7 V 0.56 (for constant G) (13a)

    Te = constant3 (for constant G) (13b)here k7 is a constant (k7 = 0.037 K m0.56 s0.56). The expo-ent 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) 310320 317

    Fsa

    aa[r

    cT

    FV(

    av

    ig. 5. (a) Calculated average undercooling T values vs. extremum lamellarpacing e for the BiCd eutectic alloy (a) at a constant V (8.33 mm/s) and (b)t a constant G (4.74 K/mm).

    nd 0.50 obtained by Gunduz et al. [54] for AlSi eutecticlloy, Cadrl et al. [55] for PbCd eutectic alloy, Kaya et al.56] for SnZn eutectic alloy and JH eutectic theory Eq. (3.a),espectively.

    Fig. 7a and b shows the variation 2V, as function of G foronstant V and 2G, as function of V for constant G, respectively.he values of 2V decreases as the values G increase (Fig. 7a),2e V = k1G1.19 (k1 = 0.044 K1.19 m1.81 s1) (14.a)2aV = k2G1.12 (k2 = 0.071 K1.12 m1.88 s1) (14.b) ig. 6. (a) Variation of the minimum undercooling with G at a constant(V = 8.33 m/s) and (b) the variation of the undercooling with V at a G

    G = 4.74 K/mm).

    2a V = k3G0.93 (k3 = 0.570 K0.93 m2.07 s1) (14.b)nd the values of 2V decreases as the values of G increase. Thealues of 2G decreases as the values of V increase (Fig. 7b)

    2eG = k1 V0.91 (k1 = 0.110 m2.91 K s0.91) (15.a)2aG = k2 V0.96 (k2 = 0.142 m2.96 K s0.96) (15.b)2G = k V0.82 (k = 0.265 m2.82 K s0.82) (15.c)e 3 3

    From Eq. (15) a can be expressed as following,

    a = k V1/2 G1/2 (16)

  • 318 H. Kaya et al. / Journal of Materials Pro

    F(

    Ab

    toK[Lb

    4

    1

    2

    3

    4

    5

    A

    TmmCC

    ff

    ig. 7. (a) Variation of 2V as a function of G at a constant V (V = 8.33 m/s),b) variation of 2G as a function of V at a constant G (G = 4.74 K/mm).

    s can be seen from Eqs. (14.b) and (15.b) values decreasesoth increasing G and V values.

    The average value of exponent (1.12) in this work is higherhan (0.67) obtained by Toloui and Hellawell [47], (0.73)btained by Gunduz et al. [54], (0.81) obtained by Fisher andurz [60] for AlSi eutectic alloy, (0.63) obtained by Kaya et al.

    56] for SnZn eutectic alloy, (0.7) obtained by Schurman andoblich [58] for cast iron and (0.84) obtained by Fisher [59] fororneol-succinonitrile.

    . Conclusions. The change of lamellar spacings i, according to the solid-ification parameters V, G and T for the BiCd eutecticalloy was investigated and relationships between them were

    DKK

    cessing Technology 183 (2007) 310320

    obtained by binary regression analysis as follows:

    a = k3G0.47 (for a constant V ) (k3 = 0.09 m0.53 K0.47)

    a = k9V0.46 (for a constant G) (k9 = 5.03 m1.46 s0.46)

    . Effects of temperature gradients G and growth rates V onthe undercooling T were examined and the relationshipsbetween them were obtained as follows:

    T = k1G0.61 (for constant G) (k1 = 3.58 K0.39 m0.61)

    T = k7V 0.56 (for constant V )(k7 = 0.037 K m0.56 s0.56)

    It can be seen from these relationships, effects of G and V onthe T are nearly the same.

    . The bulk growth rate 2aV was obtained by using experimen-tal values and found to be 39.059 m3/s which is slightlyhigher than the JH theoretical value (29.75 m3/s). 2eV ,2mV ,

    2MV ,

    2a V and

    2aV values have been found to be con-

    stant for a given G with the increasing V according to Eqs. (2)and (3) (physical parameters [6165] used for calculationsare given in Appendix).2eG,

    2aV ,

    2a G,

    2m G and

    2M G

    values have been found to be constant with the increasing Gfor the constant V according to Eq. (10).

    . T increases with the increasing temperature gradient G fora given V, whereas e decreases, similarly, the T increaseswith the increasing growth rate V for a given G, whereas edecreases. On the other hand, Te value was found to beconstant for both cases.

    . It can be seen from exponent values of G and V, effects oftemperature gradient and growth rate on the lamellar spacingsand the undercoolings are similar.

    ppendix

    The physical parameters used for BiCd eutectic alloy

    E (K) 418.5 [61]

    (K (wt.%)1) 3.173 [61] (K (wt.%)1) 2.91 [61]E (wt.%) 39.70[61]o (wt.%) 100 [61]

    0.43 [62]

    0.57 [62]

    (K m) 0.105 [63]

    (K m) 0.106 [63]

    () 9.57 [63] () 24.38 [63]

    (m2/s) 5000 [64,65]

    1 (K s/m2) 0.004 (calculated from the physical parameters)

    2 (m K) 0.119 (calculated from the physical parameters)2V = K2/K1 = 29.75 m3/s (calculated from the physical parameters)

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    H. Kaya et al. / Journal of Materials P

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    Eutectic growth of unidirectionally solidified bismuth-cadmium alloyIntroductionExperimental procedureThe measurement of lamellar spacings lambdai, temperature gradient,G, growth rates,V and The calculation of undercooling, DeltaT

    Result and discussionThe effect of the temperature gradient and the growth rate on the lamellar spacingsEffect of temperature gradients and growth rates on the minimum undercooling

    ConclusionsAppendixReferences