Macrosegregation during steady-state arrayed growth of dendrites in directionally solidified Pb-Sn...

Macrosegregation during Steady-State Arrayed Growth of Dendrites in Directionally Solidified Pb-Sn Alloys

S.N. TEWARI and RAJESH SHAH

Macrosegregation along the length of the directionally solidified samples is produced when Pb-Sn alloys (10 to 58 wt pct Sn) are directionally solidified in a positive thermal gradient (melt on top, solid below, and gravity pointing down) with steady-state dendritic arrayed morphology (the length of the mushy zone, much smaller than the initial length of the melt column, re- maining nearly constant during growth). The extent of the macrosegregation increases with increasing tin content, becomes maximum for 33.3 wt pct Sn, and decreases with further increase in tin content. The intensity of the interdendritic thermosolutal convection responsible for the longitudinal macrosegregation can be represented by the effective partition coefficient (ke), an empirical parameter obtained from the dependence of the longitudinal macrosegregation on fraction distance solidified. The extent of the macrosegregation appears to be related to a parameter, {A ~fL.(CE -- Ct)}, where /~1 is the primary dendrite spacing, fe is the volume fraction of the interdendritic melt, and CE and C, are the eutectic composition and the melt composition ahead of the dendrite tips, respectively.

I. INTRODUCTION

THE temperature and composition profiles in the melt, both in the interdendritic mushy region and in the melt column ahead, are schematically shown in Figure 1 for a steady-state arrayed growth of dendrites obtained during directional solidification of binary alloys in a positive thermal gradient. This figure also shows the lead- rich portion of the Pb-Sn phase diagram. The solute content of the interdendritic melt decreases from C•, (eutectic composition) at the base of the array to C, at the dendrite tip. The composition in the bulk melt, ahead of the dendritic array, decreases from C, to the bulk melt composition, Co, over some distance, which, for the diffusive mass transport, is approximately equal to D JR, where D~ is the solutal diffusivity in the melt and R the growth speed. The solutal buildup at the tips (C, - Co) has been shown tl,2,31 to depend on the gradient of constitutional supercooling (1 - S), where S = D~G~/ m~RCo(k - 1). Here, Gt is the thermal gradient in the melt at the liquid-solid interface, mt is the liquidus slope, and k is the solute partition coefficient. Ignoring the rapid solidification regime, (C, - Co) is negligible for S ~ 0, where the dendritic microstructures are observed. Larger (C, - Co) are expected for growth at small gradients of constitutional supercoolings, where the array morphology changes to cellular.

For the growth conditions depicted in Figure 1, melt on the top and the solid below, with gravity pointing down, the temperature profile alone is expected to be stabilizing against natural convection. However, the solutal profile will be destabilizing for those alloys where rejection of the solute into the melt during solidification results in reduced melt density, as is expected for Pb-Sn alloys with Co < CE (Figure 1). Depending upon the temperature and composition dependence of the melt

S.N. TEWARI, Professor, and RAJESH SHAH, Graduate Student, are with the Chemical Engineering Department, Cleveland State University, Cleveland, OH 44115.

Manuscript submitted December 16, 1991.

density (schematically shown in Figure 1), thermosolutal convection tal can start in the melt, both within the interdendritic region and ahead of the dendrite array. Besides the thermophysical properties of the alloy and the growth parameters (G/, Co, R), several other factors, such as the primary arm spacings (A~), the liquid volume fraction (fl), and the length of the mushy zone (H), are expected to control the onset and extent of interdendritic thermosolutal convection. The extent of solute enrichment (C, - Co) and the thermal gradient G~ will determine the onset and intensity of the convection in the melt ahead. Because of the negligibly small (C, - Co), the composition profile in the bulk melt ahead of the array is not expected to be a major factor in thermosolutal convection for dendritic arrays, and the interdendritic convection will dominate. Thermosolutal convecion in the bulk melt will, however, be very important for cellular arrays because of the larger solutal buildup at the array tips.

These fluid flows will produce longitudinal (parallel to the growth direction) and transverse macrosegregation, the solutal inhomogeneities occurring over length scales much larger than the dendrite spacings. Even though macrosegregation has been extensively examined theoreticallyl5 J2] and experimentally, t'3-221 the mushy zone length was not kept constant during the solidification process for most of the investigations. The growth conditions for the onset of the thermosolutal convection (in the interdendritic melt or in the melt near the dendrite tips) have not been experimentally examined for the steady-state dendritic arrayed growth. The relationships between such thermosolutal convection and the resulting macrosegregation are relatively unexplored. 118'19'2°1 The dependence of the longitudinal and transverse macrosegregation during steady-state dendritic/cellular arrayed growth on A~, f/, H, and (C, - Co) has not been systematically studied. There is conflicting behavior reported in the literature. The absence of macrosegregation in space-grown A1-Cu specimens, as compared to the extensive transverse macrosegregation in specimens solidified under identical conditions on earth, has been at- tributed to interdendritic fluid flow. 1221 Yet, it has been

METALLURGICAL TRANSACTIONS A VOLUME 23A, DECEMBER 1992--3383

shown by directional solidification experiments in the presence of extensive stirring t~S1 that convection in the melt was not able to penetrate into the dendritic arrays; this observation was also supported by a recent numer- ical analysisY 2j Boettinger et al. I2°1 directionally solidified Pb-57 pct Sn and observed extensive longitudinal macrosegregation in the specimen portion containing lead dendrites. Little macrosegregation was present in the absence of dendrites (where the microstructure was aligned in situ composite), suggesting that convection was more severe in the presence of dendrites.

The objective of this research was to investigate the role of A~, f~, and (C, - Co) on the thermosolutal convection and their influence on the longitudinal and transverse macrosegregation. Experimentally, Gt, R, and Co were varied systematically during steady-state arrayed dendritic/cellular growth of Pb-Sn alloys. In this article, we will examine the influence of interdendritic thermosolutal convection on the longitudinal and transverse macrosegregation for dendritic alloys. We will also examine the growth conditions for the onset of the thermosolutal convection in the interdendritic melt and its relationship with the formation of channel segregates. Thermosolutal convection in the bulk melt because of the solutal buildup at the tips and the resulting longitudinal macrosegregation, especially for the cellular arrays, will be presented separately.

II. E X P E R I M E N T A L

The alloy preparation and directional solidification procedures have been described previously, t231 Precast Pb-Sn alloy samples (about 30-cm long) contained in quartz crucibles (0.7-cm inner diameter) were remelted (melt column about 20-cm long) in vacuum and directionally solidified by raising the furnace assembly at various speeds (0.4 to 66 /xm s -l) with respect to the stationary sample, thus avoiding any convection caused by the crucible motion. Growth conditions were ther- mally stable, i .e. , positive thermal gradient in the melt, with the melt on top and the solid below. Specimens with in situ CHROMEL-ALUMEL* thermocouples were

*CHROMEL-ALUMEL is a trademark of Hoskins Manufacturing Company, Hamburg, MI.

used to measure the thermal profiles within the melt for growth conditions which were identical to those used for the macrosegregation study. The G~ values were measured at the liquidus temperature from these thermal profiles. After 12 to 14 cm of directional growth, the specimens were quenched by directing a jet of helium gas, cooled by liquid nitrogen, onto the surface of the quartz crucibles. The furnace and crucible arrangements are such that the furnace translation rate is equal to the directional solidification speed. This was verified by correlating the longitudinal (parallel to growth direction) microstructure of the directionally solidified specimen with the furnace translation distance. The steady-state thermal profile was maintained during directional solidification, as evidenced by identical thermal profiles indicated by the two thermocouples which were located along the sample length with the separation distance of

+'3 r . 5 7 3

+ i K/C~,O Ct dE 273 II ;I ! I I I I I I

Pb 10 20 30 40 50 60 70

Weight percent Tin

U0u'0 II / Ill . . . . . . . . +' I J--

Q

Solid _~ = ' =

Temperature Composition Mel t density

Fig. 1 - -Schemat i c representation of the thermal and solutal profiles in the melt for dendritic array growth during directional solidification of binary alloys.

about 2 c m . 1231 The steady-state thermal profile ensures a constant mushy zone length throughout the directional solidification. Mushy zone length during these experiments is much smaller (less than one order of magnitude) than the length of the initial melt column.

The longitudinal and transverse microstructures were examined in the unetched condition by standard optical metallography techniques. Primary arm spacings were measured on the transverse sections by hand-counting the number (N) of cells/dendrites in an area and cal- culating the primary arm spacing, Az = BV'-A--~a/N, 1241 where B has been taken as unity. About 160 to 350 cells/ dendrites were counted for each measurement. Macro- segregation along the length of the directionally solidified bars was obtained by measuring the tin content of thin slices (~3-mm wide), cut along the length, by atomic absorption spectrometry. The distance from the tip of the mushy zone at the onset of directional solidification to its tip at the time of quench, as observed from the longitudinal microstructure, is taken as the total solidification distance. The ratio of the distance solidified, as measured from the tip of the mushy zone at the onset of directional solidification, to the total length of the initial melt column was taken as the fraction solid (fs). Spec- imens with increasing tin content (10 to 57.9 wt pct) were directionally solidified at controlled G~ and R so as to yield approximately similar primary dendrite spacings to investigate the influence of increasing volume fraction interdendritic liquid.

3384--VOLUME 23A, DECEMBER 1992 METALLURGICAL TRANSACTIONS A

III. R E S U L T S

Figure 2 shows the typical thermal profiles obtained during the directional solidification in the vicinity of the mushy zone. The thermal gradients in the liquid (G~) at the liquid-solid interface at their corresponding liquidus temperatures were obtained from such plots. The experimental parameters, such as Co, A l, G~, and R, are listed in Table I.

Figure 3(a) shows the typical alignment of primary dendrites as observed at the quenched liquid-solid interface of the directionally solidified specimens. The corresponding transverse microstructure in the directionally solidified portion is shown in Figure 3(b). The primary dendrites were observed to be uniformly distributed across the entire specimen cross section, except for some channel segregates occasionally located on the outer periphery. For the growth condition shown in Figure 3, two channel segregates, located on the outer periphery of the sample, were observed. Figure 3(c) presents a typical higher magnification view of the tin-rich channel. For all the samples examined in this study, the channels were always located on the specimen periphery, their number remained constant along the specimen length, and their spatial distributions did not show much variation (Table I compiles the solidification parameters and their corresponding number of channels observed in this study). These observations are in contrast with earlier observations, 12u where the channels were observed to merge and decrease in number along the length of the solidified in- gots. This difference is most probably because of the continuously changing mushy zone length during prior experiments, where the heat was extracted from the bot- tom of the mold during solidification of cylindrical in- gots. In the present investigation, the mushy zone length, much smaller than the length of the initial melt column, was kept constant during steady-state directional solidification.

Figure 4 shows the typical transverse microstructures of the directionally solidified samples used for examin- ing the influence of increasing interdendritic volume

873 m i n d Co = 23.2% Sn R = 6 ixms -1

C o = 57.9% Sn 773 - - " ~ ~ g - 10 i, ms -1

, , , \ " -1

4 7 3

373 0 2 4 6 8 10 12

Distance, cm

Fig. 2 - - T y p i c a l thermal profiles in the vicinity of the mushy zone during directional solidification of binary Pb-Sn alloys used to measure the thermal gradient, G~, at the liquidus temperatures (distance in this figure is from some arbitrary location).

fraction liquid on the longitudinal macrosegregation. In these figures, the primary lead dendrites appear dark, and the interdendritic phase appears light. As expected, the increased tin content results in larger interdendritic volume fraction liquid. Suitable growth conditions, Gj and R, were used for these samples to yield approximately similar primary dendrite spacings, 166 to 185/zm, in order to carry out a systematic investigation of the increasing volume fraction interdendritic liquid.

Figure 5 shows the thermal gradient and growth speed dependence of the primary dendrite spacings. It plots the primary arm spacing (Al) vs (C°25/G°SR°25), following the relationship suggested by Hunt. ~251 The broken line is the linear least-squared fit through the datapoints. The relationship can be expressed as h~ = B(C 025/(7 °5R°25), where B = 0.0084 (with a standard deviation -0.0037), ha is in centimeters, Co is in wt pct Sn, Gt is in K cm -~, and R is in cm s -~ . The quantitative predictions from the three analytical models available in the literature, which describe the growth parameter dependence of the primary dendrite spacing, Hunt, t251 Trivedi, 126] and Laxmanan, TM yield the constant B values to be 0.013, 0.027, and 0.027 cm, respectively (the physical properties listed in Reference 23 were used in the above relationships). The three models differ only in their man- ner of estimating the thermal gradient and growth-rate dependence of Ct. Otherwise, they all follow the same basic approach used by Hunt. [25j These models include only diffusive transports and do not include convection in their analysis. Considering that nearly all our samples have been grown in the presence of thermosolutal convection and there are uncertainties in the various physical properties used in estimating the theoretically predicted B values, the agreement with the model by Hunt (B = 0.008 -+ 0.004 observed vs 0.013 predicted) is good.

Figure 6 shows typical macrosegregation along the length of the directionally solidified Pb-Sn alloys. The ratio of the distance solidified, as measured from the tip of the mushy zone at the onset of directional solidification, to the total length of the initial melt column is taken as the fraction solid (fs). The open symbols represent the directionally solidified portion. The closed symbols represent the quenched liquid portion. Scatter in the composition value due to the analysis technique used in this investigation is ---0.7 wt pct (for Co 33 pet). The five data points at fs -- 0.45 indicate the compositions measured in the radial direction from the center of the sample to its outer periphery (a cylindrical surface was machined by lathe and filings corresponding to the five radial distances were analyzed). Despite large macrosegregation along the length of the directionally solidified sample, there does not appear to be any radial macrosegregation for this sample. The area under the curve was used to obtain the initial alloy content of the melt (Co). It was found to be within -+5 pct of the analysis obtained from specimens cut from the as-cast feed stock bars. The Co values thus obtained are listed in Table I.

Figure 7 shows the influence of the tin content on the longitudinal macrosegregation. In order to avoid con- fusion, the data from the quenched liquid portion are not included here; only the directionally solidified portions are depicted. Separate ordinates have been used in order to present the wide range of tin contents (10 to 58 wt


Table I. Macrosegregation and Channel Segregates in Directionally Solidified Pb-Sn Alloys

Co R Gt AI C, (Aj)2Fe(CE - Ct) Number of Sample (Wt Pct Sn) (/xm s -l) (K cm -1) (/zm) FE (Wt Pct) kE* (cm 2 Wt Pct) RE Channels

SN001 10.0 10 110 115 0.01 11.4 1.0 8.0 x 10 -5 0.016 0 1A 16.5 4 101 172 0.09 20.4 0.90-+ 0.046/0.88 1.1 x 10 3 0.075 1 3C 23.2 6 77 185 0.18 25.0 0.86 4- 0.011/0.86 2.3 × 10 -3 0.071 4 3B 23.7 24 81 164 0.19 24.3 0.92 --- 0.025/0.96 1.9 x 10 -3 0.045 0 3D 27.0 66 59 155 0.24 27.3 0.93 +- 0.012/0.95 2.0 x 10 -3 0.028 0 4A 33.3 8 75 166 0.38 34.7 0.84 -+ 0.013/0.86 2.8 X 10 -3 0.04 2 5A 57.9 10 105 234 0.93 59.0 0.93 --- 0.019/0.98 1.5 X 10 -3 0.524 0 5B 54.7 40 67 177 0.84 55.2 0.97 --- 0.046/0.98 1.8 × 10 -3 0.094 0

*First number indicates kr obtained from slope, _+ number its standard deviation, a n d / t h e third number the kE obtained from intercept.

(c)

(a) (b)

Fig. 3 - - T y p i c a l longitudinal and transverse microstructures (Co = 33.3 pct Sn, R = 8 / z m s ~, G~ = 75 K cm -~) of the directionally solidified Pb-Sn alloy specimens: (a) longitudinal, (b) transverse (there are two channel segregates on the outer periphery), and (c) high-magnification view of channel segregate.

pct). However, identical scales have been used for a convenient comparison of the extent of macrosegregation. Uncertainties in the composition values for each specimen due to the chemical analysis technique are indicated by the error bars in this figure. Macro- segregation is absent for the lowest tin content, Co = 10 wt pct. The extent of macrosegregation increases with increasing tin content, from 16.5 to 33.3 wt pct Sn. However, with further increase in tin content to 57.9 wt pct, virtually no longitudinal macrosegregation is observed.

IV. DISCUSSION

In the absence of any convective mixing of the interdendritic fluid with the bulk melt, the solutal profile along

the length of the directionally solidified sample should consist of an initial short length of inverse segregation (corresponding to the formation of the mushy zone length) followed by the region of constant solute content (Co) during the steady-state growth. Only near the end, where the mushy region finally solidifies, a positive segregation will be expected. The macrosegregation profiles shown in Figure 7 therefore clearly indicate convective solute transport from the interdendritic liquid to the bulk melt.

A. Effective Partition Coefficient

Macrosegregation caused by convection in the melt has been extensively studied for solidification with planar liquid-solid interfaces.128.29.3°1 A simple analytical


(a) (b)

(c ) . . . . . . z ~ a p m ( d )

Fig. 4 - - 1 n f l u e n c e of increasing tin content on the interdendri t ic vo lume fraction liquid. Growth condi t ions have been selected to yield approx- imate ly s imi la r pr imary dendri te spacings. (a) Co = 16.5 pct, R = 4 / x m s ~, G, = 101 K cm ~, A~ = 172 /xm; (b) Co = 23.2 pct, R = 6 / z m s -~, G~ = 77 K cm ~, A~ = 185 p.m; (c) Co = 33.3 pct, R = 8 /zm s ~, G~ = 75 K cm ~, At = 166/ . t in , and (d) Co = 54.7 pct, R = 40 p.m s ~,G~ = 6 7 K c m ~,Al = 177 /zm.


.0275 - -

E o .0225

a .

i .0175 - -

f-

~: .0125 - -

A

.0075 .9

A

A A

I I I 1.1 1.3 1.5

(Co0.25/R0.25GI0.5)

I 1.7

Fig. 5--Growth parameter (C°25/G°SR ° 25) dependence of the primary dendrite spacings.

38

35

29

41 - -

0 1.0

0 Directlonally solidified , F - l e - - • Quenched liquid I

I

I I

J op,9/'o

_ ce /

4"

I I I I .2 .4 .6 .8 Fraction distance solidified (fs)

Fig. 6--Typical macrosegregation along the length of the directionally solidified specimen. The open symbols represent the directionally solidified portion; the closed symbols represent the quenched liquid portion. (Co = 33.3 pct Sn, R = 8 /xm s t; G~ = 75 K cm t).

relationship, Cs = kEC0(l -fJE ,, where kE is the effective solutal partition coefficient, is used. to describe the longitudinal macrosegregation, psi As the extent of convection increases from purely diffusive mass transport toward the comple te mixing in the melt, kE decreases from nearly unity to k, the equilibrium solute partition coefficient. The following relationship, kE = {k/k + (1 - k) exp - (R6/DI)}, is used to relate the momentum boundary layer thickness (6) with kE and k.

The above relationship is not valid for the longitudinal macrosegregation caused by interdendritic thermosolutal convection during steady-state growth of dendritic arrays because the entire mushy zone cannot be treated as a closed volume having no solute exchange with the rest o f the melt, and therefore, 6 does not have any physical significance. However , in the absence of a rigorous analysis o f the longitudinal macrosegregation during steady- state growth of dendritic array, the above relationship

60

,58

56

t O0 0

0

0 ~ 0 0

0 0 0

0

I I

36 - -

30

t -

28 O p. P23

3 4 - -

32 - -

I°

21

19

17

m

O Directionally solidified • Quenched liquid

0

0 o O0

0

I I

[]

D DD

[]

I I

O0

[]

0 D

000

I I

15

13

llli

i , , , , , , I ,="

I I I I

1 1

9 0 .2 .4 .6 .8

Fraction distance solidified (fs)

Fig. 7--Effect of tin content (Co) on the longitudinal macrosegregation. The corresponding G~, R, and Al values are listed in Table I.

3388- -VOLUME 23A, DECEMBER 1992 METALLURGICAL TRANSACTIONS A

has been assumed 1~81 to obtain 6 in a stirred melt. We need a quantitative representation of the extent of the longitudinal macrosegregation, typically shown in Figure 7, and the effective partition coefficient appears to be a suitable empirical parameter. As typically shown in Figure 8, the relationship, Cs = keCo(l -f~)kL-~, can be used to describe the experimentally observed longitudinal macrosegregation. This figure plots loglo (Cs/Co) vs log10 (1 - f , ) data, and the linear least- squared fit to the data is indicated by the solid line. The effective partition coefficients obtained from the slope of this line, 0.843 -+ 0.013, and that obtained from the intercept on the ordinate, 0.86, are approximately the same, suggesting that the above relationship can be used to describe the experimentally observed longitudinal macrosegregation. The two ke values, one obtained from the slope and the other from the intercept, are listed in Table I for all of the growth conditions examined in this study. Their average value will be used in Section B to represent the extent of the longitudinal macrosegregation (related to the intensity of the thermosolutal convection), the kE approaching unity with the decreasing convection.

B. Thermosolutal Convection

The fluid velocity (V) parallel to the direction of the dendrite columns, as given by D'Arcy's law, is as fol- lows: V = - ( K / t ~ f t ) ( A p / A y - pg), where K is the permeability of the mushy region, A p / A y, the pressure gradient along the direction of gravity, p, the melt density, /z, the melt viscosity, and g is the gravitational ac- celeration. Using extensive experimental data on Pb-Sn and Borneol paraffin and following the Hagen-Poiseuille model, Poirier 1271 has shown the following dependence of permeability (K) on the liquid volume fraction and primary dendrite spacing, K = 3.75 × 10 4f~A~ (-+31 pct).

.03 ~

.01 - -

- .01 - -

o

~ - . 0 3 -

: 0 5 - -

- . 07 --.8

O O

I I I I - .6 - .4 - .2 0

log(1 - fs)

Fig. 8--Typical analysis of the longitudinal macrosegregation to obtain the effective solute partition coefficient, log (C,/Co) vs log (1 - f,) plot (Co = 33.3 pct Sn, R = 8 /zm s ', G~ = 75 K cm ~).

As a crude approximation, the velocity V can be expressed as V = - ( K / t i f t ) {pg( /3AC - aAT)}, where/3 (5.2 × 10-3/wt pct 1211) and a (1.2 × 10-4/K 12jl) are the solutal and thermal coefficients of volumetric expan- sions. Because of the dominance of the solutal contri- bution, the a A T term can be neglected. Introducing the above A I and ft dependence of K, the fluid velocity becomes V = ( -3 .75 × lO-4g/3p/Iz) {A~flAC}. Assum- ing that AC is proportional to (CE - Ct), one would expect that the fluid velocity will be proportional to {A~f/(C~- - C,)}. The fraction of the interdendritic liquid (ft) varies from the base of the dendrites to their tip. However, the fraction of the eutectic liquid ( f e ) at the base of the dendrites can be assumed to represent the fraction liquid, f~. The severity of the longitudinal macrosegregation, as represented by decreasing kE, will be expected to increase with the increasing interdendritic convection, as indicated by the increasing fluid velocity, V. This is supported by Figure 9, which represents all the data obtained from our dendritic Pb-Sn alloy specimens, grown under a wide range of growth conditions (G~, R, and Co). This figure plots the experimentally obtained effective partition coefficients vs {A ~E(CE - C,)}. In this analysis, the C, values calculated from Reference 3 and f e values calculated from Reference 31 (by using the corresponding C,) have been used, as listed in Table I. However, because of the negligible solutal buildup at the array tips for the dendritic morphology, the behavior will be similar if (Ce - Co) is used instead of (CE - C,). The linear regression fit through the data points indicated by the solid line in Figure 9 shows that ke decreases with increasing {A~fE(CE - C,)}. The decreasing fluid velocities indicated by the decreasing values of the parameter, {A~ fE(CE -- Ct)}, result in reduced thermosolutal convection, as indicated by kE approaching unity.

As described earlier (Figure 7), a maximum is observed in the macrosegregation vs tin content; zero macrosegregation for Co = 10 and 58 wt pct Sn, and

1.00

.95

0

~ .90

I " .85 •

.80 I I I 0 .001 .002

k 12fE(C E - Ct) , (cm2wt%)

Fig. 9--Influence of the parameter {(AO2f~(CE -- C,)} on k~.

.003


maximum for Co = 33.3 wt pct Sn. This is more clearly indicated in Figure 10, which plots the extent of longitudinal macrosegregation as represented by the change in tin content f romfs = 0.2 tof~ = 0.6 vs Co for all of the specimens examined in this study. This is contrary to the behavior expected at first glance; the increasing solute content should produce a larger volume fraction of the interdendritic liquid, i.e., more permeable mushy region, and hence, easier fluid flow and more macrosegregation. [The influence of any increase in tortuosity because of the finer secondary dendrite spacings in the mushy zone produced by increasing tin content will be much less than the influence of increased volume fraction of interdendritic liquid (Figure 4).] However, increasing tin content will also reduce the driving force (CE - C~) for the thermosolutal convection and thus result in lower V and less macrosegregation. Examination of the dependence of the longitudinal macrosegregation, Cs(f~ = 0.6) - C~(f~ = 0.2), on {,~fE(C, - CE)} shows the expected behavior (Figure 11); the extent of macrosegregation increases with increasing {A~fE(C, - CE)}.

C. Onset of lnterdendritic Flow and Channel Segregates

Thermal (Rr) and solutal (Rs) Rayleigh numbers have been defined in the literature to describe the onset of thermosolutal convection in the melt. t~2'2~'291 The two numbers are given as RT = gc~Gth4/KLt ' and Rs = g~G,.ha/DL v, where a is the thermal (volumetric) coefficient of expansion (1.15 x 10 -4 K - l ) ; KL, the thermal diffusivity (1.08 x 10 -5 m 2 s ~); ~,, the kinematic viscosity (2.47 x 10 -7 m 2 s-l); r , the solutal (volumetric) coefficient of expansion (5.2 x 10 -3 wt pct-l); G~, the solutal gradient; Dc, the solutal diffusivity (3 x 10 9 m 2 s-~); and h, a characteristic length. The physical properties for the Pb-Sn alloy melt listed above are from Reference 21. What dimension should be selected as the

characteristic length scale? For a planar liquid-solid interface, (DL/R) has been assumed to represent the characteristic length, t32] For the dendritic arrays, the primary arm spacing t21] and the mushy zone length [12] have been used as the characteristic length. Assuming A1 to be the characteristic length, the onset of interdendritic thermosolutal convection (assumed to correspond to the channel formation) in Pb-Sn alloys has been observed to occur at a critical effective Rayleigh number, Re = Rs/ (Ki/Dt) - RT, of 0.063 (calculated from Table I in Reference 21).

Table I enables us to examine the influence of Co and A~ on the formation of channels and their association with the interdendritic thermosolutal convection, as evidenced by the longitudinal macrosegregation. Examin- ing the influence of increasing Co shows that no channels were observed in alloys containing less than 16.5 wt pct tin, as compared to the 5 wt pct tin reported earlier. This observation yields a critical Re value (for 16.5 wt pct tin, Table I) of 0.075.

Approximately the same critical Re value is obtained for the 23 wt pct Sn alloy, where the primary arm spacings were systematically varied to control the interdendritic fluid flow. The 23 wt pct tin alloy showed channel segregates at 6 / zm s -~ (Re = 0.071) but not at 24 (Re = 0.045) and 66 /xm s -~ (Re = 0.028). This observation yields a critical Re value of 0.071. How- ever, the 33.3 wt pt Sn alloy with a smaller Re value of 0.039 also showed channel segregates. It is interesting to note that (except for the 33.3 wt pct Sn alloy), the critical Re values (about 0.07) observed in this study are remarkably close to that reported by Sarazin and Hellawell t2u in the Pb-Sn alloys solidified under very different growth conditions (our much larger Gt, 60 to 100 K cm -~, compared with 2 to 14 K cm -~ used by Sarazin and Hellawell, produces much smaller primary dendrite spacings and mushy zone lengths). Also, their

4 ,-& d #

|

o

N

/ \ I \ I \ \ I \ I \ I \

\ / • \

/ \

_. / I I o ; I v

20 40 60 Tin content, wt%

Fig. 1 0 - - T i n content dependence of the longitudinal macrosegregation as represented by C~(f, = 0.6) - C~(f, = 0.2).

II

|

~. 2 U

J~ Ov

/ /

/ /

/ /

/ /

/o ,/ /

/ • /

/ • /

e -.I"~/- • I I v

.001 .002 .003

feb 2(C~ - ct), cm2wt%

Fig. l l - - D e p e n d e n c e of the longitudinal macrosegregation on {(A,)2fE(CE- C,)}.


samples were much larger, about 38 mm in diameter as compared to 7 mm in the present study.

However, examination of Table I shows that while the critical effective Rayleigh number concept t2~l may be able to predict the lower limit for the onset of the channel segregates, it does not explain several observations, such as why we do not observe channel segregates for tin contents larger than 33 wt pct despite their large Re values and why our channel segregates are always located along the specimen outer periphery, as opposed to the internal channels reported by Sarazin and Hellawell. t211 It is pos- sibly because of the much smaller specimen diameter used in the present study. A recent two-dimensional numer- ical study of channel formation 1331 shows that, in narrow molds, the solute accumulates along the inner wall of the mold because of the no-slip condition. This leads to the onset of channel segregates on the outer periphery of the sample. In large diameter samples, restrictions to the fluid flow present within the mushy zone, such as foreign particles and broken dendrite fragments, also lo- cally retard the interdendritic fluid flow, resulting in solute accumulation and channel formation in the interior of the specimen cross section.

It is important to note that, in this study, the channel formation is observed to be always associated with the longitudinal macrosegregation. For the same tin content (23 wt pct), the 6/xm s -~ specimen shows channel segregates and the macrosegregation, while 24/zm s-l shows neither. Alloys grown dendritically with tin content greater than 33 wt pct show no longitudinal macrosegregation and no channel segregates (Table I). All of the dendritic specimens showing longitudinal macrosegregation in- variably also contain channel segregates. This observation is also supported from the data reported by Sarazin and Hellawell, t211 even though their mushy zone length was not constant during the experiment. It is obvious that channel formation is strongly associated with the onset of thermosolutal convection in the interdendritic melt. A better understanding of the interdendritic thermosolutal convection is therefore very important t341 in predicting the formation of channel segregates.

It should, however, be pointed out that it is possible to have large longitudinal macrosegregation without having any channel segregates, as seen in Pb-58 pct Sn grown t2°l at Gt/R values much larger than used in the -gresent study. It is proposed that the longitudinal macro- ;egregation at high Gt/R is produced by thermosolutal convection in the melt column ahead of the dendritic array because of the large solutal buildup at their tips and not from the convection in the interdendritic melt. Experi- ments are presently under progress to examine the influence of such solutal buildup on the longitudinal macrosegregation.

V. CONCLUSIONS

The following conclusions can be drawn from this study which examined the growth parameter dependence of the longitudinal macrosegregation due to interdendritic thermosolutal convection during directional solidification of Pb-Sn alloys with a steady-state dendritic arrayed morphology:

1. The experimentally observed longitudinal macrosegregation can be described by an empirical parameter, effective partition coefficient (ke), obtained from Cs = kEC0(l - f s ) kE-~, where the original tin content is Co and that corresponding to the fraction distance solidified (fs) is Cs.

2. The increasing intensity of the interdendritic thermosolutal convection, as evidenced by decreasing kE values, appears to correlate with increasing {A~fe(Ce - C,)}, where Ai is the primary dendrite spacing, fe is the volume fraction interdendritic eutectic, and CE and C, are the eutectic and the tip compositions in the melt, respectively.

3. Channel formation is strongly associated with the onset of thermosolutal convection in the interdendritic melt. The critical effective Rayleigh number (RE) for the onset of channel formation is about 0.07, a value similar to that reported in the literature (0.063) for the Pb-Sn alloys solidified under very different growth conditions) TM A better theoretical understanding of the onset of interdendritic thermosolutal convection is therefore crucial in order to predict channel formation.

ACKNOWLEDGMENTS

Appreciation is expressed to the Microgravity Science and Applications Division for the use of the Microgravity Materials Science Laboratory at the NASA-Lewis Research Center. Continuous encouragement from Thomas K. Glasgow, Chief Processing Science and Technology Branch, and help from Bruce Rosenthal, John Sedlock (deceased), and Jerry Loveland is gratefully acknowledged.

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