Undercooling-Induced macrosegregation in directional solidification
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I.
INTRODUCTION
TIrE conditions leading to nucleation, the nucleation rate, and the particular phase formed are of critical importance in determining the structure and characteristics of any casting.t 1] Control of nucleation has been the driving force behind the entire grain refinement industry, and Chiba and Spittle t2l have shown the dramatic effect of nonreciprocal nucleation on the structure of castings in various pure-eutectic systems. The objectives of this work were to show how interracial energies influence nucleation behavior and to determine some of the possible consequences of that behavior in directionally solidified ingots and to what extent these effects occur. The Pb-Sn system was used because of its nonreciprocal nucleation behavior, low melting temperatures, wellcharacterized properties, and wide use as a model alloy. Sundquist and Mondolfe t3] used the "classical" heterogeneous nucleation theory developed by Volmer, [4J Tumbull/5~ and Fisher~61 to explain their experimental results of nonreciprocal nucleation. The nonreciprocal nucleation behavior of Pb-Sn, as well as that of many other alloys, has been verified and studied extensively by others using fine droplets/7,8,9j and bulksized ingots; tl~ this nucleation behavior is explained by solid-liquid interfacial-energy barriers to nucleation, rather than by lattice misfit between the nucleating catalyst and potential solid. 13a3j Following the classical nucleation theory, it can be shown that for a spherical cap H E N R Y C. de G R O H III, Materials Research Engineer, is with the N A S A Lewis Research Center, Cleveland, OH 44135. Manuscript submitted April 5, 1993. METALLURGICAL AND MATERIALS TRANSACTIONS A
of effective radius equal to the critical radius required for stability, the nucleation activation energy is Gh
=
16~ry3f 3(ASyAT~) z
[I]
where 3' is interfacial energy,f is the spherical cap shape factor equal to (2 + cos 8) (1 - cos 8)2/4, AS/is the entropy of fusion, and AT,c is the amount of undercooling (8 is the wetting angle of the nucleating cap). To appreciate the dominance of interfacial energy on nucleation, and especially on nonreciprocal nucleation, it is helpful to consider both y cubed and f, the shape factor. The interfacial energy balances of Pb and Sn heterogeneously nucleating on an essentially flat surface of Sn and Pb, respectively, are, for Pb nucleating on Sn, 'YSn = "~PbSn + YPb COS ~
[2]
and, for Sn nucleating on Pb, yro = YPbS,+ Ysn cos 0
[3]
where YPb is the interfacial energy of Pb alloy in contact with eutectic liquid and Ys, is the interfacial energy between alloyed Sn and the eutectic liquid; the energy associated with the eutectic solid Pb-solid Sn interface is YPbsn, with q~and 0 being the wetting angles of nucleating spherical caps of Pb on Sn and Sn on Pb, respectively. The relative wetting angles can be determined from Eqs. [2] and [3] to be YPb -
-
"YSn
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(1 + cos 0)
=
(1 + COS ~p)
[4]
VOLUME 25A, NOVEMBER 1994--2507
Giindiiz and Hunt
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