Calculation of Phase Competition and Selection in Solidification Using a Combined Nucleation and Calphad Approach
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G. SHAO and P. TSAKIROPOULOS Department of Materials Science and Engineering University of Surrey, Guildford, Surrey GU2 5XH, UK
ABSTRACT Thermodynamic data derived by the CALPHAD method has been extended to undercooled melts to study phase competition and the role of time dependent and steady state nucleation in rapidly solidified materials. The theoretical predictions made about phase competition between hcp a, bcc j3 and L1 0 y in some AI(Ti,V) alloys are compared with experimental results. It is shown that high melt cooling rates introduce significant transient nucleation effects on phase nucleation.
INTRODUCTION Rapid solidification from the melt has been used for the development of new materials because of its effects on refining microstructure, extending solid solubility and forming metastable phases. The prediction of microstructure and phase selection as a function of alloy composition and processing conditions is very important for the design of new alloys. By coupling steady state homogeneous nucleation theory and thermodynamic data, the glass forming abilities in metallic alloys [1,2] and phase competition in some Al alloys [3] have been successfully predicted. However, the steady state nucleation approach cannot explain recent experimental evidence for phase competition between the hcp a, bec 1 and L10 Y phases in melt spun Ti aluminides. In this paper, it will be shown that it is necessary to include the time dependent nucleation effects into the nucleation treatment in order to predict phase selection when high melt cooling rates are involved.
METHOD In classical nucleation the behaviour of a non-equilibrium system of particles can be described by the Zeldovich-Frenkel equation [4]. The most comprehensive analytical approximation of the Zeldovich-Frenkel equation was achieved by Kashchiev [5] who related the time (t) dependent nucleation rate Jt to the steady state nucleation rate J, by 0o
(-1)mexp(-m 2 t/z)] J= = Js ([I + 2B E m=1
(1)
where -ris the incubation time and B is a constant related to pre-existing clusters. For a 45 Mat. Res. Soc. Symp. Proc. Vol. 398 0 1996 Materials Research Society
solidification process starting with superheated liquid, B= 1. A numerical solution of the Zeldovich-Frenkel equation by Kelton, Greer and Thompson [6] showed good agreement with Kashchiev's treatment. Integration of eq. (1) gives the number of nuclei N, for B=1 [5] 0o
Nt = JL [t - -E2r
-
2T E
exp(-m 2 t/l)]
(-'i
(2)
Nt predicted by eq. (2) was in excellent agreement with experiments on vapour deposition of mercury on a platinum electrode [7]. If t>5'r, the sum on the right-hand side of eq. (2) can be left out. Nt then increases linearly with time and J, approaches J, [5] (3)
Nt = IT (t - n2 -/6) In eq. (3) the transient time, ttr=5', is [5] .r
(4) (4)G
0 -8kT
p aa2 (,&G)I• =2 P* 2n
which is the same as the incubation time derived by Feder et al [8]. When tgttr, nucleation is controlled by transient nucleation process. Eq. (4) was used previously to derive the transient time for heterogen
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