Theoretical calculation of nucleation temperature and the undercooling behaviors of Fe-Cr alloys studied with the electr
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ACCORDING to the classical nucleation theory, the steady-state nucleation frequency for metallic systems can be expressed as follows:[1–5] I5
1
2
K DG* exp 2 h kT
[1]
Tl
1T 2 T 2 3.35Tm
g
where Tg is the glass transition temperature (K), Tg > 0.25 [X Al T Am 1 (1 2 X Al )T mB ], with T Am and T mB referring to the melting points of the pure components A and B, respectively, and X Al to the mole fraction of component A in the solution; k is the Boltzman constant; DG* is the nucleation barrier, DG* 5 16p /3 f (u) s3/DG2v, for spherical nuclei, where DGv refers to the free energy change per unit volume (J/m3), s to the solid-liquid interface energy (J/m2), and f(u ) to the shape factor, f(u ) 5 (2–3* cos u 1 cos3 u)/4, with u being the contact angle between the nucleus and the catalytic surface. The nucleation temperature Tn is usually defined as follows:
XUEZHI ZHANG, formerly Sci. & Tech. Agency Fellow with the Joining and Interface Research Station, National Research Institute for Metals, is Senior Programmer, Crystal Studio, Sydney, NSW 2150, Australia. SUSUMU TSUKAMOTO, Leader 5th Laboratory, is with the Joining and Interface Research Station, National Research Institute of Metals, Tsukuba, Ibaraki 305, Japan. Manuscript submitted December 18, 1997. METALLURGICAL AND MATERIALS TRANSACTIONS A
1 ˙ Iv(T )dT 5 1 for homogeneous nucleation [2] T
and ScV
#
Tn
Tl
In Eq. [1], I is the steady-state nucleation frequency in Hz/ m3; K is the nucleation frequency constant, Kv 5 1039 Pa/ m3 for volume nucleation and Ks 5 1029 Pa/m2 for nucleation on catalytic substrate surface; h is the liquid dynamic viscosity (Pa s),
h > 1024.3 exp
#
Tn
I. INTRODUCTION
1 ˙ I (T )dT 5 1 for heterogeneous nucleation T s
where V is the volume of the solidifying liquid (m3); Sc is 2 3 the catalytic surface area per ˙ unit volume (m /m ); Tl is the liquidus temperature; and T is the cooling rate. The nucleation frequency changes exponentially with the nucleation barrier, which is proportional to the cube of the interface energy between the liquid and the nuclei and inversely proportional to the square of the volume free energy change from liquid to the solid nuclei. Therefore, the interface energy and the volume free energy change are very critical in the estimation of the nucleation frequency and the nucleation temperature. Recently, we have extended[6] the Thompson and Spaepen regular solution model[5] for the calculation of the free energy change in crystal nucleation in binary melts to the general solution situation and have shown that the predicted results by the general solution model agree very well with the maximum undercooling results obtained from dispersion experiments.[7,8] In the present article, recent models for the estimation of the solid-liquid interface energy will be briefly reviewed and used together with our general solution model for free energy change in crystal nucleation from binary melts to calculate the nucleation temperature for Fe-Cr alloys. Calculated results are then compared with the experim
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